diff --git a/.gitattributes b/.gitattributes
index 6825630198..5523647754 100644
--- a/.gitattributes
+++ b/.gitattributes
@@ -18,3 +18,5 @@ research/coordinated-frames-and-entrance-banks/** -text -whitespace
 # Frozen PR210 package and round-nine evidence retain their audited bytes.
 research/five-stage-source527-banks/** -text -whitespace
 research/community-round9-audit/** -text -whitespace
+# PR346's w3 bit word with a per-point p = 10 complex supplier: MANIFEST.json and SOURCE.json pin exact bytes.
+research/w3-p10b-pointwise-complex/** -text -whitespace
diff --git a/.github/workflows/w3-p10b-pointwise-complex.yml b/.github/workflows/w3-p10b-pointwise-complex.yml
new file mode 100644
index 0000000000..1addcb77ac
--- /dev/null
+++ b/.github/workflows/w3-p10b-pointwise-complex.yml
@@ -0,0 +1,39 @@
+name: w3-p10b-pointwise-complex
+
+on:
+  pull_request:
+    paths:
+      - 'research/w3-p10b-pointwise-complex/**'
+      - '.github/workflows/w3-p10b-pointwise-complex.yml'
+  push:
+    paths:
+      - 'research/w3-p10b-pointwise-complex/**'
+      - '.github/workflows/w3-p10b-pointwise-complex.yml'
+  workflow_dispatch:
+
+permissions:
+  contents: read
+
+jobs:
+  verify:
+    runs-on: ubuntu-latest
+    timeout-minutes: 60
+    strategy:
+      fail-fast: false
+      matrix:
+        python-version: ['3.12', '3.14']
+    steps:
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1 # v7
+        with:
+          persist-credentials: false
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97 # v7
+        with:
+          python-version: ${{ matrix.python-version }}
+      - name: Install the regeneration dependencies of the pinned pipeline
+        run: python -m pip install numpy==2.3.5 scipy==1.17.0
+      - name: Pins, bit side, the committed complex program (gx.check1, gxcore, exact price) and kappa
+        run: python3 -B research/w3-p10b-pointwise-complex/verify.py --quick
+      - name: Regenerate the per-point complex program with PR327's chain and compare byte for byte
+        run: python3 -B research/w3-p10b-pointwise-complex/verify.py
+      - name: Require an unchanged tree
+        run: git diff --exit-code
diff --git a/certificates/joint-dual-kappa.json b/certificates/joint-dual-kappa.json
index b7f7b89bc2..1f35b89221 100644
--- a/certificates/joint-dual-kappa.json
+++ b/certificates/joint-dual-kappa.json
@@ -842,7 +842,7 @@
     "scripts/experiments/pin_joint_dual_sources.py": "4d31cc0e38b7902709ca5f8f714cba109760d2592346dd7f8dc0b900a2d9fe00",
     "scripts/experiments/verify_joint_dual.py": "7a31bba1e9c0ded32ae285df2ca9bc91ef20c330f6789fdba361876f3c7a1985"
   },
-  "source_manifest_sha256": "4f3c15b82a2d97335d4fe95fb8bcb96bd2a921087da289d9da16c68213a17f17",
+  "source_manifest_sha256": "698fe862a620e5b80b85fbc6034c9456fa0fa9aea638dbbbcb3f9edd24c0415e",
   "sources": [
     {
       "files": {
diff --git a/certificates/joint-dual-validation.json b/certificates/joint-dual-validation.json
index 194861d3c3..61a8d41622 100644
--- a/certificates/joint-dual-validation.json
+++ b/certificates/joint-dual-validation.json
@@ -226,7 +226,7 @@
     }
   },
   "bit_saving": "2382370177/50000000000000",
-  "certificate_sha256": "9801147b1502cf1e43d5fa39aa37f5291f03b48c30c4dd4f272e0877894a1c24",
+  "certificate_sha256": "6f684fa7f747073a132e8d0b5652cf12a8afa389f2a16ff6ade8e080d786dd22",
   "kappa": "4764513337/100000000000000",
   "margins": 7,
   "producer_words_regenerated_and_matched": true,
diff --git a/research/joint-dual/SOURCE.json b/research/joint-dual/SOURCE.json
index e3a8386b17..e1fbf9c5aa 100644
--- a/research/joint-dual/SOURCE.json
+++ b/research/joint-dual/SOURCE.json
@@ -3,9 +3,9 @@
   "base_pr57_commit": "cd350f76c9bc01489ec83568bded532cb69be938",
   "base_pr58_commit": "bc2f7ed4c20dc18898305ab17165c0c995cbb804",
   "files": {
-    "Makefile": "768131bd4ce18433f6abccdc9812a38c92e11220e3a1019c6d95334b42f529d1",
+    "Makefile": "cd0fb59dc078d1fdf54a163ea13bb3660d03c8fe69c1a7f99fc8e5f476a3d5c2",
     "NOTICE": "b49022cb4104d90b182e4c335f670faf45529025b15db77bf0291835083c28bd",
-    "README.md": "d104c31d5b7e0d3a8d011a0e3517ab615e5b499b3dc05684f7ad3cadcd28bdae",
+    "README.md": "b81f8035b859007add6693c6d860371c9100a6766e494045495a28a457a5fcb9",
     "certificates/joint-dual-compiler.json": "516c4d85158553dabd0a98216a064b2a7397957a9d5dd0df1d6e5aec53a4e805",
     "certificates/joint-dual-profiles-23.json": "159fa8fee175fc1d24f63e72d3c2f1dc9f4fcd78b938c5e7d2f735fd0fd37209",
     "certificates/joint-dual-profiles-25.json": "8d190ce42d097e828c5756918c52f2963df576cf4c6600b0fc7f2b7fdb6ad223",
diff --git a/research/w3-p10b-pointwise-complex/MANIFEST.json b/research/w3-p10b-pointwise-complex/MANIFEST.json
new file mode 100644
index 0000000000..08f47d3a8e
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/MANIFEST.json
@@ -0,0 +1,78 @@
+{
+ "description": "sha256 of every file of research/w3-p10b-pointwise-complex except this one; verify.py requires the package to be exactly these files",
+ "files": {
+  "NOTICE": "89d5e68252ca35205382e39a9e0c6b317c36f2d8be435c4faee6364d83392370",
+  "PROOF.md": "65539c02cfebca3dc93aebcf3fc228f6baae1ff6c9db05ef4362ef69db56bb71",
+  "README.md": "65ff95f63441f902912e1cbe24897bcd77fb018769ad6d184e8dc33386af0387",
+  "SOURCE.json": "8af41f59ebe6ac70cafc04d576309ee03f873c80f7275869d57c3b9d103a19c4",
+  "aligned_word_p.py": "bca04f6027c6c6696c1593767487dffa0965cbd7c9fc83493552de4569857757",
+  "aligned_word_spec.py": "6d249179796ecf5aa3e2722e65d98485510752861dc78465498f4cdce7e219ee",
+  "bit/pr346/FINAL-RECORDS.bin.gz": "fc4f1986da4585dc1721bcd954a95b3ffb2b562cc7031f40242de81664e3acb6",
+  "bit/pr346/LICENSE": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4",
+  "bit/pr346/NOTICE.md": "02b59cc0542c78f409d30ce4fd546a806b606b76e185acd9aad1e32deb705008",
+  "bit/pr346/RECORD-SHA256SUMS": "21f732671eb34468be2252d4dfb81bdec4e7d7cd6564b93a482c2dd925e9c3b2",
+  "bit/pr346/kappa-L10f.json": "5b93dee60fb047f70f0b20e3e4bb84f3992515fbc39fc9d27788cd9ae2d950a8",
+  "centre46.py": "40e36dbb4b8a6a2ddb7c947433522d31efbb1ed92969eab7b9da03dacf4e8f9b",
+  "certificate/expected.json": "3a008374da466c2885bb1cad1309c5871ab2b09f8fc0c6eb1cfcebbf63314930",
+  "certificates/gcert1-p10-b325-c2.json.gz": "5c51f5d5c70ef53526740e140d435828ae6bb2be072aae9eca3bc2c40f11265f",
+  "discovery/README.md": "f445c7bb8b026294175ef8c57a99ca56aae2fae51611c35b39411e4e808b1b8d",
+  "discovery/pipe.py": "dccba23ec014a995a65149784ca2b3208231c11a33a0c327608b0c57045406f2",
+  "discovery/pr327/add_centre.py": "891c89f436ebe9e785c98b5e48b85a875dcf18c08fe5f6aac754b59b36502512",
+  "discovery/pr327/extend_floor.py": "6762dd49b801f2419416bcf2c60b6390605a382d252b30679cf53a024aaa8126",
+  "discovery/pr327/make_layer_p.py": "ea339b774109bbfb4ca5c00ff164d7ddff4c3c97433ab064b0a8d78744c3469e",
+  "discovery/pr327/pairs_mw_p.py": "7f619d53d1109626f8f3307b1735606572645518d07ac4f423b851939ff56382",
+  "discovery/pr327/pr233/README.md": "1bd628da0afbccfd9baafc2d51cd96a51a4e46c782b783a1c16bea04fd88d98c",
+  "discovery/pr327/pr233/bundles.py": "71c81bc858e62e03e2a5230c16c4c199a027c24402cdc937bd4ded60f4ee3286",
+  "discovery/pr327/pr233/joint.py": "e4857418f6e16d6fbb1815d3c4148116e35a6b9940cb57acb2b563da5621b92c",
+  "discovery/pr327/pr233/matching.py": "59d1cd30e67d3511cf799140eb248e785e77aaccff7c47a22a0feef8fb6f021e",
+  "discovery/pr327/pr233/pairs_mw.py": "aa927718ad27bebcfa2bddeb3704d4721433c2d4e6b6e218ceb9d28a07cd6c83",
+  "discovery/pr327/pr233/physical_opt.py": "cbfd06026a44df30944781262848c88ec987ca5e80381e35ae7ad715a9db1fce",
+  "gcert_emit.py": "10d2fc3f1f3d8cda5f0b07720f51b5a6fd40cfc1490d361939ff93e5470805d3",
+  "gx/gx.py": "6f3dc8bb639933227a8dae181ec4a542e85c7276be9c9e330f49518e7b630b95",
+  "gx/gxcore.py": "eb972c06fe8c322db2d9a8d9a7d05596e8f40c4ca8e5f42eb6df3c7361b5d517",
+  "layers/c2/kernel-pairs-mw.json": "709cf1e26557c94ee86dff46265518ac903f8f691d5af273bc7f407a9db1f431",
+  "layers/c2/matching-arcs.json": "99029733d575d67ff8a69b91f37d45274478b40a6265f510b96f08c2e417f245",
+  "layers/c2/physical-frames.json": "8fc1c64d2f10ad276818b5886caad1d58a40996868e7704fd358469e279d28a2",
+  "layers/c2/physical-pairs-mw.json": "eb2903eb019c460fad111ce42bcbdf0ee19df1e5361611db12036c1ad4032951",
+  "layers/uniform/kernel-pairs-mw.json": "709cf1e26557c94ee86dff46265518ac903f8f691d5af273bc7f407a9db1f431",
+  "layers/uniform/matching-arcs.json": "99818cca6260ecd2cb010fdb5ccc4f35998ac33e0d98ae9d487e0452254abfec",
+  "layers/uniform/physical-frames.json": "85a57de777bb0e5410612807208e741f3d2688bd4c0ae05a1e356befe609f50d",
+  "layers/uniform/physical-pairs-mw.json": "9cb7988300b3d393ff42e77442382316abe06fa5c03a53ef100242f9f8c89f6b",
+  "lean/companion-7932-evidence.txt": "005f0d3a0dfc02d46999eda86f1dbe1ee0c8d3d18e958072295a4031169bd0da",
+  "module_cut.py": "4004cce8e4b392d90db02862e1d631aca586483a59532de9eca426504ff1293f",
+  "modules/pm_p10.json": "8d71f5004acb0ea902ad594a7d298f8d2099114e5d6d42c6357f09c44ab3ce87",
+  "modules/pointwise/pmod_i0.json": "7d9c0aaf02fbb91a7004821adcd9bdcc577513a4052cd21689a8e1bea7616ce5",
+  "modules/pointwise/pmod_i1.json": "867744aacd6ce119954ad2ba2de51819551e590a22eae4815ba6a111d16ef748",
+  "modules/pointwise/pmod_i2.json": "0c56f9e38be7d6c9e4968012afcc37479c114453a2dad6a0ff64b5a446123f16",
+  "modules/pointwise/pmod_i3.json": "7d9c0aaf02fbb91a7004821adcd9bdcc577513a4052cd21689a8e1bea7616ce5",
+  "modules/pointwise/pmod_i4.json": "867744aacd6ce119954ad2ba2de51819551e590a22eae4815ba6a111d16ef748",
+  "modules/pointwise/pmod_i5.json": "5e7f9239867d066850dcca94106cab8420fd04b494e7718526311ecc08950117",
+  "modules/pointwise/pmod_i6.json": "5e7f9239867d066850dcca94106cab8420fd04b494e7718526311ecc08950117",
+  "modules/pointwise/pmod_i7.json": "a10816b44292874da2d7596b044b9be876e112ca00581bde2ef9a74213b10dc3",
+  "modules/pointwise/pmod_i8.json": "5e7f9239867d066850dcca94106cab8420fd04b494e7718526311ecc08950117",
+  "modules/pointwise/pmod_i9.json": "953e80941984780dfba6952537cace9d1214270bda5735f27dc1cfe020cfcef6",
+  "modules/qmod_p10.json": "40ffb168466bc016bc18a2bbdde65d38b7fd1b406d8a4026f1a6e517fe153459",
+  "modules/tmod_t2_p10.json": "4d39faeedfb5a5283a7ad34a8a18af93072e4d155d61dec6c28d7751bc2e4e04",
+  "normalize_scatter.py": "403ef1db92bafcb973857063bc96ec9ac326838b7595af642d5f11fef3350211",
+  "paircheck.py": "acde7bef664c2b0aa9f0c99424053ff4cf5091e4eb7bba7e048f5a51e0933b31",
+  "pricing/outer.py": "c1ebb52163a7219504c728a5c2b098bf1e50bf20148a2879c5a74c3cadc7c03c",
+  "references/centre46_graph.diff": "229eea46b56d19b262d1498f489f9bca84c347e33b16553079d38c4276700280",
+  "references/pr233-layer/NOTICE": "5826b9f155b70d496d729cbaddeadb881abfb68da77f82dfc1746ddfc753313b",
+  "references/pr233-layer/SOURCE.json": "9ad0549ba2403e8d84d7c0c8e71e5ca952a0604ad626f403426164bd4bc2982c",
+  "references/pr233-layer/baseline-pr202.tar.gz": "5d08386f71c421d2b83b21020e3b74198e9d0c75884d7d81987307740d98883f",
+  "references/pr233-layer/certificate.json": "691cb0aaeb0d6b993e1e1e5116ae7a5c57af058a8c413249016347d1b40d3b90",
+  "references/pr233-layer/verify.py": "cd61a8c61462b4eee95c0858335a9a11bdf1ff8e8e63ebb1e488da29f061ca71",
+  "references/pr233-layer/witness/kernel-pairs.json": "e33b753fc4519170373207b0527b9cb53e0b552e2a27f48f72381a881c7a227f",
+  "references/pr233-layer/witness/physical-frames.json": "1a6c215cdc58216f7ec96fb898a471e25d094fd86de085f6c06e86fd8997bfad",
+  "references/pr233-layer/witness/physical-pairs.json": "e6409fffbf5f3b1ac6434acf2e426503de3a8a18537b8fee761fc4fc931fb896",
+  "references/sussman/pr193-blocks.json": "4049d066f71119fa72bbc37f7db753ce1e79ae5a69aef9cd2651d618bea5c855",
+  "scripts/moment.py": "0c7c68122cea18503e973ccf0d244adf8fa0edab95f104393528c6c57f24863a",
+  "scripts/source527/NOTICE": "c461d8aef9a0c1ec5015a300bdbf37e566610d50139d730ba677eb0df5d7ffe0",
+  "scripts/source527/base_two_moment.py": "aee70c242a2f1d225528fabe2002e1564d1dc884797c07d28a831c7878ee9098",
+  "scripts/source527/moment.py": "9d465c716c6b910ff66e1453e3bf0057608a787b5f5361be93781f42cb3370f7",
+  "spec/c2.json": "569bd70b4d945ccc42e3ff84f8901af49b65f77f97cabe2e2655443244c067b5",
+  "spec/uniform.json": "c0f1bfed8343ed2a0cbb374b175ff2dd194757116edf8c1b6de656ddc3bd042a",
+  "spec_graph.py": "8dc9a58bd6f5a4bc8e29a4550464dba65583dc1a8d3a3e4cb82b66d5489ab3c2",
+  "verify.py": "12c49aa6b943f7ecdc5380c4018b2598b882fd99d8c78f3ecd961a8011bf238a"
+ }
+}
diff --git a/research/w3-p10b-pointwise-complex/NOTICE b/research/w3-p10b-pointwise-complex/NOTICE
new file mode 100644
index 0000000000..ebd612a2be
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/NOTICE
@@ -0,0 +1,57 @@
+Prepared by Chafik Boukhalfa with substantial Anthropic Claude assistance. Apache-2.0.
+Inherited files keep their original notices and assistance disclosures. This package claims no exclusive priority
+over concurrent work.
+
+What this package contributes:
+- the per-point p = 10 complex program (claim 7940/10^7, coarse b = 794022782431403/10^18), together with its
+  pointwise pair modules, its frozen layer and its spec;
+- the per-invocation module spec and builder (spec_graph.py, aligned_word_spec.py, and anchors (e) and (f));
+- the composition of that program with PR #346's bit word, an independent recount of that word's paid children,
+  and the reproduction of PR #346's kappa;
+- verify.py, MANIFEST.json and the git-blob pins of SOURCE.json.
+
+Everything else is retained and credited:
+
+- LJH-217 / Louis Harrison (Anthropic Claude assistance), PR #346 (research/w3-p10b-complex-p10 at 11b4de5): the
+  final w3 bit word on PR #325's p10b word. It is vendored unchanged in bit/pr346/, with its records, hash list,
+  kappa receipt, LICENSE and NOTICE.md. PR #346 also composed it with a p = 10 complex supplier and computed the
+  complex-bound kappa that this package reproduces. Its bit word's own checks (Design T verifier, PR #266
+  checkers, primes, bank tiling) are PR #346's and are not re-run here.
+- utcorvusvolat-dotcom: PR #310, the w3 Design T and twin condensation, with their builders and cost model, which
+  PR #346 ported to h = 20.
+- PR #266's official cohort legality and five-stage column checkers, as run by PR #346.
+- DreamingOfClouds (Anthropic Claude assistance):
+  - PR #325 (c4f1b74): the p10b bit word under PR #346's, the re-annealed pair module pm_p10.json (the default pair
+    module of the spec and the start of the pointwise climb) and the all-but-one module qmod_p10.json;
+  - PR #315: the p = 10 five-stage package, qmod_p10.json, the centre-sharing Graph.finish patch, and the outer
+    assembly convention (complex-bound path) used for kappa;
+  - PR #327 (40ed045): the p = 10 complex supplier, its recipe and chain (aligned_word_p.py, centre46.py,
+    paircheck.py, normalize_scatter.py, module_cut.py, discovery/pr327/), the triple module tmod_t2_p10.json, and
+    verify.py, which this verify.py generalises; all vendored byte for byte;
+  - PR #304: the p = 11 recipe.
+- Jacob Sussman (Anthropic Claude assistance): gcert/1, gx.py and gxcore.py (vendored unchanged in gx/,
+  Apache-2.0, Copyright 2026 Jacob Sussman, jacobalansussman/wht-power-saving-lean at f010392c), the bridged
+  five-stage layout B_2, and the Lean generator and Compare.lean used for the companion program.
+- chafreaky / Chafik Boukhalfa (Anthropic Claude and OpenAI Codex assistance):
+  - PR #233: the maximum-weight pairing, and the layer package vendored in references/pr233-layer/;
+  - PR #200: the frame descent and the complete complex checker;
+  - PR #256: the gcert emitter gcert_emit.py;
+  - PR #225: scripts/moment.py;
+  - PR #329: pricing/outer.py, which is PR #315's unchanged.
+- icekylinx (OpenAI GPT-6 Astra and Codex assistance): PR #144 paired cubes and Graph.finish; PR #184 the
+  source-assisted construction, frame flow, exact lift and contract.
+- Avi Eisenberg (ikeboy): PR #191/193/194/202, the source-aligned pipeline and the PR #202 baseline.
+- eumemic (Anthropic Claude and OpenAI Codex assistance):
+  - PR #168 modules and frames;
+  - the source527 rational moment engines (scripts/source527/, see its NOTICE);
+  - outer.py, the 47-constraint outer assembly of PR #315 (from icekylinx's PR #144 and Zhihao Chen's PR #23,
+    with hipotures' balanced prefix).
+- DaysSky: PR #162, the extended carrier closure.
+- hcg890: PR #234, the five-stage architecture and the 10^-16 fallback envelope.
+- SovereignSteak: PR #250.
+- Evan McKinney: the completed banks of the five-stage packages.
+- Rohan Arun: the joint-dual re-pin in PR #327's head.
+- The PR #202 baseline lineage, as credited in references/pr233-layer/NOTICE.
+- OpenAI: the original complex phase network of "Integer multiplication below n log n".
+
+The credits above do not imply review or endorsement by the credited authors.
diff --git a/research/w3-p10b-pointwise-complex/PROOF.md b/research/w3-p10b-pointwise-complex/PROOF.md
new file mode 100644
index 0000000000..959247c3f6
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/PROOF.md
@@ -0,0 +1,101 @@
+# Conditional κ = 793392809291691/10¹⁸: PR #346's bit word with a per-point p = 10 complex supplier
+
+**Claim (conditional, finite).** Assume the inherited interfaces of PR #315/#325/#346. Take PR #346's final w3
+bit word (bit coarse saving c = 32069035133443/(4·10¹⁶)) together with the complex program
+`certificates/gcert1-p10-b325-c2.json.gz` (complex coarse saving b = 794022782431403/10¹⁸). Then PR #315's outer
+assembly admits κ = 793392809291691/10¹⁸ with the complex side binding, and rejects the next 10⁻¹⁸ point. The
+same computation applied to PR #346's own b = 396604388013523/(5·10¹⁷) gives PR #346's κ =
+396290046684973/(5·10¹⁷) exactly.
+
+## 1. The complex program
+
+The program is PR #327's recipe and chain, vendored at `40ed045`, applied to the modules of `spec/c2.json`. Each
+invocation has its own module:
+
+| invocations | module |
+|---|---|
+| triple module | PR #327's `tmod_t2_p10.json` |
+| 90 all-but-one invocations | PR #325's `qmod_p10.json` |
+| pair invocations (i, 0) and (i, 1) | `modules/pointwise/pmod_i{i}.json` |
+
+The layer is `layers/c2/`: 6,779 carrier arcs, 10,036 moved frames, 1,680 maximum-weight pairs and no kernel pairs.
+
+The graph is built by `spec_graph.finish_spec`, which is PR #327's `centre46.finish_centre46` with the module
+looked up per invocation. Two anchors check it against PR #327's code on the one-module-per-kind spec
+`spec/uniform.json`:
+
+- **(e)** `finish_spec` and `finish_centre46` build the same graph, in every field;
+- **(f)** `aligned_word_spec.py --spec` and PR #327's `aligned_word_p.py --centre46` write the same seven files.
+
+`aligned_word_spec.py` differs from `aligned_word_p.py` only in its graph-building branch.
+
+The full run then applies PR #327's chain to the per-point word, and every step must pass:
+
+1. the module contracts (`load_pmod46` for every pointwise module);
+2. `compile_closure` on the frozen arcs, and the forward closure returning every frozen frame;
+3. `paircheck.py` with five mutation controls;
+4. PR #200's complete complex checker;
+5. PR #184's flow (new R 6,253) and the exact lift;
+6. PR #194's contract (13 checks);
+7. PR #256's emitter, then `gx.check1` (labels, blocks, N = R·h + 2v(h − 1) + cst = 161,900, exact scalar
+   identity) and the `gxcore` mirror, both on the raw program and on its star form;
+8. the E5 controls;
+9. byte equality with the committed program (canonical sha256 `ff2d7307…5112`).
+
+**Price.** The five-stage word at h = 20 has H₅ = 5·H_inv + 2v(e₃₈ + e₁₉ + e₄₂ + e₄), m = 100, W = 4v + R =
+10,093, deficit 2,040. The exact bounds give:
+
+- PR #225's upper bound: a = 7940/10⁷, with and without the 10⁻¹⁶ fallback;
+- source527's lower bound: 7941/10⁷ is rejected;
+- both rational engines: b = 794022782431403/10¹⁸ with the fallback (794022785498833/10¹⁸ without), with the next
+  10⁻¹⁸ point excluded.
+
+The program has not been built in Lean (§4).
+
+## 2. The bit word
+
+The bit word is PR #346's and is vendored unchanged. `verify.py` checks the following:
+
+- **records.** The decompressed final records match PR #346's `RECORD-SHA256SUMS` (`TT/249-records.bin`). They
+  are 405,188 six-tuples: 47,020 MOVE, 358,128 ADD, 20 COPY and 20 ERASE.
+- **paid children.** An independent recount over the MOVE and COPY records gives H (45,884 children, rank mass
+  169,560).
+- **normalized histogram.** PR #346's histogram is 60·H + 12·2v·(e₃₈ + e₁₉ + e₄₂ + e₄) exactly.
+- **stock.** W = 125,712 = 12·(4v + 132,720/20), taken from PR #346's receipt.
+- **bit saving.** Both engines certify c = 32069035133443/(4·10¹⁶) with the 10⁻¹⁶ fallback and exclude the next
+  point. This equals PR #346's c.
+
+The bit word's own legality (Design T, PR #266's checkers, primes, banks) is PR #346's claim and is not re-run here.
+
+## 3. κ
+
+The steps are PR #315's `math_check` (as in PR #329) on the complex-bound path:
+
+1. β = 10⁻⁹, η = 10⁻¹², and the cap is (1 − β)·b.
+2. Take s = min(c, ⌊cap − 10⁻¹⁸⌋₁₀⁻¹⁸). Here s < c, so the complex side binds. s is re-certified by the moment.
+3. Bootstrap three times from 384599/10¹⁰: a ← (1 − s)·s + s·a.
+4. Set q = a(1 − 2η) and minimum = (1 − η)·q/(1 + q). κ is the last 10⁻¹⁸ grid point strictly below the minimum.
+5. `outer.assembly(a, b, bridge, κ)` passes all 47 strict constraints and the 7 margins. At κ + 10⁻¹⁸ it fails. At
+   the cap itself it fails `leaf_saving_above_bit`.
+
+The results:
+
+| complex b | κ |
+|---|---|
+| PR #346: 396604388013523/(5·10¹⁷) | 396290046684973/(5·10¹⁷) ≈ 7.92580093369946·10⁻⁴ (reproduced) |
+| this package: 794022782431403/10¹⁸ | **793392809291691/10¹⁸ ≈ 7.93392809291691·10⁻⁴** |
+
+The gain is +812715921745/10¹⁸ = +0.1025 %.
+
+## 4. What is not claimed
+
+- **No Lean build of the 7940/10⁷ program yet.** It passes `gx.check1`, `gxcore` and every Python check above. The
+  companion 7932/10⁷ program (`spec/uniform.json`) was built in Lean with Sussman's unchanged generator
+  (`wht_main_block_B2Gcu32x` at 1 − 7932084/10¹⁰, `Compare.lean` PASS; `lean/companion-7932-evidence.txt`).
+- **PR #346's bit-side obligations** are inherited unchanged: the w3 downstream chain is not ported to p = 10, the
+  bank feasibility only, and the scalar/prime/finite stages.
+- PR #315's portable complex checks, ported to p = 10 by PR #346 (labels, scalar words, splice, precision guard),
+  are not run on this program.
+- The five-stage B₂ ledger at h = 20 and the 10⁻¹⁶ fallback envelope are inherited assumptions, as in PR
+  #327/#346.
+- This is a conditional finite construction under the retained public all-size hypotheses, not a theorem about κ.
diff --git a/research/w3-p10b-pointwise-complex/README.md b/research/w3-p10b-pointwise-complex/README.md
new file mode 100644
index 0000000000..cfea704e2d
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/README.md
@@ -0,0 +1,123 @@
+κ = 7.93392809291691e-4
+
+# PR #346's w3 bit word with a per-point p = 10 complex supplier
+
+A conditional finite construction with **κ = 793392809291691/10¹⁸ ≈ 7.93392809291691 × 10⁻⁴**. That is
++8.127·10⁻⁷ (+0.1025 %) over PR #346 (κ = 396290046684973/(5·10¹⁷) ≈ 7.92580093369946·10⁻⁴).
+
+PR #346 (LJH-217) put PR #310's w3 Design T and twin condensation on PR #325's p10b bit word. That word's bit
+coarse saving lies above every known complex cap, so κ is set by the complex supplier. This package keeps PR
+#346's bit word unchanged and replaces its complex supplier with a stronger p = 10 program of the same recipe. In
+that program each point i has its own pair module, shared by both bits.
+
+| side | supplier | coarse saving (10⁻¹⁶ fallback) |
+| --- | --- | ---: |
+| bit | PR #346's final w3 word on PR #325's p10b word (vendored) | c = 32069035133443/(4·10¹⁶) ≈ 8.01725878336075·10⁻⁴ |
+| complex, PR #346 | p = 10 gcert/1 program, PR #325's pair module, flow R 6,265 | 396604388013523/(5·10¹⁷) ≈ 7.93208776027046·10⁻⁴ |
+| **complex, this package** | **p = 10 gcert/1 program, pointwise pair modules, R 6,253** | **b = 794022782431403/10¹⁸ ≈ 7.94022782431403·10⁻⁴** |
+
+Both suppliers make the complex side bind. Here the bit saving is capped one 10⁻¹⁸ grid point below the leaf cap
+(1 − 10⁻⁹)·b and fed through PR #315's outer assembly, which checks 47 strict constraints and the finite bridge.
+`verify.py` first reproduces PR #346's κ exactly from PR #346's b, then computes κ from this package's b. In both
+cases the next 10⁻¹⁸ κ and the cap itself are rejected.
+
+## Complex supplier
+
+The word, recipe and chain are PR #327's (at `40ed045`): the source-assisted complex word of PR #184/#194 at
+p = 10, with the centre-sharing graph, HK carrier arcs plus PR #162's extension, PR #200's frame descent, PR #233's
+maximum-weight pairs, PR #184's flow and exact lift, PR #194's contract and PR #256's emitter. The query modules
+come from `spec/c2.json`:
+
+- triple module: PR #327's `tmod_t2_p10.json`;
+- all-but-one module: PR #325's `qmod_p10.json` (PR #315's), the same for all 90 invocations;
+- pair module of point i (both bits): `modules/pointwise/pmod_i{i}.json`. These six distinct files were found by a
+  greedy climb that started from PR #325's `pm_p10.json` (`discovery/README.md`). Each has 36 inputs plus the
+  star-centre root, and `centre46.load_pmod46` checks its contract.
+
+| program | R | W | N | blocks | a | rejected | coarse b (/10¹⁸) | Lean |
+|---|---|---|---|---|---|---|---|---|
+| PR #327 centre-mw | 6,455 | 10,295 | 165,940 | 47,979 | 7795/10⁷ | 7796/10⁷ | 779597612622781 | 1 − 7795973/10¹⁰ |
+| shared #325 modules (`spec/uniform.json`; same b as PR #346's supplier) | 6,265 | 10,105 | 162,140 | 47,619 | 7932/10⁷ | 7933/10⁷ | 793208776027046 | 1 − 7932084/10¹⁰ |
+| **pointwise pair modules (`spec/c2.json`, this package)** | **6,253** | **10,093** | **161,900** | **47,599** | **7940/10⁷** | **7941/10⁷** | **794022782431403** | **pending** |
+
+**Status of the 7940/10⁷ program.** Sussman's reference checker `gx.check1` (exact scalar identity) and his
+`gxcore` mirror of the two Lean checks accept it, and both reject a flipped sign. It also passes the rest of the
+chain: `paircheck.py`, PR #200's checker, the exact flow lift and PR #194's contract (13/13). `verify.py`
+regenerates it byte for byte. **Its Lean run is pending**: the host with the Lean toolchain needs an interactive
+re-authentication. The companion program with one shared pair module (7932/10⁷) went through the same pipeline and
+was built in Lean (`lean/companion-7932-evidence.txt`). The per-point program uses the same generator path, with
+only its module files changed.
+
+## Bit side
+
+PR #346's final word is vendored unchanged (`bit/pr346/`). It consists of:
+
+- its final records (`FINAL-RECORDS.bin.gz`, checked against its `RECORD-SHA256SUMS`);
+- its κ receipt (the normalized five-stage histogram, the stock W = 125,712 and c);
+- its LICENSE and NOTICE.
+
+`verify.py` decodes the records with its own reader and recounts the paid children (MOVE and COPY ranks). It then
+requires PR #346's normalized histogram to equal 60·H + 12·2v·(e₃₈ + e₁₉ + e₄₂ + e₄), and certifies c from it with
+both rational engines. The stock W is taken from PR #346's receipt and is not re-derived. PR #346's own bit checks
+are not re-run here: the Design T verifier, PR #266's legality and five-stage column checkers, primes and bank
+tiling. Since the complex side binds, κ depends on the bit word only through c > (1 − 10⁻⁹)·b, and c is 0.97 %
+above that cap.
+
+## Verify
+
+    python -m pip install numpy==2.3.5 scipy==1.17.0
+    python3 -B research/w3-p10b-pointwise-complex/verify.py --quick   # about 30 s: pins, bit side, program, kappa
+    python3 -B research/w3-p10b-pointwise-complex/verify.py           # about 6 min: also regenerates the program
+
+Nothing is fetched; `-O` is refused. `--quick` covers:
+
+1. `MANIFEST.json`;
+2. `SOURCE.json`: git blob and sha256 of every vendored file at PR #327 `40ed045`, PR #325 `c4f1b74`, PR #346
+   `11b4de5` and PR #329 `c6b1a28`;
+3. the bit side;
+4. the committed program: canonical sha256 `ff2d7307…`, `gx.check1`, `gxcore`, the E5 controls, and the exact
+   price (7940/10⁷ with and without the fallback, 7941/10⁷ rejected, b by both engines);
+5. κ for PR #346 (reproduced) and for this package.
+
+The full run also regenerates the program:
+
+1. it rebuilds PR #202's 308 files from the vendored PR #233 package;
+2. it runs PR #327's anchors (a) and (b), then anchors (d) and (e) on `spec/uniform.json`;
+3. it runs anchor (f): on the companion's frozen layer, `aligned_word_spec.py --spec` writes the same seven files as
+   PR #327's `aligned_word_p.py --centre46`;
+4. it builds the per-point word with `aligned_word_spec.py --spec spec/c2.json` and runs the whole chain;
+5. it requires the committed program byte for byte, and compares the run with `certificate/expected.json`.
+
+`.github/workflows/w3-p10b-pointwise-complex.yml` runs both modes on Python 3.12 and 3.14.
+
+## Open obligations
+
+- The Lean build of the 7940/10⁷ program (pending; see above).
+- Everything PR #346 lists as open for its bit word: PR #310's native downstream chain is not ported to p = 10,
+  PR #315's scalar/prime/finite stages cannot take an edited word, and the complex guard keeps PR #315's p = 11
+  additive row constants.
+- PR #346 also ran PR #315's portable complex checks, ported to p = 10 (labels, scalar words, splice, precision
+  guard); they are not run on this program here. This package's complex checks are PR #327's chain plus
+  `gx.check1`/`gxcore`.
+- All inherited interfaces of PR #315/#325/#346 remain assumptions. This is a conditional finite construction,
+  not a formal verification of the multiplication theorem.
+
+## Files
+
+| path | origin | content |
+|---|---|---|
+| `verify.py` | new (from PR #327's verify.py and PR #329's math_check.py) | the checks above |
+| `spec_graph.py`, `aligned_word_spec.py` | new | per-invocation spec and word builder |
+| `spec/c2.json`, `spec/uniform.json` | new | the program's spec, and the companion spec used for anchors |
+| `modules/pointwise/` | new | the ten per-point pair modules |
+| `modules/pm_p10.json`, `modules/qmod_p10.json` | PR #325 `c4f1b74` | default pair module and the all-but-one module |
+| `modules/tmod_t2_p10.json` | PR #327 `40ed045` | triple module |
+| `layers/c2/`, `layers/uniform/` | new | frozen layers of the program and of the companion |
+| `certificates/gcert1-p10-b325-c2.json.gz` | new | the program, star form |
+| `certificate/expected.json` | new | the full run's record |
+| `bit/pr346/` | PR #346 `11b4de5` | bit word records, hash list, κ receipt, LICENSE, NOTICE |
+| `pricing/outer.py` | PR #329 `c6b1a28` (= PR #315's) | the 47-constraint outer assembly |
+| `lean/companion-7932-evidence.txt` | new | Lean console output of the companion program |
+| chain, checkers, engines, `references/`, `discovery/pr327/` | PR #327 `40ed045` | vendored byte for byte |
+| `discovery/` | new | how the modules and layer were found |
+| `SOURCE.json`, `MANIFEST.json`, `PROOF.md`, `NOTICE` | new | provenance, pins, claim, credits |
diff --git a/research/w3-p10b-pointwise-complex/SOURCE.json b/research/w3-p10b-pointwise-complex/SOURCE.json
new file mode 100644
index 0000000000..9385d3f960
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/SOURCE.json
@@ -0,0 +1,359 @@
+{
+ "description": "Provenance of research/w3-p10b-pointwise-complex. Every file under \"origins\" is vendored byte for byte; verify.py checks its sha256 and git blob id against the named commit (no network). MANIFEST.json pins every package file.",
+ "spec": "spec/c2.json",
+ "origins": {
+  "pr327": {
+   "repository": "CrocSwap/integer-mult-bounds",
+   "pr": 327,
+   "commit": "40ed0456ab1990beb3440e61bb883ccf95fdcea4",
+   "path": "research/complex-p10",
+   "author": "DreamingOfClouds (Anthropic Claude assistance)",
+   "note": "the p = 10 complex supplier: its recipe, chain (aligned_word_p.py, centre46.py, paircheck.py, normalize_scatter.py, module_cut.py), triple module tmod_t2_p10.json, vendored PR #256 / Sussman / source527 files and layer-discovery code; 40ed045 is the PR head (a22e63a plus Rohan Arun's joint-dual re-pin)",
+   "files": {
+    "aligned_word_p.py": {
+     "path": "research/complex-p10/aligned_word_p.py",
+     "sha256": "bca04f6027c6c6696c1593767487dffa0965cbd7c9fc83493552de4569857757",
+     "git_blob": "98da7115194287191a46736dd9e3867b6a236b28"
+    },
+    "centre46.py": {
+     "path": "research/complex-p10/centre46.py",
+     "sha256": "40e36dbb4b8a6a2ddb7c947433522d31efbb1ed92969eab7b9da03dacf4e8f9b",
+     "git_blob": "ede0d1b210979981fb228de381f587e259138d1a"
+    },
+    "gcert_emit.py": {
+     "path": "research/complex-p10/gcert_emit.py",
+     "sha256": "10d2fc3f1f3d8cda5f0b07720f51b5a6fd40cfc1490d361939ff93e5470805d3",
+     "git_blob": "fd5d56f651e384003d0b70be300296caff1e6d44"
+    },
+    "module_cut.py": {
+     "path": "research/complex-p10/module_cut.py",
+     "sha256": "4004cce8e4b392d90db02862e1d631aca586483a59532de9eca426504ff1293f",
+     "git_blob": "67ef90486676ba64b46073eeb02f7f7b2f326fc1"
+    },
+    "normalize_scatter.py": {
+     "path": "research/complex-p10/normalize_scatter.py",
+     "sha256": "403ef1db92bafcb973857063bc96ec9ac326838b7595af642d5f11fef3350211",
+     "git_blob": "4f62515895c4417d55556cea9ebd89205607e473"
+    },
+    "paircheck.py": {
+     "path": "research/complex-p10/paircheck.py",
+     "sha256": "acde7bef664c2b0aa9f0c99424053ff4cf5091e4eb7bba7e048f5a51e0933b31",
+     "git_blob": "673cc0a6844e23eb1f2670a10a403e393d8b4698"
+    },
+    "gx/gx.py": {
+     "path": "research/complex-p10/gx/gx.py",
+     "sha256": "6f3dc8bb639933227a8dae181ec4a542e85c7276be9c9e330f49518e7b630b95",
+     "git_blob": "6d14c0a976146eb881c1e3897b2476f8633d4e9c"
+    },
+    "gx/gxcore.py": {
+     "path": "research/complex-p10/gx/gxcore.py",
+     "sha256": "eb972c06fe8c322db2d9a8d9a7d05596e8f40c4ca8e5f42eb6df3c7361b5d517",
+     "git_blob": "3e3f0a3076f3c326e7833c74eceecb3436dd3185"
+    },
+    "scripts/moment.py": {
+     "path": "research/complex-p10/scripts/moment.py",
+     "sha256": "0c7c68122cea18503e973ccf0d244adf8fa0edab95f104393528c6c57f24863a",
+     "git_blob": "11c622d33c78f44743f431fd0f9b15f689f95f1b"
+    },
+    "scripts/source527/NOTICE": {
+     "path": "research/complex-p10/scripts/source527/NOTICE",
+     "sha256": "c461d8aef9a0c1ec5015a300bdbf37e566610d50139d730ba677eb0df5d7ffe0",
+     "git_blob": "520559625d953b883f755fc25bbade798849cfd6"
+    },
+    "scripts/source527/base_two_moment.py": {
+     "path": "research/complex-p10/scripts/source527/base_two_moment.py",
+     "sha256": "aee70c242a2f1d225528fabe2002e1564d1dc884797c07d28a831c7878ee9098",
+     "git_blob": "14a90fa0e788217856f070382c9e61bd7c603c61"
+    },
+    "scripts/source527/moment.py": {
+     "path": "research/complex-p10/scripts/source527/moment.py",
+     "sha256": "9d465c716c6b910ff66e1453e3bf0057608a787b5f5361be93781f42cb3370f7",
+     "git_blob": "1e43aaeaca1b9d1ff40b8805ca7ae3e6a219c392"
+    },
+    "references/centre46_graph.diff": {
+     "path": "research/complex-p10/references/centre46_graph.diff",
+     "sha256": "229eea46b56d19b262d1498f489f9bca84c347e33b16553079d38c4276700280",
+     "git_blob": "76437ab85699f999a1d50a0a384eb0bb2191d762"
+    },
+    "references/pr233-layer/NOTICE": {
+     "path": "research/complex-p10/references/pr233-layer/NOTICE",
+     "sha256": "5826b9f155b70d496d729cbaddeadb881abfb68da77f82dfc1746ddfc753313b",
+     "git_blob": "27c7d368fae8ae255b454b33c803ef01fc331aa8"
+    },
+    "references/pr233-layer/SOURCE.json": {
+     "path": "research/complex-p10/references/pr233-layer/SOURCE.json",
+     "sha256": "9ad0549ba2403e8d84d7c0c8e71e5ca952a0604ad626f403426164bd4bc2982c",
+     "git_blob": "dca4742101e5620b253ce61b0b7a84f0fcf19b91"
+    },
+    "references/pr233-layer/baseline-pr202.tar.gz": {
+     "path": "research/complex-p10/references/pr233-layer/baseline-pr202.tar.gz",
+     "sha256": "5d08386f71c421d2b83b21020e3b74198e9d0c75884d7d81987307740d98883f",
+     "git_blob": "39bdd153c4bd858dd7d9c050f500463cc9e6f8ef"
+    },
+    "references/pr233-layer/certificate.json": {
+     "path": "research/complex-p10/references/pr233-layer/certificate.json",
+     "sha256": "691cb0aaeb0d6b993e1e1e5116ae7a5c57af058a8c413249016347d1b40d3b90",
+     "git_blob": "796efbb0d4dfeb9088701ed1e7aa5ae0e93f22a6"
+    },
+    "references/pr233-layer/verify.py": {
+     "path": "research/complex-p10/references/pr233-layer/verify.py",
+     "sha256": "cd61a8c61462b4eee95c0858335a9a11bdf1ff8e8e63ebb1e488da29f061ca71",
+     "git_blob": "2ddf0d2f39b572d97fe42263e0d3e81e79f19391"
+    },
+    "references/pr233-layer/witness/kernel-pairs.json": {
+     "path": "research/complex-p10/references/pr233-layer/witness/kernel-pairs.json",
+     "sha256": "e33b753fc4519170373207b0527b9cb53e0b552e2a27f48f72381a881c7a227f",
+     "git_blob": "e889b7b95c884496361a56ceb5aec5b55a084f21"
+    },
+    "references/pr233-layer/witness/physical-frames.json": {
+     "path": "research/complex-p10/references/pr233-layer/witness/physical-frames.json",
+     "sha256": "1a6c215cdc58216f7ec96fb898a471e25d094fd86de085f6c06e86fd8997bfad",
+     "git_blob": "9044957452ee8e3620cbf24dbbcc1e5d43ba27d3"
+    },
+    "references/pr233-layer/witness/physical-pairs.json": {
+     "path": "research/complex-p10/references/pr233-layer/witness/physical-pairs.json",
+     "sha256": "e6409fffbf5f3b1ac6434acf2e426503de3a8a18537b8fee761fc4fc931fb896",
+     "git_blob": "a582ff3bea5ae605f068c6985a437be14d40dca0"
+    },
+    "references/sussman/pr193-blocks.json": {
+     "path": "research/complex-p10/references/sussman/pr193-blocks.json",
+     "sha256": "4049d066f71119fa72bbc37f7db753ce1e79ae5a69aef9cd2651d618bea5c855",
+     "git_blob": "aeb5da69b2ab6b4da834f8fcb91703c68156c98c"
+    },
+    "modules/tmod_t2_p10.json": {
+     "path": "research/complex-p10/modules/tmod_t2_p10.json",
+     "sha256": "4d39faeedfb5a5283a7ad34a8a18af93072e4d155d61dec6c28d7751bc2e4e04",
+     "git_blob": "7d2185c2bed6041866c2c3e9325d9b4ca8bd3177"
+    },
+    "discovery/pr327/add_centre.py": {
+     "path": "research/complex-p10/discovery/add_centre.py",
+     "sha256": "891c89f436ebe9e785c98b5e48b85a875dcf18c08fe5f6aac754b59b36502512",
+     "git_blob": "d9ee63aa2fb137f6242f5d5a9e457d23b2e3685e"
+    },
+    "discovery/pr327/extend_floor.py": {
+     "path": "research/complex-p10/discovery/extend_floor.py",
+     "sha256": "6762dd49b801f2419416bcf2c60b6390605a382d252b30679cf53a024aaa8126",
+     "git_blob": "b9f2bb7619409806533f1b315ecda00749f05b58"
+    },
+    "discovery/pr327/make_layer_p.py": {
+     "path": "research/complex-p10/discovery/make_layer_p.py",
+     "sha256": "ea339b774109bbfb4ca5c00ff164d7ddff4c3c97433ab064b0a8d78744c3469e",
+     "git_blob": "747db04c5666f5d8a9ef495571d580f16a4ebbf7"
+    },
+    "discovery/pr327/pairs_mw_p.py": {
+     "path": "research/complex-p10/discovery/pairs_mw_p.py",
+     "sha256": "7f619d53d1109626f8f3307b1735606572645518d07ac4f423b851939ff56382",
+     "git_blob": "0a96000bf59bc851e05995688c59eeb929716f03"
+    },
+    "discovery/pr327/pr233/README.md": {
+     "path": "research/complex-p10/discovery/pr233/README.md",
+     "sha256": "1bd628da0afbccfd9baafc2d51cd96a51a4e46c782b783a1c16bea04fd88d98c",
+     "git_blob": "339adb1c76a8cf3356c74e4361daf73111c2e8e4"
+    },
+    "discovery/pr327/pr233/bundles.py": {
+     "path": "research/complex-p10/discovery/pr233/bundles.py",
+     "sha256": "71c81bc858e62e03e2a5230c16c4c199a027c24402cdc937bd4ded60f4ee3286",
+     "git_blob": "2035223567be249869e1721b4cafb4877ef8387c"
+    },
+    "discovery/pr327/pr233/joint.py": {
+     "path": "research/complex-p10/discovery/pr233/joint.py",
+     "sha256": "e4857418f6e16d6fbb1815d3c4148116e35a6b9940cb57acb2b563da5621b92c",
+     "git_blob": "a67d18034d35038de750945d8f46c15e96546911"
+    },
+    "discovery/pr327/pr233/matching.py": {
+     "path": "research/complex-p10/discovery/pr233/matching.py",
+     "sha256": "59d1cd30e67d3511cf799140eb248e785e77aaccff7c47a22a0feef8fb6f021e",
+     "git_blob": "e84e348556fe752843c0ed9f4e09a5746aedcd90"
+    },
+    "discovery/pr327/pr233/pairs_mw.py": {
+     "path": "research/complex-p10/discovery/pr233/pairs_mw.py",
+     "sha256": "aa927718ad27bebcfa2bddeb3704d4721433c2d4e6b6e218ceb9d28a07cd6c83",
+     "git_blob": "7e0db9777e8d906411d8218af1eae5eb72f44ae7"
+    },
+    "discovery/pr327/pr233/physical_opt.py": {
+     "path": "research/complex-p10/discovery/pr233/physical_opt.py",
+     "sha256": "cbfd06026a44df30944781262848c88ec987ca5e80381e35ae7ad715a9db1fce",
+     "git_blob": "a2e8e7d053074137d319be4735bfc904c54ced74"
+    }
+   }
+  },
+  "pr325": {
+   "repository": "CrocSwap/integer-mult-bounds",
+   "pr": 325,
+   "commit": "c4f1b74bd580fc5a18fec6f12ab498dca01b5533",
+   "path": "research/five-stage-p10b-banks/bitword/producer/data",
+   "author": "DreamingOfClouds (Anthropic Claude assistance)",
+   "note": "the p = 10 bit word's pair module pm_p10.json (re-annealed for #325: 36 inputs plus the star-centre root, 273 additions) and all-but-one module qmod_p10.json (#315's, 24 additions, kept unchanged by #325); used here, unchanged, as the complex word's query modules",
+   "files": {
+    "modules/pm_p10.json": {
+     "path": "research/five-stage-p10b-banks/bitword/producer/data/pm_p10.json",
+     "sha256": "8d71f5004acb0ea902ad594a7d298f8d2099114e5d6d42c6357f09c44ab3ce87",
+     "git_blob": "66d54f6efbdc6e273111dbfbfe6aabac9a9fd1f7"
+    },
+    "modules/qmod_p10.json": {
+     "path": "research/five-stage-p10b-banks/bitword/producer/data/qmod_p10.json",
+     "sha256": "40ffb168466bc016bc18a2bbdde65d38b7fd1b406d8a4026f1a6e517fe153459",
+     "git_blob": "b837d6e20978ba4d8c4c6b3ad8193a63c5652ea9"
+    }
+   }
+  },
+  "pr346": {
+   "repository": "CrocSwap/integer-mult-bounds (head LJH-217/integer-mult-bounds w3-p10b-complex-p10)",
+   "pr": 346,
+   "commit": "11b4de5ff8ed5fccf55d4af1043358d257308e8e",
+   "path": "research/w3-p10b-complex-p10/",
+   "author": "LJH-217 / Louis Harrison (Anthropic Claude assistance)",
+   "note": "the final w3 bit word on PR #325's p10b word (PR #310 Design T and twin condensation ported to h = 20): its final records, their hash list, its kappa receipt (normalized five-stage histogram, stock, bit coarse saving), licence and notice",
+   "files": {
+    "bit/pr346/FINAL-RECORDS.bin.gz": {
+     "path": "research/w3-p10b-complex-p10/data/bit/FINAL-RECORDS.bin.gz",
+     "sha256": "fc4f1986da4585dc1721bcd954a95b3ffb2b562cc7031f40242de81664e3acb6",
+     "git_blob": "884af231daea2b9d5fdb7d44b28e3126c42c9fab"
+    },
+    "bit/pr346/RECORD-SHA256SUMS": {
+     "path": "research/w3-p10b-complex-p10/data/bit/RECORD-SHA256SUMS",
+     "sha256": "21f732671eb34468be2252d4dfb81bdec4e7d7cd6564b93a482c2dd925e9c3b2",
+     "git_blob": "6525e19277a941f45d75ab9235c72fd9a0ac0bc8"
+    },
+    "bit/pr346/kappa-L10f.json": {
+     "path": "research/w3-p10b-complex-p10/receipts/kappa-L10f.json",
+     "sha256": "5b93dee60fb047f70f0b20e3e4bb84f3992515fbc39fc9d27788cd9ae2d950a8",
+     "git_blob": "8b3c2025a50459be6b63f0b5c78be85d19324e30"
+    },
+    "bit/pr346/LICENSE": {
+     "path": "research/w3-p10b-complex-p10/LICENSE",
+     "sha256": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4",
+     "git_blob": "261eeb9e9f8b2b4b0d119366dda99c6fd7d35c64"
+    },
+    "bit/pr346/NOTICE.md": {
+     "path": "research/w3-p10b-complex-p10/NOTICE.md",
+     "sha256": "02b59cc0542c78f409d30ce4fd546a806b606b76e185acd9aad1e32deb705008",
+     "git_blob": "6d5643cc768697bae1e5c42668492ee446eb4fb0"
+    }
+   }
+  },
+  "pr329": {
+   "repository": "CrocSwap/integer-mult-bounds (head chafreaky/integer-mult-bounds p10b-stages)",
+   "pr": 329,
+   "commit": "c6b1a28fcc3cf00c9bb129e184cc4d95035c7892",
+   "path": "research/five-stage-p10b-transcript-stages",
+   "author": "chafreaky (Anthropic Claude assistance); outer.py is PR #315's unchanged (eumemic, PR144/PR23 lineage)",
+   "note": "outer.py: the 47-constraint outer assembly; byte-identical to PR #315's and PR #346's code/pricing/outer.py",
+   "files": {
+    "pricing/outer.py": {
+     "path": "research/five-stage-p10b-transcript-stages/outer.py",
+     "sha256": "c1ebb52163a7219504c728a5c2b098bf1e50bf20148a2879c5a74c3cadc7c03c",
+     "git_blob": "43007488ad2c43187c77cc6798beff8b1cd76ad2"
+    }
+   }
+  }
+ },
+ "anchors": {
+  "pr194_certificate_sha256": "27398a0cfcbe34e5ee97360e69a2d163a417c8ae1b4fdf28ccf89a9eaf4e4cd4",
+  "note": "PR #256's certificate of PR #194's word (canonical uncompressed sha256), as pinned by PR #327's SOURCE.json"
+ },
+ "pr346": {
+  "commit": "11b4de5ff8ed5fccf55d4af1043358d257308e8e",
+  "complex_coarse": "396604388013523/500000000000000000",
+  "bit_coarse": "32069035133443/40000000000000000",
+  "binding": "complex",
+  "kappa": "396290046684973/500000000000000000",
+  "note": "PR #346's complex supplier (gcert1-p10-L10f, flow R 6265) has the same coarse saving as the Lean-checked 7932/10^7 program of spec/uniform.json"
+ },
+ "lean": {
+  "this_program": "not built in Lean (pending: the Lean host needs an interactive re-authentication)",
+  "companion": "spec/uniform.json program (canonical sha256 56e1139aa4a420219d37b9af00d8894d4c21df83b3d9c266d36a787676e9ad4e, 7932/10^7): OAI.PowerSaving.WHT.wht_main_block_B2Gcu32x at 1 - 7932084/10^10, wht-power-saving-lean f010392, Compare.lean PASS"
+ },
+ "prerequisites": {
+  "pr202": {
+   "commit": "8d8d67bcf69c5ea67d3a29dbc64ba588156d6e8d",
+   "note": "the 308 files rebuilt by references/pr233-layer/verify.py: PR #184 flow/lift (icekylinx), PR #194 v4 scripts (ikeboy), PR #168 modules (eumemic), PR #200 checker (chafreaky)",
+   "repository": "CrocSwap/integer-mult-bounds"
+  },
+  "pr225": {
+   "note": "scripts/moment.py: exact rational moment bounds (chafreaky; adapted from eumemic/exact-dft-bounds scripts/verify_moment.py, Apache-2.0)",
+   "path": "research/exact-dft-transfer/scripts/moment.py",
+   "pr": 225,
+   "repository": "CrocSwap/integer-mult-bounds"
+  },
+  "pr233": {
+   "author": "chafreaky (Anthropic Claude and OpenAI Codex assistance); PR #202 baseline by ikeboy et al., see NOTICE",
+   "commit": "109a857",
+   "complex_saving": "3549537/5000000000",
+   "path": "research/source-assisted-v4-layer",
+   "pr": 233,
+   "repository": "CrocSwap/integer-mult-bounds",
+   "vendored": "references/pr233-layer (baseline-pr202.tar.gz, SOURCE.json, verify.py, certificate.json, NOTICE, witness/), as vendored by PR #256"
+  },
+  "pr256": {
+   "author": "chafreaky (Anthropic Claude and OpenAI Codex assistance)",
+   "commit": "db3f75cdc5f03f3131d48fb220f6bf1958404ff3",
+   "path": "research/gcert-program-233",
+   "pr": 256,
+   "pr194_certificate_sha256": "27398a0cfcbe34e5ee97360e69a2d163a417c8ae1b4fdf28ccf89a9eaf4e4cd4",
+   "pr233_certificate_sha256": "0c026244591a0b6737993aaac5fa8ac4a2bd27aa28e469f2567c698f57ed10a8",
+   "pr233_five_stage": "7547/10^7",
+   "repository": "CrocSwap/integer-mult-bounds",
+   "vendored": "gcert_emit.py, gx/gx.py, scripts/moment.py, references/pr233-layer/, references/sussman/pr193-blocks.json (byte for byte; the certificate sha256 values are of the canonical uncompressed JSON, from its certificate/expected.json)"
+  },
+  "pr304": {
+   "pr": 304,
+   "repository": "CrocSwap/integer-mult-bounds",
+   "commit": "7ac169c58c96cebdb5cd490842331dff3610af62",
+   "path": "research/complex-module-anneal",
+   "author": "DreamingOfClouds (Anthropic Claude assistance)",
+   "claims": {
+    "cmod-mw": "7635/10^7",
+    "cmod-parity": "7586/10^7",
+    "pr168-v4-mw": "7522/10^7",
+    "pr168-v4-parity": "7469/10^7"
+   },
+   "vendored": "aligned_word_p.py (= its aligned_word.py with --p and --centre46), paircheck.py, normalize_scatter.py, discovery/make_layer_p.py (= its discovery/make_layer.py with --p, --centre46, pairs_mw_p.py), discovery/extend_floor.py and discovery/pr233/ (byte for byte); verify.py follows its verify.py",
+   "note": "the p = 11 programs whose recipe this package applies at p = 10: the re-annealed modules tmod_cmod_v2.json, pmod_cmod_dropin.json and qmod_cmod.json are the parents of three of the four p = 10 modules"
+  },
+  "pr315": {
+   "pr": 315,
+   "repository": "CrocSwap/integer-mult-bounds",
+   "commit": "243899975d0fbe888d5deda399060885d609e0a1",
+   "path": "research/five-stage-p10-banks",
+   "author": "DreamingOfClouds (Anthropic Claude assistance)",
+   "complex_supplier": "inputs/complex/gcert1-p11-cmod-centre-mw-flow.json.gz: the centre-sharing p = 11 program, 7727/10^7 (with the 10^-16 fallback 772714351296671/10^18)",
+   "vendored": "references/centre46_graph.diff (its inputs/complex/centre-mw/centre46_graph.diff, byte for byte: the 10-line Graph.finish patch whose effect centre46.py reproduces)"
+  },
+  "source527": {
+   "repository": "CrocSwap/integer-mult-bounds",
+   "commit": "b07cafe23202d22d832961a04b07d48d9b245e10",
+   "path": "research/five-stage-source527-banks",
+   "author": "eumemic (OpenAI Codex assistance), PR #210 lineage; see scripts/source527/NOTICE",
+   "vendored": "scripts/source527/moment.py and scripts/source527/base_two_moment.py, byte for byte: the two rational moment engines with which the five-stage packages certify a coarse saving with the 10^-16 fallback envelope"
+  },
+  "sussman": {
+   "commit": "f010392c",
+   "files": {
+    "tools/certificate/gcert1-p11-pr193.json.gz": "15251deb0c11e0504d7b297c9786818888a93f08b92b03c6c122784b70488633 (its block histogram and counts in references/sussman/pr193-blocks.json)",
+    "tools/gx/gx.py": "6f3dc8bb639933227a8dae181ec4a542e85c7276be9c9e330f49518e7b630b95 (vendored unchanged as gx/gx.py)",
+    "tools/gx/gxcore.py": "eb972c06fe8c322db2d9a8d9a7d05596e8f40c4ca8e5f42eb6df3c7361b5d517 (vendored unchanged as gx/gxcore.py: his Python mirror of the two Lean checks, independent of gx.py)"
+   },
+   "licence": "Apache-2.0, Copyright 2026 Jacob Sussman",
+   "repository": "jacobalansussman/wht-power-saving-lean"
+  },
+  "pr327": {
+   "repository": "CrocSwap/integer-mult-bounds",
+   "pr": 327,
+   "commit": "40ed0456ab1990beb3440e61bb883ccf95fdcea4",
+   "path": "research/complex-p10"
+  },
+  "pr325": {
+   "repository": "CrocSwap/integer-mult-bounds",
+   "pr": 325,
+   "commit": "c4f1b74bd580fc5a18fec6f12ab498dca01b5533",
+   "path": "research/five-stage-p10b-banks"
+  },
+  "pr346": {
+   "repository": "CrocSwap/integer-mult-bounds",
+   "pr": 346,
+   "commit": "11b4de5ff8ed5fccf55d4af1043358d257308e8e",
+   "path": "research/w3-p10b-complex-p10/"
+  }
+ }
+}
diff --git a/research/w3-p10b-pointwise-complex/aligned_word_p.py b/research/w3-p10b-pointwise-complex/aligned_word_p.py
new file mode 100644
index 0000000000..98da711519
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/aligned_word_p.py
@@ -0,0 +1,265 @@
+#!/usr/bin/env python3
+"""aligned_word_p.py: PR #194's source-aligned complex word for given query modules and a frozen physical layer, at any
+cube size p.
+
+PR #304's aligned_word.py with three changes: (1) --p (default 11): Graph(p) and the module sizes p, p - 1, p - 2
+replace the literals 11, 10, 9; (2) --centre46: the pair module carries a 37th root (the sum of all its inputs) that
+supplies the star centre (centre46.py, the effect of PR #315's Graph.finish patch); (3) when select() chooses gauges
+that the script then drops (rank other than h - 4, or unpaired), the record's gauge ledger (selected_roles,
+selected_rank_histogram, remaining_internal_histogram) is corrected to the kept gauges, so that PR #200's checker can
+recount it; nothing is dropped at p = 11 or on the annealed p = 10 modules, so the output is then unchanged. At
+p = 11 without --centre46 it is aligned_word.py.
+
+This is research/source-assisted-v4/source_aligned_local_v4.py of PR #202 (ikeboy, PR #193/#194; derived from
+icekylinx's research/source-assisted/decision/source_aligned_local.py, GPT-6 Astra assistance, Apache-2.0) with
+its four discovery steps replaced by inputs:
+  * the three query modules are --tmod/--pmod/--qmod (the script hard-codes PR #168 v4's three files); they are
+    loaded and contract-checked by the tree's own loaders (triple_module_from, pair_module_from, all_but_one_from);
+  * the carrier arcs are --arcs (the script maps PR #168's frozen matching through the PR #168 v4 word); the tree's
+    compile_closure() checks them (one donor per use, acyclic generalized dependencies, every value span inside its
+    frame, exact positive-rank ledger);
+  * the preferred frame of every query operation is its frame in --frames (the script inherits PR #168's physical
+    frames). The forward closure, the clipping to the exact future cap and the minimal-frame check of every local
+    pair are the script's, unchanged, and the closure must return every frozen frame unchanged;
+  * the reuse pairs are --pairs (as the script's own --pairs option). Each pair must be an edge of the script's
+    candidate graph (donor ungauged and not a root, its last frame inside the recipient's gauge frame, its last
+    operation before the recipient's first, deadline as the script assigns it) and the list must already be a
+    fixed point of the script's last-late-control reordering.
+Everything else, in the script's order: local configuration (G all 's', A all 'fd'), Graph.finish, merge
+f8:00111100, compile_closure, select, the rank-at-most-two local phase-one prefix with its role predecessors, gauge
+filtering, the target data histogram, the exact physical checks of scripts/paired_cube_physical.py (nested chains,
+pair legality, spliced recount, exact numeric replay with mutations rejected) and the files written: cache/
+{graph,frames,selection,record}.json, references/paired-cube/physical/{frames,pairs}.json and
+certificates/paired-cube-sinks-input.json, byte for byte in the script's encoding. Given PR #168's modules and the
+script's own arcs, frames and pairs it reproduces the script's output byte for byte (verify.py checks this).
+
+usage: aligned_word_p.py [--p P] [--centre46] --tree TREE --tmod F --pmod F --qmod F --arcs F --frames F --pairs F --out DIR
+Run with the rebuilt PR #202 tree as working directory, as the upstream scripts are; -O is refused.
+"""
+from collections import Counter, defaultdict
+from pathlib import Path
+import argparse
+import json
+import sys
+import time
+
+if sys.flags.optimize:
+    raise SystemExit('Assertions must remain enabled: run without -O')
+
+
+def read(p):
+    return json.loads(Path(p).read_text())
+
+
+def write(p, d):
+    p.parent.mkdir(parents=True, exist_ok=True)
+    p.write_text(json.dumps(d, separators=(',', ':')) + '\n')
+
+
+def main():
+    ap = argparse.ArgumentParser(description=__doc__, formatter_class=argparse.RawDescriptionHelpFormatter)
+    ap.add_argument('--tree', type=Path, required=True, help='the rebuilt PR #202 tree (its scripts/ are imported)')
+    ap.add_argument('--tmod', type=Path, required=True, help='disjoint-triple module (triple_module_from, p = 11)')
+    ap.add_argument('--pmod', type=Path, required=True, help='pair module (pair_module_from, n = 10)')
+    ap.add_argument('--qmod', type=Path, required=True, help='all-but-one module (all_but_one_from, n = 9)')
+    ap.add_argument('--arcs', type=Path, required=True, help='{"matching_arcs": [[donor node, use code], ...]}')
+    ap.add_argument('--frames', type=Path, required=True, help='{"frames": [[operation, frame rows], ...]}')
+    ap.add_argument('--pairs', type=Path, required=True, help='{"pairs": [[donor, recipient, deadline], ...]}')
+    ap.add_argument('--out', type=Path, required=True)
+    ap.add_argument('--p', type=int, default=11, help='cube size p (cp10 generalisation; 11 = the original script)')
+    ap.add_argument('--centre46', action='store_true', help='cp10: pair module with an extra full-sum root supplies the star centre (agent-cmod finish_centre46)')
+    a = ap.parse_args()
+    start = time.monotonic()
+    sys.path.insert(0, str(a.tree / 'scripts'))
+    from paired_cube.graph import Graph
+    from paired_cube.modules import triple_module_from, pair_module_from, all_but_one_from, merge_outputs
+    from paired_cube.closure import compile_closure
+    from paired_cube.gauges import select
+    from paired_cube.frames import basis, perp, contained
+    import paired_cube_physical as physical
+    s = a.tree / 'references/paired-cube/sources'
+    local = read(s / 'local_L1.json')
+    local['G'] = {k: 's' for k in local['G']}
+    local['A'] = {k: 'fd' for k in local['A']}
+    builder = Graph(a.p, local=local)
+    if a.centre46:
+        sys.path.insert(0, str(Path(__file__).resolve().parent))
+        from centre46 import load_pmod46, finish_centre46
+        g = finish_centre46(builder, triple_module_from(a.tmod, a.p), load_pmod46(a.pmod, a.p - 1), all_but_one_from(a.qmod, a.p - 2))
+    else:
+        g = builder.finish(triple_module_from(a.tmod, a.p), pair_module_from(a.pmod, a.p - 1), all_but_one_from(a.qmod, a.p - 2))
+    g = merge_outputs(g, builder, 'f8:00111100')
+    g['matching_frames'] = 'coordinate'
+    h, v = g['h'], g['v']
+    newcut = v + g['counts']['local_channel_additions']
+    newargs = [None] + [None if x is None else [t + 1 for t in x] for x in g['args']]
+    newroots = [dict(r, node=r['node'] + 1) for r in g['roots']]
+    # frozen carrier arcs (in place of the arcs mapped from PR #168's matching)
+    arcs = read(a.arcs)['matching_arcs']
+    assert len({x for x, _ in arcs}) == len(arcs), 'one arc per donor'
+    p, w = compile_closure(g, arcs)
+    assert w['matching_arcs'] == arcs, 'compile_closure did not keep the frozen arcs'
+    p, word = select(g, p, w)
+    assert all(ca in (-1, 1) and cb in (-1, 1) for ca, cb in word['opcoeff'])
+    print(json.dumps(dict(stage='word', local=g['counts']['local_channel_additions'], arcs=len(arcs), R=p['R'],
+                          selected=p['selected_rank_histogram'], seconds=time.monotonic() - start)), flush=True)
+    ops = word['ops']
+    # frozen preferred frames (in place of the frames inherited from PR #168's physical layer)
+    frames_in = read(a.frames)['frames']
+    desired = [perp(tuple(w['annihilators'][x]), h) for _, _, x in ops]
+    for i, F in frames_in:
+        desired[i] = tuple(F)
+    spans = [()] + [(q,) for q in g['inputs']]
+    for aa, bb in newargs[v + 1:]:
+        spans.append(basis(spans[aa] + spans[bb]))
+    root_frames = {}
+    for r, slot in zip(newroots, word['rootroles']):
+        root_frames[slot] = spans[r['node']] if r['kind'] == 'center' else perp(basis(g['inputs'][t] for t in r['targets']), h)
+    R = p['R']
+    pset = set(word['phase1'])
+    # Both signs of a pair must occur before the center cut (source_aligned_local_v4.py, unchanged).
+    previous = [-1] * R
+    parents = []
+    for i, (aa, bb, x) in enumerate(ops):
+        parents.append((previous[aa], previous[bb]))
+        previous[aa] = previous[bb] = i
+        if x <= newcut and len(spans[x]) <= 2:
+            pset.add(i)
+    todo = list(pset)
+    while todo:
+        i = todo.pop()
+        for j in parents[i]:
+            if j >= 0 and j not in pset:
+                pset.add(j)
+                todo.append(j)
+    word['phase1'] = sorted(pset)
+    p['phase1_operations'] = len(pset)
+    chron = sorted(pset) + [i for i in range(len(ops)) if i not in pset]
+    cap = [perp(root_frames[s], h) if s in root_frames else () for s in range(R)]
+    maxima = [None] * len(ops)
+    for i in reversed(chron):
+        aa, bb, x = ops[i]
+        A = basis(cap[aa] + cap[bb])
+        maxima[i] = perp(A, h)
+        cap[aa] = cap[bb] = A
+        assert contained(spans[x], maxima[i])
+    selected = [z for z in word['selected'] if z['rank'] == h - 4]
+    gauge = {z['role']: perp(tuple(z['annihilator']), h) for z in selected}
+    current = [() for _ in range(R)]
+    for x, slot in word['sources'].items():
+        current[slot] = (g['inputs'][int(x) - 1],)
+    for slot, U in gauge.items():
+        current[slot] = U
+    frames = [None] * len(ops)
+    minimal_pairs = query_ops = 0
+    for i in chron:
+        aa, bb, x = ops[i]
+        if x <= newcut:
+            want = spans[x]
+        else:
+            want = desired[i]
+            query_ops += 1
+            want = perp(basis(perp(want, h) + perp(maxima[i], h)), h)
+        U = basis(spans[x] + want + current[aa] + current[bb])
+        assert contained(U, maxima[i]), ('Forward frame outside future cap', i)
+        frames[i] = current[aa] = current[bb] = U
+        if x <= newcut and len(spans[x]) == 2:
+            assert U == spans[x], ('Local pair failed its minimal physical frame', i)
+            minimal_pairs += 1
+    for slot, U in root_frames.items():
+        assert contained(current[slot], U)
+    bad = [i for i in range(len(ops)) if frames[i] != desired[i]]
+    assert not bad, ('the forward closure changes %d frozen frames, first at operation %d' % (len(bad), bad[0]))
+    role_ops = defaultdict(list)
+    for i, (aa, bb, _) in enumerate(ops):
+        role_ops[aa].append(i)
+        role_ops[bb].append(i)
+    pos = {i: k for k, i in enumerate(chron)}
+    for slot in role_ops:
+        role_ops[slot].sort(key=pos.__getitem__)
+    recipients = sorted(gauge)
+    recipient_set = set(recipients)
+    donors = {slot for slot in range(R) if slot not in recipient_set and slot not in root_frames and slot in role_ops}
+    for slot in recipients:
+        assert role_ops[slot][0] not in pset
+    # frozen reuse pairs (in place of the script's matching; the script's own --pairs option)
+    pairs = read(a.pairs)['pairs']
+    assert len({d for d, _, _ in pairs}) == len({b for _, b, _ in pairs}) == len(pairs), 'one pair per donor and recipient'
+    for d, b, t in pairs:
+        assert d in donors and b in recipient_set, ('pair outside the candidate roles', d, b)
+        last, first = role_ops[d][-1], role_ops[b][0]
+        assert contained(frames[last], gauge[b]), ('donor last frame outside the recipient gauge', d, b)
+        assert pos[last] < pos[first], ('donor not dead before the recipient', d, b)
+        assert t == (None if last in pset else first), ('deadline differs from the script\'s rule', d, b, t)
+    # The inherited negative replay moves its last late recipient to the phase cut (unchanged); the frozen list
+    # must already be in that order.
+    last_write = {}
+    for i, (aa, bb, x) in enumerate(ops):
+        last_write[aa] = max(last_write.get(aa, -1), pos[i])
+    controls = [q for q in pairs if q[2] is not None and last_write.get(q[0], -1) >= len(pset)]
+    if controls:
+        chosen = controls[-1]
+        assert pairs[-1] == chosen, 'frozen pairs are not in the script\'s control order'
+    kept = {b for _, b, _ in pairs}
+    chosen = word['selected']
+    word['selected'] = [z for z in selected if z['role'] in kept]
+    # complex-p10: keep the record's gauge ledger equal to the gauges the word keeps. select() may choose gauges of
+    # rank other than h - 4 (or unpaired ones); the script drops them from the selection, but its record still counts
+    # their births, which PR #200's complex checker (verify_fused.py) recounts and rejects. Each dropped birth is
+    # reversed (H[r - d] -= 1, H[r] += 1 with r the rank of the role's first frame, as select() booked it). Nothing is
+    # dropped at p = 11 on PR #168 v4's or PR #304's modules, nor on this package's annealed modules: the record is then
+    # untouched (byte for byte). Only the verifiers read these three fields.
+    keep_roles = {z['role'] for z in word['selected']}
+    dropped = [z for z in chosen if z['role'] not in keep_roles]
+    if dropped:
+        first_x = {}
+        for aa, bb, x in ops:
+            first_x.setdefault(aa, x)
+            first_x.setdefault(bb, x)
+        H = list(p['remaining_internal_histogram'])
+        ranks = dict(p['selected_rank_histogram'])
+        for z in dropped:
+            r, d = h - len(w['annihilators'][first_x[z['role']]]), z['rank']
+            H[r - d] -= 1
+            H[r] += 1
+            ranks[d] -= 1
+            if not ranks[d]:
+                del ranks[d]
+        p['remaining_internal_histogram'], p['selected_rank_histogram'], p['selected_roles'] = H, ranks, len(word['selected'])
+    moved = [[i, list(U)] for i, U in enumerate(frames) if U != perp(tuple(w['annihilators'][ops[i][2]]), h)]
+    assert moved == frames_in, 'frozen frames are not in the script\'s canonical form'
+    deadline = {b: t for _, b, t in pairs}
+    Y = [() for _ in range(v)]
+    target = Counter()
+    for z in sorted(word['selected'], key=lambda z: len(pset) if deadline[z['role']] is None else pos[deadline[z['role']]]):
+        U = gauge[z['role']]
+        for t in z['targets']:
+            assert contained(Y[t], U)
+            target[len(U) - len(Y[t])] += 1
+            Y[t] = U
+    for r, slot in zip(newroots, word['rootroles']):
+        if r['kind'] == 'side':
+            U = root_frames[slot]
+            for t in r['targets']:
+                assert contained(Y[t], U)
+                target[len(U) - len(Y[t])] += 1
+                Y[t] = U
+    for t in range(v):
+        target[h - 1 - len(Y[t])] += 1
+    p['target_data_histogram'] = dict(target)
+    print(json.dumps(dict(stage='physical', query_ops=query_ops, moved_frames=len(moved), minimal_pair_ops=minimal_pairs,
+                          pairs=len(pairs), gauges=len(word['selected']), seconds=time.monotonic() - start,
+                          dropped_gauges=dict(sorted(Counter(z['rank'] for z in dropped).items())))), flush=True)
+    out = a.out
+    for name, data in [('graph.json', g), ('frames.json', w), ('selection.json', word), ('record.json', p)]:
+        write(out / 'cache' / name, data)
+    write(out / 'references/paired-cube/physical/frames.json', {'frames': moved})
+    write(out / 'references/paired-cube/physical/pairs.json', {'pairs': pairs})
+    result = physical.physical(g, w, word, p, moved, pairs)
+    write(out / 'certificates/paired-cube-sinks-input.json', result)
+    print(json.dumps(dict(stage='checked', R=p['R'], physical_R=result['physical_R'], arcs=len(arcs), moved_frames=len(moved),
+                          pairs=len(pairs), seconds=time.monotonic() - start)), flush=True)
+
+
+if __name__ == '__main__':
+    main()
diff --git a/research/w3-p10b-pointwise-complex/aligned_word_spec.py b/research/w3-p10b-pointwise-complex/aligned_word_spec.py
new file mode 100644
index 0000000000..f3cbf6789e
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/aligned_word_spec.py
@@ -0,0 +1,295 @@
+#!/usr/bin/env python3
+"""aligned_word_spec.py: PR #327's aligned_word_p.py with one change: the graph may be built from a per-invocation
+module spec (--spec SPEC.json, resolved against --spec-root): spec_graph.finish_spec, i.e. PR #327's
+centre46.finish_centre46 with the pair module of every invocation (i, bit) and the all-but-one module of every
+invocation (i, j, mode) looked up in the spec's tables, followed by the same merge_outputs(f8:00111100). With --spec
+the options --tmod/--pmod/--qmod are ignored (the spec names them) and --centre46 is implied. Without --spec it is
+aligned_word_p.py byte for byte in behaviour (verify.py checks the --spec path against aligned_word_p.py on a spec
+with one module per kind, and the source differs from aligned_word_p.py only in this docstring, the two options and
+the graph-building branch).
+Prepared by Chafik Boukhalfa with substantial Anthropic Claude assistance; Apache-2.0. aligned_word_p.py is
+DreamingOfClouds' (PR #327, from PR #304), on PR #202's source_aligned_local_v4.py (ikeboy) and icekylinx's
+source_aligned_local.py. Its docstring follows.
+
+aligned_word_p.py: PR #194's source-aligned complex word for given query modules and a frozen physical layer, at any
+cube size p.
+
+PR #304's aligned_word.py with three changes: (1) --p (default 11): Graph(p) and the module sizes p, p - 1, p - 2
+replace the literals 11, 10, 9; (2) --centre46: the pair module carries a 37th root (the sum of all its inputs) that
+supplies the star centre (centre46.py, the effect of PR #315's Graph.finish patch); (3) when select() chooses gauges
+that the script then drops (rank other than h - 4, or unpaired), the record's gauge ledger (selected_roles,
+selected_rank_histogram, remaining_internal_histogram) is corrected to the kept gauges, so that PR #200's checker can
+recount it; nothing is dropped at p = 11 or on the annealed p = 10 modules, so the output is then unchanged. At
+p = 11 without --centre46 it is aligned_word.py.
+
+This is research/source-assisted-v4/source_aligned_local_v4.py of PR #202 (ikeboy, PR #193/#194; derived from
+icekylinx's research/source-assisted/decision/source_aligned_local.py, GPT-6 Astra assistance, Apache-2.0) with
+its four discovery steps replaced by inputs:
+  * the three query modules are --tmod/--pmod/--qmod (the script hard-codes PR #168 v4's three files); they are
+    loaded and contract-checked by the tree's own loaders (triple_module_from, pair_module_from, all_but_one_from);
+  * the carrier arcs are --arcs (the script maps PR #168's frozen matching through the PR #168 v4 word); the tree's
+    compile_closure() checks them (one donor per use, acyclic generalized dependencies, every value span inside its
+    frame, exact positive-rank ledger);
+  * the preferred frame of every query operation is its frame in --frames (the script inherits PR #168's physical
+    frames). The forward closure, the clipping to the exact future cap and the minimal-frame check of every local
+    pair are the script's, unchanged, and the closure must return every frozen frame unchanged;
+  * the reuse pairs are --pairs (as the script's own --pairs option). Each pair must be an edge of the script's
+    candidate graph (donor ungauged and not a root, its last frame inside the recipient's gauge frame, its last
+    operation before the recipient's first, deadline as the script assigns it) and the list must already be a
+    fixed point of the script's last-late-control reordering.
+Everything else, in the script's order: local configuration (G all 's', A all 'fd'), Graph.finish, merge
+f8:00111100, compile_closure, select, the rank-at-most-two local phase-one prefix with its role predecessors, gauge
+filtering, the target data histogram, the exact physical checks of scripts/paired_cube_physical.py (nested chains,
+pair legality, spliced recount, exact numeric replay with mutations rejected) and the files written: cache/
+{graph,frames,selection,record}.json, references/paired-cube/physical/{frames,pairs}.json and
+certificates/paired-cube-sinks-input.json, byte for byte in the script's encoding. Given PR #168's modules and the
+script's own arcs, frames and pairs it reproduces the script's output byte for byte (verify.py checks this).
+
+usage: aligned_word_p.py [--p P] [--centre46] --tree TREE --tmod F --pmod F --qmod F --arcs F --frames F --pairs F --out DIR
+Run with the rebuilt PR #202 tree as working directory, as the upstream scripts are; -O is refused.
+"""
+from collections import Counter, defaultdict
+from pathlib import Path
+import argparse
+import json
+import sys
+import time
+
+if sys.flags.optimize:
+    raise SystemExit('Assertions must remain enabled: run without -O')
+
+
+def read(p):
+    return json.loads(Path(p).read_text())
+
+
+def write(p, d):
+    p.parent.mkdir(parents=True, exist_ok=True)
+    p.write_text(json.dumps(d, separators=(',', ':')) + '\n')
+
+
+def main():
+    ap = argparse.ArgumentParser(description=__doc__, formatter_class=argparse.RawDescriptionHelpFormatter)
+    ap.add_argument('--tree', type=Path, required=True, help='the rebuilt PR #202 tree (its scripts/ are imported)')
+    ap.add_argument('--tmod', type=Path, required=True, help='disjoint-triple module (triple_module_from, p = 11)')
+    ap.add_argument('--pmod', type=Path, required=True, help='pair module (pair_module_from, n = 10)')
+    ap.add_argument('--qmod', type=Path, required=True, help='all-but-one module (all_but_one_from, n = 9)')
+    ap.add_argument('--arcs', type=Path, required=True, help='{"matching_arcs": [[donor node, use code], ...]}')
+    ap.add_argument('--frames', type=Path, required=True, help='{"frames": [[operation, frame rows], ...]}')
+    ap.add_argument('--pairs', type=Path, required=True, help='{"pairs": [[donor, recipient, deadline], ...]}')
+    ap.add_argument('--out', type=Path, required=True)
+    ap.add_argument('--p', type=int, default=11, help='cube size p (cp10 generalisation; 11 = the original script)')
+    ap.add_argument('--centre46', action='store_true', help='cp10: pair module with an extra full-sum root supplies the star centre (agent-cmod finish_centre46)')
+    ap.add_argument('--spec', type=Path, default=None, help='per-invocation module spec (spec_graph.py); implies --centre46')
+    ap.add_argument('--spec-root', type=Path, default=None, help='directory the spec paths are relative to')
+    a = ap.parse_args()
+    start = time.monotonic()
+    sys.path.insert(0, str(a.tree / 'scripts'))
+    from paired_cube.graph import Graph
+    from paired_cube.modules import triple_module_from, pair_module_from, all_but_one_from, merge_outputs
+    from paired_cube.closure import compile_closure
+    from paired_cube.gauges import select
+    from paired_cube.frames import basis, perp, contained
+    import paired_cube_physical as physical
+    s = a.tree / 'references/paired-cube/sources'
+    local = read(s / 'local_L1.json')
+    local['G'] = {k: 's' for k in local['G']}
+    local['A'] = {k: 'fd' for k in local['A']}
+    builder = Graph(a.p, local=local)
+    if a.spec is not None:
+        sys.path.insert(0, str(Path(__file__).resolve().parent))
+        from centre46 import load_pmod46
+        import spec_graph
+        spec = read(a.spec)
+        assert spec['p'] == a.p and spec.get('centre46', True), 'spec is not a centre-sharing p = %d spec' % a.p
+        table = spec_graph.expand(spec, a.spec_root or a.spec.parent)
+        loaded = {}
+
+        def get(path, loader, n):
+            if path not in loaded:
+                loaded[path] = loader(path, n)
+            return loaded[path]
+        g = spec_graph.finish_spec(builder, triple_module_from((a.spec_root or a.spec.parent) / spec['tmod'], a.p),
+                                   lambda i, bit: get(table['p', i, bit], load_pmod46, a.p - 1),
+                                   lambda i, j, mode: get(table['q', i, j, mode], all_but_one_from, a.p - 2))
+    elif a.centre46:
+        sys.path.insert(0, str(Path(__file__).resolve().parent))
+        from centre46 import load_pmod46, finish_centre46
+        g = finish_centre46(builder, triple_module_from(a.tmod, a.p), load_pmod46(a.pmod, a.p - 1), all_but_one_from(a.qmod, a.p - 2))
+    else:
+        g = builder.finish(triple_module_from(a.tmod, a.p), pair_module_from(a.pmod, a.p - 1), all_but_one_from(a.qmod, a.p - 2))
+    g = merge_outputs(g, builder, 'f8:00111100')
+    g['matching_frames'] = 'coordinate'
+    h, v = g['h'], g['v']
+    newcut = v + g['counts']['local_channel_additions']
+    newargs = [None] + [None if x is None else [t + 1 for t in x] for x in g['args']]
+    newroots = [dict(r, node=r['node'] + 1) for r in g['roots']]
+    # frozen carrier arcs (in place of the arcs mapped from PR #168's matching)
+    arcs = read(a.arcs)['matching_arcs']
+    assert len({x for x, _ in arcs}) == len(arcs), 'one arc per donor'
+    p, w = compile_closure(g, arcs)
+    assert w['matching_arcs'] == arcs, 'compile_closure did not keep the frozen arcs'
+    p, word = select(g, p, w)
+    assert all(ca in (-1, 1) and cb in (-1, 1) for ca, cb in word['opcoeff'])
+    print(json.dumps(dict(stage='word', local=g['counts']['local_channel_additions'], arcs=len(arcs), R=p['R'],
+                          selected=p['selected_rank_histogram'], seconds=time.monotonic() - start)), flush=True)
+    ops = word['ops']
+    # frozen preferred frames (in place of the frames inherited from PR #168's physical layer)
+    frames_in = read(a.frames)['frames']
+    desired = [perp(tuple(w['annihilators'][x]), h) for _, _, x in ops]
+    for i, F in frames_in:
+        desired[i] = tuple(F)
+    spans = [()] + [(q,) for q in g['inputs']]
+    for aa, bb in newargs[v + 1:]:
+        spans.append(basis(spans[aa] + spans[bb]))
+    root_frames = {}
+    for r, slot in zip(newroots, word['rootroles']):
+        root_frames[slot] = spans[r['node']] if r['kind'] == 'center' else perp(basis(g['inputs'][t] for t in r['targets']), h)
+    R = p['R']
+    pset = set(word['phase1'])
+    # Both signs of a pair must occur before the center cut (source_aligned_local_v4.py, unchanged).
+    previous = [-1] * R
+    parents = []
+    for i, (aa, bb, x) in enumerate(ops):
+        parents.append((previous[aa], previous[bb]))
+        previous[aa] = previous[bb] = i
+        if x <= newcut and len(spans[x]) <= 2:
+            pset.add(i)
+    todo = list(pset)
+    while todo:
+        i = todo.pop()
+        for j in parents[i]:
+            if j >= 0 and j not in pset:
+                pset.add(j)
+                todo.append(j)
+    word['phase1'] = sorted(pset)
+    p['phase1_operations'] = len(pset)
+    chron = sorted(pset) + [i for i in range(len(ops)) if i not in pset]
+    cap = [perp(root_frames[s], h) if s in root_frames else () for s in range(R)]
+    maxima = [None] * len(ops)
+    for i in reversed(chron):
+        aa, bb, x = ops[i]
+        A = basis(cap[aa] + cap[bb])
+        maxima[i] = perp(A, h)
+        cap[aa] = cap[bb] = A
+        assert contained(spans[x], maxima[i])
+    selected = [z for z in word['selected'] if z['rank'] == h - 4]
+    gauge = {z['role']: perp(tuple(z['annihilator']), h) for z in selected}
+    current = [() for _ in range(R)]
+    for x, slot in word['sources'].items():
+        current[slot] = (g['inputs'][int(x) - 1],)
+    for slot, U in gauge.items():
+        current[slot] = U
+    frames = [None] * len(ops)
+    minimal_pairs = query_ops = 0
+    for i in chron:
+        aa, bb, x = ops[i]
+        if x <= newcut:
+            want = spans[x]
+        else:
+            want = desired[i]
+            query_ops += 1
+            want = perp(basis(perp(want, h) + perp(maxima[i], h)), h)
+        U = basis(spans[x] + want + current[aa] + current[bb])
+        assert contained(U, maxima[i]), ('Forward frame outside future cap', i)
+        frames[i] = current[aa] = current[bb] = U
+        if x <= newcut and len(spans[x]) == 2:
+            assert U == spans[x], ('Local pair failed its minimal physical frame', i)
+            minimal_pairs += 1
+    for slot, U in root_frames.items():
+        assert contained(current[slot], U)
+    bad = [i for i in range(len(ops)) if frames[i] != desired[i]]
+    assert not bad, ('the forward closure changes %d frozen frames, first at operation %d' % (len(bad), bad[0]))
+    role_ops = defaultdict(list)
+    for i, (aa, bb, _) in enumerate(ops):
+        role_ops[aa].append(i)
+        role_ops[bb].append(i)
+    pos = {i: k for k, i in enumerate(chron)}
+    for slot in role_ops:
+        role_ops[slot].sort(key=pos.__getitem__)
+    recipients = sorted(gauge)
+    recipient_set = set(recipients)
+    donors = {slot for slot in range(R) if slot not in recipient_set and slot not in root_frames and slot in role_ops}
+    for slot in recipients:
+        assert role_ops[slot][0] not in pset
+    # frozen reuse pairs (in place of the script's matching; the script's own --pairs option)
+    pairs = read(a.pairs)['pairs']
+    assert len({d for d, _, _ in pairs}) == len({b for _, b, _ in pairs}) == len(pairs), 'one pair per donor and recipient'
+    for d, b, t in pairs:
+        assert d in donors and b in recipient_set, ('pair outside the candidate roles', d, b)
+        last, first = role_ops[d][-1], role_ops[b][0]
+        assert contained(frames[last], gauge[b]), ('donor last frame outside the recipient gauge', d, b)
+        assert pos[last] < pos[first], ('donor not dead before the recipient', d, b)
+        assert t == (None if last in pset else first), ('deadline differs from the script\'s rule', d, b, t)
+    # The inherited negative replay moves its last late recipient to the phase cut (unchanged); the frozen list
+    # must already be in that order.
+    last_write = {}
+    for i, (aa, bb, x) in enumerate(ops):
+        last_write[aa] = max(last_write.get(aa, -1), pos[i])
+    controls = [q for q in pairs if q[2] is not None and last_write.get(q[0], -1) >= len(pset)]
+    if controls:
+        chosen = controls[-1]
+        assert pairs[-1] == chosen, 'frozen pairs are not in the script\'s control order'
+    kept = {b for _, b, _ in pairs}
+    chosen = word['selected']
+    word['selected'] = [z for z in selected if z['role'] in kept]
+    # complex-p10: keep the record's gauge ledger equal to the gauges the word keeps. select() may choose gauges of
+    # rank other than h - 4 (or unpaired ones); the script drops them from the selection, but its record still counts
+    # their births, which PR #200's complex checker (verify_fused.py) recounts and rejects. Each dropped birth is
+    # reversed (H[r - d] -= 1, H[r] += 1 with r the rank of the role's first frame, as select() booked it). Nothing is
+    # dropped at p = 11 on PR #168 v4's or PR #304's modules, nor on this package's annealed modules: the record is then
+    # untouched (byte for byte). Only the verifiers read these three fields.
+    keep_roles = {z['role'] for z in word['selected']}
+    dropped = [z for z in chosen if z['role'] not in keep_roles]
+    if dropped:
+        first_x = {}
+        for aa, bb, x in ops:
+            first_x.setdefault(aa, x)
+            first_x.setdefault(bb, x)
+        H = list(p['remaining_internal_histogram'])
+        ranks = dict(p['selected_rank_histogram'])
+        for z in dropped:
+            r, d = h - len(w['annihilators'][first_x[z['role']]]), z['rank']
+            H[r - d] -= 1
+            H[r] += 1
+            ranks[d] -= 1
+            if not ranks[d]:
+                del ranks[d]
+        p['remaining_internal_histogram'], p['selected_rank_histogram'], p['selected_roles'] = H, ranks, len(word['selected'])
+    moved = [[i, list(U)] for i, U in enumerate(frames) if U != perp(tuple(w['annihilators'][ops[i][2]]), h)]
+    assert moved == frames_in, 'frozen frames are not in the script\'s canonical form'
+    deadline = {b: t for _, b, t in pairs}
+    Y = [() for _ in range(v)]
+    target = Counter()
+    for z in sorted(word['selected'], key=lambda z: len(pset) if deadline[z['role']] is None else pos[deadline[z['role']]]):
+        U = gauge[z['role']]
+        for t in z['targets']:
+            assert contained(Y[t], U)
+            target[len(U) - len(Y[t])] += 1
+            Y[t] = U
+    for r, slot in zip(newroots, word['rootroles']):
+        if r['kind'] == 'side':
+            U = root_frames[slot]
+            for t in r['targets']:
+                assert contained(Y[t], U)
+                target[len(U) - len(Y[t])] += 1
+                Y[t] = U
+    for t in range(v):
+        target[h - 1 - len(Y[t])] += 1
+    p['target_data_histogram'] = dict(target)
+    print(json.dumps(dict(stage='physical', query_ops=query_ops, moved_frames=len(moved), minimal_pair_ops=minimal_pairs,
+                          pairs=len(pairs), gauges=len(word['selected']), seconds=time.monotonic() - start,
+                          dropped_gauges=dict(sorted(Counter(z['rank'] for z in dropped).items())))), flush=True)
+    out = a.out
+    for name, data in [('graph.json', g), ('frames.json', w), ('selection.json', word), ('record.json', p)]:
+        write(out / 'cache' / name, data)
+    write(out / 'references/paired-cube/physical/frames.json', {'frames': moved})
+    write(out / 'references/paired-cube/physical/pairs.json', {'pairs': pairs})
+    result = physical.physical(g, w, word, p, moved, pairs)
+    write(out / 'certificates/paired-cube-sinks-input.json', result)
+    print(json.dumps(dict(stage='checked', R=p['R'], physical_R=result['physical_R'], arcs=len(arcs), moved_frames=len(moved),
+                          pairs=len(pairs), seconds=time.monotonic() - start)), flush=True)
+
+
+if __name__ == '__main__':
+    main()
diff --git a/research/w3-p10b-pointwise-complex/bit/pr346/FINAL-RECORDS.bin.gz b/research/w3-p10b-pointwise-complex/bit/pr346/FINAL-RECORDS.bin.gz
new file mode 100644
index 0000000000..884af231da
Binary files /dev/null and b/research/w3-p10b-pointwise-complex/bit/pr346/FINAL-RECORDS.bin.gz differ
diff --git a/research/w3-p10b-pointwise-complex/bit/pr346/LICENSE b/research/w3-p10b-pointwise-complex/bit/pr346/LICENSE
new file mode 100644
index 0000000000..261eeb9e9f
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/bit/pr346/LICENSE
@@ -0,0 +1,201 @@
+                                 Apache License
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+      You may add Your own copyright statement to Your modifications and
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+      the conditions stated in this License.
+
+   5. Submission of Contributions. Unless You explicitly state otherwise,
+      any Contribution intentionally submitted for inclusion in the Work
+      by You to the Licensor shall be under the terms and conditions of
+      this License, without any additional terms or conditions.
+      Notwithstanding the above, nothing herein shall supersede or modify
+      the terms of any separate license agreement you may have executed
+      with Licensor regarding such Contributions.
+
+   6. Trademarks. This License does not grant permission to use the trade
+      names, trademarks, service marks, or product names of the Licensor,
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+      Contributor provides its Contributions) on an "AS IS" BASIS,
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+      PARTICULAR PURPOSE. You are solely responsible for determining the
+      appropriateness of using or redistributing the Work and assume any
+      risks associated with Your exercise of permissions under this License.
+
+   8. Limitation of Liability. In no event and under no legal theory,
+      whether in tort (including negligence), contract, or otherwise,
+      unless required by applicable law (such as deliberate and grossly
+      negligent acts) or agreed to in writing, shall any Contributor be
+      liable to You for damages, including any direct, indirect, special,
+      incidental, or consequential damages of any character arising as a
+      result of this License or out of the use or inability to use the
+      Work (including but not limited to damages for loss of goodwill,
+      work stoppage, computer failure or malfunction, or any and all
+      other commercial damages or losses), even if such Contributor
+      has been advised of the possibility of such damages.
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+   9. Accepting Warranty or Additional Liability. While redistributing
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+   you may not use this file except in compliance with the License.
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+   Unless required by applicable law or agreed to in writing, software
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diff --git a/research/w3-p10b-pointwise-complex/bit/pr346/NOTICE.md b/research/w3-p10b-pointwise-complex/bit/pr346/NOTICE.md
new file mode 100644
index 0000000000..6d5643cc76
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/bit/pr346/NOTICE.md
@@ -0,0 +1,28 @@
+# Attribution and provenance
+
+Prepared with substantial Anthropic Claude assistance. Apache-2.0 (see `LICENSE`) applies to the new code and to the
+inherited Apache-2.0 code below; third-party notices are preserved where files are vendored.
+
+- **Base bit word, re-annealed pair module, p = 10 five-stage pipeline and its moment/outer/prime engines**
+  (`data/bit/BASE-*`, `data/complex/pm_p10b.json`, `code/pricing/{moment,base_two_moment,outer,prime_witnesses}.py`):
+  DreamingOfClouds, PR #325 at `0eca9340a3df6141b8e71a41638c3937b3522888` on PR #315 at
+  `64bccfb67494f7c4549bfb2f186b400a3cfefb42` (#285/#276 gen5/gen4 lineage on eumemic's PR168 generator and source527
+  package, PR #200's physical rules, hcg890 and Jacob Sussman's five-stage architecture, Evan McKinney's completed
+  banks). The engines are copied unchanged from PR #315's package.
+- **Design T, twin condensation, the Design T verifier and the cost model** (`code/builders/{interface,gather,build_v5,twin,addcomp}.py`,
+  `code/checkers/{verifyT.cpp,sub.hpp,exactnd_h.py}`, `code/pricing/cost.py`): utcorvusvolat-dotcom's w3 supplier work
+  (Agents A/B) packaged in PR #310 at `484b85f1d1fada27fda2ddab3ec952c1314d84c7`, ported here to h = 20 (literals 24/23
+  replaced by h and h − 1, a `drop` case for gauge-bound partial carriers, configurable paths).
+- **Official checkers** (`code/checkers/{legality_h,columns_h}.cpp`): PR #266's `cohort-legality-independent` and
+  `cohort-five-stage-columns` as retained in PR #310, with only h, v, R and the copy count generalized.
+  `json.hpp`: Niels Lohmann, MIT license in its header.
+- **Triple module** (`data/complex/tmod_t2_p10.json`): PR #327 (complex-p10) at `40ed0456ab1990beb3440e61bb883ccf95fdcea4`.
+- **Complex chain and checks** (`producer/`, `code/complex/`): PR #304's recipe and PR #315's portable complex checks
+  (#256/#233/#202/#200/#194/#184 lineage: icekylinx, Chafik Boukhalfa, Avi Eisenberg, eumemic and others), ported to
+  p = 10. Jacob Sussman's `gx.check1`, `gxcore` and pinned Lean sources are vendored unchanged under
+  `code/complex/inputs/complex/sources` with their own notices.
+- **New here** (Claude-assisted): the exporter, the h = 20 ports, the selection of the complex modules with a
+  one-instance compile proxy, the p = 10 complex program and its pins, the tiling/prime/certification helpers and
+  `verify.sh`.
+
+Credits do not imply endorsement or review by the credited authors.
diff --git a/research/w3-p10b-pointwise-complex/bit/pr346/RECORD-SHA256SUMS b/research/w3-p10b-pointwise-complex/bit/pr346/RECORD-SHA256SUMS
new file mode 100644
index 0000000000..6525e19277
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/bit/pr346/RECORD-SHA256SUMS
@@ -0,0 +1,2 @@
+c6a9311acfcf85f4a5775ed0187e31f84531614bb08434ef887ae1e63dda99c4  TT/249-records.bin
+f1933dbf6f82303bb0cc6e42d4e292a652d99fb44286648452db374178e60c6f  snap/249-records.bin
diff --git a/research/w3-p10b-pointwise-complex/bit/pr346/kappa-L10f.json b/research/w3-p10b-pointwise-complex/bit/pr346/kappa-L10f.json
new file mode 100644
index 0000000000..8b3c2025a5
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/bit/pr346/kappa-L10f.json
@@ -0,0 +1,32 @@
+{
+ "word": "TT",
+ "normalized_histogram": {
+  "1": 1255200,
+  "2": 520140,
+  "17": 152160,
+  "3": 363900,
+  "5": 65520,
+  "6": 60360,
+  "4": 104580,
+  "8": 9840,
+  "7": 38400,
+  "16": 57600,
+  "10": 14880,
+  "12": 60360,
+  "9": 17580,
+  "11": 4800,
+  "14": 20640,
+  "18": 30000,
+  "13": 120,
+  "38": 23040,
+  "19": 23040,
+  "42": 23040
+ },
+ "W": 125712,
+ "m": 100,
+ "bit_coarse": "32069035133443/40000000000000000",
+ "complex_coarse": "396604388013523/500000000000000000",
+ "binding": "complex",
+ "kappa": "396290046684973/500000000000000000",
+ "kappa_float": 0.000792580093369946
+}
\ No newline at end of file
diff --git a/research/w3-p10b-pointwise-complex/centre46.py b/research/w3-p10b-pointwise-complex/centre46.py
new file mode 100644
index 0000000000..ede0d1b210
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/centre46.py
@@ -0,0 +1,91 @@
+"""centre46.py: the centre-sharing variant of PR #144's Graph.finish used for PR #315's complex program (the research
+copy, unchanged): the pair module supplies the star centre as an extra root (sum of all its inputs) instead of
+Graph.sum over the A's. load_pmod46(path, n) checks the contract (C(n,2) roots + one full-sum root).
+Prepared by DreamingOfClouds with Anthropic Claude assistance; Apache-2.0. Graph.finish is icekylinx's (PR #144)."""
+import json
+from pathlib import Path
+from itertools import combinations
+
+
+# ---- centre-sharing variant (agent-cmod): the pair module supplies the star centre as a 46th root -------------
+def load_pmod46(path, n=10):
+    """Pair module JSON with 45 contract roots plus a 46th root = sum of all 45 inputs (checked)."""
+    from itertools import combinations as _c
+    d = json.loads(Path(path).read_text())
+    pairs = list(_c(range(n), 2))
+    assert d['input_count'] == len(pairs) and len(d['roots']) == len(pairs) + 1
+    support = []
+    for x, a in enumerate(d['args']):
+        if x < len(pairs):
+            assert a is None; support.append(1 << x)
+        else:
+            assert a is not None and 0 <= a[0] < x and 0 <= a[1] < x and not support[a[0]] & support[a[1]]
+            support.append(support[a[0]] | support[a[1]])
+    for (i, j), r in zip(pairs, d['roots']):
+        assert support[r] == sum(1 << k for k, (a, b) in enumerate(pairs) if i not in (a, b) and j not in (a, b))
+    assert support[d['roots'][-1]] == (1 << len(pairs)) - 1
+    return dict(kind='pair_disjoint_centre', n=n, input_count=len(pairs), input_labels=pairs, output_labels=pairs,
+                args=[None if a is None else list(a) for a in d['args']], roots=list(d['roots']))
+
+
+def finish_centre46(G, triple, pair46, allbut):
+    """Graph.finish (star centre) with centre[2i+e] = the pair module (i, e)'s 46th root instead of G.sum(...)."""
+    from itertools import product as _prod
+    from fractions import Fraction
+    G.local_channels()
+    base_nodes = len(G.a)
+    D = dict(zip(G.cubes, G.module(triple, [G.F[I] for I in G.cubes])))
+    P, Q, CEN = {}, {}, {}
+    for i in range(G.p):
+        others = [a for a in range(G.p) if a != i]
+        pairs = list(combinations(others, 2))
+        for bit in range(2):
+            inputs = [G.A[tuple(sorted((i, *K))), i, bit] for K in pairs]
+            outs = G.module(pair46, inputs)
+            for K, node in zip(pairs, outs[:len(pairs)]):
+                P[tuple(sorted((i, *K))), i, bit] = node
+            CEN[i, bit] = outs[len(pairs)]
+    for i, j in combinations(range(G.p), 2):
+        others = [a for a in range(G.p) if a not in (i, j)]
+        for mode in range(2):
+            inputs = [G.G[tuple(sorted((i, j, k))), i, j, mode] for k in others]
+            for k, node in zip(others, G.module(allbut, inputs)):
+                Q[tuple(sorted((i, j, k))), i, j, mode] = node
+    query_nodes = len(G.a)
+    def emit(I, bits_list, node, numerator=1, channel=''):
+        G.root.append(dict(node=node, targets=[G.source[I, b] for b in bits_list],
+                           coefficients=[f'{numerator}/2'] * len(bits_list), kind='side', channel=channel))
+    for I in G.cubes:
+        all_bits = list(_prod(range(2), repeat=3))
+        emit(I, all_bits, D[I], channel='disjoint')
+        for a in range(2):
+            emit(I, [b for b in all_bits if b[0] == a], P[I, I[0], 1 - a], channel='face0')
+        for a, b in _prod(range(2), repeat=2):
+            emit(I, [(a, b, c) for c in range(2)], P[I, I[1], 1 - b], channel='face1')
+        for a, b in _prod(range(2), repeat=2):
+            emit(I, [(a, b, c) for c in range(2)], Q[I, I[0], I[1], a ^ b], 2 * a - 1, 'edge01')
+        for bits in all_bits:
+            emit(I, [bits], P[I, I[2], 1 - bits[2]], channel='face2')
+        for ii, jj in [(0, 2), (1, 2)]:
+            for bits in all_bits:
+                emit(I, [bits], Q[I, I[ii], I[jj], bits[ii] ^ bits[jj]], 2 * bits[ii] - 1, f'edge{ii}{jj}')
+    for i in range(G.p):
+        for e in range(2):
+            G.centers.append(CEN[i, e])
+    for coordinate, node in enumerate(G.centers):
+        coefficients = ['1/3' if coordinate in t else '-1/6' for t in G.labels]
+        G.root.append(dict(node=node, targets=list(range(len(G.labels))), coefficients=coefficients, kind='center',
+                           coordinate=coordinate))
+    return dict(p=G.p, h=G.h, v=len(G.labels), labels=G.labels, inputs=[sum(1 << x for x in t) for t in G.labels],
+                args=G.a, signs=G.signs, roots=G.root, centers=G.centers, center_kind='star',
+                center_scatter='1/2 sum_(i in T) S_i - 1/6 sum_i S_i',
+                local_K='Two 4x4 Hadamard/2 blocks on opposite cube parities, using original X destructively',
+                source_data_histogram={2: len(G.labels), G.h - 4: len(G.labels), 1: len(G.labels)},
+                target_data_histogram={G.h - 4: len(G.labels), 1: 3 * len(G.labels)},
+                counts=dict(local_channel_additions=base_nodes - len(G.labels), query_additions=query_nodes - base_nodes,
+                            center_additions=0, center_tree_incidences=0,
+                            signed_additions=sum(x < 0 for x in G.signs),
+                            side_root_uses=sum(r['kind'] == 'side' for r in G.root), center_roots=len(G.centers)),
+                status='agent-cmod centre-sharing variant')
+
+
diff --git a/research/w3-p10b-pointwise-complex/certificate/expected.json b/research/w3-p10b-pointwise-complex/certificate/expected.json
new file mode 100644
index 0000000000..5cc51d3239
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/certificate/expected.json
@@ -0,0 +1,498 @@
+{
+ "bit": {
+  "W": 125712,
+  "calls": 2845200,
+  "coarse": "32069035133443/40000000000000000",
+  "coarse_decimal": "0.000801725878336075",
+  "deficit": 24480,
+  "equals_pr346": true,
+  "m": 100,
+  "moment_interval": [
+   "26959946667150639772888688255444757631748930553794591157949039712185/26959946667150639794667015087019630673637144422540572481103610249216",
+   "26959946667150639772888688255444757631748930553794591157949039712187/26959946667150639794667015087019630673637144422540572481103610249216"
+  ],
+  "next_moment_lower_bound": "13479973333575319919224719865443015256277351504205423299504221172901/13479973333575319897333507543509815336818572211270286240551805124608",
+  "normalization": 12,
+  "paid_children": {
+   "1": 20920,
+   "10": 248,
+   "11": 80,
+   "12": 1006,
+   "13": 2,
+   "14": 344,
+   "16": 960,
+   "17": 2536,
+   "18": 500,
+   "2": 8669,
+   "3": 6065,
+   "4": 1359,
+   "5": 1092,
+   "6": 1006,
+   "7": 640,
+   "8": 164,
+   "9": 293
+  },
+  "rank_mass": 12546720,
+  "record_kinds": {
+   "0": 47020,
+   "1": 358128,
+   "2": 20,
+   "3": 20
+  },
+  "records": 405188,
+  "records_sha256": "c6a9311acfcf85f4a5775ed0187e31f84531614bb08434ef887ae1e63dda99c4",
+  "stock_from": "PR #346 receipts/kappa-L10f.json (W = 125712 = 12 (4v + 132720/20)); not re-derived here"
+ },
+ "comparison": {
+  "complex_coarse": "794022782431403/1000000000000000000",
+  "gain": "162543184349/200000000000000000",
+  "gain_percent": "0.102541",
+  "kappa": "793392809291691/1000000000000000000",
+  "pr346_complex_coarse": "396604388013523/500000000000000000",
+  "pr346_kappa": "396290046684973/500000000000000000"
+ },
+ "kappa": {
+  "adjacent_kappa_rejected": true,
+  "binding": "complex",
+  "bit_coarse": "32069035133443/40000000000000000",
+  "bit_moment_interval": [
+   "6739860413027678912150056938220764388592407908481648050558994398091/6739986666787659948666753771754907668409286105635143120275902562304",
+   "13479720826055357824300113876441528777184815816963296101117988796183/13479973333575319897333507543509815336818572211270286240551805124608"
+  ],
+  "bit_saving_used": "794022781637379/1000000000000000000",
+  "bootstrap_chain": [
+   "384599/10000000000",
+   "793422847496399334578933850010359/1000000000000000000000000000000000000",
+   "794022305276003580094575072041433848410374131609061/1000000000000000000000000000000000000000000000000000000",
+   "794022781259137215624478914637951714816385034762096642473649492691119/1000000000000000000000000000000000000000000000000000000000000000000000000"
+  ],
+  "complex_coarse": "794022782431403/1000000000000000000",
+  "complex_leaf_cap": "794022781637380217568597/1000000000000000000000000000",
+  "kappa": "793392809291691/1000000000000000000",
+  "kappa_decimal": "0.000793392809291691",
+  "kappa_scientific": "7.933928092916910e-04",
+  "leaf_cap_itself_rejected": true,
+  "minimum_margin": "397011390628377573640351545518320398173451214287738381636396417766883787857605795149492691119/500397011390628774585030980320103351378493554565666231936202062649703085850507308881000000000000",
+  "smallest_constraint": "compact_phase_layer_above_kappa",
+  "strict_constraints": 47
+ },
+ "kappa_pr346_reproduced": {
+  "adjacent_kappa_rejected": true,
+  "binding": "complex",
+  "bit_coarse": "32069035133443/40000000000000000",
+  "bit_moment_interval": [
+   "13479694143261354696660370526812988752897863386217766150017088418661/13479973333575319897333507543509815336818572211270286240551805124608",
+   "6739847071630677348330185263406494376448931693108883075008544209331/6739986666787659948666753771754907668409286105635143120275902562304"
+  ],
+  "bit_saving_used": "198302193808459/250000000000000000",
+  "bootstrap_chain": [
+   "384599/10000000000",
+   "49538131362681415853118745045319/62500000000000000000000000000000000",
+   "12393879693137766942528665479140841169585160553421/15625000000000000000000000000000000000000000000000000",
+   "3098471776785791227633987058086904306924394147549122127268111188239/3906250000000000000000000000000000000000000000000000000000000000000000"
+  ],
+  "complex_coarse": "396604388013523/500000000000000000",
+  "complex_leaf_cap": "396604387616918611986477/500000000000000000000000000",
+  "kappa": "396290046684973/500000000000000000",
+  "kappa_decimal": "0.000792580093369946",
+  "kappa_scientific": "7.925800933699460e-04",
+  "leaf_cap_itself_rejected": true,
+  "minimum_margin": "1549235888388247906151817940673777958666294351301232559317038835322745382339768768111188239/1954674235888389797142040207737815818166404110169467636669486506471992231888811761000000000000",
+  "smallest_constraint": "compact_phase_layer_above_kappa",
+  "strict_constraints": 47
+ },
+ "program": {
+  "N": 161900,
+  "R": 6253,
+  "blocks": {
+   "c": {
+    "18": 20
+   },
+   "s": {
+    "1": 17306,
+    "10": 294,
+    "11": 131,
+    "12": 1110,
+    "13": 938,
+    "14": 19,
+    "17": 513,
+    "2": 7448,
+    "3": 6557,
+    "4": 2936,
+    "5": 1074,
+    "6": 1481,
+    "7": 341,
+    "8": 481,
+    "9": 230
+   },
+   "x": {
+    "1": 960,
+    "16": 960,
+    "2": 960
+   },
+   "y": {
+    "1": 2880,
+    "16": 960
+   }
+  },
+  "blocks_total": 47599,
+  "certificate_sha256": "ff2d7307b09aceb8776654f21099b10a68be547b562b7f4a7fc918f6c5f51112",
+  "controls": {
+   "flipped_scalar_sign": [
+    "gx.check1: REJECTED (E5)",
+    "gxcore.mirror: REJECTED (E5)"
+   ]
+  },
+  "cst": 360,
+  "distinct_frame_steps": 38233,
+  "five_stage": {
+   "W": 10093,
+   "a": "7940/10^7",
+   "child_histogram": {
+    "1": 105730,
+    "10": 1470,
+    "11": 655,
+    "12": 5550,
+    "13": 4690,
+    "14": 95,
+    "16": 9600,
+    "17": 2565,
+    "18": 100,
+    "19": 1920,
+    "2": 42040,
+    "3": 32785,
+    "38": 1920,
+    "4": 16600,
+    "42": 1920,
+    "5": 5370,
+    "6": 7405,
+    "7": 1705,
+    "8": 2405,
+    "9": 1150
+   },
+   "coarse": {
+    "fallback_cost": "3067430/10^18",
+    "scope": "10^-16 and 32 m^2 rank-one fallback calls per paid child are the five-stage packages' inherited envelope, applied here at m = 5h = 100; not a new m = 5h admission",
+    "with_fallback": {
+     "b": "794022782431403/10^18",
+     "b_decimal": "0.000794022782431403",
+     "base_two_next_lower": "10000000000000000007999899862287045877472503189/10000000000000000000000000000000000000000000000",
+     "base_two_upper": "4999999999999999991256879841178513862151442081/5000000000000000000000000000000000000000000000",
+     "moment_interval": [
+      "26959946667150639747524204449733897099011822196769159093932302451783/26959946667150639794667015087019630673637144422540572481103610249216",
+      "26959946667150639747524204449733897099011822196769159093932302451785/26959946667150639794667015087019630673637144422540572481103610249216"
+     ],
+     "next_grid_point": "794022782431404/10^18",
+     "next_moment_lower_bound": "6739986666787659954058675612505938664834592203296758881010968728609/6739986666787659948666753771754907668409286105635143120275902562304"
+    },
+    "without_fallback": {
+     "b": "794022785498833/10^18",
+     "b_decimal": "0.000794022785498833",
+     "base_two_next_lower": "1000000000000000001031805468121/1000000000000000000000000000000",
+     "base_two_upper": "999999999999999998483191450139/1000000000000000000000000000000",
+     "moment_interval": [
+      "26959946667150639753773937478476767035357143740361587952724413045859/26959946667150639794667015087019630673637144422540572481103610249216",
+      "26959946667150639753773937478476767035357143740361587952724413045861/26959946667150639794667015087019630673637144422540572481103610249216"
+     ],
+     "next_grid_point": "794022785498834/10^18",
+     "next_moment_lower_bound": "6739986666787659955621108869610146446631149682208070221939950731711/6739986666787659948666753771754907668409286105635143120275902562304"
+    }
+   },
+   "deficit": 2040,
+   "h": 20,
+   "idle_ranks": [
+    38,
+    19,
+    42,
+    4
+   ],
+   "m": 100,
+   "margin_lower_bound": "586109425736993520318040717/10^30",
+   "moment_upper_bound": "10092999413890574263006479681959283/10^30",
+   "next_grid_point": "7941/10^7",
+   "next_grid_point_normalized_moment_lower_bound": "1684996998287380885745342311702642250830299335767163994542041551085/1684996666696914987166688442938726917102321526408785780068975640576",
+   "next_grid_point_rejected": true,
+   "rank": 1007260,
+   "with_fallback": {
+    "fallback_rank_one_calls": "32 m^2 x 245675 x 10^-16",
+    "margin_lower_bound": "586030521750995934311378546/10^30",
+    "moment_upper_bound": "10092999413969478249004065688621454/10^30"
+   }
+  },
+  "frames": 12516,
+  "gates_A": 10622,
+  "gates_B": 24913,
+  "registers": 8173,
+  "scalar_denominators": {
+   "s": 2,
+   "x": 2,
+   "y": 6
+  },
+  "scalar_max_abs": {
+   "s": "2",
+   "x": "3/2",
+   "y": "2"
+  }
+ },
+ "regeneration": {
+  "anchor_centre_patch": {
+   "center_additions": 0,
+   "centres_from_pair_module": 20,
+   "differing_fields": [
+    "status"
+   ],
+   "fields_compared": 16,
+   "hunks": 2,
+   "patch_sha256": "229eea46b56d19b262d1498f489f9bca84c347e33b16553079d38c4276700280"
+  },
+  "anchor_pr194": {
+   "N": 262944,
+   "R": 9412,
+   "blocks_total": 70169,
+   "certificate_sha256": "27398a0cfcbe34e5ee97360e69a2d163a417c8ae1b4fdf28ccf89a9eaf4e4cd4",
+   "equals_pr256_certificate": true,
+   "equals_sussman_pr193_blocks": true
+  },
+  "anchor_spec_builder": {
+   "files_compared": 7,
+   "identical": true,
+   "layer": "layers/uniform",
+   "physical_R": 9165,
+   "spec": "spec/uniform.json"
+  },
+  "anchor_spec_graph": {
+   "differing_fields": [],
+   "distinct_module_files": {
+    "all_but_one": 1,
+    "pair": 1
+   },
+   "fields_compared": 16,
+   "invocations": {
+    "all_but_one": 90,
+    "pair": 20
+   },
+   "query_additions": 8942
+  },
+  "chain": {
+   "admission": {
+    "formal": {
+     "absolute_coefficient_bound": 686010,
+     "base_exceeds_twice_bound": true,
+     "columns": [
+      {
+       "K_and_inverse": true,
+       "all_aliases_and_deadline_reads": true,
+       "all_physical_dirty_slots_restored": true,
+       "direction": 1,
+       "dirty_columns": 9141,
+       "every_formal_column": true,
+       "source_columns": 960,
+       "target_columns": 960
+      },
+      {
+       "K_and_inverse": true,
+       "all_aliases_and_deadline_reads": true,
+       "all_physical_dirty_slots_restored": true,
+       "direction": -1,
+       "dirty_columns": 9141,
+       "every_formal_column": true,
+       "source_columns": 960,
+       "target_columns": 960
+      }
+     ],
+     "packed_digit_bits": 24
+    },
+    "mutations": {
+     "early-read": "REJECTED",
+     "missing-read": "REJECTED"
+    },
+    "source_sha256": {
+     "notes/general-clifford-frames.tex": "c9bb9655b594583c6b4bab8cb485233f1d1e588f83858cf1e4c3530cee8fed54",
+     "research/paired-cube-diagonal-bit-168/complex/physical.py": "01961fad4b04aaa940d2dc99c87690bc861b3861200296974c5e065ce3446659",
+     "research/paired-cube-diagonal-bit-168/complex/verify_fused.py": "75c3dcd80fcdbc461651fa2d2fa463e6b41a569d695fc74700d240223ef198e9",
+     "scripts/paired_cube/frames.py": "7d897418dbfee47de0723ea4983098a982610ccba45e1bac9000a79626d8421c",
+     "scripts/paired_cube_physical.py": "9c9ea1ea29e551ab6116d7c8ca09f4703ff9e2fa102032d5055327fde07ba319"
+    },
+    "status": "PASS exact fused physical candidate, all formal source/target/dirty columns"
+   },
+   "contract_checks": {
+    "all_centers_in_phase1": true,
+    "all_fresh_columns_equal_identity": true,
+    "all_target_cap_reads_in_phase2": true,
+    "arithmetic": "integers, endpoint scaled by 6",
+    "checked_source_columns": 960,
+    "checked_target_rows": 960,
+    "controls_before_original_K": true,
+    "original_source_V_before_all_controls": true,
+    "positive_target_data_histogram": {
+     "1": 2880,
+     "16": 960
+    },
+    "read_counts": {
+     "center": 20,
+     "deferred": 1680,
+     "side": 2280
+    },
+    "source_controls_at_paid_parity_frames": true,
+    "target_chains_nested": true,
+    "target_data_histogram": {
+     "0": 4560,
+     "1": 2880,
+     "16": 960
+    }
+   },
+   "flow": {
+    "child_histogram": {
+     "1": 63438,
+     "10": 882,
+     "11": 393,
+     "12": 3330,
+     "13": 2814,
+     "14": 57,
+     "16": 5760,
+     "17": 1539,
+     "18": 60,
+     "2": 27144,
+     "3": 19671,
+     "4": 8808,
+     "5": 3222,
+     "6": 4443,
+     "7": 1023,
+     "8": 1443,
+     "9": 690
+    },
+    "deficit": 840,
+    "kernel_reuses": 0,
+    "loss": 360,
+    "new_R": 6253,
+    "new_W": 8173,
+    "original_physical_roles": 9141,
+    "source_control_erasures": 0
+   },
+   "layer": {
+    "R": 10821,
+    "arcs": 6779,
+    "kernel_pairs": 0,
+    "moved_frames": 10036,
+    "pairs": 1680,
+    "physical_R": 9141
+   },
+   "lift": {
+    "all_actual_coefficients_dyadic": true,
+    "inverse_elementary_gates": 38997,
+    "max_denominator": "2",
+    "nodes": 12572,
+    "physical_R": 6253,
+    "status": "Exact rational local lifts and inverse gate programs verified; monotone flow and zero-fresh kernel reuse DAG verified."
+   },
+   "paircheck": {
+    "controls": {
+     "donor_frame_outside_gauge": "REJECTED (donor frame outside gauge)",
+     "duplicate_pair": "REJECTED (not one-to-one)",
+     "late_read_before_donor_death": "REJECTED (donor frame outside gauge, late chronology)",
+     "parity_erasable_donor": "REJECTED (erasable)",
+     "unpaired_gauge": "REJECTED (live external gauges)"
+    },
+    "early": 766,
+    "erasable": 0,
+    "first_violations": [],
+    "gauges": 1680,
+    "kinds": [],
+    "late": 914,
+    "pairs": 1680,
+    "status": "PASS",
+    "violations": 0
+   },
+   "scatter": {
+    "changed_coefficients": 0,
+    "changed_gates": 0,
+    "flipped_coordinates": [],
+    "flipped_registers": [],
+    "gx_check1": "ACCEPTED (raw and committed program)",
+    "gxcore_mirror": "ACCEPTED (raw and committed program; label and scalar checks)",
+    "raw_certificate_sha256": "ff2d7307b09aceb8776654f21099b10a68be547b562b7f4a7fc918f6c5f51112",
+    "raw_scatter": "star"
+   }
+  },
+  "modules": {
+   "pmod": {
+    "additions": 273,
+    "outputs": 37,
+    "sha256": "8d71f5004acb0ea902ad594a7d298f8d2099114e5d6d42c6357f09c44ab3ce87"
+   },
+   "qmod": {
+    "additions": 24,
+    "outputs": 8,
+    "sha256": "40ffb168466bc016bc18a2bbdde65d38b7fd1b406d8a4026f1a6e517fe153459"
+   },
+   "tmod": {
+    "additions": 1322,
+    "outputs": 120,
+    "sha256": "4d39faeedfb5a5283a7ad34a8a18af93072e4d155d61dec6c28d7751bc2e4e04"
+   }
+  },
+  "pointwise_pair_modules": {
+   "0": {
+    "additions": 273,
+    "file": "modules/pointwise/pmod_i0.json",
+    "outputs": 37,
+    "sha256": "7d9c0aaf02fbb91a7004821adcd9bdcc577513a4052cd21689a8e1bea7616ce5"
+   },
+   "1": {
+    "additions": 273,
+    "file": "modules/pointwise/pmod_i1.json",
+    "outputs": 37,
+    "sha256": "867744aacd6ce119954ad2ba2de51819551e590a22eae4815ba6a111d16ef748"
+   },
+   "2": {
+    "additions": 273,
+    "file": "modules/pointwise/pmod_i2.json",
+    "outputs": 37,
+    "sha256": "0c56f9e38be7d6c9e4968012afcc37479c114453a2dad6a0ff64b5a446123f16"
+   },
+   "3": {
+    "additions": 273,
+    "file": "modules/pointwise/pmod_i3.json",
+    "outputs": 37,
+    "sha256": "7d9c0aaf02fbb91a7004821adcd9bdcc577513a4052cd21689a8e1bea7616ce5"
+   },
+   "4": {
+    "additions": 273,
+    "file": "modules/pointwise/pmod_i4.json",
+    "outputs": 37,
+    "sha256": "867744aacd6ce119954ad2ba2de51819551e590a22eae4815ba6a111d16ef748"
+   },
+   "5": {
+    "additions": 273,
+    "file": "modules/pointwise/pmod_i5.json",
+    "outputs": 37,
+    "sha256": "5e7f9239867d066850dcca94106cab8420fd04b494e7718526311ecc08950117"
+   },
+   "6": {
+    "additions": 273,
+    "file": "modules/pointwise/pmod_i6.json",
+    "outputs": 37,
+    "sha256": "5e7f9239867d066850dcca94106cab8420fd04b494e7718526311ecc08950117"
+   },
+   "7": {
+    "additions": 272,
+    "file": "modules/pointwise/pmod_i7.json",
+    "outputs": 37,
+    "sha256": "a10816b44292874da2d7596b044b9be876e112ca00581bde2ef9a74213b10dc3"
+   },
+   "8": {
+    "additions": 273,
+    "file": "modules/pointwise/pmod_i8.json",
+    "outputs": 37,
+    "sha256": "5e7f9239867d066850dcca94106cab8420fd04b494e7718526311ecc08950117"
+   },
+   "9": {
+    "additions": 273,
+    "file": "modules/pointwise/pmod_i9.json",
+    "outputs": 37,
+    "sha256": "953e80941984780dfba6952537cace9d1214270bda5735f27dc1cfe020cfcef6"
+   }
+  }
+ },
+ "status": "PASS: conditional kappa = 793392809291691/1000000000000000000 (complex-bound) with PR #346's w3 bit word and the per-point p = 10 complex program (7940/10^7, b = 794022782431403/10^18); PR #346's kappa 396290046684973/500000000000000000 reproduced"
+}
diff --git a/research/w3-p10b-pointwise-complex/certificates/gcert1-p10-b325-c2.json.gz b/research/w3-p10b-pointwise-complex/certificates/gcert1-p10-b325-c2.json.gz
new file mode 100644
index 0000000000..b714959e9a
Binary files /dev/null and b/research/w3-p10b-pointwise-complex/certificates/gcert1-p10-b325-c2.json.gz differ
diff --git a/research/w3-p10b-pointwise-complex/discovery/README.md b/research/w3-p10b-pointwise-complex/discovery/README.md
new file mode 100644
index 0000000000..5c7f2a99b8
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/discovery/README.md
@@ -0,0 +1,36 @@
+# Discovery (not run by verify.py)
+
+`verify.py` never searches. It reads `spec/c2.json`, `modules/` and `layers/c2/` and checks them. This note says
+how they were found.
+
+**Recipe.** The layer is PR #327's recipe (`discovery/pr327/make_layer_p.py`, steps 1–6, the `mw` recipe only).
+`pipe.py` is our re-implementation of those steps for screening module sets, about 90 s per set at p = 10:
+
+1. build the centre-sharing source-aligned graph from the per-invocation spec;
+2. find carrier arcs with the Hopcroft–Karp carrier matching (ordering `reverse`, local donors dropped), then
+   extend them with PR #162's greedy extension restricted to query donors;
+3. run `compile_closure` and `select`, then PR #200's operation-frame descent (up to three cycles);
+4. take the forward closure, with the descended frames as preferred frames;
+5. pair donors with PR #233's maximum-weight late compensated pairing (`pairs_mw_p.py`, f = h − 4);
+6. compute kernel pairs with PR #184's `complex_frame_flow.py`. It found none here.
+
+`pipe.py` built the graph with our per-invocation builder (`CXUP_SPEC`). `spec_graph.finish_spec` is the same
+builder without its orientation option, which was left at the identity.
+
+**Modules.** The search ran in two steps:
+
+1. **Shared modules.** PR #325's bit-word pair and all-but-one modules replaced PR #327's in the complex word. The
+   result is the companion program `spec/uniform.json` + `layers/uniform/`, at 7932/10⁷ and Lean-checked. PR #346
+   independently ships a complex supplier with the same coarse saving. Float five-stage roots: PR #327's set
+   7.795976e-4, #325's pair module only 7.8754e-4, #325's all-but-one module only 7.8515e-4, both 7.9321e-4.
+2. **Pointwise pair modules.** Starting from step 1, a greedy hill climb ran on the exact compile proxy. Its
+   moves replace the pair module of one point i, for both bits at once, by a random local rewrite of it
+   (re-derive one addition, or re-root one output, keeping the module contract); a move is kept if the proxy
+   improves. The best
+   state, `spec/c2.json` (`modules/pointwise/pmod_i0..9.json`, six distinct files), gives float root 7.94023e-4,
+   R 6,253 and 6,779 arcs. Its layer was then re-found by `pipe.py`, and the program was certified with the steps of
+   PR #327's `chain()`. The pointwise modules are data for the proof: their contracts are checked by `load_pmod46`,
+   and the whole word is checked by the chain.
+
+**Not shipped.** The climb driver, the module annealers and the compile proxies are research code and are not
+included.
diff --git a/research/w3-p10b-pointwise-complex/discovery/pipe.py b/research/w3-p10b-pointwise-complex/discovery/pipe.py
new file mode 100644
index 0000000000..816123ee2e
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/discovery/pipe.py
@@ -0,0 +1,198 @@
+#!/usr/bin/env python3
+"""cxup pipe.py: p=10 complex word for given modules -> layer (arcs, descended frames, mw pairs) -> aligned word ->
+PR #184 flow -> five-stage float price.  Recipe = #327 make_layer_p.py steps 1-6 (mw only), re-written for screening.
+usage: pipe.py --tree T --tmod F --pmod F --qmod F [--centre46] --out DIR [--keep]
+Prints one JSON line {R, new_R, five, three, ...}.  Discovery only (NOT run by verify.py).
+Layout: run it from a flat directory that also holds aligned_word_p.py, centre46.py, discovery/pr327/*.py and
+discovery/pr327/pr233/*.py of this package (it imports them from its own directory); --tree is the rebuilt PR #202
+tree. The package's layer was produced with CXUP_SPEC set to the spec (graph built by the spec builder, whose graph
+for this spec equals centre46.finish_centre46's; verify.py anchor (e)); without CXUP_SPEC, --centre46 builds the
+same graph. Floating-point scores choose frames and pairs; nothing here is a proof.
+Prepared by Chafik Boukhalfa with substantial Anthropic Claude assistance; Apache-2.0."""
+import argparse, bisect, json, math, subprocess, sys, time, os, shutil
+from collections import defaultdict, Counter
+from pathlib import Path
+HERE = Path(__file__).resolve().parent
+
+
+def dump(path, data):
+    path.parent.mkdir(parents=True, exist_ok=True)
+    path.write_text(json.dumps(data, separators=(',', ':')) + '\n')
+
+
+def five_price(child, v, h, R):
+    Hinv = Counter()
+    for r, n in child.items():
+        r = int(r); n = n - (2 * v if r == 2 else 0)
+        assert n % 3 == 0, (r, n)
+        if n: Hinv[r] += n // 3
+    H5 = Counter({r: 5 * n for r, n in Hinv.items()})
+    for r in (2 * h - 2, h - 1, 2 * h + 2, 4): H5[r] += 2 * v
+    m, W = 5 * h, 4 * v + R
+    lo, hi = 0.0, 0.01
+    for _ in range(100):
+        a = (lo + hi) / 2
+        if math.fsum(n * (r / m) ** (1 - a) for r, n in H5.items()) < W: lo = a
+        else: hi = a
+    return lo, H5, m, W
+
+
+def build_graph(root, mods, centre, P):
+    sys.path.insert(0, str(root / 'scripts')); sys.path.insert(0, str(HERE))
+    from paired_cube.graph import Graph
+    from paired_cube.modules import triple_module_from, pair_module_from, all_but_one_from, merge_outputs
+    local = json.loads((root / 'references/paired-cube/sources/local_L1.json').read_text())
+    local['G'] = {k: 's' for k in local['G']}
+    local['A'] = {k: 'fd' for k in local['A']}
+    builder = Graph(P, local=local)
+    if centre:
+        from centre46 import load_pmod46, finish_centre46
+        g = finish_centre46(builder, triple_module_from(mods['tmod'], P), load_pmod46(mods['pmod'], P - 1), all_but_one_from(mods['qmod'], P - 2))
+    else:
+        g = builder.finish(triple_module_from(mods['tmod'], P), pair_module_from(mods['pmod'], P - 1), all_but_one_from(mods['qmod'], P - 2))
+    g = merge_outputs(g, builder, 'f8:00111100')
+    g['matching_frames'] = 'coordinate'
+    return g
+
+
+def layer(root, g, cycles=3, log=print):
+    from paired_cube.closure import compile_closure
+    from paired_cube.gauges import select
+    from paired_cube.frames import basis, perp, contained
+    from matching import carrier_matching
+    from extend_floor import extended_arcs
+    import physical_opt, bundles, joint
+    from pairs_mw_p import pairs_maxweight
+    h, v = g['h'], g['v']
+    newcut = v + g['counts']['local_channel_additions']
+    arcs, stats = carrier_matching(g, os.environ.get('ORDERING', 'reverse'))
+    arcs = [x for x in arcs if x[0] > newcut]
+    arcs = extended_arcs(g, arcs, min_donor=newcut + 1)
+    base, w = compile_closure(g, arcs)
+    row, word = select(g, base, w)
+    compile_R = base['R']
+    o = physical_opt.Physical(g, w, word, row)
+    for k in range(cycles):
+        r1 = o.descend(6); r2 = bundles.descend(o, 3); r3 = joint.descend(o, 3, 'forward', k)
+        if not any(x['changed'] for x in r1) and not any(x['moved_components'] for x in r2) and not any(x['moved'] for x in r3):
+            break
+    dframes = o.frames
+    p, w = compile_closure(g, arcs)
+    p, word = select(g, p, w)
+    ops = word['ops']
+    newargs = [None] + [None if x is None else [t + 1 for t in x] for x in g['args']]
+    newroots = [dict(r, node=r['node'] + 1) for r in g['roots']]
+    spans = [()] + [(q,) for q in g['inputs']]
+    for aa, bb in newargs[v + 1:]:
+        spans.append(basis(spans[aa] + spans[bb]))
+    root_frames = {}
+    for r, slot in zip(newroots, word['rootroles']):
+        root_frames[slot] = spans[r['node']] if r['kind'] == 'center' else perp(basis(g['inputs'][t] for t in r['targets']), h)
+    R = p['R']
+    pset = set(word['phase1'])
+    previous = [-1] * R
+    parents = []
+    for i, (aa, bb, x) in enumerate(ops):
+        parents.append((previous[aa], previous[bb]))
+        previous[aa] = previous[bb] = i
+        if x <= newcut and len(spans[x]) <= 2:
+            pset.add(i)
+    todo = list(pset)
+    while todo:
+        i = todo.pop()
+        for j in parents[i]:
+            if j >= 0 and j not in pset:
+                pset.add(j); todo.append(j)
+    word['phase1'] = sorted(pset)
+    p['phase1_operations'] = len(pset)
+    chron = sorted(pset) + [i for i in range(len(ops)) if i not in pset]
+    cap = [perp(root_frames[s], h) if s in root_frames else () for s in range(R)]
+    maxima = [None] * len(ops)
+    for i in reversed(chron):
+        aa, bb, x = ops[i]
+        A = basis(cap[aa] + cap[bb])
+        maxima[i] = perp(A, h)
+        cap[aa] = cap[bb] = A
+    selected = [z for z in word['selected'] if z['rank'] == h - 4]
+    gauge = {z['role']: perp(tuple(z['annihilator']), h) for z in selected}
+    current = [() for _ in range(R)]
+    for x, slot in word['sources'].items():
+        current[slot] = (g['inputs'][int(x) - 1],)
+    for slot, U in gauge.items():
+        current[slot] = U
+    frames = [None] * len(ops)
+    for i in chron:
+        aa, bb, x = ops[i]
+        want = spans[x] if x <= newcut else perp(basis(perp(tuple(dframes[i]), h) + perp(maxima[i], h)), h)
+        U = basis(spans[x] + want + current[aa] + current[bb])
+        assert contained(U, maxima[i]), ('Forward frame outside future cap', i)
+        frames[i] = current[aa] = current[bb] = U
+    moved = [[i, list(U)] for i, U in enumerate(frames) if U != perp(tuple(w['annihilators'][ops[i][2]]), h)]
+    o = physical_opt.Physical(g, w, dict(word, selected=selected), p)
+    for i, U in enumerate(frames):
+        o.frames[i] = tuple(U)
+    pairs_mw, mwstats = pairs_maxweight(o)
+    good = [k for k, (dd, b, t) in enumerate(pairs_mw) if t is not None and any(i not in o.phase and o.ops[i][0] == dd for i in o.role_ops[dd])]
+    if good:
+        pairs_mw.append(pairs_mw.pop(good[-1]))
+    dropped = len(word['selected']) - len(selected)
+    return dict(arcs=arcs, frames=moved, pairs=pairs_mw, compile_R=compile_R, R=R, gauges=len(selected),
+                dropped=dropped, unpaired=len(selected) - len(pairs_mw), selected_hist=p['selected_rank_histogram'])
+
+
+def run(root, *args):
+    r = subprocess.run([sys.executable, '-B', *map(str, args)], cwd=root, capture_output=True, text=True)
+    if r.returncode:
+        raise RuntimeError('FAILED: %s\n%s%s' % (args[0], r.stdout[-2000:], r.stderr[-2000:]))
+    return r.stdout
+
+
+def main():
+    ap = argparse.ArgumentParser()
+    ap.add_argument('--tree', type=Path, required=True)
+    ap.add_argument('--tmod', type=Path, required=True); ap.add_argument('--pmod', type=Path, required=True)
+    ap.add_argument('--qmod', type=Path, required=True)
+    ap.add_argument('--centre46', action='store_true')
+    ap.add_argument('--p', type=int, default=10)
+    ap.add_argument('--out', type=Path, required=True)
+    ap.add_argument('--keep', action='store_true')
+    a = ap.parse_args()
+    t0 = time.time()
+    root = a.tree.resolve(); out = a.out.resolve(); out.mkdir(parents=True, exist_ok=True)
+    mods = dict(tmod=a.tmod.resolve(), pmod=a.pmod.resolve(), qmod=a.qmod.resolve())
+    if os.environ.get('CXUP_SPEC'):
+        import spec
+        g = spec.graph(root, os.environ['CXUP_SPEC'])
+    else:
+        g = build_graph(root, mods, a.centre46, a.p)
+    L = layer(root, g)
+    t1 = time.time()
+    dump(out / 'matching-arcs.json', dict(matching_arcs=L['arcs']))
+    dump(out / 'physical-frames.json', dict(frames=L['frames']))
+    dump(out / 'physical-pairs-mw.json', dict(pairs=L['pairs']))
+    tree = out / 'tree'
+    run(root, HERE / 'aligned_word_p.py', '--p', a.p, *(['--centre46'] if a.centre46 else []), '--tree', root,
+        '--tmod', mods['tmod'], '--pmod', mods['pmod'], '--qmod', mods['qmod'], '--arcs', out / 'matching-arcs.json',
+        '--frames', out / 'physical-frames.json', '--pairs', out / 'physical-pairs-mw.json', '--out', tree)
+    run(root, root / 'research/source-assisted/decision/complex_frame_flow.py', '--tree', tree, '--cache', tree / 'cache',
+        '--out', out / 'flow.json', '--witness', '--purify-source-donors', '--recycle-kernels')
+    fd = json.loads((out / 'flow.json').read_text())
+    kern = json.loads((out / 'flow.witness.json').read_text())['kernel_pairs']
+    dump(out / 'kernel-pairs-mw.json', dict(kernel_pairs=kern))
+    v, h = g['v'], g['h']
+    five, H5, m, W = five_price(fd['child_histogram'], v, h, fd['new_R'])
+    res = dict(R=L['R'], compile_R=L['compile_R'], new_R=fd['new_R'], gauges=L['gauges'], dropped=L['dropped'],
+               unpaired=L['unpaired'], pairs=len(L['pairs']), arcs=len(L['arcs']), moved=len(L['frames']),
+               kernel=len(kern), three=fd['numerical_local_root'], five=five, W=W,
+               layer_s=round(t1 - t0, 1), total_s=round(time.time() - t0, 1))
+    dump(out / 'result.json', res)
+    if not a.keep:
+        shutil.rmtree(tree, ignore_errors=True)
+        for f in ('flow.witness.json',):
+            try: (out / f).unlink()
+            except FileNotFoundError: pass
+    print(json.dumps(res), flush=True)
+
+
+if __name__ == '__main__':
+    main()
diff --git a/research/w3-p10b-pointwise-complex/discovery/pr327/add_centre.py b/research/w3-p10b-pointwise-complex/discovery/pr327/add_centre.py
new file mode 100644
index 0000000000..d9ee63aa2f
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/discovery/pr327/add_centre.py
@@ -0,0 +1,30 @@
+"""Append a centre root (sum of all inputs) to a pair module by a greedy disjoint cover of existing nodes.
+usage: add_centre.py SRC.json DST.json   (made pmod46_p1 from pmod_p1, the seed of the centre-sharing anneal)
+Prepared by DreamingOfClouds with Anthropic Claude assistance (Apache-2.0)."""
+import json,sys
+src,dst=sys.argv[1],sys.argv[2]
+d=json.load(open(src)); n=d['input_count']; args=d['args']
+sup=[]
+for x,a in enumerate(args):
+    sup.append(1<<x if a is None else sup[a[0]]|sup[a[1]])
+full=(1<<n)-1
+assert len(d['roots'])==n
+order=sorted(range(len(args)),key=lambda x:-bin(sup[x]).count('1'))
+cov=0;parts=[]
+for x in order:
+    if sup[x]&cov==0:
+        parts.append(x);cov|=sup[x]
+        if cov==full:break
+assert cov==full
+# balanced pairwise sum
+cur=parts[:]
+while len(cur)>1:
+    nxt=[]
+    for i in range(0,len(cur)-1,2):
+        args.append([cur[i],cur[i+1]]);nxt.append(len(args)-1)
+    if len(cur)%2:nxt.append(cur[-1])
+    cur=nxt
+d['roots']=d['roots']+[cur[0]]
+d['provenance']=dict(source=src,extra_root='greedy disjoint cover centre, %d parts'%len(parts))
+json.dump(d,open(dst,'w'))
+print('parts',len(parts),'additions',len(args)-n,'roots',len(d['roots']))
diff --git a/research/w3-p10b-pointwise-complex/discovery/pr327/extend_floor.py b/research/w3-p10b-pointwise-complex/discovery/pr327/extend_floor.py
new file mode 100644
index 0000000000..b9f2bb7619
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/discovery/pr327/extend_floor.py
@@ -0,0 +1,99 @@
+# agent-cmod copy of PR233 discovery extend.py (PR162 extension, DaysSky et al., Apache-2.0) with a donor floor.
+"""PR162 generalized-carrier construction helper, DaysSky et al. Apache-2.0.
+Extracted unchanged function only; no foreign producer invocation.
+"""
+from collections import defaultdict
+from functools import lru_cache
+from paired_cube.frames import basis,perp,contained
+@lru_cache(maxsize=None)
+def join(A,B):
+ if A==B or not B:return A
+ if not A:return B
+ return basis(A+B)
+@lru_cache(maxsize=None)
+def orthogonal(S,A):return all((s&a).bit_count()%2==0 for s in S for a in A)
+def extended_arcs(g, frozen, min_donor=0):
+    """Greedy extension of the frozen arcs.  gspan[x]: span of every value that reaches x through DAG or arc edges;
+    gann[x]: annihilator of the intersection of every root frame reachable from x.  An arc x -> t is legal iff
+    gspan[x] is orthogonal to gann[t] and t does not reach x."""
+    h = g['h']; inputs = g['inputs']
+    args = [None] + [None if a is None else [x + 1 for x in a] for a in g['args']]
+    roots = [dict(r, node=r['node'] + 1) for r in g['roots']]
+    v, n = len(inputs), len(args)
+    spans = [()] * n
+    for i, u in enumerate(inputs): spans[i + 1] = (u,)
+    for x in range(v + 1, n): spans[x] = basis(spans[args[x][0]] + spans[args[x][1]])
+    rann = [perp(spans[r['node']], h) if r.get('kind', 'side') == 'center' else basis(inputs[t] for t in r['targets'])
+            for r in roots]
+    active = set(range(1, v + 1)); todo = [r['node'] for r in roots]
+    while todo:
+        x = todo.pop()
+        if x in active: continue
+        active.add(x)
+        if args[x]: todo.extend(args[x])
+    succ = [set() for _ in args]; pred = [set() for _ in args]; direct = [[] for _ in args]
+    uses = defaultdict(list)
+    for x in sorted(active):
+        if args[x]:
+            for j, y in enumerate(args[x]): succ[y].add(x); pred[x].add(y); uses[y].append(2 * x + j)
+    for j, r in enumerate(roots): direct[r['node']].extend(rann[j]); uses[r['node']].append((1 << 31) | j)
+    gann = [None] * n; gspan = list(spans)
+    for x in sorted(active, reverse=True): gann[x] = basis(direct[x] + [z for y in succ[x] for z in gann[y]])
+    plain = list(gann)                                       # DAG-only intersections, for the candidate screen
+    ancestors = {}
+    def above(x):
+        if x not in ancestors:
+            seen = set(); stack = [x]
+            while stack:
+                y = stack.pop()
+                if args[y]:
+                    for z in args[y]:
+                        if z not in seen: seen.add(z); stack.append(z)
+            ancestors[x] = seen
+        return ancestors[x]
+    arcs = {}; used = set()
+    def value(code): return roots[code & 0x7fffffff]['node'] if code >> 31 else args[code // 2][code & 1]
+    def add(x, code, check):
+        t = None if code >> 31 else code // 2
+        At = rann[code & 0x7fffffff] if t is None else gann[t]
+        if check:
+            if not orthogonal(gspan[x], At): return False
+            if t is not None:
+                stack = [t]; seen = {t}
+                while stack:
+                    y = stack.pop()
+                    if y == x: return False
+                    for z in succ[y]:
+                        if z not in seen: seen.add(z); stack.append(z)
+        arcs[x] = code; used.add(code)
+        work = [(x, At)]
+        while work:
+            y, extra = work.pop(); new = join(gann[y], extra)
+            if new == gann[y]: continue
+            gann[y] = new
+            for z in pred[y]: work.append((z, new))
+        if t is not None:
+            succ[x].add(t); pred[t].add(x); work = [(t, gspan[x])]
+            while work:
+                y, extra = work.pop(); new = join(gspan[y], extra)
+                if new == gspan[y]: continue
+                gspan[y] = new
+                for z in succ[y]: work.append((z, new))
+        return True
+    for x, code in frozen: assert add(x, code, True), ('frozen arc rejected', x, code)
+    for x in sorted(active):
+        if x < min_donor: continue
+        if not args[x]: continue
+        for j, y in enumerate(args[x]):
+            other = args[x][1 - j]
+            for code in uses[y]:
+                if code == 2 * x + j or x in arcs or code in used: continue
+                if code >> 31: screen = rann[code & 0x7fffffff]
+                else:
+                    t = code // 2
+                    if t < x and t in above(x): continue
+                    screen = plain[t]
+                if not orthogonal(spans[other], screen): continue
+                if sum(1 for u in uses[value(code)] if u not in used) <= 1: continue    # every value keeps a free use
+                add(x, code, True)
+    return sorted([x, code] for x, code in arcs.items())
diff --git a/research/w3-p10b-pointwise-complex/discovery/pr327/make_layer_p.py b/research/w3-p10b-pointwise-complex/discovery/pr327/make_layer_p.py
new file mode 100644
index 0000000000..747db04c56
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/discovery/pr327/make_layer_p.py
@@ -0,0 +1,318 @@
+#!/usr/bin/env python3
+"""make_layer_p.py: discovery of a frozen physical layer for three query modules at cube size p (NOT run by verify.py).
+
+PR #304's discovery/make_layer.py with the cube size as a parameter (Graph(p), triple module at p, pair module at
+p - 1, all-but-one module at p - 2 instead of the literals 11, 10, 9), the centre-sharing graph as an option
+(--centre46: the pair module's 37th root is the star centre, centre46.py), the maximum-weight pairing with the gauge
+rank f = h - 4 (pairs_mw_p.py; PR #233's pairs_mw.py hard-codes f = 18, which is h - 4 only at h = 22) and
+aligned_word_p.py for the kernel-pair step. Nothing else changed.
+
+verify.py never searches: it reads layers/<set>/ and checks it. This script is how those files were found, kept for
+provenance and for re-pinning after a module changes. Recipe (the same for the control and the results):
+  1. the source-aligned graph of PR #194's pipeline with the given modules (aligned_word.py's construction);
+  2. fresh carrier arcs: the deterministic Hopcroft-Karp carrier matching of PR #200/#233's discovery harness
+     (pr233/matching.py, ordering 'reverse'), arcs of local donors dropped (as the mapped arcs of PR #193/#194 drop
+     them), then PR #162's greedy extension restricted to query donors (extend_floor.py);
+  3. compile_closure + select, then PR #200's operation-frame descent on that word (pr233/physical_opt.py single
+     endpoint moves, bundles.py equal-frame components, joint.py adjacent components; up to three cycles);
+  4. source_aligned_local_v4.py's forward closure with the descended frames as preferred frames (clipped to the
+     future cap; local operations keep their literal spans);
+  5. reuse pairs, two recipes on the same frames: 'parity' = source_aligned_local_v4.py --avoid-parity-erasure (PR
+     #193's minimum-weight full matching avoiding PR #184's parity erasures), 'mw' = PR #200/#233's maximum-weight
+     late compensated pairing (pairs_mw_p.py = pr233/pairs_mw.py with f = h - 4) with harness.descend's
+     last-late-pair order;
+  6. kernel pairs: PR #184's complex_frame_flow.py --witness --purify-source-donors --recycle-kernels on the
+     aligned tree written by aligned_word_p.py (its own maximum matching), frozen as found.
+Writes layers/<set>/{matching-arcs,physical-frames,physical-pairs-parity,physical-pairs-mw,kernel-pairs-parity,
+kernel-pairs-mw}.json and, for a set of package modules, records the module files used in SOURCE.json's
+module_sets (a module given outside modules/ is first copied there). Then run verify.py --write (re-pins
+SOURCE.json, sets the claims, regenerates the certificates) and verify.py.
+
+usage: python3 -B discovery/make_layer_p.py --set anneal [--p 10] [--tmod F] [--pmod F] [--qmod F] [--centre46]
+A module not given (and --centre46 when not given) defaults to the set's current entry in SOURCE.json. Needs
+numpy/scipy as verify.py; about two minutes at p = 10. Floating-point scores choose the frames and pairs; nothing
+here is a proof.
+Adapted from PR #304 (DreamingOfClouds with Anthropic Claude assistance, Apache-2.0) by DreamingOfClouds with
+Anthropic Claude assistance (Apache-2.0).
+"""
+import argparse
+import bisect
+import importlib.util
+import json
+import subprocess
+import sys
+import tempfile
+import time
+from collections import defaultdict
+from pathlib import Path
+
+if sys.flags.optimize:
+    raise SystemExit('Assertions must remain enabled: run without -O')
+sys.dont_write_bytecode = True
+HERE = Path(__file__).resolve().parent
+PKG = HERE.parent
+KINDS = ('tmod', 'pmod', 'qmod')
+DESCRIPTIONS = dict(
+    arcs='Frozen carrier arcs of the source-aligned word (discovery/make_layer.py step 2): deterministic HK carrier '
+         'matching (PR200/233 harness, ordering reverse) on query donors plus PR162 extension; checked by compile_closure.',
+    frames='Frozen operation frames (source_aligned_local_v4.py canonical form: operations whose frame differs from the '
+           'closure frame), from PR200 descent on the aligned word and the script\'s forward closure.',
+    parity='Frozen reuse pairs [donor, recipient, deadline]: PR193 recipe (source_aligned_local_v4.py '
+           '--avoid-parity-erasure minimum-weight full matching), in the script\'s control order.',
+    mw='Frozen reuse pairs [donor, recipient, deadline]: PR200/233 maximum-weight late compensated pairing '
+       '(pairs_mw.py) with the last late pair moved to the phase cut.',
+    kernel='Frozen kernel pairs of the frame flow on this layer (complex_frame_flow.py --witness --recycle-kernels).')
+
+
+def dump(path, data):
+    path.parent.mkdir(parents=True, exist_ok=True)
+    path.write_text(json.dumps(data, separators=(',', ':')) + '\n')
+
+
+def log(msg, t0=time.monotonic()):
+    print('[%6.1fs] %s' % (time.monotonic() - t0, msg), flush=True)
+
+
+def rebuild_tree(dest):
+    spec = importlib.util.spec_from_file_location('pr233_layer_verify', PKG / 'references/pr233-layer/verify.py')
+    mod = importlib.util.module_from_spec(spec)
+    spec.loader.exec_module(mod)
+    mod.rebuild(dest)
+    return dest
+
+
+def main():
+    ap = argparse.ArgumentParser(description=__doc__, formatter_class=argparse.RawDescriptionHelpFormatter)
+    ap.add_argument('--set', required=True, help='layer name: writes layers/<set>/')
+    ap.add_argument('--p', type=int, default=10, help='cube size (PR #304: 11)')
+    ap.add_argument('--centre46', action='store_true', default=None,
+                    help='centre-sharing graph: the pair module supplies the star centre as an extra root (centre46.py)')
+    ap.add_argument('--tmod', type=Path); ap.add_argument('--pmod', type=Path); ap.add_argument('--qmod', type=Path)
+    ap.add_argument('--tree', type=Path, help='an already rebuilt PR #202 tree (default: rebuild into a temporary directory)')
+    ap.add_argument('--out', type=Path, help='output directory (default: layers/<set>)')
+    a = ap.parse_args()
+    out = (a.out or PKG / 'layers' / a.set).resolve()
+    source = json.loads((PKG / 'SOURCE.json').read_text())
+    current = source['module_sets'].get(a.set, {})
+    centre = bool(current.get('centre46', False) if a.centre46 is None else a.centre46)
+    P = a.p
+    with tempfile.TemporaryDirectory(prefix='make-layer-') as d:
+        root = a.tree.resolve() if a.tree else rebuild_tree(Path(d).resolve())
+        src = root / 'references/paired-cube/sources'
+        mods, record = {}, {}
+        for k in KINDS:
+            given = getattr(a, k)
+            if given is None:
+                rel = current[k]
+                mods[k] = root / rel[5:] if rel.startswith('tree:') else PKG / rel
+                record[k] = rel
+                continue
+            given = given.resolve()
+            if (PKG / 'modules') not in given.parents:
+                dest = PKG / 'modules' / given.name
+                assert not dest.exists() or dest.read_bytes() == given.read_bytes(), 'a different module file exists: %s' % dest
+                dest.write_bytes(given.read_bytes())
+                log('copied %s into modules/' % given.name)
+                given = dest
+            mods[k] = given
+            record[k] = given.relative_to(PKG).as_posix()
+        sys.path.insert(0, str(root / 'scripts'))
+        sys.path.insert(0, str(HERE / 'pr233'))
+        sys.path.insert(0, str(HERE))
+        from paired_cube.graph import Graph
+        from paired_cube.modules import triple_module_from, pair_module_from, all_but_one_from, merge_outputs
+        from paired_cube.closure import compile_closure
+        from paired_cube.gauges import select
+        from paired_cube.frames import basis, perp, contained
+        from matching import carrier_matching
+        from extend_floor import extended_arcs
+        import physical_opt
+        import bundles
+        import joint
+        from pairs_mw_p import pairs_maxweight   # PR #233's pairs_mw.py with f = h - 4
+        import numpy as np
+        from scipy.sparse import csr_matrix
+        from scipy.sparse.csgraph import min_weight_full_bipartite_matching
+        # 1. graph (aligned_word.py / source_aligned_local_v4.py construction)
+        local = json.loads((src / 'local_L1.json').read_text())
+        local['G'] = {k: 's' for k in local['G']}
+        local['A'] = {k: 'fd' for k in local['A']}
+        builder = Graph(P, local=local)
+        if centre:
+            sys.path.insert(0, str(PKG))
+            from centre46 import load_pmod46, finish_centre46
+            g = finish_centre46(builder, triple_module_from(mods['tmod'], P), load_pmod46(mods['pmod'], P - 1), all_but_one_from(mods['qmod'], P - 2))
+        else:
+            g = builder.finish(triple_module_from(mods['tmod'], P), pair_module_from(mods['pmod'], P - 1), all_but_one_from(mods['qmod'], P - 2))
+        g = merge_outputs(g, builder, 'f8:00111100')
+        g['matching_frames'] = 'coordinate'
+        h, v = g['h'], g['v']
+        newcut = v + g['counts']['local_channel_additions']
+        # 2. fresh arcs
+        arcs, stats = carrier_matching(g, 'reverse')
+        arcs = [x for x in arcs if x[0] > newcut]
+        arcs = extended_arcs(g, arcs, min_donor=newcut + 1)
+        assert all(x[0] > newcut for x in arcs)
+        log('arcs: %d (HK matched %d of %d donors)' % (len(arcs), stats['matched'], stats['donors']))
+        # 3. descent on the aligned word
+        base, w = compile_closure(g, arcs)
+        row, word = select(g, base, w)
+        o = physical_opt.Physical(g, w, word, row)
+        for k in range(3):
+            r1 = o.descend(6); r2 = bundles.descend(o, 3); r3 = joint.descend(o, 3, 'forward', k)
+            if not any(x['changed'] for x in r1) and not any(x['moved_components'] for x in r2) and not any(x['moved'] for x in r3):
+                break
+        dframes = o.frames
+        log('descent done')
+        # 4. forward closure (source_aligned_local_v4.py, preferred frames = descended frames)
+        p, w = compile_closure(g, arcs)
+        p, word = select(g, p, w)
+        ops = word['ops']
+        assert len(ops) == len(dframes)
+        newargs = [None] + [None if x is None else [t + 1 for t in x] for x in g['args']]
+        newroots = [dict(r, node=r['node'] + 1) for r in g['roots']]
+        spans = [()] + [(q,) for q in g['inputs']]
+        for aa, bb in newargs[v + 1:]:
+            spans.append(basis(spans[aa] + spans[bb]))
+        root_frames = {}
+        for r, slot in zip(newroots, word['rootroles']):
+            root_frames[slot] = spans[r['node']] if r['kind'] == 'center' else perp(basis(g['inputs'][t] for t in r['targets']), h)
+        R = p['R']
+        pset = set(word['phase1'])
+        previous = [-1] * R
+        parents = []
+        for i, (aa, bb, x) in enumerate(ops):
+            parents.append((previous[aa], previous[bb]))
+            previous[aa] = previous[bb] = i
+            if x <= newcut and len(spans[x]) <= 2:
+                pset.add(i)
+        todo = list(pset)
+        while todo:
+            i = todo.pop()
+            for j in parents[i]:
+                if j >= 0 and j not in pset:
+                    pset.add(j)
+                    todo.append(j)
+        word['phase1'] = sorted(pset)
+        p['phase1_operations'] = len(pset)
+        chron = sorted(pset) + [i for i in range(len(ops)) if i not in pset]
+        cap = [perp(root_frames[s], h) if s in root_frames else () for s in range(R)]
+        maxima = [None] * len(ops)
+        for i in reversed(chron):
+            aa, bb, x = ops[i]
+            A = basis(cap[aa] + cap[bb])
+            maxima[i] = perp(A, h)
+            cap[aa] = cap[bb] = A
+        selected = [z for z in word['selected'] if z['rank'] == h - 4]
+        gauge = {z['role']: perp(tuple(z['annihilator']), h) for z in selected}
+        current = [() for _ in range(R)]
+        for x, slot in word['sources'].items():
+            current[slot] = (g['inputs'][int(x) - 1],)
+        for slot, U in gauge.items():
+            current[slot] = U
+        frames = [None] * len(ops)
+        for i in chron:
+            aa, bb, x = ops[i]
+            want = spans[x] if x <= newcut else perp(basis(perp(tuple(dframes[i]), h) + perp(maxima[i], h)), h)
+            U = basis(spans[x] + want + current[aa] + current[bb])
+            assert contained(U, maxima[i]), ('Forward frame outside future cap', i)
+            frames[i] = current[aa] = current[bb] = U
+        moved = [[i, list(U)] for i, U in enumerate(frames) if U != perp(tuple(w['annihilators'][ops[i][2]]), h)]
+        log('frames: %d of %d operations moved from their closure frames' % (len(moved), len(ops)))
+        # 5. pairs
+        role_ops = defaultdict(list)
+        for i, (aa, bb, _) in enumerate(ops):
+            role_ops[aa].append(i)
+            role_ops[bb].append(i)
+        pos = {i: k for k, i in enumerate(chron)}
+        for slot in role_ops:
+            role_ops[slot].sort(key=pos.__getitem__)
+        last_write = {}
+        for i, (aa, bb, x) in enumerate(ops):
+            last_write[aa] = max(last_write.get(aa, -1), pos[i])
+
+        def control_order(pairs):
+            controls = [q for q in pairs if q[2] is not None and last_write.get(q[0], -1) >= len(pset)]
+            if controls:
+                chosen = controls[-1]
+                pairs.remove(chosen)
+                pairs.append(chosen)
+            return pairs
+        recipients = sorted(gauge)
+        recipient_set = set(recipients)
+        donors = [slot for slot in range(R) if slot not in recipient_set and slot not in root_frames and slot in role_ops]
+        groups = defaultdict(list)
+        for j, slot in enumerate(recipients):
+            groups[gauge[slot]].append((pos[role_ops[slot][0]], j))
+        group_data = []
+        for U, ls in groups.items():
+            ls.sort()
+            group_data.append((U, [x for x, _ in ls], [j for _, j in ls]))
+        cache, indices, indptr = {}, [], [0]
+        for slot in donors:
+            last = role_ops[slot][-1]
+            U = frames[last]
+            eligible = cache.get(U)
+            if eligible is None:
+                eligible = cache[U] = [(ps, js) for V, ps, js in group_data if contained(U, V)]
+            for ps, js in eligible:
+                indices.extend(js[bisect.bisect_right(ps, pos[last]):])
+            indptr.append(len(indices))
+        parity = []
+        for cube in range(v // 8):
+            for bit in range(2):
+                parity.append(basis(tuple(g['inputs'][8 * cube + j] for j in range(8) if j.bit_count() % 2 == bit)))
+        erasable = {}
+
+        def weight(slot, j):
+            key = (frames[role_ops[slot][-1]], gauge[recipients[j]])
+            if key not in erasable:
+                U, V = key
+                erasable[key] = len(U) <= 3 and any(contained(U, S) and contained(S, V) for S in parity)
+            return 2.0 if erasable[key] else 1.0
+        weights = np.array([weight(slot, indices[k]) for r, slot in enumerate(donors) for k in range(indptr[r], indptr[r + 1])])
+        wmatrix = csr_matrix((weights, np.array(indices, dtype=np.int32), np.array(indptr, dtype=np.int32)), shape=(len(donors), len(recipients)))
+        rows, cols = min_weight_full_bipartite_matching(wmatrix.T.tocsr())
+        matched = np.full(len(donors), -1)
+        matched[cols] = rows
+        pairs_parity = control_order([[slot, recipients[j], None if role_ops[slot][-1] in pset else role_ops[recipients[j]][0]]
+                                      for slot, j in zip(donors, matched) if j >= 0])
+        o = physical_opt.Physical(g, w, dict(word, selected=selected), p)
+        for i, U in enumerate(frames):
+            o.frames[i] = tuple(U)
+        pairs_mw, mwstats = pairs_maxweight(o)
+        good = [k for k, (dd, b, t) in enumerate(pairs_mw) if t is not None and any(i not in o.phase and o.ops[i][0] == dd for i in o.role_ops[dd])]
+        if good:
+            pairs_mw.append(pairs_mw.pop(good[-1]))
+        log('pairs: parity %d, mw %d (of %d gauges)' % (len(pairs_parity), len(pairs_mw), len(recipients)))
+        dump(out / 'matching-arcs.json', dict(description=DESCRIPTIONS['arcs'], matching_arcs=arcs))
+        dump(out / 'physical-frames.json', dict(description=DESCRIPTIONS['frames'], frames=moved))
+        dump(out / 'physical-pairs-parity.json', dict(description=DESCRIPTIONS['parity'], pairs=pairs_parity))
+        dump(out / 'physical-pairs-mw.json', dict(description=DESCRIPTIONS['mw'], pairs=pairs_mw))
+        # 6. kernel pairs: aligned tree with aligned_word.py, flow with its own kernel matching
+        for recipe in ('parity', 'mw'):
+            tree = Path(d) / ('tree-' + recipe)
+            for cmd in ([PKG / 'aligned_word_p.py', '--p', P, *(['--centre46'] if centre else []), '--tree', root,
+                         '--tmod', mods['tmod'], '--pmod', mods['pmod'], '--qmod', mods['qmod'],
+                         '--arcs', out / 'matching-arcs.json', '--frames', out / 'physical-frames.json',
+                         '--pairs', out / ('physical-pairs-%s.json' % recipe), '--out', tree],
+                        [root / 'research/source-assisted/decision/complex_frame_flow.py', '--tree', tree, '--cache', tree / 'cache',
+                         '--out', tree / 'flow.json', '--witness', '--purify-source-donors', '--recycle-kernels']):
+                r = subprocess.run([sys.executable, '-B', *map(str, cmd)], cwd=root, capture_output=True, text=True)
+                if r.returncode:
+                    raise SystemExit('FAILED: %s\n%s%s' % (cmd[0], r.stdout[-3000:], r.stderr[-3000:]))
+            kernel = json.loads((tree / 'flow.witness.json').read_text())['kernel_pairs']
+            flow = json.loads((tree / 'flow.json').read_text())
+            dump(out / ('kernel-pairs-%s.json' % recipe), dict(description=DESCRIPTIONS['kernel'], kernel_pairs=kernel))
+            log('%s: flow new_R %d, kernel reuses %d, numerical three-stage root %.7e' % (recipe, flow['new_R'], len(kernel), flow['numerical_local_root']))
+    log('wrote %s' % out)
+    if centre:
+        record['centre46'] = True
+    if out == (PKG / 'layers' / a.set).resolve() and source['module_sets'].get(a.set) != record:
+        source['module_sets'][a.set] = dict(sorted(record.items()))
+        (PKG / 'SOURCE.json').write_text(json.dumps(source, indent=1) + '\n')
+        log('recorded the modules of %s in SOURCE.json: %s; now run verify.py --write, then verify.py' % (a.set, record))
+
+
+if __name__ == '__main__':
+    main()
diff --git a/research/w3-p10b-pointwise-complex/discovery/pr327/pairs_mw_p.py b/research/w3-p10b-pointwise-complex/discovery/pr327/pairs_mw_p.py
new file mode 100644
index 0000000000..0a96000bf5
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/discovery/pr327/pairs_mw_p.py
@@ -0,0 +1,47 @@
+# pairs_mw_p.py: PR #233's discovery/pairs_mw.py (Chafik Boukhalfa with Anthropic Claude and OpenAI Codex
+# assistance, Apache-2.0; vendored unchanged in pr233/pairs_mw.py) with ONE line changed: the gauge rank f in the
+# donor weight is h - 4 (the rank of the selected gauges) instead of the literal 18, which equals h - 4 only at
+# h = 22. At h = 18 the literal makes every weight 0 (no pairs); see README.md. Identical to pairs_mw.py at h = 22.
+# Change by DreamingOfClouds with Anthropic Claude assistance (Apache-2.0).
+"""Maximum-weight donor pairing for gauged recipients (weights depend on the donor only, so the
+transversal-matroid greedy with augmenting paths is optimal). Explicit legality: donor ungauged non-root with ops,
+last frame contained in the recipient gauge, donor dead before the recipient's first operation (or in phase one)."""
+from collections import defaultdict
+import sys
+from paired_cube.frames import contained
+def pairs_maxweight(o):
+ sys.setrecursionlimit(100000)
+ donors=[s for s in range(o.R) if s not in o.gauge and s not in o.roots and o.role_ops[s]]
+ recipients=sorted(o.gauge,key=lambda b:(o.position[o.role_ops[b][0]],b));rindex={b:i for i,b in enumerate(recipients)}
+ rtime=[o.position[o.role_ops[b][0]] for b in recipients]
+ byframe=defaultdict(list)
+ for s in donors:byframe[o.frames[o.role_ops[s][-1]]].append(s)
+ groups={}
+ adj=defaultdict(list)  # donor -> recipient indices
+ for b in recipients:
+  F=o.gauge[b]
+  if F not in groups:groups[F]=[s for D,ss in byframe.items() if contained(D,F) for s in ss]
+  t=rtime[rindex[b]]
+  for s in groups[F]:
+   if o.position[o.role_ops[s][-1]]<t:adj[s].append(rindex[b])
+ def weight(s):
+  dd=len(o.frames[o.role_ops[s][-1]]);f=o.h-4  # cp10: PR #233 hard-codes f=18 (= h-4 at h=22)
+  return 3*o.phi[o.h-dd]+o.phi[3*f]-3*o.phi[f-dd] if f>=dd else -1e9
+ order=sorted(adj,key=lambda s:(-weight(s),o.position[o.role_ops[s][-1]],s))
+ mr=[-1]*len(recipients);md={}
+ def dfs(s,seen):
+  for r in adj[s]:
+   if r in seen:continue
+   seen.add(r)
+   if mr[r]<0 or dfs(mr[r],seen):mr[r]=s;md[s]=r;return True
+  return False
+ for s in order:
+  if weight(s)<=0:break
+  dfs(s,set())
+ result=[]
+ for r,s in enumerate(mr):
+  if s>=0:
+   b=recipients[r];first=o.role_ops[b][0]
+   result.append([s,b,None if o.role_ops[s][-1] in o.phase else first])
+ stats=dict(donors=len(donors),recipients=len(recipients),edges=sum(map(len,adj.values())),matched=len(result),total_weight=sum(weight(s) for s,_,_ in result))
+ return result,stats
diff --git a/research/w3-p10b-pointwise-complex/discovery/pr327/pr233/README.md b/research/w3-p10b-pointwise-complex/discovery/pr327/pr233/README.md
new file mode 100644
index 0000000000..339adb1c76
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/discovery/pr327/pr233/README.md
@@ -0,0 +1,18 @@
+# Discovery (not run by verify.py)
+
+The physical layer of the certified package is PR #168's operation frames with PR #200's late compensated reuse
+pairing (`physical_opt.py:pairs`, the maximum-weight pairing of `pairs_mw.py` (transversal-matroid greedy with augmenting paths, donor weight the gain heuristic of PR #200)). `flow_frames.py` runs `harness.descend` on the source-aligned cache of
+PR #194's pipeline with all frame moves frozen (`DESCENT_LIGHT_ROUNDS=1 DESCENT_FREEZE_FRAC=1.0 --light
+--seed-frames`), recomputes the pairs and prices the layer with PR #184's `complex_frame_flow.py`:
+
+    SA_TREE=<PR202 tree> SA_PY=<python with numpy/scipy> DESCENT_LIGHT_ROUNDS=1 DESCENT_FREEZE_FRAC=1.0 \
+      python3 -B flow_frames.py --aligned <aligned tree> --seed-frames --light --mw 1 --keep out/
+
+Float flow roots: PR #168's pairing 7.00918e-4; PR #200's default pairing (donors ordered by the gain heuristic)
+7.09765e-4; donors ordered by widest end frame with recipients in reverse chronology (`PAIR_ORDER=wide
+PAIR_RECIPIENTS=reverse`) 7.09882e-4; the maximum-weight pairing (`--mw 1`, `pairs_mw.py`, every donor-weight variant
+tried) 7.09907e-4, the certified layer; random donor orders 6.4e-4 to 6.6e-4. With frames also descended: full cycle
+7.08613e-4, one light round 7.0950e-4, 98% of operations frozen 7.09767e-4; a screen over every single endpoint move
+on top of PR #168's frames (`flow_moves.py`) finds no improving move, so the frames are left untouched. The matching
+itself is PR #168's frozen carrier matching (`matching.py`, `extend.py` are needed only to import `harness`).
+Prepared by Chafik Boukhalfa with Anthropic Claude and OpenAI Codex assistance; Apache-2.0.
diff --git a/research/w3-p10b-pointwise-complex/discovery/pr327/pr233/bundles.py b/research/w3-p10b-pointwise-complex/discovery/pr327/pr233/bundles.py
new file mode 100644
index 0000000000..2035223567
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/discovery/pr327/pr233/bundles.py
@@ -0,0 +1,50 @@
+"""Connected equal-frame endpoint moves, motivated by PR164/168 bundle descent.
+
+Own explicit boundary implementation: all source spans, incoming and outgoing
+role frames are recomputed before a move. No physical legality is inferred from
+an optimization score; the independent physical verifier remains mandatory.
+"""
+from collections import defaultdict
+from physical_opt import join,meet
+from paired_cube.frames import contained
+
+def descend(o,rounds=4):
+ report=[]
+ for rnd in range(rounds):
+  n=len(o.ops);parent=list(range(n))
+  def root(i):
+   while parent[i]!=i:parent[i]=parent[parent[i]];i=parent[i]
+   return i
+  def union(i,j):
+   a,b=root(i),root(j)
+   if a!=b:parent[max(a,b)]=min(a,b)
+  for i in range(n):
+   for j in o.prev[i]:
+    if j is not None and o.frames[i]==o.frames[j]:union(i,j)
+  groups=defaultdict(list)
+  for i in range(n):groups[root(i)].append(i)
+  groups=[l for l in groups.values() if len(l)>1];groups.sort(key=lambda l:min(l),reverse=not rnd%2)
+  moved=changed=0;gain=0.;big=0
+  for ids in groups:
+   inside=set(ids);F=o.frames[ids[0]];assert all(o.frames[i]==F for i in ids)
+   L=();U=o.full;previous=[];after=[]
+   for i in ids:
+    a,b,x=o.ops[i];L=join(L,o.spans[x])
+    for role,j in zip((a,b),o.prev[i]):
+     if j not in inside:
+      A=o.frames[j] if j is not None else o.start[role];previous.append(A);L=join(L,A)
+    for role,j in zip((a,b),o.next[i]):
+     if j not in inside:
+      A=o.frames[j] if j is not None else o.end[role];after.append(A);U=meet(U,A,o.h)
+   assert contained(L,F) and contained(F,U)
+   def score(A):
+    d=len(A);return sum(o.phi[d-len(P)] for P in previous)+sum(o.phi[len(N)-d] for N in after)
+   old=score(F);candidate=min((L,U),key=score);delta=old-score(candidate)
+   if delta>1e-7:
+    for i in ids:o.frames[i]=candidate
+    moved+=1;changed+=len(ids);gain+=delta;big=max(big,len(ids))
+  rec=dict(round=rnd,components=len(groups),moved_components=moved,changed_operations=changed,numerical_excess_gain=gain,largest_moved_component=big);report.append(rec);print('bundles',rec,flush=True)
+  if not moved:break
+ for i in range(n):
+  P,N=o.neighbors(i);assert contained(o.spans[o.ops[i][2]],o.frames[i]);assert all(contained(F,o.frames[i]) for F in P);assert all(contained(o.frames[i],F) for F in N)
+ return report
diff --git a/research/w3-p10b-pointwise-complex/discovery/pr327/pr233/joint.py b/research/w3-p10b-pointwise-complex/discovery/pr327/pr233/joint.py
new file mode 100644
index 0000000000..a67d18034d
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/discovery/pr327/pr233/joint.py
@@ -0,0 +1,67 @@
+"""Joint collapse of adjacent operation-frame components.
+
+All incoming/outgoing chains and actual value supports bound each common
+frame. Old internal rank increments are charged once per role edge. A move
+must reduce the full boundary-plus-internal numerical objective. Floating
+scores select a witness only; admission requires the independent exact audit.
+"""
+from collections import defaultdict
+import random,time
+from physical_opt import join,meet
+from paired_cube.frames import contained
+
+def components(o):
+ n=len(o.ops);parent=list(range(n))
+ def root(i):
+  while parent[i]!=i:parent[i]=parent[parent[i]];i=parent[i]
+  return i
+ for i in range(n):
+  for j in o.prev[i]:
+   if j is not None and o.frames[i]==o.frames[j]:
+    a,b=root(i),root(j)
+    if a!=b:parent[max(a,b)]=min(a,b)
+ groups=defaultdict(list)
+ for i in range(n):groups[root(i)].append(i)
+ adjacent=set()
+ for i in range(n):
+  for j in o.prev[i]:
+   if j is not None and root(i)!=root(j):adjacent.add(tuple(sorted((root(i),root(j)))))
+ return groups,sorted(adjacent)
+
+def descend(o,rounds=5,order='forward',seed=0):
+ result=[];rng=random.Random(seed)
+ for rnd in range(rounds):
+  t=time.monotonic();groups,adjacent=components(o)
+  if order=='random':rng.shuffle(adjacent)
+  elif (order=='reverse')^(rnd%2==1):adjacent.reverse()
+  tried=feasible=moved=changed=largest=0;gain=0.
+  for a,b in adjacent:
+   ids=groups[a]+groups[b];inside=set(ids);tried+=1
+   # Each original component remains uniform even after overlapping moves.
+   assert all(o.frames[i]==o.frames[a] for i in groups[a])
+   assert all(o.frames[i]==o.frames[b] for i in groups[b])
+   L=();U=o.full;previous=[];after=[];old=0.
+   for i in ids:
+    x,y,node=o.ops[i];F=o.frames[i];L=join(L,o.spans[node])
+    for role,j in zip((x,y),o.prev[i]):
+     P=o.frames[j] if j is not None else o.start[role]
+     assert contained(P,F);old+=o.phi[len(F)-len(P)]
+     if j not in inside:previous.append(P);L=join(L,P)
+    for role,j in zip((x,y),o.next[i]):
+     if j not in inside:
+      N=o.frames[j] if j is not None else o.end[role]
+      assert contained(F,N);after.append(N);U=meet(U,N,o.h);old+=o.phi[len(N)-len(F)]
+   if not contained(L,U):continue
+   feasible+=1
+   def score(F):
+    d=len(F);return sum(o.phi[d-len(P)] for P in previous)+sum(o.phi[len(N)-d] for N in after)
+   best=min((L,U),key=lambda F:(score(F),len(F),F));delta=old-score(best)
+   if delta>1e-7:
+    for i in ids:o.frames[i]=best
+    moved+=1;changed+=len(ids);gain+=delta;largest=max(largest,len(ids))
+  row=dict(round=rnd,groups=len(groups),tried=tried,feasible=feasible,moved=moved,changed=changed,largest=largest,numerical_excess_gain=gain,seconds=time.monotonic()-t)
+  print('joint',row,flush=True);result.append(row)
+  if not moved:break
+ for i in range(len(o.ops)):
+  P,N=o.neighbors(i);assert contained(o.spans[o.ops[i][2]],o.frames[i]);assert all(contained(F,o.frames[i]) for F in P);assert all(contained(o.frames[i],F) for F in N)
+ return result
diff --git a/research/w3-p10b-pointwise-complex/discovery/pr327/pr233/matching.py b/research/w3-p10b-pointwise-complex/discovery/pr327/pr233/matching.py
new file mode 100644
index 0000000000..e84e348556
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/discovery/pr327/pr233/matching.py
@@ -0,0 +1,131 @@
+"""Own explicit legal-edge Hopcroft-Karp witness; from our r13 search."""
+from collections import deque
+import sys
+from paired_cube.frames import basis,perp,contained
+def carrier_matching(g,ordering='node'):
+    """Exact legal edge list, then deterministic Hopcroft--Karp matching.
+
+    Frame and chronological predicates are the retained compiler's predicates;
+    the resulting explicit arcs are checked anew by compile_graph.
+    """
+    h=g['h'];v=g['v'];args=[None]+[None if a is None else [y+1 for y in a] for a in g['args']]
+    roots=[dict(r,node=r['node']+1) for r in g['roots']];n=len(args)
+    spans=[()]*n
+    for i,u in enumerate(g['inputs']):spans[i+1]=(u,)
+    for x in range(v+1,n):spans[x]=basis(spans[args[x][0]]+spans[args[x][1]])
+    rann=[];rframe=[]
+    for r in roots:
+        if r['kind']=='center':U=spans[r['node']];A=perp(U,h)
+        else:A=basis(g['inputs'][t] for t in r['targets']);U=perp(A,h)
+        rann.append(A);rframe.append(U)
+    active=set(range(1,v+1));todo=[r['node'] for r in roots]
+    while todo:
+        x=todo.pop()
+        if x in active:continue
+        active.add(x);todo.extend(args[x] or [])
+    initial=spans
+    if g['matching_frames'] in ('maximal','coordinate'):
+        initial=[()]*n;ann=[None]*n;successors=[[] for _ in args];constraints=[[] for _ in args]
+        for x in sorted(active):
+            for y in args[x] or []:successors[y].append(x)
+        for j,r in enumerate(roots):constraints[r['node']].extend(rann[j])
+        for x in sorted(active,reverse=True):
+            ann[x]=basis(constraints[x]+[z for y in successors[x] for z in ann[y]])
+            initial[x]=perp(ann[x],h)
+            if g['matching_frames']=='coordinate':
+                cover=0
+                for u in spans[x]:cover|=u
+                initial[x]=perp(ann[x]+tuple(1<<j for j in range(h) if not cover>>j&1),h)
+            assert contained(spans[x],initial[x])
+    order=sorted(active,key=lambda x:(len(initial[x]),x));position={x:i for i,x in enumerate(order)}
+    uses=[[] for _ in args];uf=[];ut=[];uc=[]
+    for x in order:
+        for j,y in enumerate(args[x] or []):uses[y].append(len(uc));uf.append(initial[x]);ut.append(x);uc.append(2*x+j)
+    for j,r in enumerate(roots):uses[r['node']].append(len(uc));uf.append(rframe[j]);ut.append(n+j);uc.append((1<<31)|j)
+    donors=[x for x in order if args[x]]
+    if ordering=='reverse':donors.reverse()
+    elif ordering=='narrow':donors.sort(key=lambda x:(len(initial[x]),x))
+    elif ordering=='wide':donors.sort(key=lambda x:(-len(initial[x]),x))
+    elif ordering=='node':donors.sort()
+    elif ordering.startswith('random:'):
+        import random
+        rnd=random.Random(int(ordering.split(':',1)[1]));rnd.shuffle(donors)
+    adjacency=[]
+    for x in donors:
+        row=[u for y in args[x] for u in uses[y] if (ut[u]>=n or position[ut[u]]>position[x]) and contained(initial[x],uf[u])]
+        if ordering.startswith('late'):   # prefer the latest use in the chronology (root uses last), then the largest growth
+            row.sort(key=lambda u:(-(position[ut[u]] if ut[u]<n else 10**9),-(len(uf[u])-len(initial[x])),uc[u]))
+        elif ordering.startswith('latefirst'):
+            row.sort(key=lambda u:(-(position[ut[u]] if ut[u]<n else -1),uc[u]))
+        else:
+            row.sort(key=lambda u:(len(uf[u])-len(initial[x]),uc[u]))
+        if ordering.startswith('random:') and ordering.count(':')==2:rnd.shuffle(row)   # 'random:SEED:rows' also shuffles use order
+        adjacency.append(row)
+    if ordering=='struct':
+        return dict(donors=donors,adjacency=adjacency,uc=uc,uf=uf,ut=ut,initial=initial,n=n,position=position),None
+    L=len(donors);ml=[-1]*L;mr=[-1]*len(uc);dist=[0]*L;iterations=0
+    def bfs():
+        queue=deque();found=False
+        for u in range(L):
+            dist[u]=0 if ml[u]<0 else -1
+            if ml[u]<0:queue.append(u)
+        while queue:
+            u=queue.popleft()
+            for v2 in adjacency[u]:
+                z=mr[v2]
+                if z<0:found=True
+                elif dist[z]<0:dist[z]=dist[u]+1;queue.append(z)
+        return found
+    def dfs(u):
+        for v2 in adjacency[u]:
+            z=mr[v2]
+            if z<0 or dist[z]==dist[u]+1 and dfs(z):ml[u]=v2;mr[v2]=u;return True
+        dist[u]=-1;return False
+    sys.setrecursionlimit(max(10000,2*L))
+    while bfs():
+        iterations+=1
+        for u in range(L):
+            if ml[u]<0:dfs(u)
+    arcs=[[donors[i],uc[v2]] for i,v2 in enumerate(ml) if v2>=0]
+    stats=dict(donors=L,uses=len(uc),edges=sum(map(len,adjacency)),matched=len(arcs),augment_rounds=iterations)
+    if ordering.startswith('frozen:'):
+        import json as _j
+        given=_j.loads(open(ordering.split(':',1)[1]).read());pos_of={uc[u]:u for u in range(len(uc))}
+        growth=0;seen=0
+        for x,code in given:
+            u=pos_of.get(code)
+            if u is None:continue
+            seen+=1;growth+=len(uf[u])-len(initial[x])
+        stats.update(frozen_arcs=len(given),frozen_seen=seen,frozen_growth=growth,hk_growth=int(sum(len(uf[v2])-len(initial[donors[i]]) for i,v2 in enumerate(ml) if v2>=0)))
+        import os as _os
+        if _os.environ.get('MATCH_FEATURES'):
+            chosen={x:pos_of[c] for x,c in given if c in pos_of};hk={donors[i]:v2 for i,v2 in enumerate(ml) if v2>=0}
+            rec=[]
+            for i,x in enumerate(donors):
+                cands=[dict(u=u,growth=len(uf[u])-len(initial[x]),udim=len(uf[u]),root=int(ut[u]>=n),gap=(position[ut[u]]-position[x]) if ut[u]<n else -1,frozen=int(chosen.get(x)==u),hk=int(hk.get(x)==u)) for u in adjacency[i]]
+                if cands:rec.append(dict(x=x,ddim=len(initial[x]),pos=position[x],cands=cands))
+            open(_os.environ['MATCH_FEATURES'],'w').write(_j.dumps(rec))
+        return [list(a) for a in given],stats
+    if ordering.startswith('mincost'):
+        # Among matchings that saturate the Hopcroft-Karp matched donor set, pick the one minimizing the total frame
+        # growth (dim use frame - dim donor frame), via scipy's sparse LAPJV. 'mincost:max' maximizes it instead.
+        import numpy as np
+        from scipy.sparse import csr_matrix
+        from scipy.sparse.csgraph import min_weight_full_bipartite_matching
+        parts=ordering.split(':');sign=-1 if len(parts)>1 and parts[1]=='max' else 1
+        noise=None
+        if len(parts)>2:
+            import random as _r;noise=_r.Random(int(parts[2]))
+        rows=[i for i,v2 in enumerate(ml) if v2>=0];rid={i:k for k,i in enumerate(rows)}
+        data=[];ri=[];ci=[]
+        for i in rows:
+            x=donors[i]
+            for u in adjacency[i]:
+                growth=len(uf[u])-len(initial[x]);ri.append(rid[i]);ci.append(u);data.append(1+sign*growth+(0 if sign>0 else 30)+(noise.random()*float(parts[3]) if noise else 0.0))
+        M=csr_matrix((np.array(data,dtype=float),(np.array(ri),np.array(ci))),shape=(len(rows),len(uc)))
+        r_ind,c_ind=min_weight_full_bipartite_matching(M)
+        arcs=[[donors[rows[r]],uc[c]] for r,c in zip(r_ind,c_ind)]
+        stats.update(matched=len(arcs),total_growth=int(sum(len(uf[c])-len(initial[donors[rows[r]]]) for r,c in zip(r_ind,c_ind))),mincost=True)
+        hk_growth=int(sum(len(uf[v2])-len(initial[donors[i]]) for i,v2 in enumerate(ml) if v2>=0));stats['hk_growth']=hk_growth
+    return arcs,stats
+
diff --git a/research/w3-p10b-pointwise-complex/discovery/pr327/pr233/pairs_mw.py b/research/w3-p10b-pointwise-complex/discovery/pr327/pr233/pairs_mw.py
new file mode 100644
index 0000000000..7e0db9777e
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/discovery/pr327/pr233/pairs_mw.py
@@ -0,0 +1,42 @@
+"""Maximum-weight donor pairing for gauged recipients (weights depend on the donor only, so the
+transversal-matroid greedy with augmenting paths is optimal). Explicit legality: donor ungauged non-root with ops,
+last frame contained in the recipient gauge, donor dead before the recipient's first operation (or in phase one)."""
+from collections import defaultdict
+import sys
+from paired_cube.frames import contained
+def pairs_maxweight(o):
+ sys.setrecursionlimit(100000)
+ donors=[s for s in range(o.R) if s not in o.gauge and s not in o.roots and o.role_ops[s]]
+ recipients=sorted(o.gauge,key=lambda b:(o.position[o.role_ops[b][0]],b));rindex={b:i for i,b in enumerate(recipients)}
+ rtime=[o.position[o.role_ops[b][0]] for b in recipients]
+ byframe=defaultdict(list)
+ for s in donors:byframe[o.frames[o.role_ops[s][-1]]].append(s)
+ groups={}
+ adj=defaultdict(list)  # donor -> recipient indices
+ for b in recipients:
+  F=o.gauge[b]
+  if F not in groups:groups[F]=[s for D,ss in byframe.items() if contained(D,F) for s in ss]
+  t=rtime[rindex[b]]
+  for s in groups[F]:
+   if o.position[o.role_ops[s][-1]]<t:adj[s].append(rindex[b])
+ def weight(s):
+  dd=len(o.frames[o.role_ops[s][-1]]);f=18
+  return 3*o.phi[o.h-dd]+o.phi[3*f]-3*o.phi[f-dd] if f>=dd else -1e9
+ order=sorted(adj,key=lambda s:(-weight(s),o.position[o.role_ops[s][-1]],s))
+ mr=[-1]*len(recipients);md={}
+ def dfs(s,seen):
+  for r in adj[s]:
+   if r in seen:continue
+   seen.add(r)
+   if mr[r]<0 or dfs(mr[r],seen):mr[r]=s;md[s]=r;return True
+  return False
+ for s in order:
+  if weight(s)<=0:break
+  dfs(s,set())
+ result=[]
+ for r,s in enumerate(mr):
+  if s>=0:
+   b=recipients[r];first=o.role_ops[b][0]
+   result.append([s,b,None if o.role_ops[s][-1] in o.phase else first])
+ stats=dict(donors=len(donors),recipients=len(recipients),edges=sum(map(len,adj.values())),matched=len(result),total_weight=sum(weight(s) for s,_,_ in result))
+ return result,stats
diff --git a/research/w3-p10b-pointwise-complex/discovery/pr327/pr233/physical_opt.py b/research/w3-p10b-pointwise-complex/discovery/pr327/pr233/physical_opt.py
new file mode 100644
index 0000000000..a2e8e7d053
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/discovery/pr327/pr233/physical_opt.py
@@ -0,0 +1,127 @@
+#!/usr/bin/env python3
+"""New deterministic exact frame endpoints and legal late handoff discovery.
+
+Chafik Boukhalfa with OpenAI Codex assistance. Apache-2.0.
+Builds on PR144 binary frame/word contract and PR157 physical reuse theorem.
+Numerical excess weights only select witnesses; exact containment, histogram
+and scalar checks must be independently replayed on the selected candidate.
+"""
+from collections import defaultdict,deque,Counter
+from functools import lru_cache
+import math
+from paired_cube.frames import basis,perp,contained
+
+@lru_cache(maxsize=300000)
+def join(A,B):
+    if contained(A,B):return B
+    if contained(B,A):return A
+    return basis(A+B)
+@lru_cache(maxsize=300000)
+def meet(A,B,h):
+    if contained(A,B):return A
+    if contained(B,A):return B
+    return perp(join(perp(A,h),perp(B,h)),h)
+
+class Physical:
+    def __init__(self,g,w,word,row):
+        self.g,self.w,self.word,self.row=g,w,word,row;self.h=g['h'];self.R=row['R'];self.v=g['v'];h=self.h
+        self.full=basis(1<<i for i in range(h));self.ops=[tuple(op) for op in word['ops']]
+        args=[None]+[None if a is None else [x+1 for x in a] for a in g['args']];self.spans=[()]*len(args)
+        for x in range(1,len(args)):
+            self.spans[x]=(g['inputs'][x-1],) if args[x] is None else join(self.spans[args[x][0]],self.spans[args[x][1]])
+        self.frames=[perp(tuple(w['annihilators'][x]),h) for _,_,x in self.ops];self.original=self.frames[:]
+        self.start=[()]*self.R;self.end=[self.full]*self.R
+        self.sources={int(x):s for x,s in word['sources'].items()}
+        for x,s in self.sources.items():self.start[s]=(g['inputs'][x-1],)
+        self.gauge={z['role']:perp(tuple(z['annihilator']),h) for z in word['selected']}
+        for s,F in self.gauge.items():self.start[s]=F
+        self.roots=set(word['rootroles'])
+        for r,s in zip(g['roots'],word['rootroles']):
+            self.end[s]=self.spans[r['node']+1] if r['kind']=='center' else perp(basis(g['inputs'][t] for t in r['targets']),h)
+        self.role_ops=[[] for _ in range(self.R)]
+        self.prev=[];last=[None]*self.R
+        for i,(a,b,x) in enumerate(self.ops):
+            self.prev.append([last[a],last[b]])
+            for s in (a,b):last[s]=i;self.role_ops[s].append(i)
+        self.next=[None]*len(self.ops);nxt=[None]*self.R
+        for i in range(len(self.ops)-1,-1,-1):
+            a,b,x=self.ops[i];self.next[i]=[nxt[a],nxt[b]];nxt[a]=nxt[b]=i
+        phase=set(word['phase1']);self.order=list(word['phase1'])+[i for i in range(len(self.ops)) if i not in phase]
+        self.order.sort(key=lambda i:(i not in phase,i));self.position={i:j for j,i in enumerate(self.order)};self.phase=phase
+        import os
+        _a=float(os.environ.get('PHI_ALPHA','0.00057'));self.rng=__import__('random').Random(int(os.environ.get('DESCENT_SEED','0')))
+        self.phi=[0.]+[t*math.expm1(_a*math.log(3*h/t))/_a for t in range(1,3*h+1)]
+
+    def neighbors(self,i):
+        a,b,_=self.ops[i]
+        P=[self.frames[j] if j is not None else self.start[s] for s,j in zip((a,b),self.prev[i])]
+        N=[self.frames[j] if j is not None else self.end[s] for s,j in zip((a,b),self.next[i])]
+        return P,N
+
+    def descend(self,rounds=4,lower_only=False):
+        changes=[]
+        for rnd in range(rounds):
+            changed=0;gain=0.
+            order=list(range(len(self.ops))) if rnd%2==0 else list(range(len(self.ops)-1,-1,-1))
+            if __import__('os').environ.get('DESCENT_SEED'):self.rng.shuffle(order)
+            for i in order:
+                P,N=self.neighbors(i);old=self.frames[i];x=self.ops[i][2]
+                L=join(join(P[0],P[1]),self.spans[x]);assert contained(L,old)
+                U=meet(N[0],N[1],self.h);assert contained(old,U)
+                def cost(r):return sum(self.phi[r-len(F)] for F in P)+sum(self.phi[len(F)-r] for F in N)
+                choices=[L] if lower_only else [L,U]
+                best=min(choices,key=lambda F:(cost(len(F)),len(F),F))
+                delta=cost(len(old))-cost(len(best))
+                if best!=old and (lower_only or delta>1e-10 or abs(delta)<=1e-10 and len(best)<len(old)):
+                    self.frames[i]=best;changed+=1;gain+=delta
+            changes.append(dict(round=rnd,changed=changed,numerical_excess_gain=gain));print('descent',changes[-1],flush=True)
+            if not changed:break
+        for i in range(len(self.ops)):
+            P,N=self.neighbors(i);assert contained(self.spans[self.ops[i][2]],self.frames[i]);assert all(contained(F,self.frames[i]) for F in P);assert all(contained(self.frames[i],F) for F in N)
+        return changes
+
+    def pairs(self):
+        donors=[s for s in range(self.R) if s not in self.gauge and s not in self.roots and self.role_ops[s]]
+        byframe=defaultdict(list)
+        for s in donors:byframe[self.frames[self.role_ops[s][-1]]].append(s)
+        recipients=sorted(self.gauge,key=lambda b:(self.position[self.role_ops[b][0]],b))
+        groups={};adj=[];gains={}
+        for b in recipients:
+            F=self.gauge[b]
+            if F not in groups:
+                groups[F]=[s for D,ss in byframe.items() if contained(D,F) for s in ss]
+            time=self.position[self.role_ops[b][0]]
+            row=[s for s in groups[F] if self.position[self.role_ops[s][-1]]<time]
+            def gain(s):
+                dd=len(self.frames[self.role_ops[s][-1]]);f=len(F)
+                return 3*self.phi[self.h-dd]+self.phi[3*f]-3*self.phi[f-dd]
+            row.sort(key=lambda s:(-gain(s),self.position[self.role_ops[s][-1]],s));adj.append(row)
+        L=len(recipients);ml=[-1]*L;mr={};dist=[0]*L
+        def bfs():
+            q=deque();found=False
+            for u in range(L):
+                dist[u]=0 if ml[u]<0 else -1
+                if ml[u]<0:q.append(u)
+            while q:
+                u=q.popleft()
+                for d in adj[u]:
+                    z=mr.get(d,-1)
+                    if z<0:found=True
+                    elif dist[z]<0:dist[z]=dist[u]+1;q.append(z)
+            return found
+        def dfs(u):
+            for d in adj[u]:
+                z=mr.get(d,-1)
+                if z<0 or dist[z]==dist[u]+1 and dfs(z):ml[u]=d;mr[d]=u;return True
+            dist[u]=-1;return False
+        while bfs():
+            for u in range(L):
+                if ml[u]<0:dfs(u)
+        result=[]
+        for j,d in enumerate(ml):
+            if d>=0:
+                b=recipients[j];first=self.role_ops[b][0]
+                result.append([d,b,None if self.role_ops[d][-1] in self.phase else first])
+        return result,dict(donors=len(donors),recipients=L,edges=sum(map(len,adj)),matched=len(result),gauge_frames=len(groups),donor_frames=len(byframe))
+
+    def changed(self):return [[i,F] for i,F in enumerate(self.frames) if F!=self.original[i]]
diff --git a/research/w3-p10b-pointwise-complex/gcert_emit.py b/research/w3-p10b-pointwise-complex/gcert_emit.py
new file mode 100644
index 0000000000..fd5d56f651
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/gcert_emit.py
@@ -0,0 +1,318 @@
+#!/usr/bin/env python3
+"""gcert_emit.py: lower a source-aligned complex flow to Jacob Sussman's certificate format gcert/1.
+
+Inputs (all produced by the integer-mult-bounds tool chain of PR #184 / #194 on an aligned tree):
+  --cache     the aligned tree's cache (graph.json, selection.json)
+  --flow      flow.json          (complex_frame_flow.py --witness)
+  --witness   flow.witness.json  (same run)
+  --lift      lift.certificate.json.gz (exact_complex_flow_lift.py: local invertible completions as elementary gates)
+Output: a gcert/1 JSON(.gz) as read by wht-power-saving-lean/tools/gx/gx.py (check1) and gxgen.py.
+
+Registers: x_j = j (sources, start at the port line), y_t = v+t (targets, start at frame 0, end at port^perp),
+slots 2v.. (one per physical flow coordinate: v source copies + births; kernel reuse continues a slot).
+Every node of the flow DAG (phase, frame) becomes, at its frame: the source copies (x -> slot), the reads
+(slot -> targets, or a retained total for the scatter), then the lift's elementary gates.  A 'swap' relabels
+positions, a 'scale' changes the unit of a slot (wanted = lam * stored, as gxconv.py), an 'add' is one 'out'
+gate.  After phase 2: the paired-cube K blocks (1 - J/2) on each parity class of a cube, the direct reads of the
+mixed sources into their antipodal targets at the target cap frames, and the inverse K blocks at the full frame.
+"""
+import argparse, gzip, json, heapq, sys, time
+from collections import Counter, defaultdict
+from fractions import Fraction as F
+
+
+def load(p):
+    p = str(p)
+    if p.endswith('.gz'):
+        return json.loads(gzip.decompress(open(p, 'rb').read()))
+    return json.load(open(p))
+
+
+def rref(rows):
+    piv = {}
+    for x in rows:
+        for k in sorted(piv, reverse=True):
+            if x >> k & 1:
+                x ^= piv[k]
+        if x:
+            k = x.bit_length() - 1
+            for kk in list(piv):
+                if piv[kk] >> k & 1:
+                    piv[kk] ^= x
+            piv[k] = x
+    return tuple(piv[k] for k in sorted(piv, reverse=True))
+
+
+def perp(rows, h):
+    """basis of the orthogonal complement (standard dot product mod 2) of span(rows)"""
+    B = rref(rows)
+    piv = [b.bit_length() - 1 for b in B]
+    free = [k for k in range(h) if k not in piv]
+    out = []
+    for k in free:
+        u = 1 << k
+        for b in B:
+            if b >> k & 1:
+                u |= 1 << (b.bit_length() - 1)
+        out.append(u)
+    return rref(out)
+
+
+def enc(x):
+    x = F(x)
+    return [x.numerator, x.denominator]
+
+
+def main():
+    ap = argparse.ArgumentParser(description=__doc__)
+    ap.add_argument('--cache', required=True)
+    ap.add_argument('--flow', required=True)
+    ap.add_argument('--witness', required=True)
+    ap.add_argument('--lift', required=True)
+    ap.add_argument('--out', required=True)
+    ap.add_argument('--name', default='flow word')
+    a = ap.parse_args()
+    t0 = time.time()
+    g = load(a.cache + '/graph.json'); word = load(a.cache + '/selection.json')
+    prof = load(a.flow); wit = load(a.witness); lift = load(a.lift)
+    h, v = g['h'], g['v']
+    ports = g['inputs']
+    assert ports == wit['original_source_ports']
+    nodes, edges, vectors = wit['nodes'], wit['edges'], wit['vectors']
+    maps, order = lift['maps'], lift['topological_order']
+    assert len(maps) == len(nodes) == len(order)
+    roots = g['roots']
+    # ---- frames
+    frames = [(), tuple(1 << i for i in range(h - 1, -1, -1))]
+    fid = {frames[0]: 0, frames[1]: 1}
+
+    def frame(rows):
+        key = rref(rows)
+        if key not in fid:
+            fid[key] = len(frames); frames.append(key)
+        return fid[key]
+    xstart = [frame((q,)) for q in ports]
+    caps = [frame(perp((q,), h)) for q in ports]
+    # ---- signed side rows of the deferred roles (as contract_v4.validate)
+    ops, coeff = word['ops'], word['opcoeff']
+    Rw = max(max(x, y) for x, y, _ in ops) + 1
+    selected = {s['role']: s for s in word['selected']}
+    side = [dict() for _ in range(Rw)]
+    for r, s in zip(roots, word['rootroles']):
+        if r['kind'] != 'center':
+            side[s] = {t: (1 if c == '1/2' else -1) for t, c in zip(r['targets'], r['coefficients'])}
+
+    def add_scaled(dst, src, c):
+        for k, x in src.items():
+            y = dst.get(k, 0) + c * x
+            if y:
+                dst[k] = y
+            else:
+                dst.pop(k, None)
+    for (x, y, _), (ca, cb) in zip(reversed(ops), reversed(coeff)):
+        add_scaled(side[y], side[x], cb)
+        if ca != 1:
+            side[x] = {j: ca * c for j, c in side[x].items()}
+    for s, z in selected.items():
+        assert set(side[s]) == set(z['targets'])
+
+    def responses(rd):
+        if rd['kind'] == 'side':
+            r = roots[rd['ref']]
+            return {t: F(1, 2) if c == '1/2' else F(-1, 2) for t, c in zip(r['targets'], r['coefficients'])}
+        assert rd['kind'] == 'deferred'
+        return {t: F(-c, 2) for t, c in side[rd['ref']].items()}
+    # ---- edge coordinates: (target node, input position) -> (source node, output position)
+    by_frame = {(n['frame'][0], tuple(n['frame'][1])): i for i, n in enumerate(nodes)}
+    out_off, in_off = Counter(), Counter()
+    coord_in = {}
+    for e in edges:
+        i = by_frame[(e['source'][0], tuple(e['source'][1]))]; j = by_frame[(e['target'][0], tuple(e['target'][1]))]
+        for k in range(len(e['basis'])):
+            coord_in[j, in_off[j] + k] = (i, out_off[i] + k)
+        out_off[i] += len(e['basis']); in_off[j] += len(e['basis'])
+    kernel_birth = {}
+    for i, aa, j, bb in wit['kernel_pairs']:
+        kernel_birth[j, bb] = (i, aa)
+    # ---- emission
+    A, B, ret = [], [], []
+    nslot = 0
+    lam = {}
+
+    def new_slot():
+        nonlocal nslot
+        s = 2 * v + nslot; nslot += 1; lam[s] = F(1)
+        return s
+    slot_out, kernel_slot = {}, {}
+    center_lam = {}
+    stats = Counter()
+    phase_of_node = {}
+    for i in order:
+        node, mp = nodes[i], maps[i]
+        ph, U = node['frame']
+        f = frame(tuple(U))
+        gates = A if ph == 1 else B
+        n, t, b, d, r = node['n'], node['t'], node['new_dirty'], node['d'], node['r']
+        N = n + b
+        assert mp['size'] == N and not mp['source_erase'] and not mp['source_inject'], 'source controls not supported yet'
+        pos = [None] * N
+        nin = in_off[i]
+        g0_node = len(gates)
+        for p in range(n):
+            if p < nin:
+                pos[p] = slot_out[coord_in[i, p]]
+            else:                                   # original source copy at its port line
+                row = vectors[node['inputs'][p]]
+                assert len(row) == 1 and row[0][1] == 1 and ph == 1 and f == xstart[row[0][0]]
+                s = new_slot(); pos[p] = s
+                gates.append(['out', f, row[0][0], [[s, 1, 1]], []])
+        for q in range(b):
+            if (i, q) in kernel_birth:
+                s = kernel_slot[kernel_birth[i, q]]; lam[s] = F(1)
+            else:
+                s = new_slot()
+            pos[n + q] = s
+        # reads (in terms of the input positions, before the completion)
+        for rd, rc in zip(node['reads'], mp['read_coefficients']):
+            if rd['kind'] == 'center':
+                assert ph == 1
+                k = roots[rd['ref']]['coordinate']
+                val = rd['value']
+                if val in node['outputs']:
+                    assert len(rc) == 1
+                    center_lam[k] = (i, 'out', node['outputs'].index(val), F(rc[0][1], rc[0][2]))
+                else:
+                    # the total is formed in place in one of its (retiring) input registers
+                    assert t == 0 and not mp['inverse_gates'], ('center total neither leaves nor rests', i)
+                    p0, n0, d0 = rc[0]; s0 = pos[p0]; c0 = F(n0, d0) * lam[s0]
+                    for p, num, den in rc[1:]:
+                        s = pos[p]
+                        gates.append(['out', f, s, [[s0, *enc(F(num, den) * lam[s] / c0)]], []])
+                    center_lam[k] = (i, 'pos', p0, c0)
+                continue
+            assert ph == 2
+            resp = responses(rd)
+            for p, num, den in rc:
+                s = pos[p]
+                c = F(num, den) * lam[s]
+                gates.append(['out', f, s, [[v + tt, *enc(c * w)] for tt, w in sorted(resp.items())], []])
+        # completion gates: the lift stores the row operations taking Q to I; the registers need Q itself,
+        # i.e. the inverse operations in reverse order (exact_complex_flow_lift.inverse_program, 'forward').
+        g0 = len(gates)
+        for kind, x, y, num, den in reversed(mp['inverse_gates']):
+            if kind == 'swap':
+                pos[x], pos[y] = pos[y], pos[x]
+            elif kind == 'scale':
+                assert x == y
+                lam[pos[x]] /= F(num, den)
+            else:
+                assert kind == 'add' and x != y
+                sx, sy = pos[x], pos[y]
+                c = -F(num, den) * lam[sy] / lam[sx]
+                gates.append(['out', f, sy, [[sx, *enc(c)]], []])
+        # every coordinate that passes through the node is moved to its frame (the flow charges the edge):
+        # registers of inputs and births that no gate of this node names ride as extra registers
+        named = set()
+        for gg in gates[g0_node:]:
+            named.add(gg[2]); named.update(x_ for x_, _, _ in gg[3])
+            if gg[0] == 'out':
+                named.update(gg[4])
+        idle = [s_ for s_ in pos if s_ not in named]
+        if idle:
+            outs = [gg for gg in gates[g0_node:] if gg[0] == 'out']
+            if outs:
+                outs[-1][4].extend(idle)
+            else:
+                stats['idle_nodes'] += 1
+                gates.append(['out', f, idle[0], [], idle[1:]])
+        for q in range(t):
+            slot_out[i, q] = pos[q]
+        for aa in range(node['retired'] - (d - r)):
+            kernel_slot[i, aa] = pos[t + (d - r) + aa]
+        # retained totals: the coordinate must survive the completion in one register
+        for k, z in list(center_lam.items()):
+            if isinstance(z, tuple) and z[0] == i:
+                ii, how, p, c = z
+                s = pos[p]
+                ret.append([k, s, f]); center_lam[k] = c * lam[s] if how == 'out' else c
+    R = nslot
+    assert R == prof['new_R'], (R, prof['new_R'])
+    ret.sort()
+    assert [z[0] for z in ret] == list(range(h)), 'one retained total per coordinate expected'
+    # ---- K blocks, direct reads, inverse K blocks
+    for cube in range(v // 8):
+        ids8 = list(range(8 * cube, 8 * cube + 8))
+        for parity in range(2):
+            ids = [j for j in ids8 if j.bit_count() % 2 == parity]
+            S = frame(tuple(ports[j] for j in ids))
+            assert len(frames[S]) == 3
+            x1, rest = ids[0], ids[1:]
+            B.append(['in', S, x1, [[r_, 1, 1] for r_ in rest]])
+            B.append(['out', S, x1, [[r_, -1, 2] for r_ in rest], []])
+            B.append(['in', S, x1, [[r_, 1, 1] for r_ in rest]])
+    for cube in range(v // 8):
+        ids8 = list(range(8 * cube, 8 * cube + 8))
+        for j in ids8:
+            anti = [t for t in ids8 if (ports[t] ^ ports[j]).bit_count() == 6]
+            assert len(anti) == 1
+            t = anti[0]
+            x1 = [jj for jj in ids8 if jj.bit_count() % 2 == j.bit_count() % 2][0]
+            B.append(['out', caps[t], j, [[v + t, -1 if j == x1 else 1, 1]], []])
+    for cube in range(v // 8):
+        ids8 = list(range(8 * cube, 8 * cube + 8))
+        for parity in range(2):
+            ids = [j for j in ids8 if j.bit_count() % 2 == parity]
+            x1, rest = ids[0], ids[1:]
+            B.append(['in', 1, x1, [[r_, -1, 1] for r_ in rest]])
+            B.append(['out', 1, x1, [[r_, 1, 2] for r_ in rest], []])
+            B.append(['in', 1, x1, [[r_, -1, 1] for r_ in rest]])
+    # ---- scatter
+    sin, sout = F(1, 3), F(-1, 6)
+    if all(center_lam[k] == 1 for k in range(h)):
+        scat = dict(inside=[1, 3], outside=[-1, 6])
+    else:
+        scat = dict(table=[[[k, *enc((sin if ports[t] >> k & 1 else sout) * center_lam[k])] for k in range(h)] for t in range(v)])
+    nreg = 2 * v + R
+    start = [0] * nreg; final = [1] * nreg
+    for j in range(v):
+        start[j] = xstart[j]; final[v + j] = caps[j]
+    cst = sum(len(frames[f]) for _, _, f in ret)
+    # ---- blocks (the climb count of gx.check1)
+    dim = [len(f) for f in frames]
+    cur = list(start)
+    H = {'x': Counter(), 'y': Counter(), 's': Counter(), 'c': Counter()}
+    cls = lambda r_: 'x' if r_ < v else 'y' if r_ < 2 * v else 's'
+
+    def climb(r_, f):
+        o = cur[r_]
+        if o != f:
+            assert dim[o] < dim[f], ('step not increasing', r_, o, f)
+            H[cls(r_)][dim[f] - dim[o]] += 1; cur[r_] = f
+    for ph, gates in (('A', A), ('B', B)):
+        if ph == 'B':
+            for k, s, f in ret:
+                assert cur[s] == f, ('retained slot moved before the cut', k, s)
+                H['c'][dim[f]] += 1
+        for gg in gates:
+            regs = [gg[2]] + ([x for x, _, _ in gg[3]] + list(gg[4]) if gg[0] == 'out' else [x for x, _, _ in gg[3]] if gg[0] == 'in' else [])
+            for r_ in regs:
+                climb(r_, gg[1])
+    for r_ in range(nreg):
+        climb(r_, final[r_])
+    blocks = {k: {str(r_): n_ for r_, n_ in sorted(w.items())} for k, w in H.items()}
+    Nn = sum(r_ * n_ for w in H.values() for r_, n_ in w.items())
+    assert Nn == R * h + 2 * v * (h - 1) + cst, (Nn, R * h + 2 * v * (h - 1) + cst)
+    c1 = dict(format='gcert/1', derived_from='%s (gcert_emit.py from complex_frame_flow.py witness + exact_complex_flow_lift.py completions)' % a.name,
+              p=g['p'], h=h, v=v, R=R, cst=cst, N=Nn, ports=ports, frames=[list(f) for f in frames], start=start, final=final,
+              ext=[], ret=ret, scat=scat, A=A, B=B, blocks=blocks)
+    data = json.dumps(c1, separators=(',', ':')).encode()
+    if a.out.endswith('.gz'):
+        open(a.out, 'wb').write(gzip.compress(data))
+    else:
+        open(a.out, 'wb').write(data)
+    print(json.dumps(dict(R=R, N=Nn, cst=cst, idle_nodes=stats['idle_nodes'], gates_A=len(A), gates_B=len(B), frames=len(frames), blocks=blocks,
+                          scatter='table' if 'table' in scat else 'star', seconds=round(time.time() - t0, 1))))
+
+
+if __name__ == '__main__':
+    main()
diff --git a/research/w3-p10b-pointwise-complex/gx/gx.py b/research/w3-p10b-pointwise-complex/gx/gx.py
new file mode 100644
index 0000000000..6d14c0a976
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/gx/gx.py
@@ -0,0 +1,296 @@
+#!/usr/bin/python3 -I
+"""gx.py.  OWN code (imports nothing of port-producer, nothing outside).
+(1) convert(c0): gcert/0 (port-producer's event list) -> gcert/1, the LEAN-FACING NORMAL FORM:
+    only the main phase (no read0 / undo / uninj: the invocation theorem derives the dirt gate and the undo
+    gate from the product), every gate an elementary fan-out or fan-in add, the 4x4 K block written as three
+    such adds plus a sign carried by one register, the star scatter as a table.
+(2) check1(c1): the local decidable conditions of EXTENDED-CERT-SPEC.md on a gcert/1 file:
+    E0 sizes, E1 labels (nested subspaces by containment, equal frame at a gate), E2 scatter cut,
+    E3 shape (kinds of gates; X1; S1), E4 bracket rule S2, E5 scalar (x-content replay: hid and X3),
+    E6 price data (block histograms, N), E7 exterior gauges only (Q2).  Raises AssertionError("<rule>: ...").
+"""
+from fractions import Fraction as Fr
+from collections import Counter
+from math import lcm
+
+
+def need(c, msg):
+    if not c:
+        raise AssertionError(msg)
+
+
+def reduce(u, basis):
+    """basis: reduced echelon, pivot = leading bit, descending.  Returns u modulo the span."""
+    for b in basis:
+        if u >> (b.bit_length() - 1) & 1:
+            u ^= b
+    return u
+
+
+def echelon_ok(f):
+    piv = [b.bit_length() - 1 for b in f]
+    if any(b == 0 for b in f) or piv != sorted(piv, reverse=True) or len(set(piv)) != len(piv):
+        return False
+    return all(not (b >> p & 1) for i, b in enumerate(f) for j, p in enumerate(piv) if i != j)
+
+
+def inside(a, b):
+    return all(reduce(u, b) == 0 for u in a)
+
+
+def par(x):
+    return bin(x).count("1") & 1
+
+
+KJ = [[[1, 2] if i == j else [-1, 2] for j in range(4)] for i in range(4)]
+
+
+def convert(c0):
+    need(c0["format"] == "gcert/0", "input is not gcert/0")
+    v, h, R = c0["v"], c0["h"], c0["R"]
+    ev = c0["events"]
+    cen = [n for n, e in enumerate(ev) if e[0] == "centre"]
+    need(cen == list(range(cen[0], cen[0] + len(cen))), "centres are not contiguous: no single scatter cut")
+    A, B, ret = [], [], []
+    neg = set()            # x registers currently holding MINUS their value (between the two K blocks)
+    openq = {}
+    for n, e in enumerate(ev):
+        k = e[0]
+        out = A if n < cen[0] else B
+        if k in ("read0", "undo", "uninj"):
+            continue
+        if k == "inj":
+            need(e[2] not in neg, "inj reads a negated x register")
+            out.append(["out", e[3], e[2], [[e[1], 1, 1]], []])
+        elif k == "op":
+            if e[3] == -1:
+                out.append(["neg", e[5], e[1]])
+            else:
+                need(e[3] == 1, "op ca")
+            out.append(["out", e[5], e[2], [[e[1], e[4], 1]], []])
+        elif k == "centre":
+            ret.append([e[2], e[1], e[3]])
+        elif k == "gread":
+            hit = set(t for t, _, _ in e[3])
+            out.append(["out", e[2], e[1], [[v + t, a, b] for t, a, b in e[3]], [v + t for t in e[4] if t not in hit]])
+        elif k == "rread":
+            out.append(["out", e[2], e[1], [[v + t, a, b] for t, a, b in e[3]], []])
+        elif k == "ysh":
+            out.append(["out", e[4], e[2], [[e[1], e[3], 1]], []])
+        elif k == "yw":
+            out.append(["out", e[4], e[2], [[e[1], e[3][0], e[3][1]]], []])
+        elif k == "kd":
+            out.append(["out", e[3], e[2], [[e[1], -1 if e[2] in neg else 1, 1]], []])
+        elif k == "ktr":
+            q = tuple(e[1])
+            need(e[2] == KJ, "ktr matrix is not 1 - J/2")
+            x1, rest = q[0], list(q[1:])
+            if q not in openq:          # first block: x := M x, register x1 keeps the sign
+                out.append(["in", e[3], x1, [[r, 1, 1] for r in rest]])
+                out.append(["out", e[3], x1, [[r, -1, 2] for r in rest], []])
+                out.append(["in", e[3], x1, [[r, 1, 1] for r in rest]])
+                openq[q] = n
+                neg.add(x1)
+            else:                       # second block: the three adds undone in reverse order
+                out.append(["in", e[3], x1, [[r, -1, 1] for r in rest]])
+                out.append(["out", e[3], x1, [[r, 1, 2] for r in rest], []])
+                out.append(["in", e[3], x1, [[r, -1, 1] for r in rest]])
+                del openq[q]
+                neg.discard(x1)
+        else:
+            raise AssertionError("unknown event kind %r" % k)
+    need(not openq and not neg, "a K block is not closed")
+    ext = sorted([z[3], z[1]] for z in c0["gauge"] if z[5])
+    c1 = dict(format="gcert/1", derived_from=c0["name"] + " (gcert/0)", p=c0["p"], h=h, v=v, R=R, cst=c0["cst"],
+              N=c0["N"], ports=c0["ports"], frames=c0["frames"], start=c0["start"], final=c0["final"],
+              ext=ext, ret=sorted(ret), scat=dict(inside=c0["scat"]["inside"], outside=c0["scat"]["outside"]),
+              A=A, B=B, blocks=c0["blocks"])
+    return c1
+
+
+def scat_row(c, t):
+    """the scatter of target t: [(k, coefficient)], from the table or from the star rule"""
+    sc = c["scat"]
+    if "table" in sc:
+        return [(k, Fr(a, b)) for k, a, b in sc["table"][t]]
+    sin, sout = Fr(*sc["inside"]), Fr(*sc["outside"])
+    return [(k, sin if c["ports"][t] >> k & 1 else sout) for k in range(c["h"])]
+
+
+def gate_regs(g):
+    if g[0] == "neg":
+        return [g[2]], [], []
+    if g[0] == "out":
+        return [g[2]] + [t for t, _, _ in g[3]] + list(g[4]), [(t, g[2], Fr(a, b)) for t, a, b in g[3]], g[4]
+    if g[0] == "in":
+        return [g[2]] + [s for s, _, _ in g[3]], [(g[2], s, Fr(a, b)) for s, a, b in g[3]], []
+    raise AssertionError("E0: unknown gate kind %r" % (g[0],))
+
+
+def check1(c, scalar=True, stats=None):
+    need(c.get("format") == "gcert/1", "E0: format")
+    h, v, R = c["h"], c["v"], c["R"]
+    nreg = 2 * v + R
+    ports, fr = c["ports"], [tuple(f) for f in c["frames"]]
+    need(len(ports) == v == len(set(ports)) and all(par(q) and 0 < q < 1 << h for q in ports), "E0: ports")
+    need(fr[0] == () and fr[1] == tuple(1 << i for i in range(h - 1, -1, -1)), "E0: frames 0 and 1")
+    need(all(echelon_ok(f) for f in fr) and len(set(fr)) == len(fr), "E0: frame table not reduced echelon / repeats")
+    dim = [len(f) for f in fr]
+    start, final = c["start"], c["final"]
+    need(len(start) == len(final) == nreg, "E0: start / final")
+    ext = {r: f for r, f in c["ext"]}
+    for t in range(v):
+        need(fr[start[t]] == (ports[t],) and final[t] == 1, "E1: x start / final")
+        f = fr[final[v + t]]
+        need(start[v + t] == 0 and len(f) == h - 1 and all(par(u & ports[t]) == 0 for u in f), "E1: y start / final")
+    for r in range(2 * v, nreg):
+        need(start[r] == ext.get(r, 0) and final[r] == 1, "E1: slot start / final")
+    cls = lambda r: "x" if r < v else "y" if r < 2 * v else "s"
+    cur = list(start)
+    H = {"x": Counter(), "y": Counter(), "s": Counter(), "c": Counter()}
+    steps = set()
+
+    def climb(r, f, where):
+        o = cur[r]
+        if o != f:
+            if (o, f) not in steps:
+                need(dim[o] < dim[f] and inside(fr[o], fr[f]), "E1: step not nested (%s, register %d)" % (where, r))
+                steps.add((o, f))
+            H[cls(r)][dim[f] - dim[o]] += 1
+            cur[r] = f
+    okind = {("x", "s"), ("s", "s"), ("s", "y"), ("x", "y"), ("x", "x"), ("y", "y")}
+    ysum, piv, ytg = Counter(), set(), set()
+    inc = 0
+    for ph, gates in (("A", c["A"]), ("B", c["B"])):
+        if ph == "B":      # ---- E2: the scatter cut
+            need(sorted(k for k, _, _ in c["ret"]) == list(range(len(c["ret"]))), "E2: retained totals numbered 0..")
+            if "table" in c["scat"]:
+                need(len(c["scat"]["table"]) == v and all(0 <= k < len(c["ret"]) for row in c["scat"]["table"] for k, _, _ in row),
+                     "E2: scatter table")
+            else:
+                need(len(c["ret"]) == h, "E2: one retained total per coordinate (star rule)")
+            for k, s, f in c["ret"]:
+                need(cls(s) == "s" and cur[s] == f, "E2: retained slot %d is not at its frame" % s)
+                H["c"][dim[f]] += 1
+            need(all(cur[v + t] == 0 for t in range(v)), "E2: a y role left frame 0 before the scatter")
+        for i, g in enumerate(gates):
+            regs, pairs, _ = gate_regs(g)
+            need(all(0 <= r < nreg for r in regs) and len(set(regs)) == len(regs), "E3: registers of gate %s%d" % (ph, i))
+            inc += len(regs)
+            for r in regs:
+                climb(r, g[1], "%s%d" % (ph, i))
+            for t, s, co in pairs:
+                kd = (cls(s), cls(t))
+                need(kd in okind, "E3: gate kind %s -> %s (gate %s%d)" % (kd + (ph, i)))
+                need(ph == "B" or kd in {("x", "s"), ("s", "s")}, "E3: bank gate before the scatter (gate A%d)" % i)
+                if kd == ("y", "y"):
+                    ysum[s, t] += co
+                    piv.add(s)
+                    ytg.add(t)
+            need(ph == "B" or all(cls(r) != "y" for r in regs), "E2: y role named in phase A")
+    for r in range(nreg):
+        climb(r, final[r], "final")
+    need(not (piv & ytg), "E4: a pivot is the target of a gate reading y")
+    need(all(x == 0 for x in ysum.values()), "E4: y-reads of a (pivot, target) pair do not cancel")
+    Hs = {k: {str(r): n for r, n in sorted(w.items())} for k, w in H.items()}
+    need(Hs == c["blocks"], "E6: block histograms")
+    N = sum(r * n for w in H.values() for r, n in w.items())
+    tails = sum(dim[f] for f in ext.values())
+    need(N == c["N"] and N + tails == R * h + 2 * v * (h - 1) + c["cst"], "E6: N")
+    if stats is not None:
+        stats.update(steps=len(steps), incidences=inc, blocks=sum(sum(w.values()) for w in H.values()),
+                     step_work=sum(dim[a] * dim[b] for a, b in steps), ysh_pairs=len(ysum))
+    if scalar:
+        scal(c, stats)
+        q2(c)
+    return True
+
+
+def scal(c, stats=None):
+    """E5: exact replay of the x-content of every register through A, scatter, B."""
+    h, v, R = c["h"], c["v"], c["R"]
+    nreg = 2 * v + R
+    reg = [({r: Fr(1)} if r < v else {}) for r in range(nreg)]
+    den = {"x": 1, "y": 1, "s": 1}
+    mx = {"x": Fr(0), "y": Fr(0), "s": Fr(0)}
+    cls = lambda r: "x" if r < v else "y" if r < 2 * v else "s"
+
+    def axpy(t, src, co):
+        d = reg[t]
+        for k, u in src.items():
+            w = d.get(k, 0) + co * u
+            if w:
+                d[k] = w
+                kk = cls(t)
+                if w.denominator != 1:
+                    den[kk] = lcm(den[kk], w.denominator)
+                if abs(w) > mx[kk]:
+                    mx[kk] = abs(w)
+            else:
+                d.pop(k, None)
+
+    def run(gates):
+        for g in gates:
+            if g[0] == "neg":
+                reg[g[2]] = {k: -u for k, u in reg[g[2]].items()}
+            elif g[0] == "out":
+                src = dict(reg[g[2]])
+                for t, a, b in g[3]:
+                    axpy(t, src, Fr(a, b))
+            else:
+                for s, a, b in g[3]:
+                    axpy(g[2], reg[s], Fr(a, b))
+    run(c["A"])
+    tot = {k: dict(reg[s]) for k, s, _ in c["ret"]}
+    for t in range(v):
+        for k, co in scat_row(c, t):
+            axpy(v + t, tot[k], co)
+    run(c["B"])
+    need(all(reg[t] == {t: 1} for t in range(v)), "E5: an x role is not restored (X3)")
+    need(all(reg[v + t] == {t: 1} for t in range(v)), "E5: a target does not receive exactly its source (hid)")
+    if stats is not None:
+        stats.update(den=dict(den), maxabs={k: str(x) for k, x in mx.items()},
+                     gates_A=len(c["A"]), gates_B=len(c["B"]))
+
+
+def q2(c):
+    """E7 (only for files with exterior gauges, rule (2)): for every slot q that starts off the zero frame, the
+    column of q in the y rows of the product is zero (its dirt is removed in line, the derived dirt gate at
+    frame 0 does not touch it).  Replay with the gauged slots as variables, x = 0."""
+    h, v, R = c["h"], c["v"], c["R"]
+    ext = [r for r, _ in c["ext"]]
+    if not ext:
+        return True
+    reg = [dict() for _ in range(2 * v + R)]
+    for r in ext:
+        reg[r] = {r: Fr(1)}
+
+    def axpy(t, src, co):
+        d = reg[t]
+        for k, u in src.items():
+            w = d.get(k, 0) + co * u
+            if w:
+                d[k] = w
+            else:
+                d.pop(k, None)
+
+    def run(gates):
+        for g in gates:
+            if g[0] == "neg":
+                reg[g[2]] = {k: -u for k, u in reg[g[2]].items()}
+            elif g[0] == "out":
+                src = dict(reg[g[2]])
+                for t, a, b in g[3]:
+                    axpy(t, src, Fr(a, b))
+            else:
+                for s, a, b in g[3]:
+                    axpy(g[2], reg[s], Fr(a, b))
+    run(c["A"])
+    tot = {k: dict(reg[s]) for k, s, _ in c["ret"]}
+    for t in range(v):
+        for k, co in scat_row(c, t):
+            if tot[k]:
+                axpy(v + t, tot[k], co)
+    run(c["B"])
+    need(all(not reg[v + t] for t in range(v)), "E7: the dirt of an exterior-gauged slot reaches a y role (Q2)")
+    return True
diff --git a/research/w3-p10b-pointwise-complex/gx/gxcore.py b/research/w3-p10b-pointwise-complex/gx/gxcore.py
new file mode 100644
index 0000000000..3e3f0a3076
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/gx/gxcore.py
@@ -0,0 +1,375 @@
+"""gxcore.py: gcert/1 JSON -> normal data D, PYTHON MIRROR of the two Lean checks
+(label side: E0 frames, E1 pairs, E1/E2/E6 replay in segments; scalar side: E3 kinds, E4 bracket, E5 replay
+in source batches, E7), derived data (distinct climbs, cut states, block histogram) and kernel sizes.
+Own code; the reference is gx.py (check1), not imported.  Python 3, standard library only."""
+import json, gzip
+from collections import Counter
+from math import gcd
+
+
+class Reject(Exception):
+    pass
+
+
+def need(c, msg):
+    if not c:
+        raise Reject(msg)
+
+
+def load(path):
+    c = json.load(gzip.open(path, "rt") if path.endswith(".gz") else open(path))
+    need(c.get("format") == "gcert/1", "E0: format is not gcert/1")
+    return c
+
+
+def co(num, den):
+    need(den > 0, "E0: denominator")
+    return (num < 0, abs(num), den)
+
+
+def normal(c):
+    """the Lean-facing data: gates as tuples, coefficients (neg, num, den), ret by total"""
+    def gate(g):
+        if g[0] == "out":
+            return ("out", g[1], g[2], [(t, co(a, b)) for t, a, b in g[3]], list(g[4]))
+        need(g[0] == "in", "E0: unknown gate kind %r" % (g[0],))
+        return ("in", g[1], g[2], [(s, co(a, b)) for s, a, b in g[3]])
+    ret = sorted(c["ret"])
+    need([k for k, _, _ in ret] == list(range(len(ret))), "E2: retained totals numbered 0..")
+    sc = c["scat"]
+    scat = ("table", [[(k, co(a, b)) for k, a, b in row] for row in sc["table"]]) if "table" in sc else \
+        ("star", co(*sc["inside"]), co(*sc["outside"]))
+    return dict(h=c["h"], v=c["v"], R=c["R"], N=c["N"], cst=c["cst"], ports=list(c["ports"]),
+                frames=[list(f) for f in c["frames"]], start=list(c["start"]), final=list(c["final"]),
+                ext=[tuple(e) for e in c.get("ext", [])], ret=[(s, f) for _, s, f in ret], scat=scat,
+                A=[gate(g) for g in c["A"]], B=[gate(g) for g in c["B"]],
+                blocks={k: {int(r): n for r, n in c["blocks"][k].items()} for k in "xysc"})
+
+
+def regs(g):
+    return [g[2]] + [t for t, _ in g[3]] + (g[4] if g[0] == "out" else [])
+
+
+def adds(g):
+    """single adds (tgt, src, coef) in order"""
+    return [(t, g[2], k) for t, k in g[3]] if g[0] == "out" else [(g[2], s, k) for s, k in g[3]]
+
+
+def reduce(u, basis):
+    for b in basis:
+        if u >> (b.bit_length() - 1) & 1:
+            u ^= b
+    return u
+
+
+def echelon_ok(f):
+    piv = [b.bit_length() - 1 for b in f]
+    if any(b == 0 for b in f) or any(piv[i] <= piv[i + 1] for i in range(len(piv) - 1)):
+        return False
+    return all(not (b >> p & 1) for i, b in enumerate(f) for j, p in enumerate(piv) if i != j)
+
+
+def par(x):
+    return bin(x).count("1") & 1
+
+
+def cls(D, r):
+    return "x" if r < D["v"] else "y" if r < 2 * D["v"] else "s"
+
+
+# ---- label side ------------------------------------------------------------------------------------------
+def chk_frames(D):
+    """E0: ports; frame table reduced echelon, no repeats, zero first, full second"""
+    h, v, fr = D["h"], D["v"], D["frames"]
+    p = D["ports"]
+    need(len(p) == v == len(set(p)) and all(par(q) and 0 < q < 1 << h for q in p), "E0: ports")
+    need(len(fr) >= 2 and fr[0] == [] and fr[1] == [1 << i for i in range(h - 1, -1, -1)], "E0: frames 0 and 1")
+    need(all(echelon_ok(f) and all(b < 1 << h for b in f) for f in fr), "E0: frame table not reduced echelon")
+    need(len(set(map(tuple, fr))) == len(fr), "E0: frame table has repeats")
+    return [len(f) for f in fr]
+
+
+def derive(D):
+    """distinct climbs in order of first use (NOT a check: plain replay of the frame ids)"""
+    cur, seen, pairs = list(D["start"]), set(), []
+
+    def go(r, f):
+        o = cur[r]
+        if o != f:
+            if (o, f) not in seen:
+                seen.add((o, f))
+                pairs.append((o, f))
+            cur[r] = f
+    for g in D["A"] + D["B"]:
+        for r in regs(g):
+            go(r, g[1])
+    for r, f in enumerate(D["final"]):
+        go(r, f)
+    return pairs
+
+
+def chk_pairs(D, dim, pairs):
+    """E1, level 2: every listed pair is a nested pair of the table; returns the mask-operation count"""
+    fr, n, work = D["frames"], len(D["frames"]), 0
+    for a, b in pairs:
+        need(a < n and b < n and dim[a] < dim[b], "E1: pair (%d, %d) does not climb" % (a, b))
+        need(all(reduce(u, fr[b]) == 0 for u in fr[a]), "E1: pair (%d, %d) is not nested" % (a, b))
+        work += dim[a] * dim[b]
+    return work
+
+
+def cut_points(D, gmax, wmax=210000):
+    """replay segments: each phase (A, B) is cut into k pieces of equal gate count, k = the least number with at
+    most `gmax` gates AND at most `wmax` work per piece on average (work = sum over gates of the dimension of the
+    gate's frame; measured at h = 22: about 1 ms of kernel time per unit, NOTES.md).
+    Cut points = gate counts in A ++ B (always 0, |A| and the end)."""
+    nA, nB = len(D["A"]), len(D["B"])
+    cuts = [0]
+    for off, gs in ((0, D["A"]), (nA, D["B"])):
+        n = len(gs)
+        k = max(1, -(-n // gmax), -(-sum(len(D["frames"][g[1]]) for g in gs) // wmax))
+        cuts += [off + (n * i) // k for i in range(1, k + 1)]
+    return sorted(set(cuts))
+
+
+def label_data(D, dim, cuts):
+    """for gx-labels (gcert-interface.md LABELS v1): ordered block ranks per replay segment, the final climbs as
+    gates without adds (same final frame, at most 32 registers, only registers that still climb), their ranks,
+    the largest number of registers named by one gate.  Plain replay (the mirror has accepted the file)."""
+    cur, G = list(D["start"]), D["A"] + D["B"]
+    rs, most = [], 0
+    for a, b in zip(cuts, cuts[1:]):
+        r = []
+        for g in G[a:b]:
+            most = max(most, len(regs(g)))
+            for x in regs(g):
+                if cur[x] != g[1]:
+                    r.append(dim[g[1]] - dim[cur[x]])
+                    cur[x] = g[1]
+        rs.append(r)
+    by = {}
+    for x, f in enumerate(D["final"]):
+        if cur[x] != f:
+            by.setdefault(f, []).append(x)
+    fins, rF = [], []
+    for f in sorted(by):
+        for i in range(0, len(by[f]), 32):
+            xs = by[f][i:i + 32]
+            fins.append(("out", f, xs[0], [], xs[1:]))
+            rF += [dim[f] - dim[cur[x]] for x in xs]
+    return dict(rs=rs, fins=fins, rF=rF, most=max(most, 32 if fins else 0))
+
+
+def chk_replay(D, dim, pairs, cuts):
+    """E1 (labels), E2 (scatter cut), E6 (price), in the segments given by `cuts`.
+    Returns (states at the cuts, histogram by class, visits per segment)."""
+    h, v, R, fr = D["h"], D["v"], D["R"], D["frames"]
+    n = 2 * v + R
+    start, final, ext = D["start"], D["final"], dict(D["ext"])
+    need(len(start) == len(final) == n, "E0: start / final")
+    for t in range(v):
+        need(fr[start[t]] == [D["ports"][t]] and final[t] == 1, "E1: x start / final")
+        f = fr[final[v + t]]
+        need(start[v + t] == 0 and len(f) == h - 1 and all(par(u & D["ports"][t]) == 0 for u in f), "E1: y start / final")
+    for r in range(2 * v, n):
+        need(start[r] == ext.get(r, 0) and final[r] == 1, "E1: slot start / final")
+    ok = set(pairs)
+    need(len(ok) == len(pairs), "E1: pair list has repeats")
+    cur = list(start)
+    H = {k: Counter() for k in "xysc"}
+
+    def climb(r, f, where):
+        o = cur[r]
+        if o != f:
+            need((o, f) in ok, "E1: climb (%d, %d) is not a tested pair (%s, register %d)" % (o, f, where, r))
+            H[cls(D, r)][dim[f] - dim[o]] += 1
+            seg[dim[f] - dim[o]] += 1
+            cur[r] = f
+    seg, segs = Counter(), []
+    G, nA = D["A"] + D["B"], len(D["A"])
+    states, visits = [], []
+    need(cuts[0] == 0 and cuts[-1] == len(G) and cuts == sorted(cuts), "cut points")
+    for a, b in zip(cuts, cuts[1:]):
+        states.append(list(cur))
+        vs = 0
+        for i in range(a, b):
+            if i == nA:
+                scatter_cut(D, cur, dim, H, seg)
+            g = G[i]
+            rs = regs(g)
+            need(all(0 <= r < n for r in rs) and len(set(rs)) == len(rs), "E3: registers of gate %d" % i)
+            need(i >= nA or all(r < v or r >= 2 * v for r in rs), "E2: y role named in phase A (gate %d)" % i)
+            vs += len(rs)
+            for r in rs:
+                climb(r, g[1], "gate %d" % i)
+        visits.append(vs)
+        if b < len(G):
+            segs.append(seg)
+            seg = Counter()
+    if nA == len(G):
+        scatter_cut(D, cur, dim, H, seg)
+    states.append(list(cur))
+    for r in range(n):
+        climb(r, final[r], "final")
+    segs.append(seg)
+    Hb = {k: dict(sorted(w.items())) for k, w in H.items()}
+    need(Hb == D["blocks"], "E6: block histogram differs from `blocks`")
+    N = sum(r * c for w in H.values() for r, c in w.items())
+    need(N == D["N"] and N + sum(dim[f] for f in ext.values()) == R * h + 2 * v * (h - 1) + D["cst"], "E6: N")
+    need(sum(r * c for r, c in H["c"].items()) == D["cst"], "E6: cst")
+    return states, Hb, visits, [[w[r] for r in range(h + 1)] for w in segs]
+
+
+def scatter_cut(D, cur, dim, H, seg):
+    v = D["v"]
+    if D["scat"][0] == "table":
+        need(len(D["scat"][1]) == v and all(k < len(D["ret"]) for row in D["scat"][1] for k, _ in row), "E2: scatter table")
+    else:
+        need(len(D["ret"]) == D["h"], "E2: one retained total per coordinate (star rule)")
+    for s, f in D["ret"]:
+        need(s >= 2 * v and cur[s] == f, "E2: retained slot %d is not at its frame" % s)
+        H["c"][dim[f]] += 1
+        seg[dim[f]] += 1
+    need(all(cur[v + t] == 0 for t in range(v)), "E2: a y role left frame 0 before the scatter")
+
+
+# ---- scalar side -----------------------------------------------------------------------------------------
+SC = 6            # all scalar entries are kept as integers in units of 1/6
+KINDS = {("x", "s"), ("s", "s"), ("s", "y"), ("x", "y"), ("x", "x"), ("y", "y")}
+
+
+def chk_shape(D):
+    """E3 kinds, E4 bracket"""
+    ysum, piv, ytg = Counter(), set(), set()
+    for ph in "AB":
+        for i, g in enumerate(D[ph]):
+            for t, s, (ng, a, b) in adds(g):
+                kd = (cls(D, s), cls(D, t))
+                need(kd in KINDS and t != s, "E3: gate kind %s -> %s (gate %s%d)" % (kd + (ph, i)))
+                need(ph == "B" or kd in {("x", "s"), ("s", "s")}, "E3: bank gate before the scatter (gate A%d)" % i)
+                if kd == ("y", "y"):
+                    need(SC % b == 0, "E4: denominator beyond sixths")
+                    ysum[s, t] += (-a if ng else a) * (SC // b)
+                    piv.add(s)
+                    ytg.add(t)
+    need(not (piv & ytg), "E4: a pivot is the target of a gate reading y")
+    need(all(x == 0 for x in ysum.values()), "E4: y-reads of a (pivot, target) pair do not cancel")
+    return len(ysum)
+
+
+def scat_row(D, t):
+    sc = D["scat"]
+    if sc[0] == "table":
+        return sc[1][t]
+    return [(k, sc[1] if D["ports"][t] >> k & 1 else sc[2]) for k in range(D["h"])]
+
+
+
+def replay(D, init, count=None):
+    """exact replay through A, scatter, B of sparse vectors with entries in sixths; `init` = register -> {key: 6}"""
+    v = D["v"]
+    reg = [dict(init.get(r, ())) for r in range(2 * v + D["R"])]
+
+    def axpy(t, src, k):
+        ng, a, b = k
+        d = reg[t]
+        for key, u in src.items():
+            w = u * a
+            need(w % b == 0, "E5: an entry needs a denominator beyond sixths (register %d)" % t)
+            w = w // b
+            w = d.get(key, 0) + (-w if ng else w)
+            if count is not None:
+                count[key] += 1
+            if w:
+                d[key] = w
+            else:
+                d.pop(key, None)
+
+    def run(gates):
+        for g in gates:
+            if g[0] == "out":
+                src = dict(reg[g[2]])
+                for t, k in g[3]:
+                    axpy(t, src, k)
+            else:
+                for s, k in g[3]:
+                    axpy(g[2], reg[s], k)
+    run(D["A"])
+    afterA = sum(len(d) for d in reg)
+    tot = [dict(reg[s]) for s, _ in D["ret"]]
+    for t in range(v):
+        for k, cf in scat_row(D, t):
+            if tot[k]:
+                axpy(v + t, tot[k], cf)
+    run(D["B"])
+    return reg, afterA
+
+
+def chk_scalar(D, batches=1):
+    """E5 (+ E7 when `ext` is non-empty).  Returns stats: denominators and largest numerators per class in the
+    units of the spec (x halves, y sixths, slots integers), updates per source batch, entries after A."""
+    v = D["v"]
+    count = Counter()
+    reg, afterA = replay(D, {t: {t: SC} for t in range(v)}, count)
+    need(all(reg[t] == {t: SC} for t in range(v)), "E5: an x role is not restored (X3)")
+    need(all(reg[v + t] == {t: SC} for t in range(v)), "E5: a target does not receive exactly its source (hid)")
+    if D["ext"]:
+        rg, _ = replay(D, {r: {r: SC} for r, _ in D["ext"]})
+        need(all(not rg[v + t] for t in range(v)), "E7: the dirt of an exterior-gauged slot reaches a y role")
+    step = -(-v // batches)
+    per = [sum(count[k] for k in range(lo, min(lo + step, v))) for lo in range(0, v, step)]
+    return dict(updates=sum(count.values()), per_batch=per, afterA=afterA,
+                batches=[(lo, min(lo + step, v)) for lo in range(0, v, step)])
+
+
+def digits(D):
+    """largest |entry| met and denominators per class, by a second replay that watches every write (slow path
+    kept separate from the check): returns {class: (lcm of denominators, largest |numerator| in units 1/den)}"""
+    v = D["v"]
+    den = {"x": 1, "y": 1, "s": 1}
+    mx = {"x": 0, "y": 0, "s": 0}
+    reg = [({r: SC} if r < v else {}) for r in range(2 * v + D["R"])]
+
+    def axpy(t, src, k):
+        ng, a, b = k
+        d, kk = reg[t], cls(D, t)
+        for key, u in src.items():
+            w = d.get(key, 0) + (-(u * a // b) if ng else u * a // b)
+            if w:
+                d[key] = w
+                q = SC // gcd(w, SC)
+                if den[kk] % q:
+                    den[kk] = den[kk] * q // gcd(den[kk], q)
+                if abs(w) > mx[kk]:
+                    mx[kk] = abs(w)
+            else:
+                d.pop(key, None)
+    for g in D["A"]:
+        for t, s, k in adds(g):
+            axpy(t, reg[s], k)
+    tot = [dict(reg[s]) for s, _ in D["ret"]]
+    for t in range(v):
+        for k, cf in scat_row(D, t):
+            axpy(v + t, tot[k], cf)
+    for g in D["B"]:
+        for t, s, k in adds(g):
+            axpy(t, reg[s], k)
+    return {k: (den[k], mx[k] * den[k] // SC if (mx[k] * den[k]) % SC == 0 else "%d/6" % mx[k]) for k in "xys"}
+
+
+def mirror(D, gmax=12000, batches=1, pairs=None, cuts=None):
+    """both checks; returns everything the generator and the dry run need"""
+    dim = chk_frames(D)
+    pairs = derive(D) if pairs is None else pairs
+    work = chk_pairs(D, dim, pairs)
+    cuts = cut_points(D, gmax) if cuts is None else cuts
+    states, H, visits, segH = chk_replay(D, dim, pairs, cuts)
+    ysh = chk_shape(D)
+    S = chk_scalar(D, batches)
+    hist = Counter()
+    for k in "xysc":
+        for r, n in H[k].items():
+            hist[r] += n
+    return dict(dim=dim, pairs=pairs, pair_work=work, cuts=cuts, states=states, H=H, visits=visits, ysh=ysh,
+                scal=S, hist=dict(sorted(hist.items())), segH=segH,
+                lab=label_data(D, dim, cuts))
diff --git a/research/w3-p10b-pointwise-complex/layers/c2/kernel-pairs-mw.json b/research/w3-p10b-pointwise-complex/layers/c2/kernel-pairs-mw.json
new file mode 100644
index 0000000000..2572ce148f
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/layers/c2/kernel-pairs-mw.json
@@ -0,0 +1 @@
+{"kernel_pairs":[]}
diff --git a/research/w3-p10b-pointwise-complex/layers/c2/matching-arcs.json b/research/w3-p10b-pointwise-complex/layers/c2/matching-arcs.json
new file mode 100644
index 0000000000..b2a49894d2
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/layers/c2/matching-arcs.json
@@ -0,0 +1 @@
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diff --git a/research/w3-p10b-pointwise-complex/layers/c2/physical-frames.json b/research/w3-p10b-pointwise-complex/layers/c2/physical-frames.json
new file mode 100644
index 0000000000..c484eaf570
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/layers/c2/physical-frames.json
@@ -0,0 +1 @@
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diff --git a/research/w3-p10b-pointwise-complex/layers/c2/physical-pairs-mw.json b/research/w3-p10b-pointwise-complex/layers/c2/physical-pairs-mw.json
new file mode 100644
index 0000000000..4c5fb2ec72
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/layers/c2/physical-pairs-mw.json
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diff --git a/research/w3-p10b-pointwise-complex/layers/uniform/kernel-pairs-mw.json b/research/w3-p10b-pointwise-complex/layers/uniform/kernel-pairs-mw.json
new file mode 100644
index 0000000000..2572ce148f
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/layers/uniform/kernel-pairs-mw.json
@@ -0,0 +1 @@
+{"kernel_pairs":[]}
diff --git a/research/w3-p10b-pointwise-complex/layers/uniform/matching-arcs.json b/research/w3-p10b-pointwise-complex/layers/uniform/matching-arcs.json
new file mode 100644
index 0000000000..6a4b016d06
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/layers/uniform/matching-arcs.json
@@ -0,0 +1 @@
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diff --git a/research/w3-p10b-pointwise-complex/layers/uniform/physical-frames.json b/research/w3-p10b-pointwise-complex/layers/uniform/physical-frames.json
new file mode 100644
index 0000000000..9ee6e137b3
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/layers/uniform/physical-frames.json
@@ -0,0 +1 @@
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diff --git a/research/w3-p10b-pointwise-complex/layers/uniform/physical-pairs-mw.json b/research/w3-p10b-pointwise-complex/layers/uniform/physical-pairs-mw.json
new file mode 100644
index 0000000000..fc5a59c47e
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/layers/uniform/physical-pairs-mw.json
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diff --git a/research/w3-p10b-pointwise-complex/lean/companion-7932-evidence.txt b/research/w3-p10b-pointwise-complex/lean/companion-7932-evidence.txt
new file mode 100644
index 0000000000..4bc00e22bc
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/lean/companion-7932-evidence.txt
@@ -0,0 +1,102 @@
+Lean evidence for the COMPANION program (spec/uniform.json, 7932/10^7; tag CU32, bank B2Gcu32)
+==============================================================================================
+
+This is NOT the program this package prices (certificates/gcert1-p10-b325-c2.json.gz, 7940/10^7), whose Lean run is
+pending. The companion is the same recipe with one shared PR #325 pair module; its certificate is not in this
+package (sha256 below; verify.py regenerates its word only as the builder anchor (f)). It is the Lean-checked counterpart of PR #346's supplier (same coarse saving b).
+
+What this file is. These are the console outputs of the Lean run, copied from the session log of the run
+(2026-10-11, host "spark", Ubuntu; a checkout of jacobalansussman/wht-power-saving-lean at commit f010392 with the
+Lean toolchain and Mathlib that commit pins, the same commit as PR #327's Lean runs). The generated Lean modules and the full build log stayed
+on that host; they are not in this package, and nothing here is re-run by verify.py. The run used Sussman's
+generator unchanged. Lines are verbatim; "..." marks omitted lines and [ ] marks notes added here.
+
+Program built: the committed certificate, gz sha256
+aff3f5d1445bacfbb4f0b22fabe4c18fc18ef3e9a0e15f3122600832e7577509, canonical uncompressed sha256
+56e1139aa4a420219d37b9af00d8894d4c21df83b3d9c266d36a787676e9ad4e (the program was emitted under the name
+p10-b325-pmqm).
+
+
+1. Generate the Lean modules (Sussman's tools/gx/regen_check.py, unchanged)
+---------------------------------------------------------------------------
+
+$ cd ~/wht-lean && GX_OUT=$HOME/cxup/lean/gen python3 -B tools/gx/regen_check.py \
+      $HOME/cxup/lean/gcert1-p10-b325-pmqm.json.gz CU32 B2Gcu32 --keep $HOME/cxup/lean/gen > ~/cxup/lean/regen.log 2>&1
+
+$ grep -i -m8 "accept\|rate\|1 - \|exponent\|segments" ~/cxup/lean/regen.log
+CU32: mirror accepts; h=20 v=960 R=6265 N=162140; pairs=38262; replay segments=4; scalar batches=2
+$ GX_OUT=<fresh directory> python3 tools/gx/gxrate.py B2Gcu32 CU32
+RateB2Gcu32: block 79320/10^8 (evaluated 79320, fails 79321), rank 43876 (evaluated 43876, fails 43883), s=13, slack 1.62e-08, rank slack 1.53e-05
+RateB2Gcu32x: block 7932084/10^10 (evaluated 7932087, fails 7932088), rank 4388203 (evaluated 4388205, fails 4388209), s=40, slack 2.24e-10, rank slack 1.23e-07
+packed B2Gcu32x wht_main_block_B2Gcu32x 1 - 7932084/10^10 challenge code lines changed (0-based): [321, 322, 323]
+packed B2Gcu32 wht_main_block_B2Gcu32 1 - 79320/10^8 challenge code lines changed (0-based): [321, 322, 323]
+(not in the repository)        Work/GCert/Data/RateB2Gcu32.lean
+(not in the repository)        Work/GCert/Data/RateB2Gcu32x.lean
+
+$ tail -30 ~/cxup/lean/regen.log
+(not in the repository)        Work/GCert/Data/Gen/CU32.lean
+(not in the repository)        Work/GCert/Data/Gen/CU32/S0.lean
+...
+(not in the repository)        Work/GCert/Data/SolutionB2Gcu32x.lean
+(not in the repository)        comparator/B2Gcu32.json
+(not in the repository)        comparator/B2Gcu32x.json
+generated files kept in /home/cfky/cxup/lean/gen
+RESULT: FAIL (0 repository files reproduced byte for byte, 0 different, 35 generated files not in the repository)
+
+[regen_check.py's own RESULT line compares the generated files with files of the same name committed in Sussman's
+ repository. For a new program none exist, so it reports 35 files "not in the repository", 0 different, and prints
+ FAIL. That line does not judge the program; the mirror line above ("CU32: mirror accepts") and the Lean build
+ below do. PR #327's CC95/CP37 runs and our CX97 run went through the same step.]
+
+$ ls ~/cxup/lean/gen/Work/GCert/Data
+B2Gcu32.lean  B2Gcu32Main.lean  ChallengeB2Gcu32.lean  ChallengeB2Gcu32x.lean  CU32Cert.lean  CU32Price.lean
+CU32ScalOK.lean  Gen  InstB2Gcu32.lean  RateB2Gcu32.lean  RateB2Gcu32x.lean  SolutionB2Gcu32.lean  SolutionB2Gcu32x.lean
+
+
+2. Kernel-check the solution module
+-----------------------------------
+
+$ cd ~/wht-lean && cp -R ~/cxup/lean/gen/Work/GCert/Data/. Work/GCert/Data/
+$ lake build Work.GCert.Data.SolutionB2Gcu32x > ~/cxup/lean/build.log 2>&1; echo BUILD_EXIT $? >> ~/cxup/lean/build.log
+
+$ grep -c "error" ~/cxup/lean/build.log; grep BUILD_EXIT ~/cxup/lean/build.log; grep -i "Build completed\|jobs" ~/cxup/lean/build.log | tail -2
+0
+BUILD_EXIT 0
+Build completed successfully (9220 jobs).
+
+$ grep "wht_main_block_B2Gcu32x" ~/cxup/lean/build.log | tail -2
+info: Work/GCert/Data/SolutionB2Gcu32x.lean:38:0: 'OAI.PowerSaving.WHT.wht_main_block_B2Gcu32x' depends on axioms: [propext, Classical.choice, Quot.sound]
+
+
+3. Compare the solution with the challenge statement (Sussman's tools/Compare.lean)
+-----------------------------------------------------------------------------------
+
+[The first Compare.lean attempt stopped because the challenge module had not been built yet ("object file ...
+ ChallengeB2Gcu32x.olean ... does not exist", CMP_EXIT 1). The challenge module was then built and the comparison
+ re-run:]
+
+$ lake build Work.GCert.Data.ChallengeB2Gcu32x 2>&1 | tail -2
+warning: Work/GCert/Data/ChallengeB2Gcu32x.lean:323:8: declaration uses `sorry`
+Build completed successfully (8924 jobs).
+
+$ lake env lean --run tools/Compare.lean Work.GCert.Data.ChallengeB2Gcu32x Work.GCert.Data.SolutionB2Gcu32x \
+      OAI.PowerSaving.WHT.wht_main_block_B2Gcu32x 2>&1 | tail -8
+...
+  identical: OAI.PowerSaving.WHT.wht   [solution: WHTCheck.Solution]
+  identical: OAI.PowerSaving.WHT.whtInKind   [solution: WHTCheck.Solution]
+  identical: OAI.PowerSaving.WHT.whtOutKind   [solution: WHTCheck.Solution]
+OK: every statement dependency is identical in the solution
+solution proof reaches 37839 constants; axioms: #[Classical.choice, Quot.sound, propext]
+OK: only permitted axioms
+
+RESULT: PASS
+
+[The `sorry` is in the challenge module, which states the theorem without proof (as in Sussman's own challenge
+ files); the solution module proves it, and Compare.lean checks that the two statements agree.]
+
+
+Scope
+-----
+The theorem is the Walsh-Hadamard power-saving statement of Sussman's repository at exponent 1 - 7932084/10^10,
+built from this program with his unchanged generator. It is not an integer-multiplication κ assembly. The official
+comparator (landrun sandbox) and a clean build from an empty cache were not run for this program.
diff --git a/research/w3-p10b-pointwise-complex/module_cut.py b/research/w3-p10b-pointwise-complex/module_cut.py
new file mode 100644
index 0000000000..67ef904866
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/module_cut.py
@@ -0,0 +1,114 @@
+#!/usr/bin/env python3
+"""module_cut.py: zero restriction of a query module to fewer points (p = 11 module files -> p = 10).
+
+Setting the inputs that touch the dropped points to zero turns a module for n points into a module for n - k points:
+an addition with one zero operand becomes its other operand, an addition of two live operands is kept (merged with an
+equal-support node if one exists), and nodes that no kept root reaches are removed. The roots of the dropped labels
+are discarded. The result satisfies the smaller module's contract whenever the original satisfies its own (every
+root of a kept label still sums exactly its kept disjoint inputs); the loaders of the rebuilt tree check this.
+min_cut() drops the points that leave the fewest additions (ties: lexicographically first drop), the rule by which the
+control modules of this package (PR #168 v4's modules cut to p = 10) were made; verify.py re-derives them byte for byte.
+
+Prepared by DreamingOfClouds with Anthropic Claude assistance (Apache-2.0); the cut is cp10's mkmods.py, unchanged
+apart from file layout.
+usage: module_cut.py KIND N_FROM K MODULE_JSON OUT_JSON      (KIND tmod: N = p; pmod: N = p - 1; qmod: N = p - 2)
+"""
+import json
+import sys
+from itertools import combinations
+from pathlib import Path
+
+
+def restrict(args, inputs_image, nin, roots_old, one_based):
+    """generic zero restriction: inputs_image[k] = new input index or None (zero); returns (args, roots) pruned."""
+    off = 1 if one_based else 0
+    n_new = sum(1 for x in inputs_image if x is not None)
+    support = [1 << i for i in range(n_new)]
+    new_args = [None] * n_new
+    by = {s: i for i, s in enumerate(support)}
+    image = list(inputs_image)
+    for x in range(nin + off, len(args)):
+        a, b = args[x]
+        A, B = image[a - off], image[b - off]
+        if A is None or B is None:
+            image.append(A if B is None else B)
+            continue
+        assert not support[A] & support[B]
+        s = support[A] | support[B]
+        if s not in by:
+            by[s] = len(new_args); new_args.append([A, B]); support.append(s)
+        image.append(by[s])
+    roots = [image[r - off] for r in roots_old]
+    active = set(range(n_new)); stack = list(roots)
+    while stack:
+        x = stack.pop()
+        if x in active:
+            continue
+        active.add(x)
+        if new_args[x] is not None:
+            stack.extend(new_args[x])
+    ids = sorted(active); ren = {x: i for i, x in enumerate(ids)}
+    args2 = [None if new_args[x] is None else [ren[y] for y in new_args[x]] for x in ids]
+    return args2, [ren[x] for x in roots], support, ren, ids
+
+
+def cut_triple(d, P, keep):
+    labels = list(combinations(range(P), 3))
+    keep = sorted(keep); rn = {x: j for j, x in enumerate(keep)}; p2 = len(keep)
+    new_labels = list(combinations(range(p2), 3)); nid = {t: i for i, t in enumerate(new_labels)}
+    img = [nid[tuple(rn[x] for x in t)] if all(x in rn for x in t) else None for t in labels]
+    roots_old = [r for t, r in zip(labels, d['roots']) if all(x in rn for x in t)]
+    args, roots, support, ren, ids = restrict(d['args'], img, len(labels), roots_old, one_based=True)
+    # back to the one-based layout of triple_module_from
+    n = len(new_labels)
+    out_args = [[0, 0]] * (n + 1) + [[x + 1, y + 1] for x, y in (a for a in args[n:])]
+    return dict(kind='disjoint_triples', p=p2, input_labels=new_labels, target_labels=new_labels, input_count=n,
+                args=out_args, roots=[r + 1 for r in roots], source='cp10 zero restriction to points %s' % keep)
+
+
+def cut_pair(d, n, keep):
+    pairs = list(combinations(range(n), 2))
+    keep = sorted(keep); rn = {x: j for j, x in enumerate(keep)}; n2 = len(keep)
+    new_pairs = list(combinations(range(n2), 2)); nid = {t: i for i, t in enumerate(new_pairs)}
+    img = [nid[tuple(rn[x] for x in t)] if all(x in rn for x in t) else None for t in pairs]
+    roots_old = [r for t, r in zip(pairs, d['roots']) if all(x in rn for x in t)]
+    extra = d['roots'][len(pairs):]
+    args, roots, *_ = restrict(d['args'], img, len(pairs), roots_old + extra, one_based=False)
+    return dict(input_count=len(new_pairs), args=args, roots=roots, source='cp10 zero restriction to points %s' % keep)
+
+
+def cut_q(d, n, keep):
+    keep = sorted(keep); rn = {x: j for j, x in enumerate(keep)}
+    img = [rn.get(i) for i in range(n)]
+    roots_old = [d['roots'][i] for i in keep]
+    args, roots, *_ = restrict(d['args'], img, n, roots_old, one_based=False)
+    return dict(input_count=len(keep), args=args, roots=roots, source='cp10 zero restriction to points %s' % keep)
+
+
+def additions(d):
+    return sum(1 for a in d['args'] if a is not None and list(a) != [0, 0])
+
+
+CUT = dict(tmod=cut_triple, pmod=cut_pair, qmod=cut_q)
+
+
+def min_cut(kind, d, n, k=1):
+    """the zero restriction to n - k points with the fewest additions (ties: lexicographically first drop)"""
+    cands = []
+    for drop in combinations(range(n), k):
+        keep = [x for x in range(n) if x not in drop]
+        e = CUT[kind](d, n, keep)
+        cands.append((additions(e), drop, e))
+    cands.sort(key=lambda c: (c[0], c[1]))
+    return cands[0]
+
+
+def encode(d):
+    return (json.dumps(d, separators=(',', ':')) + '\n').encode()
+
+
+if __name__ == '__main__':
+    kind, n, k, src, dst = sys.argv[1], int(sys.argv[2]), int(sys.argv[3]), Path(sys.argv[4]), Path(sys.argv[5])
+    adds, drop, e = min_cut(kind, json.loads(src.read_text()), n, k)
+    dst.write_bytes(encode(e))
+    print('%s: %d -> %d points, dropped %s, %d additions' % (kind, n, n - k, list(drop), adds))
diff --git a/research/w3-p10b-pointwise-complex/modules/pm_p10.json b/research/w3-p10b-pointwise-complex/modules/pm_p10.json
new file mode 100644
index 0000000000..66d54f6efb
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/modules/pm_p10.json
@@ -0,0 +1 @@
+{"input_count": 36, "args": [null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, [31, 33], [24, 36], [28, 37], [19, 38], [13, 39], [0, 1], [4, 41], [2, 3], [9, 10], [15, 16], [44, 45], [21, 46], [24, 28], [47, 48], [43, 49], [1, 6], [19, 51], [50, 52], [17, 18], [19, 20], [1, 55], [4, 5], [30, 57], [32, 34], [58, 59], [36, 60], [6, 7], [35, 62], [61, 63], [56, 64], [54, 65], [22, 23], [25, 67], [66, 68], [9, 15], [25, 70], [23, 27], [5, 33], [56, 73], [18, 34], [74, 75], [72, 76], [25, 43], [10, 27], [16, 26], [79, 80], [29, 81], [8, 42], [14, 20], [7, 84], [83, 85], [59, 86], [82, 87], [0, 2], [3, 57], [21, 72], [26, 30], [22, 92], [91, 93], [90, 94], [89, 95], [11, 12], [44, 97], [13, 98], [37, 99], [96, 100], [25, 29], [32, 102], [34, 103], [0, 8], [13, 105], [52, 106], [14, 35], [99, 108], [15, 51], [8, 46], [34, 111], [48, 112], [21, 102], [113, 114], [2, 24], [15, 116], [69, 117], [18, 72], [12, 119], [13, 14], [0, 121], [48, 114], [35, 123], [62, 124], [43, 125], [122, 126], [61, 127], [22, 26], [11, 17], [129, 130], [111, 131], [128, 132], [120, 133], [56, 134], [33, 107], [2, 90], [94, 97], [54, 138], [2, 95], [30, 97], [17, 141], [18, 142], [5, 143], [3, 144], [88, 145], [91, 104], [35, 147], [71, 116], [7, 149], [23, 150], [7, 104], [20, 152], [96, 97], [54, 111], [154, 155], [50, 119], [9, 12], [24, 68], [158, 159], [63, 122], [61, 161], [11, 162], [160, 163], [2, 164], [13, 108], [20, 166], [19, 167], [36, 59], [8, 143], [169, 170], [168, 171], [29, 172], [114, 168], [2, 71], [42, 130], [38, 93], [147, 177], [109, 178], [31, 32], [129, 180], [4, 181], [1, 17], [182, 183], [4, 11], [34, 73], [23, 186], [3, 29], [28, 188], [10, 189], [4, 144], [27, 44], [12, 192], [27, 54], [9, 154], [7, 14], [10, 196], [104, 197], [22, 32], [16, 54], [15, 200], [140, 201], [28, 80], [79, 203], [31, 204], [3, 205], [52, 105], [53, 129], [194, 203], [188, 209], [65, 210], [167, 207], [7, 212], [29, 78], [213, 214], [26, 79], [97, 216], [6, 28], [111, 139], [40, 219], [84, 111], [87, 143], [5, 175], [222, 223], [48, 174], [1, 156], [8, 47], [214, 227], [32, 129], [85, 229], [176, 230], [228, 231], [136, 191], [16, 213], [190, 234], [27, 235], [92, 201], [20, 237], [169, 191], [110, 140], [5, 12], [4, 130], [31, 106], [242, 243], [1, 202], [153, 245], [33, 241], [75, 247], [212, 248], [151, 249], [132, 225], [31, 242], [213, 252], [32, 253], [190, 254], [107, 247], [206, 233], [80, 255], [44, 127], [181, 259], [185, 260], [148, 177], [62, 262], [39, 200], [240, 264], [33, 168], [115, 120], [266, 267], [43, 45], [125, 269], [55, 270], [184, 271], [34, 221], [103, 273], [34, 120], [213, 239], [49, 215], [41, 85], [236, 248], [22, 39], [148, 280], [149, 253], [137, 263], [67, 117], [9, 284], [31, 233], [228, 278], [275, 287], [238, 281], [173, 204], [5, 288], [127, 193], [101, 218], [157, 256], [139, 274], [285, 286], [199, 282], [187, 292], [67, 224], [180, 251], [70, 168], [171, 301], [189, 217], [162, 303], [77, 270], [195, 198], [159, 302], [208, 244]], "roots": [289, 179, 290, 307, 268, 300, 295, 220, 283, 211, 118, 305, 272, 246, 265, 304, 165, 298, 261, 306, 293, 276, 279, 258, 146, 257, 250, 297, 299, 296, 277, 291, 294, 232, 308, 226, 135], "provenance": "p10next: pair module annealed at p = 10 with a debt-penalised five-stage objective from the PR #315 module (nbsa_debt.py, LAMBDA 150-300)"}
\ No newline at end of file
diff --git a/research/w3-p10b-pointwise-complex/modules/pointwise/pmod_i0.json b/research/w3-p10b-pointwise-complex/modules/pointwise/pmod_i0.json
new file mode 100644
index 0000000000..5e4b0940aa
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/modules/pointwise/pmod_i0.json
@@ -0,0 +1 @@
+{"input_count": 36, "args": [null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, [31, 33], [24, 36], [28, 37], [19, 38], [13, 39], [0, 1], [4, 41], [2, 3], [9, 10], [15, 16], [44, 45], [21, 46], [24, 28], [47, 48], [43, 49], [1, 6], [19, 51], [50, 52], [17, 18], [19, 20], [1, 55], [4, 5], [30, 57], [32, 34], [58, 59], [36, 60], [6, 7], [35, 62], [61, 63], [56, 64], [54, 65], [22, 23], [25, 67], [66, 68], [9, 15], [25, 70], [23, 27], [5, 33], [56, 73], [18, 34], [74, 75], [72, 76], [25, 43], [10, 27], [16, 26], [79, 80], [29, 81], [8, 42], [14, 20], [7, 84], [83, 85], [59, 86], [82, 87], [0, 2], [3, 57], [21, 72], [26, 30], [22, 92], [91, 93], [90, 94], [89, 95], [11, 12], [44, 97], [13, 98], [37, 99], [96, 100], [25, 29], [32, 102], [34, 103], [0, 8], [13, 105], [52, 106], [14, 35], [99, 108], [15, 51], [8, 46], [34, 111], [48, 112], [21, 102], [113, 114], [2, 24], [15, 116], [69, 117], [18, 72], [12, 119], [13, 14], [0, 121], [48, 114], [35, 123], [62, 124], [43, 125], [122, 126], [61, 127], [22, 26], [11, 17], [129, 130], [111, 131], [128, 132], [120, 133], [56, 134], [33, 107], [2, 90], [94, 97], [54, 138], [2, 95], [30, 97], [17, 141], [18, 142], [5, 143], [3, 144], [88, 145], [91, 104], [35, 147], [71, 116], [7, 149], [23, 150], [7, 104], [20, 152], [96, 97], [54, 111], [154, 155], [50, 119], [9, 12], [24, 68], [158, 159], [63, 122], [61, 161], [11, 162], [160, 163], [2, 164], [13, 108], [20, 166], [19, 167], [36, 59], [8, 143], [169, 170], [168, 171], [29, 172], [114, 168], [2, 71], [42, 130], [38, 93], [147, 177], [109, 178], [31, 32], [129, 180], [4, 181], [1, 17], [182, 183], [4, 11], [34, 73], [23, 186], [3, 29], [28, 188], [10, 189], [4, 144], [27, 44], [12, 192], [27, 54], [9, 154], [7, 14], [10, 196], [104, 197], [22, 32], [16, 54], [15, 200], [140, 201], [28, 80], [79, 203], [31, 204], [3, 205], [52, 105], [53, 129], [194, 203], [188, 209], [65, 210], [167, 207], [7, 212], [29, 78], [213, 214], [26, 79], [97, 216], [6, 28], [111, 139], [40, 219], [84, 111], [87, 143], [5, 175], [222, 223], [48, 174], [1, 156], [111, 21], [214, 227], [32, 129], [85, 229], [176, 230], [228, 231], [136, 191], [16, 213], [190, 234], [27, 235], [92, 201], [20, 237], [169, 191], [110, 140], [5, 12], [4, 130], [31, 106], [242, 243], [1, 202], [153, 245], [33, 241], [75, 247], [212, 248], [151, 249], [132, 225], [31, 242], [213, 252], [32, 253], [190, 254], [107, 247], [206, 233], [80, 255], [44, 127], [181, 259], [185, 260], [148, 177], [62, 262], [39, 200], [240, 264], [33, 168], [115, 120], [266, 267], [43, 45], [125, 269], [55, 270], [184, 271], [34, 221], [103, 273], [34, 120], [213, 239], [49, 215], [41, 85], [236, 248], [22, 39], [148, 280], [149, 253], [137, 263], [67, 117], [9, 284], [31, 233], [228, 278], [275, 287], [238, 281], [173, 204], [5, 288], [127, 193], [101, 218], [157, 256], [139, 274], [285, 286], [199, 282], [187, 292], [67, 224], [180, 251], [70, 168], [171, 301], [189, 217], [162, 303], [77, 270], [195, 198], [159, 302], [208, 244]], "roots": [289, 179, 290, 307, 268, 300, 295, 220, 283, 211, 118, 305, 272, 246, 265, 304, 165, 298, 261, 306, 293, 276, 279, 258, 146, 257, 250, 297, 299, 296, 277, 291, 294, 232, 308, 226, 135]}
\ No newline at end of file
diff --git a/research/w3-p10b-pointwise-complex/modules/pointwise/pmod_i1.json b/research/w3-p10b-pointwise-complex/modules/pointwise/pmod_i1.json
new file mode 100644
index 0000000000..fe8c82b41c
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/modules/pointwise/pmod_i1.json
@@ -0,0 +1 @@
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diff --git a/research/w3-p10b-pointwise-complex/modules/pointwise/pmod_i2.json b/research/w3-p10b-pointwise-complex/modules/pointwise/pmod_i2.json
new file mode 100644
index 0000000000..ca279e90f8
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/modules/pointwise/pmod_i2.json
@@ -0,0 +1 @@
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diff --git a/research/w3-p10b-pointwise-complex/modules/pointwise/pmod_i3.json b/research/w3-p10b-pointwise-complex/modules/pointwise/pmod_i3.json
new file mode 100644
index 0000000000..5e4b0940aa
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/modules/pointwise/pmod_i3.json
@@ -0,0 +1 @@
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diff --git a/research/w3-p10b-pointwise-complex/modules/pointwise/pmod_i4.json b/research/w3-p10b-pointwise-complex/modules/pointwise/pmod_i4.json
new file mode 100644
index 0000000000..fe8c82b41c
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/modules/pointwise/pmod_i4.json
@@ -0,0 +1 @@
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diff --git a/research/w3-p10b-pointwise-complex/modules/pointwise/pmod_i5.json b/research/w3-p10b-pointwise-complex/modules/pointwise/pmod_i5.json
new file mode 100644
index 0000000000..d9009baa49
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/modules/pointwise/pmod_i5.json
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diff --git a/research/w3-p10b-pointwise-complex/modules/pointwise/pmod_i6.json b/research/w3-p10b-pointwise-complex/modules/pointwise/pmod_i6.json
new file mode 100644
index 0000000000..d9009baa49
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/modules/pointwise/pmod_i6.json
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diff --git a/research/w3-p10b-pointwise-complex/modules/pointwise/pmod_i7.json b/research/w3-p10b-pointwise-complex/modules/pointwise/pmod_i7.json
new file mode 100644
index 0000000000..e52591a0a0
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/modules/pointwise/pmod_i7.json
@@ -0,0 +1 @@
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diff --git a/research/w3-p10b-pointwise-complex/modules/pointwise/pmod_i8.json b/research/w3-p10b-pointwise-complex/modules/pointwise/pmod_i8.json
new file mode 100644
index 0000000000..d9009baa49
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/modules/pointwise/pmod_i8.json
@@ -0,0 +1 @@
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diff --git a/research/w3-p10b-pointwise-complex/modules/pointwise/pmod_i9.json b/research/w3-p10b-pointwise-complex/modules/pointwise/pmod_i9.json
new file mode 100644
index 0000000000..c679b14331
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/modules/pointwise/pmod_i9.json
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diff --git a/research/w3-p10b-pointwise-complex/modules/qmod_p10.json b/research/w3-p10b-pointwise-complex/modules/qmod_p10.json
new file mode 100644
index 0000000000..b837d6e209
--- /dev/null
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diff --git a/research/w3-p10b-pointwise-complex/modules/tmod_t2_p10.json b/research/w3-p10b-pointwise-complex/modules/tmod_t2_p10.json
new file mode 100644
index 0000000000..7d2185c2be
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/modules/tmod_t2_p10.json
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indexed by J sums input I exactly when I and J are disjoint.","source":"agent-cmod ctsa annealing of the complex compile ledger"}
diff --git a/research/w3-p10b-pointwise-complex/normalize_scatter.py b/research/w3-p10b-pointwise-complex/normalize_scatter.py
new file mode 100644
index 0000000000..4f62515895
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/normalize_scatter.py
@@ -0,0 +1,106 @@
+#!/usr/bin/env python3
+"""normalize_scatter.py: write the scatter of a gcert/1 program as the star rule by re-signing retained totals.
+
+gcert_emit.py (PR #256, vendored unchanged) keeps every slot in a unit lam (wanted = lam * stored, the convention of
+Sussman's gxconv.py). When the exact lift leaves a retained total k in a register whose unit lam_k is not 1, the
+emitter writes the scatter as a table: row t, column k is the star coefficient (1/3 if coordinate k lies in port t,
+-1/6 otherwise) times lam_k. The five-stage packages' complex code reads the star rule only. If every lam_k is +1 or
+-1, flipping the sign convention of the registers with lam_k = -1 over the whole program (P -> D P D with
+D = diag(+-1)) gives the same map from sources to targets with the star rule: every gate coefficient on an edge
+with exactly one endpoint in a flipped register is negated, and the scatter column of a flipped total becomes the
+star column. Registers, frames, gates, start and final labels, retained totals, blocks and N are unchanged; only
+coefficient signs, the scatter field and the derived_from note change. A program already in star form is returned
+unchanged. verify.py runs gx.check1 (and Sussman's gxcore mirror) on the result and compares it with the input.
+
+Prepared by DreamingOfClouds with Anthropic Claude assistance (Apache-2.0).
+usage: normalize_scatter.py IN.json[.gz] OUT.json[.gz]
+"""
+import copy
+import gzip
+import json
+import sys
+from fractions import Fraction as Q
+
+if sys.flags.optimize:
+    raise SystemExit('Assertions must remain enabled: run without -O')
+STAR = dict(inside=[1, 3], outside=[-1, 6])
+
+
+def star(c, t, k):
+    return Q(1, 3) if c['ports'][t] >> k & 1 else Q(-1, 6)
+
+
+def units(c):
+    """lam_k of every retained total, read off a table scatter (None for a star scatter)"""
+    sc = c['scat']
+    if 'table' not in sc:
+        assert sc == STAR, 'a star scatter must be 1/3 inside, -1/6 outside'
+        return None
+    h, v = c['h'], c['v']
+    assert len(sc['table']) == v
+    lam = {}
+    for t, row in enumerate(sc['table']):
+        assert [k for k, _, _ in row] == list(range(h)), 'table row %d does not list the coordinates 0..h-1 in order' % t
+        for k, a, b in row:
+            q = Q(a, b) / star(c, t, k)
+            assert lam.setdefault(k, q) == q, 'column %d is not the star column times one unit' % k
+    return lam
+
+
+def normalize(c):
+    lam = units(c)
+    if lam is None:
+        return c, dict(raw_scatter='star', flipped_coordinates=[], flipped_registers=[], changed_gates=0, changed_coefficients=0)
+    assert all(q in (1, -1) for q in lam.values()), 'units other than +-1 cannot be normalized by a sign flip: %s' % lam
+    v = c['v']
+    ret = {k: s for k, s, _ in c['ret']}
+    assert sorted(ret) == list(range(c['h'])) and len(set(ret.values())) == len(ret), 'one retained total per coordinate'
+    flip = {ret[k] for k, q in lam.items() if q == -1}
+    assert all(s >= 2 * v for s in flip), 'only slots are re-signed'
+    c2 = copy.deepcopy(c)
+    gates = coefficients = 0
+    for ph in ('A', 'B'):
+        for g in c2[ph]:
+            assert g[0] in ('out', 'in'), 'gate kind %r' % (g[0],)
+            pivot = g[2] in flip
+            hit = False
+            for term in g[3]:
+                if pivot != (term[0] in flip):       # exactly one endpoint of this edge is re-signed
+                    term[1] = -term[1]
+                    coefficients += 1
+                    hit = True
+            gates += hit
+    c2['scat'] = dict(STAR)
+    c2['derived_from'] = c['derived_from'] + ' + normalize_scatter.py (registers %s re-signed; star scatter)' % sorted(flip)
+    return c2, dict(raw_scatter='table', flipped_coordinates=sorted(k for k, q in lam.items() if q == -1),
+                    flipped_registers=sorted(flip), changed_gates=gates, changed_coefficients=coefficients)
+
+
+def same_shape(a, b):
+    """True when two programs differ at most in coefficient signs, the scatter and derived_from"""
+    keys = set(a) | set(b)
+    for k in keys - {'A', 'B', 'scat', 'derived_from'}:
+        if a.get(k) != b.get(k):
+            return False
+    for ph in ('A', 'B'):
+        if len(a[ph]) != len(b[ph]):
+            return False
+        for g, h in zip(a[ph], b[ph]):
+            if g[:3] != h[:3] or g[4:] != h[4:] or len(g[3]) != len(h[3]):
+                return False
+            if any(x[0] != y[0] or x[2] != y[2] or abs(x[1]) != abs(y[1]) for x, y in zip(g[3], h[3])):
+                return False
+    return True
+
+
+def main():
+    src, dst = sys.argv[1:3]
+    raw = gzip.decompress(open(src, 'rb').read()) if src.endswith('.gz') else open(src, 'rb').read()
+    c2, report = normalize(json.loads(raw))
+    data = json.dumps(c2, separators=(',', ':')).encode()
+    open(dst, 'wb').write(gzip.compress(data) if dst.endswith('.gz') else data)
+    print(json.dumps(report))
+
+
+if __name__ == '__main__':
+    main()
diff --git a/research/w3-p10b-pointwise-complex/paircheck.py b/research/w3-p10b-pointwise-complex/paircheck.py
new file mode 100644
index 0000000000..673cc0a684
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/paircheck.py
@@ -0,0 +1,165 @@
+#!/usr/bin/env python3
+"""paircheck.py: standalone re-check of the compensated reuse pairs of a written aligned tree.
+
+Reads only the written word (cache/{graph,frames,selection}.json, references/paired-cube/physical/{frames,pairs}.json)
+and re-verifies the pairs independently of the code that chose them and of aligned_word.py:
+  kinds:      the donor is an ungauged, non-root role with operations; the recipient is a selected (rank h-4) gauge;
+              pairs are one-to-one and every selected gauge is paired (no live external gauge);
+  chronology: early pair (deadline None): the donor's last operation in phase 1, the recipient's first after the cut;
+              late pair: deadline = the recipient's first operation (phase 2), the donor's last operation strictly
+              before it in the executed order (phase 1, then phase 2 in index order), i.e. the donor is dead before the
+              recipient's old-value read;
+  frames:     the donor's last physical frame lies inside the recipient's gauge frame;
+  parity:     no pair is erasable by PR #184's --purify-source-donors (donor last frame of rank <= 3 inside a cube
+              parity 3-space inside the recipient gauge).
+Five mutation controls must each be rejected for the condition they break (every violated condition is listed):
+a late recipient given an unused donor still alive at its read; an unused donor with legal timing whose last frame
+lies outside the gauge (frame containment alone); a duplicated pair; an unpaired gauge; an unused donor with legal
+timing whose last frame is parity-erasable (erasability alone).
+Prepared by DreamingOfClouds with Anthropic Claude assistance (Apache-2.0); uses the rebuilt tree's
+scripts/paired_cube/frames.py only for basis/perp/containment.
+usage: paircheck.py --root PR202_TREE WORD_TREE [WORD_TREE ...]      (prints one JSON line per word tree)"""
+import argparse
+import json
+import sys
+from pathlib import Path
+
+if sys.flags.optimize:
+    raise SystemExit('Assertions must remain enabled: run without -O')
+
+
+def check(tree, mutate=None):
+    from paired_cube.frames import basis, perp, contained
+    tree = Path(tree)
+    rd = lambda p: json.loads((tree / p).read_text())
+    g, w, word = rd('cache/graph.json'), rd('cache/frames.json'), rd('cache/selection.json')
+    moved = rd('references/paired-cube/physical/frames.json')['frames']
+    pairs = rd('references/paired-cube/physical/pairs.json')['pairs']
+    h, v = g['h'], g['v']
+    ops = word['ops']
+    frames = [perp(tuple(w['annihilators'][x]), h) for _, _, x in ops]
+    for i, F in moved:
+        frames[i] = tuple(F)
+    pset = set(word['phase1'])
+    order = sorted(pset) + [i for i in range(len(ops)) if i not in pset]
+    pos = {i: k for k, i in enumerate(order)}
+    role_ops = {}
+    for i in order:
+        a, b, _ = ops[i]
+        role_ops.setdefault(a, []).append(i)
+        role_ops.setdefault(b, []).append(i)
+    gauge = {z['role']: perp(tuple(z['annihilator']), h) for z in word['selected']}
+    assert all(z['rank'] == h - 4 for z in word['selected']), 'non rank-(h-4) gauge kept'
+    roots = set(word['rootroles'])
+    parity = []
+    for cube in range(v // 8):
+        for bit in range(2):
+            parity.append(basis(tuple(g['inputs'][8 * cube + j] for j in range(8) if j.bit_count() % 2 == bit)))
+    erasable_pair = lambda U, V: len(U) <= 3 and any(contained(U, S) and contained(S, V) for S in parity)
+    if mutate is not None:
+        pairs = mutate([list(p) for p in pairs], dict(frames=frames, pos=pos, role_ops=role_ops, gauge=gauge, roots=roots,
+                                                       pset=pset, contained=contained, erasable=erasable_pair))
+    bad = []
+    donors, recips = set(), set()
+    late = early = erasable = 0
+    for d, b, t in pairs:
+        if d in donors or b in recips:
+            bad.append(('not one-to-one', d, b))
+        donors.add(d)
+        recips.add(b)
+        if d in gauge or d in roots or d not in role_ops:
+            bad.append(('donor kind', d))
+        if b not in gauge:
+            bad.append(('recipient kind', b))
+            continue
+        last, first = role_ops[d][-1], role_ops[b][0]
+        if t is None:
+            early += 1
+            if not (last in pset and first not in pset):
+                bad.append(('early chronology', d, b))
+        else:
+            late += 1
+            if not (t == first and t not in pset and pos[last] < pos[t]):
+                bad.append(('late chronology', d, b, t))
+        if not contained(frames[last], gauge[b]):
+            bad.append(('donor frame outside gauge', d, b))
+        if erasable_pair(frames[last], gauge[b]):
+            erasable += 1
+    unpaired = set(gauge) - recips
+    if unpaired:
+        bad.append(('live external gauges', len(unpaired)))
+    return dict(pairs=len(pairs), early=early, late=late, gauges=len(gauge), erasable=erasable,
+                violations=len(bad), kinds=sorted({x[0] for x in bad}), first_violations=bad[:5],
+                status='PASS' if not bad and not erasable else 'FAIL')
+
+
+def controls(tree):
+    """five mutations; each must be rejected for the condition it breaks (all violated conditions are listed)"""
+    def free_donors(pairs, c):
+        used = {p[0] for p in pairs}
+        return [(d, ops[-1]) for d, ops in sorted(c['role_ops'].items()) if d not in used and d not in c['gauge'] and d not in c['roots']]
+
+    def legal_timing(last, b, t, c):
+        first = c['role_ops'][b][0]
+        return last in c['pset'] if t is None else (t == first and c['pos'][last] < c['pos'][t])
+
+    def swap(pairs, c, *wants):
+        # replace the donor of some pair by an unused donor chosen by the first satisfiable predicate
+        # want(last frame, gauge, last op, recipient, deadline)
+        free = free_donors(pairs, c)
+        for want in wants:
+            for k, (d, b, t) in enumerate(pairs):
+                for d2, last in free:
+                    if want(c['frames'][last], c['gauge'][b], last, b, t):
+                        pairs[k] = [d2, b, t]
+                        return pairs
+        raise RuntimeError('no control instance')
+
+    def late_before_death(pairs, c):
+        # a late recipient gets an unused donor that is still alive at the recipient's read; a donor whose frame also
+        # fits the gauge is preferred (none exists on the committed words, so the frame check fires as well)
+        alive = lambda U, V, last, b, t: t is not None and c['pos'][last] > c['pos'][t]
+        return swap(pairs, c, lambda U, V, last, b, t: alive(U, V, last, b, t) and c['contained'](U, V) and not c['erasable'](U, V),
+                    alive)
+
+    def frame_outside(pairs, c):
+        # an unused donor with legal timing whose last frame is outside the recipient's gauge
+        return swap(pairs, c, lambda U, V, last, b, t: legal_timing(last, b, t, c) and not c['contained'](U, V))
+
+    def duplicate(pairs, c):
+        pairs.append(list(pairs[0]))
+        return pairs
+
+    def drop(pairs, c):
+        return pairs[1:]
+
+    def erasable(pairs, c):
+        # an unused donor with legal timing whose last frame (rank <= 3) lies in a cube parity 3-space inside the gauge
+        return swap(pairs, c, lambda U, V, last, b, t: legal_timing(last, b, t, c) and c['erasable'](U, V))
+    out = {}
+    for name, f, intended in [('late_read_before_donor_death', late_before_death, 'late chronology'),
+                              ('donor_frame_outside_gauge', frame_outside, 'donor frame outside gauge'),
+                              ('duplicate_pair', duplicate, 'not one-to-one'), ('unpaired_gauge', drop, 'live external gauges'),
+                              ('parity_erasable_donor', erasable, 'erasable')]:
+        try:
+            r = check(tree, f)
+        except RuntimeError as e:
+            out[name] = 'no instance: %s' % e
+            continue
+        found = r['kinds'] + (['erasable'] if r['erasable'] else [])
+        out[name] = ('REJECTED (%s)' if intended in found else 'ACCEPTED (control failed: %s)') % ', '.join(found)
+    return out
+
+
+def main():
+    ap = argparse.ArgumentParser(description=__doc__, formatter_class=argparse.RawDescriptionHelpFormatter)
+    ap.add_argument('--root', type=Path, required=True, help='the rebuilt PR #202 tree (scripts/paired_cube/frames.py)')
+    ap.add_argument('trees', type=Path, nargs='+')
+    a = ap.parse_args()
+    sys.path.insert(0, str(a.root / 'scripts'))
+    for t in a.trees:
+        print(json.dumps(dict(check(t), controls=controls(t))), flush=True)
+
+
+if __name__ == '__main__':
+    main()
diff --git a/research/w3-p10b-pointwise-complex/pricing/outer.py b/research/w3-p10b-pointwise-complex/pricing/outer.py
new file mode 100644
index 0000000000..43007488ad
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/pricing/outer.py
@@ -0,0 +1,58 @@
+#!/usr/bin/env python3
+"""Exact semantic assembly with the balanced prefix, for the paired-cube network.
+
+Prepared by eumemic with Anthropic Claude assistance. Apache-2.0. The 47 strict constraints, the seven margins and
+the finite bridge are unchanged from icekylinx's scripts/structured_bulk_assembly.py (PR144, adapted from Zhihao
+Chen's PR23). The prefix is the RaD project's (hipotures) balanced positional transform, reviewed in
+references/semantic-bulk/rad20/reports/review-balanced-transform.md: the prefix margin becomes 1-eps instead of
+1-eps(1+c), and eps(1+c)<1 remains a strict geometric condition (K_geometry) whose gap need not exceed kappa.
+The parameters c=q+eta/4 and eps=(1-eta)/(1+q) are those of the reviewed balanced assembly in
+research/copied-fixed/balanced_assembly.py, with its backoff h equal to eta. Nothing else differs from
+structured_bulk_assembly.assembly.
+"""
+from fractions import Fraction as Q
+
+
+def assembly(bit_saving,complex_saving,bridge,kappa,eta=Q(1,10**8),beta=Q(1,4)):
+    a,b=bit_saving,complex_saving
+    tau,sigma=1-a,1-b
+    q=a*(1-2*eta); lp=1-q; lam=(tau+lp)/2
+    c=q+eta/4; eps=(1-eta)/(1+q)
+    minimum=eps*q; r=(minimum+1-eps)/2; delta=eta/8
+    internal=tau+(1-beta)*max(sigma-tau,Q(0));leaf=sigma+beta*(1-sigma)
+    margins=dict(balanced_prefix=1-eps,coordinate_movement=a,
+        compact_phase_layer=minimum,bulk_exposure=a,
+        Gaussian_arithmetic=min(1-eps-delta,r-delta),
+        scalar_work=1-eps-delta,dimension=eps)
+    slacks=dict(bit_positive=a,complex_above_bit=b-a,
+        complex_below_one_over32=Q(1,32)-b,beta_positive=beta,
+        beta_below_one=1-beta,leaf_saving_above_bit=(1-beta)*b-a,
+        q_positive=q,q_below_internal=1-internal-q,q_below_leaf=1-leaf-q,
+        c_positive=c,c_below_one=1-c,q_below_reservations=c-q,
+        lambda_above_tau=lam-tau,lambda_above_sigma=lam-sigma,
+        lambda_above_internal=lam-internal,lambda_prime_above_lambda=lp-lam,
+        compact_leaf=lp-leaf,compact_reservations=lp-(1-c),
+        lambda_prime_below_one=q,epsilon_positive=eps,epsilon_below_one=1-eps,
+        guard_width=1-eps,K_geometry=1-eps*(1+c),K_dominates_log=eps*c,
+        record_suffix=1-eps,phase_local=1-eps-delta,phase_boundary=r-delta,
+        gamma_sublinear=1-eps-r,cell_above_band=eps-(1-r)/2,
+        prime_interval_packing=1-eps,alpha_positive=r,alpha_below_one=1-r,
+        alpha_below_one_fourth=Q(1,4)-r,delta_positive=delta,
+        delta_below_one_eighth=Q(1,8)-delta,short_record_fallback=eps-a,
+        small_field_exposure=1-eps-minimum,
+        artificial_boundary=8-eps+r-delta-minimum,
+        literal_scalar_guard=Q(bridge['semantic']['strict_literal_gap']),
+        row_product_gap=bridge['rows']['degree_gap'])
+    slacks.update({name+'_above_kappa':val-kappa for name,val in margins.items()})
+    assert len(slacks)==47 and len(margins)==7
+    assert all(v>0 for v in slacks.values()),{k:str(v) for k,v in slacks.items() if v<=0}
+    assert min(margins.values())==minimum
+    assert margins['balanced_prefix']-minimum==eta and 1-eps-r==eta/2
+    assert 1-eps*(1+c)==eta-eps*eta/4
+    return dict(parameters=dict(a_bit=a,a_complex=b,tau=tau,sigma=sigma,
+            eta=eta,beta=beta,q=q,c=c,epsilon=eps,lambda_=lam,
+            lambda_prime=lp,alpha_squared_power=r,delta=delta,
+            C0=bridge['semantic']['C0'],C1=1,kappa=kappa),
+        strict_constraints=slacks,margins=margins,minimum_margin=minimum,
+        absorption_gap=minimum-kappa,
+        recurrence=dict(internal=internal,leaf=leaf,reservations=1-c))
diff --git a/research/w3-p10b-pointwise-complex/references/centre46_graph.diff b/research/w3-p10b-pointwise-complex/references/centre46_graph.diff
new file mode 100644
index 0000000000..76437ab856
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/references/centre46_graph.diff
@@ -0,0 +1,33 @@
+--- graph_orig.py	2026-10-10 03:55:31
++++ graph.py	2026-10-10 03:55:31
+@@ -190,14 +190,18 @@
+         self.local_channels()
+         base_nodes = len(self.a)
+         D = dict(zip(self.cubes, self.module(triple, [self.F[I] for I in self.cubes])))
+-        P, Q = {}, {}
++        P, Q, star = {}, {}, {}
+         for i in range(self.p):
+             others = [a for a in range(self.p) if a != i]
+             pairs = list(combinations(others, 2))
+             for bit in range(2):
+                 inputs = [self.A[tuple(sorted((i, *K))), i, bit] for K in pairs]
+-                for K, node in zip(pairs, self.module(pair, inputs)):
++                outs = self.module(pair, inputs)
++                for K, node in zip(pairs, outs[:len(pairs)]):
+                     P[tuple(sorted((i, *K))), i, bit] = node
++                if len(outs) > len(pairs):
++                    # centre-sharing pair module: its extra root is the sum of all inputs = star(2i+bit)
++                    star[i, bit] = outs[len(pairs)]
+         for i, j in combinations(range(self.p), 2):
+             others = [a for a in range(self.p) if a not in (i, j)]
+             for mode in range(2):
+@@ -231,7 +235,8 @@
+             # Their scatter is 1/2 sum_(i in T) S_i - 1/6 sum_i S_i.
+             for i in range(self.p):
+                 for e in range(2):
+-                    self.centers.append(self.sum(self.A[I, i, e] for I in self.cubes if i in I))
++                    self.centers.append(star[i, e] if (i, e) in star else
++                                        self.sum(self.A[I, i, e] for I in self.cubes if i in I))
+             incidences = 0
+             center_scatter = '1/2 sum_(i in T) S_i - 1/6 sum_i S_i'
+         else:
diff --git a/research/w3-p10b-pointwise-complex/references/pr233-layer/NOTICE b/research/w3-p10b-pointwise-complex/references/pr233-layer/NOTICE
new file mode 100644
index 0000000000..27c7d368fa
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/references/pr233-layer/NOTICE
@@ -0,0 +1,43 @@
+Prepared by Chafik Boukhalfa with substantial Anthropic Claude and OpenAI
+Codex assistance. Apache-2.0. Inherited files retain their original notices
+and assistance disclosures. This package claims no exclusive priority over
+concurrent work and reports no new kappa.
+
+What this package contributes: the physical layer (operation frames and reuse
+pairs) of PR #200's descent applied to the source-aligned complex word, the
+admission of that layer with PR #200's checker on that word, and the
+observation that PR #184's frame flow prices it at 7.0861e-4. Everything
+else is retained and credited:
+
+- icekylinx (OpenAI GPT-6 Astra and Codex assistance): PR144 paired cubes,
+  signed decoder, shared-core accounting, original compiler, carrier
+  matching (the frozen round-nine matching reused here), chronological
+  gauges, PR130 generalized Clifford frames; PR184 source-assisted
+  construction: source-parity local factorization, frame flow with
+  independent dirty lifts and source controls, exact lifts, contract,
+  global assembly (research/source-assisted, vendored byte for byte).
+- ikeboy: PR191/193/194, the rebuild of PR184's construction on the PR168 v4
+  modules (research/source-assisted-v4: source_aligned_local_v4.py,
+  contract_v4.py, assemble.py, the parity-erasure-avoiding matching), and the
+  composition with PR189's/PR200's bit supplier.
+- eumemic (Anthropic Claude and OpenAI Codex assistance): PR117 module
+  infrastructure; PR155/157 scalar modules, physical frames and reuse;
+  PR161 output merging; PR168 annealed modules, nested-prefix schedules,
+  local L1 circuit, fused outputs, bundle descent, both physical layers
+  (the physical layer replaced here is PR168's, used as the descent's seed),
+  pinned at 4a3c769e5c5430e7114c4d3e099ff34664677f17; PR131/143 frame/reuse
+  and late-read lineage; PR154 isolated snapshot verification.
+- jamesyc / James Chang: PR124 compensated birth cuts, PR34 balanced
+  assembly, PR166 terminal-output accumulation. DaysSky: PR162 extended
+  carrier closure, PR192 frame-layout ceiling. GamingPuzzled: PR160.
+  gupt1156 PR148 and Abhinav Ramachandran/geckods PR158 (Hermes assistance).
+  jacklightChen / Zhihao Chen: PR23/29. Rohan Arun: PR100/103. an664: PR128.
+  Earlier lineage: Zhihao Chen PR97, Swapnil Jain, DaysSky PR150, hpst3r
+  PR147, SovereignSteak PR122, Aurel Prosz, dleen, Douglas Colkitt.
+- Chafik Boukhalfa (Anthropic Claude and OpenAI Codex assistance): PR175
+  fast checker; PR179/181/189/196/200 local circuits, operation-frame descent
+  (discovery/), independent complete complex admission (complex/physical.py,
+  verify_fused.py, vendored from PR200 at a1175449) and the PR200 bit
+  supplier the assembly reads; PR202 (the pinned tree, 8d8d67b).
+- OpenAI: the original complex phase network of "Integer multiplication
+  below n log n".
diff --git a/research/w3-p10b-pointwise-complex/references/pr233-layer/SOURCE.json b/research/w3-p10b-pointwise-complex/references/pr233-layer/SOURCE.json
new file mode 100644
index 0000000000..dca4742101
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/references/pr233-layer/SOURCE.json
@@ -0,0 +1,323 @@
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+  "scripts/partial_swap/match_exported_dag.cpp": "7901b0fb1b80b628a24cfd7fda57bbc32f58771594d08b318b90e4b8cdf0ef82",
+  "scripts/partial_swap/match_positive_dag.cpp": "8a6d64e2d0ac2c69210fb0b8a5bd0d714ad40b3609f75a5fddfcba3291a3d552",
+  "scripts/partial_swap/paired.py": "ddcecace3a8e04d690914cbd64873ab8dde8c0c3477a255767b0ed6f639bb0c7",
+  "scripts/partial_swap/positive.py": "1dadf771226d6e4f3bf9ce3aae2e6a2ad7173f4d1c900299d0663fc4c432f3e8",
+  "scripts/partial_swap/shared.py": "81e9af92cdf38243a540cdcff2062778e449467848d444da37a9553c12ca4c34",
+  "scripts/partial_swap_network.py": "acdd81b4dea2815f8838a0c3ea8ac4968f2bc13fd7c99173473d865779096782",
+  "scripts/partial_swap_producer.py": "dba07d742c68e8e4aedd1d93fc4c47a0481778268f7480997463e2b269fdd0a0",
+  "scripts/prepare_layers.py": "a6e2adc612f264b1c0e51d18ce6315de9ae5e2ae2420aa22e99a22d8bf304947",
+  "scripts/prime_field_checks.py": "be21a411cdca5fa331fb9f36352ce98c055cb5e973553255e4530d07e99ae5c4",
+  "scripts/prime_field_circuit.py": "50485ac5a65b2707aab2b870477fbf503f4776699f6dd5ce38fb2006cc95d0f6",
+  "scripts/prime_field_network.py": "16629edd6aa8357b3fabcd5e1a70027860864a64475e8c0e7c822ee81dcd1ab9",
+  "scripts/prime_field_supports.cpp": "927dbb2ac2a411f06d8d9e0c440548f7ab965311fd83f4a573668d94d0cff1fe",
+  "scripts/rank_obstructions.py": "af0bb5d7c21409bf0318ec4ec7d5f391163b7ad9564ed9e35de92ada14bca975",
+  "scripts/rational_span_frames.py": "ac3b9ab462b790d195628f0e5aa4f95143596a51ce7df1a99360b98594c29932",
+  "scripts/rational_tensor.py": "629b3f00a1aaab32fe908a73ca8f8412918209a300e6b1e5afe6325b9cfed3c3",
+  "scripts/research_networks.py": "3db82e1aa5faec2f76bcfd19a996a447048af56b25000af3324884a751e8e864",
+  "scripts/reuse_network.py": "f345f4f015a17497c173ed331aad2133908e2a67d683455ffd60dbb3b3ed1aa1",
+  "scripts/routing_smt.py": "25d2dfb09a34c0c0ecfdf870abe2d3883c95677189836dcce25421b9443e8a44",
+  "scripts/run_isolated_tests.py": "b5c38b4f4f5e967ccda5f95702d18d6f01c18d43f056bb419aaf1628043c6ecc",
+  "scripts/search_incidence_partitions.py": "d2e5a98d07dee702064671d9d081b8dc18e7a1bfeb45e4217905141732abd1bf",
+  "scripts/search_network.py": "3a4faa5037d9b8e7ca5828afac193a96b6966a8b114f14937921795b24ec4f6a",
+  "scripts/search_network_variants.py": "069a80a1ac504e83c5da60f4737440552df1ca4c57218ab2e32523f88199cbbc",
+  "scripts/shared_point_circuit.py": "c9bb7a5df92d260a4e3ad532b53e26936730d7c93f7facfb0784957c1e81c8ba",
+  "scripts/shared_point_network.py": "a85e7f0d8c42d6a27e3846308571ca324a1824ac5c2d59e4daa5c7673c33c697",
+  "scripts/small_block_witness.py": "a8bff53b5e6a13808305cab3d53c2532b1fcb5ff172653d02761b028eb60d79b",
+  "scripts/source_assisted/complex_allocated_replay.py": "831a8a2a008dd9f07af5f7f906947649c8c6faca2bc4ef90d5b0bc1491af24cc",
+  "scripts/source_assisted_verify.py": "2a2ea22dedbf0b7e18e103ca5100887d98bac41dcb7fc4ecdb7cd3f62d037015",
+  "scripts/stopped_product/README.md": "6c1805dda45b01ed0ef420c674ec2cd0048fb9d7a1b015ab6cd837067fe66176",
+  "scripts/stopped_product/__init__.py": "7b794c22bccc00bb6ae8ba4d3379f469570565bbcb1a86090699133b8cf1b7d3",
+  "scripts/stopped_product/complex.py": "407650d56dceb5cb09d74928658cebed1bb42f4d7e28ca9e3fd36eea075f1ad0",
+  "scripts/stopped_product_network.py": "a8baaa87dd7679db4f2dd1a18a720397ee6f443262ad04fa26cf51b39909b74c",
+  "scripts/stopped_product_producer.py": "2c5c71399136ebcfebb8e5277317d5f334eb8df5716ad287391d0996fcc1492c",
+  "scripts/structured_bulk/README.md": "7d6a91d324019628c842db7ef2deb1fa035fbe39dd7c22877c83f190516fc0bf",
+  "scripts/structured_bulk/__init__.py": "9c6d0e9c71caa218263b636359b7d6c88f8428cba3cbd27b9f549045d4609b7d",
+  "scripts/structured_bulk/complex.py": "ee769039ea2111de5488fdc4abba28d50ed8885fa7e8dcd12728c5dbe6170b92",
+  "scripts/structured_bulk/exactness.py": "2e04898ee4463bf4d2f7ad6b50dd0801e4ebe195d06a33ce428ffd1c0beeb67f",
+  "scripts/structured_bulk/rankone_profiles.cpp": "5e003fde6e4779078988ce5ba1adf988cb846afbd8fd44faa920a3baa4126848",
+  "scripts/structured_bulk_assembly.py": "e99467e14211fba699878ed036e1e2362eb8e395dc22f9ecdcecc7fbf2cc736d",
+  "scripts/structured_bulk_network.py": "82608e2e7eb622f3ff78ce9bdc3e6c97087c6785aecd1a0eab947e389856600f",
+  "scripts/structured_bulk_producer.py": "c20e04b5216b04b9343cb28f1b1d164c6ac42ceaac067d82d2850f908dd7d5f6",
+  "scripts/ternary_assembly.py": "38d12b09dd9074ca56cb658e25f4e0b9e50fad9c36d0d837fc5c059ae4ddb5b2",
+  "scripts/three_stage_cover/README.md": "b39a28fa8da5d7d3188dfed4c0e0af1e2397f0f5c8f69cfb69671ed27ddca4bb",
+  "scripts/three_stage_cover/__init__.py": "b964b59af038e70cc15af655a22bcf6aaafc9c842b213765781bc8bea7166430",
+  "scripts/three_stage_cover/lift.py": "ae5f4c56cd32b5232d6dcdcaed1e24efb81fe20467a2d915ee51479bced06cfc",
+  "scripts/three_stage_cover/select.py": "457e636ced9bbd368198f78e015660a929ba1aaccbe34122d18da09ebb488f33",
+  "scripts/three_stage_cover/verify.py": "4bd2347e87e4edb699968bc31dfb94f2ab89bf1227576e59f43c779ba74e4e06",
+  "scripts/three_stage_cover_network.py": "7e0ad6fd1921b6905a53ce1eeaa03d3bb3a8a8bbaa0382516ea98b636f46335e",
+  "scripts/three_stage_cover_producer.py": "ea6dd4f06173304bbcb74b8e5bcc02357b7bb162a6d26b4a6f7a9e886158123f",
+  "scripts/tune_routing.py": "5728c848e23c905f37a9d77928283e64ace5e7b69c3572780c1291800d005e6e"
+ },
+ "note": "The files of PR #202 (commit 8d8d67b: PR #194's source-assisted-v4 pipeline on the PR #189/#200 lineage) that verify.py runs: scripts/, the PR168 v4 references and inherited witnesses, the Clifford frames note, the source-assisted decision/global scripts, the v4 package, and the PR #200 package certificate, arithmetic and complex checker; byte for byte; reproducible with: git archive --format=tar 8d8d67b <paths> | gzip -n -9",
+ "pr": 202,
+ "repository": "CrocSwap/integer-mult-bounds",
+ "witness": {
+  "kernel-pairs.json": "e33b753fc4519170373207b0527b9cb53e0b552e2a27f48f72381a881c7a227f",
+  "physical-frames.json": "1a6c215cdc58216f7ec96fb898a471e25d094fd86de085f6c06e86fd8997bfad",
+  "physical-pairs.json": "e6409fffbf5f3b1ac6434acf2e426503de3a8a18537b8fee761fc4fc931fb896"
+ }
+}
diff --git a/research/w3-p10b-pointwise-complex/references/pr233-layer/baseline-pr202.tar.gz b/research/w3-p10b-pointwise-complex/references/pr233-layer/baseline-pr202.tar.gz
new file mode 100644
index 0000000000..39bdd153c4
Binary files /dev/null and b/research/w3-p10b-pointwise-complex/references/pr233-layer/baseline-pr202.tar.gz differ
diff --git a/research/w3-p10b-pointwise-complex/references/pr233-layer/certificate.json b/research/w3-p10b-pointwise-complex/references/pr233-layer/certificate.json
new file mode 100644
index 0000000000..796efbb0d4
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/references/pr233-layer/certificate.json
@@ -0,0 +1,151 @@
+{
+ "assembly": {
+  "bit_effective_saving": "338670433250477381/500000000000000000000",
+  "complex_saving": "3549537/5000000000",
+  "kappa": "6768823/10000000000",
+  "status": "Exact arithmetic; construction and finite bridge are explicit proof dependencies"
+ },
+ "binding": "bit",
+ "complex_profile": {
+  "W_per_vertex": 12052,
+  "child_histogram": {
+   "1": 87534,
+   "10": 2415,
+   "11": 1059,
+   "12": 3423,
+   "13": 504,
+   "14": 2175,
+   "15": 1815,
+   "16": 243,
+   "17": 117,
+   "18": 7920,
+   "19": 4323,
+   "2": 40641,
+   "20": 66,
+   "3": 29964,
+   "4": 11517,
+   "5": 5928,
+   "6": 4473,
+   "7": 1926,
+   "8": 4434,
+   "9": 1212
+  },
+  "deficit_per_vertex": 1320,
+  "m": 66,
+  "rank_per_vertex": 794112
+ },
+ "complex_saving": "3549537/5000000000",
+ "contract_checks": {
+  "all_centers_in_phase1": true,
+  "all_fresh_columns_equal_identity": true,
+  "all_target_cap_reads_in_phase2": true,
+  "arithmetic": "integers, endpoint scaled by 6",
+  "checked_source_columns": 1320,
+  "checked_target_rows": 1320,
+  "controls_before_original_K": true,
+  "original_source_V_before_all_controls": true,
+  "positive_target_data_histogram": {
+   "1": 3960,
+   "18": 1320
+  },
+  "read_counts": {
+   "center": 22,
+   "deferred": 2310,
+   "side": 3135
+  },
+  "source_controls_at_paid_parity_frames": true,
+  "target_chains_nested": true,
+  "target_data_histogram": {
+   "0": 6270,
+   "1": 3960,
+   "18": 1320
+  }
+ },
+ "flow": {
+  "child_histogram": {
+   "1": 87534,
+   "10": 2415,
+   "11": 1059,
+   "12": 3423,
+   "13": 504,
+   "14": 2175,
+   "15": 1815,
+   "16": 243,
+   "17": 117,
+   "18": 7920,
+   "19": 4323,
+   "2": 40641,
+   "20": 66,
+   "3": 29964,
+   "4": 11517,
+   "5": 5928,
+   "6": 4473,
+   "7": 1926,
+   "8": 4434,
+   "9": 1212
+  },
+  "deficit": 1320,
+  "loss": 440,
+  "new_R": 9412,
+  "new_W": 12052,
+  "original_physical_roles": 13372
+ },
+ "kappa": "6768823/10000000000",
+ "layer": {
+  "audit_source_sha256": {
+   "notes/general-clifford-frames.tex": "c9bb9655b594583c6b4bab8cb485233f1d1e588f83858cf1e4c3530cee8fed54",
+   "research/paired-cube-diagonal-bit-168/complex/physical.py": "01961fad4b04aaa940d2dc99c87690bc861b3861200296974c5e065ce3446659",
+   "research/paired-cube-diagonal-bit-168/complex/verify_fused.py": "75c3dcd80fcdbc461651fa2d2fa463e6b41a569d695fc74700d240223ef198e9",
+   "scripts/paired_cube/frames.py": "7d897418dbfee47de0723ea4983098a982610ccba45e1bac9000a79626d8421c",
+   "scripts/paired_cube_physical.py": "9c9ea1ea29e551ab6116d7c8ca09f4703ff9e2fa102032d5055327fde07ba319"
+  },
+  "audit_status": "PASS exact fused physical candidate, all formal source/target/dirty columns",
+  "formal": {
+   "absolute_coefficient_bound": 603018,
+   "base_exceeds_twice_bound": true,
+   "columns": [
+    {
+     "K_and_inverse": true,
+     "all_aliases_and_deadline_reads": true,
+     "all_physical_dirty_slots_restored": true,
+     "direction": 1,
+     "dirty_columns": 13372,
+     "every_formal_column": true,
+     "source_columns": 1320,
+     "target_columns": 1320
+    },
+    {
+     "K_and_inverse": true,
+     "all_aliases_and_deadline_reads": true,
+     "all_physical_dirty_slots_restored": true,
+     "direction": -1,
+     "dirty_columns": 13372,
+     "every_formal_column": true,
+     "source_columns": 1320,
+     "target_columns": 1320
+    }
+   ],
+   "packed_digit_bits": 24
+  },
+  "frames": 17789,
+  "frames_changed_from_pr168": 0,
+  "mutations": {
+   "early-read": "REJECTED",
+   "missing-read": "REJECTED"
+  },
+  "pairs": 2310
+ },
+ "lift": {
+  "all_actual_coefficients_dyadic": true,
+  "max_denominator": "2",
+  "nodes": 18532,
+  "physical_R": 9412,
+  "status": "Exact rational local lifts and inverse gate programs verified; monotone flow and zero-fresh kernel reuse DAG verified."
+ },
+ "prerequisites": {
+  "pr194_complex_saving": "219037/312500000",
+  "pr200_bit": "research/paired-cube-diagonal-bit-168/certificate.json",
+  "pr202": "8d8d67bcf69c5ea67d3a29dbc64ba588156d6e8d"
+ },
+ "status": "PASS source-assisted v4 word with this package's physical layer"
+}
diff --git a/research/w3-p10b-pointwise-complex/references/pr233-layer/verify.py b/research/w3-p10b-pointwise-complex/references/pr233-layer/verify.py
new file mode 100644
index 0000000000..2ddf0d2f39
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/references/pr233-layer/verify.py
@@ -0,0 +1,223 @@
+#!/usr/bin/env python3
+"""A stronger complex supplier: our physical layer on the source-assisted v4 complex word.
+
+The source-assisted complex word of PR #184 (icekylinx) on the PR #168 v4 modules (PR #193/#194, ikeboy) is
+compiled by a frame flow whose child histogram depends on the physical layer (operation frames and reuse pairs)
+that the flow receives. PR #194 uses PR #168's physical layer and certifies the complex saving 7.00918e-4. This
+package keeps every other input fixed and supplies the physical layer of PR #200's descent (endpoint moves of
+single operations, equal-frame components and connected bundles, seeded from PR #168's frames); the certified
+complex saving becomes 3543063/5000000000 = 7.0861e-4.
+
+verify.py (Python 3.11+, numpy/scipy for the regeneration steps; -O refused):
+  1. checks baseline-pr202.tar.gz and every file in it against SOURCE.json (bytes of PR #202 at 8d8d67b, which
+     carries PR #194's pipeline, the PR #168 v4 scripts and references, and PR #200's certificate and checker),
+     and the three witness files;
+  2. in a temporary directory: regenerates the PR #168 v4 complex cache (scripts/paired_cube_producer.py) and the
+     source-aligned word (source_aligned_local_v4.py, with its own physical layer and checks);
+  3. installs this package's physical frames and reuse pairs as the word's physical layer and admits that layer
+     with PR #200's complete complex checker (complex/physical.py: all descended frames, nested chains, aliases,
+     target read chronology, exact adjoint, every formal source/target/dirty column in both shear signs);
+  4. runs PR #184's frame flow (complex_frame_flow.py), exact lift (exact_complex_flow_lift.py) and contract
+     (contract_v4.py) on that layer, with the package's kernel pairs and physical pairs as the frozen witnesses;
+  5. runs PR #194's assembly (assemble.py) against PR #200's bit certificate and compares the canonical result
+     with certificate.json (--write regenerates it).
+Nothing outside the temporary directory is read or written, except certificate.json with --write.
+"""
+import argparse
+import gzip
+import hashlib
+import io
+import json
+import shutil
+import subprocess
+import sys
+import tarfile
+import tempfile
+import time
+from fractions import Fraction as Q
+from pathlib import Path
+
+if sys.flags.optimize:
+    raise SystemExit('Assertions must remain enabled: run without -O')
+sys.set_int_max_str_digits(0)
+sys.dont_write_bytecode = True
+HERE = Path(__file__).resolve().parent
+T0 = time.time()
+
+
+def log(*a):
+    print('[%4.0fs]' % (time.time() - T0), *a, flush=True)
+
+
+def sha(data):
+    return hashlib.sha256(data).hexdigest()
+
+
+def read(p):
+    raw = Path(p).read_bytes()
+    return json.loads(gzip.decompress(raw) if str(p).endswith('.gz') else raw)
+
+
+def write_gz(p, data):
+    with gzip.GzipFile(filename='', fileobj=open(p, 'wb'), mode='wb', mtime=0) as f:
+        f.write((json.dumps(data, separators=(',', ':')) + '\n').encode())
+
+
+def rebuild(dest):
+    src = read(HERE / 'SOURCE.json')
+    data = (HERE / src['archive']).read_bytes()
+    assert sha(data) == src['archive_sha256'], 'baseline archive differs from SOURCE.json'
+    with tarfile.open(fileobj=io.BytesIO(data), mode='r:gz') as archive:
+        members = [x for x in archive.getmembers() if x.isfile()]
+        assert sorted(x.name for x in members) == sorted(src['files']), 'archive lists other files than SOURCE.json'
+        for x in members:
+            part = Path(x.name)
+            assert not part.is_absolute() and '..' not in part.parts, 'unsafe path in archive: ' + x.name
+            body = archive.extractfile(x).read()
+            assert sha(body) == src['files'][x.name], 'baseline file differs from SOURCE.json: ' + x.name
+            (dest / part).parent.mkdir(parents=True, exist_ok=True)
+            (dest / part).write_bytes(body)
+    for name, digest in src['witness'].items():
+        assert sha((HERE / 'witness' / name).read_bytes()) == digest, 'witness differs from SOURCE.json: ' + name
+    return src
+
+
+def run(root, *args):
+    r = subprocess.run([sys.executable, '-B', *map(str, args)], cwd=root, capture_output=True, text=True)
+    if r.returncode:
+        raise SystemExit('FAILED: ' + ' '.join(map(str, args)) + '\n' + r.stdout[-3000:] + r.stderr[-3000:])
+    return r.stdout
+
+
+def strip_seconds(d):
+    if isinstance(d, dict):
+        return {k: strip_seconds(v) for k, v in d.items() if k not in ('seconds', 'maxrss')}
+    if isinstance(d, list):
+        return [strip_seconds(v) for v in d]
+    return d
+
+
+def main():
+    ap = argparse.ArgumentParser(description=__doc__, formatter_class=argparse.RawDescriptionHelpFormatter)
+    ap.add_argument('--write', action='store_true', help='rewrite certificate.json from this run')
+    ap.add_argument('--temp-root', type=Path, default=None)
+    a = ap.parse_args()
+    with tempfile.TemporaryDirectory(prefix='sa-v4-layer-', dir=a.temp_root) as d:
+        root = Path(d).resolve()
+        src = rebuild(root)
+        log('rebuilt %d pinned files of PR #202 (commit %s) and checked the witnesses' % (len(src['files']), src['commit'][:7]))
+        PKG = root / 'research/source-assisted-v4'
+        SA = root / 'research/source-assisted'
+        BIT = root / 'research/paired-cube-diagonal-bit-168'
+        work = root / '.work'
+        work.mkdir()
+        base, aligned = work / 'base', work / 'aligned'
+        run(root, root / 'scripts/paired_cube_producer.py', '--work-dir', base, '--output', work / 'producer.json')
+        log('regenerated the PR #168 v4 complex cache')
+        run(root, PKG / 'source_aligned_local_v4.py', '--tree', root, '--cache', base, '--out', aligned)
+        log('regenerated the source-aligned word with its own physical layer and checks')
+        # 3. install this package's physical layer and admit it with PR #200's complete complex checker
+        phys = aligned / 'references/paired-cube/physical'
+        frames = read(HERE / 'witness/physical-frames.json')['frames']
+        pairs = read(HERE / 'witness/physical-pairs.json')['pairs']
+        theirs = read(phys / 'frames.json')['frames']
+        (phys / 'frames.json').write_text(json.dumps(dict(frames=frames), separators=(',', ':')) + '\n')
+        (phys / 'pairs.json').write_text(json.dumps(dict(pairs=pairs), separators=(',', ':')) + '\n')
+        sys.path.insert(0, str(root / 'scripts'))
+        from paired_cube.closure import compile_closure            # noqa: E402
+        from paired_cube_physical import physical                  # noqa: E402
+        cache = aligned / 'cache'
+        g, w, word, row = (read(cache / n) for n in ('graph.json', 'frames.json', 'selection.json', 'record.json'))
+        baseline, _ = compile_closure(g, [[x, c] for x, c in w['matching_arcs']])
+        profile = physical(g, w, word, row, frames, pairs)
+        cand = work / 'candidate'
+        cand.mkdir()
+        for name, data in [('graph', g), ('baseline', baseline), ('frames', w), ('word', word), ('profile-before', row),
+                           ('physical-frames', frames), ('physical-pairs', pairs), ('profile', profile)]:
+            write_gz(cand / (name + '.json.gz'), data)
+        pins = {p.name: sha(p.read_bytes()) for p in cand.glob('*.json.gz')}
+        (cand / 'result.json').write_text(json.dumps(dict(tag='source-assisted-v4-layer', pins=pins)) + '\n')
+        run(root, BIT / 'complex/physical.py', '--candidate', cand, '--source', root, '--out', work / 'audit.json')
+        audit = read(work / 'audit.json')
+        assert audit['status'].startswith('PASS'), audit['status']
+        # the checker keys its source hashes by absolute path; record them relative to the rebuilt tree
+        audit['source_sha256'] = {Path(k).resolve().relative_to(root).as_posix(): v for k, v in audit['source_sha256'].items()}
+        log('admitted this package\'s physical layer with PR #200\'s complex checker: %s; %d frames, %d pairs, %d of PR #168\'s %d frames changed'
+            % (audit['status'][:60], len(frames), len(pairs), sum(1 for (i, F), (j, G) in zip(sorted(frames), sorted(theirs)) if tuple(F) != tuple(G)), len(theirs)))
+        # 4. frame flow, exact lift and contract on this layer
+        flow = work / 'flow.json'
+        run(root, SA / 'decision/complex_frame_flow.py', '--tree', aligned, '--cache', cache, '--out', flow, '--witness',
+            '--purify-source-donors', '--recycle-kernels', '--kernel-pairs', HERE / 'witness/kernel-pairs.json')
+        flow_data = read(flow)
+        witness = flow.with_suffix('.witness.json')
+        log('frame flow: new_R %d, new_W %d, deficit %d, numerical root %.7e' % (flow_data['new_R'], flow_data['new_W'], flow_data['deficit'], flow_data['numerical_local_root']))
+        lift = work / 'lift.json'
+        run(root, SA / 'decision/exact_complex_flow_lift.py', '--witness', witness, '--profile', flow, '--out', lift)
+        lift_data = read(lift)
+        lift_data.update(certificate_path=(work / 'lift.certificate.json.gz').relative_to(root).as_posix(),
+                         witness_path=witness.relative_to(root).as_posix())
+        lift.write_text(json.dumps(lift_data) + '\n')
+        log('exact lift: %s' % lift_data['status'][:80])
+        shutil.copy2(HERE / 'witness/physical-pairs.json', PKG / 'data/physical-pairs.json')
+        shutil.copy2(HERE / 'witness/kernel-pairs.json', PKG / 'data/kernel-pairs.json')
+        cprof = work / 'complex-profile.json'
+        run(root, PKG / 'contract_v4.py', '--tree', aligned, '--cache', cache, '--witness', witness,
+            '--flow-profile', flow, '--lift-profile', lift, '--out', cprof)
+        cdata = read(cprof)
+        for name, check in cdata['contract_checks'].items():
+            assert check not in (False, 'FAIL'), 'contract check %s failed' % name
+        log('contract: %d checks, W %d, rank %d, deficit %d' % (len(cdata['contract_checks']), cdata['W_per_vertex'], cdata['rank_per_vertex'], cdata['deficit_per_vertex']))
+        # 5. assembly against PR #200's bit certificate
+        final = work / 'global.json'
+        run(root, PKG / 'assemble.py', '--complex', cprof, '--output', final)
+        fdata = read(final)
+        for branch in ('complex', 'bit'):
+            fdata[branch].pop('numerical_root_for_discovery_only', None)
+        complex_saving = Q(fdata['complex']['saving'])
+        kappa = Q(fdata['kappa'])
+        log('assembly: complex saving %s = %.7e, bit effective %.7e, kappa %s = %.7e' % (
+            complex_saving, float(complex_saving), float(Q(fdata['bit']['effective_saving'])), kappa, float(kappa)))
+        result = dict(status='PASS source-assisted v4 word with this package\'s physical layer',
+                      complex_saving=str(complex_saving), complex_saving_float=float(complex_saving),
+                      kappa=str(kappa), kappa_float=float(kappa), binding='bit' if Q(fdata['bit']['effective_saving']) < complex_saving else 'complex',
+                      complex_profile=dict(m=cdata['m'], W_per_vertex=cdata['W_per_vertex'], rank_per_vertex=cdata['rank_per_vertex'],
+                                           deficit_per_vertex=cdata['deficit_per_vertex'], child_histogram=cdata['child_histogram']),
+                      layer=dict(frames=len(frames), pairs=len(pairs), audit=strip_seconds(audit)),
+                      flow=strip_seconds({k: v for k, v in flow_data.items() if k not in ('source_sha256', 'cache_sha256')}),
+                      lift={k: lift_data[k] for k in ('status', 'nodes', 'physical_R', 'all_actual_coefficients_dyadic', 'max_denominator', 'exact_scalar_program_sha256')},
+                      contract_checks=cdata['contract_checks'], assembly=strip_seconds(fdata),
+                      prerequisites=dict(pr202=src['commit'], pr194_complex_saving='219037/312500000', pr200_bit='research/paired-cube-diagonal-bit-168/certificate.json'))
+        # certificate.json holds exact, platform-independent fields only (no floats, no hashes of generated gzip
+        # artifacts); the full run record goes to report.json (informational, written only with --write)
+        canon = dict(status=result['status'], complex_saving=result['complex_saving'], kappa=result['kappa'], binding=result['binding'],
+                     complex_profile=result['complex_profile'],
+                     layer=dict(frames=len(frames), pairs=len(pairs), frames_changed_from_pr168=sum(1 for (i, F), (j, G) in zip(sorted(frames), sorted(theirs)) if tuple(F) != tuple(G)),
+                                audit_status=audit['status'], audit_source_sha256=audit['source_sha256'],
+                                formal=json.loads(json.dumps(audit.get('formal'))), mutations=audit.get('mutations')),
+                     flow=dict(new_R=flow_data['new_R'], new_W=flow_data['new_W'], deficit=flow_data['deficit'], loss=flow_data['loss'],
+                               original_physical_roles=flow_data['original_physical_roles'], child_histogram=flow_data['child_histogram']),
+                     lift=dict(status=lift_data['status'], nodes=lift_data['nodes'], physical_R=lift_data['physical_R'],
+                               all_actual_coefficients_dyadic=lift_data['all_actual_coefficients_dyadic'], max_denominator=lift_data['max_denominator']),
+                     contract_checks=cdata['contract_checks'],
+                     assembly=dict(kappa=fdata['kappa'], complex_saving=fdata['complex']['saving'], bit_effective_saving=fdata['bit']['effective_saving'], status=fdata.get('status')),
+                     prerequisites=result['prerequisites'])
+        canon = json.loads(json.dumps(canon, sort_keys=True))
+        if a.write:
+            (HERE / 'certificate.json').write_text(json.dumps(canon, indent=1, sort_keys=True) + '\n')
+            (HERE / 'report.json').write_text(json.dumps(json.loads(json.dumps(result, sort_keys=True)), indent=1, sort_keys=True) + '\n')
+            log('wrote certificate.json and report.json')
+        committed = read(HERE / 'certificate.json')
+        if committed != canon:
+            for k in sorted(set(committed) | set(canon)):
+                if committed.get(k) != canon.get(k):
+                    log('DIFFERS: %s' % k)
+                    if isinstance(canon.get(k), dict):
+                        for kk in sorted(set(committed.get(k, {})) | set(canon[k])):
+                            if committed.get(k, {}).get(kk) != canon[k].get(kk):
+                                log('  DIFFERS: %s/%s: %s vs %s' % (k, kk, str(committed.get(k, {}).get(kk))[:80], str(canon[k].get(kk))[:80]))
+        assert committed == canon, 'result differs from certificate.json'
+        log('PASS certificate.json reproduced: complex saving %s (PR #194: 219037/312500000), kappa %s' % (complex_saving, kappa))
+
+
+if __name__ == '__main__':
+    main()
diff --git a/research/w3-p10b-pointwise-complex/references/pr233-layer/witness/kernel-pairs.json b/research/w3-p10b-pointwise-complex/references/pr233-layer/witness/kernel-pairs.json
new file mode 100644
index 0000000000..e889b7b95c
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/references/pr233-layer/witness/kernel-pairs.json
@@ -0,0 +1 @@
+{"description":"Kernel pairs of the frame flow on this layer (from complex_frame_flow.py --witness)","kernel_pairs":[]}
\ No newline at end of file
diff --git a/research/w3-p10b-pointwise-complex/references/pr233-layer/witness/physical-frames.json b/research/w3-p10b-pointwise-complex/references/pr233-layer/witness/physical-frames.json
new file mode 100644
index 0000000000..9044957452
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/references/pr233-layer/witness/physical-frames.json
@@ -0,0 +1 @@
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diff --git a/research/w3-p10b-pointwise-complex/references/pr233-layer/witness/physical-pairs.json b/research/w3-p10b-pointwise-complex/references/pr233-layer/witness/physical-pairs.json
new file mode 100644
index 0000000000..a582ff3bea
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/references/pr233-layer/witness/physical-pairs.json
@@ -0,0 +1 @@
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792],[8567,15479,33793],[8564,15480,33794],[13267,15481,33796],[12407,15482,33797],[11521,15483,33798],[11791,15484,33803],[5616,15485,33804],[11541,15486,33805],[11809,15487,33807],[12871,15488,33808],[11854,15489,33809],[2928,15490,33811],[11718,15491,33812],[8955,15492,33813],[11596,15493,33816],[11393,15494,33817],[11305,15495,33818],[12446,15496,33821],[12512,15497,33822],[8491,15498,33823],[11149,15499,33825],[11303,15500,33826],[12865,15501,33827],[10028,15502,33829],[9480,15503,33830],[9478,15504,33831],[8546,15505,33835],[4012,15506,33836],[12495,15507,33837],[2976,15508,33841],[11435,15509,33842],[8986,15510,33843],[11831,15511,33845],[11520,15512,33846],[11920,15513,33847],[11659,15514,33850],[12399,15515,33851],[11865,15516,33852],[11604,15517,33854],[11545,15518,33855],[12444,15519,33856],[12417,15520,33858],[11789,15521,33859],[12479,15522,33860],[12445,15523,33863],[11098,15524,33864],[8947,15525,33865],[11439,15526,33867],[11441,15527,33868],[11437,15528,33869],[11101,15529,33872],[11352,15530,33873],[11355,15531,33874],[11624,15532,33877],[11539,15533,33878],[11494,15534,33879],[11961,15535,33884],[11641,15536,33885],[12391,15537,33886],[12471,15538,33888],[11868,15539,33889],[11847,15540,33890],[8803,15541,33893],[11042,15542,33894],[11040,15543,33895],[11825,15544,33898],[8818,15545,33899],[8815,15546,33900],[4289,15547,33903],[3989,15548,33904],[3988,15549,33905],[11630,15550,33907],[12460,15551,33908],[12472,15552,33909],[11546,15553,33911],[11664,15554,33912],[11650,15555,33913],[8943,15556,33918],[12402,15557,33919],[8989,15558,33920],[10703,15559,33922],[10399,15560,33923],[10397,15561,33924],[9435,15562,33926],[9433,15563,33927],[12867,15564,33928],[11845,15565,33931],[9987,15566,33932],[9985,15567,33933],[13107,15568,33936],[11636,15569,33937],[11632,15570,33938],[7488,15571,33941],[11614,15572,33942],[11593,15573,33943],[11737,15574,33945],[13110,15575,33946],[11775,15576,33947],[11633,15577,33949],[11757,15578,33950],[11761,15579,33951],[2400,15580,33953],[11804,15581,33954],[11502,15582,33955],[13055,15583,33958],[11780,15584,33959],[13163,15585,33960],[11788,15586,33963],[11736,15587,33964],[13266,15588,33965],[12365,15589,33967],[12416,15590,33968],[11634,15591,33969],[11740,15592,33974],[11748,15593,33975],[13108,15594,33976],[11792,15595,33978],[11661,15596,33979],[6384,15597,33980],[7200,15598,33982],[11544,15599,33983],[12864,15600,33984],[7056,15601,33987],[11642,15602,33988],[11777,15603,33989],[12476,15604,33992],[11527,15605,33993],[11755,15606,33994],[2736,15607,33996],[11873,15608,33997],[11859,15609,33998],[12903,15610,34000],[12485,15611,34001],[11628,15612,34002],[12366,15613,34006],[2304,15614,34007],[11818,15615,34008],[11489,15616,34012],[13277,15617,34013],[816,15618,34014],[11765,15619,34016],[11763,15620,34017],[13102,15621,34018],[13153,15622,34021],[11800,15623,34022],[13234,15624,34023],[11512,15625,34025],[11743,15626,34026],[11525,15627,34027],[11898,15628,34029],[11734,15629,34030],[13157,15630,34031],[11817,15631,34034],[11668,15632,34035],[12401,15633,34036],[12845,15634,34038],[12854,15635,34039],[12917,15636,34040],[11639,15637,34043],[1536,15638,34044],[11754,15639,34045],[11629,15640,34048],[11519,15641,34049],[11975,15642,34050],[12421,15643,34055],[11776,15644,34056],[13124,15645,34057],[12383,15646,34059],[11773,15647,34060],[11745,15648,34061],[11760,15649,34064],[288,15650,34065],[3408,15651,34066],[13089,15652,34069],[11767,15653,34070],[5856,15654,34071],[13281,15655,34074],[11742,15656,34075],[12852,15657,34076],[11948,15658,34078],[11783,15659,34079],[13095,15660,34080],[624,15661,34082],[11501,15662,34083],[11733,15663,34084],[11608,15664,34089],[3024,15665,34090],[12423,15666,34091],[11739,15667,34093],[12386,15668,34094],[11746,15669,34095],[11867,15670,34097],[11547,15671,34098],[11957,15672,34099],[11851,15673,34102],[13117,15674,34103],[11655,15675,34104],[12905,15676,34107],[11706,15677,34108],[11626,15678,34109],[12901,15679,34112],[720,15680,34113],[11802,15681,34114]]}
diff --git a/research/w3-p10b-pointwise-complex/references/sussman/pr193-blocks.json b/research/w3-p10b-pointwise-complex/references/sussman/pr193-blocks.json
new file mode 100644
index 0000000000..aeb5da69b2
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/references/sussman/pr193-blocks.json
@@ -0,0 +1,49 @@
+{
+ "N": 262944,
+ "R": 9412,
+ "blocks": {
+  "c": {
+   "20": 22
+  },
+  "s": {
+   "1": 23898,
+   "10": 930,
+   "11": 354,
+   "12": 1197,
+   "13": 127,
+   "14": 742,
+   "15": 1879,
+   "16": 88,
+   "17": 81,
+   "18": 13,
+   "19": 167,
+   "2": 11078,
+   "3": 9923,
+   "4": 4209,
+   "5": 1950,
+   "6": 1605,
+   "7": 706,
+   "8": 1531,
+   "9": 429
+  },
+  "x": {
+   "1": 1320,
+   "18": 1320,
+   "2": 1320
+  },
+  "y": {
+   "1": 3960,
+   "18": 1320
+  }
+ },
+ "cst": 440,
+ "file_sha256": "15251deb0c11e0504d7b297c9786818888a93f08b92b03c6c122784b70488633",
+ "frames": 18248,
+ "gates_A": 12158,
+ "gates_B": 30878,
+ "h": 22,
+ "source": "jacobalansussman/wht-power-saving-lean tools/certificate/gcert1-p11-pr193.json.gz at commit f010392c (Jacob Sussman, Apache-2.0): the rebuilt paired-cube circuit of PR #193",
+ "uncompressed_sha256": "c64d2a95431c3cf24452de7391cce780bd71521da44ce23923ebb235f6c7962c",
+ "v": 1320,
+ "whole_block_saving_lean": "7474547/10^10 (Sussman, proved in Lean; gxdry.py: 7474549 holds, 7474550 fails)"
+}
\ No newline at end of file
diff --git a/research/w3-p10b-pointwise-complex/scripts/moment.py b/research/w3-p10b-pointwise-complex/scripts/moment.py
new file mode 100644
index 0000000000..11c622d33c
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/scripts/moment.py
@@ -0,0 +1,45 @@
+#!/usr/bin/env python3
+"""Exact upper bound of the batched tensor moment  sum_rho n_rho (rho/m)^theta,  theta = 1 - a.
+
+Adapted from scripts/verify_moment.py of eumemic/exact-dft-bounds (eumemic, Claude assistance, Apache-2.0),
+commit 6f87d1a0c9ae462cdcb2034a1361c87623221e66, which this package credits for the batched tensor recursion
+(Theorem 3.1 there) and these bounds:
+    (rho/m)^theta = (rho/m) * exp(a * ln(m/rho)),
+    ln(x) <= 2*sum_{j<J} y^(2j+1)/(2j+1) + 2*y^(2J+1)/((2J+1)(1-y^2)),  y = (x-1)/(x+1),
+    exp(x) <= 1 + x + x^2/(2(1 - x/3))  for 0 <= x < 3.
+All arithmetic is exact rational; the result is an upper bound of the moment, so a positive margin is a proof.
+"""
+from fractions import Fraction
+
+
+def ln_upper(x, terms=4000):
+    assert x >= 1
+    y = (x - 1) / (x + 1)
+    y2 = y * y
+    grid = 10**60
+    s = 0
+    p = y
+    for j in range(terms):
+        k = 2 * j + 1
+        t = p / k
+        s += -((-t.numerator * grid) // t.denominator)
+        p *= y2
+        if p < Fraction(1, 10**70):
+            break
+    k = 2 * (j + 1) + 1
+    tail = 2 * p / (k * (1 - y2))
+    return Fraction(2 * s, grid) + tail
+
+
+def exp_upper(x):
+    assert 0 <= x < 3
+    return 1 + x + x * x / (2 * (1 - x / 3))
+
+
+def moment_margin(m, W, hist, a):
+    """W minus an exact upper bound of sum_rho n_rho (rho/m)^(1-a); positive means the moment is below W."""
+    assert all(1 <= r < m for r in hist), 'every child must be strictly narrower than a role'
+    total = Fraction(0)
+    for r, n in sorted(hist.items()):
+        total += n * Fraction(r, m) * exp_upper(a * ln_upper(Fraction(m, r)))
+    return W - total, total
diff --git a/research/w3-p10b-pointwise-complex/scripts/source527/NOTICE b/research/w3-p10b-pointwise-complex/scripts/source527/NOTICE
new file mode 100644
index 0000000000..520559625d
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/scripts/source527/NOTICE
@@ -0,0 +1,21 @@
+scripts/source527/moment.py and scripts/source527/base_two_moment.py are vendored byte for byte from
+CrocSwap/integer-mult-bounds, research/five-stage-source527-banks/ at commit
+b07cafe23202d22d832961a04b07d48d9b245e10 ("Import audited source527 checkpoint from PR210", eumemic).
+
+sha256
+  9d465c716c6b910ff66e1453e3bf0057608a787b5f5361be93781f42cb3370f7  moment.py
+  aee70c242a2f1d225528fabe2002e1564d1dc884797c07d28a831c7878ee9098  base_two_moment.py
+
+Licence: Apache-2.0 (that package's LICENSE). Attribution as stated in that package's NOTICE.md: "The source527
+construction builds on eumemic's PR210 source reuse and target transforms, sennemmi's PR230 operation-frame changes,
+Dugongue's PR216 and PR207, Rohan Arun's PR211, Chafik Boukhalfa's PR200, Evan McKinney's completed bank work, and
+their retained dependencies ... prepared by eumemic with substantial OpenAI Codex assistance. All earlier contributor
+notices and assistance disclosures remain in the supplied source material." The five-stage architecture is Henry
+Grant / hcg890's PR234.
+
+Only moment.moment, moment.certify, moment.decimal and base_two_moment.moment are called (verify.py, coarse()): two
+independent exact rational enclosures of the normalized five-stage moment, the first with the inherited 10^-16
+bad-class fallback envelope (32 m^2 rank-one calls per paid child) as its `bad` option, the second without it (the
+envelope is added as a separate rank-one term, as PR315's math_check.py does). moment.price (p = 12 constants) is
+not called. Both files are unchanged; PR315 vendors the same base_two_moment.py and a moment.py whose only change is
+in price().
diff --git a/research/w3-p10b-pointwise-complex/scripts/source527/base_two_moment.py b/research/w3-p10b-pointwise-complex/scripts/source527/base_two_moment.py
new file mode 100644
index 0000000000..14a90fa0e7
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/scripts/source527/base_two_moment.py
@@ -0,0 +1,63 @@
+#!/usr/bin/env python3
+"""Maintainer arithmetic cross-check of the pinned PR39 certificate.
+
+Uses no producer/checker imports. Counts are inputs, not proved by this script.
+80-term rational logarithms and 12-term exponentials independently enclose
+the two moments. The accompanying written review addresses their realization.
+"""
+import argparse
+from fractions import Fraction as Q
+from hashlib import sha256
+import json
+from pathlib import Path
+
+ROOT = Path(__file__).resolve().parents[1]
+GRID = 10**30
+
+
+def require(condition, message):
+    if not condition:
+        raise ValueError(message)
+
+
+def log_unit(x):
+    require(1 <= x <= 2, "logarithm range")
+    z = (x-1)/(x+1)
+    lo = 2*sum((z**(2*j+1)/(2*j+1) for j in range(80)), Q(0))
+    return lo, lo+2*z**161/(161*(1-z*z))
+
+
+def log_bounds(x):
+    k = 0
+    while x > 2:
+        x /= 2
+        k += 1
+    lo, hi = log_unit(x)
+    l2, u2 = log_unit(Q(2))
+    lo, hi = lo+k*l2, hi+k*u2
+    # Outward rational rounding keeps later certificates compact.
+    return Q((lo*GRID).__floor__(), GRID), Q((hi*GRID).__ceil__(), GRID)
+
+
+def exp_bounds(x):
+    require(0 <= x < 1, "exponential range")
+    term = total = Q(1)
+    for j in range(1, 13):
+        term *= x/j
+        total += term
+    tail = term*x/13/(1-x/14)
+    return total, total+tail
+
+
+def moment(m, W, entries, saving):
+    lo = hi = Q(0)
+    for t, count in entries:
+        t, count = int(t), int(count)
+        require(0 < t < m and count > 0, "improper child")
+        l, u = log_bounds(Q(m, t))
+        lower = exp_bounds(saving*l)[0]
+        upper = exp_bounds(saving*u)[1]
+        lo += Q(t*count, m*W)*lower
+        hi += Q(t*count, m*W)*upper
+    return Q((lo*GRID).__floor__(), GRID), Q((hi*GRID).__ceil__(), GRID)
+
diff --git a/research/w3-p10b-pointwise-complex/scripts/source527/moment.py b/research/w3-p10b-pointwise-complex/scripts/source527/moment.py
new file mode 100644
index 0000000000..1e43aaeaca
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/scripts/source527/moment.py
@@ -0,0 +1,114 @@
+"""Independent prospective ordinary five-stage cost, never supplier admission.
+
+Consumes a raw one-invocation ledger made from the actual source/internal/target
+paths.  No packed profile is used as an input.  Missing physical/finite/compiler
+obligations must remain explicit in the accompanying report.
+"""
+from collections import Counter
+from decimal import Decimal, localcontext, ROUND_FLOOR
+from fractions import Fraction as F
+from functools import lru_cache
+from math import factorial
+import hashlib, json, sys
+from pathlib import Path
+
+DYAD = 1 << 224
+
+def floor(x): return F(x.numerator*DYAD//x.denominator, DYAD)
+def ceil(x): return -floor(-x)
+
+@lru_cache(None)
+def logarithm(x):
+    """Mercator bounds after binary reduction; ln(2) via two Mercator sums.
+
+    For 1 <= y < 2, log(y) is an alternating series in z=y-1.
+    We instead reduce to 1 <= y < 3/2, and bound log(3/2) and
+    log(2) by the same alternating series at z=1/2 and z=1/3,
+    using log(2)=log(3/2)+log(4/3). All sums are exact rationals.
+    """
+    assert x >= 1
+    k = 0
+    while x >= 2: x /= 2; k += 1
+    j = int(x >= F(3, 2))
+    if j: x /= F(3, 2)
+    def mercator(y):
+        z=y-1
+        assert 0 <= z <= F(1,2)
+        # An even partial sum is below log; the next term bounds its tail.
+        t=z; s=F(0)
+        for n in range(1, 225):
+            s += (t if n & 1 else -t)/n
+            t *= z
+        return s, s+t/225
+    a,b=mercator(x); c,d=mercator(F(3,2)); e,f=mercator(F(4,3))
+    return floor(a+(k+j)*c+k*e), ceil(b+(k+j)*d+k*f)
+
+def exponential(l,u):
+    assert 0 <= l <= u < F(1,10)
+    def poly(x):
+        s=t=F(1)
+        for j in range(1,25): t=t*x/j; s+=t
+        return s,t
+    lo,_=poly(l);hi,last=poly(u)
+    return floor(lo),ceil(hi+last*u/25/(1-u/26))
+
+def moment(H,m,W,a,bad=False):
+    lo=hi=F(0)
+    for r,n in H.items():
+        assert 0 < r < m and n > 0
+        l,u=logarithm(F(m,r)); e,f=exponential(a*l,a*u)
+        p=F(r*n,m*W);lo+=p*e;hi+=p*f
+    if bad:
+        # Existing envelope shape, only conditionally portable to the new cover.
+        l,u=logarithm(F(m));e,f=exponential(a*l,a*u)
+        p=F(32*m*sum(H.values()),10**16*W)
+        lo+=p*e;hi+=p*f
+    return floor(lo),ceil(hi)
+
+def decimal(x):
+    with localcontext() as c:
+        c.prec=70
+        return str(Decimal(x.numerator)/Decimal(x.denominator))
+
+def certify(H,m,W,bad=False):
+    with localcontext() as c:
+        c.prec=70
+        terms=[(Decimal(r*n)/Decimal(m*W),(Decimal(m)/Decimal(r)).ln()) for r,n in H.items()]
+        fallback=Decimal(32*m*sum(H.values()))/Decimal(10**16*W) if bad else Decimal(0)
+        lm=Decimal(m).ln();lo=Decimal(0);hi=Decimal('.003')
+        for _ in range(210):
+            a=(lo+hi)/2
+            if sum(p*(a*l).exp() for p,l in terms)+fallback*(a*lm).exp()<1:lo=a
+            else:hi=a
+        q=int(((lo+hi)/2*10**24).to_integral_value(rounding=ROUND_FLOOR))
+    lo=F(q,10**24);hi=lo+F(1,10**24)
+    lv=moment(H,m,W,lo,bad);hv=moment(H,m,W,hi,bad)
+    assert lv[1]<1<hv[0],(lo,hi,lv,hv)
+    k=F(694782719690783,10**18);required=k/(1-k)
+    at=moment(H,m,W,required,bad)
+    return dict(lower=decimal(lo),upper=decimal(hi),lower_endpoint_residual_upper=decimal(lv[1]-1),upper_endpoint_residual_lower=decimal(hv[0]-1),strict_rational_signs=True,moment_minus_one_at_current_k_required_rate=[decimal(x-1) for x in at],nominal_k_ceiling_interval=[decimal(lo/(1+lo)),decimal(hi/(1+hi))])
+
+def price(core,gauges,R,v=1760,h=24):
+    core=Counter({int(k):n for k,n in core.items() if n})
+    gauges=Counter({int(k):n for k,n in gauges.items() if n})
+    m=5*h;W=4*v+R;ell=24*22
+    assert sum(r*n for r,n in core.items())==h*R+2*v*(h-1)+ell-sum(r*n for r,n in gauges.items())
+    idle=Counter({r:2*v for r in (2*h-2,h-1,2*h+2,4)})
+    ans={}
+    for name,bundled in [('separate_rank_a_corrections',False),('bundled_rank_5a_corrections',True)]:
+        H=Counter({r:5*n for r,n in core.items()});H.update(idle)
+        for a,n in gauges.items():H[5*a if bundled else a]+=n if bundled else 5*n
+        mass=sum(r*n for r,n in H.items());assert m*W-mass==4*v-5*ell==4400
+        ans[name]=dict(histogram=dict(sorted(H.items())),calls=sum(H.values()),rank_mass=mass,deficit=m*W-mass,maxchild=max(H),root=certify(H,m,W),root_with_inherited_bad_envelope=certify(H,m,W,True))
+    return dict(status='PROSPECTIVE_COST_ONLY_NOT_PHYSICAL_OR_ASSEMBLY_ADMISSION',m=m,W=W,R=R,v=v,h=h,copy_calls_per_invocation=24,copy_rank=22,ell=ell,raw_core=dict(sorted(core.items())),auxiliary_entrances=dict(sorted(gauges.items())),idle_histogram=dict(sorted(idle.items())),profiles=ans,interval_method='Independent exact rational alternating Mercator log bounds (224 terms), 24-degree exponential with geometric tail, 224-bit outward dyadic rounding; Decimal only proposes endpoints.',bad_envelope_scope='10^-16 and 32*m^2 rank-one fallback calls per paid child are inherited assumptions, not a new m120 admission.')
+
+if __name__=='__main__':
+    path=Path(sys.argv[1]);raw=path.read_bytes();data=json.loads(raw)
+    core=data['one_stage_helper_histogram_including_copies']
+    gauges=data['auxiliary_entrance_rank_histogram']
+    result=price(core,gauges,data['physical_R'])
+    result['input_ledger_sha256']=hashlib.sha256(raw).hexdigest()
+    result['input_head']=data['head']
+    result['one_invocation_parts']={key:data[key] for key in ('physical_internal_excluding_center_copies','paid_center_copy_histogram','physical_source_histogram','physical_target_histogram')}
+    result['source_owned_gauges_not_banked']={19:3,2:26,4:2}
+    print(json.dumps(result,indent=2))
diff --git a/research/w3-p10b-pointwise-complex/spec/c2.json b/research/w3-p10b-pointwise-complex/spec/c2.json
new file mode 100644
index 0000000000..d6efa4020c
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/spec/c2.json
@@ -0,0 +1,36 @@
+{
+ "name": "p10-b325-c2",
+ "p": 10,
+ "centre46": true,
+ "orientation": "sorted",
+ "recipe": "mw",
+ "tmod": "modules/tmod_t2_p10.json",
+ "pmod": "modules/pm_p10.json",
+ "qmod": "modules/qmod_p10.json",
+ "pmods": {
+  "0,0": "modules/pointwise/pmod_i0.json",
+  "0,1": "modules/pointwise/pmod_i0.json",
+  "1,0": "modules/pointwise/pmod_i1.json",
+  "1,1": "modules/pointwise/pmod_i1.json",
+  "2,0": "modules/pointwise/pmod_i2.json",
+  "2,1": "modules/pointwise/pmod_i2.json",
+  "3,0": "modules/pointwise/pmod_i3.json",
+  "3,1": "modules/pointwise/pmod_i3.json",
+  "4,0": "modules/pointwise/pmod_i4.json",
+  "4,1": "modules/pointwise/pmod_i4.json",
+  "5,0": "modules/pointwise/pmod_i5.json",
+  "5,1": "modules/pointwise/pmod_i5.json",
+  "6,0": "modules/pointwise/pmod_i6.json",
+  "6,1": "modules/pointwise/pmod_i6.json",
+  "7,0": "modules/pointwise/pmod_i7.json",
+  "7,1": "modules/pointwise/pmod_i7.json",
+  "8,0": "modules/pointwise/pmod_i8.json",
+  "8,1": "modules/pointwise/pmod_i8.json",
+  "9,0": "modules/pointwise/pmod_i9.json",
+  "9,1": "modules/pointwise/pmod_i9.json"
+ },
+ "qmods": {},
+ "layer": "layers/c2",
+ "certificate": "certificates/gcert1-p10-b325-c2.json.gz",
+ "claim": "7940/10^7"
+}
diff --git a/research/w3-p10b-pointwise-complex/spec/uniform.json b/research/w3-p10b-pointwise-complex/spec/uniform.json
new file mode 100644
index 0000000000..803eef432b
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/spec/uniform.json
@@ -0,0 +1,16 @@
+{
+ "name": "uniform",
+ "p": 10,
+ "centre46": true,
+ "orientation": "sorted",
+ "recipe": "mw",
+ "tmod": "modules/tmod_t2_p10.json",
+ "pmod": "modules/pm_p10.json",
+ "qmod": "modules/qmod_p10.json",
+ "pmods": {},
+ "qmods": {},
+ "note": "PR #325 modules, one per kind (the Lean-checked 7932/10^7 program of the same recipe); used only for the builder anchors",
+ "layer": "layers/uniform",
+ "certificate_sha256": "56e1139aa4a420219d37b9af00d8894d4c21df83b3d9c266d36a787676e9ad4e",
+ "claim": "7932/10^7"
+}
diff --git a/research/w3-p10b-pointwise-complex/spec_graph.py b/research/w3-p10b-pointwise-complex/spec_graph.py
new file mode 100644
index 0000000000..3c60099054
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/spec_graph.py
@@ -0,0 +1,103 @@
+"""spec_graph.py: the per-invocation module spec of the p = 10 complex word, and the centre-sharing graph it builds.
+
+A spec names the triple module (tmod), a default pair module with the star centre as its last root (pmod, 37 roots
+at p = 10; centre46.load_pmod46) and a default all-but-one module (qmod), and may override them per invocation:
+  pmods {"i,bit": path}        the pair module of invocation (i, bit), i < p, bit in {0, 1}   (2p invocations)
+  qmods {"i,j,mode": path}     the all-but-one module of invocation (i, j, mode), i < j < p   (p(p-1) invocations)
+expand() resolves the spec into one module file per invocation. finish_spec() is centre46.finish_centre46 with
+the module of each invocation looked up in those tables (same node order, same roots, same centres); with one
+module per kind it builds finish_centre46's graph, which verify.py checks field by field (anchor (e)) and through
+the written word files (anchor (f)); the status string is finish_centre46's.
+Cube orientation is the sorted one throughout (position k of cube {a < b < c} is its k-th element).
+Prepared by Chafik Boukhalfa with substantial Anthropic Claude assistance; Apache-2.0. finish_centre46 is
+DreamingOfClouds' (PR #327, from PR #315), Graph.finish is icekylinx's (PR #144)."""
+from itertools import combinations, product
+from pathlib import Path
+
+
+def expand(spec, base):
+    """{('p', i, bit): Path, ('q', i, j, mode): Path} for every invocation of the p-cube word"""
+    P = spec['p']
+    base = Path(base)
+    out = {}
+    extra = set(spec.get('pmods', {})) - {'%d,%d' % (i, b) for i in range(P) for b in range(2)}
+    extra |= set(spec.get('qmods', {})) - {'%d,%d,%d' % (i, j, m) for i, j in combinations(range(P), 2) for m in range(2)}
+    assert not extra, 'spec names invocations that do not exist: %s' % sorted(extra)
+    for i in range(P):
+        for b in range(2):
+            out['p', i, b] = base / spec.get('pmods', {}).get('%d,%d' % (i, b), spec['pmod'])
+    for i, j in combinations(range(P), 2):
+        for m in range(2):
+            out['q', i, j, m] = base / spec.get('qmods', {}).get('%d,%d,%d' % (i, j, m), spec['qmod'])
+    return out
+
+
+def shared(table):
+    """the single module file per kind if every invocation of that kind uses the same file, else None"""
+    res = {}
+    for kind in ('p', 'q'):
+        files = {f for k, f in table.items() if k[0] == kind}
+        res[kind] = files.pop() if len(files) == 1 else None
+    return res
+
+
+def finish_spec(G, triple, pmod_of, qmod_of):
+    """finish_centre46 with per-invocation modules: pmod_of(i, bit) a load_pmod46 module, qmod_of(i, j, mode) an
+    all_but_one_from module"""
+    G.local_channels()
+    base_nodes = len(G.a)
+    D = dict(zip(G.cubes, G.module(triple, [G.F[I] for I in G.cubes])))
+    P, Q, CEN = {}, {}, {}
+    for i in range(G.p):
+        others = [a for a in range(G.p) if a != i]
+        pairs = list(combinations(others, 2))
+        for bit in range(2):
+            inputs = [G.A[tuple(sorted((i, *K))), i, bit] for K in pairs]
+            outs = G.module(pmod_of(i, bit), inputs)
+            for K, node in zip(pairs, outs[:len(pairs)]):
+                P[tuple(sorted((i, *K))), i, bit] = node
+            CEN[i, bit] = outs[len(pairs)]
+    for i, j in combinations(range(G.p), 2):
+        others = [a for a in range(G.p) if a not in (i, j)]
+        for mode in range(2):
+            inputs = [G.G[tuple(sorted((i, j, k))), i, j, mode] for k in others]
+            for k, node in zip(others, G.module(qmod_of(i, j, mode), inputs)):
+                Q[tuple(sorted((i, j, k))), i, j, mode] = node
+    query_nodes = len(G.a)
+
+    def emit(I, bits_list, node, numerator=1, channel=''):
+        G.root.append(dict(node=node, targets=[G.source[I, b] for b in bits_list],
+                           coefficients=[f'{numerator}/2'] * len(bits_list), kind='side', channel=channel))
+    for I in G.cubes:
+        all_bits = list(product(range(2), repeat=3))
+        emit(I, all_bits, D[I], channel='disjoint')
+        for a in range(2):
+            emit(I, [b for b in all_bits if b[0] == a], P[I, I[0], 1 - a], channel='face0')
+        for a, b in product(range(2), repeat=2):
+            emit(I, [(a, b, c) for c in range(2)], P[I, I[1], 1 - b], channel='face1')
+        for a, b in product(range(2), repeat=2):
+            emit(I, [(a, b, c) for c in range(2)], Q[I, I[0], I[1], a ^ b], 2 * a - 1, 'edge01')
+        for bits in all_bits:
+            emit(I, [bits], P[I, I[2], 1 - bits[2]], channel='face2')
+        for ii, jj in [(0, 2), (1, 2)]:
+            for bits in all_bits:
+                emit(I, [bits], Q[I, I[ii], I[jj], bits[ii] ^ bits[jj]], 2 * bits[ii] - 1, f'edge{ii}{jj}')
+    for i in range(G.p):
+        for e in range(2):
+            G.centers.append(CEN[i, e])
+    for coordinate, node in enumerate(G.centers):
+        coefficients = ['1/3' if coordinate in t else '-1/6' for t in G.labels]
+        G.root.append(dict(node=node, targets=list(range(len(G.labels))), coefficients=coefficients, kind='center',
+                           coordinate=coordinate))
+    return dict(p=G.p, h=G.h, v=len(G.labels), labels=G.labels, inputs=[sum(1 << x for x in t) for t in G.labels],
+                args=G.a, signs=G.signs, roots=G.root, centers=G.centers, center_kind='star',
+                center_scatter='1/2 sum_(i in T) S_i - 1/6 sum_i S_i',
+                local_K='Two 4x4 Hadamard/2 blocks on opposite cube parities, using original X destructively',
+                source_data_histogram={2: len(G.labels), G.h - 4: len(G.labels), 1: len(G.labels)},
+                target_data_histogram={G.h - 4: len(G.labels), 1: 3 * len(G.labels)},
+                counts=dict(local_channel_additions=base_nodes - len(G.labels), query_additions=query_nodes - base_nodes,
+                            center_additions=0, center_tree_incidences=0,
+                            signed_additions=sum(x < 0 for x in G.signs),
+                            side_root_uses=sum(r['kind'] == 'side' for r in G.root), center_roots=len(G.centers)),
+                status='agent-cmod centre-sharing variant')   # finish_centre46's status, so that both write the same graph.json
+
diff --git a/research/w3-p10b-pointwise-complex/verify.py b/research/w3-p10b-pointwise-complex/verify.py
new file mode 100644
index 0000000000..7fe1ffd22e
--- /dev/null
+++ b/research/w3-p10b-pointwise-complex/verify.py
@@ -0,0 +1,788 @@
+#!/usr/bin/env python3
+"""PR #346's w3 bit word with a per-point p = 10 complex supplier (7940/10^7): conditional κ = 793392809291691/10^18.
+
+PR #346 (LJH-217) composes PR #310's w3 Design T and twin condensation on PR #325's p10b bit word (bit coarse
+c = 32069035133443/(4·10^16) ≈ 8.01726e-4) with a p = 10 complex supplier of coarse saving
+b = 396604388013523/(5·10^17) ≈ 7.93209e-4; the complex side binds, κ = 396290046684973/(5·10^17). This package keeps
+PR #346's bit word (vendored) and replaces the complex supplier by a stronger p = 10 program of the same recipe
+(PR #327's chain at 40ed045): PR #325's modules with a separate pair module for each point i (shared by both bits),
+claim 7940/10^7, b = 794022782431403/10^18. The complex side still binds and κ rises to 793392809291691/10^18.
+
+This script (Python 3.11+; numpy/scipy for the pinned regeneration and the checkers; -O refused)
+  1. checks MANIFEST.json (every package file, nothing missing or extra) and SOURCE.json (sha256 and git blob id of
+     every vendored file at its origin commit: PR #327 40ed045, PR #325 c4f1b74, PR #346 11b4de5, PR #329 c6b1a28);
+  2. bit side (PR #346's word, not re-derived): decodes PR #346's final records (FINAL-RECORDS.bin.gz, hash pinned by
+     its RECORD-SHA256SUMS), recounts the paid child histogram (MOVE and COPY ranks) independently, requires PR #346's
+     normalized five-stage histogram to be 60·H + 12·2v·(e38 + e19 + e42 + e4), takes its normalized stock W = 125,712
+     from PR #346's receipt, and certifies the bit coarse saving c with both rational engines and the 10^-16
+     fallback (adjacent 10^-18 point excluded), equal to PR #346's;
+  3. complex side: the committed program (canonical sha256 ff2d7307...): gx.check1 (labels, blocks, N, exact scalar
+     identity), Sussman's gxcore mirror, a flipped scalar sign rejected by both (E5), the exact five-stage price at
+     h = 20 (7940/10^7 with and without the fallback, 7941/10^7 rejected, coarse b by both engines);
+  4. κ: the complex-bound outer assembly of PR #315 / PR #329 (the bit saving capped one grid point below the leaf
+     cap (1 - β) b, β = 10^-9; three bootstrap steps from 384599/10^10; η = 10^-12; κ the last 10^-18 grid point below
+     the minimum margin; PR #315's outer.assembly with its 47 strict constraints and the finite bridge; the adjacent
+     κ and the cap itself rejected). It first reproduces PR #346's κ exactly from PR #346's b, then computes κ with
+     this package's b;
+  5. (full run, not --quick) regenerates the complex program: rebuilds PR #202's 308 files from the vendored PR #233
+     package, PR #327's anchors (a) and (b), anchors (d) and (e) on the one-module-per-kind spec (e: spec_graph.
+     finish_spec = centre46.finish_centre46 in every field), a builder anchor (f): on that spec's frozen layer
+     (layers/uniform/, the Lean-checked 7932/10^7 companion), aligned_word_spec.py --spec writes the same seven files
+     as PR #327's aligned_word_p.py --centre46; then builds the per-point word with aligned_word_spec.py --spec
+     spec/c2.json on the frozen layer,
+     paircheck.py, PR #200's checker, PR #184's flow and exact lift, PR #194's contract, PR #256's emitter, gx.check1
+     and gxcore, and requires the committed program byte for byte;
+  6. compares the run with certificate/expected.json (--quick: without the regeneration entries).
+The 7940/10^7 program has NOT been built in Lean (the Lean host was unavailable); the companion 7932/10^7 program of
+the same recipe with one pair module (spec/uniform.json; PR #346's supplier has the same b) was built in Lean.
+usage: python3 -B research/w3-p10b-pointwise-complex/verify.py [--quick] [--write] [--temp-root DIR]
+--write rewrites certificate/expected.json and then MANIFEST.json; it never rewrites the program, modules, layer or
+bit data (a different regenerated program is an error).
+Prepared by Chafik Boukhalfa with substantial Anthropic Claude assistance (Apache-2.0), from PR #327's verify.py
+(DreamingOfClouds with Anthropic Claude assistance) and PR #329's math_check.py.
+"""
+import argparse
+import copy
+import gzip
+import hashlib
+import importlib.util
+import json
+import math
+import shutil
+import struct
+import subprocess
+import sys
+import tempfile
+import time
+from collections import Counter
+from fractions import Fraction as Q
+from math import comb
+from pathlib import Path
+
+if sys.flags.optimize:
+    raise SystemExit('Assertions must remain enabled: run without -O')
+sys.dont_write_bytecode = True
+HERE = Path(__file__).resolve().parent
+sys.path.insert(0, str(HERE / 'gx'))
+sys.path.insert(0, str(HERE / 'scripts'))
+sys.path.insert(0, str(HERE))
+import gx                      # noqa: E402  (Jacob Sussman's reference checker, unchanged)
+import gxcore                  # noqa: E402  (Jacob Sussman's Python mirror of the two Lean checks, unchanged)
+import normalize_scatter       # noqa: E402
+import module_cut              # noqa: E402
+import spec_graph              # noqa: E402
+from moment import moment_margin  # noqa: E402  (PR #225's exact upper bound, as PR #304 and #327 price)
+T0 = time.monotonic()
+P = 10
+PROGRAM_SHA256 = 'ff2d7307b09aceb8776654f21099b10a68be547b562b7f4a7fc918f6c5f51112'
+SEVEN = ('cache/graph.json', 'cache/frames.json', 'cache/selection.json', 'cache/record.json',
+         'references/paired-cube/physical/frames.json', 'references/paired-cube/physical/pairs.json',
+         'certificates/paired-cube-sinks-input.json')
+PR168 = dict(tmod='tmod_TE_TD_TB3_1_1_4_full_6.0617964e-4.json', pmod='pmod_J0_full_6.0666810e-4.json',
+             qmod='qmod_climb3u_best.json')
+GRID7, GRID18, FALLBACK_WEIGHT = Q(1, 10**7), Q(1, 10**18), Q(1, 10**16)
+SKIP = ('__pycache__',)
+
+
+def log(msg):
+    print('[%6.1fs] %s' % (time.monotonic() - T0, msg), flush=True)
+
+
+def sha(data):
+    return hashlib.sha256(data).hexdigest()
+
+
+def git_blob(data):
+    return hashlib.sha1(b'blob %d\0' % len(data) + data).hexdigest()
+
+
+def read(p):
+    p = Path(p)
+    return json.loads(gzip.decompress(p.read_bytes()) if p.suffix == '.gz' else p.read_bytes())
+
+
+def write_gz(p, data):
+    with open(p, 'wb') as raw, gzip.GzipFile(filename='', fileobj=raw, mode='wb', mtime=0) as f:
+        f.write((json.dumps(data, separators=(',', ':')) + '\n').encode())
+
+
+def run(root, *args):
+    r = subprocess.run([sys.executable, '-B', *map(str, args)], cwd=root, capture_output=True, text=True)
+    if r.returncode:
+        raise SystemExit('FAILED: ' + ' '.join(map(str, args)) + '\n' + r.stdout[-3000:] + r.stderr[-3000:])
+    return r.stdout
+
+
+def load_module(name, path):
+    spec = importlib.util.spec_from_file_location(name, path)
+    mod = importlib.util.module_from_spec(spec)
+    spec.loader.exec_module(mod)
+    return mod
+
+
+S527 = load_module('source527_moment', HERE / 'scripts/source527/moment.py')          # eumemic, unchanged
+B2M = load_module('source527_base_two_moment', HERE / 'scripts/source527/base_two_moment.py')   # unchanged
+
+
+def canonical(cert):
+    return (json.dumps(cert, separators=(',', ':'), sort_keys=True) + '\n').encode()
+
+
+def grid(a):
+    return '%d/10^7' % (a * 10**7)
+
+
+def grid18(a):
+    return '%d/10^18' % (a * 10**18)
+
+
+def parse_grid(s):
+    num, den = s.split('/')
+    assert den == '10^7'
+    return Q(int(num), 10**7)
+
+
+# ---------------------------------------------------------------------------------------------------- 1. pins
+def package_files():
+    return sorted(p.relative_to(HERE).as_posix() for p in HERE.rglob('*')
+                  if p.is_file() and not any(part in SKIP for part in p.relative_to(HERE).parts))
+
+
+def manifest(write=False):
+    files = {rel: sha((HERE / rel).read_bytes()) for rel in package_files() if rel != 'MANIFEST.json'}
+    for rel in files:
+        data = (HERE / rel).read_bytes()
+        assert b'\r\n' not in data or rel.endswith('.gz'), 'CRLF line ending in ' + rel
+    if write:
+        (HERE / 'MANIFEST.json').write_text(json.dumps(dict(
+            description='sha256 of every file of research/w3-p10b-pointwise-complex except this one; verify.py '
+                        'requires the package to be exactly these files', files=files), indent=1, sort_keys=True) + '\n')
+    pinned = read(HERE / 'MANIFEST.json')['files']
+    missing = sorted(set(pinned) - set(files))
+    extra = sorted(set(files) - set(pinned))
+    changed = sorted(k for k in set(pinned) & set(files) if pinned[k] != files[k])
+    assert not (missing or extra or changed), 'MANIFEST.json: missing %s, unpinned %s, changed %s' % (missing, extra, changed)
+    return len(files)
+
+
+def check_source(src):
+    n = Counter()
+    for key, origin in src['origins'].items():
+        for rel, pin in origin['files'].items():
+            data = (HERE / rel).read_bytes()
+            assert sha(data) == pin['sha256'] and git_blob(data) == pin['git_blob'], \
+                '%s differs from %s at %s (%s)' % (rel, origin['repository'], origin['commit'][:7], pin['path'])
+            n[key] += 1
+    return n
+
+
+# ------------------------------------------------------------------------------------------ 2. the bit side
+def bit_side(src):
+    """PR #346's final bit word: recount the paid children from its records, check its normalized five-stage
+    histogram against them, and certify the bit coarse saving c from it"""
+    B = HERE / 'bit/pr346'
+    raw = gzip.decompress((B / 'FINAL-RECORDS.bin.gz').read_bytes())
+    sums = dict(reversed(line.split()) for line in (B / 'RECORD-SHA256SUMS').read_text().splitlines() if line.strip())
+    assert sha(raw) == sums['TT/249-records.bin'], 'final records differ from PR #346\'s RECORD-SHA256SUMS'
+    assert len(raw) % 24 == 0
+    kinds, H = Counter(), Counter()
+    for t, a, b, c, rank5, rank6 in struct.iter_unpack('<6i', raw):
+        kinds[t] += 1
+        if t == 0 and rank5 > 0:        # MOVE (0, role, old frame, new frame, rank, z): a paid child of that rank
+            H[rank5] += 1
+        elif t == 2 and rank6 > 0:      # COPY (2, centre, temp, copy frame, target frame, rank): a paid copy
+            H[rank6] += 1
+    receipt = read(B / 'kappa-L10f.json')
+    h, v, m, T = 20, 960, 100, 12      # T: PR #346's normalization (60 replicas / 5)
+    norm = {int(r): n for r, n in receipt['normalized_histogram'].items()}
+    idle = (2 * h - 2, h - 1, 2 * h + 2, 4)
+    expect = Counter({r: 5 * T * n for r, n in H.items()})
+    for r in idle:
+        expect[r] += T * 2 * v
+    assert norm == dict(expect), 'PR #346\'s normalized histogram is not 60 H + 12 * 2v * idle on its own records'
+    W = receipt['W']
+    assert receipt['m'] == m and W % T == 0 and (W // T - 4 * v) * h == 132720, 'unexpected stock'
+    mass = sum(r * n for r, n in norm.items())
+    root = S527.certify(norm, m, W, True)
+    c = Q(int(Q(root['lower']) * 10**18), 10**18)
+    bm, nextbm = S527.moment(norm, m, W, c, True), S527.moment(norm, m, W, c + GRID18, True)
+    assert bm[1] < 1 < nextbm[0], 'bit coarse saving not certified by moment.py'
+    fallback = 32 * m * m * sum(norm.values())
+    _, upper = B2M.moment(m, W, list(norm.items()), c)
+    _, bad = B2M.moment(m, W, [(1, fallback)], c)
+    lower, _ = B2M.moment(m, W, list(norm.items()), c + GRID18)
+    badlower, _ = B2M.moment(m, W, [(1, fallback)], c + GRID18)
+    assert upper + FALLBACK_WEIGHT * bad < 1 < lower + FALLBACK_WEIGHT * badlower, 'bit coarse saving not certified by base_two_moment.py'
+    assert c == Q(receipt['bit_coarse']), 'bit coarse saving differs from PR #346\'s'
+    log('bit side (PR #346): %d records (%s), paid children recounted (%d, rank mass %d); normalized m = %d, W = %d, '
+        'calls %d, rank mass %d, deficit %d; bit coarse c = %s ≈ %.12e (both engines, 10^-16 fallback, next point excluded) = PR #346\'s'
+        % (len(raw) // 24, dict(sorted(kinds.items())), sum(H.values()), sum(r * n for r, n in H.items()), m, W,
+           sum(norm.values()), mass, m * W - mass, c, float(c)))
+    return dict(records=len(raw) // 24, record_kinds={str(k): n for k, n in sorted(kinds.items())},
+                records_sha256=sha(raw), paid_children={str(r): n for r, n in sorted(H.items())},
+                normalization=T, m=m, W=W, calls=sum(norm.values()), rank_mass=mass, deficit=m * W - mass,
+                stock_from='PR #346 receipts/kappa-L10f.json (W = 125712 = 12 (4v + 132720/20)); not re-derived here',
+                coarse=c, coarse_decimal=S527.decimal(c), moment_interval=[str(x) for x in bm],
+                next_moment_lower_bound=str(nextbm[0]), equals_pr346=True)
+
+
+# ------------------------------------------------------------------------------------------- 2. the program
+def idle_ranks(h):
+    """the idle climbs of Sussman's bridged word B_2 per port: 2 blocks each of rank 2h-2, h-1, 2h+2, 4"""
+    return (2 * h - 2, h - 1, 2 * h + 2, 4)
+
+
+def five_stage(blocks, v, h, R):
+    Hinv = Counter()
+    for w in blocks.values():
+        for r, n in w.items():
+            Hinv[int(r)] += n
+    H5 = {r: 5 * n for r, n in Hinv.items()}
+    for r in idle_ranks(h):
+        H5[r] = H5.get(r, 0) + 2 * v
+    return 5 * h, 4 * v + R, H5
+
+
+def certify(name, cert, v, h):
+    st = {}
+    gx.check1(cert, scalar=True, stats=st)
+    R, N, cst = cert['R'], cert['N'], cert['cst']
+    assert N == R * h + 2 * v * (h - 1) + cst and not cert['ext']
+    m, W, H5 = five_stage(cert['blocks'], v, h, R)
+    rank = sum(r * n for r, n in H5.items())
+    assert W * m - rank == 4 * v - 5 * cst, '%s: five-stage deficit is not 4v - 5 cst' % name
+    return dict(R=R, N=N, cst=cst, gates_A=len(cert['A']), gates_B=len(cert['B']), frames=len(cert['frames']),
+                registers=2 * v + R, blocks_total=sum(sum(w.values()) for w in cert['blocks'].values()),
+                scalar_denominators=st['den'], scalar_max_abs=st['maxabs'], distinct_frame_steps=st['steps'],
+                five_stage=dict(h=h, m=m, W=W, idle_ranks=list(idle_ranks(h)), rank=rank, deficit=W * m - rank,
+                                child_histogram={str(r): n for r, n in sorted(H5.items())}))
+
+
+def price(name, five, a):
+    """PR #327's price(): PR #225's exact upper bound below W at a, with and without the 10^-16 fallback envelope;
+    the next 10^-7 grid point fails the upper bound and is rejected by source527's exact lower bound"""
+    hist = {int(r): n for r, n in five['child_histogram'].items()}
+    m, W = five['m'], five['W']
+    margin, total = moment_margin(m, W, hist, a)
+    assert margin > 0, '%s: five-stage moment not below W at a = %s' % (name, grid(a))
+    calls = sum(hist.values())
+    with_fallback = dict(hist)
+    with_fallback[1] = with_fallback.get(1, 0) + Q(32 * m * m * calls) * FALLBACK_WEIGHT
+    margin_fb, total_fb = moment_margin(m, W, with_fallback, a)
+    assert margin_fb > 0, '%s: five-stage moment with the 10^-16 fallback not below W at a = %s' % (name, grid(a))
+    nxt = a + GRID7
+    margin_next, _ = moment_margin(m, W, hist, nxt)
+    assert margin_next <= 0, '%s: the next grid point %s also passes; raise the claim' % (name, grid(nxt))
+    lower_next, _ = S527.moment(hist, m, W, nxt, False)
+    assert lower_next > 1, '%s: the next grid point %s is not excluded by an exact lower bound' % (name, grid(nxt))
+    scale = 10**30
+    up = lambda t: -((-t.numerator * scale) // t.denominator)
+    return dict(a=grid(a), moment_upper_bound='%d/10^30' % up(total), margin_lower_bound='%d/10^30' % (W * scale - up(total)),
+                with_fallback=dict(moment_upper_bound='%d/10^30' % up(total_fb), margin_lower_bound='%d/10^30' % (W * scale - up(total_fb)),
+                                   fallback_rank_one_calls='32 m^2 x %d x 10^-16' % calls),
+                next_grid_point=grid(nxt), next_grid_point_rejected=True,
+                next_grid_point_normalized_moment_lower_bound=str(lower_next))
+
+
+def coarse(name, five, a):
+    """PR #327's coarse(): the largest 10^-18 grid point at which the normalized moment with the 10^-16 fallback
+    envelope is below 1, by both exact engines, the next point excluded by both; likewise without the fallback"""
+    H = {int(r): n for r, n in five['child_histogram'].items()}
+    m, W = five['m'], five['W']
+    fallback = 32 * m * m * sum(H.values())
+    out = {}
+    for label, bad in (('with_fallback', True), ('without_fallback', False)):
+        root = S527.certify(H, m, W, bad)
+        b = Q(int(Q(root['lower']) * 10**18), 10**18)
+        cm = S527.moment(H, m, W, b, bad)
+        nextcm = S527.moment(H, m, W, b + GRID18, bad)
+        assert cm[1] < 1 < nextcm[0], '%s: coarse %s not certified by moment.py' % (name, label)
+        _, upper = B2M.moment(m, W, list(H.items()), b)
+        lower, _ = B2M.moment(m, W, list(H.items()), b + GRID18)
+        if bad:
+            _, badupper = B2M.moment(m, W, [(1, fallback)], b)
+            badlower, _ = B2M.moment(m, W, [(1, fallback)], b + GRID18)
+            upper, lower = upper + FALLBACK_WEIGHT * badupper, lower + FALLBACK_WEIGHT * badlower
+        assert upper < 1 < lower, '%s: coarse %s not certified by base_two_moment.py' % (name, label)
+        out[label] = dict(b=grid18(b), b_decimal=S527.decimal(b), moment_interval=[str(cm[0]), str(cm[1])],
+                          next_grid_point=grid18(b + GRID18), next_moment_lower_bound=str(nextcm[0]),
+                          base_two_upper=str(upper), base_two_next_lower=str(lower))
+    b = Q(out['with_fallback']['b'].split('/')[0]) / 10**18
+    b0 = Q(out['without_fallback']['b'].split('/')[0]) / 10**18
+    assert a <= b < b0 < a + GRID7, '%s: the coarse savings do not bracket the 10^-7 claim' % name
+    out['fallback_cost'] = grid18(b0 - b)
+    out['scope'] = ('10^-16 and 32 m^2 rank-one fallback calls per paid child are the five-stage packages\' inherited '
+                    'envelope, applied here at m = 5h = 100; not a new m = 5h admission')
+    return out
+
+
+def flip_one_sign(cert):
+    """negate the first coefficient of the first phase-A add gate with a coefficient list (PR #315's control)"""
+    gate = next(g for g in cert['A'] if g[0] in ('out', 'in') and g[3])
+    t, a, b = gate[3][0]
+    gate[3][0] = [t, -a, b]
+    return cert
+
+
+def histogram(cert):
+    Hinv = Counter()
+    for w in cert['blocks'].values():
+        for r, n in w.items():
+            Hinv[int(r)] += n
+    return Hinv
+
+
+def mirror(name, cert, hinv):
+    m = gxcore.mirror(gxcore.normal(copy.deepcopy(cert)))
+    assert m['hist'] == dict(hinv), '%s: gxcore histogram differs from the blocks' % name
+
+
+def controls(name, cert):
+    """a flipped scalar sign must be rejected by gx.check1 and by the gxcore mirror on the scalar identity (E5)"""
+    out = []
+    try:
+        gx.check1(flip_one_sign(copy.deepcopy(cert)), scalar=True)
+    except AssertionError as error:
+        assert str(error).startswith('E5'), ('%s: flipped sign rejected for the wrong reason' % name, str(error))
+        out.append('gx.check1: REJECTED (%s)' % str(error)[:2])
+    else:
+        raise AssertionError('%s: gx.check1 accepted a flipped scalar sign' % name)
+    try:
+        gxcore.mirror(gxcore.normal(flip_one_sign(copy.deepcopy(cert))))
+    except gxcore.Reject as error:
+        assert str(error).startswith('E5'), ('%s: flipped sign rejected for the wrong reason' % name, str(error))
+        out.append('gxcore.mirror: REJECTED (%s)' % str(error)[:2])
+    else:
+        raise AssertionError('%s: gxcore mirror accepted a flipped scalar sign' % name)
+    return dict(flipped_scalar_sign=out)
+
+
+# -------------------------------------------------------------------------------------- 3./4. regeneration
+def admit(root, tree, cand):
+    """PR #233's step 3: admit the word's physical layer with PR #200's complete complex checker (PR #327's admit)"""
+    from paired_cube.closure import compile_closure            # noqa: E402  (the rebuilt tree's scripts/)
+    from paired_cube_physical import physical                  # noqa: E402
+    cache = tree / 'cache'
+    g, w, word, row = (read(cache / n) for n in ('graph.json', 'frames.json', 'selection.json', 'record.json'))
+    frames = read(tree / 'references/paired-cube/physical/frames.json')['frames']
+    pairs = read(tree / 'references/paired-cube/physical/pairs.json')['pairs']
+    baseline, _ = compile_closure(g, [[x, c] for x, c in w['matching_arcs']])
+    profile = physical(g, w, word, row, frames, pairs)
+    assert json.loads(json.dumps(profile)) == read(tree / 'certificates/paired-cube-sinks-input.json'), 'physical profile differs from the sinks certificate'
+    cand.mkdir()
+    for name, data in [('graph', g), ('baseline', baseline), ('frames', w), ('word', word), ('profile-before', row),
+                       ('physical-frames', frames), ('physical-pairs', pairs), ('profile', profile)]:
+        write_gz(cand / (name + '.json.gz'), data)
+    pins = {p.name: sha(p.read_bytes()) for p in cand.glob('*.json.gz')}
+    (cand / 'result.json').write_text(json.dumps(dict(tag='complex-p10', pins=pins)) + '\n')
+    out = cand.with_suffix('.audit.json')
+    run(root, root / 'research/paired-cube-diagonal-bit-168/complex/physical.py', '--candidate', cand, '--source', root, '--out', out)
+    audit = read(out)
+    assert audit['status'].startswith('PASS'), audit['status']
+    return dict(status=audit['status'], formal=json.loads(json.dumps(audit.get('formal'))), mutations=audit.get('mutations'),
+                source_sha256={Path(k).resolve().relative_to(root).as_posix(): x for k, x in audit['source_sha256'].items()})
+
+
+def chain(root, work, name, tree, kernel, title, contract=True, admission=True, p=P):
+    """PR #327's chain(): flow, exact lift, contract, gcert/1 emission, gx.check1 and gxcore on one aligned tree"""
+    PKG, SA = root / 'research/source-assisted-v4', root / 'research/source-assisted'
+    cache = tree / 'cache'
+    g = read(cache / 'graph.json')
+    v, h = g['v'], g['h']
+    assert (g['p'], h, v) == (p, 2 * p, 8 * comb(p, 3)), '%s: not a p = %d graph' % (name, p)
+    rec = {}
+    if admission:
+        rec['admission'] = admit(root, tree, work / (name + '.candidate'))
+        log('%s: PR #168\'s physical() profile recomputed; layer admitted by PR #200\'s complex checker: %s'
+            % (name, rec['admission']['status'][:60]))
+    flow = work / (name + '.flow.json')
+    run(root, SA / 'decision/complex_frame_flow.py', '--tree', tree, '--cache', cache, '--out', flow, '--witness',
+        '--purify-source-donors', '--recycle-kernels', '--kernel-pairs', kernel)
+    witness = flow.with_suffix('.witness.json')
+    fd = read(flow)
+    lift = work / (name + '.lift.json')
+    run(root, SA / 'decision/exact_complex_flow_lift.py', '--witness', witness, '--profile', flow, '--out', lift)
+    lift_data = read(lift)
+    log('%s: flow new_R %d, deficit %d, numerical three-stage root %.7e; lift: %s' % (
+        name, fd['new_R'], fd['deficit'], fd['numerical_local_root'], lift_data['status'][:60]))
+    rec['flow'] = dict(new_R=fd['new_R'], new_W=fd['new_W'], deficit=fd['deficit'], loss=fd['loss'], kernel_reuses=fd['kernel_reuses'],
+                       source_control_erasures=fd['source_control_erasures'], original_physical_roles=fd['original_physical_roles'],
+                       child_histogram=fd['child_histogram'])
+    rec['lift'] = {k: lift_data[k] for k in ('status', 'nodes', 'physical_R', 'inverse_elementary_gates', 'all_actual_coefficients_dyadic', 'max_denominator')}
+    if contract:
+        (PKG / 'data/physical-pairs.json').write_bytes((tree / 'references/paired-cube/physical/pairs.json').read_bytes())
+        shutil.copyfile(kernel, PKG / 'data/kernel-pairs.json')
+        cprof = work / (name + '.complex-profile.json')
+        run(root, PKG / 'contract_v4.py', '--tree', tree, '--cache', cache, '--witness', witness, '--flow-profile', flow,
+            '--lift-profile', lift, '--out', cprof)
+        cdata = read(cprof)
+        for k, check in cdata['contract_checks'].items():
+            assert check not in (False, 'FAIL'), '%s: contract check %s failed' % (name, k)
+        assert cdata['physical_R'] == fd['new_R'] and cdata['child_histogram'] == fd['child_histogram']
+        rec['contract_checks'] = cdata['contract_checks']
+        log('%s: PR #194 contract: %d checks pass' % (name, len(cdata['contract_checks'])))
+    out = work / ('gcert1-p%d-%s-flow.json' % (p, name))
+    run(root, HERE / 'gcert_emit.py', '--cache', cache, '--flow', flow, '--witness', witness, '--lift', lift_data['certificate_path'],
+        '--out', out, '--name', title)
+    raw = read(out)
+    rec.update(certify(name, raw, v, h))
+    Hinv = histogram(raw)
+    mirror(name, raw, Hinv)
+    log('%s: emitted program: gx.check1 ACCEPTED (labels, blocks, N = %d, exact scalar identity) and gxcore mirror ACCEPTED; %d gates, %d registers'
+        % (name, rec['N'], rec['gates_A'] + rec['gates_B'], rec['registers']))
+    child = Counter({r: 3 * n for r, n in Hinv.items()})
+    child[2] += 2 * v
+    assert {int(r): n for r, n in fd['child_histogram'].items()} == dict(child), '%s: ledger differs from the flow profile' % name
+    cert, norm = normalize_scatter.normalize(raw)
+    st = {}
+    gx.check1(cert, scalar=True, stats=st)
+    assert cert['scat'] == normalize_scatter.STAR and normalize_scatter.same_shape(raw, cert), '%s: normalization changed more than signs' % name
+    assert (cert['blocks'], cert['N'], cert['frames'], len(cert['A']), len(cert['B'])) == (raw['blocks'], raw['N'], raw['frames'], len(raw['A']), len(raw['B']))
+    assert (st['den'], st['maxabs'], st['steps']) == (rec['scalar_denominators'], rec['scalar_max_abs'], rec['distinct_frame_steps'])
+    mirror(name, cert, Hinv)
+    norm.update(raw_certificate_sha256=sha(canonical(raw)), gx_check1='ACCEPTED (raw and committed program)',
+                gxcore_mirror='ACCEPTED (raw and committed program; label and scalar checks)')
+    rec['scatter'] = norm
+    log('%s: star form (%s): gx.check1 and gxcore ACCEPTED, blocks and N unchanged'
+        % (name, 'registers %s re-signed, %d coefficients' % (norm['flipped_registers'], norm['changed_coefficients'])
+           if norm['flipped_registers'] else 'emitted in star form, normalization not needed'))
+    rec['blocks'] = cert['blocks']
+    rec['certificate_sha256'] = sha(canonical(cert))
+    return rec, cert
+
+
+def install(root, rel):
+    """install a package module file next to PR #168 v4's in the rebuilt tree"""
+    dest = root / 'references/paired-cube/sources' / Path(rel).name
+    if dest.exists():
+        assert dest.read_bytes() == (HERE / rel).read_bytes(), 'module file name collides with a different tree file: ' + rel
+    else:
+        shutil.copyfile(HERE / rel, dest)
+    return dest
+
+
+def module_record(path, kind):
+    """the module's contract (the rebuilt tree's loaders; centre46.load_pmod46 for the 37-root pair module)"""
+    from paired_cube.modules import triple_module_from, all_but_one_from  # noqa: E402
+    from centre46 import load_pmod46                                     # noqa: E402
+    if kind == 'tmod':
+        triple_module_from(path, P)
+    elif kind == 'pmod':
+        load_pmod46(path, P - 1)
+    else:
+        all_but_one_from(path, P - 2)
+    d = read(path)
+    outputs = {'tmod': comb(P, 3), 'pmod': comb(P - 1, 2) + 1, 'qmod': P - 2}[kind]
+    assert len(d['roots']) == outputs, (path, len(d['roots']))
+    return dict(sha256=sha(Path(path).read_bytes()), additions=module_cut.additions(d), outputs=outputs)
+
+
+def apply_diff(text, diff):
+    """apply a unified diff (exact context, every hunk once) to text"""
+    lines = text.split('\n')
+    hunks, cur = [], None
+    for ln in diff.split('\n'):
+        if ln.startswith('@@'):
+            cur = ([], [])
+            hunks.append(cur)
+        elif cur is not None and ln[:1] in (' ', '-', '+'):
+            if ln[0] in ' -':
+                cur[0].append(ln[1:])
+            if ln[0] in ' +':
+                cur[1].append(ln[1:])
+    for old, new in hunks:
+        hits = [i for i in range(len(lines)) if lines[i:i + len(old)] == old]
+        assert len(hits) == 1, 'patch hunk does not apply exactly once'
+        lines[hits[0]:hits[0] + len(old)] = new
+    return '\n'.join(lines), len(hunks)
+
+
+def local_design(root):
+    local = json.loads((root / 'references/paired-cube/sources/local_L1.json').read_text())
+    local['G'] = {k: 's' for k in local['G']}
+    local['A'] = {k: 'fd' for k in local['A']}
+    return local
+
+
+def graph_anchors(root, spec, table, files):
+    """anchor (d): PR #315's Graph.finish patch builds centre46.finish_centre46's graph on these modules; anchor (e):
+    spec_graph.finish_spec with the spec's per-invocation tables builds the same graph"""
+    graph_py = root / 'scripts/paired_cube/graph.py'
+    diff = (HERE / 'references/centre46_graph.diff').read_text()
+    patched, hunks = apply_diff(graph_py.read_text(), diff)
+    (root / 'scripts/paired_cube/graph_centre46_patch.py').write_text(patched)
+    import paired_cube.graph as G0                                                  # noqa: E402
+    import paired_cube.graph_centre46_patch as G1                                   # noqa: E402
+    (root / 'scripts/paired_cube/graph_centre46_patch.py').unlink()                # leave the rebuilt tree as pinned
+    from paired_cube.modules import triple_module_from, all_but_one_from            # noqa: E402
+    from centre46 import load_pmod46, finish_centre46                               # noqa: E402
+    local = local_design(root)
+    t = triple_module_from(HERE / spec['tmod'], P)
+    pm46, q = load_pmod46(files['pmod'], P - 1), all_but_one_from(files['qmod'], P - 2)
+    a = finish_centre46(G0.Graph(P, local=copy.deepcopy(local)), t, pm46, q)
+    b = G1.Graph(P, local=copy.deepcopy(local)).finish(t, pm46, q)
+    differ = sorted(k for k in set(a) | set(b) if a.get(k) != b.get(k))
+    assert differ == ['status'], 'centre46.py and the patched Graph.finish differ in %s' % differ
+    cache = {}
+
+    def get(path, loader, n):
+        if path not in cache:
+            cache[path] = loader(path, n)
+        return cache[path]
+    c = spec_graph.finish_spec(G0.Graph(P, local=copy.deepcopy(local)), t,
+                               lambda i, bit: get(table['p', i, bit], load_pmod46, P - 1),
+                               lambda i, j, mode: get(table['q', i, j, mode], all_but_one_from, P - 2))
+    differ_e = sorted(k for k in set(a) | set(c) if a.get(k) != c.get(k))
+    assert differ_e == [], 'spec_graph.finish_spec and centre46.finish_centre46 differ in %s' % differ_e
+    return dict(d=dict(patch_sha256=sha(diff.encode()), hunks=hunks, centres_from_pair_module=len(a['centers']),
+                       fields_compared=len(a), differing_fields=differ, center_additions=a['counts']['center_additions']),
+                e=dict(invocations=dict(pair=sum(k[0] == 'p' for k in table), all_but_one=sum(k[0] == 'q' for k in table)),
+                       distinct_module_files=dict(pair=len({f for k, f in table.items() if k[0] == 'p'}),
+                                                  all_but_one=len({f for k, f in table.items() if k[0] == 'q'})),
+                       fields_compared=len(a), differing_fields=differ_e,
+                       query_additions=c['counts']['query_additions']))
+
+
+def program(spec, cert):
+    """step 3: the committed complex program on its own"""
+    name = spec['name']
+    assert sha(canonical(cert)) == PROGRAM_SHA256, 'the committed program is not the pinned one'
+    assert (cert['format'], cert['p'], cert['h'], cert['v']) == ('gcert/1', P, 2 * P, 8 * comb(P, 3))
+    assert cert['scat'] == normalize_scatter.STAR, 'the committed program is not in star form'
+    rec = certify(name, cert, cert['v'], cert['h'])
+    mirror(name, cert, histogram(cert))
+    log('complex %s: gx.check1 ACCEPTED (labels, blocks, N = %d, exact scalar identity) and gxcore mirror ACCEPTED; R %d, %d gates, %d blocks'
+        % (name, rec['N'], rec['R'], rec['gates_A'] + rec['gates_B'], rec['blocks_total']))
+    rec['controls'] = controls(name, cert)
+    log('complex %s: flipped scalar sign rejected by gx.check1 and gxcore (E5)' % name)
+    assert rec['cst'] == 20 * 18 and rec['five_stage']['deficit'] == 4 * 960 - 5 * 360 == 2040
+    five = rec['five_stage']
+    claim = parse_grid(spec['claim'])
+    five.update(price(name, five, claim))
+    five['coarse'] = coarse(name, five, claim)
+    log('complex %s: five-stage word at h = 20 (m = %d, W = %d, deficit %d) certified at a = %s with and without the 10^-16 fallback, '
+        '%s rejected; coarse b = %s with the fallback, %s without (both engines)'
+        % (name, five['m'], five['W'], five['deficit'], five['a'], five['next_grid_point'],
+           five['coarse']['with_fallback']['b'], five['coarse']['without_fallback']['b']))
+    rec['blocks'] = cert['blocks']
+    rec['certificate_sha256'] = sha(canonical(cert))
+    return rec
+
+
+# ------------------------------------------------------------------------------------------------ 4. kappa
+OUTER = load_module('pr315_outer', HERE / 'pricing/outer.py')      # PR #315's outer assembly, unchanged
+BRIDGE = dict(proof='PROOF.md', representation='Exact powers with source-bound finite overcharges',
+              semantic=dict(G=dict(base=2, exponent=30000), E=dict(base=2, exponent=100000),
+                            B_upper=dict(base=2, exponent=100001), C0=dict(base=2, exponent=210000), C1=1,
+                            strict_literal_gap=1, induction_gap_lower=1),
+              rows=dict(coefficient=20161, degree=10**6, suffix_slope=4 * 10**6, degree_gap=Q(10**6) - Q(51 * 20161, 25)))
+
+
+def kappa(bit, b):
+    """the outer assembly of PR #315's math_check (as in PR #329): bit saving c capped below the complex leaf cap"""
+    c = bit['coarse']
+    hist = {int(r): n for r, n in bit['histogram'].items()}
+    m, W = bit['m'], bit['W']
+    grid, eta, beta = GRID18, Q(1, 10**12), Q(1, 10**9)
+    cap = (1 - beta) * b
+    s = min(c, Q(((cap - grid) * 10**18).__floor__(), 10**18))
+    binding = 'bit' if s == c else 'complex'
+    sm = S527.moment(hist, m, W, s, True)
+    assert 0 < s <= c and sm[1] < 1
+    if binding == 'complex':
+        assert s + grid > cap - grid
+    chain = [Q(384599, 10**10)]
+    for _ in range(3):
+        a = (1 - s) * s + s * chain[-1]
+        assert chain[-1] < a < s < 1 - a
+        chain.append(a)
+    a_bit = chain[-1]
+    assert a_bit < s <= cap - grid < cap and BRIDGE['rows']['degree_gap'] > 0
+    q = a_bit * (1 - 2 * eta)
+    minimum = (1 - eta) * q / (1 + q)
+    ticks = minimum * 10**18
+    k = Q((ticks.numerator - 1) // ticks.denominator, 10**18)
+    asm = OUTER.assembly(a_bit, b, BRIDGE, k, eta=eta, beta=beta)
+    assert len(asm['strict_constraints']) == 47 and len(asm['margins']) == 7 and min(asm['strict_constraints'].values()) > 0
+    try:
+        OUTER.assembly(a_bit, b, BRIDGE, k + grid, eta=eta, beta=beta)
+    except AssertionError:
+        pass
+    else:
+        raise AssertionError('adjacent kappa admitted')
+    if binding == 'complex':
+        try:
+            OUTER.assembly(cap, b, BRIDGE, k, eta=eta, beta=beta)
+        except AssertionError as error:
+            assert 'leaf_saving_above_bit' in str(error)
+        else:
+            raise AssertionError('complex leaf cap admitted')
+    tight = min(asm['strict_constraints'], key=lambda n: asm['strict_constraints'][n])
+    return dict(complex_coarse=b, complex_leaf_cap=cap, bit_coarse=c, bit_saving_used=s, binding=binding,
+                bit_moment_interval=[str(x) for x in sm], bootstrap_chain=[str(x) for x in chain], minimum_margin=minimum,
+                kappa=k, kappa_decimal=S527.decimal(k), kappa_scientific=format(float(k), '.15e'),
+                strict_constraints=47, smallest_constraint=tight, adjacent_kappa_rejected=True,
+                leaf_cap_itself_rejected=binding == 'complex')
+
+
+# -------------------------------------------------------------------------------------- 5. regeneration
+def regenerate(src, spec, uniform, temp_root):
+    result = {}
+    sussman = read(HERE / 'references/sussman/pr193-blocks.json')
+    layer = load_module('pr233_layer_verify', HERE / 'references/pr233-layer/verify.py')
+    utable = spec_graph.expand(uniform, HERE)
+    one = spec_graph.shared(utable)
+    assert one['p'] and one['q']
+    table = spec_graph.expand(spec, HERE)
+    with tempfile.TemporaryDirectory(prefix='w3-p10b-pointwise-', dir=temp_root) as d:
+        root = Path(d).resolve()
+        pins = layer.rebuild(root)
+        log('rebuilt %d pinned files of PR #202 (commit %s) from the vendored PR #233 package' % (len(pins['files']), pins['commit'][:7]))
+        sys.path.insert(0, str(root / 'scripts'))
+        PKG = root / 'research/source-assisted-v4'
+        work = root / '.work'
+        work.mkdir()
+        base, aligned = work / 'base', work / 'aligned'
+        run(root, root / 'scripts/paired_cube_producer.py', '--work-dir', base, '--output', work / 'producer.json')
+        run(root, PKG / 'source_aligned_local_v4.py', '--tree', root, '--cache', base, '--out', aligned)
+        log('regenerated the PR #168 v4 producer word and the source-aligned word at p = 11 (PR #194\'s pipeline)')
+        src_dir = root / 'references/paired-cube/sources'
+        pr168 = {k: src_dir / f for k, f in PR168.items()}
+        mods = ['--tmod', pr168['tmod'], '--pmod', pr168['pmod'], '--qmod', pr168['qmod']]
+        anchor = work / 'anchor'
+        phys = aligned / 'references/paired-cube/physical'
+        run(root, HERE / 'aligned_word_p.py', '--tree', root, *mods, '--arcs', aligned / 'cache/frames.json',
+            '--frames', phys / 'frames.json', '--pairs', phys / 'pairs.json', '--out', anchor)
+        for f in SEVEN:
+            assert (anchor / f).read_bytes() == (aligned / f).read_bytes(), 'aligned_word_p.py differs from source_aligned_local_v4.py: ' + f
+        log('anchor (a): aligned_word_p.py at p = 11 reproduces the 7 output files of source_aligned_local_v4.py byte for byte')
+        tree = work / 'pr194'
+        run(root, HERE / 'aligned_word_p.py', '--tree', root, *mods, '--arcs', aligned / 'cache/frames.json',
+            '--frames', phys / 'frames.json', '--pairs', PKG / 'data/physical-pairs.json', '--out', tree)
+        rec, _ = chain(root, work, 'pr194', tree, PKG / 'data/kernel-pairs.json', 'pr194 flow word (PR168 frames, PR193 pairs)',
+                       contract=False, admission=False, p=11)
+        assert rec['blocks'] == sussman['blocks'] and rec['N'] == sussman['N'], 'pr194: not the histogram of Sussman\'s certificate of #193'
+        assert rec['certificate_sha256'] == src['anchors']['pr194_certificate_sha256'], 'pr194: differs from PR #256\'s certificate'
+        result['anchor_pr194'] = dict(N=rec['N'], R=rec['R'], blocks_total=rec['blocks_total'], certificate_sha256=rec['certificate_sha256'],
+                                      equals_pr256_certificate=True, equals_sussman_pr193_blocks=True)
+        log('anchor (b): PR #194\'s word through this chain is PR #256\'s certificate (sha256 %s...) with Sussman\'s #193 histogram' % rec['certificate_sha256'][:12])
+        files = dict(tmod=HERE / uniform['tmod'], pmod=one['p'], qmod=one['q'])
+        installed = {k: install(root, Path(f).relative_to(HERE).as_posix()) for k, f in files.items()}
+        anchors = graph_anchors(root, uniform, utable, installed)
+        result['anchor_centre_patch'], result['anchor_spec_graph'] = anchors['d'], anchors['e']
+        log('anchor (d): PR #315\'s Graph.finish patch builds centre46.py\'s graph on PR #325\'s modules; anchor (e): spec_graph.finish_spec '
+            'on spec/uniform.json builds the same graph (every field)')
+        U = HERE / uniform['layer']
+        uargs = ['--arcs', U / 'matching-arcs.json', '--frames', U / 'physical-frames.json', '--pairs', U / 'physical-pairs-mw.json']
+        fa, fb = work / 'builder-p', work / 'builder-spec'
+        run(root, HERE / 'aligned_word_p.py', '--p', P, '--centre46', '--tree', root, '--tmod', installed['tmod'],
+            '--pmod', installed['pmod'], '--qmod', installed['qmod'], *uargs, '--out', fa)
+        run(root, HERE / 'aligned_word_spec.py', '--p', P, '--spec', HERE / 'spec/uniform.json', '--spec-root', HERE, '--tree', root,
+            '--tmod', 'x', '--pmod', 'x', '--qmod', 'x', *uargs, '--out', fb)
+        for f in SEVEN:
+            assert (fa / f).read_bytes() == (fb / f).read_bytes(), 'aligned_word_spec.py --spec differs from aligned_word_p.py --centre46: ' + f
+        result['anchor_spec_builder'] = dict(spec='spec/uniform.json', layer=uniform['layer'], files_compared=len(SEVEN), identical=True,
+                                             physical_R=read(fa / 'certificates/paired-cube-sinks-input.json')['physical_R'])
+        L = HERE / spec['layer']
+        layer_args = ['--arcs', L / 'matching-arcs.json', '--frames', L / 'physical-frames.json', '--pairs', L / 'physical-pairs-mw.json']
+        log('anchor (f): on the companion layer, aligned_word_spec.py --spec spec/uniform.json writes the 7 files of aligned_word_p.py --centre46 byte for byte')
+        result['modules'] = {kind: module_record(installed[kind], kind) for kind in ('pmod', 'qmod', 'tmod')}
+        result['pointwise_pair_modules'] = {}
+        for i in range(P):
+            assert table['p', i, 0] == table['p', i, 1], 'pair module of point %d differs between the two bits' % i
+            result['pointwise_pair_modules'][str(i)] = dict(file=table['p', i, 0].relative_to(HERE).as_posix(),
+                                                            **module_record(table['p', i, 0], 'pmod'))
+        assert spec_graph.shared(table)['q'] == one['q']
+        key = spec['name']
+        tree = work / key
+        built = run(root, HERE / 'aligned_word_spec.py', '--p', P, '--spec', HERE / 'spec/c2.json', '--spec-root', HERE, '--tree', root,
+                    '--tmod', 'x', '--pmod', 'x', '--qmod', 'x', *layer_args, '--out', tree)
+        dropped = [json.loads(x) for x in built.splitlines() if x.startswith('{')]
+        dropped = next(x for x in dropped if x.get('stage') == 'physical')['dropped_gauges']
+        assert not dropped, '%s: gauges dropped %s' % (key, dropped)
+        sinks = read(tree / 'certificates/paired-cube-sinks-input.json')
+        log('%s: word on the frozen layer (%d arcs, %d moved frames, %d pairs), physical checks pass, physical R %d'
+            % (key, len(read(L / 'matching-arcs.json')['matching_arcs']), len(read(L / 'physical-frames.json')['frames']),
+               len(read(L / 'physical-pairs-mw.json')['pairs']), sinks['physical_R']))
+        pc = json.loads(run(root, HERE / 'paircheck.py', '--root', root, tree))
+        assert pc['status'] == 'PASS' and not pc['violations'] and not pc['erasable'], '%s: paircheck %s' % (key, pc)
+        assert all(x.startswith('REJECTED') for x in pc['controls'].values()), '%s: a paircheck control was not rejected: %s' % (key, pc['controls'])
+        assert pc['controls']['donor_frame_outside_gauge'] == 'REJECTED (donor frame outside gauge)' and \
+            pc['controls']['parity_erasable_donor'] == 'REJECTED (erasable)', '%s: a paircheck control is not isolated: %s' % (key, pc['controls'])
+        log('%s: paircheck PASS (%d early, %d late pairs, none parity-erasable); %d mutation controls rejected'
+            % (key, pc['early'], pc['late'], len(pc['controls'])))
+        rec, cert = chain(root, work, key, tree, L / 'kernel-pairs-mw.json', key)
+        rec['paircheck'] = pc
+        rec['layer'] = dict(R=read(tree / 'cache/record.json')['R'], physical_R=sinks['physical_R'],
+                            arcs=len(read(L / 'matching-arcs.json')['matching_arcs']),
+                            moved_frames=len(read(L / 'physical-frames.json')['frames']),
+                            pairs=len(read(L / 'physical-pairs-mw.json')['pairs']),
+                            kernel_pairs=len(read(L / 'kernel-pairs-mw.json')['kernel_pairs']))
+        result['regenerated'] = rec
+    return result, cert
+
+
+def main():
+    ap = argparse.ArgumentParser(description=__doc__, formatter_class=argparse.RawDescriptionHelpFormatter)
+    ap.add_argument('--quick', action='store_true', help='steps 1-4 and 6 only: no regeneration of the complex program (about two minutes)')
+    ap.add_argument('--write', action='store_true', help='rewrite certificate/expected.json, then MANIFEST.json')
+    ap.add_argument('--temp-root', type=Path, default=None)
+    a = ap.parse_args()
+    assert not (a.quick and a.write), '--write needs the full run'
+    src = read(HERE / 'SOURCE.json')
+    spec, uniform = read(HERE / 'spec/c2.json'), read(HERE / 'spec/uniform.json')
+    if not a.write:
+        log('MANIFEST.json: %d package files present and unchanged' % manifest())
+    log('SOURCE.json: vendored files equal their git blobs at their origin commits: %s' % dict(check_source(src)))
+    bit = bit_side(src)
+    committed_path = HERE / spec['certificate']
+    committed = read(committed_path)
+    rec = program(spec, committed)
+    b = Q(rec['five_stage']['coarse']['with_fallback']['b'].split('/')[0]) / 10**18
+    bitk = dict(coarse=bit['coarse'], m=bit['m'], W=bit['W'],
+                histogram=read(HERE / 'bit/pr346/kappa-L10f.json')['normalized_histogram'])
+    ref = src['pr346']
+    k346 = kappa(bitk, Q(ref['complex_coarse']))
+    assert k346['binding'] == ref['binding'] and k346['kappa'] == Q(ref['kappa']), 'PR #346\'s kappa not reproduced'
+    log('PR #346 reproduced: complex b = %s binds, kappa = %s = %s' % (ref['complex_coarse'], ref['kappa'], k346['kappa_scientific']))
+    k = kappa(bitk, b)
+    assert k['binding'] == 'complex'
+    gain = k['kappa'] / k346['kappa'] - 1
+    log('this package: complex b = %s binds (bit c = %s above the leaf cap); kappa = %s = %s, %+.4f%% over PR #346; 47 strict constraints, '
+        'adjacent kappa and the leaf cap itself rejected' % (grid18(b), grid18(bit['coarse']),
+                                                            k['kappa'], k['kappa_scientific'], 100 * float(gain)))
+    canon = dict(bit=bit, program=rec, kappa_pr346_reproduced=k346, kappa=k,
+                 comparison=dict(pr346_kappa=str(k346['kappa']), kappa=str(k['kappa']), gain=str(k['kappa'] - k346['kappa']),
+                                 gain_percent='%.6f' % float(100 * gain),
+                                 pr346_complex_coarse=str(Q(ref['complex_coarse'])), complex_coarse=str(b)))
+    if not a.quick:
+        result, cert = regenerate(src, spec, uniform, a.temp_root)
+        assert sha(canonical(cert)) == rec['certificate_sha256'] == PROGRAM_SHA256, 'the regenerated program differs from the committed one'
+        assert canonical(cert) == gzip.decompress(committed_path.read_bytes()), 'committed program differs from the regenerated one'
+        reg = result.pop('regenerated')
+        for key in ('R', 'N', 'cst', 'gates_A', 'gates_B', 'frames', 'registers', 'blocks_total', 'scalar_denominators',
+                    'scalar_max_abs', 'distinct_frame_steps', 'blocks', 'certificate_sha256'):
+            assert reg[key] == rec[key], key
+        log('regenerated program = committed program byte for byte (uncompressed canonical JSON, sha256 %s...)' % PROGRAM_SHA256[:12])
+        canon['regeneration'] = dict(result, chain={key: reg[key] for key in ('admission', 'flow', 'lift', 'contract_checks', 'scatter', 'paircheck', 'layer')})
+    canon['status'] = ('PASS: conditional kappa = %s (complex-bound) with PR #346\'s w3 bit word and the per-point p = 10 complex program '
+                       '(%s, b = %s); PR #346\'s kappa %s reproduced' % (k['kappa'], rec['five_stage']['a'], grid18(b), k346['kappa']))
+    canon = json.loads(json.dumps(canon, sort_keys=True, default=str))
+    if a.write:
+        (HERE / 'certificate/expected.json').write_text(json.dumps(canon, indent=1, sort_keys=True) + '\n')
+        log('wrote certificate/expected.json; MANIFEST.json: %d files pinned' % manifest(write=True))
+    expected = read(HERE / 'certificate/expected.json')
+    if a.quick:
+        expected = {key: x for key, x in expected.items() if key != 'regeneration'}
+    if expected != canon:
+        for key in sorted(set(expected) | set(canon)):
+            if expected.get(key) != canon.get(key):
+                log('DIFFERS: %s' % key)
+    assert expected == canon, 'result differs from certificate/expected.json'
+    log('PASS %s: conditional kappa = %s ≈ %s (complex-bound; PR #346: %s, %+.4f%%)'
+        % ('certificate/expected.json reproduced' + (' (without the regeneration entry)' if a.quick else ''),
+           k['kappa'], k['kappa_scientific'], k346['kappa'], 100 * float(gain)))
+
+
+if __name__ == '__main__':
+    main()