diff --git a/.github/workflows/native-complex-moment552.yml b/.github/workflows/native-complex-moment552.yml
new file mode 100644
index 0000000000..04d6a60aff
--- /dev/null
+++ b/.github/workflows/native-complex-moment552.yml
@@ -0,0 +1,21 @@
+name: Verify native complex moment on PR552
+on:
+  pull_request:
+    paths:
+      - 'research/native-complex-moment552/**'
+      - '.github/workflows/native-complex-moment552.yml'
+  workflow_dispatch:
+jobs:
+  verify:
+    runs-on: ubuntu-latest
+    strategy:
+      matrix:
+        python: ['3.11', '3.14']
+    steps:
+      - uses: actions/checkout@v4
+      - uses: actions/setup-python@v5
+        with:
+          python-version: ${{ matrix.python }}
+      - run: python -B research/native-complex-moment552/verify.py --work "$RUNNER_TEMP/native-complex-moment552"
+        env:
+          PYTHONDONTWRITEBYTECODE: '1'
diff --git a/research/native-complex-moment552/EXPECTED.json b/research/native-complex-moment552/EXPECTED.json
new file mode 100644
index 0000000000..9fc6616a33
--- /dev/null
+++ b/research/native-complex-moment552/EXPECTED.json
@@ -0,0 +1,8 @@
+{
+  "kappa": "1591968905569359/1000000000000000000",
+  "kappa_decimal": "0.001591968905569359",
+  "complex_lower": "398626827907293921411/250000000000000000000000",
+  "complex_decimal": "0.001594507311629175685644",
+  "old_kappa": "795984451416673/500000000000000000",
+  "absolute_gain": "2736013/1000000000000000000"
+}
diff --git a/research/native-complex-moment552/EXPECTED558.json b/research/native-complex-moment552/EXPECTED558.json
new file mode 100644
index 0000000000..0d4ec1740e
--- /dev/null
+++ b/research/native-complex-moment552/EXPECTED558.json
@@ -0,0 +1,8 @@
+{
+  "kappa": "796258946026667/500000000000000000",
+  "kappa_decimal": "0.001592517892053334",
+  "complex_lower": "797529025268839550469/500000000000000000000000",
+  "complex_decimal": "0.001595058050537679100938",
+  "old_kappa": "796258944657507/500000000000000000",
+  "absolute_gain": "34229/12500000000000000"
+}
diff --git a/research/native-complex-moment552/LICENSE b/research/native-complex-moment552/LICENSE
new file mode 100644
index 0000000000..261eeb9e9f
--- /dev/null
+++ b/research/native-complex-moment552/LICENSE
@@ -0,0 +1,201 @@
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diff --git a/research/native-complex-moment552/MANIFEST.json b/research/native-complex-moment552/MANIFEST.json
new file mode 100644
index 0000000000..643510d841
--- /dev/null
+++ b/research/native-complex-moment552/MANIFEST.json
@@ -0,0 +1,40 @@
+{
+  "EXPECTED.json": "a79dc7eec1301a69560a2e7fd6ed69634463272696ee96f09e15db1aad8ab0b0",
+  "EXPECTED558.json": "1e922eef350748956f4cd785d7fc5f73297b0ed6f53b130aca2da29d26777c34",
+  "LICENSE": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4",
+  "NOTICE.md": "02e74770785e31234d863b36ad2291e04267eaf32a4ba82f67cb1c804da1de44",
+  "PROOF-REVIEW-STATUS.json": "f54fcf048c5e24f5d3ccf308259b4b1b291b736e014fee81b4518980d358786c",
+  "PROOF.md": "d5d498edc56fdef8c5798af7f5ec66948bd2f66c553b9efe366cdaab01bd3453",
+  "README.md": "74f5cd377861bdff57a2825b19ac6c2b0bb6aa62f3b0354c7ceec0a76866c61c",
+  "REVIEW.json": "518fd773b1287c9be19563316f55acfa443667946d1f5bc7cc7b884bb9005094",
+  "SOURCES.json": "32e1466dd29be472e63e11e8ad76b228c0a1e3ec33596ce4319a64b89bd6d912",
+  "engines/base_two_moment.py": "aee70c242a2f1d225528fabe2002e1564d1dc884797c07d28a831c7878ee9098",
+  "engines/moment.py": "81cbc864e106418fc7854322746f72180cb8110ff39f74450a3b8d55a82c1482",
+  "engines/outer.py": "c1ebb52163a7219504c728a5c2b098bf1e50bf20148a2879c5a74c3cadc7c03c",
+  "evidence/PR552-INDEPENDENT-REPLAY.json": "6db0dfeedce73cf9a6eb31bea0a499f06dc4961da40a60d91059bb9ab48fd73c",
+  "evidence/PR552-PUBLISHED-INVOICE.json": "a21fd7e02d3001b1e8065e9eb6084d80e1f64c44035733ed1115a38675a01fd0",
+  "evidence/PR552-packaged311.json": "8a06db71ab002a9e080f0d60d22ee32a3a6a848a0cc633f37fc8fde22d21694b",
+  "evidence/PR552-packaged314.json": "53142566b590fdde9258e36e6fe62991c2f39331fc2822bf0105f812e52d6aa6",
+  "evidence/PR552-regression-final.json": "0c8bb51c2234a3190fab3e0e33187dee1bd4f2aab32483a69452366abb3b2567",
+  "evidence/PR558-COMPILER-LINEAGE.json": "db1e3390ade85f2299fc1ff0652e12fe1384c35bdcaf5c49e5403e4eb6b255b9",
+  "evidence/PR558-INDEPENDENT-RECEIPT.json": "c121cb83baeeb950a42eac4bec73676119dd836386b822e03bf7820fa1bf5262",
+  "evidence/PR558-INDEPENDENT-REPLAY.json": "b32113b56745cd5c142c982426ce31f0fe8d4f1a4ba219541e8c8befc69f017c",
+  "evidence/PR558-packaged311.json": "fed5ee89ea9f6b3c3ffb150898fc95f9a057a0c598b18de6df59e66908342b2e",
+  "evidence/PR558-packaged314.json": "71251eee88024f14982a4e7328377f50eb9d04314f692b891b53b5911498e3c2",
+  "inputs/PR543-EXPECTED.json": "06eccee9f6b77e140833df83407e7996dc948c17d49b0779ab171fbe463fd28d",
+  "inputs/PR552-EXACT-ASSEMBLY.json": "c7495a6787f48a9c5184a00daa841f4a972e37c580c6ff3dacc07bd947914f99",
+  "inputs/PR552-EXPECTED.json": "00c462828be76f6d65c1311ad9ef102f18f088029e039cffa599eaeb41668fb6",
+  "inputs/PR552-INVOICE.json": "4ac060dbf2ea2624dd1f4bda23bdb81c0a3013dfe49b77c092b631885529947f",
+  "inputs/PR552-OWNERS.json": "ecac65c65418641abf598db74d588ba8f53b3f358505bd31e215f09562922e3f",
+  "inputs/PR558-BASELINE-ASSEMBLY.json": "9f5e050486c3c337793d09e4ce96769336a3653de0a3a84a77c39431cf062f16",
+  "inputs/PR558-EXPECTED.json": "6c16d41d9c10bc04c65b548ff34422ee2d90d63c37e9bdea4adbdf06fa959f44",
+  "inputs/PR558-INVOICE.json": "4b6977a049663ab958bd9a48c2864f7a7bddb26aef03e14f3068aede020ef76f",
+  "inputs/PR558-OWNERS.json": "0bcaa62bf4723672113f2acdc3ae30c6faa18e5aefa3992e0ec7dc46b9cdd552",
+  "inputs/PR558-RESULT.json": "81618cefc1df42f827d0a7da838b13b9167d3500ca5cd598963c3e09e62ceb4e",
+  "inputs/UPSTREAM-LINEAGE.md": "2ec1c9b1f25e2f3d2556bee7369069ec2340cf126b54fa112e0bff9e269ef0e0",
+  "notes/general-clifford-frames.tex": "c9bb9655b594583c6b4bab8cb485233f1d1e588f83858cf1e4c3530cee8fed54",
+  "notes/three-stage-cover-assembly.tex": "ccf34a6d37c4af14bebeab15be50ab204eb1caebe546385520050ec30089441d",
+  "notes/three-stage-cover-bit.tex": "47123a756f823c83a0d48cca2bf5add6472cd88410b13aed345e70c8834fb511",
+  "notes/three-stage-cover-complex.tex": "1c5b638aa9b75365b8addec9c8eeee6bc5e499722901079b94c65fce033fde8a",
+  "verify.py": "6b444c686891b20b3a92385d025818fc86d9a6ddd2f77504d9223afaa47acd21"
+}
diff --git a/research/native-complex-moment552/NOTICE.md b/research/native-complex-moment552/NOTICE.md
new file mode 100644
index 0000000000..f173f442dc
--- /dev/null
+++ b/research/native-complex-moment552/NOTICE.md
@@ -0,0 +1 @@
+Apache-2.0; original notices retained in source files. PR552: Dugongue. PR543: eumemic. Original Clifford/complex-cover notes and PR531/535: icekylinx with stated assistance. PR534 frame work: rohanarun/Claude. Inherited PR541/510/503/493/484/463/444/442/386 compiler and hosting lineage remains credited in inputs/UPSTREAM-LINEAGE.md. New audit and arithmetic composition by Rohan Arun with OpenAI Codex assistance.
diff --git a/research/native-complex-moment552/PROOF-REVIEW-STATUS.json b/research/native-complex-moment552/PROOF-REVIEW-STATUS.json
new file mode 100644
index 0000000000..30032f58d8
--- /dev/null
+++ b/research/native-complex-moment552/PROOF-REVIEW-STATUS.json
@@ -0,0 +1,7 @@
+{
+  "status": "INDEPENDENT_SOURCE_SCOPE_REVIEW_ACCEPTED",
+  "packaged_verification": "PASS on Python 3.11 and 3.14 for PR558 composition, identical mathematical fields; PR552 regression PASS",
+  "initial_preflight_failure": "Published source invoice was accidentally paired with independently replayed owner receipt; hash guard rejected it. Fixed by pairing both receipts from the same completed independent replay. Original published invoice retained in evidence. No mathematical result accepted from mismatched receipts.",
+  "independent_provider_replay": "PR558 full physical/admission replay PASS at 060b137c8a6f9d73fd7864c7f8cf2736d6441774",
+  "draft_history": "Initial PR552 draft and subsequent PR558 update were published before their respective packaged checks, as requested. Receipts supersede pending status."
+}
diff --git a/research/native-complex-moment552/PROOF.md b/research/native-complex-moment552/PROOF.md
new file mode 100644
index 0000000000..9adf304048
--- /dev/null
+++ b/research/native-complex-moment552/PROOF.md
@@ -0,0 +1,47 @@
+# Independent review: the PR552 complex bad-class surcharge
+
+**Assessment: removal is justified under the same inherited exact complex-cover/compiler assumptions.** This review supports omitting the additional bad-residue fallback from the complex recurrence only. It does not certify a numerical improvement, weaken the bit recurrence, or discharge any inherited all-size theorem.
+
+## Original recurrence distinction
+
+The pinned `notes/three-stage-cover-assembly.tex` at commit12a4934d15006e8c10f9b2264f13f8f341be802b gives the complex moment as the exact child-profile sum, divided by the complex live stock. It then separately introduces the entire32*m_b^2-child fallback on a10^-16 fraction of every ideal edge of the **bit** ledger. Its precision/router guard explicitly retains the full finite orthogonal group and Clifford routers.
+
+The adjacent pinned `notes/general-clifford-frames.tex` constructs exact Gaussian-dyadic representatives for every binary subspace, exact nested transitions, rank-zero adapters, and dirty-tail/reverse identities. It gives exact algebraic operators, not good-class operators accompanied by a failed residue-class fraction. Fixed binary GL matrices that choose these representatives are invertible address maps over F2; they are not the q-adic selectors responsible for bit-side fallback.
+
+Both original note hashes and Git-blob identities were independently checked against the parent acquisition pins; see REVIEW.json.
+
+## Actual PR552 compiler audit
+
+This conclusion follows from the emitted program and its exact controller, not merely from an absence of keywords:
+
+1. `emit_local.py` emits both orientations from the selected source. Every positive-rank frame climb and center descent is explicitly a recursive child; all scalar operations, coefficient work, phases, and residual adapters remain separate finite work. It checks all exact scalar columns and paid-profile equality.
+2. `emit_supplier_hosted.py` assigns each entire helper invocation to an explicit physical family and orthogonal chart. Its profile is the sum over all72 complete local invocations plus every nonempty idle remainder. The emitted asynchronous schedule retains every source event on each replica and the final correction.
+3. `audit_supplier_hosted.py` independently checks all owner windows, one-owner exclusivity, residual/spectator axes, phase bindings, and every explicit remainder. It reconstructs actual orthogonal73-column charts and the same global histogram. It rejects omitted idle children and invalid owner assignments.
+4. `invoice_final.py` recounts all scalar/Horner work, wrappers, adapters, movement, and router costs. It charges the full finite orthogonal group and preserves the precision, tape, external-row and halving guards. Its prefix argument uses exact right-translation bijections on the cover group, so every cover vertex obeys the same operator construction.
+
+The prior independent physical replay receipts have been checked for PASS and matching template/owner hashes. This review does not rerun that full physical verifier. The actual2,972 owner-chart types are finitely checked; the enormous entire orthogonal group is not enumerated element-by-element. The translation-bijection argument supplies the uniform passage to the full cover.
+
+Consequently the complex controller has an exact exhaustive recursive-child profile. There is no further width-dependent good/bad q-adic selector in this construction requiring a fraction of additional recursive fallback children. The finite Clifford routers and adapters are already paid in the unchanged finite invoice. Removing them, or removing other precision/row guards, would not follow from this argument.
+
+## Where the extra term enters
+
+`vendor535/code/assemble_combined.py` uses one generic `two_moments` function for both suppliers. It calls the moment engine with `bad=True` and explicitly adds the fallback through the second engine for the complex profile as well as the bit profile. The original moment engine labels that option an inherited envelope and makes the default `bad=False`. Thus the correction is a recurrence-model choice at assembly, not a changed scalar word, owner schedule, or hidden physical optimization.
+
+For complex profile H, dimension m, and live stock W, retain exactly
+
+    sum_r H[r] * (r/m)^(1-b) / W.
+
+Omit only the additional complex term proportional to32*m^2*sum(H)/10^16. Keep the entire bit bad-class surcharge and all its mutable-source/selector assumptions.
+
+## Scope and relation to PR557
+
+This remains conditional on the retained all-size realization, common-label/compiler, completed-call hosting, semantic precision, fixed-tape/exact-recovery, and analytic-transfer interfaces. The review does not replace these assumptions with a new theorem. Under those same interfaces, a uniform exact complex cover uses its pure child moment, matching the original complex assembly argument. A port to a different complex implementation that actually has exceptional residue classes would require its own analysis.
+
+PR557 explicitly fixes its conservative bad-class charge and excludes alternative charges from its ceiling. Changing that unnecessary complex surcharge therefore lies outside that certificate's stated model; it does not contradict the verified fixed-charge ceiling. Separately, PR557 fixes the PR551 scalar word and does not directly cover final PR552.
+
+No numerical value is claimed by this review. Both rational engines, the next endpoint, all strict outer inequalities, and a publication package must be checked by the parent before announcing a new conditional bound. No PR or provider source was edited, and all artifacts remain on the mounted SSD.
+
+
+## Transfer to PR558
+
+PR558 changes eight kernel entrances and repacks complete owners, while retaining byte-identical local emission, scalar audit, phase/adapters, complete-owner emission/audit and invoice code from reviewed PR552. Its new exact finite witness uses the same native binary-Clifford/orthogonal-cover contract and introduces no q-adic selector. The same pure-complex recurrence therefore applies after its new finite admission. The published receipts are pinned; an independent full replay passed at head 060b137c8a6f9d73fd7864c7f8cf2736d6441774. Fresh admission and compiler-lineage receipts are preserved in evidence/. The bit supplier and its bad-class charge remain unchanged.
diff --git a/research/native-complex-moment552/README.md b/research/native-complex-moment552/README.md
new file mode 100644
index 0000000000..3e5e80b917
--- /dev/null
+++ b/research/native-complex-moment552/README.md
@@ -0,0 +1,20 @@
+# Native complex moment: PR558 plus the exact recurrence correction
+
+Conditional **kappa = 0.001592517892053334**, above PR558's0.001592517889315014 by **2.738320e-12** (+0.00000017194909%). This is a tiny accounting correction on eumemic's structural result; it is not another geometry or circuit improvement.
+
+The exact binary-Clifford complex cover has a complete recursive-child histogram. Generic pricing adds the bit supplier's GL local-ring bad-residue surcharge to both sides. This package returns the complex side to its native moment. The bit charge, selected scalar word, owner schedule and finite invoice remain unchanged. [Independent proof-scope review](PROOF.md) gives the uniform chart/right-translation argument.
+
+Complex b becomes0.001595058050537679100938. Cited PR543 c stays0.001653826988146798944268. Both rational discovery engines pass the new endpoint and reject the next10^-24 tick; seven bootstrap levels and47 strict outer inequalities give the displayed kappa.
+
+## Verification
+
+The full independent PR558 replay passed in 126.92 seconds, including fresh phase, owner, invoice and assembly checks. The updated packaged checks passed on Python 3.11 and 3.14 with identical mathematical outputs, and the PR552 regression passed. Receipts are in `evidence/`. CI status must be read separately from GitHub; local success is not a green CI claim. This arithmetic-only verifier cites unchanged physical operators rather than rebuilding them.
+
+    python -B verify.py --work /absolute/new/output
+    python -B verify.py --provider 552 --work /absolute/new/regression-output
+
+Python3.11+, standard library, offline, assertions enabled. Inputs/owner/invoice and exact baseline are bound; the original bit contamination is checked in both engines. The new complex endpoint and next tick,47 outer inequalities, seven bootstrap cutoffs and parameter controls are checked. The552 regression retains six bootstrap levels and its original candidate0.001591968905569359.
+
+All inherited all-size compiler, common-label/mutable-source, complete-call hosting, precision/tape and analytic assumptions remain, including PR543 H1-H5. Explicit checks cover chart types; full-group validity uses right-translation bijections. No unconditional theorem, new Lean proof or practical speedup. PR557 fixes the old surcharge and different boundaries, so this is outside its scope.
+
+Credits: eumemic/Codex PR558; Dugongue PR552; eumemic/Claude PR543; icekylinx original Clifford/complex-cover notes and PR531/535; all inherited compiler/hosting authors. New source audit and arithmetic by Rohan Arun with OpenAI Codex assistance. Original notices remain preserved.
diff --git a/research/native-complex-moment552/REVIEW.json b/research/native-complex-moment552/REVIEW.json
new file mode 100644
index 0000000000..381201476a
--- /dev/null
+++ b/research/native-complex-moment552/REVIEW.json
@@ -0,0 +1,99 @@
+{
+  "status": "CONDITIONALLY_JUSTIFIED_ZERO_COMPLEX_BADCLASS_SURCHARGE",
+  "source_pr552_head": "33c33af0b7697155be1ea708e08cbd3f0707d94c",
+  "notes_pins": [
+    {
+      "path": "notes/general-clifford-frames.tex",
+      "commit": "12a4934d15006e8c10f9b2264f13f8f341be802b",
+      "blob": "0367eeb005298601f28a673861d6f71bf1d0ebbb",
+      "sha256": "c9bb9655b594583c6b4bab8cb485233f1d1e588f83858cf1e4c3530cee8fed54"
+    },
+    {
+      "path": "notes/three-stage-cover-assembly.tex",
+      "commit": "12a4934d15006e8c10f9b2264f13f8f341be802b",
+      "blob": "983ffb6ac8fb9117360cd0797c00c05dfa071c9d",
+      "sha256": "ccf34a6d37c4af14bebeab15be50ab204eb1caebe546385520050ec30089441d"
+    }
+  ],
+  "read_source_bindings": [
+    {
+      "path": "complex552-fallback-audit/general-clifford-frames.tex",
+      "sha256": "c9bb9655b594583c6b4bab8cb485233f1d1e588f83858cf1e4c3530cee8fed54"
+    },
+    {
+      "path": "complex552-fallback-audit/three-stage-cover-assembly.tex",
+      "sha256": "ccf34a6d37c4af14bebeab15be50ab204eb1caebe546385520050ec30089441d"
+    },
+    {
+      "path": "source552-review/complex-global/vendor531/code/emit_local.py",
+      "sha256": "eefc76e0bee003015fe21ac685c2cce658033d5444add404d2d509c7d6f39bac"
+    },
+    {
+      "path": "source552-review/complex-global/vendor531/code/emit_supplier_hosted.py",
+      "sha256": "53c0b653b3a5100756d6a9473c90101b8c3ab72723d7047983906ab997bdf920"
+    },
+    {
+      "path": "source552-review/complex-global/vendor531/code/audit_supplier_hosted.py",
+      "sha256": "301e9b5c593b6d583065337eb7b3ef207cf039e265f099f89080a4bf25856b43"
+    },
+    {
+      "path": "source552-review/complex-global/vendor531/code/invoice_final.py",
+      "sha256": "96b6bed96ca8b60dfc064cd1c7962c45a5cb53c4e52cd90aaea6ccf8d109db97"
+    },
+    {
+      "path": "source552-review/complex-global/vendor535/code/assemble_combined.py",
+      "sha256": "41216e8da3b647fe6eead9079858f25b2592cab090154aca43f3cc73ff36dc9b"
+    },
+    {
+      "path": "source552-review/engines/moment.py",
+      "sha256": "81cbc864e106418fc7854322746f72180cb8110ff39f74450a3b8d55a82c1482"
+    },
+    {
+      "path": "source552-review/README.md",
+      "sha256": "79cc63fc45ea52c04f32555676704a1664a3eafdd917b25689b1acd390a4636d"
+    },
+    {
+      "path": "pr552-independent314/complex-local/local/INDEPENDENT-LOCAL-REPLAY.json",
+      "sha256": "e871d8d56014146c19963430e4822714aadbf57367f783dc1dd42fa3f5b0a0ce"
+    },
+    {
+      "path": "pr552-independent314/complex-global/audit/OWNERS-AND-PROFILE.json",
+      "sha256": "ecac65c65418641abf598db74d588ba8f53b3f358505bd31e215f09562922e3f"
+    },
+    {
+      "path": "pr552-independent314/complex-global/audit/ASYNC-AUDIT.json",
+      "sha256": "01d37b06d35ffb1995533cc618b0baaa093f55d288786153406fc46c94ab9926"
+    },
+    {
+      "path": "pr552-independent314/complex-global/audit/PREFIX-AND-INVOICE.json",
+      "sha256": "4ac060dbf2ea2624dd1f4bda23bdb81c0a3013dfe49b77c092b631885529947f"
+    },
+    {
+      "path": "pr552-independent314/complex-global/audit/GLOBAL-PREFIX.json",
+      "sha256": "789d003bce7c6b6fcc6dad6b2bfb2d1bd05e8bba68aa87139a05c6752b861529"
+    }
+  ],
+  "independent_owner_status": "PASS_INDEPENDENT_ASYNC_OWNERS_AND_PROFILE",
+  "independent_invoice_status": "PASS_INDEPENDENT_ASYNC_SCALAR_PREFIX_AND_FINITE_INVOICE",
+  "independent_local_status": "PASS_INDEPENDENT_LOCAL_SCALAR_FRAME_AND_WRAPPER_REPLAY",
+  "complex_live": 7168,
+  "physical": 7682,
+  "profile": {
+    "1": 165600,
+    "2": 32128,
+    "3": 35907,
+    "4": 13248,
+    "5": 4032,
+    "6": 4032,
+    "7": 208,
+    "8": 576,
+    "39": 1,
+    "48": 1660
+  },
+  "actual_73column_sigmas": 2972,
+  "template_sha256": "1ddb317665797be007dfcc06e295f46bf9115ddeb308a21a60f98b9c28c6aff0",
+  "decision": "The actual complete finite complex controller meets the exact-cover alternative in the retained complex recurrence, under the same inherited all-size compiler and semantic interfaces. No exceptional bad-residue fraction is introduced by its fixed exact Clifford maps or orthogonal cover translations. The bit bad-class fallback must remain.",
+  "new_numerical_claim": false,
+  "new_allsize_proof": false,
+  "unconditional_multiplication_claim": false
+}
diff --git a/research/native-complex-moment552/SOURCES.json b/research/native-complex-moment552/SOURCES.json
new file mode 100644
index 0000000000..12ace451da
--- /dev/null
+++ b/research/native-complex-moment552/SOURCES.json
@@ -0,0 +1,47 @@
+{
+  "pr552": {
+    "head": "33c33af0b7697155be1ea708e08cbd3f0707d94c",
+    "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/552"
+  },
+  "pr543": {
+    "head": "2065fe4f72a914d907f1046603f5a36f17018535",
+    "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/543"
+  },
+  "original_notes": [
+    {
+      "path": "notes/general-clifford-frames.tex",
+      "commit": "12a4934d15006e8c10f9b2264f13f8f341be802b",
+      "blob": "0367eeb005298601f28a673861d6f71bf1d0ebbb",
+      "sha256": "c9bb9655b594583c6b4bab8cb485233f1d1e588f83858cf1e4c3530cee8fed54"
+    },
+    {
+      "path": "notes/three-stage-cover-assembly.tex",
+      "commit": "12a4934d15006e8c10f9b2264f13f8f341be802b",
+      "blob": "983ffb6ac8fb9117360cd0797c00c05dfa071c9d",
+      "sha256": "ccf34a6d37c4af14bebeab15be50ab204eb1caebe546385520050ec30089441d"
+    }
+  ],
+  "other_notes": {
+    "commit": "12a4934d15006e8c10f9b2264f13f8f341be802b",
+    "files": [
+      {
+        "path": "notes/three-stage-cover-complex.tex",
+        "sha": "c54cb3d5ff13281fe2aa44ac3baeaac7fce75469"
+      },
+      {
+        "path": "notes/three-stage-cover-bit.tex",
+        "sha": "f05225d21f92cf8cef21b6e12601d5b4d61991f0"
+      },
+      {
+        "path": "notes/three-stage-cover-note.tex",
+        "sha": "f5ad09a450cc5dd31a5c6aa07ff08523064d7281"
+      }
+    ]
+  },
+  "input_receipts": "PR552 owner and invoice are from the same completed independent replay. Published original invoice is preserved separately; profile/template match pinned EXPECTED.",
+  "pr558": {
+    "head": "060b137c8a6f9d73fd7864c7f8cf2736d6441774",
+    "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/558",
+    "inputs": "Exact published python314 validation receipts; normalized EXPECTED copies result plus profile alias and invoice template, checked at runtime."
+  }
+}
diff --git a/research/native-complex-moment552/engines/base_two_moment.py b/research/native-complex-moment552/engines/base_two_moment.py
new file mode 100644
index 0000000000..14a90fa0e7
--- /dev/null
+++ b/research/native-complex-moment552/engines/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/native-complex-moment552/engines/moment.py b/research/native-complex-moment552/engines/moment.py
new file mode 100644
index 0000000000..7d697cb452
--- /dev/null
+++ b/research/native-complex-moment552/engines/moment.py
@@ -0,0 +1,113 @@
+"""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,h):
+    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=h*(h-2)  # p = 10 port: was the p = 12 literal 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
+        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=h,copy_rank=h-2,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 m = 5h 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'],data['v'],data['h'])
+    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')}
+    print(json.dumps(result,indent=2))
diff --git a/research/native-complex-moment552/engines/outer.py b/research/native-complex-moment552/engines/outer.py
new file mode 100644
index 0000000000..43007488ad
--- /dev/null
+++ b/research/native-complex-moment552/engines/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/native-complex-moment552/evidence/PR552-INDEPENDENT-REPLAY.json b/research/native-complex-moment552/evidence/PR552-INDEPENDENT-REPLAY.json
new file mode 100644
index 0000000000..8117926d22
--- /dev/null
+++ b/research/native-complex-moment552/evidence/PR552-INDEPENDENT-REPLAY.json
@@ -0,0 +1,49 @@
+{
+  "status": "PASS_FILTRATION_WAVEFRONT",
+  "kappa": "795984451416673/500000000000000000",
+  "kappa_decimal": "0.001591968902833346",
+  "bit_lower": "413456747036699736067/250000000000000000000000",
+  "complex_lower": "1594507308884431019637/1000000000000000000000000",
+  "complex_template_sha256": "1ddb317665797be007dfcc06e295f46bf9115ddeb308a21a60f98b9c28c6aff0",
+  "complex_canonical_word_sha256": "995e46ed6dc8377f02bf05a3370171796364cedefdfeaeae47c2a207b5a8b173",
+  "complex_global_profile": {
+    "1": 165600,
+    "2": 32128,
+    "3": 35907,
+    "4": 13248,
+    "5": 4032,
+    "6": 4032,
+    "7": 208,
+    "8": 576,
+    "39": 1,
+    "48": 1660
+  },
+  "complex_prefix_bits": 225,
+  "forward_events": 21483,
+  "backward_events": 21483,
+  "waves": 8,
+  "wave_omission_controls": 8,
+  "coset_rewrites": 12,
+  "compensation_additions_saved": 624,
+  "outer_constraints": 47,
+  "bit_operator": "Pinned PR543 cited dependency, not physically replayed here",
+  "stages": [
+    {
+      "name": "complex-reconstruction-and-local-checks",
+      "exit": 0,
+      "seconds": 13.987790917046368
+    },
+    {
+      "name": "complete-owner-physical-and-invoice",
+      "exit": 0,
+      "seconds": 7.874530042055994
+    },
+    {
+      "name": "exact-two-supplier-assembly",
+      "exit": 0,
+      "seconds": 0.24127279093954712
+    }
+  ],
+  "elapsed_seconds": 22.136513874982484,
+  "scope": "Exact composition of the freshly rebuilt and physically audited wave complex supplier with the pinned PR543 published bit receipt. The unchanged bit operator is cited, not rebuilt in this package. Inherited all-size compiler, common-label/mutable-source, hosting, precision/tape and analytic interfaces remain conditional. R420 interleaved phase-B source writes rely on its stated H1-H5 interface argument."
+}
diff --git a/research/native-complex-moment552/evidence/PR552-PUBLISHED-INVOICE.json b/research/native-complex-moment552/evidence/PR552-PUBLISHED-INVOICE.json
new file mode 100644
index 0000000000..2cd1365bdd
--- /dev/null
+++ b/research/native-complex-moment552/evidence/PR552-PUBLISHED-INVOICE.json
@@ -0,0 +1,106 @@
+{
+  "status": "PASS_INDEPENDENT_ASYNC_SCALAR_PREFIX_AND_FINITE_INVOICE",
+  "template_sha256": "1ddb317665797be007dfcc06e295f46bf9115ddeb308a21a60f98b9c28c6aff0",
+  "parent_template_sha256": "7eddf21720ce26ff72069a0e94fa7f0e692cfe3bea2defb1a3769bea352cf471",
+  "owner_audit_sha256": "6572e69d9df9a72a600d7ee88404f96c525fea760b76cb6607273d85681579d1",
+  "controller_audit_sha256": "01d37b06d35ffb1995533cc618b0baaa093f55d288786153406fc46c94ab9926",
+  "independent_local_replay_sha256": "e871d8d56014146c19963430e4822714aadbf57367f783dc1dd42fa3f5b0a0ce",
+  "canonical_word_sha256": "995e46ed6dc8377f02bf05a3370171796364cedefdfeaeae47c2a207b5a8b173",
+  "local_programs_sha256": "7b191ec016ab99a0e1415206b898a8a6c5485fe9e8d91e59539e7c3710725563",
+  "local_checks_sha256": "2c51cdb9b282aadb99ea00d05b6c4bee140577111bdfd5c675b038373e309a17",
+  "global_prefix_bits": 225,
+  "charged_prefix_bits": 512,
+  "complete_invocation_sweeps": 72,
+  "bridge_sweeps": 72,
+  "row_prefix_extra": 1024,
+  "local": {
+    "backward": {
+      "events": 21483,
+      "macros": 16290,
+      "real_component_primitives": 161356,
+      "prefix_l1": "1044730572396",
+      "prefix_bits": 40,
+      "coefficient_work_primitive_count": 79906,
+      "all_exact_columns": 728,
+      "dirty_helper_columns": 504,
+      "all_dirty_helper_columns_restored": true,
+      "private_centres_and_coefficient_work_cleared": true,
+      "pre_wrappers": 212,
+      "post_wrappers": 212,
+      "endpoint_evidence": "Hash-bound fresh INDEPENDENT-LOCAL-REPLAY.json"
+    },
+    "forward": {
+      "events": 21483,
+      "macros": 16290,
+      "real_component_primitives": 161308,
+      "prefix_l1": "88876257700",
+      "prefix_bits": 37,
+      "coefficient_work_primitive_count": 79874,
+      "all_exact_columns": 728,
+      "dirty_helper_columns": 504,
+      "all_dirty_helper_columns_restored": true,
+      "private_centres_and_coefficient_work_cleared": true,
+      "pre_wrappers": 212,
+      "post_wrappers": 212,
+      "endpoint_evidence": "Hash-bound fresh INDEPENDENT-LOCAL-REPLAY.json"
+    }
+  },
+  "helpers": 504,
+  "centres": 8,
+  "live": 7168,
+  "physical": 7682,
+  "physical_stock_identity": {
+    "data_and_owner_families": 7168,
+    "local_helpers": 504,
+    "private_centres": 8,
+    "coefficient_work": 2
+  },
+  "entering_helpers": 212,
+  "entrance_rank": 380,
+  "width_inventory": {
+    "5": 56,
+    "8": 156,
+    "9": 292
+  },
+  "local_adapters": 333344,
+  "local_primitive_bound": 8645055460940,
+  "wrapper_elementary_factors_charged": 4396032,
+  "factors_per_wrapper": 144,
+  "compiler": 25934336,
+  "unhosted_pervertex": 1574554051175192,
+  "hosted_rebased_pervertex": 1574554050875816,
+  "conservative_pervertex_used": 4452501040899760,
+  "retained_prior_overcharge": {
+    "sha256": "ce1bfa49be88ae67a404f606a11db2232b7e1f2610d4418058e39749d86d8478",
+    "previous_pervertex_allowance": 2877946989724568,
+    "added_complete_new_pervertex": 1574554051175192,
+    "policy": "Retain old allowance and add the entire fresh recount."
+  },
+  "finite": {
+    "physical_stock_bits": 2641,
+    "router_bits": 7985,
+    "literal_bits": 35282,
+    "physical_row_coefficient": 16467,
+    "retained_row_coefficient": 20161,
+    "required_C0_bits": 200012,
+    "halving_integer_inequality": [
+      5329,
+      4608
+    ],
+    "G_exponent": 30000,
+    "E_exponent": 100000,
+    "C0_exponent": 210000
+  },
+  "strict_checks": {
+    "local_primitive_bound": true,
+    "logical": true,
+    "router": true,
+    "literal": true,
+    "mass": true,
+    "precision": true,
+    "tape": true,
+    "row": true,
+    "halving": true
+  },
+  "scope": "Fresh emitted primitive/Horner, prefix, wrapper and finite recount, using the hash-bound independent local all-column/frame replay and actual whole-owner audit. No old INTEGRATION receipt is used to certify the changed word. The inherited all-size compiler, precision/tape and analytic-transfer contracts remain conditional. Retained payload 512 and physical row allowance 20161 are unchanged."
+}
diff --git a/research/native-complex-moment552/evidence/PR552-packaged311.json b/research/native-complex-moment552/evidence/PR552-packaged311.json
new file mode 100644
index 0000000000..b964eea61a
--- /dev/null
+++ b/research/native-complex-moment552/evidence/PR552-packaged311.json
@@ -0,0 +1,146 @@
+{
+  "status": "PASS_NATIVE_COMPLEX_MOMENT_AND_CONDITIONAL_COMPOSITION",
+  "baseline_kappa": "795984451416673/500000000000000000",
+  "baseline_kappa_decimal": "0.001591968902833346",
+  "candidate_kappa": "1591968905569359/1000000000000000000",
+  "candidate_kappa_decimal": "0.001591968905569359",
+  "old_complex_lower": "1594507308884431019637/1000000000000000000000000",
+  "complex_lower": "398626827907293921411/250000000000000000000000",
+  "complex_decimal": "0.001594507311629175685644",
+  "absolute_gain": "2736013/1000000000000000000",
+  "relative_gain_percent": "1.718634701425708280641836122474865049015926763915375461899680701380779E-7",
+  "profile": {
+    "1": 165600,
+    "2": 32128,
+    "3": 35907,
+    "4": 13248,
+    "5": 4032,
+    "6": 4032,
+    "7": 208,
+    "8": 576,
+    "39": 1,
+    "48": 1660
+  },
+  "live": 7168,
+  "m": 73,
+  "rank_deficit": 2560,
+  "bit_lower": "413456747036699736067/250000000000000000000000",
+  "moments": [
+    {
+      "endpoint": "398626827907293921411/250000000000000000000000",
+      "accept": true,
+      "mercator": [
+        "-4895688882012886766331792451077040638497877/26959946667150639794667015087019630673637144422540572481103610249216",
+        "-4895688882012886766331792451077040638497875/26959946667150639794667015087019630673637144422540572481103610249216"
+      ],
+      "base_two": [
+        "-22699/125000000000000000000000000000",
+        "-181591/1000000000000000000000000000000"
+      ]
+    },
+    {
+      "endpoint": "318901462325835137129/200000000000000000000000",
+      "accept": false,
+      "mercator": [
+        "78062267921507236210488055889752793689635265/26959946667150639794667015087019630673637144422540572481103610249216",
+        "78062267921507236210488055889752793689635267/26959946667150639794667015087019630673637144422540572481103610249216"
+      ],
+      "base_two": [
+        "289549/100000000000000000000000000000",
+        "2895491/1000000000000000000000000000000"
+      ]
+    }
+  ],
+  "outer": {
+    "bit_positive": "97321002907054178373747480789122492190356235870531375332775927943680026964774615115237938228273004247294924183783013314400416614253200879298737647634957420329181801397927397860839186346258781529921029804922527732346197665829089340870779927279248849120940961483433812034433324267039784971/61035156250000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "complex_above_bit": "1658234941085877507809643764129468624667224072056319973035225384884762061771726995752705075816216986685599583385746799120701262352365042579670818198602072602139160813653741218470078970195077472267653802334170910659129220072720751150879059038516566187965566675732960215029/61035156250000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "complex_below_one_over32": "7413873172092706078589/250000000000000000000000",
+    "beta_positive": "1/1000000000000000000",
+    "beta_below_one": "999999999999999999/1000000000000000000",
+    "leaf_saving_above_bit": "1560913938178823327777661342254468624667224072056319973035225384884762061771726995752705075816216986685599583385746799120701262352365042579670818198602072602139160813653741218470078970195077472267653802334170910659129220072720751150879059038516566187965566675732960215029/61035156250000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "q_positive": "48660501453527089089552737487507067721430637146143195476031728101308638149611379613938942149361887008409523863618502409905284123343587125248952209564277830865853253064006278601237791775201992904121328556202482336252069027992016938089192297810535083689690553462468056896275700650086080451066675732960215029/30517578125000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "q_below_internal": "97321002907054178373747480789122492190356235870531375332775927943680026964774615115237938228273004247294924183783013314400416614253200879298737647634957420329181801397927397860839186346258781529921029804922527732346197665829089340870779927279248849120940961483433812034433324267039784971/30517578125000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "q_below_leaf": "877777971996465842262578151916356804523968271898691361850388620386061057850638112991590476136381497590094715876656412874751047790435722169134146746935993721398762208224798007095878671443797517663747930972007983061910807702189464916310309446537531943103724299349913919548933324267039784971/30517578125000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "c_positive": "48660501453527096718947268737507067721430637146143195476031728101308638149611379613938942149361887008409523863618502409905284123343587125248952209564277830865853253064006278601237791775201992904121328556202482336252069027992016938089192297810535083689690553462468056896275700650086080451066675732960215029/30517578125000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "c_below_one": "30468917623546472903281052731262492932278569362853856804523968271898691361850388620386061057850638112991590476136381497590094715876656412874751047790435722169134146746935993721398762208224798007095878671443797517663747930972007983061910807702189464916310309446537531943103724299349913919548933324267039784971/30517578125000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "q_below_reservations": "1/4000000000000000000",
+    "lambda_above_tau": "97321002907054178373747480789122492190356235870531375332775927943680026964774615115237938228273004247294924183783013314400416614253200879298737647634957420329181801397927397860839186346258781529921029804922527732346197665829089340870779927279248849120940961483433812034433324267039784971/61035156250000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "lambda_above_sigma": "1755555943992931686183391244918591116857580307926851348368001312828442088736501610867943014044489990932894507569529812435101678966618243458969555846237030022468342615051668616330918156541336253797574832139093438391475417738549840491749838965795815037086507637216394027063433324267039784971/61035156250000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "lambda_above_internal": "97321002907054178373747480789122492190356235870531375332775927943680026964774615115237938228273004247294924183783013314400416614253200879298737647634957420329181801397927397860839186346258781529921029804922527732346197665829089340870779927279248849120940961483433812034433324267039784971/61035156250000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "lambda_prime_above_lambda": "97321002907054178373747480789122492190356235870531375332775927943680026964774615115237938228273004247294924183783013314400416614253200879298737647634957420329181801397927397860839186346258781529921029804922527732346197665829089340870779927279248849120940961483433812034433324267039784971/61035156250000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "compact_leaf": "877777971996465842262578151916356804523968271898691361850388620386061057850638112991590476136381497590094715876656412874751047790435722169134146746935993721398762208224798007095878671443797517663747930972007983061910807702189464916310309446537531943103724299349913919548933324267039784971/30517578125000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "compact_reservations": "1/4000000000000000000",
+    "lambda_prime_below_one": "48660501453527089089552737487507067721430637146143195476031728101308638149611379613938942149361887008409523863618502409905284123343587125248952209564277830865853253064006278601237791775201992904121328556202482336252069027992016938089192297810535083689690553462468056896275700650086080451066675732960215029/30517578125000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "epsilon_positive": "30517578124999999969482421875000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000/30566238626453527089089552737487507067721430637146143195476031728101308638149611379613938942149361887008409523863618502409905284123343587125248952209564277830865853253064006278601237791775201992904121328556202482336252069027992016938089192297810535083689690553462468056896275700650086080451066675732960215029",
+    "epsilon_below_one": "48660501453527119607130862487507067721430637146143195476031728101308638149611379613938942149361887008409523863618502409905284123343587125248952209564277830865853253064006278601237791775201992904121328556202482336252069027992016938089192297810535083689690553462468056896275700650086080451066675732960215029/30566238626453527089089552737487507067721430637146143195476031728101308638149611379613938942149361887008409523863618502409905284123343587125248952209564277830865853253064006278601237791775201992904121328556202482336252069027992016938089192297810535083689690553462468056896275700650086080451066675732960215029",
+    "guard_width": "48660501453527119607130862487507067721430637146143195476031728101308638149611379613938942149361887008409523863618502409905284123343587125248952209564277830865853253064006278601237791775201992904121328556202482336252069027992016938089192297810535083689690553462468056896275700650086080451066675732960215029/30566238626453527089089552737487507067721430637146143195476031728101308638149611379613938942149361887008409523863618502409905284123343587125248952209564277830865853253064006278601237791775201992904121328556202482336252069027992016938089192297810535083689690553462468056896275700650086080451066675732960215029",
+    "K_geometry": "22936844095203527096718947268737507067721430637146143195476031728101308638149611379613938942149361887008409523863618502409905284123343587125248952209564277830865853253064006278601237791775201992904121328556202482336252069027992016938089192297810535083689690553462468056896275700650086080451066675732960215029/30566238626453527089089552737487507067721430637146143195476031728101308638149611379613938942149361887008409523863618502409905284123343587125248952209564277830865853253064006278601237791775201992904121328556202482336252069027992016938089192297810535083689690553462468056896275700650086080451066675732960215029000000000000000000",
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+  "controls": [
+    "new_endpoint_rejected_by_old_charge",
+    "next_grid",
+    "zero_eta",
+    "zero_beta",
+    "unit_beta"
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+  "seconds": 0.3144757089903578,
+  "python": "3.11.15 (main, Mar  3 2026, 00:52:57) [Clang 17.0.0 (clang-1700.6.3.2)]",
+  "manifest_sha256": "c18faa56483c79b6d9472a1222b9661bec25818a68e89e8e54c03e608011cb57",
+  "scope": "Arithmetic repricing of the unchanged pinned PR552 exact binary-Clifford complex supplier, with cited PR543 bit supplier and unchanged bit bad-class charge. Original source distinguishes exact complex recursion from GL local-ring bit selectors. All inherited all-size compiler, common-label/mutable-source, whole-call hosting, precision/tape and analytic assumptions remain conditional. This verifier does not rerun unchanged physical operators."
+}
diff --git a/research/native-complex-moment552/evidence/PR552-packaged314.json b/research/native-complex-moment552/evidence/PR552-packaged314.json
new file mode 100644
index 0000000000..09ac046a73
--- /dev/null
+++ b/research/native-complex-moment552/evidence/PR552-packaged314.json
@@ -0,0 +1,146 @@
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+  "controls": [
+    "new_endpoint_rejected_by_old_charge",
+    "next_grid",
+    "zero_eta",
+    "zero_beta",
+    "unit_beta"
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+  "seconds": 0.32349079195410013,
+  "python": "3.14.8 (main, Sep 30 2026, 17:55:09) [Clang 21.0.0 (clang-2100.3.34.2)]",
+  "manifest_sha256": "c18faa56483c79b6d9472a1222b9661bec25818a68e89e8e54c03e608011cb57",
+  "scope": "Arithmetic repricing of the unchanged pinned PR552 exact binary-Clifford complex supplier, with cited PR543 bit supplier and unchanged bit bad-class charge. Original source distinguishes exact complex recursion from GL local-ring bit selectors. All inherited all-size compiler, common-label/mutable-source, whole-call hosting, precision/tape and analytic assumptions remain conditional. This verifier does not rerun unchanged physical operators."
+}
diff --git a/research/native-complex-moment552/evidence/PR552-regression-final.json b/research/native-complex-moment552/evidence/PR552-regression-final.json
new file mode 100644
index 0000000000..dcb83cefca
--- /dev/null
+++ b/research/native-complex-moment552/evidence/PR552-regression-final.json
@@ -0,0 +1,147 @@
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+      "cutoff_log2": "55047635511618551378662307484419944359189"
+    }
+  ],
+  "controls": [
+    "new_endpoint_rejected_by_old_charge",
+    "next_grid",
+    "zero_eta",
+    "zero_beta",
+    "unit_beta"
+  ],
+  "seconds": 0.3205774159869179,
+  "python": "3.14.8 (main, Sep 30 2026, 17:55:09) [Clang 21.0.0 (clang-2100.3.34.2)]",
+  "manifest_sha256": "0fa9fe4d6ed0dcacaccdbd0c53e816d00b4bda84c72589759b6ec53f51c22a77",
+  "scope": "Arithmetic repricing of the unchanged pinned selected exact binary-Clifford complex supplier, with cited PR543 bit supplier and unchanged bit bad-class charge. Original source distinguishes exact complex recursion from GL local-ring bit selectors. All inherited all-size compiler, common-label/mutable-source, whole-call hosting, precision/tape and analytic assumptions remain conditional. This verifier does not rerun unchanged physical operators."
+}
diff --git a/research/native-complex-moment552/evidence/PR558-COMPILER-LINEAGE.json b/research/native-complex-moment552/evidence/PR558-COMPILER-LINEAGE.json
new file mode 100644
index 0000000000..1a58a75b96
--- /dev/null
+++ b/research/native-complex-moment552/evidence/PR558-COMPILER-LINEAGE.json
@@ -0,0 +1,41 @@
+{
+  "status": "IDENTICAL_EXACT_COMPLEX_COMPILER_LINEAGE",
+  "files": [
+    {
+      "path": "complex-global/vendor531/code/emit_local.py",
+      "sha256": "eefc76e0bee003015fe21ac685c2cce658033d5444add404d2d509c7d6f39bac",
+      "byte_identical_to_reviewed552": true
+    },
+    {
+      "path": "complex-global/vendor531/code/audit_local.py",
+      "sha256": "69b12c45d0a961175a824c1967353c7f3b94818a7dd6ac7bbcf56a6d1a5ed8e8",
+      "byte_identical_to_reviewed552": true
+    },
+    {
+      "path": "complex-global/vendor531/code/adapters_local.py",
+      "sha256": "4e89a5c517cd87ff6a5aa9b7334c0d74821179ae5777d31284da25b9d8abf113",
+      "byte_identical_to_reviewed552": true
+    },
+    {
+      "path": "complex-global/vendor531/code/emit_supplier_hosted.py",
+      "sha256": "53c0b653b3a5100756d6a9473c90101b8c3ab72723d7047983906ab997bdf920",
+      "byte_identical_to_reviewed552": true
+    },
+    {
+      "path": "complex-global/vendor531/code/audit_supplier_hosted.py",
+      "sha256": "301e9b5c593b6d583065337eb7b3ef207cf039e265f099f89080a4bf25856b43",
+      "byte_identical_to_reviewed552": true
+    },
+    {
+      "path": "complex-global/vendor531/code/invoice_final.py",
+      "sha256": "96b6bed96ca8b60dfc064cd1c7962c45a5cb53c4e52cd90aaea6ccf8d109db97",
+      "byte_identical_to_reviewed552": true
+    },
+    {
+      "path": "complex-global/vendor531/vendor/clifford/clifford_exact.py",
+      "sha256": "2982ead07721e83358dc67c6f49c2b4c16a236a6bebdfd72f58221be1f5d0962",
+      "byte_identical_to_reviewed552": true
+    }
+  ],
+  "assessment": "The existing zero-complex-badclass scope argument carries to the promoted supplier conditional on fresh phase/owner/invoice admission. Changing starts, widths and owners does not introduce a probabilistic/q-adic selector; it changes the exact profile and requires the fresh admission now running."
+}
diff --git a/research/native-complex-moment552/evidence/PR558-INDEPENDENT-RECEIPT.json b/research/native-complex-moment552/evidence/PR558-INDEPENDENT-RECEIPT.json
new file mode 100644
index 0000000000..2064abce71
--- /dev/null
+++ b/research/native-complex-moment552/evidence/PR558-INDEPENDENT-RECEIPT.json
@@ -0,0 +1,71 @@
+{
+  "status": "PASS_COMPLEX_EIGHT_RANK5_PIVOTS_AND_CITED_BIT",
+  "steps": [
+    {
+      "name": "reconstruct552-and-promote8",
+      "exit_code": 0,
+      "seconds": 14.96228679199703,
+      "log_sha256": "b7b035547780f64e33ea2432fd2cc79ed5795bf88c9791cbb913d9ec2a7c6a2b"
+    },
+    {
+      "name": "all-columns-native-frames",
+      "exit_code": 0,
+      "seconds": 0.4768510829890147,
+      "log_sha256": "b3b7b19f1d9c7baa563cfa169302daa4486727169334c8102bc9b6b606a3c3e5"
+    },
+    {
+      "name": "emit-both-reflected-ledgers",
+      "exit_code": 0,
+      "seconds": 0.44516691705211997,
+      "log_sha256": "f382385a67f705b446978a73fc6492ce127cb2ae76011e87e36fdbe9977f4347"
+    },
+    {
+      "name": "independent-local-and-controls",
+      "exit_code": 0,
+      "seconds": 1.4429402919486165,
+      "log_sha256": "b694b4b22abb9d1de9a70bbbdf9e9b0e030ec00a7eddc264e4c18376dd859cee"
+    },
+    {
+      "name": "normalized-wrappers",
+      "exit_code": 0,
+      "seconds": 0.8871639160206541,
+      "log_sha256": "284cce3149354d057b4e2f04163535c0d8b0a924898e5b771ba8af2740314154"
+    },
+    {
+      "name": "all-phase-recipes",
+      "exit_code": 0,
+      "seconds": 100.04125804197975,
+      "log_sha256": "33aa90a041ba29ebb44f21206dc7990a45b29297ef134780940f608844a4722c"
+    },
+    {
+      "name": "literal-owners-full-invoice",
+      "exit_code": 0,
+      "seconds": 8.275277458014898,
+      "log_sha256": "2fee1e317b9601b87b6e96b23d8f6077abf7ea38b7154aa215c46c1a9bde9a53"
+    },
+    {
+      "name": "two-engine-exact-assembly",
+      "exit_code": 0,
+      "seconds": 0.1878548749955371,
+      "log_sha256": "b50e71a1c084e221ca32b43001f5383c8cffb09d72e66b1436f22aa240795a9c"
+    },
+    {
+      "name": "assembly-admission-controls",
+      "exit_code": 0,
+      "seconds": 0.19738204102031887,
+      "log_sha256": "9cc0577eb648247a02f4352eaf5187000a8378657f388e8ffa8cbeb55a43b9ec"
+    }
+  ],
+  "artifacts": {
+    "RESULT.json": "b32113b56745cd5c142c982426ce31f0fe8d4f1a4ba219541e8c8befc69f017c",
+    "ASSEMBLY.json": "de60471def08f776e5de1b79bb80d871a4f528ecda8fb8eaf99f3757173c520e",
+    "ASSEMBLY-CONTROLS.json": "a97f22aeb8ca732f9bfce074c30be73564da8892118395136456ec7533654331",
+    "reconstruction/promoted/PROMOTION.json": "9ad34faa8611a6ea7f161d2e1fe331ab1ba5ccec8fca3978e31c3e7bf789bc31",
+    "complex-local/INDEPENDENT-LOCAL-REPLAY.json": "ad3f2b941127688f98ce6872d00f110644f1c2b156f8adad1b6700e838ed0e26",
+    "complex-local/CLIFFORD-REVIEW.json": "a1d49f9272a5427d3d34086f7cc3c3e6c6fd24655b600277c1e816970e8cfb66",
+    "complex-global/audit/ASYNC-AUDIT.json": "177cfc604cd7085019b615945f33c04197eee181e8c30ec13831cf253c138e23",
+    "complex-global/audit/OWNERS-AND-PROFILE.json": "33bf86f3968796cb2f98214674846c9d51e4f6bb55c6d4dd7159b0a8cdf6a21b",
+    "complex-global/audit/PREFIX-AND-INVOICE.json": "0518f59792aaf017af7669fa3d617aaadb391a4d4ffd8498916dcd7a3d2bfbbf"
+  },
+  "manifest_sha256": "f28c04eecbea7599fa9e0867dd8d0311a463af12cab79f90a43733b162527164"
+}
diff --git a/research/native-complex-moment552/evidence/PR558-INDEPENDENT-REPLAY.json b/research/native-complex-moment552/evidence/PR558-INDEPENDENT-REPLAY.json
new file mode 100644
index 0000000000..2521c93be8
--- /dev/null
+++ b/research/native-complex-moment552/evidence/PR558-INDEPENDENT-REPLAY.json
@@ -0,0 +1,31 @@
+{
+  "status": "PASS_COMPLEX_EIGHT_RANK5_PIVOTS_AND_CITED_BIT",
+  "kappa": "796258944657507/500000000000000000",
+  "kappa_decimal": "0.001592517889315014",
+  "complex_lower": "199382255973827096291/125000000000000000000000",
+  "bit_lower": "413456747036699736067/250000000000000000000000",
+  "word_sha256": "375bd23001cd805707968005a9342827d5464e3f8e3a754197b31c46fc3f63c2",
+  "local_profile": {
+    "1": 2316,
+    "4": 192,
+    "8": 8,
+    "3": 464,
+    "2": 424,
+    "6": 56,
+    "5": 56
+  },
+  "global_profile": {
+    "1": 166880,
+    "2": 30528,
+    "3": 35779,
+    "4": 13824,
+    "5": 4032,
+    "6": 4032,
+    "7": 208,
+    "8": 576,
+    "39": 1,
+    "48": 1660
+  },
+  "seconds": 126.92180670797825,
+  "scope": "Conditional finite complex witness composed with cited unchanged PR543 bit supplier. All-size compiler, common-label/mutable-source realization, whole-call hosting, precision/tape and analytic transfer remain assumptions. No unconditional theorem or full Lean certification."
+}
diff --git a/research/native-complex-moment552/evidence/PR558-packaged311.json b/research/native-complex-moment552/evidence/PR558-packaged311.json
new file mode 100644
index 0000000000..463a2b1174
--- /dev/null
+++ b/research/native-complex-moment552/evidence/PR558-packaged311.json
@@ -0,0 +1,152 @@
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+  "python": "3.11.15 (main, Mar  3 2026, 00:52:57) [Clang 17.0.0 (clang-1700.6.3.2)]",
+  "manifest_sha256": "0fa9fe4d6ed0dcacaccdbd0c53e816d00b4bda84c72589759b6ec53f51c22a77",
+  "scope": "Arithmetic repricing of the unchanged pinned selected exact binary-Clifford complex supplier, with cited PR543 bit supplier and unchanged bit bad-class charge. Original source distinguishes exact complex recursion from GL local-ring bit selectors. All inherited all-size compiler, common-label/mutable-source, whole-call hosting, precision/tape and analytic assumptions remain conditional. This verifier does not rerun unchanged physical operators."
+}
diff --git a/research/native-complex-moment552/evidence/PR558-packaged314.json b/research/native-complex-moment552/evidence/PR558-packaged314.json
new file mode 100644
index 0000000000..3f4f93f70c
--- /dev/null
+++ b/research/native-complex-moment552/evidence/PR558-packaged314.json
@@ -0,0 +1,152 @@
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+    },
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+      "cutoff_log2": "54781202327468026783894182173401441017136"
+    },
+    {
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+      "gap": "159724669638872638181979211719709024193870661556461524836074006960253501145838669324165798463425942660987350955192816966993070590009343134274334948862002596361065027121783723946602780949394897691927160711225005909739498303387244795851338828342064202978541685153096883149281751278938569336702231264610156481193441/3906250000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+      "cutoff_log2": "21531712536714952515574694935268683045804843382"
+    }
+  ],
+  "controls": [
+    "new_endpoint_rejected_by_old_charge",
+    "next_grid",
+    "zero_eta",
+    "zero_beta",
+    "unit_beta"
+  ],
+  "seconds": 0.32867266691755503,
+  "python": "3.14.8 (main, Sep 30 2026, 17:55:09) [Clang 21.0.0 (clang-2100.3.34.2)]",
+  "manifest_sha256": "0fa9fe4d6ed0dcacaccdbd0c53e816d00b4bda84c72589759b6ec53f51c22a77",
+  "scope": "Arithmetic repricing of the unchanged pinned selected exact binary-Clifford complex supplier, with cited PR543 bit supplier and unchanged bit bad-class charge. Original source distinguishes exact complex recursion from GL local-ring bit selectors. All inherited all-size compiler, common-label/mutable-source, whole-call hosting, precision/tape and analytic assumptions remain conditional. This verifier does not rerun unchanged physical operators."
+}
diff --git a/research/native-complex-moment552/inputs/PR543-EXPECTED.json b/research/native-complex-moment552/inputs/PR543-EXPECTED.json
new file mode 100644
index 0000000000..b61d859826
--- /dev/null
+++ b/research/native-complex-moment552/inputs/PR543-EXPECTED.json
@@ -0,0 +1,239 @@
+{
+ "word": {
+  "file": "data/bitE8mut-r420.json.gz",
+  "file_sha256": "9721a3819efd68307f66f00d84a952eb28eb7c3ad2ce79653ce295c98a764784",
+  "content_sha256": "e597e3885c0a8daa262afa5985ee2930443329126aece76d33159d81882bdcc2",
+  "plan_sha256": "d5c2494b4baae801143049e172db303b3c8e25a002e9d9715a1d79dcb2d74c58",
+  "R": 420,
+  "retained_totals": 9,
+  "gates": {
+   "D": 120,
+   "A": 2064,
+   "B": 837,
+   "C": 2661
+  },
+  "local_histogram": {
+   "1": 1320,
+   "2": 1440,
+   "4": 60,
+   "5": 120,
+   "8": 9
+  },
+  "legality": {
+   "records": 29874,
+   "common_frame_ADDs": 26916,
+   "rank_mass": 5112
+  },
+  "five_stage_columns": 900,
+  "five_stage_payload_bits": 100,
+  "nine_stage_columns": 1380,
+  "nine_stage_payload_bits": 100,
+  "eleven_stage_columns": 1620,
+  "eleven_stage_payload_bits": 100
+ },
+ "interface": {
+  "hypotheses": {
+   "H1_no_target_feeds_z": true,
+   "H2_D_frame0_targets_from_helpers": true,
+   "H3_A_z_only_scatter_targets_B_split": true,
+   "H4_C_literal_reverse_at_full": true,
+   "H5_V_identity": true
+  },
+  "conclusion": {
+   "U_block_form": true,
+   "J_identity": true,
+   "K_equals_D": true,
+   "C_is_U_inverse": true
+  },
+  "interleaving": {
+   "z_gates_A": 2064,
+   "source_writes_A": 0,
+   "z_gates_B": 597,
+   "source_writes_B": 597,
+   "helper_writes_B": 0,
+   "sources_rewritten_B": 120,
+   "B_z_write_frame_dims": {
+    "4": 477,
+    "6": 120
+   },
+   "B_z_operand_kinds": {
+    "source<-helper": 120,
+    "source<-source": 477
+   },
+   "B_z_operand_kinds_by_dim": {
+    "dim4:source<-source": 477,
+    "dim6:source<-helper": 120
+   },
+   "B_z_writes_after_a_target_read_of_the_same_register": 597,
+   "B_target_reads_of_registers_rewritten_earlier_in_B": 120,
+   "target_reads_of_targets": 0,
+   "cleanup_gates": 2661
+  }
+ },
+ "nine": {
+  "pipeline": "PR #510 (PR #463 nine-stage template, 32 replicas, PR #442 hosting, PR #493 outer assembly)",
+  "bit_lower": "413456747036699736067/250000000000000000000000",
+  "bit_lower_decimal": "0.001653826988146798944268",
+  "profile": {
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+   "2": 426624,
+   "4": 32640,
+   "5": 34560,
+   "6": 15360,
+   "7": 23040,
+   "8": 2592,
+   "42": 3456
+  },
+  "bit": {
+   "live": 30720,
+   "physical": 30729,
+   "maxchild": 42,
+   "rank_mass": 1956096,
+   "hosted_residuals": 120960,
+   "bank_assignments": 0,
+   "prefix_bits": 189,
+   "charged_prefix_bits": 192,
+   "coefficient_bits": 189,
+   "two_level_coefficient_bits": 416,
+   "charged_two_level_coefficient_bits": 512,
+   "row_coefficient": 18355,
+   "halving_depth": 2
+  },
+  "kappa": {
+   "pr535": {
+    "kappa": "793733736881443/500000000000000000",
+    "kappa_decimal": "0.001587467473762886",
+    "complex_lower": "1589991533606044174031/1000000000000000000000000",
+    "binding_component": "complex"
+   },
+   "pr531": {
+    "kappa": "1517760977820331/1000000000000000000",
+    "kappa_decimal": "0.001517760977820331",
+    "complex_lower": "304013615566499291631/200000000000000000000000",
+    "binding_component": "complex"
+   }
+  }
+ },
+ "eleven": {
+  "pipeline": "PR #493 (PR #445 geometry, PR #472 emitter, audit and invoice, PR #442 hosting)",
+  "bit_lower": "82015816228313440961/50000000000000000000000",
+  "bit_lower_decimal": "0.00164031632456626881922",
+  "profile": {
+   "1": 116160,
+   "2": 130560,
+   "4": 9120,
+   "5": 10560,
+   "6": 3839,
+   "7": 7680,
+   "8": 792,
+   "14": 1,
+   "56": 3497
+  },
+  "bit": {
+   "live": 9600,
+   "physical": 9609,
+   "maxchild": 56,
+   "rank_mass": 745536,
+   "hosted_residuals": 36960,
+   "prefix_bits": 4710,
+   "charged_prefix_bits": 4736,
+   "coefficient_bits": 4710,
+   "three_level_coefficient_bits": 14185,
+   "charged_three_level_coefficient_bits": 14208,
+   "row_coefficient": 19431,
+   "halving_depth": 3
+  },
+  "emitted_program_sha256": "3de4325e0df33ca85116d05cbcc6810f025d6083e5134d128236769c97699f51",
+  "kappa": {
+   "pr535": {
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+    "kappa_decimal": "0.001587467473762886",
+    "complex_lower": "1589991533606044174031/1000000000000000000000000",
+    "binding_component": "complex"
+   },
+   "pr531": {
+    "kappa": "1517760977820331/1000000000000000000",
+    "kappa_decimal": "0.001517760977820331",
+    "complex_lower": "304013615566499291631/200000000000000000000000",
+    "binding_component": "complex"
+   }
+  }
+ },
+ "controls": {
+  "pr510_r480_nine": {
+   "word": "vendor/pr510/e8-r480-dead-totals/data/bitE8mut-r480.json.gz",
+   "bit_lower": "396861776523137034277/250000000000000000000000",
+   "bit_lower_decimal": "0.001587447106092548137108",
+   "kappa": {
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+     "kappa_decimal": "0.001584931111785787",
+     "complex_lower": "1589991533606044174031/1000000000000000000000000",
+     "binding_component": "bit"
+    },
+    "pr531": {
+     "kappa": "1517760977820331/1000000000000000000",
+     "kappa_decimal": "0.001517760977820331",
+     "complex_lower": "304013615566499291631/200000000000000000000000",
+     "binding_component": "complex"
+    },
+    "pr484": {
+     "kappa": "662515748707749/500000000000000000",
+     "kappa_decimal": "0.001325031497415498",
+     "complex_lower": "663394767670133/500000000000000000",
+     "binding_component": "complex"
+    },
+    "pr471": {
+     "kappa": "260706019542937/200000000000000000",
+     "kappa_decimal": "0.001303530097714685",
+     "complex_lower": "1305231506267591/1000000000000000000",
+     "binding_component": "complex"
+    },
+    "pr458": {
+     "kappa": "323880830669511/250000000000000000",
+     "kappa_decimal": "0.001295523322678044",
+     "complex_lower": "1297203880559577927527/1000000000000000000000000",
+     "binding_component": "complex"
+    }
+   }
+  },
+  "pr493_r489_eleven": {
+   "word": "../e8-bit-carrier-skeleton/data/bit/bitE8mut-r489.json.gz",
+   "bit_lower": "391360506745414227149/250000000000000000000000",
+   "bit_lower_decimal": "0.001565442026981656908596",
+   "emitted_program_sha256": "e37b52e4861c39154b5891460fb45df8651c92cd68961078f8f06596f057c822",
+   "kappa": {
+    "pr535": {
+     "kappa": "97687203033227/62500000000000000",
+     "kappa_decimal": "0.001562995248531632",
+     "complex_lower": "1589991533606044174031/1000000000000000000000000",
+     "binding_component": "bit"
+    },
+    "pr531": {
+     "kappa": "1517760977820331/1000000000000000000",
+     "kappa_decimal": "0.001517760977820331",
+     "complex_lower": "304013615566499291631/200000000000000000000000",
+     "binding_component": "complex"
+    },
+    "pr484": {
+     "kappa": "662515748707749/500000000000000000",
+     "kappa_decimal": "0.001325031497415498",
+     "complex_lower": "663394767670133/500000000000000000",
+     "binding_component": "complex"
+    },
+    "pr471": {
+     "kappa": "260706019542937/200000000000000000",
+     "kappa_decimal": "0.001303530097714685",
+     "complex_lower": "1305231506267591/1000000000000000000",
+     "binding_component": "complex"
+    },
+    "pr458": {
+     "kappa": "323880830669511/250000000000000000",
+     "kappa_decimal": "0.001295523322678044",
+     "complex_lower": "1297203880559577927527/1000000000000000000000000",
+     "binding_component": "complex"
+    }
+   }
+  }
+ }
+}
diff --git a/research/native-complex-moment552/inputs/PR552-EXACT-ASSEMBLY.json b/research/native-complex-moment552/inputs/PR552-EXACT-ASSEMBLY.json
new file mode 100644
index 0000000000..f5bf532a11
--- /dev/null
+++ b/research/native-complex-moment552/inputs/PR552-EXACT-ASSEMBLY.json
@@ -0,0 +1,176 @@
+{
+  "status": "PASS_ASYNC_COMPLEX_AND_CITED_BIT_EXACT_ASSEMBLY",
+  "complex_lower": "1594507308884431019637/1000000000000000000000000",
+  "complex_decimal": "0.001594507308884431019637",
+  "next_b_rejected": "797253654442215509819/500000000000000000000000",
+  "exact_moment_receipts": [
+    {
+      "endpoint": "1594507308884431019637/1000000000000000000000000",
+      "accept": true,
+      "mercator": [
+        "-40186902685703231311821970003273532124160065/13479973333575319897333507543509815336818572211270286240551805124608",
+        "-627920354464112989247218281301148939440001/210624583337114373395836055367340864637790190801098222508621955072"
+      ],
+      "base_two": [
+        "-1863269341316411144019/625000000000000000000000000000000000000000000",
+        "-3726537432632822288021/1250000000000000000000000000000000000000000000"
+      ]
+    },
+    {
+      "endpoint": "797253654442215509819/500000000000000000000000",
+      "accept": false,
+      "mercator": [
+        "2584151432270445003531030214735456403866167/26959946667150639794667015087019630673637144422540572481103610249216",
+        "2584151432270445003531030214735456403866169/26959946667150639794667015087019630673637144422540572481103610249216"
+      ],
+      "base_two": [
+        "958510539299785465391/10000000000000000000000000000000000000000000000",
+        "958520539299785465527/10000000000000000000000000000000000000000000000"
+      ]
+    }
+  ],
+  "bit_lower": "396861776523137034277/250000000000000000000000",
+  "bit_decimal": "0.001587447106092548137108",
+  "bit_exact_moment_receipts": [
+    {
+      "endpoint": "396861776523137034277/250000000000000000000000",
+      "accept": true,
+      "mercator": [
+        "-16022724396259728143779813573647160538228917/26959946667150639794667015087019630673637144422540572481103610249216",
+        "-16022724396259728143779813573647160538228915/26959946667150639794667015087019630673637144422540572481103610249216"
+      ],
+      "base_two": [
+        "-5943162573288295739953/10000000000000000000000000000000000000000000000",
+        "-1188630514657659147971/2000000000000000000000000000000000000000000000"
+      ]
+    },
+    {
+      "endpoint": "1587447106092548137109/1000000000000000000000000",
+      "accept": false,
+      "mercator": [
+        "70448808687833500409705881909924095280013277/26959946667150639794667015087019630673637144422540572481103610249216",
+        "70448808687833500409705881909924095280013279/26959946667150639794667015087019630673637144422540572481103610249216"
+      ],
+      "base_two": [
+        "3266363428371004299993/1250000000000000000000000000000000000000000000",
+        "26130917426968034400043/10000000000000000000000000000000000000000000000"
+      ]
+    }
+  ],
+  "bit_result_sha256": "19b861f69f7602e47b390d472b487d044bff4230b250520807537c213ad0249f",
+  "bit_dependency": {
+    "pull_request": 510,
+    "commit": "64edee18af02aab0c1696b677a0287dd0146381b",
+    "path": "research/e8-r480-nine-stage",
+    "expected_sha256": "19b861f69f7602e47b390d472b487d044bff4230b250520807537c213ad0249f"
+  },
+  "bit_prefix_bits": 256,
+  "bit_charged_two_level_coefficient_bits": 512,
+  "external_row_allowance": 20161,
+  "kappa": "1584931111785787/1000000000000000000",
+  "kappa_decimal": "0.001584931111785787",
+  "binding_component": "bit",
+  "improvements": {
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+      "relative_gain": "67170133965456/1517760977820331",
+      "percent_display": "4.42560686"
+    },
+    "rejected_POST208_checkpoint": {
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+      "relative_gain": "41290330067045/1543640781718742",
+      "percent_display": "2.67486649"
+    },
+    "PR533": {
+      "baseline": "381051987698311/250000000000000000",
+      "relative_gain": "60723160992543/1524207950793244",
+      "percent_display": "3.98391577"
+    }
+  },
+  "strict_outer_constraints": {
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+    "complex_below_one_over32": "29655492691115568980363/1000000000000000000000000",
+    "beta_positive": "1/1000000000000000000",
+    "beta_below_one": "999999999999999999/1000000000000000000",
+    "leaf_saving_above_bit": "7060202791882905723602508024811241233500225499627723644532489569312798749735575207324411482233239583707631312250354296798781303533719966836301725924238770077838843/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
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+    "q_below_reservations": "1/4000000000000000000",
+    "lambda_above_tau": "1587447106092548112318890183090757739129499774500372276355467510430687201250264424792675588517766760416292368687749645703201218696466280033163698274075761229922161157/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "lambda_above_sigma": "7060202791882908905556923001790373189390408590385462774032264069685075105203085638011612732497664376383219830017114713091149991283365670037520422390518803241537117075761229922161157/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
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+    "lambda_prime_above_lambda": "1587447106092548112318890183090757739129499774500372276355467510430687201250264424792675588517766760416292368687749645703201218696466280033163698274075761229922161157/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "compact_leaf": "3530101395941454449248360104953732935640295840571600951766019285028675730335298034349406991381044584529404173891937564691759339516505686619369559428399418202617695575761229922161157/500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "compact_reservations": "1/4000000000000000000",
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+    "delta_positive": "1/8000000000000000000",
+    "delta_below_one_eighth": "999999999999999999/8000000000000000000",
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+    "small_field_exposure": "1/1000000000000000000",
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+    "literal_scalar_guard": "1",
+    "row_product_gap": "23971789/25",
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+    "dimension_above_kappa": "499206276446953725803778699156052906516528828324756692513467019206410954553289688063888332210504621558043889235058850438192715939568445670965056852111683916008162838192180988331381964648015661900676075559/500793723553046274054571997985452830757245859704159428399048233980714971324269664701965650593008618955415470595826108062435308240660483494313380630440571600581797382304424238770077838843000000000000000000"
+  },
+  "outer_controls": [
+    "next_grid",
+    "zero_eta",
+    "zero_beta",
+    "unit_beta"
+  ],
+  "bootstrap": [
+    {
+      "gap": "2458935257685996988742640373065976070329449/1000000000000000000000000000000000000000000000000",
+      "cutoff_log2": "5953992899983"
+    },
+    {
+      "gap": "3903429658882570053869779063890487076693439864938224920711764043/1000000000000000000000000000000000000000000000000000000000000000000000000",
+      "cutoff_log2": "2362706551212503884"
+    },
+    {
+      "gap": "6196488115828958169115639209590415499556136073588052115988731692262446100679396643601/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+      "cutoff_log2": "937586312398646736663238"
+    },
+    {
+      "gap": "9836597127409545868283314560200714400615051377525575546685064142160089262527033828463395195128020962202307/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+      "cutoff_log2": "372059785734359967876272695321"
+    },
+    {
+      "gap": "15615077643704555604994425885436776413026753051081759442584659149990102084077935377970027573490939593604820736101011172907705849/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+      "cutoff_log2": "147643456746455004602217631773182069"
+    },
+    {
+      "gap": "24788109816909242260870500225499627723644532489569312798749735575207324411482233239583707631312250354296798781303533719966836301725924238770077838843/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+      "cutoff_log2": "58588944991775862180494665844538845859209"
+    }
+  ],
+  "complex_template_sha256": "1ddb317665797be007dfcc06e295f46bf9115ddeb308a21a60f98b9c28c6aff0",
+  "owner_audit_sha256": "ecac65c65418641abf598db74d588ba8f53b3f358505bd31e215f09562922e3f",
+  "invoice_sha256": "4ac060dbf2ea2624dd1f4bda23bdb81c0a3013dfe49b77c092b631885529947f",
+  "bit_scope": "Pinned PR510 published profile and operator are cited dependencies. Both exact moments are freshly repriced here; the unchanged physical bit operator is not cold-reconstructed. See PR510-MUTABLE-SOURCE-CARRYOVER.md.",
+  "scope": "Conditional finite witness under the retained all-size compiler, common-label/mutable-basis, whole-call hosting, restored-work, precision/tape and analytic-transfer contracts. Fresh complex admission and finite invoice are separately hash-bound; no unconditional theorem is claimed."
+}
diff --git a/research/native-complex-moment552/inputs/PR552-EXPECTED.json b/research/native-complex-moment552/inputs/PR552-EXPECTED.json
new file mode 100644
index 0000000000..7ee2fb720a
--- /dev/null
+++ b/research/native-complex-moment552/inputs/PR552-EXPECTED.json
@@ -0,0 +1,30 @@
+{
+  "status": "PASS_FILTRATION_WAVEFRONT",
+  "kappa": "795984451416673/500000000000000000",
+  "kappa_decimal": "0.001591968902833346",
+  "bit_lower": "413456747036699736067/250000000000000000000000",
+  "complex_lower": "1594507308884431019637/1000000000000000000000000",
+  "complex_template_sha256": "1ddb317665797be007dfcc06e295f46bf9115ddeb308a21a60f98b9c28c6aff0",
+  "complex_canonical_word_sha256": "995e46ed6dc8377f02bf05a3370171796364cedefdfeaeae47c2a207b5a8b173",
+  "complex_global_profile": {
+    "1": 165600,
+    "2": 32128,
+    "3": 35907,
+    "4": 13248,
+    "5": 4032,
+    "6": 4032,
+    "7": 208,
+    "8": 576,
+    "39": 1,
+    "48": 1660
+  },
+  "complex_prefix_bits": 225,
+  "forward_events": 21483,
+  "backward_events": 21483,
+  "waves": 8,
+  "wave_omission_controls": 8,
+  "coset_rewrites": 12,
+  "compensation_additions_saved": 624,
+  "outer_constraints": 47,
+  "bit_operator": "Pinned PR543 cited dependency, not physically replayed here"
+}
diff --git a/research/native-complex-moment552/inputs/PR552-INVOICE.json b/research/native-complex-moment552/inputs/PR552-INVOICE.json
new file mode 100644
index 0000000000..fc99981f77
--- /dev/null
+++ b/research/native-complex-moment552/inputs/PR552-INVOICE.json
@@ -0,0 +1,106 @@
+{
+  "status": "PASS_INDEPENDENT_ASYNC_SCALAR_PREFIX_AND_FINITE_INVOICE",
+  "template_sha256": "1ddb317665797be007dfcc06e295f46bf9115ddeb308a21a60f98b9c28c6aff0",
+  "parent_template_sha256": "7eddf21720ce26ff72069a0e94fa7f0e692cfe3bea2defb1a3769bea352cf471",
+  "owner_audit_sha256": "ecac65c65418641abf598db74d588ba8f53b3f358505bd31e215f09562922e3f",
+  "controller_audit_sha256": "01d37b06d35ffb1995533cc618b0baaa093f55d288786153406fc46c94ab9926",
+  "independent_local_replay_sha256": "e871d8d56014146c19963430e4822714aadbf57367f783dc1dd42fa3f5b0a0ce",
+  "canonical_word_sha256": "995e46ed6dc8377f02bf05a3370171796364cedefdfeaeae47c2a207b5a8b173",
+  "local_programs_sha256": "7b191ec016ab99a0e1415206b898a8a6c5485fe9e8d91e59539e7c3710725563",
+  "local_checks_sha256": "2c51cdb9b282aadb99ea00d05b6c4bee140577111bdfd5c675b038373e309a17",
+  "global_prefix_bits": 225,
+  "charged_prefix_bits": 512,
+  "complete_invocation_sweeps": 72,
+  "bridge_sweeps": 72,
+  "row_prefix_extra": 1024,
+  "local": {
+    "backward": {
+      "events": 21483,
+      "macros": 16290,
+      "real_component_primitives": 161356,
+      "prefix_l1": "1044730572396",
+      "prefix_bits": 40,
+      "coefficient_work_primitive_count": 79906,
+      "all_exact_columns": 728,
+      "dirty_helper_columns": 504,
+      "all_dirty_helper_columns_restored": true,
+      "private_centres_and_coefficient_work_cleared": true,
+      "pre_wrappers": 212,
+      "post_wrappers": 212,
+      "endpoint_evidence": "Hash-bound fresh INDEPENDENT-LOCAL-REPLAY.json"
+    },
+    "forward": {
+      "events": 21483,
+      "macros": 16290,
+      "real_component_primitives": 161308,
+      "prefix_l1": "88876257700",
+      "prefix_bits": 37,
+      "coefficient_work_primitive_count": 79874,
+      "all_exact_columns": 728,
+      "dirty_helper_columns": 504,
+      "all_dirty_helper_columns_restored": true,
+      "private_centres_and_coefficient_work_cleared": true,
+      "pre_wrappers": 212,
+      "post_wrappers": 212,
+      "endpoint_evidence": "Hash-bound fresh INDEPENDENT-LOCAL-REPLAY.json"
+    }
+  },
+  "helpers": 504,
+  "centres": 8,
+  "live": 7168,
+  "physical": 7682,
+  "physical_stock_identity": {
+    "data_and_owner_families": 7168,
+    "local_helpers": 504,
+    "private_centres": 8,
+    "coefficient_work": 2
+  },
+  "entering_helpers": 212,
+  "entrance_rank": 380,
+  "width_inventory": {
+    "5": 56,
+    "8": 156,
+    "9": 292
+  },
+  "local_adapters": 333344,
+  "local_primitive_bound": 8645055460940,
+  "wrapper_elementary_factors_charged": 4396032,
+  "factors_per_wrapper": 144,
+  "compiler": 25934336,
+  "unhosted_pervertex": 1574554051175192,
+  "hosted_rebased_pervertex": 1574554050875816,
+  "conservative_pervertex_used": 4452501040899760,
+  "retained_prior_overcharge": {
+    "sha256": "ce1bfa49be88ae67a404f606a11db2232b7e1f2610d4418058e39749d86d8478",
+    "previous_pervertex_allowance": 2877946989724568,
+    "added_complete_new_pervertex": 1574554051175192,
+    "policy": "Retain old allowance and add the entire fresh recount."
+  },
+  "finite": {
+    "physical_stock_bits": 2641,
+    "router_bits": 7985,
+    "literal_bits": 35282,
+    "physical_row_coefficient": 16467,
+    "retained_row_coefficient": 20161,
+    "required_C0_bits": 200012,
+    "halving_integer_inequality": [
+      5329,
+      4608
+    ],
+    "G_exponent": 30000,
+    "E_exponent": 100000,
+    "C0_exponent": 210000
+  },
+  "strict_checks": {
+    "local_primitive_bound": true,
+    "logical": true,
+    "router": true,
+    "literal": true,
+    "mass": true,
+    "precision": true,
+    "tape": true,
+    "row": true,
+    "halving": true
+  },
+  "scope": "Fresh emitted primitive/Horner, prefix, wrapper and finite recount, using the hash-bound independent local all-column/frame replay and actual whole-owner audit. No old INTEGRATION receipt is used to certify the changed word. The inherited all-size compiler, precision/tape and analytic-transfer contracts remain conditional. Retained payload 512 and physical row allowance 20161 are unchanged."
+}
diff --git a/research/native-complex-moment552/inputs/PR552-OWNERS.json b/research/native-complex-moment552/inputs/PR552-OWNERS.json
new file mode 100644
index 0000000000..fbb8ae601b
--- /dev/null
+++ b/research/native-complex-moment552/inputs/PR552-OWNERS.json
@@ -0,0 +1,40 @@
+{
+  "status": "PASS_INDEPENDENT_ASYNC_OWNERS_AND_PROFILE",
+  "template_sha256": "1ddb317665797be007dfcc06e295f46bf9115ddeb308a21a60f98b9c28c6aff0",
+  "paid_profile": {
+    "1": 165600,
+    "2": 32128,
+    "3": 35907,
+    "4": 13248,
+    "5": 4032,
+    "6": 4032,
+    "7": 208,
+    "8": 576,
+    "39": 1,
+    "48": 1660
+  },
+  "live": 7168,
+  "physical": 7682,
+  "rank_mass": 520704,
+  "deficit": 2560,
+  "hosted_residuals": 36288,
+  "banked_residuals": 0,
+  "all_live_owners_per_call_distinct": true,
+  "owners_per_call": 728,
+  "actual_helper_owner_charts": 36288,
+  "actual_73column_sigmas": 2972,
+  "cross_replica_helper_roles": 28243,
+  "explicit_idle_remainder_children": 4240,
+  "source_idle_children_charged": 16128,
+  "source_artifacts": {
+    "split-centres/CANONICAL-WORD.json.gz": "995e46ed6dc8377f02bf05a3370171796364cedefdfeaeae47c2a207b5a8b173",
+    "split-centres/LOCAL-PROGRAMS.json.gz": "7b191ec016ab99a0e1415206b898a8a6c5485fe9e8d91e59539e7c3710725563",
+    "split-centres/LOCAL-CHECKS.json": "2c51cdb9b282aadb99ea00d05b6c4bee140577111bdfd5c675b038373e309a17",
+    "split-centres/RESIDUAL-ADAPTERS.json": "e4fce8ca73ad20a86711837dd6fe1efad5a335689352791a635969e4d1be4391",
+    "split-centres/CLIFFORD-RECIPES.json": "b8e0e89fdb9d140468fa63d18c073664aa6911d790f6cbcb49b238ebba117533",
+    "split-centres/CLIFFORD-REVIEW.json": "abc348faf7f7177b36712ed43ae3d2cc5dafea8f16bf8a1a27391fb49dc02806",
+    "split-centres/INDEPENDENT-LOCAL-REPLAY.json": "e871d8d56014146c19963430e4822714aadbf57367f783dc1dd42fa3f5b0a0ce",
+    "work/complex-nine-native/DATA-PHASES.json": "c5916ff312678f12aba403847643fbbc29f643acec60bdf836fa56cd3e8251fa"
+  },
+  "controller_audit_sha256": "01d37b06d35ffb1995533cc618b0baaa093f55d288786153406fc46c94ab9926"
+}
diff --git a/research/native-complex-moment552/inputs/PR558-BASELINE-ASSEMBLY.json b/research/native-complex-moment552/inputs/PR558-BASELINE-ASSEMBLY.json
new file mode 100644
index 0000000000..3f005b6df6
--- /dev/null
+++ b/research/native-complex-moment552/inputs/PR558-BASELINE-ASSEMBLY.json
@@ -0,0 +1,187 @@
+{
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+  "controls": [
+    "next_grid",
+    "zero_eta",
+    "zero_beta",
+    "unit_beta"
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+  "scope": "Conditional finite complex witness composed with cited unchanged PR543 bit supplier. All-size compiler, common-label/mutable-source realization, whole-call hosting, precision/tape and analytic transfer remain assumptions. No unconditional theorem or full Lean certification."
+}
diff --git a/research/native-complex-moment552/inputs/PR558-EXPECTED.json b/research/native-complex-moment552/inputs/PR558-EXPECTED.json
new file mode 100644
index 0000000000..1f6a020cb1
--- /dev/null
+++ b/research/native-complex-moment552/inputs/PR558-EXPECTED.json
@@ -0,0 +1,44 @@
+{
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+  "scope": "Conditional finite complex witness composed with cited unchanged PR543 bit supplier. All-size compiler, common-label/mutable-source realization, whole-call hosting, precision/tape and analytic transfer remain assumptions. No unconditional theorem or full Lean certification.",
+  "complex_global_profile": {
+    "1": 166880,
+    "2": 30528,
+    "3": 35779,
+    "4": 13824,
+    "5": 4032,
+    "6": 4032,
+    "7": 208,
+    "8": 576,
+    "39": 1,
+    "48": 1660
+  },
+  "complex_template_sha256": "cdc651b7fd865a9b0b468c519078610ea2b6f14558168a257ac0ee3d593729ae"
+}
diff --git a/research/native-complex-moment552/inputs/PR558-INVOICE.json b/research/native-complex-moment552/inputs/PR558-INVOICE.json
new file mode 100644
index 0000000000..0a0318e6bd
--- /dev/null
+++ b/research/native-complex-moment552/inputs/PR558-INVOICE.json
@@ -0,0 +1,107 @@
+{
+  "status": "PASS_INDEPENDENT_ASYNC_SCALAR_PREFIX_AND_FINITE_INVOICE",
+  "template_sha256": "cdc651b7fd865a9b0b468c519078610ea2b6f14558168a257ac0ee3d593729ae",
+  "parent_template_sha256": "7db047a427a7979d69e163c78a1e32bd445628fb86723e131a3138361355a9c3",
+  "owner_audit_sha256": "0bcaa62bf4723672113f2acdc3ae30c6faa18e5aefa3992e0ec7dc46b9cdd552",
+  "controller_audit_sha256": "177cfc604cd7085019b615945f33c04197eee181e8c30ec13831cf253c138e23",
+  "independent_local_replay_sha256": "ad3f2b941127688f98ce6872d00f110644f1c2b156f8adad1b6700e838ed0e26",
+  "canonical_word_sha256": "66d3c3c70983224650643a0a05a21ac6c6959f84d0f4528650601016f5a76844",
+  "local_programs_sha256": "99a18eede965accf65943e490b8d2b2ffb9f1c9bc6babdb825413a80cbea2cde",
+  "local_checks_sha256": "79177449e404403964ce1b2f4a9a8bd49e1916daba3123d7fd7f5cfd1f1ac0ea",
+  "global_prefix_bits": 225,
+  "charged_prefix_bits": 512,
+  "complete_invocation_sweeps": 72,
+  "bridge_sweeps": 72,
+  "row_prefix_extra": 1024,
+  "local": {
+    "backward": {
+      "events": 21483,
+      "macros": 16290,
+      "real_component_primitives": 161356,
+      "prefix_l1": "1044730572396",
+      "prefix_bits": 40,
+      "coefficient_work_primitive_count": 79906,
+      "all_exact_columns": 728,
+      "dirty_helper_columns": 504,
+      "all_dirty_helper_columns_restored": true,
+      "private_centres_and_coefficient_work_cleared": true,
+      "pre_wrappers": 212,
+      "post_wrappers": 212,
+      "endpoint_evidence": "Hash-bound fresh INDEPENDENT-LOCAL-REPLAY.json"
+    },
+    "forward": {
+      "events": 21483,
+      "macros": 16290,
+      "real_component_primitives": 161308,
+      "prefix_l1": "88876257700",
+      "prefix_bits": 37,
+      "coefficient_work_primitive_count": 79874,
+      "all_exact_columns": 728,
+      "dirty_helper_columns": 504,
+      "all_dirty_helper_columns_restored": true,
+      "private_centres_and_coefficient_work_cleared": true,
+      "pre_wrappers": 212,
+      "post_wrappers": 212,
+      "endpoint_evidence": "Hash-bound fresh INDEPENDENT-LOCAL-REPLAY.json"
+    }
+  },
+  "helpers": 504,
+  "centres": 8,
+  "live": 7168,
+  "physical": 7682,
+  "physical_stock_identity": {
+    "data_and_owner_families": 7168,
+    "local_helpers": 504,
+    "private_centres": 8,
+    "coefficient_work": 2
+  },
+  "entering_helpers": 212,
+  "entrance_rank": 388,
+  "width_inventory": {
+    "4": 8,
+    "5": 48,
+    "8": 156,
+    "9": 292
+  },
+  "local_adapters": 333344,
+  "local_primitive_bound": 8645055460940,
+  "wrapper_elementary_factors_charged": 4396032,
+  "factors_per_wrapper": 144,
+  "compiler": 25934336,
+  "unhosted_pervertex": 1574554051175192,
+  "hosted_rebased_pervertex": 1574554050875816,
+  "conservative_pervertex_used": 4452501040899760,
+  "retained_prior_overcharge": {
+    "sha256": "ce1bfa49be88ae67a404f606a11db2232b7e1f2610d4418058e39749d86d8478",
+    "previous_pervertex_allowance": 2877946989724568,
+    "added_complete_new_pervertex": 1574554051175192,
+    "policy": "Retain old allowance and add the entire fresh recount."
+  },
+  "finite": {
+    "physical_stock_bits": 2641,
+    "router_bits": 7985,
+    "literal_bits": 35282,
+    "physical_row_coefficient": 16467,
+    "retained_row_coefficient": 20161,
+    "required_C0_bits": 200012,
+    "halving_integer_inequality": [
+      5329,
+      4608
+    ],
+    "G_exponent": 30000,
+    "E_exponent": 100000,
+    "C0_exponent": 210000
+  },
+  "strict_checks": {
+    "local_primitive_bound": true,
+    "logical": true,
+    "router": true,
+    "literal": true,
+    "mass": true,
+    "precision": true,
+    "tape": true,
+    "row": true,
+    "halving": true
+  },
+  "scope": "Fresh emitted primitive/Horner, prefix, wrapper and finite recount, using the hash-bound independent local all-column/frame replay and actual whole-owner audit. No old INTEGRATION receipt is used to certify the changed word. The inherited all-size compiler, precision/tape and analytic-transfer contracts remain conditional. Retained payload 512 and physical row allowance 20161 are unchanged."
+}
diff --git a/research/native-complex-moment552/inputs/PR558-OWNERS.json b/research/native-complex-moment552/inputs/PR558-OWNERS.json
new file mode 100644
index 0000000000..668ef2ad07
--- /dev/null
+++ b/research/native-complex-moment552/inputs/PR558-OWNERS.json
@@ -0,0 +1,40 @@
+{
+  "status": "PASS_INDEPENDENT_ASYNC_OWNERS_AND_PROFILE",
+  "template_sha256": "cdc651b7fd865a9b0b468c519078610ea2b6f14558168a257ac0ee3d593729ae",
+  "paid_profile": {
+    "1": 166880,
+    "2": 30528,
+    "3": 35779,
+    "4": 13824,
+    "5": 4032,
+    "6": 4032,
+    "7": 208,
+    "8": 576,
+    "39": 1,
+    "48": 1660
+  },
+  "live": 7168,
+  "physical": 7682,
+  "rank_mass": 520704,
+  "deficit": 2560,
+  "hosted_residuals": 36288,
+  "banked_residuals": 0,
+  "all_live_owners_per_call_distinct": true,
+  "owners_per_call": 728,
+  "actual_helper_owner_charts": 36288,
+  "actual_73column_sigmas": 3178,
+  "cross_replica_helper_roles": 28380,
+  "explicit_idle_remainder_children": 4368,
+  "source_idle_children_charged": 16128,
+  "source_artifacts": {
+    "split-centres/CANONICAL-WORD.json.gz": "66d3c3c70983224650643a0a05a21ac6c6959f84d0f4528650601016f5a76844",
+    "split-centres/LOCAL-PROGRAMS.json.gz": "99a18eede965accf65943e490b8d2b2ffb9f1c9bc6babdb825413a80cbea2cde",
+    "split-centres/LOCAL-CHECKS.json": "79177449e404403964ce1b2f4a9a8bd49e1916daba3123d7fd7f5cfd1f1ac0ea",
+    "split-centres/RESIDUAL-ADAPTERS.json": "baafef737acc16dafbe1150896feb16744f3d669fa123bb2bff19ecc4e71b275",
+    "split-centres/CLIFFORD-RECIPES.json": "534ace870cabe515987e00f50f71550030965006ea18978e0924590ac6944345",
+    "split-centres/CLIFFORD-REVIEW.json": "a0b7b6cc3b8c50d90752637640be54ccdc0604af0e8f2432a1aed8d53f3d36dc",
+    "split-centres/INDEPENDENT-LOCAL-REPLAY.json": "ad3f2b941127688f98ce6872d00f110644f1c2b156f8adad1b6700e838ed0e26",
+    "work/complex-nine-native/DATA-PHASES.json": "c5916ff312678f12aba403847643fbbc29f643acec60bdf836fa56cd3e8251fa"
+  },
+  "controller_audit_sha256": "177cfc604cd7085019b615945f33c04197eee181e8c30ec13831cf253c138e23"
+}
diff --git a/research/native-complex-moment552/inputs/PR558-RESULT.json b/research/native-complex-moment552/inputs/PR558-RESULT.json
new file mode 100644
index 0000000000..2ab78d484b
--- /dev/null
+++ b/research/native-complex-moment552/inputs/PR558-RESULT.json
@@ -0,0 +1,31 @@
+{
+  "status": "PASS_COMPLEX_EIGHT_RANK5_PIVOTS_AND_CITED_BIT",
+  "kappa": "796258944657507/500000000000000000",
+  "kappa_decimal": "0.001592517889315014",
+  "complex_lower": "199382255973827096291/125000000000000000000000",
+  "bit_lower": "413456747036699736067/250000000000000000000000",
+  "word_sha256": "375bd23001cd805707968005a9342827d5464e3f8e3a754197b31c46fc3f63c2",
+  "local_profile": {
+    "1": 2316,
+    "4": 192,
+    "8": 8,
+    "3": 464,
+    "2": 424,
+    "6": 56,
+    "5": 56
+  },
+  "global_profile": {
+    "1": 166880,
+    "2": 30528,
+    "3": 35779,
+    "4": 13824,
+    "5": 4032,
+    "6": 4032,
+    "7": 208,
+    "8": 576,
+    "39": 1,
+    "48": 1660
+  },
+  "seconds": 125.83695875003468,
+  "scope": "Conditional finite complex witness composed with cited unchanged PR543 bit supplier. All-size compiler, common-label/mutable-source realization, whole-call hosting, precision/tape and analytic transfer remain assumptions. No unconditional theorem or full Lean certification."
+}
diff --git a/research/native-complex-moment552/inputs/UPSTREAM-LINEAGE.md b/research/native-complex-moment552/inputs/UPSTREAM-LINEAGE.md
new file mode 100644
index 0000000000..48a59836b2
--- /dev/null
+++ b/research/native-complex-moment552/inputs/UPSTREAM-LINEAGE.md
@@ -0,0 +1,17 @@
+# Pinned upstream lineage
+
+Snapshot through PR547. Exact changed heads and read scope are in PUBLIC-PINS.json. The refreshed conversations cover89 recently created/updated PRs with zero fetch errors; every changed source package in that snapshot was acquired, with no cold revalidation of unchanged external operators.
+
+Direct numerical and source dependencies:
+
+- [PR543](https://github.com/CrocSwap/integer-mult-bounds/pull/543), head2065fe4f72a914d907f1046603f5a36f17018535: R420 bit supplier, its actual paid profile, finite invoice and H1-H5 interleaved source-write argument. The original word's content hash and source file hash are retained in inputs/PR543-EXPECTED.json. Physical operator admission is cited, not rerun here.
+- [PR547](https://github.com/CrocSwap/integer-mult-bounds/pull/547), head4e25588aac914732bde23ab0bd6628d25a3df0db: exact two-supplier assembly pattern and comparison kappa. This package replaces its complex operand with a freshly reconstructed improved supplier.
+- [PR534](https://github.com/CrocSwap/integer-mult-bounds/pull/534): source of the477 previously accepted frame lifts, preserved in the construction chain.
+- [PR535](https://github.com/CrocSwap/integer-mult-bounds/pull/535) and [PR531](https://github.com/CrocSwap/integer-mult-bounds/pull/531): source complex word,parent controller,phase/local verification,complete-owner admission and invoice implementations.
+- [PR541](https://github.com/CrocSwap/integer-mult-bounds/pull/541): predecessor preload/curvature and static44 work. The prior delivered wave input includes its subsequent frame/syndrome/owner composition; we do not attribute those new collective contractions to PR541.
+- [PR510](https://github.com/CrocSwap/integer-mult-bounds/pull/510), [PR503](https://github.com/CrocSwap/integer-mult-bounds/pull/503), [PR493](https://github.com/CrocSwap/integer-mult-bounds/pull/493): retained nine-stage bit and mutable-carrier ancestry.
+- [PR484](https://github.com/CrocSwap/integer-mult-bounds/pull/484), [PR463](https://github.com/CrocSwap/integer-mult-bounds/pull/463), [PR444](https://github.com/CrocSwap/integer-mult-bounds/pull/444), [PR442](https://github.com/CrocSwap/integer-mult-bounds/pull/442), [PR386](https://github.com/CrocSwap/integer-mult-bounds/pull/386): retained exact realization,Clifford/local and whole-owner interfaces.
+
+Read but not added as independent gains: PR542 sliced bit frames;PR544 selective/static44 composition;PR546 prior composition;PR545 punctured111 draft bit supplier;updated PR540 literal-invocation Lean scope. The stronger PR545 bit saving cannot move the minimum while this complex supplier binds, so the existing R420 conditional route is sufficient. PR539's sharper bad-class charge is not needed or used for this bound.
+
+OpenAI/math,Mathlib and the registered project DAG supplied structural research references. Each lane records the actual declarations/files or primary literature it used. Source-only inspection is labelled as such; no Lean-import or kernel-check claim is inferred from a matching theorem name.
diff --git a/research/native-complex-moment552/notes/general-clifford-frames.tex b/research/native-complex-moment552/notes/general-clifford-frames.tex
new file mode 100644
index 0000000000..0367eeb005
--- /dev/null
+++ b/research/native-complex-moment552/notes/general-clifford-frames.tex
@@ -0,0 +1,102 @@
+\subsection{Preimage frames for arbitrary binary subspaces}
+This frame construction is due to icekylinx with OpenAI GPT-6 Astra assistance;
+the present exposition and integration use Codex.
+Let $H=\mathbb F_2^h$ with the standard dot product. In the binary
+symplectic space $H\oplus H$ put
+\[
+ L_0=\{(0,z)\},\qquad
+ F=\begin{pmatrix}I&I\\0&I\end{pmatrix},\qquad
+ L_F=FL_0=\{(z,z)\}.
+\]
+We also write $F$ for the exact Gaussian-dyadic operator $C^{\otimes h}$.
+Its exact square is a Pauli translation; only its binary symplectic square
+is the identity. For an exact Clifford operator $T$ the frame label is
+$L(T)=T^{-1}L_0$. The physical transition from $T$ to $S$ is $ST^{-1}$,
+and its Fourier rank is
+$d(L(S),L(T))$, where $d(L,L')=h-\dim(L\cap L')$.
+This follows by transporting the intersection by $S$: the intersection
+with $L_0$ computes the kernel dimension of the $B$ block of $ST^{-1}$.
+
+\begin{lemma}
+For every subspace $U\subseteq H$, including degenerate subspaces,
+\[
+ L_U=\{(u,u+v):u\in U,\ v\in U^\perp\}
+\]
+is Lagrangian, and
+\[
+ d(L_U,L_V)=\dim U+\dim V-2\dim(U\cap V),\qquad
+ FL_U=L_{U^\perp}.
+\]
+\end{lemma}
+\begin{proof}
+The displayed parametrization is injective and has dimension $h$.
+Its symplectic pairing is $u\cdot v'+v\cdot u'=0$.
+In an intersection, $u$ belongs to $U\cap V$ and $v$ to
+$(U+V)^\perp$, giving the asserted dimension and distance.
+Finally $F(u,u+v)=(v,u+v)$ proves the complement identity.
+\end{proof}
+
+\subsection{Exact representatives and nested children}
+Choose a deterministic invertible binary matrix $G$ carrying the first
+$r=\dim U$ coordinate space $E_r$ to $U$, and set
+\[
+ K_G=\begin{pmatrix}G&0\\G+G^{-T}&G^{-T}\end{pmatrix},\qquad
+ T_U=K_G T_{E_r}K_G^{-1}.
+\]
+Here $T_{E_r}$ is the exact coordinate $C$ operator, and the $K_G$ are
+exact Gaussian-dyadic Clifford lifts. The matrix $K_G$ is symplectic
+because $G^TG+I$ is symmetric. It fixes both $L_0,L_F$ and maps $L_V$
+to $L_{GV}$. Thus both $K_G$ and $FK_GF^{-1}$ have Fourier rank zero;
+their lifts are fixed binary address permutations with unit quadratic
+phases. In particular,
+\[
+ L(T_U)=L_U,\quad \operatorname{rank}B(T_U)=r,\quad
+ \operatorname{rank}B(FT_UF^{-1})=r,\quad T_U^2\text{ is Pauli}.
+\]
+Set $V_U=GE_r^\perp$, a direct complement of $U$. The binary action gives
+$T_U^{-1}L_F=L_{V_U}$ and hence
+$\operatorname{rank}B(FT_U^{-1})=h-r$.
+No assertion $V_U=U^\perp$ is needed.
+For the norm-one source lines and their orthogonal target hyperplanes,
+retain the inherited exact representatives; set $T_0=I,T_H=F$.
+At every common scalar gate use one identical exact representative on all
+incident registers. If separately transported representatives have the
+same preimage, their physical ratio has rank zero and is implemented as
+an actual left adapter before the gate.
+
+For $U\subseteq V$, the exact transition $T_VT_U^{-1}$ has rank
+$\dim V-\dim U$. Clifford elimination expresses it as two rank-zero
+adapters surrounding that many coordinate Fourier factors. These factors
+are a single $C^{\otimes r}$ child: the scaled Hadamard satisfies
+\[
+ \frac{1+i}{2}\begin{pmatrix}1&1\\1&-1\end{pmatrix}
+   =SCS,\qquad S=\operatorname{diag}(1,i).
+\]
+Thus no new irrational field is introduced. All adapters are charged
+finite movement and scalar operations. Exact transitions telescope;
+commutativity between distinct $T_U$ is unnecessary.
+
+\subsection{Dirty tails and reverse/complement words}
+A dirty role entering with gauge $T_\sigma$ and whose last active frame
+is $T_U$, with $\sigma\subseteq U$, is assigned terminal gauge
+$FT_\sigma$. Its logical value is restored by the complete transparent
+scalar word, so its physical action is exactly $F$. The tail is
+$FT_\sigma T_U^{-1}$. Its rank is
+\[
+ d(L_U,L_{V_\sigma})=h-\dim U+\dim\sigma.
+\]
+Indeed $H=\sigma\oplus V_\sigma$ and $U\supseteq\sigma$ imply
+$\dim(U\cap V_\sigma)=\dim U-\dim\sigma$.
+An inactive exterior adds $m-h$, giving $m-\dim U+\dim\sigma$.
+
+For the complete time-reversed word use
+\[
+ D_U=T_UF^{-1},\qquad L(D_U)=L_{U^\perp},\qquad
+ D_UD_V^{-1}=T_UT_V^{-1}.
+\]
+The last equality is exact, so reversal preserves every inverse transition
+rank. Reverse the scalar signs and the whole transparent word, including
+old-value corrections and final cleanup. The endpoints $D_0=F^{-1}$
+and $F$ differ by a Pauli, and the inherited norm-one endpoints are
+reconciled by actual rank-zero adapters. Neither equality of binary
+matrices nor equality of frame cosets substitutes for these operators.
diff --git a/research/native-complex-moment552/notes/three-stage-cover-assembly.tex b/research/native-complex-moment552/notes/three-stage-cover-assembly.tex
new file mode 100644
index 0000000000..983ffb6ac8
--- /dev/null
+++ b/research/native-complex-moment552/notes/three-stage-cover-assembly.tex
@@ -0,0 +1,117 @@
+\subsection{Local inventory and strict supplier moments}
+The selected complex local word has
+\[
+ h=24,\quad v=2024,\quad c=91770,\quad q_{\rm roots}=8120,
+ \quad |\mathcal M|=71185,\quad R=28705,\quad\ell=552.
+\]
+Its complete scalar word, full backward subspace intersections and 4,599
+partial source gauges are regenerated from the pinned compressed DAG by
+\path{scripts/three_stage_cover_producer.py}. The source and target chains,
+remaining first transitions, inverse cleanup and copied centers remain in
+\path{certificates/three-stage-cover-complex-input.json}.
+At each cover vertex its child inventory is exactly the three-stage table
+proved above. Thus $m_c=70$, $w_c=90163$, $s_c=6309018$ and
+$w_cm_c-s_c=2392$; the largest child is 68.
+
+For $a_c=786999/2500000000$, the rational enclosure proves
+\[
+ \frac1{w_c}\sum_r n_r(r/70)^{1-a_c}
+ <1-2.3047018455\times10^{-10}.
+\]
+The bit local ledger has $h=23,v=1771,R=28866,\ell=506$.
+Three copies of its physical local increments and exteriors give
+$m_b=67,w_b=90140,s_b=6037356$, with deficit 2024.
+The contaminated moment includes the entire $32\cdot67^2$-child fallback
+on a fraction $10^{-16}$ of every ideal edge. At $a_0=747/2000000$ its
+strict gap exceeds $6.3397366396\times10^{-7}$, and its rank moment also
+contracts. The stopped ordinary saving is
+\[
+ a_b=(1-10^{-3})a_0+10^{-3}\frac{384599}{10^{10}}
+     =\frac{3731649599}{10^{13}}.
+\]
+These are exact rational certificates; the lower tolerances displayed
+here do not substitute for their saved fractions.
+
+\subsection{Full finite router and completed-child guard}
+Let $V_0=|O(70,2)|$, $N=V_0v$, $W=V_0(2v+3R)$ and $s=V_0s_c$.
+All these finite counts appear in the certificate. Define
+\[
+ L=4(c+v)+10v+4hv+4h^2+8h+8+2h
+       +8Rv(M+16),
+\]
+where $M$ is the number of actual local mixer operations.
+The source-witness bound for the positive DAG gives dirty-readout
+coefficients of magnitude at most $h+4$, with denominators dividing
+$2(h-3)$. The last term conservatively pays their binary-addition
+implementation. Include the finite group and fixed Clifford routers by
+\[
+ G=N+3V_0L+64(m_c+1)^3W^2.
+\]
+A fixed role permutation has at most $W^2$ elementary exchanges; the
+polynomial factor also covers the binary matrix and quadratic-phase
+adapters. These finite costs retain their ordinary movement charge.
+Set
+\[
+ E=64(W+m_c+G+1)^3,\quad B=s+E,\quad C_0=32m_cB^2.
+\]
+The exact calculation checks
+\[
+ 2GW^2+8s+4W+4+32m_c<E,\qquad
+ 2B(m_c-68)\ge s+E,\qquad 2B+18<C_0.
+\]
+Every completed child is the exact Gaussian-dyadic $C^{\otimes u}$
+operator on all roles, giving the inherited semantic constant $C_1=1$.
+One grid $2^{-P}21^{-K}$, with $K=G(D_c+1)$, contains every unfinished
+call's rational-center intermediates. Completed children preserve the
+incoming odd-denominator exponent. Only the completed outer operation on
+dyadic input justifies unscaling; recursive returns are not rounded.
+The existing outer exact-integer conversion costs $O(V_{\rm bits}\log d)$.
+
+\subsection{External rows and exact composition}
+The complex group has 2,415 bits and its role stock 2,431 bits.
+The least halving degree for $70\to68$ is 24, contributing 58,344.
+Including conservative previous bit and ordinary-leaf reserves gives
+\[
+ 58344+9909+252=68505,\qquad
+ 160000-\frac{51}{25}\,68505=20249.8>0.
+\]
+Thus the fixed external reserve $p^{160000}$ meets the inherited product
+stock contract. The width-dependent q-adic selectors are supplied by
+the separate internal borrowing proof, rather than multiplied into this
+external polynomial stock. One preceding prefix and one final padding
+suffice, and sequential completed calls reuse restored rows.
+
+Put $b=a_c$, $\eta=10^{-8}$, $\beta=10^{-6}$ and
+\[
+ a=(1-\beta)b-10^{-10}<a_b,\quad \tau=1-a,\quad\sigma=1-b,
+ \quad q_0=a(1-2\eta),\quad c_0=q_0(1+\eta),
+\]
+\[
+ \epsilon=\frac{1-\eta}{1+c_0+q_0},\quad
+ \lambda'=1-q_0,\quad\lambda=(\tau+\lambda')/2,
+ \quad g=\epsilon q_0,\quad r=(g+1-\epsilon)/2,\quad\delta=\eta/8.
+\]
+All 47 retained strict inequalities hold. The seven assembly savings are
+\[
+ 1-\epsilon(1+c_0),\ a,\ g,\ a,\
+ \min(1-\epsilon-\delta,r-\delta),\ 1-\epsilon-\delta,\ \epsilon.
+\]
+Their minimum $g$ exceeds $3146011/10^{10}$ by more than
+$3.4173577301\times10^{-12}$.
+\begin{theorem}
+Under the retained analytic, semantic-precision, exact-recovery and
+fixed-tape interfaces, the selected construction gives
+\[
+ T(n)=O\!\left(n(\log n)^{1-\kappa}\right),\qquad
+ \boxed{\kappa=\frac{3146011}{10000000000}=3.146011\times10^{-4}}.
+\]
+\end{theorem}
+\begin{proof}
+The weighted bit construction supplies completed ordinary interchange,
+including its internally restored rows. The signed complex cover supplies
+the exact completed tensor on every role. The literal semantic charge and
+external reserve satisfy the unchanged precision and row contracts.
+Substituting the displayed parameters in the retained semantic/bulk
+transfer gives the seven savings above. Their positive excess absorbs
+the fixed logarithmic overheads and yields the stated conditional bound.
+\end{proof}
diff --git a/research/native-complex-moment552/notes/three-stage-cover-bit.tex b/research/native-complex-moment552/notes/three-stage-cover-bit.tex
new file mode 100644
index 0000000000..f05225d21f
--- /dev/null
+++ b/research/native-complex-moment552/notes/three-stage-cover-bit.tex
@@ -0,0 +1,264 @@
+\subsection{Atom contract and common endpoint}
+Fix an odd prime $q$, chosen below, and put $R_w=\mathbb Z/q^w\mathbb Z$.
+An atom is a complete consecutive $w$-digit address field. Write $C_n$ for
+reversal of the $n$ atom positions, without reversing digits inside an atom.
+The recursive contract is the involution
+\[
+ F_A(H,D)=(AC_nD,C_nA^{-1}H),
+\]
+where $A$ is a diagonal array of units on the head atoms. Its entries may
+depend on exposed ancestor row fields and the atom index, but not on the
+fresh role fields of this invocation or on the head/data values. Only the
+polynomial-time descriptors generated by the construction are required;
+no constant-cost oracle for arbitrary functions of a prefix is assumed.
+Every call preserves all complete spectator fields and restores its row
+coordinate. The original unweighted call has $A=I$.
+
+For a split idempotent $P$ on the head coordinates set
+\[
+ D_P=\begin{pmatrix}I-P&PC_n\\C_nP&I-C_nPC_n\end{pmatrix},
+ \qquad F=D_I.
+\]
+These are involutions, and $D_P-I$ has split rank $\operatorname{rank}P$.
+Nested commuting-projector identities are polynomial identities and require
+no symmetry of $P$. For $n=mf$ and $G=g\otimes I_f$ define
+\[
+ Q_g=\operatorname{diag}(G,C_nGC_n).
+\]
+Then $Q_gF=FQ_g$ and $Q_gD_PQ_g^{-1}=D_{GPG^{-1}}$.
+Consequently conjugating every base frame by any $g\in\operatorname{GL}_m(R_w)$
+leaves the same full physical endpoint $F$.
+
+Put $Q_A=\operatorname{diag}(A,I)$. Conjugate the entire completed scalar
+word by $Q_A$. Scalar gates mix roles having the same address map, so they
+commute with $Q_A$. A current role routing also commutes with $Q_A$, since
+it changes only fresh fields. Thus only address residuals change, and the
+completed endpoint is $Q_AFQ_A^{-1}=F_A$. More explicitly, when a port
+changes charts from $g$ to $gR_s$, its local label changes from $M$ to
+$R_s^{-1}MR_s$, and
+\[
+ (gR_s)(R_s^{-1}MR_s)(gR_s)^{-1}=gMg^{-1}.
+\]
+This routing is a change of row/chart, not an uncharged dense address map.
+The adapted chart $Q_AQ_gQ_A^{-1}$ commutes with $F_A$. None of these
+conjugations is executed as an extra normalization pass.
+
+\subsection{Explicit rational three-stage geometry}
+Use the retained PR97 local producer with
+\[
+ h=23,\qquad v=1771,\qquad R_{\rm loc}=28866,\qquad \ell=506.
+\]
+Its physical projector ranks and scalar word are unchanged. On
+$\mathcal A=\mathbb Q^{23}$ take the form $H_0=(I-J/9)/2$.
+For a triple $S$ let $q_S=\chi_S$; then $q_S^tH_0q_S=1$.
+Fix $q_0=\chi_{\{1,2,3\}}$ and $K=q_0^\perp$. A rational basis is
+\[
+ k_i=e_i-(e_i^tH_0q_0)q_0,\qquad 2\le i\le23.
+\]
+Projection along $q_0$ is injective on the span of $e_2,\ldots,e_{23}$,
+since $q_0$ has nonzero first coordinate. Thus this is indeed a basis.
+Let $\mathcal B,\mathcal C$ be orthogonal copies of $K$ and put
+$\mathcal E=\mathcal A\perp\mathcal B\perp\mathcal C$, of dimension $m=67$.
+
+Choose a coordinate permutation $\pi_S$ taking $\{1,2,3\}$ to $S$.
+It preserves $H_0$. The rational isometry $R_{12,S}$ fixes $q_S$ and
+$\mathcal C$, and exchanges $\pi_Sk_i\in q_S^\perp\subset\mathcal A$
+with $k_i\in\mathcal B$. Define $R_{23,S}$ by exchanging the same vectors
+with their copies in $\mathcal C$, fixing $q_S$ and $\mathcal B$.
+Both maps are involutions. A hyperedge $(g,S)$ uses the invocations
+$gR_{12,S},g,gR_{23,S}$, with active spaces
+\[
+ g(\mathcal B+\mathbb Qq_S),\qquad g\mathcal A,
+ \qquad g(\mathcal C+\mathbb Qq_S).
+\]
+Their projectors commute; their pairwise and common intersection is the
+projector $P_{gq_S}$, and their sum is $I+2P_{gq_S}$.
+This follows from the orthogonal decomposition
+$\mathbb Qq_S\perp q_S^\perp\perp\mathcal B\perp\mathcal C$ before
+conjugation. These are the cumulative-projector three-stage transfer
+identities: the three scalar shears complete the interchange, with zero
+interstage connectors and no rank-one endpoint correction. Each port
+family visits the full cover bijectively, by right multiplication with
+its displayed involution. The physical ledger is therefore three copies
+of the local auxiliary, data and center histograms, and three copies of
+the exterior histogram $m-\text{nullity}$.
+
+\subsection{One weighted child on a generic residue class}
+Work first on one atom column. Let $g$ be its weighted residual involution,
+of split rank $r$ relative to $I$. If the cross block has generic northeast
+rank profile, unit lower row and column elimination, without normalizing
+pivots, gives
+\[
+ Lg_{XY}R=ED_pF_0^t,
+ \quad E=(e_1,\ldots,e_r),\quad
+ F_0=(e_m,\ldots,e_{m-r+1}),
+\]
+with invertible diagonal $D_p$. Conjugate by $\operatorname{diag}(L,R^{-1})$.
+Split rank and the unit pivot minor give a factorization
+\[
+ g-I=\binom EB(A_0,D_pF_0^t),\qquad
+ A_0E+D_pF_0^tB=-2I_r.
+\]
+For clarity, if $g-I=UV$, then $g^2=I$ gives
+$U(VU+2I)V=0$. The unit minor supplies a left inverse of $U$ and a right
+inverse of $V$, proving the identity over the local ring itself.
+Set
+\[
+ K=-(B+F_0D_p^{-1})E^t+F_0D_p^{-1}A_0(I-EE^t),
+ \qquad S=\begin{pmatrix}I&0\\K&I\end{pmatrix}.
+\]
+The identities $KE=-B-F_0D_p^{-1}$ and
+$D_pF_0^tK=A_0+E^t$ give
+\[
+ SgS^{-1}=\begin{pmatrix}
+ I-EE^t&ED_pF_0^t\\F_0D_p^{-1}E^t&I-F_0F_0^t
+ \end{pmatrix}.
+\]
+This is one weighted reversed interchange on the first $r$ head and last
+$r$ tail coordinates. All other factors are unit lower intrabank maps or
+lower cross-bank additive maps. Thus variable diagonal units remain in
+the child's weight array rather than becoming unsupported routing passes.
+
+For $n=mf$, group atoms by coarse coordinate $i$ and column $j$.
+If $L_P,R_P,D_P$ are unweighted normalizers for the current projector and
+$A_j=\operatorname{diag}(A[if+j])$, then
+$L_j=A_jL_PA_j^{-1}$ and $R_j=R_P$ are unit lower normalizers for
+$A_jPC_m$. The child's exact contiguous index rule is
+\[
+ A_{\rm child}[kf+j]=A[i_kf+j]D_P[k].
+\]
+The cross-bank atom pairing is reversed, and $C_r\otimes C_f=C_{rf}$.
+Hence the middle operation is exactly one $rf$-atom call of the same
+contract, with no interleaving or gathering. Its coefficients depend only
+on parent prefix fields, which become ancestor spectators at the child;
+the child's fresh roles come from disjoint unused row stock.
+
+All unit lower adapters require $O(n)$ atom passes. The retained ordered
+affine lemma implements each later-target shear in $O(V)$, including
+prefix-controlled offsets. The generated descriptors have polynomial cost
+in $w$: good-class evaluation follows one ancestral index per level;
+fallback evaluation branches by a constant per level. Since depth is
+$O(\log e)$ and $w=\lceil e^{1/1000}\rceil$, this costs $e^{O(1)}=w^{O(1)}$.
+Stored ancestor data use $O(w\log e)$ bits. The proof of the ordered affine
+lemma still applies with these controls: polynomial preparation per prefix
+is absorbed by the complete $q^w$ target fiber. Unrelated spectator counters
+are incremented by ripple counting rather than read in their entirety.
+
+\subsection{Weighted leaves and a genuine local-ring fallback}
+At one atom write $J_a=\left(\begin{smallmatrix}0&a\\a^{-1}&0\end{smallmatrix}\right)$,
+$U(a)=\left(\begin{smallmatrix}1&a\\0&1\end{smallmatrix}\right)$ and
+$L(b)=\left(\begin{smallmatrix}1&0\\b&1\end{smallmatrix}\right)$.
+Then
+\[
+ J_a=\operatorname{diag}(1,-1)U(a)L(-a^{-1})U(a),
+ \qquad U(a)=J_1L(a)J_1.
+\]
+Thus a weighted leaf uses four old ordinary atom interchanges, three
+prefix-controlled shifts and one fixed sign. With
+$a_{\rm old}=384599/10^{10}$ it costs $O(Vw^{1-a_{\rm old}})$.
+This constant occurs only at stopped leaves.
+
+A bad residue class cannot be treated by field Bruhat elimination:
+an entry zero modulo $q$ need not vanish modulo $q^w$. Instead Gaussian
+eliminate the full invertible residual of size $N_0=2m$ with unit pivots.
+Each invertible Schur complement has a unit in its next pivot column;
+row swaps and unnormalized additions clear that column. There are at most
+$N_0(N_0-1)$ additions and $N_0$ swaps. Pivot choices are determined by
+the unweighted low residue class: diagonal conjugation scales every
+Schur-complement row and column by units and preserves its zero pattern.
+The final diagonal has product a fixed sign, since the residual has
+determinant $(-1)^r$ and row swaps change only the sign.
+
+A lower addition is a paid additive adapter. An upper addition is
+conjugated into a lower addition by two ordinary coarse-field swaps.
+For a field of $f$ atoms, let $R_{\rm tail}$ reverse the later field.
+Then the ordinary weighted swap is
+\[
+ J_A=R_{\rm tail}F_AR_{\rm tail},\qquad
+ J_A(H,D)=(AD,A^{-1}H).
+\]
+Each $R_{\rm tail}$ is one unweighted reversed call on the first and last
+$\lfloor f/2\rfloor$ atoms, fixing a possible middle atom. Thus $J_A$
+costs at most three children of widths $f,\lfloor f/2\rfloor,
+\lfloor f/2\rfloor$. A paired diagonal scaling is $J_aJ_1$, at most six
+children. The full diagonal is a product of $N_0-1$ such pairs and a fixed
+sign. Across the two banks the reversed pairing already makes the swaps
+and diagonal pairs directly reversed calls; the preceding ordinary bounds
+remain conservative. Reindexing a tail weight by $j\mapsto f-1-j$ changes
+only its descriptor. Charging every addition as upper therefore gives
+\[
+ 6N_0(N_0-1)+3N_0+6(N_0-1)<32m^2.
+\]
+Every child has width at most $f$, hence ratio at most $1/m$. Coefficients
+are computed from constant-sized local-ring matrices and parent weight
+descriptors; only units are inverted. This proves the fallback cost,
+including its actual reversals, rather than assuming nonunit pivots away.
+
+\subsection{Finite covers, batching, and the strict moment}
+Take $q>2^{80}$ avoiding all fixed rational denominators and rank witnesses.
+At atom width $w$ use the finite cover $\operatorname{GL}_m(R_w)$ and its
+right-multiplication edges $g\mapsto gR_s$. Every vertex has the unique chart
+\[
+ g=g_0+qG_1,\quad g_0\in\operatorname{GL}_m(\mathbb F_q),
+ \quad G_1\in(\mathbb Z/q^{w-1}\mathbb Z)^{m^2},
+\]
+using fixed representatives for $g_0$. Right multiplication permutes $g_0$
+and acts affinely on $G_1$, with a carry determined by $g_0$. Its fixed
+dimensional rational linear part factors into fixed scalings, ordered
+shifts and a constant number of old width-$O(w)$ interchanges. Thus each
+fixed port routing costs $O(Vw^{1-a_{\rm old}})$.
+
+Let $W_0=|\operatorname{GL}_m(\mathbb F_q)|(2v+3R_{\rm loc})$.
+The full role count is $W(w)=W_0q^{m^2(w-1)}$. Split preceding rows into
+low role type, high matrix and unused stock. Batch \emph{all} high matrices
+for each low type in one stream. A frame edge is one child on this whole
+batch, of volume $V/W_0$, and routing is executed once per fixed operation
+type. Consequently neither the number of tapes nor the number of child
+programs grows with $w$. High cardinality cancels in the normalized moment,
+but its $m^2(w-1)$ digits are charged in the row reserve.
+
+For each fixed projector require the leading minors of sizes $1$ through
+$r$ to be units. Each cleared minor polynomial in $g_0$ is nonzero:
+similarity to $\operatorname{diag}(I_r,0)$ supplies a witness after the
+fixed rank is preserved modulo $q$. Their total degree is at most $m^3$.
+The polynomial zero bound and conditioning on invertibility (a factor at
+most two) give bad fraction at most $2m^3/q<10^{-16}$ for each edge type.
+No simultaneous union bound over edge types is needed, since their moments
+are averaged separately. Good classes use the whole-rank child; bad
+classes use the full $32m^2$-child fallback just proved.
+
+Let $H_r$ be the three-stage histogram described above and
+$W=2v+3R_{\rm loc}=90140$. The retained ledger gives
+\[
+ \sum_r rH_r=6037356,\qquad Wm-\sum_r rH_r=2024=2v-3\ell,
+ \qquad \max\{r:H_r>0\}=66.
+\]
+For $a=747/2000000$, $\tau=1-a$, and $b=10^{-16}$ the conservative moment is
+\[
+ \mu_\tau\le
+ \sum_r\frac{H_r}{W}(r/m)^\tau+
+ b\frac{32m^2\sum_rH_r}{W}(1/m)^\tau<1.
+\]
+The entire fallback is added; no displaced good contribution is subtracted.
+The exact script \path{scripts/three_stage_cover_network.py} and certificate
+\path{certificates/three-stage-cover-network.json}, from the retained axis
+\path{certificates/three-stage-cover-bit-input.json}, give
+$1-\mu_\tau>6.3397366396\cdot10^{-7}$.
+The certificate uses the 32-term positive atanh logarithm series with its
+geometric tail, $\exp(u)\le1+u+u^2/(2(1-u/3))$, and upward rounding to
+the grid $2^{-120}$. All inputs have $0\le u<3$; these are rational upper
+bounds, not floating-point root estimates. Since every child ratio is at
+most one and $\tau<1$, this same strict bound implies $\mu_1<1$.
+
+With $T(n,w)$ denoting time divided by logical volume, the fixed low-type
+profile $\{c_r\}$, including fallback classes, satisfies
+\[
+ T(n,w)\le\sum_rc_rT(r\lfloor n/m\rfloor,w)+Cn+Cw^{1-a_{\rm old}}
+ \quad(n\ge m),\qquad
+ T(n,w)\le Cw^{1-a_{\rm old}}\quad(n<m).
+\]
+Peeling fewer than $m$ leftover atoms uses the weighted leaf algorithm.
+The constants are independent of $w,n$ and ancestor values. Induction,
+using $\mu_1<1$ for the linear term and $\mu_\tau<1$ for the stopped term,
+gives $T(n,w)=O(n+n^\tau w^{1-a_{\rm old}})$. The maximum proper ratio
+$66/67$ has halving degree $47$, so recursion depth is $O(\log n)$.
diff --git a/research/native-complex-moment552/notes/three-stage-cover-complex.tex b/research/native-complex-moment552/notes/three-stage-cover-complex.tex
new file mode 100644
index 0000000000..c54cb3d5ff
--- /dev/null
+++ b/research/native-complex-moment552/notes/three-stage-cover-complex.tex
@@ -0,0 +1,152 @@
+\subsection{Three invocation banks and a finite orthogonal cover}
+Use the completed $h=24$ triple word contributed by eumemic with Claude
+(PR117), with its retained licenses and notices. Its local profile is
+\[
+ v=2024,\quad R=28705,\quad \ell=552.
+\]
+The global cover below retains that word, its dirty readouts and copied
+centers. Let $E=A\perp B\perp C$ be the standard binary space with
+$\dim A=h$, $\dim B=\dim C=h-1$, and $m=3h-2$.
+For a triple $S\subseteq[h]$ put $q=q_S=\sum_{i\in S}e_i$.
+Choose $d\notin S$. The orthogonal involution
+$\tau_{q+e_d}(x)=x+(x\cdot(q+e_d))(q+e_d)$ sends $e_d$ to $q$;
+its images of the other coordinate vectors give an explicit orthonormal
+basis of $A\cap q^\perp$.
+Define $R_{12,S}$ to exchange this basis with the fixed basis of $B$,
+fixing $q,C$; define $R_{23,S}$ similarly using $C$ and fixing $q,B$.
+Both are orthogonal involutions, with
+\[
+ R_{12,S}A=B\oplus\langle q\rangle,\qquad
+ R_{23,S}A=C\oplus\langle q\rangle.
+\]
+Take $G=O(E)$ and three distinct invocation sets equal to $G$.
+The data position $(g,S)$ meets vertices $gR_{12,S},g,gR_{23,S}$,
+respectively. At a fixed vertex of any stage, the map from port $S$ to its
+position is bijective. All three local source vectors equal $gq_S$.
+Thus $N=|G|v$ and the persistent-role count is
+$W=|G|(2v+3R)$, using independent auxiliary banks.
+
+\subsection{Exact chronology and phase endpoints}
+Write $A-q=A\cap q^\perp$, and transport the following table by $g$.
+The middle stage is the complete reverse/complement word with source and
+target interchanged and the scalar sign reversed.
+\[
+\begin{array}{c|c|c|c|c}
+\text{stage}&\text{logical shear}&\text{active space}&X&Y\\\hline
+1&Y\gets Y+X&B+q&q\longrightarrow B+q&0\longrightarrow B\\
+2&X\gets X-Y&A&B+q\longrightarrow A+B&B\longrightarrow(A-q)+B\\
+3&Y\gets Y+X&C+q&A+B\longrightarrow E&(A-q)+B\longrightarrow E-q
+\end{array}
+\]
+The offsets are respectively $0$, $B$, and $(A-q)+B$, each orthogonal
+to its active space. They are fixed at each invocation: if
+$k=gR_{23,S}$, the third offset is $k(B+C)$, independently of its port.
+Every exit label is precisely the next entrance label. Any difference
+between exact representatives is reconciled by an actual rank-zero
+adapter; consequently there are no positive-rank data connectors.
+The forward stages retain their two data-chain histograms; reversing,
+complementing and exchanging the middle chains preserves their combined
+rank histogram.
+
+Let $T$ be the inherited norm-one phase on the actual transported source
+$gq$, and now let $F=C^{\otimes m}$.
+The exact entrance gauges are $(T,I)$ and the exact terminal gauges are
+$(F,FT^{-1})$, after the endpoint adapters just described.
+The scalar composition is
+\[
+ (X,Y)\mapsto(X,Y+X)\mapsto(-Y,X+Y)\mapsto(-Y,X).
+\]
+Thus physical input $(x,y)$ produces
+\[
+ (-Fy,FT^{-2}x).
+\]
+There is no remaining mixed term. Since $T^2=X_{gq}$ is an exact fixed
+Pauli translation, applying $T^2$ to the input $x$ bank, negating the
+first output and exchanging the banks yields $(Fx,Fy)$ exactly.
+Equivalently one may use the inherited fixed-sign output normalization.
+The translation uses the actual support of $gq$; it assumes no fixed
+integer weight. These fixed permutations and signs incur scalar/router
+charges, without recursive Fourier children.
+
+\subsection{Arbitrary dirty roles and exterior ranks}
+The retained local word is transparent for arbitrary logical auxiliaries:
+with source injection $V$, scalar addition circuit $M$, decoder $J$ and
+$D=JM$, the signed sequence
+\[
+ y\gets y-Dz,\quad z\gets z+Vx,\quad z\gets Mz,\quad
+ y\gets y+Jz,\quad z\gets M^{-1}z,\quad z\gets z-Vx
+\]
+restores $z$ and adds $JMVx$ to $y$. Its chronological deferred readouts
+and copied-center branches are part of the completed word. The inverse
+word also restores arbitrary auxiliaries. Adjoining a fixed orthogonal
+offset applies the same exact gauge to every register at its scalar gates.
+
+A forward dirty role with selected local entrance gauge $T_\sigma$
+has exterior rank $m-h+\dim\sigma$ by the preceding tail lemma.
+In the middle reversed word its actual first local frame is $D_H=I$
+and its actual last local frame is $D_\sigma=T_\sigma F_A^{-1}$.
+Start the role with the common-offset gauge $F_B$. Its logical value is
+restored, and assign the desired terminal gauge $F F_B$, making its total
+physical action $F$. The preimage of the actual last frame is
+$B\oplus(A\cap\sigma^\perp)$, whereas the desired terminal preimage is
+$A\oplus C$. Their intersection is $A\cap\sigma^\perp$, of dimension
+$h-\dim\sigma$. The preimage distance is therefore
+\[
+ m-h+\dim\sigma.
+\]
+This proves the same exterior rank in the reversed stage without assuming
+commuting intermediate representatives. It includes the inverse of the
+complete cleanup, not just the positive producer DAG. The three auxiliary
+banks are independent; all copied centers remain and contribute
+$3|G|\ell$ to the rank loss.
+
+\subsection{Finite rank certificate and group order}
+Let $s_a$ count selected gauges of dimension $a$, let $H_r$ be the
+remaining internal histogram, and $D_r,T_r$ the literal data histograms.
+Per group vertex the complete positive-width multiset has
+\[
+ 3\Bigl(R-\sum_a s_a\Bigr)\text{ children of width }m-h,\qquad
+ 3s_a\text{ of width }m-h+a,\qquad
+ 3(H_r+D_r+T_r)\text{ of width }r.
+\]
+Its weighted sum is $(2v+3R)m-2v+3\ell$.
+At $h=24$, $m=70$, this gives
+\[
+ W/|G|=90163,\quad s/|G|=6309018,\quad
+ (Wm-s)/|G|=2392,\quad r_{\max}=68.
+\]
+The normalized rational moment certificate uses
+$a_c=786999/2500000000$. Rational upper bounds for
+$\log(m/r)$ and
+$e^x\le 1+x+x^2/(2(1-x/3))$ prove
+\[
+ \sum_r\frac{n_r r}{90163\cdot70}
+        (70/r)^{a_c}<1,
+\]
+with certified gap exceeding $2.3047\times10^{-10}$.
+The finite group multiplicity cancels only from this normalized moment.
+
+For a standard nonalternating binary space of even dimension $2n$,
+its characteristic vector $\Omega$ is isotropic and fixed by every
+orthogonal map. Choose $e$ with $e\cdot\Omega=1$ and let
+$U=\langle\Omega,e\rangle^\perp$; $U$ is a nondegenerate alternating
+space of dimension $2n-2$. The induced quotient action is symplectic.
+For each $S\in\operatorname{Sp}(U)$, all lifts are parametrized by
+$u\in U,c\in\mathbb F_2$ through
+\[
+ \Omega\mapsto\Omega,\quad e\mapsto e+u+c\Omega,\quad
+ v\mapsto Sv+(u\cdot Sv)\Omega\quad(v\in U).
+\]
+Checking the pairings verifies these are exactly the orthogonal lifts.
+Counting symplectic bases then gives
+\[
+ |O(2n,2)|=2^{2n-1}|\operatorname{Sp}(2n-2,2)|,\qquad
+ |G|=2^{69+1156}\prod_{i=1}^{34}(2^{2i}-1).
+\]
+This group order has 2415 bits and $W$ has 2431 bits. The least integer
+$d$ satisfying $70^d>2\,68^d$ is 24, so the complex row coefficient is
+$24\cdot2431=58344$. These full finite role counts are retained in the
+scalar guard and row stock. Group elements and involutions are specified
+by their binary matrices; the definition needs no numerical enumeration.
+All fixed address adapters, endpoint Paulis and invocation permutations
+are implemented inside the finite scalar/router guard of the assembly.
diff --git a/research/native-complex-moment552/verify.py b/research/native-complex-moment552/verify.py
new file mode 100644
index 0000000000..ee9399d676
--- /dev/null
+++ b/research/native-complex-moment552/verify.py
@@ -0,0 +1,48 @@
+"""Exact arithmetic screen; zero complex contamination term requires source-scope review."""
+from pathlib import Path
+from fractions import Fraction as Q
+from decimal import Decimal,localcontext
+import importlib.util,json,hashlib,time,argparse,sys
+R=Path(__file__).resolve().parent;start=time.monotonic()
+ap=argparse.ArgumentParser();ap.add_argument('--work',type=Path,required=True);ap.add_argument('--provider',choices=['552','558'],default='558');args=ap.parse_args();O=args.work.resolve();provider=args.provider;tag='PR'+provider
+assert __debug__ and not O.exists() and not O.is_relative_to(R)
+sha=lambda p:hashlib.sha256(p.read_bytes()).hexdigest()
+manifest=json.loads((R/'MANIFEST.json').read_text());assert {p.relative_to(R).as_posix():sha(p)for p in R.rglob('*')if p.is_file()and p!=R/'MANIFEST.json'and '__pycache__'not in p.parts}==manifest
+O.mkdir(parents=True)
+def load(n):
+ p=R/'engines'/(n+'.py');s=importlib.util.spec_from_file_location(n,p);m=importlib.util.module_from_spec(s);s.loader.exec_module(m);return m
+M,B,A=[load(n)for n in ['moment','base_two_moment','outer']]
+parent=json.loads((R/'inputs'/(tag+'-EXPECTED.json')).read_text());P={int(r):n for r,n in parent['complex_global_profile'].items()};m=73;W=7168;c=Q(parent['bit_lower']);oldb=Q(parent['complex_lower']);oldk=Q(parent['kappa']);assert m*W-sum(r*n for r,n in P.items())==2560
+if provider=='558':
+ raw=json.loads((R/'inputs/PR558-RESULT.json').read_text())
+ assert parent['complex_global_profile']==raw['global_profile'] and all(parent[k]==raw[k]for k in raw)
+owner=json.loads((R/'inputs'/(tag+'-OWNERS.json')).read_text());invoice=json.loads((R/'inputs'/(tag+'-INVOICE.json')).read_text());bit=json.loads((R/'inputs/PR543-EXPECTED.json').read_text())['nine']
+assert owner['template_sha256']==invoice['template_sha256']==parent['complex_template_sha256']
+assert {int(k):v for k,v in owner['paid_profile'].items()}==P and owner['live']==W
+assert invoice['owner_audit_sha256']==sha(R/'inputs'/(tag+'-OWNERS.json')) and all(invoice['strict_checks'].values())
+assert invoice['global_prefix_bits']<invoice['charged_prefix_bits']==512 and invoice['finite']['physical_row_coefficient']<20161
+BP={int(k):v for k,v in bit['profile'].items()};assert Q(bit['bit_lower'])==c
+for e,ok in [(c,True),(c+Q(1,10**24),False)]:
+ lo,hi=M.moment(BP,64,bit['bit']['live'],e,True);a,z=B.moment(64,bit['bit']['live'],list(BP.items()),e);f,g=B.moment(64,bit['bit']['live'],[(1,32*64**2*sum(BP.values()))],e);a+=f/10**16;z+=g/10**16;assert(hi<1 and z<1)if ok else(lo>1 and a>1)
+baseline=M.certify(P,m,W,True);assert Q(baseline['lower'])==oldb
+cert=M.certify(P,m,W,False);b=Q(cert['lower']);assert b>oldb
+checks=[]
+for endpoint,ok in [(b,True),(b+Q(1,10**24),False)]:
+ l,h=M.moment(P,m,W,endpoint,False);u,v=B.moment(m,W,list(P.items()),endpoint);assert(h<1 and v<1)if ok else(l>1 and u>1);checks.append(dict(endpoint=str(endpoint),accept=ok,mercator=[str(l-1),str(h-1)],base_two=[str(u-1),str(v-1)]))
+assert M.moment(P,m,W,b,True)[0]>1
+eta=beta=Q(1,10**18);grid=Q(1,10**18);bridge={'semantic':{'C0':{'base':2,'exponent':210000},'strict_literal_gap':1},'rows':{'degree_gap':Q(10**6)-Q(51*20161,25)}}
+def assemble(bb):
+ s=min(c,(1-beta)*bb)-Q(1,10**24);assert 0<s<c and s<(1-beta)*bb;chain=[Q(384599,10**10)];cuts=[]
+ for _ in range(7 if provider=='558' else 6):
+  old=chain[-1];a=(1-s)*s+s*old;gap=min(s-a,1-a-s,1-a-s*(1-old),1-s);assert old<a<s<1-a and gap>0;ceil=lambda x:-(-x.numerator//x.denominator);cut=max(ceil(36/gap**2),ceil(2*(4+512)/gap));assert cut*gap**2>=36 and cut*gap>=2*(4+512);cuts.append(dict(exponent=str(a),gap=str(gap),cutoff_log2=str(cut)));chain.append(a)
+ q=chain[-1]*(1-2*eta);margin=(1-eta)*q/(1+q);ticks=margin/grid;k=grid*((ticks.numerator-1)//ticks.denominator);a=A.assembly(chain[-1],bb,bridge,k,eta=eta,beta=beta);assert len(a['strict_constraints'])==47 and all(v>0 for v in a['strict_constraints'].values());return k,chain,cuts,a
+assert assemble(oldb)[0]==oldk
+k,chain,cuts,a=assemble(b);expected=json.loads((R/('EXPECTED558.json' if provider=='558' else 'EXPECTED.json')).read_text());assert k==Q(expected['kappa']) and b==Q(expected['complex_lower']);controls=['new_endpoint_rejected_by_old_charge']
+for name,kk,ee,bbeta in [('next_grid',k+grid,eta,beta),('zero_eta',k,Q(0),beta),('zero_beta',k,eta,Q(0)),('unit_beta',k,eta,Q(1))]:
+ try:A.assembly(chain[-1],b,bridge,kk,eta=ee,beta=bbeta)
+ except AssertionError:controls.append(name)
+ else:raise AssertionError(name)
+def dec(v):
+ with localcontext()as ctx:ctx.prec=70;return str(Decimal(v.numerator)/Decimal(v.denominator))
+r=dict(status='PASS_NATIVE_COMPLEX_MOMENT_AND_CONDITIONAL_COMPOSITION',provider=provider,baseline_kappa=str(oldk),baseline_kappa_decimal=dec(oldk),candidate_kappa=str(k),candidate_kappa_decimal=dec(k),old_complex_lower=str(oldb),complex_lower=str(b),complex_decimal=dec(b),absolute_gain=str(k-oldk),relative_gain_percent=dec(100*(k/oldk-1)),profile=P,live=W,m=m,rank_deficit=2560,bit_lower=str(c),moments=checks,outer={n:str(v)for n,v in a['strict_constraints'].items()},bootstrap=cuts,controls=controls,seconds=time.monotonic()-start,python=sys.version,manifest_sha256=sha(R/'MANIFEST.json'),scope='Arithmetic repricing of the unchanged pinned selected exact binary-Clifford complex supplier, with cited PR543 bit supplier and unchanged bit bad-class charge. Original source distinguishes exact complex recursion from GL local-ring bit selectors. All inherited all-size compiler, common-label/mutable-source, whole-call hosting, precision/tape and analytic assumptions remain conditional. This verifier does not rerun unchanged physical operators.')
+(O/'RESULT.json').write_text(json.dumps(r,indent=2)+'\n');print({key:r[key]for key in ['status','candidate_kappa_decimal','complex_decimal','relative_gain_percent','seconds']})