rohanarun · Oct 8, 2026 · Oct 8, 2026 · Oct 8, 2026 · Oct 8, 2026 · Oct 8, 2026
diff --git a/.github/workflows/verify.yml b/.github/workflows/verify.yml
@@ -16,7 +16,7 @@ jobs:
       fail-fast: false
       matrix:
         python-version: ['3.11', '3.13', '3.14']
-        group: [community, producers, certificates, ternary, research, tests, strips, clones, positive, joint, pair]
+        group: [community, producers, certificates, ternary, research, tests, strips, clones, positive, joint, pair, rank-pair, reordered-rank-pair]
     steps:
       - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1 # v7
         with:

diff --git a/Makefile b/Makefile
@@ -24,6 +24,8 @@ verify:
 	$(MAKE) verify-joint
 	$(MAKE) verify-pair
 	$(MAKE) verify-tests
+	$(MAKE) verify-rank-pair
+	$(MAKE) verify-reordered-rank-pair
 
 verify-community: community-audit-check community-followup-check copied-reversed-producer copied-reversed-check copied-fixed-reversed-producer copied-fixed-reversed-check
 	$(MAKE) copied-fixed-verify
@@ -425,6 +427,23 @@ pair-assembly-check:
 
 pair-assembly-verify: pair-assembly-producer pair-assembly-check
 
+.PHONY: verify-skip-strips verify-pair-assembly verify-rank-pair rank-pair-verify
+verify-skip-strips: skip-strips-verify
+verify-pair-assembly: pair-assembly-verify
+verify-rank-pair: rank-pair-verify
+rank-pair-verify:
+	python3 research/rank-pair/frame_compile.py
+	python3 research/rank-pair/screen.py
+	python3 -m unittest discover -s tests -p 'test_rank_pair.py' -v
+
+.PHONY: verify-reordered-rank-pair reordered-rank-pair-verify
+verify-reordered-rank-pair: reordered-rank-pair-verify
+reordered-rank-pair-verify:
+	python3 research/reordered-rank-pair/verify.py --rebuild
+	python3 research/reordered-rank-pair/test_negative.py
+	python3 research/reordered-rank-pair/arithmetic/refine.py
+	python3 research/reordered-rank-pair/arithmetic/audit.py
+	python3 -m unittest discover -s research/reordered-rank-pair/arithmetic -p 'test_arithmetic.py' -v
 .PHONY: verify-pair
 verify-pair: pair-assembly-verify
 	python3 scripts/audit_pair_candidate.py --check docs/research/community-pair-arithmetic.json
diff --git a/NOTICE b/NOTICE
@@ -204,6 +204,19 @@ frame/movement/guard lemmas and independent paired-circuit checks (#26). Origina
 source notices and historical snapshots are preserved. Neither formal package
 claims a full formal proof of the multiplication algorithm or a new kappa.
 
+Skip-prefix strip producer, 8 October 2026
+By Avi Eisenberg (ikeboy), with Anthropic Claude (Claude Opus 5.5)
+assistance. The skip-prefix leave-one-out strip layout in
+research/skip-strips/skip_graph.py is new. Everything else reuses Chafik
+Boukhalfa's PR #43/#46/#48 research/copied-fixed pipeline, its verification
+code, exact data recovery and support-order sorting, together with:
+- Rohan Arun's PR #44 weighted carrier matching optimizer;
+- RaD (hipotures) PR #41 alternating point order;
+- every predecessor credited above.
+Complete attribution and the conditional proof boundary are recorded in
+research/skip-strips/PROOF.md. Original Apache-2.0 and CC0 notices remain
+applicable.
+
 Joint frame compilation of skip-prefix strips
 
 The local compiler and exact finite witnesses were prepared by eumemic with
@@ -302,3 +315,14 @@ inherited PR #48 sources. The stacked witness in research/pair-assembly/frame
 uses eumemic's PR #57 joint frame compiler unchanged; its driver and verifier
 are adaptations of PR #57's skip_frame_compiler.py, skip_frame_compose.py and
 verify_skip_frame.py. Original notices are retained.
+
+Rank-first reclamation on interval-strip pair assembly, 8 October 2026
+Composition and exact profile witness: Dominik Scholz, with substantial OpenAI
+GPT-6 Astra / Codex assistance. The PR #60 descending-frame-rank compiler by
+Chafik Boukhalfa is reused unchanged on Avi Eisenberg's PR #62 scalar graph.
+The joint frame compiler and paid reclamation are eumemic's PR #57, developed
+with OpenAI Codex assistance. PR #62 records Anthropic Claude assistance.
+This composition improves actual fixed-basis block profiles with unchanged
+role counts; no new slot-priority rule or scalar graph is claimed. All prior
+contributors, licenses, compiler arguments and conditional analytic/tape
+interfaces remain credited. See research/rank-pair/PROOF.md.
diff --git a/README.md b/README.md
@@ -1,3 +1,34 @@
+# Reordered rank-first regions: conditional saving 5.1028985071975e-5
+
+This branch proposes **κ = 2041159402879/(4×10^16) = 5.1028985071975×10^-5**,
+about **0.00277% above [PR #63](https://github.com/CrocSwap/integer-mult-bounds/pull/63)**.
+The bit saving is **5103158916233/10^17**. Within each frame rank, use
+cover/core order at h=23 and decreasing minimum-node order at h=25. The scalar graph, rank-first
+retired-slot policy, role counts and paid endpoints remain inherited from #63;
+the changed schedule changes the physical frame transitions and their complete
+fixed-basis child profile. W remains **137,151,806**.
+
+[Construction, proof and credits](research/reordered-rank-pair/README.md) ·
+[Exact arithmetic certificate](research/reordered-rank-pair/arithmetic/certificate.json)
+
+Run `make reordered-rank-pair-verify` to rebuild both physical words and profiles,
+then check the exact moment and inherited assembly. `make verify` includes this
+new target and all inherited verification groups. Full repository validation
+passed on research commit `49e84f9`: 467 core tests, all 20 historical patch
+checks, and complete physical/profile and exact arithmetic checks. The
+[receipt](research/reordered-rank-pair/validation.json) distinguishes that run
+from subsequent upstream integration, which leaves the witness unchanged.
+
+This is a finite conditional witness under the inherited analytic, all-size
+compiler and fixed finite-alphabet multitape hypotheses. It is not a proof
+of global optimality or a measured practical speedup. The schedule gain and
+the finer arithmetic-grid gain are distinguished in the research note.
+
+Prepared by Rohan Arun with OpenAI Codex assistance. Credit Dominik Scholz
+(#63), Avi Eisenberg (#62), Chafik Boukhalfa (#60), eumemic (#57), and the
+Alejandro Zarzuelo Urdiales (#61, parameter-refinement precedent), together with all inherited contributors.
+The reviewed main-branch result below remains separately identified.
+
 # A sharper exponent for integer multiplication
 
 **Community research maintained by Douglas Colkitt — conditional on the original

diff --git a/certificates/joint-dual-kappa.json b/certificates/joint-dual-kappa.json
@@ -842,7 +842,7 @@
     "scripts/experiments/pin_joint_dual_sources.py": "4d31cc0e38b7902709ca5f8f714cba109760d2592346dd7f8dc0b900a2d9fe00",
     "scripts/experiments/verify_joint_dual.py": "7a31bba1e9c0ded32ae285df2ca9bc91ef20c330f6789fdba361876f3c7a1985"
   },
-  "source_manifest_sha256": "fd1aab7d75753b4b55b2377f26119dae74f367b0b4fab66cba42d944b3042cc9",
+  "source_manifest_sha256": "b497d3127bd62ea97e2e022b4239ad9d4fbcc374c9b0b8b9d26a18195a2c6e7d",
   "sources": [
     {
       "files": {

diff --git a/certificates/joint-dual-validation.json b/certificates/joint-dual-validation.json
@@ -226,7 +226,7 @@
     }
   },
   "bit_saving": "2382370177/50000000000000",
-  "certificate_sha256": "2ed658f05365b28825e24a6e7bac9fc9cc015d5a5d63d9579f4ac6e0640ae559",
+  "certificate_sha256": "aa44cb8cc3b071445f49d05e2e2c6c715fdca85294d01a71cd9eaed7bf569f44",
   "kappa": "4764513337/100000000000000",
   "margins": 7,
   "producer_words_regenerated_and_matched": true,

diff --git a/research/joint-dual/SOURCE.json b/research/joint-dual/SOURCE.json
@@ -3,9 +3,9 @@
   "base_pr57_commit": "cd350f76c9bc01489ec83568bded532cb69be938",
   "base_pr58_commit": "bc2f7ed4c20dc18898305ab17165c0c995cbb804",
   "files": {
-    "Makefile": "97620e0205736d9c0fce736e8ac6ab67f8ed0e490c381f6cac152cc6a003dca7",
-    "NOTICE": "1c53970fb4face631a1b331ec66d33111eccd280a129f9b64c2da6df96861c3f",
-    "README.md": "fd92de31aece76b1f5ff897e6e0dbc38dd3bf67c140579bbcb03a1a11a792668",
+    "Makefile": "caae1250a07ceb5f91c1a20379854bdd5309792363e9635fa54318c663008a4d",
+    "NOTICE": "ca6db061015c85b57ce5d2a16667a4948010e655e70a1bf2e66e519d9180c343",
+    "README.md": "fc202819e373483c015b21e2e204317434ad697f04b09a1cc621f82a4d2d746c",
     "certificates/joint-dual-compiler.json": "516c4d85158553dabd0a98216a064b2a7397957a9d5dd0df1d6e5aec53a4e805",
     "certificates/joint-dual-profiles-23.json": "159fa8fee175fc1d24f63e72d3c2f1dc9f4fcd78b938c5e7d2f735fd0fd37209",
     "certificates/joint-dual-profiles-25.json": "8d190ce42d097e828c5756918c52f2963df576cf4c6600b0fc7f2b7fdb6ad223",

diff --git a/research/rank-pair/.gitignore b/research/rank-pair/.gitignore
@@ -0,0 +1,4 @@
+profiles
+*.bin
+*.bin.json
+*.bin.profiles.json
diff --git a/research/rank-pair/PROOF.md b/research/rank-pair/PROOF.md
@@ -0,0 +1,137 @@
+# Rank-first reclamation on interval strips and pair assembly
+
+This finite conditional witness gives
+
+\[
+\kappa=\frac{5102757}{10^{11}}=0.00005102757,
+\qquad a_b=\frac{5103018}{10^{11}}.
+\]
+
+It composes Avi Eisenberg's PR #62 interval-strip and core-aware pair graph
+(`ad0f25ff7b23cff7f08ad237c2254e6ecf74257e`) with Chafik Boukhalfa's PR #60
+rank-first reclamation compiler (`e7a492dd8bee4e6f574ced784a62af2ce735edc4`),
+which derives from eumemic's PR #57 joint frame compiler. The graph is
+unchanged, and the candidate compiler is copied byte-for-byte from PR #60.
+The composition experiment and exact validation driver were prepared by
+Dominik Scholz with substantial OpenAI GPT-6 Astra assistance.
+
+The witness exceeds PR #62's stacked κ by exactly 1066/10^11. It exceeds
+PR #61 at `afb7cb67d1858641315cfbf4ac768ee64a8eff3a`, whose stated κ is
+25508460085039/500000000000000000, by exactly
+5324914961/500000000000000000 (approximately 0.0208751% relative).
+These comparisons are against pinned research claims, not independent
+acceptance of any of the full multiplication theorems.
+
+## Mechanism and paid quantities
+
+PR #57 originally visits retired physical roles in increasing slot order.
+PR #60 visits them by descending current frame rank, breaking ties by slot
+order. In either policy a retired role is eligible only if its current frame
+is contained in the requested frame. Linear dependence is witnessed by
+literal XOR operations involving eligible retired roles and the current
+anchors; those XORs clear the selected role. Every operand is raised to the
+requested frame and every resulting transition is charged. Only then is the
+cleared role reused.
+
+For the PR #62 graph, this changes the physical frame paths and their actual
+fixed-basis profiles without changing the roles or the total rank. The
+improvement is therefore established by the complete child moment, not by a
+role-count heuristic or by treating clearing as free.
+
+| h | roles | reclaimed roles | clearing XORs | all elementary XORs | distinct profile matrices | CRT-checked matrices |
+|---|---:|---:|---:|---:|---:|---:|
+| 23 | 27918 | 801 | 1612 | 149172 | 36378 | 6049 |
+| 25 | 36586 | 1090 | 2092 | 205915 | 50419 | 8127 |
+
+Both axes have zero CRT disagreements. PR #62's original clearing counts
+were 1398 and 1827, respectively: the present witness explicitly pays the
+additional clearing. The unchanged global quantities are m=575,
+N=4073300, L=2226400, W=137151806, total rank=78860441550, and deficit=1846900.
+
+## Finite validation and inherited hypotheses
+
+`frame_compile.py` checks every input and dirty basis vector in both
+orientations, then calls the independent serialized-word replay. `screen.py`
+repeats that replay, reconstructs every transition from the actual word,
+regenerates all fixed-basis profiles with the inherited bounded-minor and CRT
+checks, and applies PR #62's unchanged complete paid-profile constructor.
+The constructor verifies the total-rank identity and copied-center counts.
+
+Exact rational logarithm/exponential enclosures show the moment is below 1
+at a_b=5103018/10^11 and its lower bound is above 1 at 5103019/10^11.
+PR #62's original complete child network also has lower moment above 1 at
+the new a_b. The inherited balanced assembly passes all 47 strict constraints
+and all seven margins at κ=5102757/10^11, and rejects the next grid point.
+It uses the unchanged β=1/20, backoff h=10^-12, complex saving 717/10^7,
+semantic C1=1, and product row stock p^2000. Smaller rational grids and
+backoff were not needed for this witness and could refine its last digits.
+
+The graph supports, scalar producer, original-envelope interpretation, data
+geometry, paid endpoint copy, exact corner recovery, and all-size proofs
+remain inherited from PR #62 and its cited predecessors. The rank-first
+compiler uses precisely PR #60's mechanism. This note does not independently
+prove OpenAI's base theorem, the fixed-alphabet/multitape transfer, the
+analytic, semantic, routing or recovery interfaces, or their eventual
+thresholds. Finite success is not formal verification of the full theorem,
+and neither global optimality nor a practical speedup is claimed.
+
+## Reproduction
+
+```sh
+python3 research/rank-pair/frame_compile.py
+python3 research/rank-pair/screen.py
+python3 -m unittest discover -s tests -p test_rank_pair.py -v
+```
+
+On this development machine the default MacOSX27 SDK fails to link because
+the selected linker does not recognize its arm64e.x1 TBD architecture. The
+successful local process-scoped override was:
+
+```sh
+CXX='c++ -isysroot /Library/Developer/CommandLineTools/SDKs/MacOSX15.2.sdk' python3 research/rank-pair/screen.py
+```
+
+The candidate compiler SHA256 is
+`9984aaccf880e4840402322d76678bfddb33ce7e494af78430a43429f51b85fd`.
+The graph SHA256 is
+`3d47e89d2649c90800a276bdd0480131def50e0bdf92c122a19494b55bf17420`.
+`frame-compiler.json` pins both sources and every serialized word; wall-clock
+timing is excluded so recompilation reproduces the record deterministically.
+`screen-certificate.json` records the exact moment, assembly, eventual
+arithmetic cutoffs, complete child multiplicities, and pinned comparisons.
+Six failure controls check omitted literal XORs, understated physical width,
+omitted paid transitions, non-nested frames, exclusion of the old network,
+and refusal to disable assertions with optimized Python. The independent experimental reproduction and a fresh parent reproduction
+on the integrated main-based branch passed, including all six failure
+controls. Repository-wide validation and PR review remain pending in CI.
+
+Integration preserves main d55bcd62 and its existing verification groups.
+The imported pair/skip source manifests were updated for main's six added
+standard-library includes in `binary_io.hpp`; its functions and binary format
+are unchanged. Their arithmetic certificates were regenerated. The imported
+stacked certificate's source paths use POSIX separators for Linux CI.
+The new candidate and both imported producers have separate CI groups.
+
+## Bounded unsuccessful experiments
+
+At h=23, three further policies were tested: descending rank with decreasing
+slot tie-break; decreasing slot without rank; and increasing rank with
+increasing slot tie-break. All retained the same 27918 roles. Full independent
+dirty-word replay, frame reconstruction and profile regeneration passed for
+each, but disjoint exact enclosures prove each has a strictly worse internal
+moment than the selected rank-first policy at a_b=5103018/10^11. No claim is
+made about those policies at h=25, which was not tested in this bounded round.
+The negative-result records and exploratory compiler variants remain local
+research scratch; they are not used to generate this witness.
+
+## Attribution
+
+Retain PR #62's full attribution in `research/pair-assembly/PROOF.md`, PR #60's
+Chafik Boukhalfa rank-first contribution, and PR #57's eumemic compiler.
+The pipeline additionally depends on Avi Eisenberg, Rohan Arun, RaD /
+hipotures, icekylinx, Zhihao Chen, Aurel Prosz / Paureel, Swapnil Jain,
+James Chang, Douglas Colkitt, OpenAI and Harvey–van der Hoeven, with the
+complete inherited notices retained in the repository. The PR #62 graph was
+prepared with Claude (Anthropic) assistance; the PR #57/#60 compiler and this
+composition use substantial OpenAI assistance. Apache-2.0 applies, subject
+to the existing separately retained source notices and terms.