Mister-Meeseeks · Oct 13, 2026 · Oct 13, 2026
diff --git a/research/t21-binomial-audit/CERTIFICATE.json b/research/t21-binomial-audit/CERTIFICATE.json
@@ -0,0 +1,80 @@
+{
+  "arity": 73,
+  "assignment_seed": 69,
+  "base_commit": "007664306fff84fbb1c9a1b4053b69f439764034",
+  "child_count": 681782,
+  "conservative": {
+    "endpoint": "1664724473541313810689/1000000000000000000000000",
+    "lower_residual_interval": [
+      "-526825226012241273767532986651546614167917669673029893633726326727871/189600358521368422407583654804722585933447296973722670396029561460711028162560000000000000000",
+      "-526825226012241273767532986545324448164731068686253653307311743456799/189600358521368422407583654804722585933447296973722670396029561460711028162560000000000000000"
+    ],
+    "successor": "166472447354131381069/100000000000000000000000",
+    "successor_residual_interval": [
+      "1002513433561922366026390346187232403779166744600723795266956622711/5925011203792763200236989212647580810420228030428833449875923795647219630080000000000000000",
+      "64160859747963031425688982368427205848239873600863346517914131234241/379200717042736844815167309609445171866894593947445340792059122921422056325120000000000000000"
+    ],
+    "terms": {
+      "1": 7743,
+      "2": 3845,
+      "3": 2546,
+      "4": 1896,
+      "48": 101,
+      "5": 1506,
+      "8": 921,
+      "fallback": 7743
+    }
+  },
+  "controls": {
+    "algebraic_comparisons": 450,
+    "mathematical_controls": 5,
+    "redistributed_minus_original_interval": [
+      "-30611254125507706510269204255614715512780285837491562546149107708127/1213442294536757903408535390750224549974062700631825090534589193348550580240384",
+      "-61222508251015413020538408511229431025557852350359115694150584663389/2426884589073515806817070781500449099948125401263650181069178386697101160480768"
+    ],
+    "rejected_inputs": 11
+  },
+  "deficit": 6720,
+  "live_stock": 18816,
+  "native": {
+    "endpoint": "1664724476433282815289/1000000000000000000000000",
+    "lower_residual_interval": [
+      "-261428779639035688672010186268399165805829457776969323/808961529691171935605690260500149699982708467087883393689726128899033720160256",
+      "-32678597454879461084001273226898073758113208166698527/101120191211396491950711282562518712497838558385985424211215766112379215020032"
+    ],
+    "successor": "166472447643328281529/100000000000000000000000",
+    "successor_residual_interval": [
+      "2123235352137840560200160598288155896009491039226008749/808961529691171935605690260500149699982708467087883393689726128899033720160256",
+      "530808838034460140050040149685342617937603707919107627/202240382422792983901422565125037424995677116771970848422431532224758430040064"
+    ],
+    "terms": {
+      "1": 7743,
+      "2": 3845,
+      "3": 2546,
+      "4": 1896,
+      "48": 101,
+      "5": 1506,
+      "8": 921
+    }
+  },
+  "physical_stock": 19330,
+  "profile": {
+    "1": 481575,
+    "2": 63882,
+    "3": 84295,
+    "4": 30618,
+    "48": 6292,
+    "5": 13608,
+    "8": 1512
+  },
+  "rank_mass": 1366848,
+  "scope": "Arithmetic consequence of the pinned T21 paid profile and fallback formula. Physical construction, owner windows, invoice, outer 47 inequalities, all-size compilation and the multiplication theorem are not replayed or proved.",
+  "selected_kappa_unchanged": "1661957774647029/1000000000000000000",
+  "source_files_sha256": {
+    "inputs/ASSEMBLY.json": "190110b509dc87d38b72ad37c3290efcc245ce26df4cae3d7e2d6dcf0bac2c43",
+    "inputs/CLAIM.json": "433d7bc4a371aaffeee35c12a1db6ea974722a85b799233626e1a57dcfdcb121",
+    "inputs/PROOF.md": "78721b32fceb34b21b8e77c95381c38c60276e72e6d9c2e9c6262cb347457def",
+    "inputs/selected-result.json": "41fb066ff0f43ce472323195e9088d6d888715c13c54eae7b42e357ed5bfc726"
+  },
+  "status": "PASS_T21_BINOMIAL_MOMENT"
+}
diff --git a/research/t21-binomial-audit/CLAIM.json b/research/t21-binomial-audit/CLAIM.json
@@ -0,0 +1,85 @@
+{
+  "status": "Completed independent arithmetic audit; prepared for public submission",
+  "date": "2026-10-13",
+  "assignment_seed": 69,
+  "seed_origin": "Retained OS-random assignment from the preliminary setup session; not rerolled",
+  "track": "Adversarial mathematical audit",
+  "base_commit": "007664306fff84fbb1c9a1b4053b69f439764034",
+  "candidates": [
+    "T21 native and conservative moment accounting",
+    "T21 partial-bank coordinate complement",
+    "Retained outer assembly arithmetic"
+  ],
+  "selected_scope": "Numerical consequence of the PR890 T21 paid profile and inherited fallback formula, using a separate positive-binomial-series implementation",
+  "method_prior_art": "PR678 introduced the same enclosure method for the historical PR543 bit moment. No new numerical method is claimed.",
+  "hypothesis": "Both recorded complex lower endpoints are strictly accepted and their adjacent 10^-24 successors are strictly rejected.",
+  "result": "Confirmed for both models by exact rational enclosures. No counterexample to this numerical claim and no new kappa.",
+  "coordination": "The two open issues inspected did not identify a sprint claim thread. The scope and result were recorded locally.",
+  "stopping_rule": "One completed audit with proof, source bindings, certificate and adverse controls; no search for a stronger construction.",
+  "overlap_checked_at": "2026-10-13T14:41:40.728825+00:00",
+  "overlap_scope": "Initial complete focused search responses for T21 (3), binomial (9), and moment PRs updated on October 13 (73). Submission refresh returned four T21 PRs and identified PR957; its source at the recorded head was inspected. This is not an exhaustive duplicate audit of all open PR contents.",
+  "overlap_heads": [
+    {
+      "number": 678,
+      "title": "Independent verification of the #543 bit moment using binomial enclosures",
+      "state": "open",
+      "merged_at": null,
+      "head_sha": "bdc9317efd6d928a0f050159670d1316ba5f4c5b",
+      "author": "0x-chad",
+      "updated_at": "2026-10-13T03:47:52Z",
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/678"
+    },
+    {
+      "number": 706,
+      "title": "Independent verification: selected complex moments via integer root bounds",
+      "state": "open",
+      "merged_at": null,
+      "head_sha": "0f7dbb54948775c010b5e5bc8ab3cf92a52a9b25",
+      "author": "arkfx",
+      "updated_at": "2026-10-13T04:34:41Z",
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/706"
+    },
+    {
+      "number": 813,
+      "title": "Independent verification: Bernstein certificates for all 47 outer inequalities",
+      "state": "closed",
+      "merged_at": "2026-10-13T14:10:26Z",
+      "head_sha": "5d3b961fca28c77cec74f5237a59acd361c8e851",
+      "author": "bitplane",
+      "updated_at": "2026-10-13T14:10:26Z",
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/813"
+    },
+    {
+      "number": 942,
+      "title": "Conditional \u03ba = 0.001661960547713947: T37 whole-owner packing and all-T bound",
+      "state": "open",
+      "merged_at": null,
+      "head_sha": "1960cfa919a0945d0ca4cb14ea96237000037903",
+      "author": "jon314159",
+      "updated_at": "2026-10-13T13:52:33Z",
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/942"
+    },
+    {
+      "number": 947,
+      "title": "Scoped negative result: exact constant-row center shear optimum (no new kappa)",
+      "state": "open",
+      "merged_at": null,
+      "head_sha": "b6f6418bdcae748b267cea72a5949d551feb7bb2",
+      "author": "RossRichardson",
+      "updated_at": "2026-10-13T14:16:22Z",
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/947"
+    },
+    {
+      "number": 957,
+      "title": "Adversarial audit: frontier890 endpoint certification, reduction and surcharge stakes (no new kappa)",
+      "state": "open",
+      "head_sha": "65c931415587c444a7a1f48939842c4cf0aa0411",
+      "author": "navinor",
+      "updated_at": "2026-10-13T14:38:32Z",
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/957"
+    }
+  ],
+  "overlap_conclusion": "PR957 concurrently audits PR890 endpoints, reduction and surcharge attribution by transcribing the two retained log/exp engines plus a direct Decimal calculation. This package corroborates its complex endpoint conclusions with a different exact binomial enclosure method, without claiming priority. No other T21 binomial audit was identified. PR678 and PR706 concern historical profiles; PR813 concerns outer inequalities; PR942 and PR947 are distinct constructions or research.",
+  "assistance": "Prepared for David Breslauer with substantial OpenAI Codex assistance. Final continuation reviewed with GPT-6 Astra at Max effort; no independent human review.",
+  "license": "Apache-2.0"
+}
diff --git a/research/t21-binomial-audit/README.md b/research/t21-binomial-audit/README.md
@@ -0,0 +1,169 @@
+# Independent arithmetic audit of the selected T21 recurrence
+
+**Status: independent finite verification; no new kappa.**
+
+This confirms both the native and conservative complex moment brackets used
+by the PR890 construction selected at the pinned base. Each lower endpoint is strictly below
+the recurrence threshold, and its adjacent `10^-24` successor is strictly
+above it. The implementation uses only positive generalized-binomial terms,
+integer outward rounding and a geometric tail. It imports no production
+logarithm, exponential, supplier or owner implementation.
+
+The exact contribution base is `007664306fff84fbb1c9a1b4053b69f439764034`. The selected
+conditional kappa remains `1661957774647029/10^18`. This is an arithmetic
+audit of its paid profile, not a fresh physical replay or a proof of the
+complete multiplication theorem. The preliminary package used
+`47481c89fd30a43f8f882eb4b391e89c29273fd0`; its assembly and selected-record
+inputs are byte-identical at the contribution base.
+
+## Reproduce
+
+Python 3.11+ and the standard library suffice. From the repository root:
+
+```sh
+python3 -I -B research/t21-binomial-audit/verify.py
+python3 -I -O -B research/t21-binomial-audit/test_package.py
+```
+
+Expected first marker:
+
+```text
+PASS_T21_BINOMIAL_MOMENT four strict endpoint decisions; 450 algebraic comparisons; 5 mathematical controls; 11 rejected inputs.
+```
+
+The second command prints `PASS_T21_BINOMIAL_PACKAGE controls=10`, followed
+by successful relocation, regeneration and failed-run output-preservation markers.
+`--write-certificate` regenerates the exact certificate; the default recomputes
+it and requires an identical result. No network or third-party library is used.
+The package also runs from an unrelated current directory after relocation.
+
+In the pinned repository, add `--repository .` to the first command to compare
+all eight source files with the checkout. This checks source bytes without
+importing or executing the production arithmetic. It does not require the
+current Git head to equal the base if those source bytes are unchanged.
+An explicit check against a later, changed selection is expected to fail;
+the default command continues to audit the frozen inputs.
+
+## Exact arithmetic argument
+
+For a child rank r, ambient arity m, and rational saving a in `[0,1]`, put
+`x = 1-r/m`. The positive binomial expansion is
+
+`(m/r)^a = sum_(k>=0) t_k`, with
+`t_0=1` and `t_(k+1)=t_k*(a+k)*x/(k+1)`.
+
+For `0 <= a <= 1`, `(a+k)/(k+1) <= 1`, so every ratio is at most x.
+Consequently the tail after term k is bounded
+above by `t_(k+1)/(1-x)`. The checker stores lower and upper term values as
+integers over `2^240`, rounding every recurrence product outward. Positivity
+allows summing those bounds directly; the rounded-up geometric tail is added
+only to the upper sum. The stopping rule bounds that tail by `2^-170` per
+power factor. Each returned interval is rational and explicitly encloses
+the infinite series. Zero saving and a full-rank power factor are exact
+special cases.
+
+Dividing the inherited recurrence sum `sum_r c_r*r^(1-a)` by
+`18816*73^(1-a)` gives the native normalized moment
+
+`M(a) = sum_r c_r * r/(73*18816) * (73/r)^a`.
+
+The conservative variant charges `32*73^2/10^16` additional rank-one
+calls per paid child. Applying the same normalization gives the fallback
+
+`32*73*sum_r(c_r)/(10^16*18816) * 73^a`.
+
+The factor `73/r` is greater than one for every paid rank, and every weight
+is positive. The moment is therefore continuous and strictly increasing in a,
+so the two strict endpoint decisions bracket its unique root on the relevant
+grid. This numerical statement takes the recurrence/fallback contract as an
+input; it does not justify its construction-level realization.
+
+## Profile and checked obligations
+
+The proof's single-call profile `{1:2548,2:338,3:446,4:162,5:72,8:8}` is
+multiplied by 189. Adding idle children `{48:6292,3:1,1:3}` reconstructs the
+committed assembly profile exactly. There are 681782 children. The 21 replicas
+use `9*21=189` local calls, live stock `8*112*21=18816`, and physical stock
+`18816+504+8+2=19330`. These values are checked against the pinned construction
+claim. Its rank mass is 1366848 and its deficit
+is `73*18816-1366848=6720=21*320`.
+
+| Price | Accepted lower endpoint |
+| --- | --- |
+| Native | `1664724476433282815289/10^24` |
+| Conservative | `1664724473541313810689/10^24` |
+
+Both successors at `+10^-24` are rejected. `CERTIFICATE.json` supplies the
+exact residual intervals and term counts. `SOURCES.json` pins the assembly,
+selected record, construction claim and proof. The assembly and proof hashes
+are also checked against the selected record. Four additional source references
+identify the pricing entrypoint and formula; both arithmetic-engine hashes
+must match the assembly's recorded inputs. These source files are not imported.
+
+The 450 algebraic checks use exact integer-power inequalities to verify the
+enclosures at rational exponents with denominators at most three. They cover
+every rank through arities 2, 3, 5, 7 and the actual arity 73, including the
+slowest-decaying series at rank one. The five
+model controls cover the zero-saving mass identity, an omitted rank-48
+child, a dropped conservative surcharge, physical versus live-stock
+normalization, and a redistribution preserving both call count and rank
+mass. In the last case two rank-48 children become rank-47 and rank-49
+children. The moment strictly decreases, as expected from concavity, despite
+the unchanged totals. These are deliberately altered models; no defect in
+the selected production checker is alleged.
+
+Package controls use actual disposable copies. Corrupting any of the four
+pinned inputs or the recorded certificate rejects. Deleting the geometric
+tail or upper rounding fails the exact algebraic comparisons; undercharging
+the fallback fails the strict endpoint test. These three implementation
+mutations run with `--write-certificate`, so they cannot pass merely by
+rejecting an old receipt. Every rejection is checked for its specific error,
+and failed runs must preserve the existing certificate. Two synthetic
+repository-fixture controls reject changed and missing reference files.
+All ten controls run under `-O`; a relocated positive and exact certificate
+regeneration also pass.
+
+## Provenance, overlap and limits
+
+OS-random assignment seed **69** selects the mathematical-audit track.
+Candidate tasks considered: the complex moment accounting, the partial-bank
+complement, and the outer assembly arithmetic. The first was selected for a
+small, independent calculation with exact source binding. The scope and result
+were recorded locally; no sprint coordination issue was identified.
+
+[#678](https://github.com/CrocSwap/integer-mult-bounds/pull/678) by **0x-chad**
+previously introduced binomial enclosures for the historical PR543 bit moment.
+[#706](https://github.com/CrocSwap/integer-mult-bounds/pull/706) by **arkfx**
+checks historical PR626 complex moments using integer roots, and
+[#813](https://github.com/CrocSwap/integer-mult-bounds/pull/813) by
+**Gareth Davidson / bitplane** independently checks the outer inequalities.
+This contribution does not claim a new enclosure technique. It supplies a
+separate implementation and exact native/conservative evidence for the
+PR890 complex profile. The October 13, 2026 submission refresh identified
+**navinor**'s concurrent
+[#957](https://github.com/CrocSwap/integer-mult-bounds/pull/957), pinned at
+`65c931415587c444a7a1f48939842c4cf0aa0411`. It transcribes the two retained
+logarithm/exponential engines, adds a direct Decimal evaluation, and audits
+endpoint-to-kappa reduction and surcharge attribution. This package corroborates
+its complex endpoint conclusions using a different exact enclosure method;
+it does not claim priority for the T21 endpoint audit. The focused search found
+no other T21 binomial audit. PR942's pending T37 extension and PR947's
+constant-row center-shear bound are distinct. PR813 has been incorporated for
+its historical checkpoint. Search results are an overlap check, not an
+exhaustive certification of all open work.
+
+Construction credit belongs to **Chafik Boukhalfa** (PR890), **Jeremy Kothe**
+(PR869/870), **Hao Yan** (PR858), **DreamingOfClouds**, **eumemic**,
+**Rohan Arun**, **icekylinx**, and the repository's cumulative lineage.
+Prepared for **David Breslauer** with substantial **OpenAI Codex** assistance,
+under Apache-2.0. There is no independent human-review claim.
+
+The preliminary subset was checked against a complete source checkout with
+shallow Git history. `make selected-record-check` passes; this binds the
+selection, its original source inventory and proof supplements. The separate
+arithmetic audit and its controls also pass. See `VALIDATION.json` for the
+recorded environment, commands and scope. No full `make verify`, supplier cold replay,
+physical tape/owner audit, bit-supplier replay, 47-inequality outer replay,
+Lean run or hosted CI success is claimed. All-size compilation, common-label
+and mutable-source realization, whole-call hosting, precision/tape and
+analytic-transfer obligations remain inherited.
diff --git a/research/t21-binomial-audit/SOURCES.json b/research/t21-binomial-audit/SOURCES.json
@@ -0,0 +1,25 @@
+{
+  "base_commit": "007664306fff84fbb1c9a1b4053b69f439764034",
+  "source_repository": "CrocSwap/integer-mult-bounds",
+  "files": {
+    "inputs/ASSEMBLY.json": "190110b509dc87d38b72ad37c3290efcc245ce26df4cae3d7e2d6dcf0bac2c43",
+    "inputs/selected-result.json": "41fb066ff0f43ce472323195e9088d6d888715c13c54eae7b42e357ed5bfc726",
+    "inputs/CLAIM.json": "433d7bc4a371aaffeee35c12a1db6ea974722a85b799233626e1a57dcfdcb121",
+    "inputs/PROOF.md": "78721b32fceb34b21b8e77c95381c38c60276e72e6d9c2e9c6262cb347457def"
+  },
+  "provenance": {
+    "inputs/ASSEMBLY.json": "research/community-frontier890-audit/ASSEMBLY.json",
+    "inputs/selected-result.json": "certificates/selected-result.json",
+    "inputs/CLAIM.json": "research/complex-t21-owners/data/CLAIM.json",
+    "inputs/PROOF.md": "research/complex-t21-owners/PROOF.md"
+  },
+  "method_predecessor": "PR678 by 0x-chad: binomial arithmetic for the historical PR543 bit profile; this is a separate implementation auditing the current PR890 complex native and conservative profile.",
+  "preliminary_base_commit": "47481c89fd30a43f8f882eb4b391e89c29273fd0",
+  "references": {
+    "research/complex-t21-owners/price.py": "b4510dd0d86322be627ff500e76a22f53cafa8405a699d04646dc3039167d569",
+    "research/complex-t21-owners/vendor/pr870/independent/price_profile_t16.py": "53f8e62dcf27bee032c1bee0245d126a9c8a12ebcf24f6aaff65ea3122fffe29",
+    "research/complex-k492-owner-repair/vendor/multicold/engines/moment.py": "81cbc864e106418fc7854322746f72180cb8110ff39f74450a3b8d55a82c1482",
+    "research/complex-k492-owner-repair/vendor/multicold/engines/base_two_moment.py": "aee70c242a2f1d225528fabe2002e1564d1dc884797c07d28a831c7878ee9098"
+  },
+  "reference_scope": "Read to identify the inherited normalized moment and fallback formula; never imported or executed by this package. --repository checks their bytes."
+}