Build #208's bank-absorption target: rank-22 residual absorption — certified accounting κ = 7.00427501305159e-4 (R1–R4 open; not a completed witness)

.github/workflows/residual-bank-run1.yml

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+name: residual-bank-run1
+
+on:
+  push:
+    paths:
+      - 'research/residual-bank-run1/**'
+      - '.github/workflows/residual-bank-run1.yml'
+  pull_request:
+    paths:
+      - 'research/residual-bank-run1/**'
+      - '.github/workflows/residual-bank-run1.yml'
+
+jobs:
+  verify:
+    runs-on: ubuntu-latest
+    strategy:
+      matrix:
+        python-version: ['3.11', '3.14']
+    steps:
+      - uses: actions/checkout@v4
+      - uses: actions/setup-python@v5
+        with:
+          python-version: ${{ matrix.python-version }}
+      - name: Offline accounting replay
+        run: python3 -B research/residual-bank-run1/verify.py

research/residual-bank-run1/NOTICE.md

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+# NOTICE
+
+## Provenance
+
+Package construction and the enumeration/schedule/arithmetic/verification code:
+prepared for the CrocSwap/integer-mult-bounds queue with **OpenAI Codex
+assistance**, 2026-10-10.
+
+## Vendored sources (byte-identical, sha256-pinned in SOURCE.json)
+
+- `interval_moment.py` — PR200's exact rational interval moment engine
+  (Chafik Boukhalfa, `research/paired-cube-diagonal-bit-168/arithmetic/`,
+  package Apache-2.0).
+- `paired_cube_assembly.py` — the unchanged balanced assembly
+  (PR168-v4/PR184 lineage via ikeboy's PR193 tree; original attributions retained).
+- `banks.py` — incidence colouring helpers (Evan McKinney, PR197, Apache-2.0;
+  vendored via rohanarun's PR205 package).
+- `references/pr193-source-assisted-v4.certificate.json` — ikeboy's PR193 complex
+  supplier certificate (unchanged; its own credits and AI disclosures stand).
+- `references/pr200-bit.certificate.json` — Chafik Boukhalfa's PR200 bit word
+  certificate (unchanged).
+- `references/pr205-packed.certificate.json` — rohanarun's PR205 packed-row
+  certificate (unchanged).
+
+All mathematical content of the vendored files is the work of their authors as
+credited in the upstream packages. The new content of this package is the rank-22
+enumeration (derived from the pinned PR200 word), the absorption schedule, the
+retained-ledger arithmetic and the obligation statement.
+
+## AI assistance disclosure
+
+The new code, enumeration and text in this package were prepared with OpenAI Codex
+(GPT-class model) assistance. The exact arithmetic is machine-replayed by
+`verify.py`; the mathematical claims remain conditional as stated in PROOF.md.
+
+Apache-2.0.

research/residual-bank-run1/PROOF.md

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+# Construction argument and scope
+
+## Claim
+
+On the retained multiplication interfaces, absorbing the rank-22 residual family of
+the PR200/PR205 terminal-modified bit word into the banks yields the conditional
+exponent
+
+    kappa = 700427501305159 / 10^18 = 0.000700427501305159
+
+with the complex supplier binding. This is certified **at the accounting level**
+(Steps 1–4 below, all machine-replayed by `verify.py`) and is **conditional on the
+physical realization obligations R1–R4** (`obligations.json`).
+
+## Step 1 — the absorbed family is exactly enumerated
+
+`inputs/absorbed-occurrences.json` lists every rank-22 ledger child of the per-vertex
+histogram with its provenance, derived from the pinned PR200 word
+(`word_p12.json`, `frames_p12.json`, the bit certificate and the base checker, all
+sha256-pinned in that file): 1,320 physical-chain frame jumps, 24 copied-centre
+children, 880 source-chain frame jumps, 0 target jumps — 2,224 per vertex, i.e. the
+full `hist[22] = 6,672` bin under the row's factor-3 normalization. No other bin is
+touched.
+
+## Step 2 — whole-bank volume and retained ledger
+
+Under the three-copy integer normalization of the published packed row
+(W = 56,402, rank mass 4,055,136, deficit 5,808, 877,638 children):
+
+* absorbed volume `22 × 20,016 = 440,352` registers = `72 × 6,116` whole banks;
+* retained mass `3,614,784`, retained children `857,622`, largest child 21;
+* row identity `72·50,286 − 3,614,784 = 5,808`, deficit unchanged, stock drop
+  `56,402 − 50,286 = 6,116` exactly the bank count.
+
+This is the accounting form of "absorbing a family into the banks drops its dirt
+and, by the row identity W = (mass + D)/m, drops the stock with it" (PR #208).
+`schedule.py` also enumerates every block tiling of width 72 over the word's
+residual families {4, 24} plus the new 22-blocks and checks pattern divisibility.
+
+## Step 3 — exact paid moments
+
+`arithmetic.py` prices the absorbed profile with PR200's exact rational interval
+moment engine, including the complete worst-case envelope (bad fraction 10^-16,
+fallback `32·72²` per retained child). Before the absorption is applied the same
+pipeline reproduces PR #205's certified packed coarse saving
+`683528191056257/10^18` **to the digit**, which validates the engine against a
+published certificate.
+
+* complex coarse saving (unchanged): `700918443859411/10^18`;
+* bit coarse saving before absorption: `683528191056257/10^18`;
+* bit coarse saving after absorption: `728251090009891/10^18` — the bit word now
+  clears the complex cap `7.009184e-4`;
+* three-level finite ordinary composition from PR200's completed ordinary supplier
+  gives the bit leaf `≈ 7.282510899902281e-4`, strictly above the complex cap.
+
+## Step 4 — balanced assembly
+
+The unchanged 47-constraint balanced assembly (eta = beta = 10^-24, weak = 10^-30)
+takes `a = min(bit leaf, (1−β)·b − weak)`. The bit leaf clears the cap, so
+`a = (1−β)·b − weak` and the complex branch binds:
+
+    kappa = 700427501305159 / 10^18 = 0.000700427501305159
+
+All 47 strict constraints and 7 margins are positive; the adjacent grid point
+`kappa + 10^-18` is rejected by the same assembly. This is the exact 10^-18-grid
+sharpening of PR #208's priced rung-1 target `7004273/10^10`.
+
+## What remains conditional (R1–R4)
+
+The rank-22 bin is not the gauge-exterior species the current bank construction
+absorbs: it is frame-transition increments. Non-gauge chains already bank their
+residual projectors and keep every increment child, so the published mechanism
+cannot absorb this family — this is exactly the "genuinely different residual type"
+the suppliers' crossover notes reserve. A completed finite witness must add:
+
+* R1: the residual-type definition, block shapes and per-occurrence block
+  assignment (schedule.py supplies the admissible tilings);
+* R2: formal-column replay of the modified word (F2 identity and defining-integer
+  decoder) under bank re-assignment at the enumerated transitions;
+* R3: exact charts, incidence colouring and prime witnesses for the new blocks;
+* R4: moment-envelope parity (any completion child R1 introduces must be added to
+  the ledger and the kappa recomputed).
+
+Until then the claim is an accounting construction and certified target, in the
+same sense every repository witness is conditional on its stated interfaces — and
+strictly more limited than the published witnesses, which have their physical
+realizations built and replayed.
+
+## Inherited conditions
+
+The retained analytic, uniform-recursion, fixed-tape, all-size compiler, selector,
+routing, precision and finite-bridge hypotheses of the PR184/PR186/PR193/PR200
+lineage are inherited, not proved. This is not an unconditional multiplication
+theorem, a formal verification of the algorithm, or a runtime benchmark.

research/residual-bank-run1/README.md

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+# Building PR208's priced target: rank-22 residual bank absorption
+
+This package **builds the accounting construction** that PR #208 priced and left
+unbuilt — the first certified-shape rung of its bank-absorption ladder: absorb the
+bit word's **rank-22 residual family** into the banks — and states precisely what a
+physical realization must still supply.
+
+Under the retained multiplication interfaces, the certified accounting target is
+
+$$
+T(n)=O\left(n(\log n)^{1-\kappa}\right),\qquad
+\kappa=\frac{700427501305159}{10^{18}}=0.000700427501305159,
+$$
+
+**conditional on the physical realization obligations R1–R4** in
+[obligations.json](obligations.json). The binding branch is the unchanged complex
+supplier: the absorbed bit word clears the complex cap, so κ sits at the complex
+side's own value, exactly as PR #208's rung table predicts (`7004273/10^10` on its
+10^-10 pricing grid; this package certifies the finer 10^-18 grid point).
+
+| | κ | status |
+| --- | --- | --- |
+| main reviewed (#186) | 6.61885549259598e-4 | published |
+| #205 packed diagonal bit | 6.83061299399923e-4 | verified this queue |
+| #210 shared entrance gauges | 6.83883297459132e-4 | verified this queue |
+| **this package, rank-22 absorption** | **7.00427501305159e-4** | **accounting certified; R1–R4 open** |
+| #208's priced rung-1 target | 7.004273e-4 | model price (superseded by the exact grid above) |
+
+That is **+2.42% over #210's frontier** (+2.52% over the #207 row #208 priced) and
+**+5.82% over main** — the complex
+supplier's own cap, with no complex-side work, exactly the move #208 priced.
+
+## What is new and machine-checked
+
+1. **The absorbed family, with provenance.** The rank-22 ledger bin is enumerated
+   occurrence by occurrence from the pinned PR200 word
+   ([inputs/absorbed-occurrences.json](inputs/absorbed-occurrences.json)):
+   1,320 physical-chain frame jumps, 24 copied-centre children and 880 source-chain
+   frame jumps per vertex — 2,224 × 3 = 6,672 ledger children. This is not the
+   banked gauge-exterior species: non-gauge chains already bank their residual
+   projectors and keep every increment child, so this family is the *"future
+   genuinely different residual type"* PR #207's crossover notes reserve.
+2. **The whole-bank schedule.** Dirt volume `22 × 20,016 = 440,352 = 72 × 6,116`
+   fills whole banks; the stock drop equals the bank count exactly; the retained
+   ledger (857,622 children, largest 21) satisfies the row identity
+   `72·50,286 − 3,614,784 = 5,808` with the deficit unchanged
+   ([schedule.py](schedule.py)).
+3. **Exact arithmetic to the κ.** Both rational interval moment engines, the
+   three-level finite ordinary composition and the unchanged 47-constraint
+   balanced assembly, with the adjacent 10^-18 grid point **excluded**
+   ([arithmetic.py](arithmetic.py)). The engine reproduces PR #205's certified
+   packed coarse saving to the digit before the absorption is applied — the
+   pipeline is validated against a published certificate, not just self-consistent.
+
+## What is not built here
+
+The physical realization of the new residual type: the banked word's formal
+columns (F2 identity and defining-integer decoder) under the modified transitions,
+the block assignment, charts, incidence colouring and prime witnesses. These are
+stated one-to-one against the gauge-family precedent in
+[obligations.json](obligations.json) (R1–R4, all OPEN). Until they verify, this is
+an **accounting construction and a certified target**, not a completed finite
+witness — the same distinction PR #208 draws ("priced target, not built"), moved
+one full step forward: the target is now certified to the grid point, the family is
+enumerated, and the remaining work is a closed checklist.
+
+As every result in this repository: conditional on the retained analytic,
+uniform-recursion, fixed-tape and finite-bridge interfaces; never an unconditional
+multiplication theorem.
+
+## Reproduce
+
+```sh
+python3 -B verify.py           # read-only replay, ~1 minute
+python3 -B verify.py --write   # authoring only
+```
+
+Python 3.11+, assertions enabled. No network, no git, no scratch space.
+
+## Credits
+
+The absorbed word and its checkers: Chafik Boukhalfa (#200). Complex supplier:
+ikeboy (#193/#202 lineage). Bank mechanism: Evan McKinney (#197), realized on this
+word by rohanarun (#205); completed entrance banks: PR #186/#207 (Dugongue) and the
+BANK-PROOF-PR200.md argument. The pricing model this builds: maxime-fleury (#204,
+#208), whose rung-1 target is reproduced and sharpened here. Frame/ceiling context:
+rohanarun (#211/#213), eumemic (#210), DaysSky (#192/#201/#212). All retained
+upstream attributions of the PR168/PR184 lineage stand.
+
+Prepared with OpenAI Codex assistance. Apache-2.0.

research/residual-bank-run1/SOURCE.json

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+{
+ "files": {
+  "banks.py": "0548a1c15bd6f6dea4505f5fd2ca1bb4ecbe914c8c90769b8cd8beeef428d0cf",
+  "inputs/absorbed-occurrences.json": "1aedd8640818afe63eb7797f0332b915ed64ebeea3e9f210aaf40a06a02478a7",
+  "interval_moment.py": "9d3d1d28bfb3375958995205179b171fb65c70a3792cf9f7a21ef5ee54f5460b",
+  "obligations.json": "0268fc3628421bb930e9f56a23469a45f3b2bfae8888f6030622c2a51ce6a511",
+  "paired_cube_assembly.py": "c1ebb52163a7219504c728a5c2b098bf1e50bf20148a2879c5a74c3cadc7c03c",
+  "references/pr193-source-assisted-v4.certificate.json": "d84406d2fcc6990deba2b135509e6d86600e57123d88f9b21f7b7240e1aeb513",
+  "references/pr200-bit.certificate.json": "853afa75980e11ce16b333c28311c4f635e84065867c049fda9916c2419091ed",
+  "references/pr205-packed.certificate.json": "c11fbb9aaa99594afc8d21e33761dc6d1d6c76ce3dbc36d5b64cae872c242f5a"
+ }
+}