Conditional κ = 7.64230861320245e-4: the paired-cube bit word at cube size p = 10 (h = 20, m = 100), with a new centre-sharing complex supplier (+1.21% over #314)

.github/workflows/five-stage-p10-banks.yml

@@ -0,0 +1,39 @@
+name: Verify the p = 10 bit word in the five-stage completed banks
+
+on:
+  push:
+    paths:
+      - 'research/five-stage-p10-banks/**'
+      - '.github/workflows/five-stage-p10-banks.yml'
+  pull_request:
+    paths:
+      - 'research/five-stage-p10-banks/**'
+      - '.github/workflows/five-stage-p10-banks.yml'
+  workflow_dispatch:
+
+permissions:
+  contents: read
+
+jobs:
+  immutable-replay:
+    runs-on: ubuntu-24.04
+    timeout-minutes: 60
+    strategy:
+      fail-fast: false
+      matrix:
+        python-version: ['3.12', '3.14']
+    steps:
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1 # v7
+        with:
+          persist-credentials: false
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97 # v7
+        with:
+          python-version: ${{ matrix.python-version }}
+      - name: Install the pinned arithmetic dependency
+        run: python -m pip install --disable-pip-version-check -r research/five-stage-p10-banks/requirements.txt
+      - name: Replay every mandatory stage (virtual, raw, bit, scalar, primes, banks, complex, math, finite)
+        run: python -B research/five-stage-p10-banks/verify.py --output "$RUNNER_TEMP/p10-five-stage-verification"
+      - name: Regenerate the pinned bit word from the generator and both physical-layer producers
+        run: python -B research/five-stage-p10-banks/bitword/producer/regenerate.py --work "$RUNNER_TEMP/p10-regeneration"
+      - name: Require committed files to remain unchanged
+        run: git diff --exit-code

certificates/joint-dual-kappa.json

@@ -842,7 +842,7 @@
     "scripts/experiments/pin_joint_dual_sources.py": "4d31cc0e38b7902709ca5f8f714cba109760d2592346dd7f8dc0b900a2d9fe00",
     "scripts/experiments/verify_joint_dual.py": "7a31bba1e9c0ded32ae285df2ca9bc91ef20c330f6789fdba361876f3c7a1985"
   },
-  "source_manifest_sha256": "4f3c15b82a2d97335d4fe95fb8bcb96bd2a921087da289d9da16c68213a17f17",
+  "source_manifest_sha256": "698fe862a620e5b80b85fbc6034c9456fa0fa9aea638dbbbcb3f9edd24c0415e",
   "sources": [
     {
       "files": {

certificates/joint-dual-validation.json

@@ -226,7 +226,7 @@
     }
   },
   "bit_saving": "2382370177/50000000000000",
-  "certificate_sha256": "9801147b1502cf1e43d5fa39aa37f5291f03b48c30c4dd4f272e0877894a1c24",
+  "certificate_sha256": "6f684fa7f747073a132e8d0b5652cf12a8afa389f2a16ff6ade8e080d786dd22",
   "kappa": "4764513337/100000000000000",
   "margins": 7,
   "producer_words_regenerated_and_matched": true,

research/five-stage-p10-banks/BANK-PROOF.md

@@ -0,0 +1,153 @@
+# Five-stage stage-private completed banks
+
+This is a finite bank-allocation extension of the PR197/PR200 completed-bank construction to PR234's five-stage
+bit geometry. It is retained from eumemic's source527 package (via #285) and applied here to the independent
+helper census of the p = 10 word: h = 20, m = 5h = 100. It does not claim a new all-size compiler theorem or
+replace the scalar admission.
+
+## Endpoint of one local invocation
+
+For one complete local invocation, the source-bound physical telescope has no data/helper cross blocks. On an
+independent helper with genuine entrance projector σ, its endpoint is the exact partial swap D_(I₂₀−σ),
+supported in the active local window. This holds on arbitrary correlated dirty contents. The complemented
+inverse invocation has the same residual: its reversed endpoints are 0 and I₂₀ − σ. Source-owned gauges are
+visible-data paths and are excluded from the independent helper census.
+
+## Bank normalizers
+
+In stage i, PR234 routes the helper at local class d to physical class g = d·τᵢ. Relative to g, the residual is
+Pᵢ = τᵢ·diag(I₂₀ − σ, 0, …, 0)·τᵢ, supported only in Hᵢ. Each stage gets a disjoint collection of banks, and
+every bank is filled by disjoint coordinate blocks of total dimension 100. For the actual gauge chart B, whose
+columns are residual columns followed by gauge columns, set
+
+    N_b = (b+1)·Π_b·embed_i(B⁻¹).
+
+For an ungauged role, B = I₂₀. Π_b is the explicit coordinate permutation sending the first residual
+coordinates in Hᵢ to the assigned bank interval. Thus N_b·Pᵢ·N_b⁻¹ is exactly that coordinate projector. The
+scalar leaves conjugation unchanged.
+
+The residual columns of B are G⁻¹ applied to the frame's annihilator rows, for G = I − J/9. Up to the positive
+scalar h − 9 = 11, these are the integer rows 11a − Σa (15a − Σa at h = 24).
+
+The checker reconstructs every actual chart and checks:
+
+- both inverse identities;
+- the G-orthogonality of its two summands;
+- its residual action;
+- its elementary factorization.
+
+This is therefore a projector identity for the actual bases, not a rank substitution.
+
+## Bank routes and distinctness
+
+The bank route is g → g·N_b⁻¹, which is d·τᵢ·N_b⁻¹ in the original local invocation. If the bank class is h,
+its residual is h·N_b·Pᵢ·N_b⁻¹·h⁻¹. Every occurrence visits every bank class once, because right multiplication
+is a bijection on the complete GL cover.
+
+If two roles in one local invocation are assigned the same bank family, their distinct N_b give distinct bank
+classes. Distinctness is literal. Choose a coordinate j outside Hᵢ. The embedded inverse fixes e_j, and
+N_b·e_j = (b+1)·e_(Π_b(j)). The resulting columns differ for all b, at every eligible prime q > 2⁸⁰ and every
+local ring Z/q^w. All scalars are between 1 and 25 (the largest bank has 25 blocks), so they remain distinct
+units. The proof uses the full GL cover, not a projective quotient.
+
+## Completed endpoint
+
+Complete the entire cover sweep for each stage and replica, in fixed order, before its boundary phase. The
+complete local source endpoint separates every bank from the data and the other banks, so each subsequent
+invocation may receive arbitrary dirty bank contents.
+
+For each bank, its block projectors are disjoint and sum to I₁₀₀, so the literal partial swaps compose to
+D_I₁₀₀ on every address column. The common inherited ancestor chart is the same for each block and cancels
+between blocks. The checker verifies:
+
+- all 200 address columns;
+- the reverse schedule;
+- failures when a block is omitted or repeated.
+
+No new independence assumption on dirt is used.
+
+The five bank collections remain separate, and each is touched only in its own stage. This gives every physical
+bank its full required endpoint once. The data stage routes, bridge maps, idle children and final exchanges
+remain PR234's. Consequently there are no independent exterior rank-5a children. Their removal follows from the
+completed bank endpoint. No internal, data, compensation or copied-centre child is removed.
+
+## Families and the exact tiling
+
+The actual residual families are:
+
+| width | source | roles |
+| ---: | --- | ---: |
+| 3 | rank-17 entrances | 70 |
+| 4 | rank-16 entrances | 1,200 |
+| 20 | ungauged roles | 6,960 |
+
+A bank holds blocks of total width exactly 100. With 60 replicas the total residual width must therefore
+satisfy 60·width ≡ 0 (mod 100), i.e. width ≡ 0 (mod 5). Banks are never padded. A padded bank would not sum to
+I₁₀₀; its unused coordinates would count as stock without children and break the telescoping identity
+m·W − mass = 60·(4v − 5h(h−2)) that `bank_check.py` and `math_check.py` assert.
+
+The physical-layer producer enforces divisibility with `make_physical.py --bank-parity`. This generalizes gen5's
+parity rule (60·width ≡ 0 mod 120). It drops the fewest trailing reuse pairs that make 60·width divisible by m.
+Dropping a pair returns its rank-17 recipient to the entrances as a width-3 family, and its donor stays a plain
+helper, so each drop adds 3 to the width.
+
+For this word the maximum matching has 892 pairs and width 144,204 ≡ 4 (mod 5). Two drops are therefore minimal
+(3k ≡ 1 mod 5 gives k = 2). The dropped pairs are [3454, 9119] and [3409, 9118], and the width goes 144,204 →
+144,207 → 144,210. In the unbanked price the two drops cost about 1.5·10⁻⁸.
+
+`bank_template.tile()` then finds an exact tiling, with no padding:
+
+| Blocks | Banks per stage |
+| --- | ---: |
+| 20 of dimension 3 and 2 of dimension 20 | 210 |
+| 25 of dimension 4 | 2,880 |
+| 5 of dimension 20 | 83,436 |
+
+The tiler mixes a width that does not divide m (here 3) with a filler width that does (here 20). It backtracks
+until every remaining pure width divides out, and asserts:
+
+- every pattern sums to m;
+- every block is used exactly once.
+
+The patterns are pinned in `word-pins.json`.
+
+## Counts and selector bill
+
+There are 86,526 banks per stage, 432,630 banks in total and 230,400 data families. Literal stock is 663,030,
+and its per-replica normalized value is 132,606/12. The checker enumerates each of the 2,469,000 actual
+role/replica/stage assignments exactly once, verifies each actual role's chart rank, and hashes the complete
+allocation.
+
+Every actual chart (163 distinct entrance bases) uses at most 150 factors. At most 99 coordinate transpositions
+and 100 scalar-coordinate factors give 349 normalizer factors. A conservative added out-and-back selector bill,
+in addition to all retained PR234 route and scalar bills, is
+
+    2·5·60·((663030 − 1) + 8230·100·349) = 172,734,017,400 < 2⁴⁰.
+
+## Literal realization
+
+The literal stock, not the rational per-replica stock, must be used for physical role routing and finite
+storage. All 60 scalar replicas execute, and all recursive child counts are multiplied by 60 in a literal
+realization. Any smaller integer normalization used only in the moment arithmetic must be distinguished from
+the literal replica count.
+
+Serial execution may share an external copied-centre work stream only after erasure. The fixed bank selector
+bill is explicitly charged before the inherited positive ordinary-leaf gap absorbs it. After completion removal
+the largest child is 42 of 100, so a single level halves, and the histogram remains a one-level profile.
+
+## Scope
+
+This witness leaves the following to the integrator (verify.py):
+
+- the full scalar, integer and sign audits;
+- the frame ledgers and the exact global lowering;
+- the prime guard and the complete fallback profile;
+- the 47 outer inequalities.
+
+Its input charts and census are the independently admitted p = 10 helper. The existing compiler, common weighted
+chart, restored selectors, complete-stream routing and all-size analytic hypotheses remain conditional, as in the
+cited sources.
+
+Prepared with OpenAI Codex assistance (source527 package, eumemic). The gen5 and p = 10 numbers, the bank
+divisibility rule and the exact tiler were added with Claude assistance. Apache-2.0; inherited source
+attribution remains unchanged.