Conditional κ = 7.14026746429965·10⁻⁴ on #249's word: 890 kernel families, #263/#270 retimings, #273's target-prefix squares and 126 cleanup sandwiches (#271), composed (+0.42% over #259; above #271/#275)

.github/workflows/multicut-kernel-composition.yml

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+name: multicut-kernel-composition
+
+on:
+  pull_request:
+    paths:
+      - 'research/multicut-kernel-composition/**'
+      - '.github/workflows/multicut-kernel-composition.yml'
+  push:
+    paths:
+      - 'research/multicut-kernel-composition/**'
+      - '.github/workflows/multicut-kernel-composition.yml'
+
+jobs:
+  verify:
+    runs-on: ubuntu-latest
+    timeout-minutes: 40
+    strategy:
+      matrix:
+        python-version: ['3.11', '3.13']
+    steps:
+      - uses: actions/checkout@v4
+      - uses: actions/setup-python@v5
+        with:
+          python-version: ${{ matrix.python-version }}
+      - name: Install the native toolchain of PR 259's checkers
+        run: sudo apt-get update && sudo apt-get install -y g++ libboost-dev
+      - name: Install SymPy for the pinned upstream regeneration
+        run: python -m pip install -r research/multicut-kernel-composition/requirements.txt
+      - name: Regenerate PR 249's word, compile and run the seven checkers on the 883-pivot selection, compare receipts and price
+        run: python3 -I research/multicut-kernel-composition/verify.py --output /tmp/mkc-replay
+      - name: Require an unchanged tree
+        run: git diff --exit-code

research/multicut-kernel-composition/BANK-PROOF.md

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+# General residual-bank construction for response-kernel entrances
+
+## Sufficient construction
+
+Suppose a literal scalar word has been independently checked to restore all arbitrary helper values and has no final data/helper block. Suppose it uses the inherited nested-frame address compiler, and each newly changed helper has initial nondegenerate frame E of dimension d and final frame I24. The changed helper then has actual residual partial swap D_P with P=I-sigma_E; this is an inherited compiler conclusion, not a conclusion of rank counting. Assume the old frame admissions remain unchanged.
+
+For the present pinned PR249 family, all newly changed helpers previously belonged to the zero-entrance rank24 residual family. There are 14,287 such helpers before new changes. The unchanged nonfull families are 2,200 rank4,22 rank6,5 rank7,48 rank11,23 rank12,2 rank15 helpers. All occur in40 replicas and5 stages.
+
+Let S be the sum of dimensions of the new entrances, m the number of changed helpers. If S is divisible by3, and the required rank4 filler does not exceed84,930 blocks per stage, there is an explicit complete stage-private bank allocation with
+
+  banks_per_stage =117563-S/3,
+  literal_stock =869415-5S/3,
+  normalized_stock =173883-S/3.
+
+For each new helper of residual rank r=24-d, make ten banks containing four of its rank-r replica blocks and30-r existing rank4 blocks. Each bank has width4r+4(30-r)=120; all40 replicas are assigned. The filler consumption per changed helper is10(6+d). Keep the inherited rank11/rank4, rank7/rank4, rank6,rank12,andrank15 patterns; distribute the remaining rank4 blocks into pure30-block banks and unchanged rank24 blocks into pure5-block banks. All helper identities and intervals are explicitly enumerated. This is sufficient for arbitrary selected changed roles subject to the stated numerical conditions; it is not restricted to pair kernels.
+
+## Exact charts establish the actual projectors
+
+The checker constructs every new chart from the supplied basis U, not from its dimension. Put G=I-J/9, H=U G U^T. Compute a rational row basis V of ker(U G), and set B=[V^T | U^T]. Then B is invertible exactly when E is nondegenerate. Its first24-d columns span the actual G-orthogonal complement. The checker constructs the actual projector
+
+  sigma=U^T H^-1 U G
+
+and checks B^-1(I-sigma)B=diag(I_(24-d),0_d), both inverse products, and U G V^T=0. It emits B and B^-1, exact determinants, and elementary row-operation counts. Every elimination pivot and intermediate coefficient is checked to have numerator and denominator of magnitude below2^80. Thus every relevant nonzero divisor is a unit at each admitted prime q>2^80, including over all inherited Z/q^w. The proof does not infer an orthogonal chart from dimension alone.
+
+For each assigned block b in stage i, use the inherited literal normalizer
+
+  N_b=(b+1) Pi_b embed_i(B^-1),
+
+where Pi_b sends the first residual coordinates in the stage window to the assigned interval and matches ordered complements bijectively. The resulting conjugate of the actual projector is precisely the assigned coordinate projector. The unchanged old-chart factor ceiling548 remains valid if each checked new chart has at most548 factors; permutation/scaling retain the total787 bound.
+
+Distinct blocks in one bank have distinct route keys on a coordinate outside the current24-coordinate stage window. The embedded chart inverse fixes that coordinate. If the two permutations send it to different coordinates, the keys are distinct; if they send it to the same coordinate, the unequal scalars b+1 distinguish them. The scalars and their differences are units because their magnitudes are at most30 and q>2^80. Use the inherited full GL-cover route d*tau_i*N_b^-1; this argument does not justify replacing it by a projective quotient or removing its gates.
+
+## Completion and finite cost
+
+Complete each stage/replica cover sweep before proceeding. All assigned coordinate projectors in a bank annihilate each other and sum to I120. Their dirty partial swaps therefore compose to the full bank swap; arbitrary initial correlations are allowed. The checker independently executes all240 address basis columns in both orders for each distinct layout, and checks that each single-block omission or repetition fails on exactly twice that block's rank.
+
+The full explicit assignment has3,317,400 helper/replica/stage entries. For literal stock W the inherited conservative selector charge is
+
+  2*5*40*((W-1)+16587*120*787).
+
+The literal scalar word, its integer coefficient bill, global formal columns, and the47 strict pricing inequalities require their separate verifiers. This bank result does not delete actual scalar gates or certify an exponent by itself.
+
+## Current results
+
+Regression on the checked132 pair entrances reproduces W=869195,173839 normalized stock,and selector626938189600. The generic checker constructs all132 charts with at most30 factors and exact intermediate numerator/denominator bound4.
+
+Preflight for132 pairs and seven selected triples (dimensions4,3,9,9,9,12,14) gives S=192,139 charts,W=869095,173819 normalized stock,selector626938149600. Maximum chart cost144; every intermediate numerator and denominator has magnitude at most15. All assignments and6,240 endpoint columns plus6,240 nonzero control columns pass. This preflight installs the proposed initial frames synthetically. Re-run against the root worker's literal transformed word outputs before treating it as source-bound composition.
+
+No standalone kappa improvement or Lean certification is claimed here.
+
+## Source-bound collective result
+
+The actual collective156 outputs now pass the generic bank checker:156 actual charts, total entrance rank327, literal stock868870, normalized stock173774, selector626938059600. All3,317,400 helper/replica/stage assignments pass, as do6,720 endpoint and6,720 nonzero control columns. Every new chart uses at most156 factors and every exact intermediate numerator/denominator is at most135, preserving the inherited548/787 ceilings and2^80 prime guard. See `collective156/BANK-REVIEW.json` and `collective156/BANK-CHARTS.json`; `collective156/SOURCE-BINDING.json` binds the actual parent word and verifier inputs.
+
+## Source-bound pooled result
+
+The actual pooled word passes with252 charts and entrance rank1047. Literal stock867670; normalized stock173534; selector626937579600. Its largest chart costs200 factors; all rational intermediate numerators/denominators are at most704. All3,317,400 assignments and10,560 forward/backward endpoint columns plus10,560 nonzero controls pass. The inherited548/787 factor ceilings and2^80 prime guard are unchanged. `pooled/SOURCE-BINDING.json` pins the concrete input word and bank receipts. The original pre-admission fundamental word is superseded because it included singular frames.
+
+## Source-bound multiple-checkpoint result
+
+The actual multicut word passes with518 charts and entrance rank1344. Literal stock867175; normalized stock173435; selector626937381600. Largest chart200 factors; all exact intermediate numerators/denominators<=704. All3,317,400 assignments and10,560 endpoint columns plus10,560 nonzero controls pass. Banks depend on actual initial/final frames, so using different verified scalar checkpoint positions introduces no additional bank geometry obligation. The scalar word and global columns remain separate checks. See `multicut/SOURCE-BINDING.json`.

research/multicut-kernel-composition/COPY-PROOF.md

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+# Copy-source lifetime
+
+A copy block copies a source, reads the copy as needed, and discards the copy when it again agrees with the source. The source must stay unchanged during the block. Reversing the block then creates the copy from that same source at the reverse entry and undoes the intervening gates. A source need not be globally excluded from a basis rewrite whose operations occur outside the block.
+
+Two selected helpers, streams 10233 and 10257, are sources of copy blocks. Both are kernel pivots. A square-zero donor += pivot basis shear never writes its pivot. The old blanket audit ban on selecting a COPY source was consequently stronger than necessary. The independent arbitrary-precision chronological checker verifies that no scalar gate writes the original source while any COPY/ERASE window is open, checks both endpoint frames and all temporary lifetimes, and requires all 24 windows to close. The actual full word is additionally checked on every arbitrary input column, forward and inverse.
+
+This replaces a conservative syntactic exclusion with the precise sufficient lifetime condition. It does not remove a required check or infer correctness from matching histograms.

research/multicut-kernel-composition/FINITE-PROOF.md

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+# Finite invoice
+
+The pinned public routing, prime, recovery and analytic interfaces remain inherited. Every new chart is explicitly inverted and its residual projector identity checked; all intermediate rational coefficients and pivots are bounded by 704, below the retained prime guard. The maximum 200 factors remains below the inherited 548 chart and 787 normalizer bounds.
+
+The native finite-invoice checker rebuilds the complete histogram from the emitted word, scales through 40 replicas and five stages, and checks it against the exact pricing histogram. The literal stock is 867175 and the normalized stock is 173435. The endpoint deficit remains 176000 in literal units. All scalar gates, COPYs, routes, new selector factors and the rare-class fallback are charged.
+
+The full selector charge is 626937381600. The counted primitive coefficient is 62639910650427201 (56 bits). The cancellation-free signed prefix bound gives a 98-bit payload bound, below the inherited 104-bit guard. Positive finite cutoff witnesses and all 47 outer inequalities are checked exactly; the general cutoff formula explicitly allows inherited constants above the counted coefficient.
+
+See expected/COHORT-FINITE-INVOICE.json and code/cohort-finite-invoice.cpp. Neither this checker nor the saved expected receipts prove the retained all-size foundations. The top-level verifier executes all components and compares their fresh outputs.

research/multicut-kernel-composition/KERNEL-PROOF.md

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+# Collective response-kernel condensation: sufficient rewrite
+
+Fix a checkpoint before any chosen helper is modified, after its initial target-correction reads. Let C be the selected helpers' complete accumulated prefix response into target data, and R their full suffix response into target data, over F2. Require that target input perturbations remain confined to targets through the prefix, with an invertible target matrix A. Correctness of the old scalar word then forces the suffix target-input block to be A^-1 on targets and zero on helpers, so R=A^-1 C. The selected helper prefix columns must consist only of their own unchanged helper value plus C: in particular there are no prefix reads of these helpers into another helper. No selected helper is a copied/source-owned role. These are actual trace prerequisites, not conclusions of a response hash. Target-to-target prefix changes are allowed.
+
+Partition chosen helper coordinates into pivots P and donors D. Choose kernel vectors v_p=e_p+sum_(d in D) a_dp e_d, with Rv_p=0 and no other pivot coordinate in v_p. Put N e_p=sum_d a_dp e_d and N e_d=0. Thus N²=0 and Q=I+N is invertible. Q^-1=I-N over the integer gate presentation; over F2 signs coincide. Each nonzero a_dp is one paid ADD donor_d += a_dp*pivot_p. The gates commute because a destination is never a source pivot. Distinct pivots may use the same donor without changing this fact.
+
+Remove the old initial target-correction reads from the pivot helpers only. At the checkpoint apply Q. Run the unchanged scalar suffix, including actual copied-center lifetimes. At its final full helper frame apply Q^-1. The suffix helper block is identity, so the final helper vector is exactly the original one. Each donor's old accumulated target correction still cancels R e_d. Removing all initial pivot correction reads removes precisely C e_p at the checkpoint; each pivot now contributes zero because this correction is absent and RQe_p=Rv_p=0. Input data behavior is unchanged. This proves correctness for all scalar inputs over F2, including arbitrary dirty-helper correlations. It does not assert the same cancellation over Z; the implementation still retains actual integer ADD coefficients for address/cost analysis.
+
+For physical legality, require every forward gate's endpoints to share a nondegenerate frame E contained in every affected helper's first positive frame. Give each pivot initial frame E; the donors retain zero entrance and move to E for the paid Q gates. The original future frame chains remain legal by containment. Q^-1 occurs at the final full frame. Normalizer charts must be supplied for every actual pivot residual I-sigma_E. When all frame moves remain nested and no COPY ownership changes, telescoping dimensions give raw movement-rank saving dim(E) per pivot; histograms and ADD invoices must still be recomputed from the literal word. The exponent requires the separate exact moment and outer-assembly checks.
+
+## Eight actual Pasch components
+
+The32 saved triple relations form8 groups, each with6 helper roles and4 triples. Label three of the triples
+
+  (a,b,c), (a,d,e), (f,b,e).
+
+The fourth is their F2 sum. Choose pivots c,d,f and donors a,b,e. The six required forward gates are
+
+  a+=c; b+=c; a+=d; e+=d; b+=f; e+=f.
+
+All commute, and their integer inverse subtracts the same contributions. The three response-kernel columns have unique pivots, so all three entrances are exposed simultaneously; disjoint-support matching would expose only one.
+
+The exact C++ rank check `pasch-intersection.cpp` reads the literal first positive frames from the pinned249 event stream. For every component, it verifies that the intersection of all6 first frames has dimension equal to the union of the4 candidate frame bases: respectively4,3,5,9,9,9,12,14. Thus each component admits its full common entrance E for all three pivots. This removes a potential geometric obstruction to shared donors; it is stronger than counting independent scalar relations.
+
+Three pivots per component give total entrance rank195 rather than65. Combined with the existing132 pair pivots this is327, divisible by3, so the unchanged40-replica bank construction applies without dropping any pairs. Predicted literal stock is868870, normalized173774. These are construction formulas, not a final price receipt: the collective literal word and the combined source-bound bank checker must pass before composition.
+
+## Sufficient compatibility beyond disjoint blocks
+
+Disjoint support between selected kernel blocks is stronger than necessary. A sufficient combination rule is: every pivot is unique and belongs to no selected donor set; for each shared donor, all entrance subspaces of blocks using it form a chain under exact subspace inclusion. Each block's entrance is contained in the first positive frames of its own pivots and donors.
+
+Order the forward kernel blocks by increasing entrance dimension. On any shared donor, this visits its chain monotonically; every pivot appears in just its own block. Therefore all paid gates are physically legal and lead into the original first positive frames. The same N²=0 proof applies because no destination donor is ever a pivot source, even across blocks. Gates with the same donor commute as usual. The inverse gates take place at full frames, where their order is unrestricted. This admits compatible nested-frame donor overlaps that disjoint-support packing rejects, without requiring a new address compiler. Actual histograms and simultaneous weighted-kernel independence remain subject to literal word replay; no estimated gains are added.