Verified construction checkpoint: conditional κ = 7.110455476244e-4

research/multicut-delayed-inverse-checkpoint/BANK-PROOF.md

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+# Exact bank delta for the PR259 replacement
+
+This document explains the selected five-quartet removal, seven delayed-inverse
+interval insertion and final once-counted 25 retimings. The bundled
+[bank_check.py](bank_check.py) and [bank_template.py](bank_template.py) check the
+actual retimed source context and global ledger. The complete construction's
+evidence and inherited assumptions are stated in [PROOF.md](PROOF.md).
+
+## Literal capacity and role assignment
+
+The focused source has 16,587 independent helper roles, of which 2,827 have a
+nonzero entrance. Its residual rank counts are
+
+```
+4:2200, 6:22, 7:7, 9:4, 10:8, 11:55, 12:30, 13:14, 14:2,
+15:20, 16:1, 18:1, 19:31, 20:19, 21:3, 22:41, 23:369, 24:13760.
+```
+
+Their sum is 16,587 and their residual volume is 351,267 per replica. For each
+residual width r other than 4 and 24, use 10 times its role count banks with four
+blocks of width r and 30-r blocks of width 4. Each such bank has exactly
+4r+4(30-r)=120 coordinates. This uses 65,830 of the 88,000 available width-4
+occurrences across 40 replicas. The remaining 22,170 width-4 occurrences fill
+739 banks of 30 blocks. The 550,400 width-24 occurrences fill 110,080 banks of
+five blocks. There is no rounding, unused capacity, or fractional bank.
+
+This gives 117,089 banks per stage and 585,445 stage-private bank families across
+five stages. There are 40*4*1760=281,600 data families, for literal stock 867,045.
+The one external work family is separate from the live-stock count.
+
+Roles retain their actual original IDs. Within each residual family they are
+sorted, and occurrence q means role floor(q/40), replica q modulo 40. The segment
+mapping in bank_template.py assigns every occurrence to one contiguous bank
+block. bank_check.py independently enumerates sequential slots and compares all
+3,317,400 stage/replica/role assignments against that callable mapping. The
+bundled production census also checks the exact role set against physical ownership,
+borrowed roles, and removed roles. No source producer or ownership is changed.
+
+## Generic charts and full addresses
+
+All 770 actual entrance frame IDs are rebuilt. For an actual source basis S and
+its Euclidean annihilator row a, the residual vector is 15a-sum(a)*1, divided by
+its integer row gcd. It is orthogonal to S for the actual form 9I-J. The residual
+vectors followed by S form a 24-column chart B. The checker computes A,d with
+BA=AB=dI, checks projector action on every chart column, and replays every exact
+rational row factor to the identity. This establishes independence and exact
+conjugacy without a norm-four or rank-one special case. The checked maxima are
+576 chart factors, factor numerator 165, and denominator 1080.
+
+For a stage i and assigned block b, the normalizer is
+N=(b+1) Pi embed_i(B^-1). Pi sends the actual residual coordinate window to its
+assigned contiguous block. A column outside the stage window is fixed by the
+embedded chart; its image under N is (b+1)e_(Pi(j)). Distinct blocks have distinct
+positive scalars, so these witnesses distinguish the normalizers even when
+chart matrices repeat. All full (family, route) namespaces are injective for all
+200 stage/replica pairs, including both active data banks, all helpers, and the
+external work register. The 34,965 intentional same-family coincidences retain
+different full routes. Equality of family numbers alone is not an address test.
+
+The production callable lowering preserves every scalar, frame, projection,
+stage and reversal field and changes only actual stream operands. ADD/COPY
+operands are checked for full-address nonaliasing. Original helper class
+d*tau_i is routed to bank class d*tau_i*N^-1. The inverse-normalizer multiplication
+order and all denominators are retained. Every actual logical operand and all
+five paid stage streams are bound by the fresh checker before acceptance.
+
+## Complete banks and completion discharge
+
+Every bank pattern partitions all 120 coordinates. Its disjoint projected block
+swaps therefore give the full swap on all 240 arbitrary-dirty bank/work columns.
+The inverse word restores all columns. bank_check.py tests omission and repetition
+of every individual block in every one of the 18 patterns. The phase-major
+schedule completes all cover classes for each replica before proceeding;
+boundary phases follow all 40 replicas of their stage. Banks are stage-private.
+
+There are 2,827 independent entrance completions with width counts
+
+```
+5:369, 10:41, 15:3, 20:19, 25:31, 30:1, 40:1, 45:20,
+50:2, 55:14, 60:30, 65:55, 70:8, 75:4, 85:7, 90:22, 100:2200.
+```
+
+The checker matches the exact tagged entrance list and every completion record
+before subtracting these counts. Widths 5, 10, 15, 20 and 50 also have ordinary
+paid children. It subtracts only the tagged count and deletes a histogram bucket
+only if the remainder is zero. A whole-bucket deletion would fail the retained
+ordinary-child assertions and exact ledger comparison. The address mapper rejects
+completion opcode 6; discharge is explicit, never a silent mapper shortcut.
+
+## Retiming boundary and conservative tariff
+
+The focused source before the 25 compatible retimings has 494,670 retained
+five-stage calls per replica and rank mass 2,596,735. Retimings preserve all
+entrance gauges, charts, packing and rank mass. The final checked count is
+494,665 per replica, reconstructed by bank_check.py from the actual live ledger.
+Forty replicas have rank mass 103,869,400 and deficit
+120*867045-103869400=176000.
+
+The normalizer bound remains the conservative 838: 576+119+120=815 is below it.
+No faster route or selector tariff is claimed. The charged selector allowance is
+
+```
+2*5*40*((867045-1)+16587*120*838) = 667542305600 < 2^40.
+```
+
+## Actual retimed registry binding
+
+The final raw word is
+acf510b1377a6bd96eea50577de1aa049382499335fa7e5cac9b0300ecb377c7.
+The bundled checker reconstructs its initial frames and actual entrance bases.
+The actual 770-chart hash is
+98d3fbe41db017678b416bdffe4e79361ad08ceda99a84eed340a7dc9d1467d2.
+These chart IDs contain 459 distinct entrance bases. An exact rebuild checks
+both inverses and every rational factor for the actual registry. The maxima
+are 576 factors, numerator 165 and denominator 1080.
+
+Annihilator row sign changes do not change the represented subspace.
+Assignment and namespace separation use the actual reconstructed charts;
+the outside-column separation proof is invariant under such sign choices.
+No earlier chart presentation is an input to these checks.
+
+The live callable bank checks bind all five actual operand streams, all
+2,827 tagged completions, the charts, the assignment and the full namespaces.
+The actual global ledger confirms 494,665 retained calls and rank mass
+2,596,735 per replica after subtracting precisely those completions. Across
+forty replicas its literal histogram has 19,786,600 calls and rank mass
+103,869,400. The complete verifier also checks the current prime inventory,
+exact moments, outer inequalities and finite bill. These finite checks retain
+the conditional theorem interfaces described in PROOF.md.