Verify shared-donor kernel improvements on pinned weighted and reordered words

research/gen5b-shared-kernels/NOTICE.md

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+# Attribution and scope
+
+New material is provided under the repository's Apache-2.0 license and was
+prepared with substantial OpenAI assistance. Separate AI-assisted checks are
+not human peer review, an OpenAI endorsement or proof-assistant verification.
+
+This proposal builds on Chafik Boukhalfa's PR #299 package, prepared with
+substantial Anthropic Claude assistance, pinned at
+`5f6bd3fbc0e6796dd31263cef1f06dc968186f64`. It retains that package's source
+lineage, licensing and assistance disclosures. Its
+[full immutable notice](https://github.com/CrocSwap/integer-mult-bounds/blob/5f6bd3fbc0e6796dd31263cef1f06dc968186f64/research/five-stage-gen5b-kernels-restorations-sinks/NOTICE.md)
+remains applicable to the input data and inherited mechanisms.
+
+In particular, the construction inherits DreamingOfClouds' gen5/stagealt word,
+Rohan Arun's descent and target stages, eumemic's scalar/five-stage/kernel
+pipeline, utcorvusvolat-dotcom's early restoration mechanism, Dugongue's
+endpoint-aware banking and terminal-sink work, Evan McKinney's completed-bank
+method, and Chafik Boukhalfa's complex supplier. The full notice credits
+Henry Grant, Jacob Sussman, ikeboy, icekylinx, James Chang, sennemmi and the
+earlier contributors and dependencies.
+
+The new contribution is the shared-donor 70- and 72-pivot construction, its
+composition with the coordinated 12-helper/root/partner schedule, explicit
+role-to-bank allocation, independently authored changed-stage checks and
+conditional repricing. The co-retiming component is the prior PR #301
+contribution, retained here for standalone reproduction. It makes no exclusive
+priority claim for inherited mechanisms or a global optimum claim. PR #294's
+separate assembly-domain proposal is not modified or required.
+
+Upstream input files are fetched as immutable data. Their original notices
+and source-specific licensing remain unchanged. No upstream executable is
+imported or run by this package.
+
+The PR #300 second constructed-frame descent is Rohan Arun's contribution,
+pinned at `fd516176fd068e4fa7af14b40cd878ce8c305296`; its source and attribution
+are retained by reference to the immutable
+[full package](https://github.com/rohanarun/integer-mult-bounds/tree/fd516176fd068e4fa7af14b40cd878ce8c305296/research/five-stage-gen5b-full).
+The new composition checks do not claim authorship of that descent or reprove
+its unchanged physical/source-span admission. PR #301 itself is unchanged.

research/gen5b-shared-kernels/PROOF-CORETIME12.md

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+# Co-retiming twelve helpers with their partner deliveries
+
+All claims below concern the pinned PR #299 input and retain its unchanged
+stage and compiler interfaces. All ranks and containments are over Q unless
+the field is explicitly stated. This is a differential construction proof,
+not a new proof of the inherited all-size foundations.
+
+## 1. Selection and the obstruction to an unchanged-frame restoration
+
+The root-cut screen identifies 34 rank-21 entrance helpers `a`. Each has one
+forward write `a += b`. Their common write frame `J` has dimension 22. Their
+last root deliveries occur at different dimension-23 hyperplanes `A` and `B`,
+whose intersection is `J` and whose join is full dimension 24.
+
+Thus merely moving `a -= b` earlier, while preserving both delivery frames,
+does not improve its endpoint. The local candidate screen instead identifies
+a preceding rank-22 root delivery by a blocker `c` to the relevant target `t`.
+Two conflicting candidates are excluded. Among the remaining 32, keep exactly
+the candidates whose old-restoration pair is hit twice. This gives 12 new
+helpers and six old restorations to undo; all selected identities are derived
+from pinned data and recorded in `certificates/coretime12/local.json`.
+
+## 2. The changed schedule
+
+At each selected blocker root:
+
+1. Move `b`'s later delivery to `t` earlier, at `J`.
+2. Deliver `c` to its unaffected target at its old frame `K`.
+3. Perform the partner setup and its unaffected delivery at the inherited
+   post-descent rank-22 frame.
+4. Advance the affected target, blocker and partner carrier as required to
+   `B`, and perform their remaining deliveries there.
+5. After `a`'s original final delivery at `A`, perform `a -= b` at `A`,
+   omitting its original cleanup write.
+
+The selected preceding prefix and gauge frames lie in `J`. For each moved
+donor, the union of all uncancelled source dependencies has 180 leaves and
+lies in `J`; this check is stronger than a mod-2 cancellation test. The local
+checker verifies the integer root map on independent helper/target values.
+
+The complete suffix checker includes all unchanged side roots, partner
+deliveries, inverse cleanup, uninject operations and kernel restorations.
+It constructs both 55,985-event streams and compares all 18,954 columns over
+F2 and Z in both directions. Its entry states are arbitrary, so the equality
+composes with the same unchanged prefix. Missing restoration is a negative
+control. The inherited prefix's own correctness is not re-proved.
+
+The two overlapping kernel donors are never pivots; their pinned setup bases
+lie in `J`. Kernel setup precedes first ordinary use and inverse kernel writes
+follow cleanup under the retained stage contract. Seven inherited sinks and
+all other suffix operations remain in the comparison.
+
+## 3. Correct chronological geometry
+
+The earlier root-only proposal missed a partner delivery at the old `K` frame.
+Promoting the target to `B` and then delivering again at `K` is non-monotone.
+The schedule above explicitly co-retimes that partner read and charges its
+carrier's split climb.
+
+The frame checker starts affected helpers at their actual base-forward final
+frames. No affected helper participates in a descent change. Each of the 12
+selected partner setups does participate: its pinned descent record replaces
+a dimension-2 setup by a dimension-22 setup. The checker verifies the exact
+new basis and uses it in both schedules. It checks every affected helper,
+carrier and target path, including the six reverted endpoints, and rejects
+omitting the partner promotion or old-restoration undo.
+
+Targets use a difference census from ZERO. Their unchanged earlier prefixes
+end within the verified first precursor frame under the retained source
+admission; the local source/gauge/target-prefix checks explicitly establish
+the required new cut containments. No full untouched-prefix frame replay is
+claimed.
+
+## 4. Paid profile and endpoint demands
+
+For the new helper/blocker operations alone, one surgery changes the local
+histogram by `{1:+1, 2:-1}`. Undoing one inherited early restoration reverses
+its delta `{1:+3, 2:-2}`. Co-retiming each partner carrier splits one rank-2
+climb into two rank-1 climbs, adding `{1:+2, 2:-1}`. Therefore
+
+```
+12*{1:+1, 2:-1} - 6*{1:+3, 2:-2} + 12*{1:+2, 2:-1}
+    = {1:+18, 2:-12}.
+```
+
+The actual chronological path census independently obtains this delta; it is
+not used as an unchecked input to pricing. It adds six calls and removes six
+units of rank mass locally. Five stages therefore add 30 calls and remove 30
+units of rank mass. The reflected paid ledger has the same rank increments
+because annihilator inclusion reverses each nested frame pair.
+
+Each new helper changes residual rank 3 to 2. Each of the six reverted old
+helpers changes residual rank 3 to 4. Thus the family-count delta is
+`{2:+12, 3:-18, 4:+6}`, of total rank mass -6.
+
+For each stage's 60 replicas, replace 27 full rank-3 banks by 12 full rank-2
+banks and 12 full rank-4 banks. Each bank has width 120. Demand is exact and
+the number of banks drops by three per stage, or 15 in the five-stage literal
+stock. The bank checker reconstructs every role's entrance and endpoint, removes the
+seven sinks, and assigns all 925,620 role/replica occurrences in a stage to
+explicit bank and block addresses without collisions or omissions. Five
+disjoint stage namespaces give 4,628,100 assignments. Every bank is full,
+attaining the exact volume lower bound. The
+unreplicated stock drops by `15/60 = 1/4`; this convention differs from
+normalizing the literal stock by five.
+
+With `W = 1229735/60`, the independently repriced histogram has 482,665 calls,
+rank mass 2,455,070 and unchanged deficit `120 W - 2455070 = 4400`.
+
+## 5. Changed chart and scalar bounds
+
+Using the retained weighted form `G = I - J/9`, construct each new rank-2
+residual projector as the endpoint projector minus its rank-21 entrance
+projector. Its complementary completion has rank 22. An exact orthogonal
+block basis uses residual, entrance and exterior columns. The six reverted
+rank-4 charts are reconstructed from their original rank-20 entrances.
+
+Every changed 24-by-24 chart is invertible. A separately replayed exact
+factor program recovers it from the identity. The largest uses 142 factors;
+factor numerators are at most 11 and denominators at most 9. These fit the
+retained chart-factor bound 576 and normalizer bound 815. New pure rank-2
+banks require at most 60 block scalars, within width 120; all are units under
+the retained characteristic `q > 2^80` hypothesis.
+
+The before/after nonnegative coefficient majorant maps agree exactly in both
+directions. A second certificate transports the old scalar norms through
+172,944 commuting adjacent swaps. Only twelve destination update orders
+change; each differs by moving one unit update. At intermediate cuts the new
+row is an old row plus or minus one old source row, yielding a conservative
+factor-two bound in each direction. Its resulting 103-bit payload remains
+below the inherited 104-bit cap.
+
+The changed displayed finite coefficient is 90,386,288,803,484,701, or 57 bits,
+below the retained 80-bit coefficient cap. All three bootstrap gaps and the
+displayed cutoffs remain positive. These are checks at the displayed fixed
+bill, not a numerical assignment to the unknown inherited full constant.
+
+## 6. Exact conditional rate
+
+The independently authored rational enclosure engine certifies the coarse
+root between adjacent points `750469712644613/10^18` and
+`750469712644614/10^18`, accepting the lower point and rejecting the higher.
+Three retained ordinary bootstrap steps and the unchanged complex saving
+produce `kappa = 749906929903449/10^18`. All 47 displayed outer constraints
+are strict; increasing kappa by `10^-18` fails `g3_above_kappa`.
+
+This establishes the stated conditional arithmetic consequence of the changed
+profile. It supplies no unconditional bound, replacement of the inherited
+finite admission, or full proof-assistant verification of multiplication.
+
+Prepared with substantial OpenAI assistance. See [NOTICE.md](NOTICE.md).

research/gen5b-shared-kernels/PROOF-KERNEL72.md

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+# Shared-donor kernel validation (separate from frozen final12)
+
+## Verified candidate
+
+Pinned source: PR299 commit `5f6bd3fbc0e6796dd31263cef1f06dc968186f64`.
+The larger candidate uses the line groups `(6,7)`, `(16,17)`, `(2,3)`, dropping
+only pivot relations 10859 and 10861 from the first group. It has 72 pivots,
+93 distinct donors,165 helper roles and 473 setup/restore shear pairs.
+All three individual groups and both proposed disjoint unions were checked.
+No source program was imported or executed.
+
+Every role is plain, not reused, not removed by the sink stage, not a member
+of the frozen kernel stage, not an endpoint in the 904 descent changes, and
+not touched by final12 and not an old restored endpoint. Any restoration-donor
+uses are included in the chronological audit (none occur for these witnesses). All new helper endpoints
+remain FULL. Each group uses the common entrance line E=span(e_i-e_j).
+Its Gram entry for 9 I-J is 18, so it is nondegenerate over Q and every retained
+prime q > 2^80. Exact rational containment checks place E inside every member's
+first required frame and check each subsequent old chronological frame step.
+The 72-pivot union uses 663 distinct old frame containments.
+
+## Scalar identity on arbitrary independent dirty inputs
+
+This is a compositional proof at the bit F2 interface, not a claim that the
+entire parity-filtered decoder is an integer identity. Let Q be the actual
+prefix through all initial plain dirty compensation reads, but before source
+injections or any new member's other use. Let S be the old remaining word.
+The inherited complete decoder is U=SQ: target += source, all source/dirty
+coordinates fixed.
+
+Let K=sum e_d e_p^T over every new donor/pivot incidence and A=I+K.
+Pivot and donor sets are disjoint, hence K²=0, A^-1=A in F2, and all individual
+setup shears commute, even with a donor shared by several pivots.
+All new members remain unchanged in Q except for their target compensation
+columns. Write R for that response matrix. The independently reconstructed
+integer adjoint, reduced modulo 2, verifies R_p=sum_d R_d for each pivot.
+If Q' is Q with precisely the pivot compensation reads omitted, this gives
+A Q'=Q A. Therefore the proposed complete word is
+
+    A^-1 S A Q' = A^-1 S Q A = A^-1 U A = U.
+
+The final equality holds because U touches only source-to-target entries,
+whereas A touches only dirty helper entries. This proves every source,
+target and dirty column, in both directions, on arbitrary entry values.
+The executable checker compares the corresponding full 18,954-column formal
+matrices. Its old remainder is the inherited decoder composed with Q^-1;
+it does not pretend to be a new replay of every old physical record.
+Omitting a setup or a restoration, or corrupting one response relation,
+is rejected by regression controls.
+
+## Literal common prefix
+
+The independent signed-adjoint reconstruction agrees with all 1,720 existing
+kernel cut-read triples. The plain odd-read block has 648,088 ADDs; its last
+read is `[3503,16719,-1]` at base record 648087. The target-prefix setup cut is
+696719, later than this block. Existing kernel setups occur inside this block
+and use disjoint helper support. New setups can therefore be inserted at the
+structural boundary immediately before the first source injection. Removing
+new pivot reads does not require shifting any inherited indexed transform:
+apply the new stage after the inherited transforms, using that structural
+boundary. All target operations already before this boundary merely read
+unmodified helper columns or use disjoint frozen kernel helpers.
+
+The source witness is `kernel-witness.json` in this package; `check_shared_kernel.py` independently derives the
+same response and physical-role map. The common cut is semantic (before first
+source injection), rather than reusing an invalid post-transformation record
+number.
+
+## Paid moves, including donor sharing
+
+Each new pivot starts at E instead of ZERO, so its first old rank-d MOVE is
+replaced by rank d-1. Each distinct donor first moves ZERO→E once, performs all
+its same-E setup shears, then moves E→its old first frame, rank d-1. A donor
+used 21 times still pays that rank 1 entrance exactly once; its 21 scalar ADDs
+are all retained and counted. At the end every member is at FULL, so all 473
+inverse shears are legal at FULL and require no additional MOVE.
+The new setup matrices commute, so either exact reverse ordering or the
+source's family ordering is valid; the emitted specification uses reversal.
+
+Measured local delta:
+
+    rank 1:+93, rank 2:+144, rank 3:-144, rank 4:+19, rank 5:-17, rank 6:-2.
+
+Its paid rank mass drops by 72. Residual family demand changes by
+`rank 24:-72, rank 23:+72`. Bank assignment and exact pricing are separate
+certificates; the path checker does not infer them from volume alone.
+
+## Conservative scalar prefix bound
+
+`check_scalar_bounds.py` independently replays all source/helper updates in
+both directions, including the final12 suffix and all frozen kernel shears.
+All 14 terminal-sink writes are redirected to their actual target pivots;
+removed sink reads, deliveries and cleanup updates are absent. At each extra
+shear it uses the actual previously propagated
+nonnegative row weight, not a unit-cost scalar approximation.
+
+For an upper bound, it restores every omitted target-compression read and
+every newly removed pivot compensation read. It includes the two target-prefix
+batches AND the seven sink setup/restore batches. Their combined directed
+source-to-destination target graph has 681 edges and 908 vertices and is checked
+acyclic. Each edge has weight equal to its total number of ADD occurrences,
+two per setup/restore pair. Applying those weighted edges in topological order
+to all accumulated nonnegative target increments dominates any chronological
+placement of those ADDs: it counts every directed path with the product of
+its edge multiplicities, including mixed prefix-to-sink paths. This proof works
+in both execution directions. Target values never feed sources or helpers.
+The monotonicity of every cancellation-free ADD majorant also bounds all
+intermediate values, not only the final vectors.
+
+Old and new uncompressed majorants are compared, with only the new setup and
+restoration shears inserted. Their difference is nonnegative. The maximum
+added forward majorant is 80662; the inverse one is 1334317. Adding these to
+the inherited bounds 77985 and 2475558 yields
+
+    F_new <=158647, B_new <=3809875.
+    64 F_new^3 B_new^2 =3709343221562739318443143000000 <2^104.
+
+This closes the retained payload-size cap conditional on the inherited old
+majorant bounds. The checker is conservative about compressed target paths;
+it does not claim to reproduce a new full physical record hash.
+
+## Role-resolved integration with final12
+
+The unified bank checker reconstructs the original 15,427 helper roles, applies
+all old kernel entrances, 434 retained early restorations, twelve new final12
+endpoints, and the 72 additional rank-one kernel entrances. The seven sinks
+remain removed. New kernel membership is checked disjoint from the affected
+final12 helpers and the inherited kernel and descent selections.
+
+The residual rank changes relative to final12 are rank 24:-72 and rank 23:+72.
+The combined inventory has 1,646 rank-23 residuals and 11,401 rank-24 residuals;
+its total residual rank is 322,862 per replica. The explicit width-120 pattern
+`(23,23,23,23,24,4)` absorbs each group of four new rank-23 residuals. The
+remaining pure patterns are adjusted with exact integer counts.
+
+All 925,620 role/replica occurrences in a stage are assigned without collision
+or omission to explicit bank and block addresses. Five stages use disjoint
+bank namespaces. All 161,431 banks per stage are full, attaining the volume
+lower bound. Total bank stock is 807,155; with 422,400 data families, literal
+stock is 1,229,555. The compact address table is reproducibly generated rather
+than duplicated in the repository.
+
+The independently reproduced moment/assembly certificate uses the measured
+kernel histogram and final12 chronological delta, while the unified finite
+bill charges all 473 setup and 473 restoration shears. Removed pivot reads
+are not subtracted from that conservative scalar bill. The corrected
+sink-aware payload bound is supplied separately by `check_scalar_bounds.py`.
+The line charts, first-frame splits, changed final12 charts and retained
+normalizer allowance are checked by the chart modules.
+
+## Evidence boundary
+
+The changed prefix is proved compositionally over F2 using the inherited
+complete decoder as a hypothesis. The checker is not a newly executed
+complete upstream physical program. All unchanged PR299 and final12 compiler,
+routing, prime, reconstruction, complex-supplier and analytic contracts remain
+inherited. No unconditional multiplication theorem, exhaustive priority claim,
+or optimality claim is asserted. The separate PR300 case, when selected, uses
+its explicit additional data pin and verified non-overlap certificate.