Verify a 12-helper co-retiming improvement on pinned PR #299

research/gen5b-coretime12/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 coordinated 12-helper/root/partner schedule,
+explicit undo of six overlapping restorations, independently authored
+changed-stage checks, and its conditional repricing. 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.

research/gen5b-coretime12/PROOF.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/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-coretime12/README.md

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+# A 12-helper co-retiming proposal on pinned PR #299
+
+This is a standalone changed-stage verifier and conditional research proposal.
+It adds no changes from PR #299 to `main`, and does not depend on PR #294.
+PR #299 is an unmerged dependency, not an already available `main` package.
+
+The proposal introduces 12 early helper restorations while undoing six old
+restorations, with coordinated root and partner deliveries. Its measured local
+paid histogram delta is `{1: +18, 2: -12}` and its residual-family delta is
+`{2: +12, 3: -18, 4: +6}`. Under the retained PR #299 admission and all-size
+interfaces, the resulting exact arithmetic gives:
+
+- Bit coarse saving: `750469712644613 / 10^18`
+- Conditional kappa: `749906929903449 / 10^18 = 0.000749906929903449`
+- PR #299 comparison: `0.000749899215319498`
+- Literal stock: `1,229,735`, down by 15
+- Unreplicated paid calls: `482,665`; rank mass: `2,455,070`; deficit: `4,400`
+
+The gain is `7714583951 / 10^18`, about 0.001029% of PR #299's kappa.
+All identified changed-stage obligations pass the checks in this packet,
+conditional on the retained source/compiler interfaces. This is not an
+unconditional multiplication bound or a full replay of the inherited finite
+admission. The earlier optimistic 32-helper price omitted
+partner-carrier work and is superseded; it is not a result of this package.
+
+## Dependency and data acquisition
+
+Every input is pinned to PR #299 head
+`5f6bd3fbc0e6796dd31263cef1f06dc968186f64`, under
+`research/five-stage-gen5b-kernels-restorations-sinks`.
+[Immutable source package](https://github.com/CrocSwap/integer-mult-bounds/tree/5f6bd3fbc0e6796dd31263cef1f06dc968186f64/research/five-stage-gen5b-kernels-restorations-sinks).
+
+`inputs.json` records each exact byte count, SHA-256 and Git blob SHA-1. Large
+word, frame and selection files are deliberately not duplicated here. The
+loader rejects missing, altered or unpinned data before parsing it.
+
+From this directory, use Python 3.11 or newer, without optimization (`-O`):
+
+```sh
+python3 -B fetch_inputs.py --destination /tmp/coretime12-inputs
+export CORETIME_SOURCE_DIR=/tmp/coretime12-inputs
+python3 -B -m unittest discover -s . -p 'test_*.py' -v
+python3 -B run_checks.py --check
+```
+
+The fetcher retrieves only the immutable public files listed in `inputs.json`
+from GitHub and verifies their bytes. It never runs a downloaded producer,
+installs a dependency or imports upstream Python. It needs network access only
+for missing data. An existing directory is verified rather than overwritten.
+
+Alternatively, set `CORETIME_SOURCE_DIR` to the package directory in an existing
+checkout containing exactly those pinned files. A normal `main` checkout is
+not sufficient while PR #299 remains unmerged. The source root can be anywhere;
+all checks work independently of the current working directory.
+
+To regenerate the compact receipts, explicit bank-address table and full chart
+factor programs elsewhere:
+
+```sh
+python3 -B run_checks.py --output-dir /tmp/coretime12-results --check
+```
+
+Add `--write-streams` to save both complete scalar suffixes as compressed JSON.
+The generated streams, full bank-address table and full matrix/factor listings
+are not committed; their deterministic reproducer and digest receipts are.
+
+## Verification coverage
+
+- Exact local integer-map and source-span checks select 12 compatible helpers
+  from the independently reproduced candidate census.
+- The complete 55,985-addition changed suffix preserves all 18,954 independent
+  entry columns forward and backward over F2 and over the integers. Sources,
+  targets and dirty entries are arbitrary, rather than initialized to zero.
+- Exact nonnegative coefficient majorant operators also agree in both
+  directions. A separate 172,944-swap commutation certificate proves signed
+  equivalence and a conservative factor-two scalar-norm bound.
+- All 60 affected chronological frame paths are checked over the rationals,
+  including 212 containments and the 12 inherited post-descent partner setups.
+- All 18 changed endpoint charts are constructed and exactly factored. They
+  use at most 142 factors, within the retained 576-chart/815-normalizer bounds.
+- Explicit role-to-bank assignment covers all 15,427 helpers and 925,620
+  role/replica occurrences per stage, with five disjoint stage namespaces.
+  A full-width bank packing witness, exact moments, all 47 outer inequalities,
+  and rejection of both adjacent coarse and kappa grid points are recomputed.
+- The changed displayed finite bill is `90386288803484701`, below `2^80`;
+  the conservative payload bound has 103 bits, below the retained `2^104` cap.
+- Mutation controls reject missing restoration, missing partner promotion,
+  missing bank capacity, incompatible frames, invalid arithmetic inputs and
+  altered source data.
+
+See [PROOF.md](PROOF.md) for the surgery and accounting argument,
+`certificates/` for exact reproducible receipts, and [NOTICE.md](NOTICE.md)
+for contributor attribution and AI-assistance disclosure.
+
+## What remains conditional
+
+The pinned unchanged prefix and its original admission are inherited. The
+unchanged finite compiler, weighted compilation, routing, prime, setup and
+all-size interfaces are not rerun here. The displayed finite-bill check does
+not assign a value to the source theorem's unknown full constant `C_full`.
+Its positive bridge inputs remain conditional on the retained finite bridge.
+The bit payload contract remains F2; the stronger integer suffix identity does
+not assert that the whole parity-filtered supplier is an integer decoder.
+
+Neither repository-wide `make verify` nor the inherited full admission suites
+were run: only newly authored verifier programs execute. Passing these checks
+does not establish a full machine-checked multiplication theorem. No production
+guard, selected main certificate or inherited manuscript patch is modified.