Conditional κ = 0.001600682403612293: joint-prefix kernels and structural bounds

research/larger-gain/CHECKS.md

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+# Recorded validation
+
+The frozen width-seven result is `kappa = 0.001600067494703216`.
+Its conservative counterpart is `0.001600067491919801`.
+
+The final portable reproduction passed all 71 fresh provider stages and all
+seven global stages. It reused the complete, hash-bound PR564 baseline
+reproduction, whose 36 stages had already passed. It rebuilt the new words,
+native programs, wrappers and exact phases before global admission. See
+[the replay receipt](global/replay-evidence/REPLAY.json) and its artifact/log
+hashes. This is distinct from merely reading the retained result.
+
+The global construction passed the full physical chronology and invoice,
+sixteen invalid-owner controls, 89 invalid-assembly controls, two accepted
+phase timing changes and two rejected phase semantic changes, both exact
+moment engines, and all 47 outer inequalities under each arithmetic model.
+
+Capacity duals passed standard-library exact replay and eight corruption
+controls. Expanded stage/bank bounds passed both rational engines. After
+publication was moved onto main, the capacity verifier and the nine-stage
+bound were rerun with the external source path to confirm that the PR564
+package need not be republished in this change. Supplementary model scripts
+accept `GAIN_PR564_SOURCE=/path/to/pinned-PR564/research/complex-allcold`.
+
+The mandated aggregate checks were attempted in a separate unchanged PR564
+baseline checkout. The initial `make verify` passed its earlier targets but
+stopped when the selected bank verifier needed SymPy. Installing the pinned
+`sympy==1.14.0` in a temporary virtual environment resolved that prerequisite.
+`make gen4-bank-verify` then passed all nine fresh stages and omitted-stage
+controls in 250.4 seconds.
+
+Continuing the remaining aggregate targets stopped in `frontier531-verify`:
+the inherited C++ bit checker could not find
+`boost/multiprecision/cpp_int.hpp`. Consequently the repository-wide
+`make verify` is **not a pass**, and targets after that failure did not run.
+No unrelated scientific source was changed to bypass it. This environment
+limitation does not replace or invalidate the separately recorded, complete
+width-seven reproduction.
+
+The additional pre-PR review was canceled at the user's request. Earlier
+independent mathematical and adapter checks are retained as historical
+evidence; no new review verdict is implied by publication.

research/larger-gain/NOTICE

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+Larger-gain research, submitted by Dominik Scholz (DominikScholz), with
+substantial OpenAI Codex (GPT-6 Astra) assistance in mathematical exploration,
+implementation, verification and documentation, October 2026. New material is provided under the
+repository's Apache-2.0 license.
+
+This directory builds on PR564 at c18417b02b80594b2ca1c8c21185d28befafc935
+and its inherited complex-allcold package. Its upstream construction,
+provider, compiler, exact arithmetic, scheduling and bit-supplier notices
+remain applicable. The repository NOTICE and the notices in the unchanged
+research/complex-allcold, research/direct-delivery504-global and referenced
+vendor packages are retained and incorporated by reference.
+
+The current work contributes extra checked suffix kernels, coupled entrance
+frame promotions, whole-owner repacking, exact dual exclusions for finite
+capacity and stage models, bounded geometry/architecture screens and
+hypothetical cost-sensitivity targets. Source copies preserve their original
+headers; adaptation manifests distinguish unchanged and modified files.
+The stage nomination screen derives from Rohan Arun's retained complex501
+stage search. Provider frame and word routines retain the credited work of
+DreamingOfClouds, icekylinx and their cited contributors, including the
+earlier scalar, phase, frame, gauge, kernel and dirty-restoration ancestry.
+
+The immediate dependency is eumemic's PR564, at
+c18417b02b80594b2ca1c8c21185d28befafc935. Named inherited contributions:
+
+- eumemic: PR564 and PR561/562/558 supply the cold-kernel words, zero/four/eight
+  provider family and admission machinery; PR550 supplies the hardened
+  assembly ancestry. PR543 supplies the cited unchanged bit circuit, whose
+  profile is repriced here without a fresh bit-circuit reconstruction.
+- Rohan Arun / rohanarun: PR559 supplies the native exact-complex moment
+  argument; the retained complex501 follow-up supplies the stage-search base.
+- Dugongue: PR552 supplies the collective-wave and weighted-response
+  construction, PR505 the asynchronous whole-owner scheduling ancestry, and
+  PR563 the earlier capacity-pattern LP used as a research starting point.
+- Chafik Boukhalfa / chafreaky: PR551 supplies the full phase replay driver;
+  the earlier geometry, physical audit and invoice lineage remains credited.
+- icekylinx: PR531 supplies the 504-helper direct-delivery factory, frame
+  coalescing and compiler/controller ancestry; PR535 supplies post-center
+  rank-four kernel work.
+- DreamingOfClouds: PR484 and its successors supply shared-donor kernels,
+  entrance gauges, normalized wrappers and the credited word/frame machinery.
+- DaysSky: PR442 and PR381 supply inherited idle-axis hosting and bridged
+  column-checking work.
+- Zhihao Chen / jacklightChen: PR425 and PR386 supply inherited mutable-source
+  interface and geometry work. All earlier contributors remain credited in
+  the original notices and repository contributor record.
+
+These are inherited or cited contributions, not independently additive
+numerical gains. Original authorship and AI-assistance disclosures remain
+applicable, including DreamingOfClouds' Claude Opus 5.5 / Claude Code work,
+icekylinx's GPT-6 Astra / Codex work, and each source's own disclosure.
+
+No priority claim is made for inherited constructions, the capacity-pattern
+LP used in PR563, or graph minrank/product methods. Exact finite checks and
+model-specific dual bounds are not an unconditional multiplication theorem,
+a full Lean proof, or a worst-case constant-time indexing result.

research/larger-gain/README.md

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+# Larger-gain research after PR564
+
+The new direction is to create residual helpers of width seven, which can
+share a rank-sixteen owner with a width-nine helper. The current width-eight
+helpers cannot make that exact pair. Thirty-two coordinated entrance-frame
+promotions per selected call create this option while preserving the scalar
+program. This changes the available packing patterns, rather than extending
+the already saturated width-five-to-four promotion family.
+
+The frozen literal nomination uses 63 width-seven providers and nine ordinary
+four-cold providers. Its complete profile has native rate
+`b = 0.001602631813774414678548` and nominal grid value
+`kappa = 0.001600067494703216`, about 0.4352% above PR564. Full global admission
+passes: physical chronology and all sixteen owner mutations, the complete
+finite invoice, both exact moment engines, all 47 outer constraints for each
+model, and rejection of all 89 invalid assembly variants. The conservative
+model also passes at `kappa = 0.001600067491919801`. See the
+[frozen result and artifact bindings](global/width7/RESULT.json).
+
+The separately completed four-cold fallback is already fully admitted at
+`kappa = 0.001593675235313849`, a 0.03396% gain. See its
+[result and artifact bindings](global/fourcold/RESULT.json) and the
+[independent adapter review](geometry/FOUR-COLD-SEAM-REVIEW.md).
+
+## Latest state and scope
+
+Research ran in an isolated checkout based on PR564. Publication is based on
+main and includes only this package; PR564 is an explicit external dependency. The
+repository has an `origin` remote; its main branch and PRs through 566 were
+fetched. The [state record](STATE.json) pins every inspected head:
+
+- `origin/main`: `db0e57bc7f36b5ac0fc844ad7ce1396165ee6647`.
+- [PR564](https://github.com/CrocSwap/integer-mult-bounds/pull/564):
+  `c18417b02b80594b2ca1c8c21185d28befafc935`, numerical baseline
+  `kappa = 0.001593134158181640`.
+- [PR565](https://github.com/CrocSwap/integer-mult-bounds/pull/565) and
+  [PR566](https://github.com/CrocSwap/integer-mult-bounds/pull/566) add formal
+  compact counting and input preparation. PR566 reports a concrete six-tape
+  composition with bound `22n+17`. These are not a higher kappa or a general
+  worst-case constant-time indexing theorem.
+
+The complete unchanged PR564 verifier was freshly run: all 36 stages passed,
+including its providers, phases, physical owners, finite invoice, two moment
+engines and outer assembly. Source hashes, receipts and the stage log are
+retained in [baseline564](baseline564/BASELINE.json).
+
+Every new numerical result here has the same conditional finite-construction
+boundary as the inherited package. All-size compilation, common labels and
+mutable bases, normalized residual realization, correlated whole-call
+hosting, precision/tape and analytic transfer remain assumptions. No
+unconditional multiplication theorem, complete Lean proof, or general O(1)
+indexing theorem is claimed.
+
+## What changed locally
+
+Three additional exact suffix relations reduce three width-nine helpers to
+width eight in each zero-, four- and eight-promotion provider. Their combined
+paid change is `3 H1 + 6 H2 - 6 H3`. The relations were reconstructed for each
+provider: copying the same insertion cuts blindly was invalid. The actual
+words, all-column scalar checks, dirty-helper restoration, omitted-kernel
+controls and phase recipes are retained under [providers](providers/README.md).
+
+Next, 32 entrance-frame promotions create width-seven helpers. Their combined
+paid change is `32 H1 + 160 H2 - 128 H3`. The fully checked eight-promotion
+provider therefore has local profile
+
+```text
+H = {1:2352, 2:592, 3:328, 4:192, 5:56, 6:56, 8:8}
+helper widths = {4:8, 5:48, 7:32, 8:128, 9:288}
+```
+
+All three versions passed 728 columns in both orientations, exact restoration,
+native frames, eleven local negative controls, 3005 phase edges and 224
+endpoint adapters. Their source-bound bundles are in
+[providers/width7](providers/width7/).
+
+The packing search must respect complete helper ownership. A width-count
+solution initially failed literal expansion because it demanded incompatible
+shared owners from the same invocation. Additional subset constraints excluded
+that solution. The retained nominee checks actual invocation windows, whole
+owners, distinct roles and disjoint prefixes.
+
+## Exact bounds that keep the search focused
+
+The strongest negative result is an exact dual certificate for the current
+PR564 providers. Within its present capacities and live stock, no repacking,
+provider remix or finite cloning can reach the next `10^-18` kappa grid point.
+The proof enumerates every 4/5/8/9 pattern in capacities 8/16/32/48, relaxes
+integrality and chronology, and checks the resulting 548 dual inequalities
+with two independent rational engines. See
+[the proof and scope](patterns/PROOF.md).
+
+Additional width-five-to-four promotions are affine continuations of the
+existing cost/inventory family. Forty more pass local checks but do not
+improve the four-cold capacity ceiling. Width-seven promotions change that
+conclusion by introducing new bin patterns. The exact upper bound for the
+32-promotion family is approximately `0.001600564129863655`. Combining it
+with forty more width-four promotions raises the relaxed upper bound only to
+`0.001600616266460206`. Neither is an admitted result: these deliberately
+ignore physical conflicts. All sixty available width-seven promotions also
+passed local checks, but their additional cost yields no further relaxed gain.
+
+The larger alphabet was also tested at five, seven, nine and eleven stages,
+allowing arbitrary fractional provider mixtures, all mixed-width auxiliary
+banks, and no chronology conflicts. Exact upper bounds are approximately:
+
+| Stages | Dimension | Kappa upper bound |
+| --- | ---: | ---: |
+| 5 | 41 | 0.001395406 |
+| 7 | 57 | 0.001577285 |
+| 9 | 73 | 0.001600617 |
+| 11 | 89 | 0.001554605 |
+
+These are rounded upward for display. The nine-stage optimum uses no banks.
+The other stage counts remain below PR564 even in this enlarged model.
+The [stage proof, exact receipts and reproduction commands](geometry/README.md)
+describe the padding and provider assumptions; these are not universal bounds
+over every possible local construction.
+
+## Where a materially larger gain could come from
+
+The next useful target is the local packet factory, especially expensive
+helper paths. An exact sensitivity experiment removes each candidate's actual
+paid frame path and residual demand, with all other costs retained. Under the
+optimistic assumption that a new word replaces that work for free, saving six
+selected helper paths per call crosses a 1% target; saving 32 crosses 5%.
+This identifies the cost budget a new construction must beat. It does not
+show that the existing helpers are removable. See the measured paths and
+adjacent exact threshold checks in [HELPER-TARGETS.md](geometry/HELPER-TARGETS.md).
+
+The 5% target would also outgrow the currently composed PR543 bit supplier.
+[PR555](https://github.com/CrocSwap/integer-mult-bounds/pull/555) already reports
+bit rate `0.001733052823330162099157`, enough headroom in principle; its
+source-bound replay and new composition were not performed here.
+
+Several structural shortcuts were checked and failed within explicit scopes:
+lossless binary dimension reduction, hyperplane puncturing, seven-center
+decoding with too few deletions, retaining 120 ports at the old center bill,
+literal tensor lifts, direct clone-free ordinary packet reads, and a full
+tetrahedron Walsh output basis with unchanged packets. The geometry and
+provider notes retain exact receipts and explain what each test leaves open.
+In particular, a joint packet refactorization, a sparse mixed output basis,
+or joint reselection of entrance kernel/gauge bases remains a substantive
+direction beyond the affine local promotions tried here.
+
+## Attribution and new contribution
+
+This contribution is submitted by [Dominik Scholz](https://github.com/DominikScholz),
+with substantial OpenAI Codex (GPT-6 Astra) assistance in mathematical
+exploration, implementation, verification and documentation. The new work is
+the additional checked suffix relations, coupled width-seven entrance
+promotions, their literal whole-owner assignment, exact model exclusions and
+portable replay. It builds on the following contributions.
+
+| Contributor | Retained or cited contribution |
+| --- | --- |
+| [eumemic](https://github.com/eumemic) | [PR564](https://github.com/CrocSwap/integer-mult-bounds/pull/564) and PR561/562/558: cold-kernel and zero/four/eight providers and admission; PR550: hardened assembly ancestry; [PR543](https://github.com/CrocSwap/integer-mult-bounds/pull/543): cited unchanged bit supplier, not freshly rebuilt here. |
+| [Rohan Arun / rohanarun](https://github.com/rohanarun) | [PR559](https://github.com/CrocSwap/integer-mult-bounds/pull/559): native exact-complex moment argument; retained complex501 follow-up: stage-search base. |
+| [Dugongue](https://github.com/Dugongue) | [PR552](https://github.com/CrocSwap/integer-mult-bounds/pull/552): collective waves and weighted response; PR505: asynchronous whole-owner ancestry; PR563: earlier capacity-pattern LP. |
+| [Chafik Boukhalfa / chafreaky](https://github.com/chafreaky) | [PR551](https://github.com/CrocSwap/integer-mult-bounds/pull/551): full phase replay driver and earlier geometry/audit/invoice lineage. |
+| [icekylinx](https://github.com/icekylinx) | [PR531](https://github.com/CrocSwap/integer-mult-bounds/pull/531): 504-helper direct delivery, frame coalescing and compiler/controller ancestry; PR535: post-center rank-four kernels. |
+| [DreamingOfClouds](https://github.com/DreamingOfClouds) | [PR484](https://github.com/CrocSwap/integer-mult-bounds/pull/484) and successors: shared-donor kernels, gauges, normalized wrappers and word/frame machinery. |
+| [DaysSky](https://github.com/DaysSky) and [Zhihao Chen / jacklightChen](https://github.com/jacklightChen) | Inherited PR442/381 hosting and bridged-column checks, and PR425/386 mutable-source interface and geometry work. |
+
+All earlier contributors and source-specific AI disclosures remain credited
+in [NOTICE](NOTICE), the repository contributor record and the unchanged
+external dependency. These contributions are not independently additive
+numerical gains.
+
+## Verification and attribution
+
+The full provider and global reproduction uses an explicit immutable PR564
+source package. The output directory must be new:
+
+```sh
+python3 -B research/larger-gain/global/reproduce.py \
+  --source /path/to/pinned-PR564/research/complex-allcold \
+  --work /tmp/width7-reproduction
+```
+
+The exact capacity exclusions can be replayed without SciPy. Set the same
+source package for the supplementary model screens:
+
+```sh
+GAIN_PR564_SOURCE=/path/to/pinned-PR564/research/complex-allcold \
+  python3 -B research/larger-gain/patterns/verify_ceiling_controls.py
+```
+
+This checks every retained capacity dual and rejects deliberately corrupted
+source pins, potentials, dimensions, rates and signs. The stage receipts have
+their separate standard-library verifier. SciPy nominates solutions only;
+exact rational checks establish the stated exclusions.
+
+The new providers preserve their source credits and immutable input hashes.
+Modified global adapters retain the complete invoice and all inherited
+guards; the patch and source/adapted hashes are included. See [NOTICE](NOTICE).
+Prepared with substantial OpenAI Codex assistance. The publication adds only
+this research package; its PR564 dependency is external and pinned. No
+upstream scientific package was modified.

research/larger-gain/STATE.json

@@ -0,0 +1,46 @@
+{
+  "recorded_utc": "2026-10-12T08:08:13.481822+00:00",
+  "base": "origin/pr-564",
+  "remote": "origin",
+  "pins": [
+    {
+      "ref": "origin/main",
+      "commit": "db0e57bc7f36b5ac0fc844ad7ce1396165ee6647",
+      "pr": null,
+      "url": "https://github.com/CrocSwap/integer-mult-bounds"
+    },
+    {
+      "ref": "origin/pr-564",
+      "commit": "c18417b02b80594b2ca1c8c21185d28befafc935",
+      "pr": 564,
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/564"
+    },
+    {
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+      "pr": 565,
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/565"
+    },
+    {
+      "ref": "origin/pr-566",
+      "commit": "86d39f3247a5a1df4f040ed71d8251bed77b91ec",
+      "pr": 566,
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/566"
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+      "pr": 567,
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/567"
+    },
+    {
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+      "commit": "71c99ac935a33bb7b59a1b4a7af8725a939b0b90",
+      "pr": 568,
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/568"
+    }
+  ],
+  "scope": "Baseline PR564 freshly reproduced. Latest additional PR567 and568 fetched during publication; their new cold-kernel constructions were not replayed or composed. PR565/566 advance formal counting/preparation, not general constant-time indexing.",
+  "original_checkout_preserved": true,
+  "publication_pr": "https://github.com/CrocSwap/integer-mult-bounds/pull/569"
+}