From 16304c166f084783b5e8fa341ebca97b858d3df1 Mon Sep 17 00:00:00 2001 From: d3lerium Date: Fri, 23 Oct 2026 23:14:43 +1100 Subject: [PATCH 1/4] Extend released-control ceiling to fixed nonlinear monomial programs --- .../.gitattributes | 2 + .../EVIDENCE.json | 26 + .../HISTORY.md | 20 + .../LICENSE | 201 + .../nonlinear-monomial-control-ceiling/NOTICE | 415 + .../README.md | 109 + .../SOURCE-NOTICE | 6 + .../SOURCES.json | 86 + .../TASK.json | 90 + .../VERIFY.json | 34 + .../evidence/finite-check/FREEZE.json | 23 + .../finite-check/MATRICES-OPTIMIZED.json | 1952 ++++ .../evidence/finite-check/MATRICES.json | 1952 ++++ .../finite-check/RESULT-OPTIMIZED.json | 7981 +++++++++++++++++ .../evidence/finite-check/RESULT.json | 7981 +++++++++++++++++ .../evidence/finite-check/REVIEW.md | 39 + .../evidence/finite-check/SHA256SUMS | 8 + .../evidence/finite-check/SOURCE-FACTS.md | 9 + .../evidence/finite-check/check_support.py | 176 + .../evidence/proof-review/FILES.json | 59 + .../evidence/proof-review/RESULT.json | 10 + .../evidence/proof-review/REVIEW.md | 61 + .../evidence/proof-review/SOURCES.json | 56 + .../evidence/proof-review/frozen/CODE.json | 1 + .../proof-review/frozen/FIRST-DRAFT.md | 43 + .../proof-review/frozen/NORMALIZERS.json | 126 + .../evidence/proof-review/frozen/PHYSICAL.md | 71 + .../proof-review/frozen/PREDECESSOR-PROOF.md | 256 + .../frozen/PREDECESSOR-SOURCE-CHECKER.py | 159 + .../frozen/PREDECESSOR-SOURCES.json | 36 + .../proof-review/frozen/REVIEWED-PROOF.md | 48 + .../verify.py | 87 + 32 files changed, 22123 insertions(+) create mode 100644 research/nonlinear-monomial-control-ceiling/.gitattributes create mode 100644 research/nonlinear-monomial-control-ceiling/EVIDENCE.json create mode 100644 research/nonlinear-monomial-control-ceiling/HISTORY.md create mode 100644 research/nonlinear-monomial-control-ceiling/LICENSE create mode 100644 research/nonlinear-monomial-control-ceiling/NOTICE create mode 100644 research/nonlinear-monomial-control-ceiling/README.md create mode 100644 research/nonlinear-monomial-control-ceiling/SOURCE-NOTICE create mode 100644 research/nonlinear-monomial-control-ceiling/SOURCES.json create mode 100644 research/nonlinear-monomial-control-ceiling/TASK.json create mode 100644 research/nonlinear-monomial-control-ceiling/VERIFY.json create mode 100644 research/nonlinear-monomial-control-ceiling/evidence/finite-check/FREEZE.json create mode 100644 research/nonlinear-monomial-control-ceiling/evidence/finite-check/MATRICES-OPTIMIZED.json create mode 100644 research/nonlinear-monomial-control-ceiling/evidence/finite-check/MATRICES.json create mode 100644 research/nonlinear-monomial-control-ceiling/evidence/finite-check/RESULT-OPTIMIZED.json create mode 100644 research/nonlinear-monomial-control-ceiling/evidence/finite-check/RESULT.json create mode 100644 research/nonlinear-monomial-control-ceiling/evidence/finite-check/REVIEW.md create mode 100644 research/nonlinear-monomial-control-ceiling/evidence/finite-check/SHA256SUMS create mode 100644 research/nonlinear-monomial-control-ceiling/evidence/finite-check/SOURCE-FACTS.md create mode 100644 research/nonlinear-monomial-control-ceiling/evidence/finite-check/check_support.py create mode 100644 research/nonlinear-monomial-control-ceiling/evidence/proof-review/FILES.json create mode 100644 research/nonlinear-monomial-control-ceiling/evidence/proof-review/RESULT.json create mode 100644 research/nonlinear-monomial-control-ceiling/evidence/proof-review/REVIEW.md create mode 100644 research/nonlinear-monomial-control-ceiling/evidence/proof-review/SOURCES.json create mode 100644 research/nonlinear-monomial-control-ceiling/evidence/proof-review/frozen/CODE.json create mode 100644 research/nonlinear-monomial-control-ceiling/evidence/proof-review/frozen/FIRST-DRAFT.md create mode 100644 research/nonlinear-monomial-control-ceiling/evidence/proof-review/frozen/NORMALIZERS.json create mode 100644 research/nonlinear-monomial-control-ceiling/evidence/proof-review/frozen/PHYSICAL.md create mode 100644 research/nonlinear-monomial-control-ceiling/evidence/proof-review/frozen/PREDECESSOR-PROOF.md create mode 100644 research/nonlinear-monomial-control-ceiling/evidence/proof-review/frozen/PREDECESSOR-SOURCE-CHECKER.py create mode 100644 research/nonlinear-monomial-control-ceiling/evidence/proof-review/frozen/PREDECESSOR-SOURCES.json create mode 100644 research/nonlinear-monomial-control-ceiling/evidence/proof-review/frozen/REVIEWED-PROOF.md create mode 100644 research/nonlinear-monomial-control-ceiling/verify.py diff --git a/research/nonlinear-monomial-control-ceiling/.gitattributes b/research/nonlinear-monomial-control-ceiling/.gitattributes new file mode 100644 index 0000000000..ad1d31d8e2 --- /dev/null +++ b/research/nonlinear-monomial-control-ceiling/.gitattributes @@ -0,0 +1,2 @@ +# Immutable source and proof captures retain raw CRLF/LF bytes. +evidence/** -text whitespace=cr-at-eol diff --git a/research/nonlinear-monomial-control-ceiling/EVIDENCE.json b/research/nonlinear-monomial-control-ceiling/EVIDENCE.json new file mode 100644 index 0000000000..a347cb3ba4 --- /dev/null +++ b/research/nonlinear-monomial-control-ceiling/EVIDENCE.json @@ -0,0 +1,26 @@ +{ + "files": { + "evidence/finite-check/FREEZE.json": "090145aee55d41069406b642942f5c9feca79c6908846998df8358da9064f043", + "evidence/finite-check/MATRICES-OPTIMIZED.json": "02bfc8bd2d1fd9a38de2213d8729aca10feee634de00290f83a94f2bfb19f611", + "evidence/finite-check/MATRICES.json": "02bfc8bd2d1fd9a38de2213d8729aca10feee634de00290f83a94f2bfb19f611", + "evidence/finite-check/RESULT-OPTIMIZED.json": "4639a2721e1e90f0ab4104ed07a2395701e12e3bbcabcb760317d3a63dab7e9e", + "evidence/finite-check/RESULT.json": "4639a2721e1e90f0ab4104ed07a2395701e12e3bbcabcb760317d3a63dab7e9e", + "evidence/finite-check/REVIEW.md": "db09dfb39e4dd5bdecc00bcef32d4db77adf9c11e7f073d119c418395fc0f3de", + "evidence/finite-check/SHA256SUMS": "180502cb8356bd9c8d92bf8f06da25d89f21efeb18a8135723b5bd847256edc4", + "evidence/finite-check/SOURCE-FACTS.md": "c6e2546054ac68cbec6a579bf7558a523d1a8843c160a70250b24a10220f90fe", + "evidence/finite-check/check_support.py": "91115f8380c01d6e788acf27870991b80a6c6ee02d5363396408e4e08dad149c", + "evidence/proof-review/FILES.json": "d6b03962432cbb2880dff5825daf6eae9637f8ce614c82de15aaa07ffb90571d", + "evidence/proof-review/RESULT.json": "8bdf9491dc7e9c8ac9883e6be3845d3aa9b05930b0c1699057baf7fb33c7df09", + "evidence/proof-review/REVIEW.md": "0b0fe1c33eea85ccd92a3423a705fd086c6f2669f1a47f37fba9c1ca4ae91973", + "evidence/proof-review/SOURCES.json": "0689d9a2bd640fab5b3e71765800064ff8689484d39b975b7834aadbf579ea0e", + "evidence/proof-review/frozen/CODE.json": "8be7f3f073b7981ec91c6af43edff0ac8a89f4824ef6026ae758c287dd6c03d0", + "evidence/proof-review/frozen/FIRST-DRAFT.md": "669b9b2a3c1797c80d46ed44c119104cbbc36b893877cc8612dcc8b3e50d64fb", + "evidence/proof-review/frozen/NORMALIZERS.json": "ae8aad06aa5d1811d725895b3ab5a4f4daa9dcdd88294aa0d5f723e78bb1e0da", + "evidence/proof-review/frozen/PHYSICAL.md": "4321b159ff8d3fe705419adaaa1beef75606e869e34341bda553f17d598a6078", + "evidence/proof-review/frozen/PREDECESSOR-PROOF.md": "c20b2f13e875d9e1fb6d0743693f9deccc4f7d68dcdc55ec241324ed3d7de7bc", + "evidence/proof-review/frozen/PREDECESSOR-SOURCE-CHECKER.py": "025ba8f3f63fa4f4d7155a32f0f51de210693f6b425b047312d5f232e96fdc5f", + "evidence/proof-review/frozen/PREDECESSOR-SOURCES.json": "e810407fc34de1f3af196fc86e91e680075ff982bbc6fe19d6ebb02f2a0c0d9c", + "evidence/proof-review/frozen/REVIEWED-PROOF.md": "76bea47531ed557c0c9cb210c4324d6000c74415bf13e5747b7036567853f6ef" + }, + "schema_version": 1 +} diff --git a/research/nonlinear-monomial-control-ceiling/HISTORY.md b/research/nonlinear-monomial-control-ceiling/HISTORY.md new file mode 100644 index 0000000000..30d6075bea --- /dev/null +++ b/research/nonlinear-monomial-control-ceiling/HISTORY.md @@ -0,0 +1,20 @@ +# Scope correction and preserved evidence + +The first proof draft (frozen/FIRST-DRAFT.md inside the proof-review packet) +used a fixed-alphabet phrase too broadly. Independent review pointed out that +weighted monomial generators can have infinitely many projective products. +The final proof restricts the theorem to a fixed finite program or explicitly +fixed finite complete-product support independent of width. It permits only +outer scalar coefficients to vary with width. Both original drafts and the +independent review remain byte-exact; rejected scope wording is not silently +removed from the historical record. + +The predecessor5160 proof is retained only as its exact source and tensor- +separation context. The new proof uses support partitions instead of a +Clifford/symplectic classification and preserves the transported-class warning. +The finite16x16 checker is a separate discriminator; its complete receipts +are not promoted into a uniform theorem or physical native-phase replay. + +Source CRLF/LF bytes and all author receipts are preserved. No old package +is edited, no mathematical source is normalized, and private regenerated +outputs are compared to the immutable complete RESULT/MATRICES bytes. diff --git a/research/nonlinear-monomial-control-ceiling/LICENSE b/research/nonlinear-monomial-control-ceiling/LICENSE new file mode 100644 index 0000000000..261eeb9e9f --- /dev/null +++ b/research/nonlinear-monomial-control-ceiling/LICENSE @@ -0,0 +1,201 @@ + Apache License + Version 2.0, January 2004 + http://www.apache.org/licenses/ + + TERMS AND CONDITIONS FOR USE, REPRODUCTION, AND DISTRIBUTION + + 1. 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We also recommend that a + file or class name and description of purpose be included on the + same "printed page" as the copyright notice for easier + identification within third-party archives. + + Copyright [yyyy] [name of copyright owner] + + Licensed under the Apache License, Version 2.0 (the "License"); + you may not use this file except in compliance with the License. + You may obtain a copy of the License at + + http://www.apache.org/licenses/LICENSE-2.0 + + Unless required by applicable law or agreed to in writing, software + distributed under the License is distributed on an "AS IS" BASIS, + WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. + See the License for the specific language governing permissions and + limitations under the License. diff --git a/research/nonlinear-monomial-control-ceiling/NOTICE b/research/nonlinear-monomial-control-ceiling/NOTICE new file mode 100644 index 0000000000..59d8e0cf31 --- /dev/null +++ b/research/nonlinear-monomial-control-ceiling/NOTICE @@ -0,0 +1,415 @@ +A sharper exponent for integer multiplication +Copyright 2026 Douglas Colkitt + +This project includes unmodified source files from OpenAI's manuscript +"Integer multiplication below n log n" and its accompanying repository: +https://github.com/openai/math +Commit: adc7f1241b42e322a6451854ab7e4b4c146bf78a + +The original manuscript identifies OpenAI as its author. Its Apache-2.0 +license is retained at upstream/LICENSE. The exact source URLs and hashes +are recorded in upstream/manifest.json. Proposed modifications are supplied +as separate files under patches/; upstream source files remain unchanged. + +The original local research note, scripts, certificates, patches, and +documentation were prepared by Douglas Colkitt with assistance from OpenAI +Codex. The community release incorporates work by the contributors identified +in the notices below; it is not solely Douglas Colkitt's work. +This project is not an official OpenAI release or an OpenAI endorsement. + +Related contributions (not imported in this checkpoint): +Zhihao Chen (jacklightChen), PR #7, supplied an earlier ternary five-subset +motif, finite-alphabet transfer and a different compressed producer: +https://github.com/CrocSwap/integer-mult-bounds/pull/7 +The present checkpoint claims no priority for that motif. Related compressed +complex-network work was submitted by eumemic in PR #3: +https://github.com/CrocSwap/integer-mult-bounds/pull/3 +The pending contributions and their stronger bounds require their own review; +this notice does not represent adoption or verification of their claims. + +Additional community work, including parallel and superseded submissions, is +acknowledged in CONTRIBUTORS.md, with pinned submission metadata in +docs/research/contribution-snapshot.json. These acknowledgements do not imply +joint authorship of the checkpoint, incorporation of pending code, or verification +of submitted mathematical claims. Original source-specific notices remain in force. + +Community release (accepted by conditional maintainer audit): +The following original contributor notices apply to the imported PR #39 chain. +The preceding non-import statements describe the preserved 2^-30 checkpoint. + +Ternary five-subset construction and common-four-point resynthesis +Copyright 2026 Zhihao Chen (jacklightChen) + +This is Zhihao Chen's first submission of this specific construction to this +repository. Research, proof development, code and verification were prepared +with substantial assistance from GPT-6 Astra (OpenAI Codex), as identified by +the contributor. This is not independent review or an OpenAI endorsement. +Future research building on this contribution should explicitly acknowledge +Zhihao Chen (jacklightChen), cite the accompanying note and pull request, and +retain the preceding contributors' attribution. This notice records research +provenance; the repository's Apache-2.0 license remains unchanged. + +Dependencies retained from eumemic's Claude-assisted PRs #3 and #5: +https://github.com/CrocSwap/integer-mult-bounds/pull/3 +https://github.com/CrocSwap/integer-mult-bounds/pull/5 +PR #5 commit: d3d370c34937503bd3fa63c83d505a05e8e50893 +These include complex_circuit.py, complex_network.py, fast_gaussian.py, +their generators, construction notes and certificates. The Gaussian input +scaling proof is corrected in this derivative; see the modification notice below. +Aligned pairing also appears in Bortlesboat's PR #2. Retained totals and bank +sharing appear in dleen's PR #4 and cheaper retained centers in eumemic's PR #6. +Their results and ideas are credited separately from this new construction. + +Bulk recursion and controlled tensor basis, 8 October 2026 +Copyright 2026 icekylinx + +The new projector batching, controlled simultaneous basis, mixed-width tape +recursions, whole-residual complex recursion, and dependency-path precision +guard in the batched-23 proof package are contributed by icekylinx, developed +with substantial OpenAI GPT-6 Astra assistance. Review corrections and +repository integration were prepared with OpenAI Codex assistance. They build on the retained +PR #7 finite producers and the preceding contributions credited above. +The new package retains the repository's Apache-2.0 license and the original +authors' attribution. It is not an official OpenAI release or endorsement. + +Modification notice, 8 October 2026 (icekylinx): the retained fast Gaussian +proof now bounds shifted approximation error by F=2^ceil(1.14 alpha^2). +The P=34p precision budget and asymptotic exponent are unchanged. New batching +checks are integrated into make verify. Existing attribution and licenses +are retained. + +Partial-swap extension, 8 October 2026 +Copyright 2026 icekylinx + +The partial-swap framing, binary retained-total triple construction, +dependency-preserving positive labels and carrier reuse, unequal (25,23,57) +common basis, improved complex batching, and exact assembly are contributed +by icekylinx with substantial OpenAI GPT-6 Astra and Codex assistance. +This extension directly inherits PR #10, commit +62691e395a0458ce089a1c7b5d89e74291e95e29. The portable producer adapters +modify the retained paired-exclusion and shared-point implementations credited +above. The original licenses, attribution, and Gaussian correction are retained. + +Endpoint-gauge extension, 8 October 2026 +Copyright 2026 icekylinx + +The complementary auxiliary endpoint gauges, (32,30,40) controlled basis, +general nondegenerate binary phase residuals, retained-center complex producer, +updated path guard and exact assembly are contributed by icekylinx with +substantial OpenAI GPT-6 Astra and Codex assistance. This extension inherits +f2ab41aebad47861caf6316282c1793e5513845e and its PR #10 ancestry. New portable +adapters retain the paired-exclusion and shared-point contributions credited +above. Original licenses, notices and the Gaussian correction are retained. + +Structured-basis and semantic-bulk extension, 8 October 2026 +Copyright 2026 icekylinx + +The selected (32,30,36) bit construction, A1/A5 block refinements, fixed +middle-axis I+J profiles with bounded-minor CRT certificates, mixed dyadic +complex centers, complete data-front batching and circuit-specific assembly +are contributed by icekylinx with substantial OpenAI GPT-6 Astra and Codex +assistance. The immediate parent is ed90fd940279c336ebc45968631bdebfb087b505. + +The semantic/bulk compatibility argument, literal scalar charge, product row +stock and parameterization adapt Zhihao Chen (jacklightChen)'s PR #23, +commit 661ebabc1076c9f525d7ce4967c876b203fa9827. Translated complex endpoints +are adopted from the same contributor's PR #21, +commit 5ba6cf0bfb68f2be8d15610e7972207c50254d6a, following eumemic's source-frame +work and the earlier nested-basis contributions. + +Completed-child semantic precision, arbitrary-coordinate routing, phase-cell +Gaussian inversion and bulk tape transfer are credited to the RaD project +(hipotures), commit 45d9b60355872041f9275f77c057514def0d45bc, exposed through +PR #20 at 4f8d6c8272b5ff307a0da51df545ec3cd96a8b6e. Their selected proof +sources retain their separate CC0 license and original NOTICE under +references/semantic-bulk/rad20. The PR #23 source retains its recorded OpenAI +Codex assistance; RaD retains its recorded Codex and GPT-6.1-Sol Ultra +assistance. These are source attributions, not claims about this session's +runtime model. Related contributions by Rohan Arun and Aurel Prosz remain +acknowledged in the imported notices. No exclusive priority or additional +license condition is asserted. + +Copied retained-center extension, 8 October 2026 +Copyright 2026 icekylinx + +The copied retained-center schedule in both orientations, its complex-phase +transfer, the Wm-N+L rank budget, and the selected (25,23)/(28,28) composition +are contributed by icekylinx with substantial OpenAI GPT-6 Astra and Codex +assistance. This extension inherits PR #32 at +0ef3aeb61f55cc0b321ce6a0ef00acee25cefe52 and the preceding PR #10/#18/#24 work. + +The two-stage topology and paid copied-stream endpoint correction are due +to Aurel Prosz (Paureel), c82d09eb4781b68435e3fa3eb6beea72fa900fab, +through Zhihao Chen (jacklightChen)'s PR #29 unequal-axis composition, +9d963275075fa98f1da821e27757b238dafd6b3c. That source also credits Swapnil +Jain's two-stage development at ae405eb474d1486b2d8aef90139869f927f4836a. + +The contiguous corner method, incidence factorization and exact partition +certificates are due to Rohan Arun (PR #31), method commit +02f68f95369c39955c5ffcfddbd75cd83e26b305. The selected (25,23) certificate +specializes Dominik Scholz's parameterization (PR #33), +a499a345c040aa15b68e984a0904eadc485c7a7e. Their source notices and proof +snapshots are preserved in references/copied-centers. Historical OpenAI +Codex, GPT-6 Astra and Anthropic Claude Opus 5.5 assistance is credited to +the respective imported sources. Existing PR #21/#23 and RaD attribution, +licenses and the Gaussian correction remain in force. + +Copied centers with reversed (23,25) data corners, 2026-10-08 +Rohan Arun with substantial OpenAI Codex assistance supplies the composition, +exact dimension23 controls and cost/assembly integration. James Chang's PR34 +supplies the reversed boundary family; icekylinx's PR36 supplies copied retained +centers and the selected complex graph. The balanced assembly remains attributed +to RaD/hipotures and its PR34 specialization. Retain Dominik Scholz PR33, Rohan +Arun PR31, Zhihao Chen PR29/21/23, Paureel/Aurel Prosz, Swapnil Jain, icekylinx, +eumemic, Douglas Colkitt, OpenAI and all earlier notices and hypotheses. +This is not a claim of formal verification, new foundational rank identities, +global optimality or measured runtime speedup. + +Fixed middle basis with copied reversed corners (2026-10-08): +Rohan Arun, with substantial OpenAI Codex assistance. Composition and +simultaneous-basis proof retain icekylinx PR32/36, James Chang PR34, +Dominik Scholz PR35/38, Zhihao Chen, Paureel, Swapnil Jain, RaD/hipotures, +eumemic and all preceding attribution and licenses. See research/copied-fixed-reversed. + +Changed-graph composition with copied centers and fixed bases, 8 October 2026 +Prepared with OpenAI Codex assistance. +The new composition uses icekylinx PR36 copied centers and PR32 fixed-I+J +projectors/profiler, Dominik Scholz PR35 two fixed factors and CRT argument, +Zhihao Chen (jacklightChen) PR29/23 interfaces, Aurel Prosz (Paureel) and +Swapnil Jain's two-stage development, Rohan Arun PR25/31 corner methods, +Dominik Scholz PR33/38 dimension/composition work; James Chang (jamesyc) +PR34 reversed geometry and balanced transfer; Rohan Arun PR37/39/40 +reversed integration, modular data tests and both-fixed baseline; +RaD (hipotures) PR41 changed scalar DAGs, alternating point order and +independent role compiler; RaD balanced routing/bulk interfaces, and all +retained predecessors. +Complete attribution and the conditional proof boundary are recorded in +research/copied-fixed/PROOF.md; imported originals and source pins are retained. +Original Apache-2.0 and CC0 notices remain applicable. + +Community follow-up integration (PRs #49, #45 and #26), 8 October 2026: +Rohan Arun contributes weighted carrier matching and climbed summand/leave-one-out +orders (PRs #44/#47/#49), with Anthropic Claude assistance. The final construction +composes Chafik Boukhalfa's changed-graph checks, exact rational data recovery and +reordered exclusion sums (PRs #43/#46/#48, OpenAI Codex assistance) with RaD / +hipotures's alternating pair order and independent physical role compiler (#41). +All predecessor attribution and conditional proof interfaces remain in force. + +Alejandro Zarzuelo Urdiales contributes Gaussian parity and canonical arithmetic +proofs, finite tensor execution, profile precision accounting and mixed-center +reference checks (#45), with substantial OpenAI Codex assistance. See +research/gaussian-parity-synthesis/PROVENANCE.md for his full attribution. +Ryan S / princezuda contributes historical Lean certificate arithmetic, algebraic +frame/movement/guard lemmas and independent paired-circuit checks (#26). Original +source notices and historical snapshots are preserved. Neither formal package +claims a full formal proof of the multiplication algorithm or a new kappa. + +Joint frame compilation of skip-prefix strips + +The local compiler and exact finite witnesses were prepared by eumemic with +substantial OpenAI Codex assistance. Avi Eisenberg / ikeboy with Anthropic +Claude assistance supplied the unmodified PR #53 skip-prefix producer. +The pinned PR #48 pipeline is Chafik Boukhalfa's contribution, incorporating +Rohan Arun, RaD / hipotures, icekylinx, Dominik Scholz, James Chang, +Zhihao Chen, Aurel Prosz / Paureel, Swapnil Jain, eumemic, Douglas Colkitt, +OpenAI, David Harvey, Joris van der Hoeven and the earlier contributors +credited in the retained sources. See references/frame-compiler/pr48/NOTICE +and references/frame-compiler/pr53/NOTICE for their original attribution. +No authorship or review by those predecessors is claimed for this compiler. + +Joint dual-skip/frame-compiler composition (2026-10-08) +--------------------------------------------------- +Rohan Gupta / gupt1156 with Anthropic Claude assistance: PR55 dual-skip +scalar graph; original LICENSE and NOTICE are retained under +references/frame-compiler/pr55. eumemic with substantial OpenAI Codex +assistance: PR57 joint reversible frame compiler, serialized physical-word +replay and profile reconstruction. Chafik Boukhalfa with OpenAI Codex +assistance: this composition, independent integration and exact certificate. +Avi Eisenberg / ikeboy, Rohan Arun, RaD / hipotures, icekylinx, and all +predecessor authors retain the notices above and in pinned source files. +No original copyright or Apache-2.0 notice is removed. + +Ranked reclamation after the PR58 joint-dual composition (2026-10-08) +---------------------------------------------------------------- +Chafik Boukhalfa with OpenAI Codex assistance: prioritize retired physical +slots by descending current frame rank, retaining slot ID as the tie-breaker; +complete replay, exact profiles, comparison and adversarial controls. +The dedicated joint_dual_reclaim_compiler.py is derived from eumemic's PR57 +compiler. Its original shared compiler remains byte-identical. The PR55 +producer and all preceding attribution remain unchanged. The complete PR58 +comparison certificate and its original source/proof provenance are retained +under references/frame-compiler/pr58. + +Skip-prefix strip producer, 8 October 2026 +By Avi Eisenberg (ikeboy), with Anthropic Claude (Claude Opus 5.5) +assistance. The skip-prefix leave-one-out strip layout in +research/skip-strips/skip_graph.py is new. Everything else reuses Chafik +Boukhalfa's PR #43/#46/#48 research/copied-fixed pipeline, its verification +code, exact data recovery and support-order sorting, together with: +- Rohan Arun's PR #44 weighted carrier matching optimizer; +- RaD (hipotures) PR #41 alternating point order; +- every predecessor credited above. +Complete attribution and the conditional proof boundary are recorded in +research/skip-strips/PROOF.md. Original Apache-2.0 and CC0 notices remain +applicable. + +Local split-pair skip-prefix research continuation (2026-10-08) +Prepared with substantial OpenAI Codex assistance at the user's request. +New contribution: singleton/pair recursive partition refinement, finite summand +order and moment search, and verified composition with inherited paid clones. +Avi Eisenberg/ikeboy with Anthropic Claude assistance: PR53 skip-prefix strips. +Chafik Boukhalfa with OpenAI Codex assistance: PR54 original-envelope clone +specialization and PR43/46/48 checking interfaces. RaD/hipotures: PR51 +whole-chain/source-partition cloning and earlier producer/interfaces. +Rohan Arun: PR44 weighted matching and earlier fixed/reversed work, with all +stated historical AI assistance. All predecessor notices remain applicable. + +Original-envelope whole-chain clones on skip-prefix strips (research/skip-clones): +Chafik Boukhalfa, prepared with OpenAI Codex assistance. The complete-chain +source-partition clone operation is adapted from RaD / hipotures, PR #51, +with original envelopes retained here. The scalar base is Avi Eisenberg's +PR #53 skip-prefix layout, prepared with Anthropic Claude assistance. +All credited PR #43/#46/#48, PR #44 and earlier code/proof notices remain +in force. This extension is provided under the repository's Apache-2.0 terms. + +Community review checkpoint #50–#61, 8 October 2026 +Maintainer integration and independent arithmetic: Douglas Colkitt, with OpenAI Codex. +Avi Eisenberg / ikeboy: skip-prefix strip framework (#53), with Anthropic Claude. +Rohan Gupta / gupt1156: parallel order search (#50, stated Antigravity assistance) +and dual-suffix strips (#55, Anthropic Claude assistance). +Eumemic: equal-frame joint compilation and paid signal reclamation (#57), with Codex. +Chafik Boukhalfa: original-envelope paid clones (#54), composition (#58) and ranked +reclamation (#60), with Codex. Rohan Arun: order search (#52) and positive-frame +composition (#56), with Anthropic Claude and stated Codex validation assistance. +RaD / hipotures: enlarged frames, source partitions and paid whole-chain clones +(#51), with Codex. Rohan Garg / rohangar1: singleton/pair recursion and paid-clone +composition (#59), with Codex. Alejandro Zarzuelo Urdiales: exact parameter +refinement, scoped Lean arithmetic, matrix/parity and search tools (#61), with +his disclosed multi-tool and Codex assistance. +Parallel #50/#52 records are reviewed and credited without replacing the earlier +climbed-48 implementation; their exact commit history is retained. The separate +#51 witness is retained as research; its source-graph/negative-basis data proof +has not received the same full replay as #56's consumed compiler machinery. +See docs/research/community-round2-review.md for precise validation boundaries. +All earlier licenses, contributions and notices remain applicable. + +Interval strips and core-aware pair assembly + +The interval leave-one-out strips and the core-aware group-pair assembly in +research/pair-assembly were prepared by Avi Eisenberg / ikeboy with Anthropic +Claude assistance. They build on PR #53's skip-prefix pipeline and its +inherited PR #48 sources. The stacked witness in research/pair-assembly/frame +uses eumemic's PR #57 joint frame compiler unchanged; its driver and verifier +are adaptations of PR #57's skip_frame_compiler.py, skip_frame_compose.py and +verify_skip_frame.py. Original notices are retained. + +Stopped product-ring extension, 9 October 2026 +Copyright 2026 icekylinx + +The opposite-bank generic projector factorization, componentwise atom-ring +adapters, stopped recurrence and outer ordinary-interchange conversion, +aligned pair-star producer with rational centers, exact denominator-21 +grid and selected (23,23)/(24,24) composition are contributed by icekylinx +with substantial OpenAI GPT-6 Astra and Codex assistance. This extension +builds on merged PR #36, 11817ccacb564bb7f98789c20dc11d3fece207e3, and is +integrated on upstream main 56b66d58297deca1d7dd130247d720e960f77a37. + +The retained ordinary leaf uses the preceding projector, corner and +copied-center proofs. The selected two-stage topology retains Aurel Prosz +(Paureel), Swapnil Jain and Zhihao Chen's credits; the semantic assembly +retains PR #21/#23 and RaD's attribution. Paired producers, carrier matching, +original source notices and each imported license remain unchanged. + +Partial-gauge extension, 9 October 2026 +Copyright 2026 icekylinx + +The pair-first cube assembly, compatible backward frame enlargement, +partial source-gauge schedule and stopped whole-projector composition are +contributed by icekylinx with substantial OpenAI GPT-6 Astra and Codex +assistance, extending PR #104 at 948ce1510df750f4c18b96bdaef436a86f8bf834. + +The physical deferred bit word, reflected event accounting, fan-order +clarification and signed two-stage endpoint are adopted from Zhihao Chen +(jacklightChen), PR #97, e15350b66e4bf796153a68c240a6cc1b788abe3e, with +Codex assistance. Its deferred readouts, V leaves, lifted frames and frozen +witness originate in Swapnil Jain's 741e7aa078392553815df7926ee17ac5e25a8c38, +with the original Claude disclosure retained. Cyclic interval and core-aware +pair primitives build on Avi Eisenberg (ikeboy), PR #62, +ad0f25ff7b23cff7f08ad237c2254e6ecf74257e, with Anthropic Claude assistance. +Prior two-stage, copied-center, ordinary-leaf, semantic/bulk, routing and +precision credits remain in force. Imported source notices and licenses +are preserved under references/partial-gauge/; see SOURCES.json. + +Three-stage cover extension, 9 October 2026 +Copyright 2026 icekylinx + +The arbitrary-subspace Clifford frame theorem, three-stage Cayley geometry, +weighted q-adic GL cover, unit-lower weighted compiler, charged local-ring +fallback, uniform batching, internal row borrowing and exact composition +are contributed by icekylinx with substantial OpenAI GPT-6 Astra assistance. This +repository integration uses Codex assistance and extends +PR #115, cf60442983c153a3a0f5272ea4713f99eba13c86. + +The selected local complex DAG and support/label replay are adopted from +Copyright 2026 eumemic, PR #117, cbb05ce504d571546d9b7794c186a613c659c3bf, +with its original Apache-2.0 terms and Claude assistance disclosure. +The unchanged bit physical word retains Zhihao Chen (jacklightChen), +PR #97, and Swapnil Jain's underlying witness and assistance notices. +The prior opposite-bank, stopped ordinary-leaf, copied-center, topology, +semantic/bulk, precision and analytic credits remain applicable. +Original source notices and license bytes are preserved in +references/three-stage-cover/pr117/ and references/partial-gauge/. + +Paired-cube and shared-core extension, 9 October 2026 +Copyright 2026 icekylinx + +The paired-coordinate triple-cube identity, signed overlap channels, +original-source involution, coordinate-star centers, concrete frame/carrier +compiler, chronological source gauges, PR97 gauge-subset selection, +orthogonal cross-stage routing and exact supplier/assembly integration +are contributed by icekylinx with substantial OpenAI GPT-6 Astra assistance. +Repository integration uses Codex assistance and extends PR #130 at +6a9970a530119174507904e23592fd59ede19a5d. + +The substantial completed-core workspace-sharing principle is credited to +an664, PR #128, 530588a019b4a74f09180680c9e3961bf649ec89, with its reported +OpenAI Codex assistance. No source code, partition or serialized witness +from that PR is imported. The new routing and paired-motif integration +are described separately and claim no exclusive priority for broad sharing, +paired-coordinate, Fourier or carrier-matching ideas. + +Selected coarse queries restrict the retained eumemic PR #117 positive DAG; +its original Apache-2.0 and Claude assistance notices remain unchanged. +The bit word and original frames retain Zhihao Chen's PR #97 and Swapnil +Jain's frozen witness, notices and historical assistance disclosures. +Earlier generalized Clifford, three-stage, uniform weighted recursion, +ordinary-leaf, semantic/bulk, precision and analytic credits remain in force. +Source-specific license bytes are preserved under references/. + +October 9 recycled-bit integration: +DaysSky (#150), William Porter/hpst3r (#147), Thomas Marchand/Th0rgal (#146), +Rohan Gupta/gupt1156 (#148), James Chang/jamesyc (#124), SovereignSteak (#122). +The original notices and assistance disclosures are retained under research/, +including research/recycled-bit-integration/NOTICE. Andrew Barnes/Bortlesboat +(#101) repairs shared verifiers using rfu08's #64 counterexample. rfu08's +scoped finite and formal-transfer packages are included separately; they do +not establish the full multiplication theorem. Earlier and parallel credits +remain applicable. Douglas Colkitt integrated these with OpenAI Codex assistance. + +SovereignSteak PR151 independently supplies the matching gauge/recycling +composition and the selected 473/10^6 atom refinement; its original package +and assistance disclosure are retained under research/gauge-recycling/. + +October 9 coordinated-frame and completed-entrance-bank integration: +Dugongue PR186, 166a34d75e853e933855f0f7e940fe77c4bb6b8d, builds on +Chafik Boukhalfa/chafreaky's PR181, 7fb2194801e3a10f54772c7f0d5505035a4fc0ad, +and eumemic's PR168 v4, 4a3c769e5c5430e7114c4d3e099ff34664677f17. +The frozen package and prerequisite archive preserve all author, source-license +and assistance notices, including PR175's compiler optimization. DaysSky's +carrier closure, James Chang's terminal substitution and balanced assembly, +geckods' stopping refinements and all preceding framework credits remain. +Rohan Arun's PR185 bootstrap and chafreaky's PR182 bit refinement are reviewed +parallel contributions; their gains are not added to the selected witness. +Douglas Colkitt integrated the package and supplied the bank scheduling +supplement with OpenAI Codex assistance, under Apache-2.0. The all-size +interfaces remain conditional; finite verification is not full formalization. diff --git a/research/nonlinear-monomial-control-ceiling/README.md b/research/nonlinear-monomial-control-ceiling/README.md new file mode 100644 index 0000000000..af83ff3281 --- /dev/null +++ b/research/nonlinear-monomial-control-ceiling/README.md @@ -0,0 +1,109 @@ +# Nonlinear weighted-monomial control ceiling + +This is a new support-partition extension of [PR #5160](https://github.com/CrocSwap/integer-mult-bounds/pull/5160), +not a repetition of its Clifford alphabet. For the released four-control endpoint, +**arbitrary nonlinear weighted monomial** one-column controls still force +`t0_f=0` at every sufficiently large width, provided the program has a fixed +finite set of complete one-column operator products independent of width. +Thus no all-width invertible family exists in this declared class. + +The outer scalar owner coefficients may depend on width. **A finite generating +alphabet alone is insufficient:** weighted generators can produce infinitely +many projective products as word lengths vary with width. Such unbounded +width-dependent products, growing complete-product supports, noncolumnwise +routing, extra paid calls/axes, changed codes and altered productive targets +are outside the theorem. This distinction was found during independent review; +the rejected first draft is retained unchanged. + +## Written theorem and source binding + +The unchanged [final proof](evidence/proof-review/frozen/REVIEWED-PROOF.md), +SHA256 `76bea47531ed557c0c9cb210c4324d6000c74415bf13e5747b7036567853f6ef`, +passes the separate [independent written review](evidence/proof-review/REVIEW.md). +The author's original status text still says review pending; this README records +the subsequent final review without altering those original bytes. + +The argument uses complete support blocks. A weighted monomial intertwiner +must send support-partition blocks to blocks; no phase or Clifford +classification is assumed. The transported letter preserves the W and q+W +coset partitions. It therefore cannot carry a direction with a nonzero U +component into a direction contained in q+W. In the actual source endpoint, +monomiality forces `s=j`; the source coefficient is then `A[j,j]=0`. +Uniform tensor separation of a fixed finite set of complete operator products +supplies the sufficiently-large-width conclusion. Cancellation cannot create +a monomial class absent from every surviving source term. + +The scientific source remains `4c0a359d88f22f05189d51f48387750b801a45d4`, +`research/coded-native-caller`, on q+U8+W8 (17 axes,112 owners). +The released endpoint is `E=Q^-1 C3 t C1 Q^-1`, `t=D3 t0 D3^-1`. +The predecessor is #5160 head `7800589ce99d932bdb063aafca1f501f1a38626d`; +#5126 head `0e8a13c251a9be1029114cda8b4194f05f3a23ec` is context, not +an executable free dictionary. Publication base is +`ec2780e8aeb1c88acaf5e9e470ef8c68e7fbbb6c`. + +The proof-review packet retains the exact raw CODE/NORMALIZERS/PHYSICAL +captures, predecessor proof/source manifest/checker, final proof and rejected +first draft. [SOURCES.json](SOURCES.json) binds reachable Git blobs; +[EVIDENCE.json](EVIDENCE.json) authenticates the unchanged original packets. +CRLF source bytes are preserved with local `evidence/** -text` attributes. +Actual112/17 source facts remain inherited from the pinned predecessor; +this package does not reexecute its native source or all-port evaluator. + +## What the finite checker establishes + +The unchanged [checker](evidence/finite-check/check_support.py) uses exact +Gaussian-rational16x16 matrices on q,u1,u2,w. It checks three examples: +weighted nonlinear Toffoli on q or w, and CCZ(q,w,u1). Its108 terminal +sandwiches are nonmonomial. The exact controls demonstrate that dropping +join preservation or the nonzero-U condition changes the answer; transported +CCZ need not preserve central q-cosets; reversing D conjugation or omitting +its inverse changes literal matrices. Gaussian C squared is X, not identity. +Both RESULT and complete MATRICES regenerate byte-for-byte in normal/O modes. + +This is finite support calibration, **not** an all-width theorem checker, +17-axis source fixture, rank lower bound or physical absolute-phase campaign. +The independently reviewed written proof carries the uniform theorem. +No t0 search, native/emitter/compiler/Lean replay or complete κ claim is made. + +## Reproduce + +```sh +python3 -I -W error -B research/nonlinear-monomial-control-ceiling/verify.py +python3 -I -O -W error -B research/nonlinear-monomial-control-ceiling/verify.py +python3 -B scripts/research_package.py --package research/nonlinear-monomial-control-ceiling --check-only +python3 -B scripts/research_package.py --package research/nonlinear-monomial-control-ceiling +python3 scripts/check_research_commit.py --package research/nonlinear-monomial-control-ceiling --ref HEAD --check-source-history +``` + +CPython3.12, standard library only; direct runs take about1second. The wrapper +authenticates every original packet manifest and runs an unchanged checker +copy in private scratch with the original four source paths. The author checker +writes RESULT/MATRICES next to itself, so direct execution against its frozen +evidence directory would mutate it: use the registered wrapper. Package and +exported source bytes stay read-only. Complete RESULT/MATRICES, not just +compact summaries or container hashes, determine equality. + +```sh +python3 scripts/export_research_package.py export --package research/nonlinear-monomial-control-ceiling --ref HEAD --output /tmp/nonlinear-monomial-control-ceiling.tar.gz +python3 scripts/export_research_package.py check /tmp/nonlinear-monomial-control-ceiling.tar.gz +``` + +## Open next questions + +A constructive continuation must change a named premise: actual noncolumnwise +control, growing complete-product support, an affordable paid control, or a +changed observation/endpoint with exact arbitrary-input ownership and source +access. No such supplier is supplied here. Retained native/helper observations +remain with their existing investigators. Unknown finite costs, precision, +restoration, compiler reserve and outer admission are not zero. + +## Credit + +Root support-partition derivation and separate independent proof review; +finite nonlinear discriminator and its author review; Chris R/Clear1nsight +packaging, all with substantial OpenAI Codex assistance. Retain Dillon's +released equations, Dugongue/mdumitrean's source A and normalizers, Cinna +Davis/1337isnot1337, Cao Yifan/intqwq, zxv, Jacob Alan Sussman, Ryanson +Jonathan and the native/helper/Clifford/compiler lineage. Original Apache-2.0 +LICENSE/NOTICE and source notice remain included. No published-paper novelty, +independent human acceptance or proof-kernel verification is claimed. diff --git a/research/nonlinear-monomial-control-ceiling/SOURCE-NOTICE b/research/nonlinear-monomial-control-ceiling/SOURCE-NOTICE new file mode 100644 index 0000000000..985b6b1bc7 --- /dev/null +++ b/research/nonlinear-monomial-control-ceiling/SOURCE-NOTICE @@ -0,0 +1,6 @@ +Coded native caller research +Apache-2.0. Prepared for Dillon with substantial OpenAI Codex assistance. + +Retained sources: Cao Yifan / intqwq, joint-helper/native-map round-three word and PR4610; Dugongue / mdumitrean, PR4929 center-free coded producer; Cinna Davis / 1337isnot1337, PR4694 center-free cache, PR4698 helper basis, PR4470 controller and cumulative construction; zxv, retained exact Clifford implementation and native/compiler lineage; Jacob Alan Sussman, mixed-target recursion. Preserve all original source-specific headers. See ORIGINS.json for exact byte origins and README.md for the new result and limits. + +The new contribution is the phase-correct commuting physical port normalization, its corrected coded raw-dual convention, and scoped caller analysis. Center deletion, code A, its spectrum and helper compensation are predecessor work. diff --git a/research/nonlinear-monomial-control-ceiling/SOURCES.json b/research/nonlinear-monomial-control-ceiling/SOURCES.json new file mode 100644 index 0000000000..a238d5894c --- /dev/null +++ b/research/nonlinear-monomial-control-ceiling/SOURCES.json @@ -0,0 +1,86 @@ +{ + "schema_version": 1, + "base_commit": "ec2780e8aeb1c88acaf5e9e470ef8c68e7fbbb6c", + "files": [ + { + "path": "certificates/selected-result.json", + "usage": "provenance", + "git_blob": "c35482cd9035843b15a634cc8276a7e1e40e8395", + "sha256": "6fe0a099b89aa2aed796a13c3a21dafc63f3388ebf397464e44e0451893b9375" + }, + { + "path": "docs/research/agent-task-catalog.json", + "usage": "provenance", + "git_blob": "f126dd2e2dad2b669e75d1d0d0bd05c600725602", + "sha256": "f7a48bfa2659b88378282a8eee7d9fe52cdf505594124c4d1686b58510fffd72" + }, + { + "path": "research/coded-native-caller/CODE.json", + "usage": "provenance", + "pin": 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+ "Change the actual demanded observations and prove a cheaper legal producer/consumer contract." + ], + "acceptance_gate": "Name the changed premise of the fixed F0/B1 observation obstruction. Show exactly which query is recovered, legal native reads and restoration, and a full favorable budget including cover maps, helpers and endpoints. Local inverse identities alone do not close the global controller. A scoped local result may be retained without a complete supplier. Expanding it into further construction rounds requires the family-level budget and obligation gate in research-family-status.md; an optimistic passing relaxation alone is not acceptance.", + "first_step": { + "command": [ + "python3", + "-B", + "scripts/research_baseline.py" + ], + "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY", + "inspect": "Read #4166 COST.md and ADMISSION.json. An actual caller connection is now proposed. Its native-component profile loses; parent park/restore versus external-call costs remain unresolved. Retain the producer and recovery lineage rather than rebuild it.", + "scope": "Source-binding orientation only; no scientific replay, theorem acceptance or supplier admission. Budget any heavy replay separately." + }, + "first_experiment": "At the actual installed cut, compare matched complete parent park/restore and external input/return protocols at the same bank/packet geometry, including every later root call and service cost. Alternatively pin both bridge observations and derive a genuinely smaller source-compatible demand. Stop if the complete necessary inequality fails or an essential cost is unknown; no larger decoder scan from a local tape-action saving.", + "bottleneck_link": "The native local words exist; retained query information and its physical access are the missing link to a composed recursive interface.", + "allowed_variations": [ + "One paid query, one changed first pair, or one precisely specified affine helper state.", + "A counterexample or lower bound for that exact proposed repair.", + "One unresolved obligation of a newer local decoder, with exact source heads and a distinct claim/domain." + ], + "scope_change_triggers": [ + "Retaining the fixed first pair and ordinary schedule while allowing only homogeneous selected-only control, already excluded by the witness.", + "Treating cover-map counts as zero-cost physical address operations.", + "Recreating the pending complete local decoder or early-record/output-bank coalescing already under investigation without a distinct audit or changed premise.", + "Treating a completed-output defect inequality as a producer-interior, dirty-range or physical-cost certificate.", + "Recreating completed-output recovery or caller wiring already supplied by #4166 without new independent coverage.", + "Dropping retained later root calls or claiming native-profile dominance from the isolated publication saving." + ], + "overlap_terms": [ + "QQ16", + "joint-query", + "changing-chart controller", + "selected observation", + "1969", + "2540", + "early record", + "output-bank retention", + "complete native decoder", + "2965", + "producer defect", + "dirty-helper range", + "F17 consumer", + "4166", + "qq16-caller-install" + ], + "existing_checks": [ + { + "scope": "PR #4166: author-checked installed producer/service/return/bridge connection; native-component comparison loses.", + "evidence": "research/qq16-caller-install/README.md", + "remaining": "Complete matched physical protocol bill, or changed source-compatible demand/circuit; no affordable supplier yet.", + "status": "author-checked", + "source_commit": "0b082d22a0e9a442d264ab0633e35ef26c790630", + "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129" + } + ] + }, + "scope_status": "changed", + "actual_scope": "Extend the released four-control obstruction from fixed columnwise monomial-Clifford programs to arbitrary nonlinear weighted monomial programs with fixed finite complete-product support. Preserve independent proof review and finite support-discriminator evidence.", + "scope_change_reason": "The declared alphabet changes beyond the finite Clifford group. Fixed finite generators alone no longer guarantee finite projective support, so the accepted scope requires a fixed finite program or fixed finite complete-product support independent of width.", + "bottleneck_link": "Tests whether nonlinear weighted monomial FREE controls can evade the released two-FULL/one-query endpoint obstruction. It supplies a scoped negative interface result, not a physical decoder, complete cost improvement or compiler admission.", + "budget_minutes": 60, + "stop_rule": "No t0 search, source/native/compiler/Lean implementation, expanded word census or physical supplier claim. Preserve final written proof, rejected scope draft and exact finite discriminator; stop at independently reviewable portable evidence.", + "outcome": "completed", + "evidence": "README.md; evidence/proof-review/frozen/REVIEWED-PROOF.md; evidence/proof-review/REVIEW.md; evidence/finite-check/FREEZE.json; verify.py", + "author": "Chris R / Clear1nsight, with OpenAI Codex assistance", + "ai_assistance": "Substantial OpenAI Codex assistance in root mathematical derivation, independent proof review, exact finite discrimination and packaging. No proof-kernel or independent human acceptance is claimed.", + "open_questions": [ + { + "question": "Which explicit changed information or paid-control contract escapes the fixed finite complete-product support hypothesis affordably?", + "why_it_matters": "The theorem does not cover noncolumnwise controls, growing complete-product support, changed endpoints or actual retained native observations.", + "first_step": "Specify one actual program, source access, inverse, arbitrary-input ownership and complete paid/finite bill before construction.", + "blocked_by": "No physical supplier/caller or compiler reserve is supplied here; native observation ownership remains with its existing investigators." + } + ] +} diff --git a/research/nonlinear-monomial-control-ceiling/VERIFY.json b/research/nonlinear-monomial-control-ceiling/VERIFY.json new file mode 100644 index 0000000000..56a932e8e8 --- /dev/null +++ b/research/nonlinear-monomial-control-ceiling/VERIFY.json @@ -0,0 +1,34 @@ +{ + "schema_version": 1, + "status": "ready", + "python": "3.12", + "requirements": [], + "inputs": [], + "commands": [ + { + "argv": [ + "{python}", + "-I", + "-W", + "error", + "-B", + "{package}/verify.py" + ], + "timeout_seconds": 60, + "checks": "Authenticate unchanged proof/review/source/evidence bytes. Execute unchanged 16x16 nonlinear weighted-monomial support discriminator in private original logical layout and compare complete RESULT/MATRICES to both frozen modes. Three examples,108 terminal sandwiches and hypothesis/order controls; not an all-width theorem,17-axis native or compiler replay." + }, + { + "argv": [ + "{python}", + "-I", + "-O", + "-W", + "error", + "-B", + "{package}/verify.py" + ], + "timeout_seconds": 60, + "checks": "Authenticate unchanged proof/review/source/evidence bytes. Execute unchanged 16x16 nonlinear weighted-monomial support discriminator in private original logical layout and compare complete RESULT/MATRICES to both frozen modes. 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component. The controls discriminate omitted hypotheses and the previously invalid central-fixed inference. This does not prove the 17-axis or all-width theorem, classify the admissible monomial group, or supply a Gaussian/rank lower bound. + +The basis is (q,u1,u2,w), encoded as bits (1,2,4,8). Matrices are 16 by 16, act on column vectors, and compose rightmost first. Every scalar is an exact pair of standard-library Fractions for its real and imaginary parts; all values in these examples are Gaussian dyadics. No floating calculation decides equality, support or monomial membership. A matrix is monomial only if every row and every column has exactly one nonzero entry. The checker verifies invertibility of the positive examples and their transported matrices, so the negative support results do not arise from singular or zero matrices. + +For every nonzero translation direction v, the Gaussian is exactly + +`C_v = ((1+i) I + (1-i) X_v)/2`. + +The inverse is the actual conjugate transpose. The checker verifies C_v^2=X_v for the central instance, Q^-1 Q=Q Q^-1=I, and D^-1 D=D D^-1=I with Q=C_q and D=C_w. In particular C is not treated as self-adjoint or as an involution. The two exact intertwining equations are h Q=Q(Q^-1 h Q) and D h=(D h D^-1)D. + +The positive examples are: + +* A Toffoli permutation with target q and controls u1,u2, weighted in column x by 2^(u1*u2). +* The same weighted Toffoli with target w. +* The cubic diagonal phase CCZ(q,w,u1), followed by the required transport k=D h D^-1. + +For each weighted Toffoli, the Boolean second difference on inputs 0,u1,u2,u1+u2 is the nonzero target vector; this proves the address map is not affine. The nonunit weights are independently retained rather than projected away. Both actual conjugates equal h in these two examples. For CCZ, both actual conjugates are also monomial, but the transported k need not preserve q-cosets. All three h preserve q-cosets, W-cosets and their join; all three transported k preserve W-cosets and the join. + +The CCZ transport explicitly sends the q-edge (u1,u1+q), encoded (2,3), to (u1,u1+q+w), encoded (2,11). The address difference becomes q+w. On input q=u1=1,w=0, k has output phase -i, fixing the conjugation orientation. Reversing the D conjugation gives a different matrix; omitting the inverse also gives a different matrix. Moreover Q^-1 k Q is nonmonomial. Thus the old central-fixed inference and the unproved reuse of the original Q-conjugacy condition after transport are both rejected by actual matrices. The correct join partition is preserved. + +For each transported example, the checker uses all six directions v in span(q,u1,u2) with a nonzero U component and all three nonzero directions t in span(q,w). It verifies that the monomial address permutation cannot send v-cosets to t-cosets and computes both + +`C_t^-1 k C_v` and `C_t^-1 k C_v^-1`. + +Every one of the 3*6*3*2=108 exact matrices is nonmonomial, with an explicit branching column and exact coefficients saved in RESULT.json. The first family is the requested two-sided intertwiner discriminator; the second also checks the same-support inverse convention relevant to the two inverse outside Gaussians in the source endpoint. These finite examples do not substitute a sign change for a general theorem about arbitrary phases. + +The controls change the answer when a premise is removed: + +* The literal swap u1<->w remains monomial and satisfies Q^-1 h Q=h, but D h D^-1 is nonmonomial. Its column zero has rows 0,2,8,10 with coefficients 1/2,i/2,-i/2,1/2. It breaks the W and join partitions. +* Using that swap itself as k, C_w^-1 k C_u1 is exactly the original monomial swap. Thus a blanket nonmonomial conclusion for arbitrary monomial k, without join preservation, would be false. +* When the nonzero-U condition is removed, C_q^-1 k C_q for the weighted target-q example is monomial. The condition is substantive. + +Normal and optimized runs use `python3 -I [-O] -B -W error check_support.py`. They complete in less than one second locally and produce byte-identical RESULT.json and MATRICES.json. MATRICES.json retains exact sparse entries for the main Gaussian, positive and control matrices, while RESULT.json records every terminal digest, row/column support counts and branching witness. Source hashes are checked before and after each run; no source file is edited. No source model, large alphabet enumeration, solver, native circuit, compiler, factory or Lean process runs. + +The proposed support-partition proof is separately reviewed. These finite examples neither establish its tensor-separation argument nor discharge its finite complete-product support premise. Fixed finite generating alphabets with unbounded width-dependent products remain outside that proposed theorem, as the revised proof states. + +Credit the released four-control equations, source real normalizers, native/Clifford/helper/precision/compiler lineage, and the proof's retained contributors. This bounded exact discriminator was produced with OpenAI Codex assistance. It admits no physical implementation, paid cost, caller transfer or kappa improvement. diff --git a/research/nonlinear-monomial-control-ceiling/evidence/finite-check/SHA256SUMS b/research/nonlinear-monomial-control-ceiling/evidence/finite-check/SHA256SUMS new file mode 100644 index 0000000000..401a248bc0 --- /dev/null +++ b/research/nonlinear-monomial-control-ceiling/evidence/finite-check/SHA256SUMS @@ -0,0 +1,8 @@ +91115f8380c01d6e788acf27870991b80a6c6ee02d5363396408e4e08dad149c check_support.py +4639a2721e1e90f0ab4104ed07a2395701e12e3bbcabcb760317d3a63dab7e9e RESULT.json +4639a2721e1e90f0ab4104ed07a2395701e12e3bbcabcb760317d3a63dab7e9e RESULT-OPTIMIZED.json +02bfc8bd2d1fd9a38de2213d8729aca10feee634de00290f83a94f2bfb19f611 MATRICES.json +02bfc8bd2d1fd9a38de2213d8729aca10feee634de00290f83a94f2bfb19f611 MATRICES-OPTIMIZED.json +db09dfb39e4dd5bdecc00bcef32d4db77adf9c11e7f073d119c418395fc0f3de REVIEW.md +c6e2546054ac68cbec6a579bf7558a523d1a8843c160a70250b24a10220f90fe SOURCE-FACTS.md +090145aee55d41069406b642942f5c9feca79c6908846998df8358da9064f043 FREEZE.json diff --git a/research/nonlinear-monomial-control-ceiling/evidence/finite-check/SOURCE-FACTS.md b/research/nonlinear-monomial-control-ceiling/evidence/finite-check/SOURCE-FACTS.md new file mode 100644 index 0000000000..80ee1eadc1 --- /dev/null +++ b/research/nonlinear-monomial-control-ceiling/evidence/finite-check/SOURCE-FACTS.md @@ -0,0 +1,9 @@ +# Source facts and scope + +The extension under test is the local [support-partition proof](../nonlinear-monomial-control-ceiling-20261023/PROOF.md), SHA256 76bea47531ed557c0c9cb210c4324d6000c74415bf13e5747b7036567853f6ef. The [released predecessor proof](../released-operator-intersection-20261023/PROOF.md), SHA256 c20b2f13e875d9e1fb6d0743693f9deccc4f7d68dcdc55ec241324ed3d7de7bc, records the endpoint order Q^-1 C3 t C1 Q^-1, transport t=D3 t0 D3^-1, the actual 17-axis embedding and the warning that transport does not force the central vector to remain fixed. + +The scientific source is research/coded-native-caller at canonical 4c0a359d88f22f05189d51f48387750b801a45d4. Its [PHYSICAL.md](https://github.com/CrocSwap/integer-mult-bounds/blob/4c0a359d88f22f05189d51f48387750b801a45d4/research/coded-native-caller/PHYSICAL.md) gives the full Gaussian entries alpha*(-i)^wt(x+y), with alpha=(1+i)^9/2^9, and identifies Q as the central C gate and D as the other eight gates. The one-bit tensor factor is therefore exactly ((1+i)I+(1-i)X)/2. PHYSICAL.md also explicitly uses rightmost-first operator order. Its retained local copy is [frozen/PHYSICAL.md](../released-operator-intersection-20261023/frozen/PHYSICAL.md), SHA256 4321b159ff8d3fe705419adaaa1beef75606e869e34341bda553f17d598a6078. + +The predecessor [SOURCES.json](../released-operator-intersection-20261023/SOURCES.json), SHA256 e810407fc34de1f3af196fc86e91e680075ff982bbc6fe19d6ebb02f2a0c0d9c, preserves the original Git blob and SHA256 pins. This checker reuses those frozen source facts and records their hashes; it does not fetch or reauthenticate Git objects and does not recheck all 112 owner normalizers or the scalar code A. The four-bit matrices are a deliberately smaller, independent support calibration with U=span(u1,u2) and W=span(w), not a literal 17-axis source fixture. + +The exact 16-by-16 data establish only the displayed finite examples and controls. The proposed uniform proof, complete-product support restriction, source owner-index argument and physical execution obligations remain separate. diff --git a/research/nonlinear-monomial-control-ceiling/evidence/finite-check/check_support.py b/research/nonlinear-monomial-control-ceiling/evidence/finite-check/check_support.py new file mode 100644 index 0000000000..9cff7d7616 --- /dev/null +++ b/research/nonlinear-monomial-control-ceiling/evidence/finite-check/check_support.py @@ -0,0 +1,176 @@ +#!/usr/bin/env python3 +"""Small exact four-bit support discriminator, not an all-width proof.""" +from collections import Counter +from fractions import Fraction as F +from hashlib import sha256 +from pathlib import Path +import json + +HERE=Path(__file__).resolve().parent +ROOT=HERE.parents[1] +N=16 +ZERO=(F(0),F(0));ONE=(F(1),F(0));I=(F(0),F(1)) +Q,U1,U2,W=1,2,4,8 + +def need(ok,why): + if not ok:raise RuntimeError(why) + +def add(a,b):return(a[0]+b[0],a[1]+b[1]) +def mul(a,b):return(a[0]*b[0]-a[1]*b[1],a[0]*b[1]+a[1]*b[0]) +def conj(a):return(a[0],-a[1]) +def reciprocal(a): + d=a[0]*a[0]+a[1]*a[1];need(d!=0,'nonzero monomial weight') + return(a[0]/d,-a[1]/d) +def fresh():return[[ZERO for _ in range(N)]for _ in range(N)] +def identity():return monomial(list(range(N)),[ONE]*N) +def monomial(perm,weights): + need(sorted(perm)==list(range(N)) and all(w!=ZERO for w in weights),'invertible weighted permutation') + a=fresh() + for j,(i,w)in enumerate(zip(perm,weights)):a[i][j]=w + return a +def product(a,b): + out=fresh() + for i in range(N): + for k in range(N): + if a[i][k]==ZERO:continue + for j in range(N): + if b[k][j]!=ZERO:out[i][j]=add(out[i][j],mul(a[i][k],b[k][j])) + return out +def adjoint(a):return[[conj(a[j][i])for j in range(N)]for i in range(N)] +def chain(*matrices): + out=matrices[0] + for a in matrices[1:]:out=product(out,a) + return out +def permutation(a): + cols=[[i for i in range(N)if a[i][j]!=ZERO]for j in range(N)] + if any(len(c)!=1 for c in cols):return None + p=[c[0]for c in cols] + return p if sorted(p)==list(range(N))else None +def inverse_monomial(a): + p=permutation(a);need(p is not None,'monomial inversion') + b=fresh() + for j,i in enumerate(p):b[j][i]=reciprocal(a[i][j]) + return b +def enc(z):return[str(z[0]),str(z[1])] +def sparse(a):return[[i,j,*enc(a[i][j])]for i in range(N)for j in range(N)if a[i][j]!=ZERO] +def digest(a):return sha256(json.dumps(sparse(a),separators=(',',':')).encode()).hexdigest() +def support(a): + return dict(row_counts=[sum(z!=ZERO for z in row)for row in a], + column_counts=[sum(a[i][j]!=ZERO for i in range(N))for j in range(N)]) +def first_branch(a): + for j in range(N): + entries=[[i,*enc(a[i][j])]for i in range(N)if a[i][j]!=ZERO] + if len(entries)>1:return dict(column=j,entries=entries) + return None +def gaussian(v): + need(0>2)for j in range(N)] + swap=monomial(swap_perm,[ONE]*N) + need(chain(adjoint(cq),swap,cq)==swap,'swap still satisfies central Q condition') + bad_d=chain(d,swap,adjoint(d)) + need(permutation(bad_d)is None and not maps_partition(swap_perm,W)and not maps_partition(swap_perm,Q|W),'swap fails D-conjugate/partition hypothesis') + swap_terminal=chain(adjoint(gaussian(W)),swap,gaussian(U1)) + need(swap_terminal==swap and permutation(swap_terminal)is not None,'omitted join hypothesis genuinely changes answer') + no_U_terminal=chain(adjoint(cq),validated['weighted_Toffoli_q'],cq) + need(permutation(no_U_terminal)is not None,'nonzero U hypothesis is necessary') + controls=dict(swap_Q_condition=True,swap_D_conjugate_monomial=False, + swap_breaks_W_and_join=True,swap_terminal_is_exactly_swap=True, + swap_nonmonomial_D_conjugate_branch=first_branch(bad_d), + zero_U_terminal_monomial=True,CCZ_transported_q_edge_input=[x,x^Q], + CCZ_transported_q_edge_output=edge,CCZ_transported_q_edge_difference=Q^W, + CCZ_transported_Q_conjugate_nonmonomial=True, + reversed_D_conjugation_differs=True,missing_D_inverse_differs=True, + CCZ_k_at_q_u1_w0=enc(validated['CCZ_q_w_u1'][permutation(validated['CCZ_q_w_u1'])[Q|U1]][Q|U1])) + matrices['swap']=sparse(swap);matrices['swap_Dconjugate']=sparse(bad_d);matrices['swap_terminal']=sparse(swap_terminal) + need(before=={n:sha256((ROOT/n).read_bytes()).hexdigest()for n in source_paths},'read-only source hashes unchanged') + results=dict(status='PASS_EXACT_FOUR_BIT_SUPPORT_DISCRIMINATOR', + basis_bits=dict(q=Q,u1=U1,u2=U2,w=W),matrix_convention='M[row][column]; column inputs; rightmost applies first', + scalars='exact pairs of Fraction(real),Fraction(imaginary); no floating membership', + gaussian='C_v=((1+i)I+(1-i)X_v)/2; inverse=adjoint; C_v^2=X_v', + source_sha256=before,examples=result_examples,terminal_tests=terminal,terminal_nonmonomial_count=len(terminal), + direction_pairs_per_example=18,conjugation_orders_tested=['Qw^-1 k Qv','Qw^-1 k Qv^-1'], + controls=controls,all_source_hashes_unchanged=True, + scope='finite 16x16 four-bit exact support calibration only; no17-axis/all-width theorem or rank lower bound', + native_solver_compiler_executed=False) + (HERE/'RESULT.json').write_text(json.dumps(results,sort_keys=True,indent=2)+'\n') + (HERE/'MATRICES.json').write_text(json.dumps(matrices,sort_keys=True,indent=2)+'\n') + print(json.dumps(dict(status=results['status'],positive_examples=len(validated),terminal_nonmonomial_tests=len(terminal),controls=controls),sort_keys=True)) + +if __name__=='__main__':main() diff --git a/research/nonlinear-monomial-control-ceiling/evidence/proof-review/FILES.json b/research/nonlinear-monomial-control-ceiling/evidence/proof-review/FILES.json new file mode 100644 index 0000000000..786a79bd99 --- /dev/null +++ b/research/nonlinear-monomial-control-ceiling/evidence/proof-review/FILES.json @@ -0,0 +1,59 @@ +{ + "files": [ + { + "path": "RESULT.json", + "sha256": "8bdf9491dc7e9c8ac9883e6be3845d3aa9b05930b0c1699057baf7fb33c7df09", + "bytes": 781 + }, + { + "path": "REVIEW.md", + "sha256": "0b0fe1c33eea85ccd92a3423a705fd086c6f2669f1a47f37fba9c1ca4ae91973", + "bytes": 9223 + }, + { + "path": "SOURCES.json", + "sha256": "0689d9a2bd640fab5b3e71765800064ff8689484d39b975b7834aadbf579ea0e", + "bytes": 2771 + }, + { + "path": "frozen/CODE.json", + "sha256": "8be7f3f073b7981ec91c6af43edff0ac8a89f4824ef6026ae758c287dd6c03d0", + "bytes": 65977 + }, + { + "path": "frozen/FIRST-DRAFT.md", + "sha256": "669b9b2a3c1797c80d46ed44c119104cbbc36b893877cc8612dcc8b3e50d64fb", + "bytes": 6556 + }, + { + "path": "frozen/NORMALIZERS.json", + "sha256": "ae8aad06aa5d1811d725895b3ab5a4f4daa9dcdd88294aa0d5f723e78bb1e0da", + "bytes": 281582 + }, + { + "path": "frozen/PHYSICAL.md", + "sha256": "4321b159ff8d3fe705419adaaa1beef75606e869e34341bda553f17d598a6078", + "bytes": 10632 + }, + { + "path": "frozen/PREDECESSOR-PROOF.md", + "sha256": "c20b2f13e875d9e1fb6d0743693f9deccc4f7d68dcdc55ec241324ed3d7de7bc", + "bytes": 12155 + }, + { + "path": "frozen/PREDECESSOR-SOURCE-CHECKER.py", + "sha256": "025ba8f3f63fa4f4d7155a32f0f51de210693f6b425b047312d5f232e96fdc5f", + "bytes": 5895 + }, + { + "path": "frozen/PREDECESSOR-SOURCES.json", + "sha256": "e810407fc34de1f3af196fc86e91e680075ff982bbc6fe19d6ebb02f2a0c0d9c", + "bytes": 1538 + }, + { + "path": "frozen/REVIEWED-PROOF.md", + "sha256": "76bea47531ed557c0c9cb210c4324d6000c74415bf13e5747b7036567853f6ef", + "bytes": 7820 + } + ] +} diff --git a/research/nonlinear-monomial-control-ceiling/evidence/proof-review/RESULT.json b/research/nonlinear-monomial-control-ceiling/evidence/proof-review/RESULT.json new file mode 100644 index 0000000000..0eb6ef3ce9 --- /dev/null +++ b/research/nonlinear-monomial-control-ceiling/evidence/proof-review/RESULT.json @@ -0,0 +1,10 @@ +{ + "status": "PASS_WRITTEN_MATHEMATICAL_REVIEW", + "reviewed_proof_sha256": "76bea47531ed557c0c9cb210c4324d6000c74415bf13e5747b7036567853f6ef", + "scientific_source_pin": "4c0a359d88f22f05189d51f48387750b801a45d4", + "scope": "Fixed finite columnwise arbitrary-monomial complete-product support independent of width; outer scalar coefficients may depend on width.", + "conclusion": "At every sufficiently large width satisfying all four memberships, t0_f=0; hence no all-width invertible family.", + "found_and_repaired": "Finite generating alphabet alone does not ensure fixed finite complete-product support for weighted monomials.", + "execution": "No t0 search, enumeration, native, compiler, Lean or numerical checker execution.", + "captured_inputs_verified_unchanged": 8 +} diff --git a/research/nonlinear-monomial-control-ceiling/evidence/proof-review/REVIEW.md b/research/nonlinear-monomial-control-ceiling/evidence/proof-review/REVIEW.md new file mode 100644 index 0000000000..f6db55e8b1 --- /dev/null +++ b/research/nonlinear-monomial-control-ceiling/evidence/proof-review/REVIEW.md @@ -0,0 +1,61 @@ +# Independent review of the nonlinear monomial four-control extension + +**PASS for the revised fixed finite complete-product support theorem.** The support-partition argument excludes an all-width invertible interleaver in the declared class of fixed finite columnwise arbitrary-monomial owner programs. It permits arbitrary nonlinear permutations of the actual one-column address set and arbitrary nonzero complex weights. It does not classify all physical FREE programs, give a paid-query lower bound, or cover changed endpoints or input encodings. + +Reviewed author proof SHA256: `76bea47531ed557c0c9cb210c4324d6000c74415bf13e5747b7036567853f6ef`. Its exact bytes are `frozen/REVIEWED-PROOF.md`; eight byte-preserving captures and the inherited source identities are recorded in `SOURCES.json`. Scientific source: `4c0a359d88f22f05189d51f48387750b801a45d4`, `research/coded-native-caller`. The stated predecessor is PR #5160 head `7800589ce99d932bdb063aafca1f501f1a38626d`; this review relies on the captured predecessor proof and source records, not a fresh public-head verification. + +## 1. A real scope defect was found and repaired + +The first draft's repeated “fixed finite alphabet” description was insufficient for weighted monomial generators if their word lengths/products could vary without bound with width. For example, `diag(2,1)^n` gives infinitely many distinct projective one-column products from one generator. Finite permutations alone do not remove this weighted-product problem. + +The reviewed revision explicitly requires a fixed finite owner program, or a fixed finite set of the **complete one-column operator products** in each proposed control, independent of width. Only the outer scalar coefficients may vary with width. This is the condition needed for the finite tensor-separation argument. The first draft is preserved as `frozen/FIRST-DRAFT.md`; it is not the accepted statement. No counterexample was found to the revised theorem. + +## 2. The support-partition lemma is sound + +Write a monomial map as `h e_y = c_y e_sigma_h(y)`, with every `c_y` nonzero. If `F` is fully supported inside each block of a partition, `Fh e_y` has precisely the block containing `sigma_h(y)` as its support. If `Fh=kF`, the other side has support `sigma_k(P(y))`. A monomial multiplication cannot create cancellation inside these supports. For all `y` in a fixed input block, this equality identifies the same output block; bijectivity gives a block bijection and the same induced block map for h and k. The source coset blocks have equal sizes, as required by the proof's stated application. + +The two-sided form also has the correct order: from `F_out^-1 a F_in=m`, obtain `a F_in=F_out m`. Column supports imply that the monomial address map a takes each input-partition block to an output-partition block. No affine, orthogonal, symplectic, unitarity or finite-phase restriction is used. + +## 3. Join transport is valid; the stronger false invariant is unnecessary + +For an initial letter h, `Q^-1 h Q` monomial gives `h Q=Q m`, so h permutes q-cosets. The equation `D3 h=k D3`, where `k=D3 h D3^-1`, gives that both h and k permute W-cosets. In particular h preserves the join of the q- and W-coset partitions, namely the `(q+W)`-cosets indexed by U. + +Let B be one such join block and tau(B) its h-image. Both `D3` and its inverse operate entirely inside these join blocks. Therefore `D3 h D3^-1` maps the vector space supported in B into the vector space supported in tau(B), even if multiplication creates cancellations. Since k is invertible monomial, its underlying permutation maps B bijectively to tau(B). Thus two addresses in distinct join blocks cannot be sent into a common join block. + +This argument does **not** assert that k preserves q-cosets, fixes the central direction, or belongs to the untransported intersection. It replaces the erroneous stronger transport inference with exactly the join-block condition needed below. Nonlinear address maps and nonconstant weights do not change it. + +## 4. Terminal order and source directions check + +With rightmost operators applied first, the terminal address word is + +`Q^-1 a k b Q^-1`, where `a=K3_i^-1 K3_r` and `b=K1_s^-1 K1_j`. + +If this word is monomial m, then `a k b Q^-1=Q m`. Apply the two-sided support lemma with `F_in=Q^-1`, `F_out=Q`, and monomial intertwiner `a k b`. Both Q and its inverse have complete q-coset support. Hence `a k b` maps q-cosets to q-cosets. + +Pulling those cosets through b and through `a^-1` gives exactly + +`v=G1_s^-1 G1_j e_q=G1_s^-1 q_j`, and `w=G3_r^-1 G3_i e_q=G3_r^-1 q_i`. + +The order of the two inverse factors is correct. Signed weights do not alter these support directions. The first vector is nonzero and lies in `span(q)+U`; the second is nonzero and lies in `span(q)+W`. If v has any U component, a v-pair starts in distinct `(q+W)` blocks while its required w-pair endpoint lies in a single such block. This contradicts the transported join invariant. Over the binary address space the remaining nonzero possibility is `v=e_q`. Distinct source directions then give `s=j`, and the source coefficient contains `A[j,j]=0`. + +Thus **every terminal term with a nonzero scalar coefficient is nonmonomial**. The reasoning does not require `r=i`, projective distinctness of all terminal terms, or an absence of cancellations among nonmonomial terms. + +The inherited exact source premises are the 112-owner invertible scalar A with zero diagonal; `K_i=Z_z P_G` using G rows as rows; `G_i e_q=q_i` with distinct q_i; disjoint q/U/W embedding; and complete nonzero Q/D3 coset support. Source PHYSICAL.md gives the rightmost-first convention, Gaussian entries and columnwise lifting. The Gaussian factors and their inverses have nonzero entries in their full coset blocks. No source-specific absolute phase identity beyond these premises is discarded: in particular Q is not silently treated as an operator involution. + +## 5. The all-width conclusion has the required quantifiers + +For m pairwise nonproportional nonzero operator vectors, tensor powers are independent for every `f>=m-1` (and positive f). For each vector choose m-1 linear functionals killing the other vectors but not that one; tensoring them isolates its tensor power. Extra factors extend to larger f. This proof applies to arbitrary complex matrices, whether monomial or not. + +Take the finite union of all complete products in the four proposed implementations, their relevant conjugates, and the terminal source products. Combine projectively equal matrices first, carrying their scalar factor to its f-th power. Conjugation is projectively injective, so a surviving initial letter whose Q or D3 conjugate is nonmonomial cannot be concealed by a distinct initial projective class. Tensor separation therefore puts each surviving initial class into the hypotheses of Lemma 2, at each sufficiently large width. + +By Lemma 3, all surviving terminal source classes are outside the monomial group. Grouping equal classes or canceling their coefficients can remove these classes but cannot introduce a monomial class. The finite union with the proposed monomial implementation of E is separated at sufficiently large width. Therefore membership of E in that implementation forces `E_f=0` there. The source codes are invertible at every width: each is a conjugate of the scalar owner matrix A, not a tensor power of its owner-space sum. Invertible C1, C3, Q and D3 then give `t_f=0` and `t0_f=0`. + +The conclusion is pointwise at every sufficiently large width satisfying the four memberships. A family supported only at finitely many small widths is not excluded; an all-width invertible family is. No small practical value of the separation threshold is claimed. Actual executable inverse programs need not be inferred from algebraic invertibility because the forward memberships already give the obstruction. + +## 6. Verification boundary and optional discriminator + +This was a written mathematical review with bounded reads of the proof, predecessor and pinned source premises. It did not execute a t0 search, operator enumeration, source/native/compiler/Lean replay, or a new numerical checker. The predecessor source checker was inspected as source-premise evidence; its structural receipt is not an all-width proof. + +An optional bounded exact discriminator can use 16 addresses with bits q,u1,u2,w. The nonlinear monomial map `q -> q XOR(u1*u2)`, with nonzero weight `2^(u1*u2)`, commutes with the central Gaussian and the w Gaussian, exercising both a nonlinear permutation and nonunitary weights. A diagonal `(-1)^(q*w*u1)` phase supplies a transported example where k can fail to preserve q-cosets while retaining the join partition. Exact support and inverses must be checked using the actual Gaussian matrix, not just a binary involution. Such examples can reject accidental affine/unitary/central-fixing restrictions; they do not replace the symbolic proof and are not required for this PASS. + +No external writes were made. Original author and predecessor files were left unchanged. 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diff --git a/research/nonlinear-monomial-control-ceiling/evidence/proof-review/frozen/FIRST-DRAFT.md b/research/nonlinear-monomial-control-ceiling/evidence/proof-review/frozen/FIRST-DRAFT.md new file mode 100644 index 0000000000..a847cb3716 --- /dev/null +++ b/research/nonlinear-monomial-control-ceiling/evidence/proof-review/frozen/FIRST-DRAFT.md @@ -0,0 +1,43 @@ +# Support-partition extension of the released four-control obstruction + +DRAFT — root derivation, not independently reviewed, not published. No paid lower bound or kappa claim. + +## Changed scope and exact predecessor + +The published PR5160 head7800589ce99d932bdb063aafca1f501f1a38626d proves the obstruction for fixed finite columnwise monomial-Clifford/scalar-owner programs. This proposed extension replaces each one-column monomial-Clifford letter by an arbitrary invertible one-column monomial matrix, with arbitrary nonzero complex weights and arbitrary address permutation, still from a fixed finite alphabet lifted columnwise. It does not cover growing alphabets, noncolumnwise or general width-dependent permutations, extra paid Gaussian calls, a changed endpoint, compressed/input-coded targets or a changed source. + +Scientific source remains4c0a359d88f22f05189d51f48387750b801a45d4, research/coded-native-caller. Reuse the authenticated source facts and exact normalizer conventions in outputs/released-operator-intersection-20261023/PROOF.md and its source checker. There are112owners; A is invertible with zero diagonal. K1s/K3r are actual signed linear permutations supported on q+U/q+W, G_s e_q=q_s, and distinct sourceq_s make G_s^-1 q_j=e_q iff s=j. Q and D3 are supported on q and W respectively, with every entry inside their respective coset blocks nonzero. These are inherited source premises, not newly machine-checked here. + +## Lemma1 — a monomial intertwiner preserves support partitions + +Let F be an invertible matrix whose support is exactly a disjoint union of complete square blocks indexed by a partition P of the computational basis. If h and k are invertible monomial matrices and F h = k F, then their address permutations both map every P-block bijectively to a P-block, with the same induced permutation of blocks. + +For a column y, support(F h e_y) is the entire P-block containing sigma_h(y). Support(k F e_y) is sigma_k(P(y)). These must be equal. If y,y' belong to the same P-block, the latter set is unchanged, so sigma_h sends that block into a single block. Invertibility and the equal finite block sizes force equality. The same identity supplies the assertion for sigma_k. No phase restriction is used; multiplication by a monomial matrix creates no internal cancellation. + +For a two-sided variant F_out^-1 a F_in monomial, where F_in/F_out have complete equal-size coset blocks (possibly different partitions), write a F_in=F_out m and use the same column-support argument. The address permutation of a maps input blocks to output blocks. This variant uses no Clifford/symplectic classification. + +## Lemma2 — transported letters preserve the required partitions + +Let H be the group of all invertible monomial matrices on the actual17one-column bits. Let h in H satisfy Q^-1 h Q in H and k=D3 h D3^-1 in H. From h Q=Q(Q^-1 h Q), Lemma1 says sigma_h preserves q-cosets. From D3 h=k D3, both sigma_h and sigma_k preserve W-cosets. + +The join of the q- and W-coset partitions is the (q+W)-coset partition, indexed by U coordinates. An invertible permutation preserving both component partitions preserves their join, hence sigma_h preserves(q+W)-cosets. D3 and D3^-1 act inside these join blocks. Therefore k=D3 h D3^-1 has no nonzero matrix entries between an input join block and any output join block other than h's induced image. Because k is monomial, sigma_k also preserves the join partition. + +Consequently, sigma_k cannot map a pair lying in different(q+W)-cosets to a pair in the same(q+W)-coset. + +## Lemma3 — every nonzero terminal source term remains nonmonomial + +For the same four-control equations as PR5160, let a=K3i^-1 K3r and b=K1s^-1 K1j. A terminal term has address operator Q^-1 a k b Q^-1, coefficient A[i,r] b[r,s,k] A[s,j]. If this operator were monomial, write a k b Q^-1=Q m. The two-sided support lemma implies a k b maps q-cosets to q-cosets. Since a,b are invertible signed linear permutations, sigma_k must map cosets of v=G1s^-1 q_j to cosets of w=G3r^-1 q_i. + +Here v belongs to span(q)+U and w belongs to span(q)+W. A w-coset consists of two points in one(q+W)-coset. If v has a nonzero U component, every v-coset has its two points in different(q+W)-cosets. This contradicts Lemma2. Thus v=e_q, so s=j by the actual distinct source directions. But then A[s,j]=A[j,j]=0. Every terminal term with nonzero source coefficient is therefore nonmonomial. This does not require r=i and does not assume the invalid transported central equality from the old draft. + +## All-width fixed-alphabet conclusion + +Use the elementary tensor-separation lemma from PR5160: tensor powers of a finite collection of pairwise nonproportional nonzero operator vectors are independent for sufficiently large f. It applies to arbitrary matrices, not only Clifford operators. + +For any fixed finite columnwise owner program, combine all projectively equal letters. Requiring t0, Q^-1t0Q and D3t0D3^-1 to have finite monomial expansions forces every surviving one-column letter h of t0 into the exact intersections in Lemma2 for all sufficiently large widths. Coefficients may depend on f; the alphabet remains fixed independently of f. Lemma3 places every surviving terminal E term outside H. Cancellation may remove outside-H classes but cannot produce an H class. Separating the finite combined set against any proposed finite monomial program for E gives E_f=0 for all sufficiently large widths. + +The actual C1,C3 and Q are invertible, so E=Q^-1 C3 t C1 Q^-1=0 implies t=0 and t0=0. Hence an all-width invertible t0 is impossible in this enlarged fixed finite columnwise arbitrary-monomial class. No paid Gaussian query lower bound follows. Arbitrary width-dependent/growing/noncolumnwise FREE programs and altered four-control/target contracts remain outside this theorem. + +## Verification and credit still required + +Independent proof review must check the partition argument, the exact matrix order of every intertwiner and the source support conventions. A small exact support checker with nonaffine examples can discriminate the extension but cannot prove the all-width theorem. No operator alphabet enumeration or native/compiler replay is proposed. Primary credit: Dillon released equations, Dugongue/mdumitrean source A, real normalizer authors and Cinna Davis/1337isnot1337, Cao Yifan/intqwq, zxv and retained helper/Clifford/compiler lineage. 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+ ], + "recoding_bill": {"forward_prefix_component_l1_upper":57,"forward_update_coefficient":"-1/4","inverse_output_prefix_component_l1_upper":"69/13","inverse_prefix_component_l1_upper":56,"inverse_update_coefficient":"-1/52","monomial_field_maps":224,"output_scaled_adds":448,"owned_fields":8,"owned_initial_clears":8,"owned_terminal_clears":8,"signed_mode_adds":448}, + "schema": "coded-native-real-normalizers/1", + "scope": "Exact physical central ABI and local gauges; not full-controller or multiplication admission.", + "source_payload_sha256": "cfcd36b40326d7a5815e1db5313444379b3fb756b27d5201c561574c085cc0dd", + "status": "PASS_CODED_NATIVE_REAL_NORMALIZERS" +} diff --git a/research/nonlinear-monomial-control-ceiling/evidence/proof-review/frozen/PHYSICAL.md b/research/nonlinear-monomial-control-ceiling/evidence/proof-review/frozen/PHYSICAL.md new file mode 100644 index 0000000000..da8837ba67 --- /dev/null +++ b/research/nonlinear-monomial-control-ceiling/evidence/proof-review/frozen/PHYSICAL.md @@ -0,0 +1,71 @@ +# Real physical normalizers for the coded producer + +This is local constructive research on canonical/pr4929, not a completed supplier or exponent claim. Source code and Gaussian conventions retain the original authors' credit. + +Let n=9 and let q be any of the source's 112 port vectors (weight 3 or 7). Let e0=1. Construct an orthogonal binary matrix G taking e0 to q. If q_0=0, use the orthogonal transvection T_(e0+q). If q_0=1, select j>0 with q_j=0 and use T_(ej+q) T_(e0+ej). Here T_v(x)=x+(v dot x)v; both v have even weight, hence T_v is orthogonal. Such j exists because q is not the all-ones vector. + +For row j of G put z_j=(wt(row_j G)-1)/2 modulo 2. Define K=Z_z P_G. It is a real signed permutation. The full Gaussian R has entries alpha*(-i)^wt(x+y), with alpha=(1+i)^9/2^9. Orthogonality gives + +wt(G^-1 d)-wt(d) = 2 z dot d (mod 4). + +Indeed the degree-two coefficients vanish modulo four because distinct columns of G^-1 have even dot product; the linear coefficient is the weight of that column minus one. Therefore K R K^-1=R exactly, including absolute phase. K also maps the central line to span(q), and its orthogonal complement to q-perp, at the canonical-frame Lagrangian level. All these statements tensor to every positive digit-column count. + +Let Q be the central C gate, D the product of the other eight C gates, so R=DQ. Let E_i and S_i be the actual retained entrance-line and exit-hyperplane frame operators. With the operator convention that rightmost applies first, use + +input X: E_i K_i Q^-1 +input Y: K_i +output X: K_i^-1 +output Y: D K_i^-1 S_i^-1. + +All four wrappers are monomial. Their Lagrangian incidences prove this for every port, and the scratch verifier checks exact tableaus and absolute phases at every one of the 112 source ports. With A from PR4929, define C_ij= A_ij K_i^-1 K_j. Then the full main endpoint is + +F = diag(D,D) [[I,0],[C Q^-1,I]]. + +C is a physically implementable fixed owner recoding: apply K_i to each owner, apply A among the owners at common physical coordinates, and apply K_i^-1. Its inverse uses A^-1=(A+12I)/13. Since each K_i is real orthogonal and A is real symmetric, C^T=C. Since K_i commutes with R, C commutes with R. The polynomial C^2+12C-13I=0 follows by similarity. The 408 independent helper jobs and every paid Gaussian child are unchanged. + +For the actual raw inverse transpose, F^-T=diag(D^-1,D^-1) [[I,-Q^-1 C],[0,I]], since Q,D are symmetric. Multiply both outputs by D^2, then conjugate the physical X bank by Q^2 (input Q^-2 and output Q^2). These are explicit monomial powers. The result is + +B = diag(D,D) [[I,-Q C],[0,I]]. + +The old source convention using Y-only Q^2 conjugation cannot simply be copied for noncommuting C. The X-only convention above is exact. + +For a prior code B0 commuting with the current D, set G(B0)=diag(B0,Q^-1 B0 Q). The following two constructive local identities hold without requiring C and Q to commute: + +(post X C) F (pre Y C) = G(C B0) F_standard G(B0)^-1, +(post X C^-1) B (pre Y C^-1) = G(C^-1 B0) B_standard G(B0)^-1. + +They use [C,DQ]=0 and [B0,D]=0. Every additional operation on the left is physically cheap fixed recoding. For disjoint fresh blocks, codes from earlier stages commute with the current D. These identities do not yet settle interstage ADD/IDLE compatibility, terminal gauges, productive jobs, or the complete moment. + +## Literal recoding and complete local bill + +The source code has an exact factorization A=I-MM^T/4 with M a 112-by-8 matrix having entries 0,+1,-1 and M^T M=56I. Each mode column has 56 nonzero entries, and each owner row has four. The mode columns are emitted in NORMALIZERS.json and all 12,544 entries are checked against the pinned CODE.json. This factorization is used here as a constructive scalar program, without relying on any common physical address chart for separate Gaussian mode transformations. + +To execute A on arbitrary independent owner fields x, allocate eight owned zero fields t. Compute t=M^T x with 448 signed ADDs; update x-=M t/4 with 448 scaled ADDs; clear the eight owned temporaries. Reading x is finished before any x is updated. Hence no input field is overwritten before its last source read. For A^-1 use the same program with coefficient -1/52, since A^-1=I-MM^T/52. Both are exact rational linear programs. The helper fields of the native producer are never used as scratch, copied as free credit, or erased. + +Conjugating this scalar program by the 112 K_i and their inverses implements C or C^-1. This pays 224 monomial field maps, 896 scalar ADDs and 16 owned CLEARs per recoding, with eight owned fields reusable sequentially. The two additional recodings in either local gauge identity therefore cost at most 448 monomial maps, 1,792 scalar ADDs and 32 CLEARs, with no recursive Gaussian child. The four actual port normalizers contribute 448 monomial maps per native invocation. The displayed backward conversion contributes at most 448 further monomial maps (112 input X powers, 112 output X powers and 224 output D-squares); fusing maps can reduce this upper bound but is not assumed. + +These are field-level operation counts. A lifted K_i is an explicit binary map on nine f-bit address blocks together with signs given by block parity. Its constants, indexed movement and phase passes use the retained native monomial implementation. It is therefore a fixed finite O(V polylog p) nonrecursive operation under that caller's precision/address ABI, not a zero-work instruction. Each fixed rational ADD is likewise charged at its actual scalar precision. In particular the inverse denominator 52 must be implemented and rounded; no free exact real-division oracle is assumed. This lemma makes no independent TM tariff or outer-budget claim. + +The maximum exact component-l1 prefix bound for the scalar A program, including its temporaries, is 57. For A^-1 it is 56 (the output-only bound is 69/13). Signed monomial maps preserve component magnitude. A fixed local sequence has a fixed suffix bound obtained by multiplying elementary bounds 1+|coefficient|; owned CLEAR has bound one. This gives a finite bound for local rounding propagation. The number of local uses, compiler precision and complete outer error allocation still belong to the caller theorem. + +The normalized producer still has its exact PR4929 paid profile + + {1:2201, 2:412, 3:510, 4:127, 5:40} + +and independent helper profile + + {4:28, 7:7, 8:47, 9:326}. + +No productive main ranks are asserted merely from the coupled central ABI. Those depend on the completed caller, its initial/final gauges and every paid correction. + +## Universal scope and reproducible evidence + +PR4929 supplies the arbitrary-input logical identity over Q[FreeGroup72] for its emitted center-free word with ground-chart helper pre-compensation. Its normalized main endpoint is L(A), its raw dual is L(A)^-T, and every one of the 408 helper endpoints is autonomous. This proof is an inherited premise with exact source pins, not a consequence of this normalizer checker. The present wrappers and owner recodings act exclusively on the 224 main fields; new scratch is explicitly owned. Thus the helper block remains autonomous and its independent inputs, requested transforms and paid profile remain unchanged. + +For every f>=1, assign the shared native letters their actual consistent invertible lifted operators. Evaluation of the source free-group algebra is a homomorphism and gives the source logical identity. Independently lift each K_i as its f-fold tensor product, retaining spectator coordinates; its signed-permutation word, the two direct frame identities and its commutation with R lift by operator composition. In every compiled monomial word, P and X act columnwise on the f-bit blocks, D sums its Z4 polynomial over the f columns, and a one-column scalar phase i^gamma becomes i^(f*gamma). Reusing the one-column global phase unchanged would be incorrect. Finally form the owner matrix C_f with entries A_ij K_i,f^-1 K_j,f. One does not tensor the whole 224-by-224 owner matrix and does not assert that tensor powers preserve sums. Instead the finite scalar owner program is evaluated directly in the algebra of these lifted address operators. This proves the central ABI and local gauge identities at every positive width. + +Run `python3 -I -B research/coded-native-caller/normalizers.py` or its optimized `-O` form. The checker authenticates the decompressed actual native payload, CODE.json and retained Clifford source before executing the latter. It verifies all 112 real signed normalizers, 57,344 explicit weight-congruence addresses, 896 forward/backward monomial wrappers and eight exact frame/block identities per port, including absolute phase. Each emitted wrapper is separately lowered to a literal P/D/X/phase-only word and checked against its defining expression by exact tableau and absolute-phase state. No C gate remains in an emitted wrapper. The independent incoming-column and outgoing-row identities imply all 5,824 nonzero off-diagonal code blocks, without sampling those pairs. It also executes the eight-mode recoding on all 112 arbitrary-input basis columns in both directions. Omitting the essential signed phase is rejected at every affected port. The default command compares a regenerated mathematical receipt and performs no writes; `--write` is the explicit authoring path. + +Source pins: canonical main 624de9dc66858af0945e5f8a0942488d62280d43, `research/kappa-round3-warp-20261022/ACTUAL-WORD.json.gz`, decompressed SHA-256 cfcd36b40326d7a5815e1db5313444379b3fb756b27d5201c561574c085cc0dd; PR4929 head 361f5d4100640177d6a983dd2f336021c75bfd5e, `research/native-center-free-code/CODE.json`, SHA-256 8be7f3f073b7981ec91c6af43edff0ac8a89f4824ef6026ae758c287dd6c03d0; canonical `research/t74-native-fast/vendor/native/new/code/borrow/vendor/clifford/clifford_exact.py`, SHA-256 2982ead07721e83358dc67c6f49c2b4c16a236a6bebdfd72f58221be1f5d0962. Gzip container bytes are not substituted for decompressed source identity. + +Credit Dugongue's PR4929 for the coded producer and stable code; Cinna Davis's PR4694/4698 and the original cache/helper construction; Cao Yifan's PR4610 native word; the retained zxv, Jacob Alan Sussman, Cinna Davis and Clifford/native/compiler lineage. The exact canonical-frame implementation and phase conventions are retained source material. The constructive real centralizing normalizers, their explicit lifted words, X-only raw-dual convention and costed local code gauges are the additions here. New work is Apache-2.0 and has substantial OpenAI Codex assistance; no formal-kernel or independent human acceptance is asserted. diff --git a/research/nonlinear-monomial-control-ceiling/evidence/proof-review/frozen/PREDECESSOR-PROOF.md b/research/nonlinear-monomial-control-ceiling/evidence/proof-review/frozen/PREDECESSOR-PROOF.md new file mode 100644 index 0000000000..08b35b275a --- /dev/null +++ b/research/nonlinear-monomial-control-ceiling/evidence/proof-review/frozen/PREDECESSOR-PROOF.md @@ -0,0 +1,256 @@ +# A source-bound obstruction for the explicit Clifford control class + +Status: local derivation awaiting separate review. No public theorem, solver, +native/emitter/compiler/Lean replay, complete caller, or new kappa is claimed. + +This answers a precise part of Dillon's released two-FULL/one-query handoff, +Discussion 4930 comment 19143357. The new bounded investigation is rooted at +comment 19143614. The public handoff is a sufficient construction target; no +successful t0 was supplied. Its physical FREE class is not fully enumerated. +The theorem below explicitly restricts that class to fixed finite columnwise +monomial Clifford words and scalar owner programs. It does not silently replace +physical FREE by parity-even operators or by every unit of an abstract algebra. + +## Pinned input + +The scientific source is canonical commit +`4c0a359d88f22f05189d51f48387750b801a45d4`, package +`research/coded-native-caller`. Exact retained files and hashes are in +`SOURCES.json` and `frozen/`. Governing guidance was read at canonical +`20a1e31bc6009b9b3f4d084bbaadc6350e34fcba`. + +There are 112 owners. The real scalar code A is invertible, has zero diagonal, +and has 52 nonzero entries in every row and column. Each source normalizer is +K_i = Z_(z_i) P_(G_i), with G_i an orthogonal binary 9 by 9 matrix and +G_i e_q = q_i. The 112 vectors q_i are distinct, of weight 3 or 7. These exact +properties were checked directly from CODE.json and NORMALIZERS.json; the +check did not invoke the source operator evaluator. + +Embed one copy K_(1,i) on the central axis q and the eight first-stage axes U, +and another K_(3,i) on q and the eight third-stage axes W. The full address +space has exactly 17 axes. The sets {q}, U, W are disjoint. G rows are used as +rows, not columns. Every operator is lifted columnwise; scalar owner matrices +are applied once, rather than tensoring an owner sum. + +Let Q be the central Gaussian, D3 the product of the eight third-stage +Gaussians, and + + C_s[i,j] = A[i,j] K_(s,i)^-1 K_(s,j). + +Dillon's four required controls are + + t0, Q^-1 t0 Q, t = D3 t0 D3^-1, + E = Q^-1 C3 t C1 Q^-1, + +including their explicitly implementable inverses. The main endpoint is + + F3 J(t0) F1 = [[0,t],[-E,0]] diag(G,G), + G = D1 D3 Q. + +## Exact scope of the obstruction + +Let H be the projective group of monomial Clifford operators on the actual +17 one-column address bits. A control in the tested class is a fixed finite +scalar owner program in columnwise lifts h^(tensor f), h in H. Owned copies, +temporary fields and clears are allowed as long as the resulting main block +has such a finite linear expansion. Coefficients may depend on f; the set of +letters is fixed independently of f. Projective phases are absorbed into the +scalar coefficients, including their correct f dependence. + +This is a generous algebraic model of the source's explicit P/D/X/phase +Clifford words and owner ADD programs. It gives no automatic physical tariff +or inverse implementation to an arbitrary algebra element. The completed +obstruction needs neither tariff nor inverse sufficiency: it rules out every +nonzero t0 in this declared class satisfying the four memberships at all +sufficiently large f. Section 4 also gives a direct scalar-owner proof which +does not require the transported-class argument. + +It does not cover arbitrary nonlinear monomial address maps, general +non-Clifford phases, or a number/alphabet of letters growing with f. It also +does not cover a changed FULL controller, extra paid calls, extra axes, a +different productive endpoint, or a different source code. + +## 1. Why all-width membership becomes support membership + +For any finite collection of pairwise nonproportional nonzero operator +vectors v_1,...,v_m, their tensor powers are linearly independent for every +f >= m-1. For each i and j != i choose a linear functional ell_(i,j) that +kills v_j but not v_i. The tensor product of these m-1 functionals kills +every v_j^(tensor(m-1)) except i. Additional factors nonzero on v_i extend +the argument to any larger f. This elementary argument also allows +width-dependent scalar coefficients. + +Apply the lemma to the finite projective Clifford alphabet containing all +letters in the fixed proposed programs and their conjugates. If t0 and its +Q and D3 conjugates are all represented by monomial Clifford programs at +every positive width, then at sufficiently large width every surviving +letter of t0 lies in + + H0 = H intersect Q H Q^-1 intersect D3^-1 H D3. + +Conjugate t0 by D3. Its support is in Ht = D3 H0 D3^-1. The following +observations do not identify Ht with H0: + +* t is monomial by hypothesis; +* D3^-1 t D3 = t0 is monomial, and conjugation by D3^2 shows that + D3 t D3^-1 is monomial as well; +* Q^-1 t Q = D3 (Q^-1 t0 Q) D3^-1 is not automatically monomial from + those statements alone. + +The last bullet is an essential warning: it cannot be inferred merely from +the three displayed controls. Consequently, the proof below uses the exact +support class for t, not an unjustified assertion that t is again in H0. + +Define instead + + Ht = D3 H0 D3^-1. + +Every surviving letter of t lies in Ht. The next section derives the actual +invariants of this transported class directly. + +## 2. Lagrangian invariants, including the D3 transport + +Use binary symplectic coordinates (x,z), with the computational Lagrangian +L0 = {(0,z)}. Let Lq = Q L0 and Lw = D3 L0. Their X projections are +span(e_q) and W respectively. Q and D3 have order two on binary symplectic +space even though their operator squares retain phases/Paulis. + +A monomial Clifford h has symplectic form + + x' = R x, + z' = S x + R^-T z, + +where R is invertible. For h in H0, it preserves L0, Lq and Lw. +Therefore R e_q = e_q and R W = W. This is the source-fixed flag constraint; +it does not identify every parity-even word with a physical free operation. + +For a transported letter k = D3 h D3^-1 in Ht: + +* k preserves L0 because h preserves Lw; +* k preserves Lw because h preserves L0 and D3^2 L0 = L0; +* k preserves Lqw = D3 Q L0 because h preserves Lq. + +Thus its invertible X block R_k preserves W. The X projection of Lqw is +span(e_q) plus W. Hence R_k preserves span(e_q) plus W as well. On the +quotient by W, R_k therefore fixes the nonzero central class [e_q]. +This is weaker than R_k e_q=e_q, and is exactly the distinction required +by the released warning about transport through D3. + +For nonzero v in span(e_q) plus U and nonzero w in span(e_q) plus W, define the rank-one +Gaussian chart L(v) = {(v v^T z,z)}. If k L(v) = L(w), its X projections +give R_k v = w. Passing to the quotient by W and using invertibility and +the fixed central class implies v=e_q. It does NOT imply w=e_q: w may +equal e_q plus a vector in W. + +This one-sided conclusion is sufficient for a useful obstruction below. + +## 3. No monomial endpoint component + +Expand an endpoint block E[i,j]. Every term is a scalar multiple of + + Q^-1 K_(3,i)^-1 K_(3,r) k K_(1,s)^-1 K_(1,j) Q^-1, + k in Ht, + +with scalar A[i,r] b[r,s,k] A[s,j], where b are the coefficients of t. + +For this word to be monomial, k must carry + + L(G_(1,s)^-1 q_j) to L(G_(3,r)^-1 q_i). + +Section 2 forces G_(1,s)^-1 q_j=e_q, so s=j by distinctness of the source +q vectors. But A[j,j]=0. Every term with a nonzero source coefficient is +therefore outside H. This conclusion does not require r=i. + +No uniqueness assertion for different intermediate owners is needed. +Group together any projectively equal nonmonomial terms. Their sum still +has support outside H; cancellation can remove classes but cannot create +a class in H. At all sufficiently large widths, section 1 makes the span +of these remaining nonmonomial classes disjoint from the span of the +finite monomial alphabet in any proposed FREE implementation of E. + +Consequently, if E is also FREE, E=0 at every such width. The exact source +codes C1,C3 and Q are invertible, so E=Q^-1 C3 t C1 Q^-1=0 implies t=0, +then t0=0. In particular the handoff's required invertible t0 cannot exist +inside this fixed finite columnwise monomial-Clifford control class. + +This result does not mistake Ht for H0. In fact Ht permits R_k e_q=e_q+w +with nonzero w in W. The proof needs only the induced quotient action and +the resulting one-sided equality s=j. It also does not need a uniform +inverse bound: the source code inverses are exact, and all four requested +inverse programs are rendered moot by the forward obstruction. + +For arbitrary width-dependent scalar coefficients, the exact conclusion +is t0_f=0 at every sufficiently large width satisfying the four +memberships. A family nonzero only at finitely many small widths is not +excluded, but cannot satisfy the all-width invertibility contract. + +## 4. Completed scoped theorem: scalar owner interleavers + +For t0 = B tensor I, all first three controls equal B tensor I and t=t0. +Its inverses, when supplied by an explicit scalar program, are physically +legal in the same scalar class. Each term of E[i,j] is now + + A[i,r] B[r,s] A[s,j] + * Q^-1 K_(3,i)^-1 K_(3,r) K_(1,s)^-1 K_(1,j) Q^-1. + +The signed linear permutation between the two Q^-1 factors has binary map + + M = G_(3,i)^-1 G_(3,r) G_(1,s)^-1 G_(1,j). + +For a signed orthogonal permutation M, Q^-1 M Q^-1 is monomial exactly +when M e_q=e_q. If the directions differ, the two central Gaussian factors +produce four distinct translations, all with nonzero coefficient; if they +agree, their equal or opposite signs give a single monomial. This includes +absolute-sign choices and does not tensor a scalar sum. + +Here M e_q=e_q is equivalent to + + G_(3,r) G_(1,s)^-1 q_j = q_i. + +The first-stage component can vanish only if G_(1,s)^-1 q_j=e_q, +so s=j; then distinct third-stage q vectors force r=i. Both source diagonal +entries are zero. Thus every nonzero term is nonmonomial. + +For fixed i,j, terms indexed by distinct pairs (r,s) are projectively +distinct. Equality would imply + + G_(3,r')^-1 G_(3,r) = G_(1,s')^-1 G_(1,s). + +The left map is supported on q+W and the right on q+U. Their intersection +contains only the identity binary linear map: both must fix U and W, and +an invertible map on the single common bit fixes q. Distinct q vectors then +give r=r' and s=s'. Conjugating by the fixed outside factors preserves +projective distinctness. Signed phases cannot change this argument. + +For any nonzero B[r,s], choose i with A[i,r] nonzero and j with A[s,j] +nonzero (there are 52 choices each). The corresponding nonmonomial term has +nonzero coefficient and cannot cancel. Uniform tensor separation against +any proposed fixed finite columnwise monomial alphabet rules out a FREE +representation of E at all sufficiently large widths. + +Hence no fixed nonzero scalar owner matrix B, including a dense one, +satisfies the fourth condition uniformly in that model. If coefficients +depend on f, the precise conclusion is B_f=0 at every sufficiently large +width where the endpoint belongs to the declared monomial span. This +excludes every all-width invertible family B_f; it does not exclude a family +supported only at finitely many small widths. +This theorem does not rely on classifying the whole physical FREE algebra. + +## Credit and open next question + +Credit Dillon for the released construction equations, the real normalizers +and their charged recoding; Dugongue/mdumitrean for PR4929 and A; Cao Yifan +for the native word; Cinna Davis, zxv, Jacob Alan Sussman, Ryanson Jonathan, +and the full retained native/helper/Clifford/compiler lineage. The elementary +uniform tensor-separation device is already used in the preceding projective +operator work; it is not claimed as a new theorem. New local analysis has +substantial OpenAI Codex assistance and no independent human or kernel +acceptance. + +The useful continuation must change at least one declared hypothesis: +the fixed columnwise monomial-Clifford control class, the sufficient four +FREE conditions (replacing some with an explicitly paid and affordable +operator), the source code, the productive target, or the controller. +Merely adding more Clifford owner/address couplings cannot create a +monomial component. A test that assumes R_k e_q=e_q after D3 transport +repeats the released error and must be rejected. diff --git a/research/nonlinear-monomial-control-ceiling/evidence/proof-review/frozen/PREDECESSOR-SOURCE-CHECKER.py b/research/nonlinear-monomial-control-ceiling/evidence/proof-review/frozen/PREDECESSOR-SOURCE-CHECKER.py new file mode 100644 index 0000000000..9b9f57090d --- /dev/null +++ b/research/nonlinear-monomial-control-ceiling/evidence/proof-review/frozen/PREDECESSOR-SOURCE-CHECKER.py @@ -0,0 +1,159 @@ +#!/usr/bin/env python3 +"""Read-only source/flag checks, not a universal operator theorem checker.""" +import hashlib +import json +from fractions import Fraction +from pathlib import Path + +ROOT = Path(__file__).resolve().parent + + +def require(condition, message): + if not condition: + raise RuntimeError(message) + + +def linear(rows, vector): + return sum(((row & vector).bit_count() & 1) << j + for j, row in enumerate(rows)) + + +def image(columns, vector): + answer = 0 + while vector: + low = vector & -vector + answer ^= columns[low.bit_length() - 1] + vector ^= low + return answer + + +def compose(left, right): + return [image(left, column) for column in right] + + +def span_basis(vectors): + pivots = {} + for vector in vectors: + while vector: + p = vector.bit_length() - 1 + if p in pivots: + vector ^= pivots[p] + else: + pivots[p] = vector + break + return pivots + + +def belongs(vector, pivots): + while vector: + p = vector.bit_length() - 1 + if p not in pivots: + return False + vector ^= pivots[p] + return True + + +def preserves(operator, space): + pivots = span_basis(space) + return all(belongs(image(operator, vector), pivots) for vector in space) + + +def main(): + manifest = json.loads((ROOT / 'SOURCES.json').read_text()) + for item in manifest['files']: + path = ROOT / 'frozen' / Path(item['path']).name + require(hashlib.sha256(path.read_bytes()).hexdigest() == item['sha256'], + 'Frozen source hash mismatch: ' + str(path)) + code = json.loads((ROOT / 'frozen/CODE.json').read_text()) + normalizers = json.loads((ROOT / 'frozen/NORMALIZERS.json').read_text()) + matrix = [[Fraction(x) for x in row] for row in code['A']] + ports = normalizers['ports'] + require(len(matrix) == 112 and all(len(row) == 112 for row in matrix), + 'Owner matrix shape changed') + require(len(ports) == 112, 'Port inventory changed') + qs = [linear(p['G_rows'], 1) for p in ports] + checks = { + 'zero_diagonal': all(matrix[i][i] == 0 for i in range(112)), + 'row_and_column_nonzero_counts_52': all( + sum(x != 0 for x in row) == 52 for row in matrix) and all( + sum(matrix[i][j] != 0 for i in range(112)) == 52 + for j in range(112)), + '112_distinct_q': len(set(qs)) == 112, + 'G_rows_match_literal_q': all(q == p['q'] for q, p in zip(qs, ports)), + 'q_weight_3_or_7': all(q.bit_count() in (3, 7) for q in qs), + 'all_G_orthogonal': all( + ((p['G_rows'][i] & p['G_rows'][j]).bit_count() & 1) == (i == j) + for p in ports for i in range(9) for j in range(9)), + 'A2_plus_12A_equals_13I': all( + sum(matrix[i][k] * matrix[k][j] for k in range(112)) + + 12 * matrix[i][j] == (13 if i == j else 0) + for i in range(112) for j in range(112)), + } + require(all(checks.values()), 'Source structural check failed') + + # Binary symplectic witness on the actual 17 axes; global phases are + # deliberately outside this finite flag check's scope. + n = 17 + q, w = 0, 9 + xmask = (1 << n) - 1 + ident = [1 << j for j in range(2*n)] + Q = ident.copy() + Q[n + q] ^= 1 << q + D = ident.copy() + for j in range(9, 17): + D[n + j] ^= 1 << j + H = ident.copy() # CZ(q,w) + H[q] ^= 1 << (n + w) + H[w] ^= 1 << (n + q) + K = compose(D, compose(H, D)) + L0 = [1 << (n+j) for j in range(n)] + Lq = [image(Q, v) for v in L0] + Lw = [image(D, v) for v in L0] + Lqw = [image(D, v) for v in Lq] + monomial = lambda op: all((image(op, v) & xmask) == 0 for v in L0) + flag = { + 'h_CZ_in_H0': all(preserves(H, space) for space in (L0, Lq, Lw)), + 'Qinv_h_Q_monomial': monomial(compose(Q, compose(H, Q))), + 'D3_h_D3inv_monomial': monomial(K), + 'k_preserves_correct_transported_flags': all( + preserves(K, space) for space in (L0, Lw, Lqw)), + 'k_central_X_image_is_q_plus_w': + (image(K, 1 << q) & xmask) == ((1 << q) | (1 << w)), + 'k_does_not_preserve_Lq': not preserves(K, Lq), + 'Qinv_k_Q_not_monomial': not monomial(compose(Q, compose(K, Q))), + 'quotient_flag_excludes_noncentral_U_vectors': all( + (image(K, v) & ((1 << 9)-2)) != 0 + for v in range(1, 1 << 9) if v != 1), + } + require(all(flag.values()), 'Transported-flag witness failed') + corrupted_qs = qs.copy() + corrupted_qs[1] = corrupted_qs[0] + bad_G = ports[0]['G_rows'].copy() + bad_G[0] ^= 1 + controls = { + 'reject_nonzero_code_diagonal': not all( + (Fraction(1) if i == 0 else matrix[i][i]) == 0 for i in range(112)), + 'reject_duplicated_port_direction': len(set(corrupted_qs)) != 112, + 'reject_corrupted_orthogonal_map': not all( + ((bad_G[i] & bad_G[j]).bit_count() & 1) == (i == j) + for i in range(9) for j in range(9)), + 'reject_untransported_H0_substitution': not preserves(K, Lq), + 'reject_central_shift_erasure': + (image(K, 1 << q) & xmask) != (1 << q), + } + require(all(controls.values()), 'A rejection control failed') + print(json.dumps({ + 'status': 'PASS_SOURCE_AND_TRANSPORTED_FLAG_ONLY', + 'scope': 'Exact pinned source structure and one 17-axis binary symplectic ' + 'transport witness; not an all-width theorem replay, phase ' + 'checker, t0 search, native/compiler replay, or supplier admission.', + 'source_checks': checks, + 'transported_flag_witness': flag, + 'rejection_controls': controls, + 'axis_embedding': {'central': 0, 'stage1': list(range(1, 9)), + 'stage3': list(range(9, 17))}, + }, indent=2, sort_keys=True)) + + +if __name__ == '__main__': + main() diff --git a/research/nonlinear-monomial-control-ceiling/evidence/proof-review/frozen/PREDECESSOR-SOURCES.json b/research/nonlinear-monomial-control-ceiling/evidence/proof-review/frozen/PREDECESSOR-SOURCES.json new file mode 100644 index 0000000000..ef42d1e3d1 --- /dev/null +++ b/research/nonlinear-monomial-control-ceiling/evidence/proof-review/frozen/PREDECESSOR-SOURCES.json @@ -0,0 +1,36 @@ +{ + "canonical_gate_pin": "20a1e31bc6009b9b3f4d084bbaadc6350e34fcba", + "scientific_source_pin": "4c0a359d88f22f05189d51f48387750b801a45d4", + "files": [ + { + "commit": "4c0a359d88f22f05189d51f48387750b801a45d4", + "path": "research/coded-native-caller/CODE.json", + "git_blob": "03170f4a31f7a8a3db7228f1cd5a6fdcb32720ae", + "sha256": "8be7f3f073b7981ec91c6af43edff0ac8a89f4824ef6026ae758c287dd6c03d0" + }, + { + "commit": "4c0a359d88f22f05189d51f48387750b801a45d4", + "path": "research/coded-native-caller/NORMALIZERS.json", + "git_blob": "bca92c222c395d3d75c047fbb678f3cd0947c60f", + "sha256": "ae8aad06aa5d1811d725895b3ab5a4f4daa9dcdd88294aa0d5f723e78bb1e0da" + }, + { + "commit": "4c0a359d88f22f05189d51f48387750b801a45d4", + "path": "research/coded-native-caller/PHYSICAL.md", + "git_blob": "da8837ba67423ef08ef1afb8e8297a8032aafe9f", + "sha256": "4321b159ff8d3fe705419adaaa1beef75606e869e34341bda553f17d598a6078" + }, + { + "commit": "4c0a359d88f22f05189d51f48387750b801a45d4", + "path": "research/coded-native-caller/ORIGINS.json", + "git_blob": "e18a033d4de1d997f55e84d1535fc6cde20424c4", + "sha256": "10cf47173ca238c32adb923daae7c1137c2e9c86c7066301d68fc74442b3f4d7" + }, + { + "commit": "4c0a359d88f22f05189d51f48387750b801a45d4", + "path": "research/coded-native-caller/NOTICE", + "git_blob": "985b6b1bc76dacd54a9cbe9897f3b659ae93c3a4", + "sha256": "317b1c5e040bab50ebfc3d0cdb3a47fc30e2f872a00895f563d7931749b1cc7b" + } + ] +} diff --git a/research/nonlinear-monomial-control-ceiling/evidence/proof-review/frozen/REVIEWED-PROOF.md b/research/nonlinear-monomial-control-ceiling/evidence/proof-review/frozen/REVIEWED-PROOF.md new file mode 100644 index 0000000000..277a32368a --- /dev/null +++ b/research/nonlinear-monomial-control-ceiling/evidence/proof-review/frozen/REVIEWED-PROOF.md @@ -0,0 +1,48 @@ +# Support-partition extension of the released four-control obstruction + +Root derivation revised after independent review; final review pending. Local-only. No paid lower bound or kappa claim. + +## Changed scope and exact predecessor + +The published PR #5160 head 7800589ce99d932bdb063aafca1f501f1a38626d proves the obstruction for fixed finite columnwise monomial-Clifford/scalar-owner programs. This extension permits arbitrary invertible one-column monomial matrices, including nonlinear address permutations and arbitrary nonzero complex weights, in a fixed finite columnwise owner program. More generally, it suffices that the complete one-column operator products occurring in each proposed control have a fixed finite support independent of width f; only their outer scalar coefficients may depend on f. Merely fixing a finite generating alphabet is insufficient if word products or lengths vary without bound with f: weighted monomial generators can generate infinitely many projective products. That larger class is not covered. It does not cover growing alphabets, noncolumnwise or general width-dependent permutations, extra paid Gaussian calls, a changed endpoint, compressed/input-coded targets or a changed source. + +Scientific source remains 4c0a359d88f22f05189d51f48387750b801a45d4, research/coded-native-caller. Reuse the authenticated source facts and exact normalizer conventions in outputs/released-operator-intersection-20261023/PROOF.md and its source checker. There are 112 owners; A is invertible with zero diagonal. K1s/K3r are actual signed linear permutations supported on q+U / q+W, G_s e_q=q_s, and distinct source q_s make G_s^-1 q_j=e_q iff s=j. Q and D3 are supported on q and W respectively, with every entry inside their respective coset blocks nonzero. These are inherited source premises, not newly machine-checked here. + +## Lemma 1 — a monomial intertwiner preserves support partitions + +Let F be an invertible matrix whose support is exactly a disjoint union of complete square blocks indexed by a partition P of the computational basis. If h and k are invertible monomial matrices and F h = k F, then their address permutations both map every P-block bijectively to a P-block, with the same induced permutation of blocks. + +For a column y, support(F h e_y) is the entire P-block containing sigma_h(y). Support(k F e_y) is sigma_k(P(y)). These must be equal. If y,y' belong to the same P-block, the latter set is unchanged, so sigma_h sends that block into a single block. Invertibility and the equal finite block sizes force equality. The same identity supplies the assertion for sigma_k. No phase restriction is used; multiplication by a monomial matrix creates no internal cancellation. + +For a two-sided variant F_out^-1 a F_in monomial, where F_in/F_out have complete equal-size coset blocks (possibly different partitions), write a F_in=F_out m and use the same column-support argument. The address permutation of a maps input blocks to output blocks. This variant uses no Clifford/symplectic classification. + +## Lemma 2 — transported letters preserve the required partitions + +Let H be the group of all invertible monomial matrices on the actual 17 one-column bits. Let h in H satisfy Q^-1 h Q in H and k=D3 h D3^-1 in H. From h Q=Q(Q^-1 h Q), Lemma 1 says sigma_h preserves q-cosets. From D3 h=k D3, both sigma_h and sigma_k preserve W-cosets. + +The join of the q- and W-coset partitions is the (q+W)-coset partition, indexed by U coordinates. An invertible permutation preserving both component partitions preserves their join, hence sigma_h preserves (q+W)-cosets. D3 and D3^-1 act inside these join blocks. Therefore k=D3 h D3^-1 has no nonzero matrix entries between an input join block and any output join block other than h's induced image. Because k is monomial, sigma_k also preserves the join partition. + +Consequently, sigma_k cannot map a pair lying in different(q+W)-cosets to a pair in the same(q+W)-coset. + +## Lemma 3 — every nonzero terminal source term remains nonmonomial + +For the same four-control equations as PR #5160, let a=K3i^-1 K3r and b=K1s^-1 K1j. A terminal term has address operator Q^-1 a k b Q^-1, coefficient A[i,r] b[r,s,k] A[s,j]. If this operator were monomial, write a k b Q^-1=Q m. The two-sided support lemma implies a k b maps q-cosets to q-cosets. Since a,b are invertible signed linear permutations, sigma_k must map cosets of v=G1s^-1 q_j to cosets of w=G3r^-1 q_i. + +Here v belongs to span(q)+U and w belongs to span(q)+W. A w-coset consists of two points in one(q+W)-coset. If v has a nonzero U component, every v-coset has its two points in different(q+W)-cosets. This contradicts Lemma 2. Thus v=e_q, so s=j by the actual distinct source directions. But then A[s,j]=A[j,j]=0. Every terminal term with nonzero source coefficient is therefore nonmonomial. This does not require r=i and does not assume the invalid transported central equality from the old draft. + +## All-width conclusion for fixed operator support + +Use the elementary tensor-separation lemma from PR #5160: tensor powers of a finite collection of pairwise nonproportional nonzero operator vectors are independent for sufficiently large f. It applies to arbitrary matrices, not only Clifford operators. + +For any fixed finite columnwise owner program, take the finite set of complete one-column operator products in its expansion and combine projectively equal products. The same applies to all four proposed control implementations. This finite-support premise is essential. Requiring t0, Q^-1t0Q and D3t0D3^-1 to have finite monomial expansions forces every surviving one-column letter h of t0 into the exact intersections in Lemma 2 for all sufficiently large widths. Coefficients may depend on f; the complete operator support remains fixed independently of f. Lemma 3 places every surviving terminal E term outside H. Cancellation may remove outside-H classes but cannot produce an H class. Separating the finite combined set against any proposed finite monomial program for E gives E_f=0 for all sufficiently large widths. + +The actual C1,C3 and Q are invertible, so E=Q^-1 C3 t C1 Q^-1=0 implies t=0 and t0=0. Hence an all-width invertible t0 is impossible in this enlarged class of fixed finite columnwise arbitrary-monomial programs, or explicitly fixed finite complete-product supports. No paid Gaussian query lower bound follows. Programs with unbounded width-dependent products even from finite weighted generators, growing supports, or noncolumnwise FREE programs and altered four-control/target contracts remain outside this theorem. + +## Verification and credit still required + +Independent proof review must check the partition argument, the exact matrix order of every intertwiner and the source support conventions. A small exact support checker with nonaffine examples can discriminate the extension but cannot prove the all-width theorem. No operator alphabet enumeration or native/compiler replay is proposed. Primary credit: Dillon released equations, Dugongue/mdumitrean source A, real normalizer authors and Cinna Davis/1337isnot1337, Cao Yifan/intqwq, zxv and retained helper/Clifford/compiler lineage. This extension is a root support-partition derivation with substantial OpenAI Codex assistance; it is not a published-paper theorem or formal-kernel proof. + + +## Revision note + +The first draft used “fixed alphabet” too loosely outside its fixed-program paragraph. Independent review identified that finite weighted monomial generators need not have finitely many projective products. This revision explicitly requires a fixed finite program or fixed finite set of complete operator products; it excludes unbounded width-dependent word products. The first draft remains in history/PROOF-DRAFT-v1.md. No source or predecessor proof was changed. diff --git a/research/nonlinear-monomial-control-ceiling/verify.py b/research/nonlinear-monomial-control-ceiling/verify.py new file mode 100644 index 0000000000..500e06d119 --- /dev/null +++ b/research/nonlinear-monomial-control-ceiling/verify.py @@ -0,0 +1,87 @@ +#!/usr/bin/env python3 +"""Portable finite support discriminator; all-width theorem remains written proof.""" +from hashlib import sha256 +import json +from pathlib import Path +import shutil +import subprocess +import sys +import tempfile + +ROOT=Path(__file__).resolve().parent + +def need(ok, why): + if not ok: + raise ValueError(why) + +def authenticate(): + manifest=json.loads((ROOT/'EVIDENCE.json').read_text()) + actual={str(f.relative_to(ROOT)) for f in (ROOT/'evidence').rglob('*') if f.is_file()} + need(actual==set(manifest['files']), 'Evidence file inventory differs') + for name,digest in manifest['files'].items(): + p=ROOT/name + need(not p.is_symlink() and sha256(p.read_bytes()).hexdigest()==digest, + 'Frozen evidence mismatch: '+name) + review=ROOT/'evidence/proof-review' + for row in json.loads((review/'FILES.json').read_text())['files']: + p=review/row['path'] + need(p.stat().st_size==row['bytes'] and sha256(p.read_bytes()).hexdigest()==row['sha256'], + 'Original proof-review capture mismatch: '+row['path']) + finite=ROOT/'evidence/finite-check' + for name,digest in json.loads((finite/'FREEZE.json').read_text())['files'].items(): + need(sha256((finite/name).read_bytes()).hexdigest()==digest, + 'Original finite-check freeze mismatch: '+name) + proof=review/'frozen/REVIEWED-PROOF.md' + need(sha256(proof.read_bytes()).hexdigest()==json.loads((review/'RESULT.json').read_text())['reviewed_proof_sha256'], + 'Written review binds a different proof') + return manifest + +def main(): + manifest=authenticate() + review=ROOT/'evidence/proof-review' + finite=ROOT/'evidence/finite-check' + # Unchanged checker derives ROOT from its own location; mirror exactly the + # four original source paths in private scratch. All generated files stay there. + with tempfile.TemporaryDirectory(prefix='nonlinear-support-') as temporary: + temp=Path(temporary) + logical={ + 'outputs/nonlinear-monomial-control-ceiling-20261023/PROOF.md':review/'frozen/REVIEWED-PROOF.md', + 'outputs/released-operator-intersection-20261023/PROOF.md':review/'frozen/PREDECESSOR-PROOF.md', + 'outputs/released-operator-intersection-20261023/SOURCES.json':review/'frozen/PREDECESSOR-SOURCES.json', + 'outputs/released-operator-intersection-20261023/frozen/PHYSICAL.md':review/'frozen/PHYSICAL.md'} + for name,source in logical.items(): + target=temp/name + target.parent.mkdir(parents=True,exist_ok=True) + shutil.copyfile(source,target) + work=temp/'outputs/nonlinear-monomial-support-check-20261023' + work.mkdir() + shutil.copyfile(finite/'check_support.py',work/'check_support.py') + command=[sys.executable,'-I'] + if sys.flags.optimize: + command.append('-O') + command+=['-W','error','-B',str(work/'check_support.py')] + replay=subprocess.run(command,cwd=temp,check=True,capture_output=True,timeout=45) + for stem in ['RESULT','MATRICES']: + data=(work/(stem+'.json')).read_bytes() + for suffix in ['', '-OPTIMIZED']: + need(data==(finite/(stem+suffix+'.json')).read_bytes(), + 'Complete generated '+stem+' differs from frozen '+suffix+' receipt') + need(all((temp/name).read_bytes()==source.read_bytes() for name,source in logical.items()), + 'Private source capture changed') + result=json.loads((work/'RESULT.json').read_text()) + matrices=(work/'MATRICES.json').read_bytes() + compact=json.loads(replay.stdout) + print(json.dumps({ + 'status':'PASS_PORTABLE_NONLINEAR_FINITE_SUPPORT_ONLY', + 'immutable_evidence_files':len(manifest['files']), + 'written_proof_sha256':sha256((review/'frozen/REVIEWED-PROOF.md').read_bytes()).hexdigest(), + 'independent_review_sha256':sha256((review/'REVIEW.md').read_bytes()).hexdigest(), + 'finite_result_sha256':sha256((finite/'RESULT.json').read_bytes()).hexdigest(), + 'finite_matrices_sha256':sha256(matrices).hexdigest(), + 'positive_examples':compact['positive_examples'], + 'terminal_sandwiches':result['terminal_nonmonomial_count'], + 'scope':'Finite exact16x16 support examples and controls only. Independently reviewed written proof supplies sufficiently-large-width fixed finite complete-product support theorem. No17-axis/all-width phase/native/compiler/caller or kappa replay.' + },indent=2,sort_keys=True)) + +if __name__=='__main__': + main() From c9160c04bd4efc8bb8a112f0f894e5f1cf2b6e64 Mon Sep 17 00:00:00 2001 From: d3lerium Date: Fri, 23 Oct 2026 23:19:43 +1100 Subject: [PATCH 2/4] Preserve separately reviewed translation-valued paid-control appendix --- .../EVIDENCE.json | 14 + .../README.md | 37 ++ .../evidence/paid-control/AUTHOR-INPUTS.json | 31 ++ .../evidence/paid-control/AUTHOR-REPORT.md | 263 ++++++++++++ .../evidence/paid-control/CAPTURES.json | 91 ++++ .../paid-control/INDEPENDENT-REVIEW.md | 111 +++++ .../evidence/paid-control/REVIEW-SOURCES.json | 63 +++ .../captures/01-PAID-SHEAR-SCREEN.md | 120 ++++++ ...03-operator-valued-gather-rank-boundary.md | 47 +++ .../paid-control/captures/04-PROOF.md | 395 ++++++++++++++++++ .../paid-control/captures/05-REPORT.md | 230 ++++++++++ .../paid-control/captures/06-REVIEW.md | 190 +++++++++ .../paid-control/captures/07-REPORT.md | 82 ++++ .../paid-control/captures/08-RESULT.json | 164 ++++++++ .../09-operator-valued-paid-theorem-fit.md | 171 ++++++++ .../captures/RAW-GIT-5126-PROOF.md | 394 +++++++++++++++++ 16 files changed, 2403 insertions(+) create mode 100644 research/nonlinear-monomial-control-ceiling/evidence/paid-control/AUTHOR-INPUTS.json create mode 100644 research/nonlinear-monomial-control-ceiling/evidence/paid-control/AUTHOR-REPORT.md create mode 100644 research/nonlinear-monomial-control-ceiling/evidence/paid-control/CAPTURES.json create mode 100644 research/nonlinear-monomial-control-ceiling/evidence/paid-control/INDEPENDENT-REVIEW.md create mode 100644 research/nonlinear-monomial-control-ceiling/evidence/paid-control/REVIEW-SOURCES.json create mode 100644 research/nonlinear-monomial-control-ceiling/evidence/paid-control/captures/01-PAID-SHEAR-SCREEN.md create mode 100644 research/nonlinear-monomial-control-ceiling/evidence/paid-control/captures/03-operator-valued-gather-rank-boundary.md create mode 100644 research/nonlinear-monomial-control-ceiling/evidence/paid-control/captures/04-PROOF.md create mode 100644 research/nonlinear-monomial-control-ceiling/evidence/paid-control/captures/05-REPORT.md create mode 100644 research/nonlinear-monomial-control-ceiling/evidence/paid-control/captures/06-REVIEW.md create mode 100644 research/nonlinear-monomial-control-ceiling/evidence/paid-control/captures/07-REPORT.md create mode 100644 research/nonlinear-monomial-control-ceiling/evidence/paid-control/captures/08-RESULT.json create mode 100644 research/nonlinear-monomial-control-ceiling/evidence/paid-control/captures/09-operator-valued-paid-theorem-fit.md create mode 100644 research/nonlinear-monomial-control-ceiling/evidence/paid-control/captures/RAW-GIT-5126-PROOF.md diff --git a/research/nonlinear-monomial-control-ceiling/EVIDENCE.json b/research/nonlinear-monomial-control-ceiling/EVIDENCE.json index a347cb3ba4..c867e8e5f3 100644 --- a/research/nonlinear-monomial-control-ceiling/EVIDENCE.json +++ b/research/nonlinear-monomial-control-ceiling/EVIDENCE.json @@ -9,6 +9,20 @@ "evidence/finite-check/SHA256SUMS": "180502cb8356bd9c8d92bf8f06da25d89f21efeb18a8135723b5bd847256edc4", "evidence/finite-check/SOURCE-FACTS.md": "c6e2546054ac68cbec6a579bf7558a523d1a8843c160a70250b24a10220f90fe", "evidence/finite-check/check_support.py": "91115f8380c01d6e788acf27870991b80a6c6ee02d5363396408e4e08dad149c", + "evidence/paid-control/AUTHOR-INPUTS.json": "228c3da4d04e20607bdefabd86d682a8d6dd4c09e35c6c0a786bc1e18d3facd4", + "evidence/paid-control/AUTHOR-REPORT.md": "19932a49cda97acb2e3f4933f95d4bed5598e441d7593132bd07d15a63c23731", + "evidence/paid-control/CAPTURES.json": "7b4206e960c709bec809731f8de1698b65d3335ebb207d5ae4f90cc9c04e0ca3", + "evidence/paid-control/INDEPENDENT-REVIEW.md": "3a255cfb2efb874fcf3cc111dacd145901e2b1018a7cfa60f12d251d3239b4e5", + "evidence/paid-control/REVIEW-SOURCES.json": "769e777c6dce6567a77c6aa4f89fd0120955b56e2554001d912d26669b492306", + "evidence/paid-control/captures/01-PAID-SHEAR-SCREEN.md": "81fefcf4a1be0be04076e7cba0e27ba4afe732d4810a13b0a76dbe827967c10c", + "evidence/paid-control/captures/03-operator-valued-gather-rank-boundary.md": "847dccf3cc3a0938b67a22e3074cd06e825f7881e0a90dfad6c29f3963d41f58", + "evidence/paid-control/captures/04-PROOF.md": "96c617b306d50fce2b0f4f88afe1f6103d11326b402c2e4980c522c9c48092db", + "evidence/paid-control/captures/05-REPORT.md": "363f66bb655dd9100438aa4ae405025baafeddcaaaad0b8d0a43eecaa43ee784", + "evidence/paid-control/captures/06-REVIEW.md": "23b5aa5df049cb4a8fe3349c054e27d84ca151e29e263c658670639bad082428", + "evidence/paid-control/captures/07-REPORT.md": "b8b785a815852069b27ee6b18337b6ecd64127c7a22985d0e4221a2fc11bd53a", + "evidence/paid-control/captures/08-RESULT.json": "b11a56c453c6f174780edd0cd19c638da7d1552ac9f3f0ff6f8340d8ada382ae", + "evidence/paid-control/captures/09-operator-valued-paid-theorem-fit.md": "660225cc59b75eaff65f241a10373f3a4f312981dc386b5833d653dba3d14460", + "evidence/paid-control/captures/RAW-GIT-5126-PROOF.md": "495a1cd85cc45020adf2205f196b57dc4f65f1eb763fc4951facdb4ccbafc768", "evidence/proof-review/FILES.json": "d6b03962432cbb2880dff5825daf6eae9637f8ce614c82de15aaa07ffb90571d", "evidence/proof-review/RESULT.json": "8bdf9491dc7e9c8ac9883e6be3845d3aa9b05930b0c1699057baf7fb33c7df09", "evidence/proof-review/REVIEW.md": "0b0fe1c33eea85ccd92a3423a705fd086c6f2669f1a47f37fba9c1ca4ae91973", diff --git a/research/nonlinear-monomial-control-ceiling/README.md b/research/nonlinear-monomial-control-ceiling/README.md index af83ff3281..8a579276ca 100644 --- a/research/nonlinear-monomial-control-ceiling/README.md +++ b/research/nonlinear-monomial-control-ceiling/README.md @@ -65,6 +65,43 @@ This is finite support calibration, **not** an all-width theorem checker, The independently reviewed written proof carries the uniform theorem. No t0 search, native/emitter/compiler/Lean replay or complete κ claim is made. +## Separate appendix: translation-valued paid-control gate + +The unchanged [author report](evidence/paid-control/AUTHOR-REPORT.md) and +[separate independent review](evidence/paid-control/INDEPENDENT-REVIEW.md) +record a narrower paid-control test. The explicit operator-valued shear +`T=I+U Q V`, with `U_i=X_(q_i)`, `V_i=c_i X_(q_i)^-1` and56 signs of each +kind, has `VU=0`, inverse `I-U Q V` and raw dual +`T^-T=I-V^T Q U^T`. All allowed translations and X-Gaussian children commute +with Q and D3, so the declared T circuit also commutes with them. +The one-child algebraic word does not supply a physical tape/precision emitter. + +Its direct whole-control policy adds228 H1, including the paid terminal +inverse. That is an upper construction bill, not a minimum for every paid +E decoder, and it loses against the inherited conditional raw44H1 allowance +before ordinary costs or reserves. Original224 main and816 independent +helper jobs remain counted once; the complete physical helper/caller remains +unrealized. Source recoding, /52, raw-dual corrections and finite work remain +charged obligations. + +A separate written kernel argument proves `r>=112` only for a **fresh +standalone T circuit with no initially supplied nonfree observations**, +translation-algebra free coefficients, commuting same-width X-Gaussian +children, and a FREE terminal E in the declared fixed finite endpoint class, +at sufficiently large faithful widths. Under exactly these conditions112 +exceeds the conditional44 allowance. This is **not** a whole-caller amortized +barrier: already-paid QY/native/IDLE/centre observations violate that initial- +information premise. Arbitrary nontranslation/nonlinear weighted gathers, +noncommuting children, wider channels, a paid E or changed targets are outside +it. The primary nonlinear-monomial theorem is not enlarged by this appendix. + +No new proof code, source evaluator or native/compiler replay checks this +written appendix; the hook only authenticates its immutable bytes. Captures +reuse primary inputs by digest. [CAPTURES.json](evidence/paid-control/CAPTURES.json) +records the monitor5126 proof's extra LF relative to raw Git; both byte versions +are retained without replacing the raw scientific pin. The separate p19 +compiler-contracts reviewer supplied the final scoped PASS. + ## Reproduce ```sh diff --git a/research/nonlinear-monomial-control-ceiling/evidence/paid-control/AUTHOR-INPUTS.json b/research/nonlinear-monomial-control-ceiling/evidence/paid-control/AUTHOR-INPUTS.json new file mode 100644 index 0000000000..c788fcfc76 --- /dev/null +++ b/research/nonlinear-monomial-control-ceiling/evidence/paid-control/AUTHOR-INPUTS.json @@ -0,0 +1,31 @@ +{ + "scientific_pin": "4c0a359d88f22f05189d51f48387750b801a45d4", + "active_query_proof_pin": "0e8a13c251a9be1029114cda8b4194f05f3a23ec", + "inputs": [ + { + "path": "outputs/released-operator-intersection-20261023/PROOF.md", + "sha256": "c20b2f13e875d9e1fb6d0743693f9deccc4f7d68dcdc55ec241324ed3d7de7bc" + }, + { + "path": "outputs/released-operator-intersection-20261023/PAID-SHEAR-SCREEN.md", + "sha256": "81fefcf4a1be0be04076e7cba0e27ba4afe732d4810a13b0a76dbe827967c10c" + }, + { + "path": "outputs/released-operator-intersection-20261023/SOURCES.json", + "sha256": "e810407fc34de1f3af196fc86e91e680075ff982bbc6fe19d6ebb02f2a0c0d9c" + }, + { + "path": "outputs/mathematics-paper-research-20261023/notes/operator-valued-gather-rank-boundary.md", + "sha256": "847dccf3cc3a0938b67a22e3074cd06e825f7881e0a90dfad6c29f3963d41f58" + }, + { + "path": "outputs/all-upstream-pr-monitor-20261023/pr5126-source-read/PROOF.md", + "sha256": "96c617b306d50fce2b0f4f88afe1f6103d11326b402c2e4980c522c9c48092db" + }, + { + "path": "coded-port-adapter-ceiling-20261023/outputs/released-operator-inventory-20261023/REPORT.md", + "sha256": "363f66bb655dd9100438aa4ae405025baafeddcaaaad0b8d0a43eecaa43ee784" + } + ], + "scope": "Read and derive only; no source execution, enumeration, solver, native replay, compiler, Lean or publication." +} diff --git a/research/nonlinear-monomial-control-ceiling/evidence/paid-control/AUTHOR-REPORT.md b/research/nonlinear-monomial-control-ceiling/evidence/paid-control/AUTHOR-REPORT.md new file mode 100644 index 0000000000..3d6499a9fe --- /dev/null +++ b/research/nonlinear-monomial-control-ceiling/evidence/paid-control/AUTHOR-REPORT.md @@ -0,0 +1,263 @@ +# Operator-valued paid control: explicit one-child word and a scoped stop + +Local read/derive gate, 2026-10-23. No enumeration, solver, source evaluator, +native replay, emitter, compiler, Lean or public post was run. Source pins +and predecessor hashes are in INPUTS.json. This does not alter #5160 or any +previous frozen packet. Substantial OpenAI Codex assistance; no independent +human or formal-kernel acceptance is claimed. + +## Decision + +The explicit one-H1 operator-valued shear below is invertible, source-bound +and phase-correct at every width. Its first three controls coincide. The +direct fourth-control decoder costs too much: the complete additional +control price is 228 H1, against the conditional 44-H1 raw allowance before +finite work and compiler reserve. + +A new scoped obstruction stops attempts to make that fourth control FREE +using arbitrary translation-valued gathers, scatters, bypasses and active +feedback around commuting X-Gaussian children: an invertible t0 then needs +at least 112 paid unary children. This includes genuine operator-valued +owner maps and the predecessor counterexample to scalar coefficient rank. +The proof uses a chronological free-input kernel, not coefficient rank. + +This is NOT an obstruction for arbitrary monomial gathers, noncommuting +paid children, or an affordably PAID fourth control. No within-budget +construction for that broader class is supplied. The gate closes with +those explicit reopening conditions. + +## Exact source and endpoint + +Keep the 112-owner physical source at +4c0a359d88f22f05189d51f48387750b801a45d4, the 17 axes q+U+W, and + + C_s[i,j] = A[i,j] K_(s,i)^-1 K_(s,j). + +The source A is invertible, has zero diagonal and 52 nonzeros per row and +column. The 112 directions q_i=G_i e_q are distinct. G_i are the actual +orthogonal binary normalizer matrices, read as rows, with their signed K_i +words retained. Each word is lifted columnwise; owner coefficients are +applied once. The four controls and exact ordered inverse formulas remain + + T=t0; S0=Q^-1 T Q; t=D3 T D3^-1; + E=Q^-1 C3 t C1 Q^-1, + E^-1=Q C1^-1 t^-1 C3^-1 Q. + +The released identity is + + F3 J(T) F1 = [[0,t],[-E,0]] diag(G,G), + J(T)=[[-T C1,T Q],[0,-C3 S0]], G=D1 D3 Q. + +The monitor confirmed #5126 already allows operator-valued monomial +gathers, active feedback and free recodings for its named fresh-input +J, QP, PQ, Q U_mode and V_mode Q endpoints. Its 112-call theorem is not +automatically a theorem about arbitrary T. Dillon's chronological native +helper/centre lane remains excluded. The monitor found no matching current +claim on this bounded fresh four-control construction; this is not an +unpublished-novelty guarantee. + +## One actual H1 child, with all-width maps and lifecycle + +Embed the actual source q_i in q+U and let P_i=X_(q_i) be the Pauli-X +translation of that address, identically on all f columns. Put c_i=+1 for +owners 0 through 55 and c_i=-1 for owners 56 through 111. Define + + U_i=P_i, V_i=c_i P_i^-1, H=Q, + T=I+U H V. + +These are literal bit translations on existing axes, not a full-operator +oracle or a new address coordinate. P_i^-1=P_i. Since sum_i c_i=0, + + VU=0, (U H V)^2=0, T^-1=I-U H V. + +The latter identity is ordered and needs no assumption that H is free. +All P_i commute with Q and D3, so S0=t=T and their inverses are T^-1. +Q is the actual rank-one Gaussian on the f-bit central row, Q^(tensor f) +in the one-column notation. It is not replaced by a scalar two-term +expansion after tensor lifting; Q^2 is the actual Pauli translation. + +One application of T uses an owned zero field g: + +1. Gather g=sum_i c_i P_i^-1 x_i before overwriting any x_i. +2. Apply one physical H1 child Q to g in place. +3. Scatter x_i <- x_i+P_i g for all 112 owners. +4. CLEAR/reset the owned g after its last read. + +Original independent owners are not scratch. The inverse repeats the same +gather and Q, then subtracts the scatter. The cache earns no productive +credit. A conservative elementary implementation uses one additional +reusable zero field h for each translated COPY/ADD, then clears h. Per +application this gives one H1, 224 monomial translation passes, 224 COPYs, +224 signed ADDs, 225 owned CLEAR/reset operations and two peak scratch +fields. These are field-pass counts, not a bit-operation or precision tariff. +An indexed translated ADD can fuse some passes only with a separately +specified physical implementation. All initialization, movement, bit height, +rounding, stock and cleanup remain actual charged work. + +For raw dual use transpose, not adjoint. Q^T=Q and P_i^T=P_i, so + + T^-T=I-V^T Q U^T. + +Its literal word gathers sum_i P_i x_i, applies one Q child, then scatters +-c_i P_i g. T^T uses the plus sign. Thus both inverse and raw-dual control +words cost one H1 and retain all arbitrary inputs. No Q=Q^-1 shortcut is +used. These words apply on the source's normalized physical contact fields; +the native backward X-only Q^2 convention and D_s^2 corrections remain. + +## Fourth control and complete conditional recursive invoice + +No FREE program for E or E^-1 exists for this T by the obstruction below. +A literal paid program for E^-1, rightmost first, is + + Q on 112 fields; + C3^-1; + T^-1 using one H1; + C1^-1; + Q on 112 fields. + +This implements the exact ordered inverse and costs 225 H1. E itself uses +the analogous two Q^-1 banks and T; inverse Gaussian banks have the same +recursive price, with their actual Q^2 Pauli correction charged ordinarily. +No input field or independent helper is silently replaced by a cache. + +The main caller uses T twice in J, t^-1 once at the positive endpoint, and +the displayed E^-1 once at the negative endpoint. Its incremental Gaussian +price is therefore + + 2 H1 + 1 H1 + 225 H1 = 228 H1. + +The existing central query bank of 112 H1 is already in the released base +and is not added a second time. There are four source C/C^-1 recoding +macros in this direct control schedule, with the inherited macro cost each +224 K maps, 896 scalar ADDs and 16 CLEARs using eight reusable mode fields; +the inverse's division by 52 and precision work remain charged. The four +shear uses contribute their explicit ordinary work above. The two fresh +112-field Q banks can execute owner by owner where dependencies permit; +their full field passes, inverse-Gaussian Pauli passes and routing remain. + +For two such shears, the same literal policy costs 232 H1. Its hypothetical +FREE-E price would be 3m H1 for m=1 or 2, but the theorem below excludes +that premise. The direct policy is not claimed to minimize a PAID E. + +Conditioned on the inherited FULL and helper-completion contracts, the +one-child caller census is + + paid P={1:4836,2:838,3:1020,4:254,5:136}, + target T={9:816,17:224}. + +All 1,040 original inputs remain: 224 main jobs and 816 distinct completed +helper jobs, counted once. No dependent query or scratch output is credited. +The 816-owner complement words, finite costs, compiler reserve and outer +admission are inherited/open obligations, not independently certified here. +The raw dual uses the explicit transpose/inverse control words and the same +Gaussian ranks. Positive inverse-Gaussian targets require their actual +Pauli correction passes. This invoice already loses before those costs. + +## Scoped theorem: translation-valued controls still need 112 children + +Let B_f be the phase-correct algebra of the fixed one-column Pauli-X +translations, lifted to f columns. Let A_f be the group algebra generated +by those translations and a fixed finite collection of X-Gaussians C_v. +Every group element commutes with every X translation. These groups are +finite projectively on the fixed 17 axes. Use the width-dependent cocycle +of the actual tensor lifts; scalar owner coefficients are never tensorized. + +Allow arbitrary chronological linear circuits for T with every FREE bypass, +gather, scatter and feedback coefficient in B_f, including dense +operator-valued rows/columns. Permit arbitrary owned copies, saved fields, +fanout and resets. Every paid child is a same-width X-Gaussian in A_f; its +input may depend on previous responses. Require T invertible on all fresh +112-owner inputs at every width and E to have a fixed finite monomial +Clifford implementation. Then the circuit for T requires at least 112 paid +unary calls at every sufficiently large width. + +### Source step: E FREE forces every translation component of T to vanish + +A monomial operator commuting with all X translations must be a scalar +translation: commutation forces its permutation p(x+a)=p(x)+a and its +multiplier to be constant. Thus the monomial intersection of A is exactly +the translation subgroup, up to phase. + +Consider a nonzero translation coefficient X_b of T[r,s]. Choose i,j with +A[i,r] A[s,j] nonzero. The corresponding endpoint word is + + Q^-1 K3_i^-1 K3_r X_b K1_s^-1 K1_j Q^-1. (1) + +Its symplectic linear part is the same as the translation-free word. +The disjoint q+U and q+W source argument says it can be monomial only if +s=j: otherwise G1_s^-1 q_j has a nonzero U component, which the stage-3 +map cannot remove. But s=j would require a zero source diagonal coefficient. +Hence (1) is nonmonomial. Translation and phase changes do not remove this +witness. + +If an A-letter H at another pair (r',s') has the same projective endpoint +word, then + + H = phase * K3_r'^-1 K3_r X_b K1_s^-1 K1_s'. (2) + +The right side is monomial, so H must be a translation. Its linear binary +part must be identity. Consequently + + G3_r'^-1 G3_r = G1_s'^-1 G1_s. + +The left map is supported on q+W, the right on q+U. Their intersection is +identity on the single common binary axis; distinct source q_i then force +r'=r and s'=s. Equation (2) now forces H=X_b projectively. Thus no other +letter or owner pair cancels the chosen coefficient. + +The fixed finite projective tensor-separation lemma from #5160 makes these +source words independent of every monomial endpoint word at sufficiently +large width. Therefore E FREE forces all translation coefficients of T to +be zero: projection_B(T)=0. This is stronger than vanishing only the scalar +identity coefficient and uses no scalar owner-rank claim. + +### Chronological input cut + +Normalize an r-call circuit by saving every paid response y_k. Its kth +input is b_k x+sum_(s B_f^r has a nonzero kernel +column w. Set the formal input x=w z. Induction makes every paid response +zero, regardless of active feedback, so + + T w = F0 w in B_f^112. + +But projection_B(T)=0 means T is supported on nontrivial right cosets of +the translation subgroup in A. Right multiplication by w over B_f stays +in those nontrivial cosets. Hence T w is also in their direct sum, disjoint +from B_f^112, and T w=0. + +At sufficiently large width the fixed group/coset support evaluates +faithfully. Thus w is a nonzero actual field embedding while T w=0, +contradicting actual invertibility of T. Therefore r>=112. This uses finite +vector-space dimension and coset direct sums; B_f need not be a division +ring and may have zero divisors. + +Since every nontrivial paid Gaussian has price at least H1=1 on the stated +comparison interval, T alone exceeds the 44-H1 allowance. The bound covers +the one/two-child proposals and all larger affordable circuits within this +scope. It does not rely on rank of an extracted scalar coefficient, which +the predecessor operator-valued example correctly disproves. + +## Reopening boundary and credit + +To continue, supply an actual noncommuting monomial gather/scatter or a +paid E/E^-1 implementation with all four complete control uses, arbitrary +input restoration, raw dual and finite work within the remaining allowance. +An internal C_s or K_i outside the translation algebra changes the circuit +scope and requires a new source-specific kernel argument. Merely inserting +such a recoding does not establish a cheap conjugate control. A smaller +PAID-E implementation is also a distinct open route; the 228-H1 direct +policy is an upper bound, not a lower bound for that route. + +Credit Dillon's released template and #5126 chronological input-kernel +argument; Dugongue/mdumitrean's #4929 source; the actual #4933 normalizers; +Cinna Davis/4694/4698, Cao Yifan, zxv, Jacob Sussman, Ryanson Jonathan and +all retained native/helper/compiler lineage. The paper worker checked the +module-cut logic under its stated scope; the source isolation and full +report still require separate review. The ordered nilpotent inverse is +elementary and consistent with the primary quasideterminant/Schur source +already documented by that worker; no paper theorem supplies a query saving. diff --git a/research/nonlinear-monomial-control-ceiling/evidence/paid-control/CAPTURES.json b/research/nonlinear-monomial-control-ceiling/evidence/paid-control/CAPTURES.json new file mode 100644 index 0000000000..a2ceca100e --- /dev/null +++ b/research/nonlinear-monomial-control-ceiling/evidence/paid-control/CAPTURES.json @@ -0,0 +1,91 @@ +{ + "scope": "Exact written paid-control/review input captures; no added science replay. Primary captures reused by digest.", + "inputs": [ + { + "original_path": "outputs/released-operator-intersection-20261023/PROOF.md", + "retained_path": "evidence/proof-review/frozen/PREDECESSOR-PROOF.md", + "sha256": "c20b2f13e875d9e1fb6d0743693f9deccc4f7d68dcdc55ec241324ed3d7de7bc", + "reuses_primary_capture": true + }, + { + "original_path": "outputs/released-operator-intersection-20261023/PAID-SHEAR-SCREEN.md", + "retained_path": "evidence/paid-control/captures/01-PAID-SHEAR-SCREEN.md", + "sha256": "81fefcf4a1be0be04076e7cba0e27ba4afe732d4810a13b0a76dbe827967c10c", + "reuses_primary_capture": false + }, + { + "original_path": "outputs/released-operator-intersection-20261023/SOURCES.json", + "retained_path": "evidence/proof-review/frozen/PREDECESSOR-SOURCES.json", + "sha256": "e810407fc34de1f3af196fc86e91e680075ff982bbc6fe19d6ebb02f2a0c0d9c", + "reuses_primary_capture": true + }, + { + "original_path": "outputs/mathematics-paper-research-20261023/notes/operator-valued-gather-rank-boundary.md", + "retained_path": "evidence/paid-control/captures/03-operator-valued-gather-rank-boundary.md", + "sha256": "847dccf3cc3a0938b67a22e3074cd06e825f7881e0a90dfad6c29f3963d41f58", + "reuses_primary_capture": false + }, + { + "original_path": "outputs/all-upstream-pr-monitor-20261023/pr5126-source-read/PROOF.md", + "retained_path": "evidence/paid-control/captures/04-PROOF.md", + "sha256": "96c617b306d50fce2b0f4f88afe1f6103d11326b402c2e4980c522c9c48092db", + "reuses_primary_capture": false + }, + { + "original_path": "coded-port-adapter-ceiling-20261023/outputs/released-operator-inventory-20261023/REPORT.md", + "retained_path": "evidence/paid-control/captures/05-REPORT.md", + "sha256": "363f66bb655dd9100438aa4ae405025baafeddcaaaad0b8d0a43eecaa43ee784", + "reuses_primary_capture": false + }, + { + "original_path": "outputs/released-operator-independent-review-20261023/REVIEW.md", + "retained_path": "evidence/paid-control/captures/06-REVIEW.md", + "sha256": "23b5aa5df049cb4a8fe3349c054e27d84ca151e29e263c658670639bad082428", + "reuses_primary_capture": false + }, + { + "original_path": "outputs/released-operator-optimistic-budget-20261023/REPORT.md", + "retained_path": "evidence/paid-control/captures/07-REPORT.md", + "sha256": "b8b785a815852069b27ee6b18337b6ecd64127c7a22985d0e4221a2fc11bd53a", + "reuses_primary_capture": false + }, + { + "original_path": "outputs/released-operator-optimistic-budget-20261023/RESULT.json", + "retained_path": "evidence/paid-control/captures/08-RESULT.json", + "sha256": "b11a56c453c6f174780edd0cd19c638da7d1552ac9f3f0ff6f8340d8ada382ae", + "reuses_primary_capture": false + }, + { + "original_path": "outputs/mathematics-paper-research-20261023/notes/operator-valued-paid-theorem-fit.md", + "retained_path": "evidence/paid-control/captures/09-operator-valued-paid-theorem-fit.md", + "sha256": "660225cc59b75eaff65f241a10373f3a4f312981dc386b5833d653dba3d14460", + "reuses_primary_capture": false + }, + { + "original_path": "coded-port-adapter-ceiling-20261023/outputs/released-operator-inventory-20261023/source/research__coded-native-caller__NORMALIZERS.json", + "retained_path": "evidence/proof-review/frozen/NORMALIZERS.json", + "sha256": "ae8aad06aa5d1811d725895b3ab5a4f4daa9dcdd88294aa0d5f723e78bb1e0da", + "reuses_primary_capture": true + }, + { + "original_path": "coded-port-adapter-ceiling-20261023/outputs/released-operator-inventory-20261023/source/research__native-center-free-code__CODE.json", + "retained_path": "evidence/proof-review/frozen/CODE.json", + "sha256": "8be7f3f073b7981ec91c6af43edff0ac8a89f4824ef6026ae758c287dd6c03d0", + "reuses_primary_capture": true + }, + { + "original_path": "coded-port-adapter-ceiling-20261023/outputs/released-operator-inventory-20261023/source/research__coded-native-caller__PHYSICAL.md", + "retained_path": "evidence/proof-review/frozen/PHYSICAL.md", + "sha256": "4321b159ff8d3fe705419adaaa1beef75606e869e34341bda553f17d598a6078", + "reuses_primary_capture": true + } + ], + "raw_git_capture_distinction": { + "commit": "0e8a13c251a9be1029114cda8b4194f05f3a23ec", + "source_path": "research/coded-active-query-bound/PROOF.md", + "raw_path": "evidence/paid-control/captures/RAW-GIT-5126-PROOF.md", + "raw_sha256": "495a1cd85cc45020adf2205f196b57dc4f65f1eb763fc4951facdb4ccbafc768", + "monitor_capture_sha256": "96c617b306d50fce2b0f4f88afe1f6103d11326b402c2e4980c522c9c48092db", + "relation": "monitor capture equals raw Git bytes plus one LF; both retained unchanged, source provenance keeps raw Git pin" + } +} diff --git a/research/nonlinear-monomial-control-ceiling/evidence/paid-control/INDEPENDENT-REVIEW.md b/research/nonlinear-monomial-control-ceiling/evidence/paid-control/INDEPENDENT-REVIEW.md new file mode 100644 index 0000000000..92f63e0553 --- /dev/null +++ b/research/nonlinear-monomial-control-ceiling/evidence/paid-control/INDEPENDENT-REVIEW.md @@ -0,0 +1,111 @@ +# Independent review of the translation-valued paid-control gate + +**PASS within the stated fresh-input, translation-valued, commuting-X-Gaussian, FREE-terminal scope.** The ordered shear/inverse/raw-dual identities and the direct 228-H1 additional invoice are correct. The new r>=112 theorem uses a valid whole-translation-module cut; it does not repeat the invalid scalar identity-coefficient rank argument. No blocking mathematical gap was found under the retained source facts and explicit scope below. + +Reviewed author report: `outputs/operator-valued-paid-control-gate-20261023/REPORT.md`, SHA256 **`19932a49cda97acb2e3f4933f95d4bed5598e441d7593132bd07d15a63c23731`**. Scientific source pin: `4c0a359d88f22f05189d51f48387750b801a45d4`; #5126 proof pin: `0e8a13c251a9be1029114cda8b4194f05f3a23ec`. SOURCES.json authenticates the report's six declared inputs and seven supporting captures. This was a source-reading and exact algebra review. No operator evaluator, source-pair enumeration, solver, native, compiler, Lean or numerical experiment ran. + +## Literal one-child shear, inverse and raw dual + +Let U be a 112-by-1 operator column and V a 1-by-112 operator row, with `U_i=P_i`, `V_i=c_i P_i^-1`, `P_i=X_(q_i)` and 56 signs of each kind. Matrix multiplication gives `VU=(sum c_i)I=0` exactly at every column width. Therefore, in the noncommuting operator ring, + +``` +(UQV)^2 = UQ(VU)QV = 0, +(I+UQV)^-1 = I-UQV. +``` + +No assumption that Q is free or involutive enters this inverse. Source Pauli translations commute with the actual central Q and with D3, so `Q^-1 T Q=D3 T D3^-1=T`, including their ordered inverses. The actual Q square remains a Pauli translation. The fixed owner sum is evaluated on lifted address operators; it is not tensor-powered as a whole owner matrix. + +The literal schedule reads all original owners into an owned gather before updating them, applies one Q child, scatters and resets only owned fields. Its inverse reverses the scatter sign while using the same Q. Thus original arbitrary inputs are not silently replaced by scratch. For transpose rather than Hermitian adjoint, Q^T=Q and P_i^T=P_i, giving + +``` +T^-T = I-V^T Q U^T. +``` + +The raw-dual gather/scatter described by the author has the correct indices and signs: gather `sum P_i x_i`, then scatter `-c_i P_i g`. T^T changes that last sign. The source native backward X-only Q²/Ds² corrections remain separate. This proves the algebraic word on the declared normalized fields; it does not certify a physical Tape implementation, finite precision or a full controller emitter. The ordinary COPY/translation/ADD/reset counts are field-level allowances; initial zero-bank provisioning and all scans/precision remain nonzero work. + +## Direct whole-control recursive bill and original owners + +The middle block is implemented as `X <- T(-C1 X+QY)` and `Y <- -C3 T Y`, with QY obtained before overwriting old Y. It therefore uses two T applications plus the already-priced 112-H1 central query bank. At the main endpoint, t^-1 contributes one more T^-1 application. The exact ordered terminal inverse + +``` +E^-1 = Q C1^-1 T^-1 C3^-1 Q +``` + +runs rightmost first: Q on 112 fields, C3^-1, T^-1, C1^-1, then Q on 112 fields. Its price is 224+1=225 H1. Thus the direct additional control price is **2+1+225=228 H1**. The original 112-H1 query bank is already included in the base invoice and is not charged again. Two one-child shears use 4m+224=232 H1 under the same direct policy. These are upper construction bills, not optimality or paid-E lower bounds. + +The two middle recodings C1/C3 and two terminal inverse recodings are the stated four native C/C^-1 macros. Their scalar/monomial/odd-division costs, signs, query ownership and raw-dual normalizers remain separate nonzero obligations. Reusing a helper as scratch, deleting a complement or dropping a required inverse is not part of this bill. + +The original 224 main owners still receive rank-17 targets after the declared decoder, and all 816 distinct helper owners remain completed once to rank nine. The promised base price decomposes as `2P0+112H1+(94H1+14H2+56H5)`; those helper complements are retained. Adding the direct 228 H1 gives `P1=4836`, with other coefficients unchanged. It adds no productive cache/query/scratch targets. This is a complete **conditional recursive** census, not a complete physical tariff or a new admission of the helper-complement contacts. + +The independently frozen raw budget gives M(C) between approximately 44.67696 and 52.13312 for C in `[0.0030217758,0.00339999]`, with maximal uniform integer extra-H1 allowance 44. The direct 228-H1 policy therefore fails before ordinary work or compiler reserve. No root or compiler arithmetic was rerun for this review. That budget remains conditional on the unrealized unchanged T/P, not actual caller funding. + +## Source-pair coefficient isolation + +The essential new implication is `E FREE => projection_B(T)=0`, for the **entire translation subgroup algebra B**, not merely its scalar identity coefficient. I checked its ordered relative-word calculation. + +First make explicit the general commutation link: every allowed free coefficient is a sum of X translations, and every allowed paid child is an X-Gaussian. They all commute with Q and D3. Consequently every circuit entry of T also commutes with those maps, so t=T in the theorem's endpoint. This is why the displayed endpoint expansion applies to the general circuit, not only to the one-child example. + +Choose a nonzero coefficient of translation X_b in T[r,s], and choose i,j with A[i,r]A[s,j] nonzero. Such rows/columns exist in the pinned source. Its endpoint word is + +``` +W = Q^-1 K3_i^-1 K3_r X_b K1_s^-1 K1_j Q^-1. +``` + +X_b changes affine translation/phase data, not the symplectic linear obstruction. A monomial endpoint would require the first-stage central direction to have no U component after the intermediate source map: `G1_s^-1 q_j=e_q`, hence s=j by the distinct source directions. Its coefficient would then contain A[j,j]=0. Thus every supported chosen translation term is nonmonomial. This uses the actual disjoint q+U / q+W embeddings and signed source K words; it does not substitute scalar A for the physical C_s. + +If an A-group letter H at another owner pair (r',s') produced the same projective endpoint in this fixed i,j block, cancellation of the outside invertible factors gives exactly + +``` +H = phase * K3_r'^-1 K3_r X_b K1_s^-1 K1_s'. +``` + +The right side is monomial. H commutes with every X translation because it belongs to the commuting-X group. A monomial commuting with all translations must have constant multiplier and affine permutation x->x+a, hence is projectively a translation. Its binary linear part is identity, forcing + +``` +G3_r'^-1 G3_r = G1_s'^-1 G1_s. +``` + +The left map is supported on q+W, the right on q+U. Their intersection fixes U and W pointwise and can only fix the single common nonzero binary axis q; hence it is identity. Distinct source q_i then force r'=r and s'=s, and the full operator equation forces H=X_b projectively. Therefore neither another owner pair nor another address letter cancels the chosen coefficient. + +At a sufficiently large faithful width, that nonzero isolated nonmonomial coefficient is independent of every monomial word in a proposed fixed FREE endpoint implementation. E cannot then be FREE. Applying this to each nonzero translation coefficient proves the claimed projection condition. Signed phases are retained as scalar cocycles and do not change the linear-part or monomiality arguments. + +This review authenticates and uses the source facts—zero diagonal, nonempty 52-entry row/column supports, orthogonal embedded normalizers, distinct q_i—from their pinned predecessor checks. It does not newly replay all ports/pairs, the normalizer evaluator or the native all-input theorem. + +## Finite projective algebra and faithful widths + +The declared algebra must be fixed before width varies. Pauli-X translations have finite order; the actual X-Gaussians C_v commute and have finite projective order (their squares are translations). A fixed finite list therefore generates a finite projective address group A. The fixed monomial-Clifford endpoint alphabet is projectively finite on the same 17 one-column axes. Include all A letters, source endpoint products and proposed free endpoint letters in the fixed finite set used for separation. + +Choose representatives by projective class and group equal classes first. Multiplication at width f has the representative cocycle raised to f, while owner coefficients are applied once. Pairwise nonproportional operator vectors have linearly independent literal tensor powers at every sufficiently large f, by the retained separating-functionals or Gram argument. This gives faithful evaluation of A_f, B_f and the endpoint comparison spaces uniformly over width-dependent scalar coefficients. It does not assert a 112-call bound at every small width; width-one identities can have extra linear dependencies. + +No closure assertion that arbitrary monomial maps with arbitrary paid operators generate a finite group is being used. No tensor power is distributed over an owner sum, and no width-growing/noncolumnwise free alphabet is admitted. + +## Chronological module cut + +Saving all paid responses puts a data-independent linear r-call circuit into + +``` +y_k = H_k(b_k x + sum_(s B_f^r` has a nonzero complex-vector-space kernel column w, since its domain dimension is 112 dim(B_f) and codomain dimension is r dim(B_f). No division-ring rank, determinant, semisimplicity or absence of zero divisors is required. Choosing x=w z makes every paid input and response zero inductively, including feedback, so `T w=F0 w` lies in B_f^112. + +But `projection_B(T)=0` places T's entries in nontrivial right cosets of the translation subgroup in A. Right multiplication by entries of w in B preserves those cosets. Thus Tw also lies in their direct sum, disjoint from B_f^112, and Tw=0. At the chosen faithful width w evaluates nontrivially; one can choose z so w z is a nonzero actual input tuple. Actual invertibility of T then contradicts Tw=0. Correlated values in that tuple are permitted because correctness is required on every arbitrary tuple of distinct input fields. + +This proves **r>=112** for the declared model. It uses no extracted scalar coefficient-rank inequality; the earlier operator-valued counterexample remains valid and is not contradicted. It also supplies the new source-specific projection premise rather than merely applying #5126's named-endpoint theorem to an unknown T. + +## Scope and pricing boundary to retain in any follow-up + +Every nontrivial paid recursive Gaussian has rank at least one and price at least one on the declared interval. Hence a **fresh standalone T implementation in this model** already costs at least 112, exceeding the conditional 44-H1 allowance. This rules out the one-/two-child FREE-E proposals and every smaller-than-112-call implementation under these premises. + +It is **not** a lower bound for amortizing a control inside the complete released caller using already-paid QY/native/IDLE/centre observations. For example, a bank of previously supplied QY is nonfree initial information for a T(Y) subproblem and breaks the normal-form starting premise. The complete caller would need its own source access and full invoice. This exclusion should remain explicit beside the budget conclusion, not only implicit in the fresh-input theorem. + +The proof also does not cover arbitrary/nontranslation monomial gathers or C/K recodings inside the T circuit, noncommuting paid children, wider packed channels, approximate endpoints, data-dependent nonlinear controls, a paid E/E^-1, changed targets, or another native/controller contract. The 225-H1 direct E^-1 construction is not a lower bound for a cheaper paid decoder. A translation-only negative result must not be promoted to those broader classes. + +Two clarity additions are recommended for a focused follow-up, without changing the mathematics: state the general T/Q/D3 commutation link and finite-order generators explicitly; and repeat the fresh-standalone/no-already-paid-observation boundary beside the 44-versus-112 conclusion. The former is already implicit in the declared X alphabet; the latter prevents an invalid complete-caller inference. The ordinary field-pass counts should retain their stated separation from initialization, precision and complete Tape tariffs. + +A focused follow-up could publish the ordered one-child word, its direct conditional bill and the scoped translation-module theorem with these hypotheses and inherited source facts. It should retain paid E and nontranslation/noncommuting or correlated-native access as open routes, preserve predecessor credit, and make no native/physical/compiler or κ admission claim. No such PR or public post was created in this review. + +Credit the released Dillon template and #5126 input-kernel method, the source #4929/#4933 normalizers and code, the earlier #5160 intersection review, the scalar-rank counterexample and paper-worker conceptual check, and all retained native/helper/query/compiler/precision contributors. 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No source evaluator, native, solver, compiler, Lean or numerical experiment." +} diff --git a/research/nonlinear-monomial-control-ceiling/evidence/paid-control/captures/01-PAID-SHEAR-SCREEN.md b/research/nonlinear-monomial-control-ceiling/evidence/paid-control/captures/01-PAID-SHEAR-SCREEN.md new file mode 100644 index 0000000000..296523437f --- /dev/null +++ b/research/nonlinear-monomial-control-ceiling/evidence/paid-control/captures/01-PAID-SHEAR-SCREEN.md @@ -0,0 +1,120 @@ +# Stop certificate for scalar-owner X-Gaussian shears + +Local read/derive screen, requested after CONTRACT.md. No operator or +parameter search was run. This note is separate from the already reviewed +Clifford FREE proof and requires its own independent review. + +## Decision and scope + +The proposed affordable class cannot work: it requires at least 112 scalar +owner rank-one shears, whereas its optimistic complete query allowance +permits at most seven. This includes dense scalar owner vectors, not just +single E_(a,b) updates. A paid term creating one monomial endpoint component +does not eliminate the old nonmonomial identity component. + +The result covers + + t0 = S_m ... S_1, + S_k = I + u_k v_k^T tensor H_k, + +where u_k,v_k are scalar 112-owner vectors and H_k belongs to the group +generated by the Pauli-X Gaussians C_v used in CONTRACT.md. Coefficients +are applied once on owners. For the explicit inverse program one assumes +v_k^T u_k=0, but the necessary lower bound does not need that assumption. + +It does not cover operator-valued owner/address gather and scatter maps, +other intervening noncommuting control alphabets, retained native/query +observations, or a changed endpoint/controller. No universal fresh-query +lower bound is asserted. + +## 1. The address group has a restrictive monomial intersection + +Every C_v is a polynomial in the X-translation X_v, so every product H +commutes with every X-translation on the 17 address bits. If such an H is +monomial, write its action as a permutation p with nonzero multipliers c. +Commutation with all translations implies + + p(x+a)=p(x)+a, c(x+a)=c(x). + +Hence p(x)=x+b and c is constant: H is a scalar times X_b. This also +follows directly from the source Gaussian words. It is not an assertion +about all monomial Clifford operators. + +The C_v group is finite up to phase because these are Clifford operators +on a fixed 17-axis space. The all-width lift and projective phase convention +are the same as in PROOF.md. Let B0 be the 112 by 112 scalar coefficient of +the projective identity address word in t0 after equal classes are grouped. + +## 2. FREE terminal output forces B0=0 + +The endpoint is E=Q^-1 C3 t0 C1 Q^-1, since every H_k commutes with D3. +Fix an identity-word contribution at intermediate owner pair (r,s). +Before the fixed outer factors, its address word is + + K_(3,r) K_(1,s)^-1. + +A contribution from any H at any other pair (r',s') could coincide with +this class only if + + H = phase * K_(3,r')^-1 K_(3,r) K_(1,s)^-1 K_(1,s'). + +The right side is a signed binary linear permutation with no translation. +By section 1 the left side, if monomial at all, is a scalar X-translation. +Therefore its translation must vanish, so H is projectively the identity. +The disjoint-stage binary-map argument in PROOF.md then gives r'=r and +s'=s. Thus the identity-word contribution at (r,s) cannot cancel against +any nonidentity H or any different owner pair. + +For any nonzero B0[r,s], choose output i and input j with A[i,r] A[s,j] +nonzero, using the actual 52 nonzeros per source row/column. The associated +endpoint word + + Q^-1 K_(3,i)^-1 K_(3,r) K_(1,s)^-1 K_(1,j) Q^-1 + +is nonmonomial by the scalar theorem: it could be monomial only if r=i +and s=j, which would require zero source diagonal entries. Its coefficient +is nonzero and, by the preceding argument, cannot cancel against any +other contribution of t0. Uniform tensor-word separation therefore excludes +a fixed finite columnwise monomial implementation of E at sufficiently +large widths. Consequently a FREE E requires B0=0. + +## 3. At least 112 paid shears are necessary + +Let T_0=I and T_k=S_k T_(k-1). The exact telescoping identity is + + t0-I = sum_k u_k (v_k^T tensor H_k) T_(k-1). + +Take the coefficient of the projective identity address word. Because each +u_k is scalar in owner space, this gives + + B0-I = sum_k u_k w_k^T + +for some scalar row vectors w_k^T. In particular rank(B0-I)<=m. But section +2 requires B0=0, so + + 112 = rank(I) <= m. + +This statement holds at each sufficiently large width even if the scalar +coefficients vary with width; it excludes an all-width invertible solution +with fewer than 112 shears. It does not assert that 112 shears suffice. + +## Invoice consequence and reopening boundary + +The explicit middle/exit execution in CONTRACT.md costs 3m H2 in addition +to the released base profile. Since m>=112, even using H2>1 for C<1 gives +an added moment greater than 336, against the proven optimistic allowance +44.67695955358 at the comparison interval's right endpoint. The class is +decisively unaffordable before finite work, compiler or stock costs. + +Stop this class; do not enumerate seven-shear words or tune coefficients. +A continuation must change a mathematical premise, for example literal +operator-valued owner/address gather/scatter with a paid mixed sandwich, +or a named already-paid native/query observation. Such a changed program +needs its own source access, ownership, phase, inverse and full cost proof. +The present rank argument cannot be transferred to it merely by calling a +whole operator-valued query a scalar rank-one shear. + +Credit the released Dillon construction, Dugongue's source code and the +full native/Clifford/helper lineage recorded in PROOF.md. This additional +screen has substantial OpenAI Codex assistance; no human, proof-kernel, +compiler or scientific admission is claimed. diff --git a/research/nonlinear-monomial-control-ceiling/evidence/paid-control/captures/03-operator-valued-gather-rank-boundary.md b/research/nonlinear-monomial-control-ceiling/evidence/paid-control/captures/03-operator-valued-gather-rank-boundary.md new file mode 100644 index 0000000000..d5dd564d8e --- /dev/null +++ b/research/nonlinear-monomial-control-ceiling/evidence/paid-control/captures/03-operator-valued-gather-rank-boundary.md @@ -0,0 +1,47 @@ +# Why the scalar shear rank bound does not extend to operator-valued gathers + +2026-10-23. Elementary read/derive boundary, not a new paper result, a successful released endpoint, or a paid-query saving. No source/native program or search was executed. The main packed candidate is unchanged. + +The source-bound scalar screen is `outputs/released-operator-intersection-20261023/PAID-SHEAR-SCREEN.md`, SHA256 `81fefcf4a1be0be04076e7cba0e27ba4afe732d4810a13b0a76dbe827967c10c`. It proves at least 112 updates for products of scalar-owner rank-one X-Gaussian shears because extracting the identity address coefficient from the telescoping sum preserves scalar owner rank at most one per update. Its stated exclusion of operator-valued gather/scatter is essential. + +## A one-update counterexample to the proposed rank extension + +Use n=112 owners and choose n distinct fixed Pauli-X translations P_i on the existing 17 one-column address bits. There are more than n available; no new axis is needed. Use their exact columnwise lifts at every positive width. Distinct translations remain nonproportional, and P_i P_j^-1 is projectively the identity exactly when i=j. + +Define an operator column U and row V by + + U_i=P_i, V_i=c_i P_i^-1, + +where 56 coefficients c_i are +1 and 56 are -1. Then + + VU=(sum_i c_i) I=0. + +Consequently (UV)^2=0 and the single operator-valued shear + + S=I+UV + +has exact inverse I-UV. The projective-identity coefficient of its (i,j) entry is + + B0(S-I)[i,j] = c_i if i=j, and 0 otherwise. + +Thus rank(B0(S-I))=112, despite there being only one operator-valued update. In particular the scalar inequality rank(B0-I)<=m cannot be applied with m=1 here, even with the nilpotence condition that makes the inverse explicit. + +The ordinary physical interpretation is gather g=sum_i c_i P_i^-1 x_i before overwriting any x_i, then scatter x_i <- x_i+P_i g. All reads, copied source fields, Pauli maps, additions, owner routing and cleanup must be charged. No Gaussian call appears in this example. Since Vx'=Vx, the gathered value can also be recomputed from the final bank for an exact reversible cleanup outline. This statement supplies no measured finite Tape bill or precision/width implementation. + +## Even B0=0 alone does not identify a solution + +For a separate algebraic example let V_i=P_i^-1, so VU=112 I. Then + + T=I-UV, T^-1=I-(UV)/111. + +The inverse identity follows from (UV)^2=112 UV. Here B0(T)=0 exactly: every diagonal identity contribution cancels and all off-diagonal translations are nonidentity. The inverse coefficient 1/111 adds a real fixed-precision/ordinary-cost obligation; it is not declared free by this algebraic formula. + +This example shows that removing the identity coefficient is only a necessary endpoint condition in the scalar screen. It does not show that all other nonmonomial endpoint classes disappear. + +## Why neither example is a released construction + +Both examples are finite scalar sums of columnwise monomial Clifford operators. They commute with the Pauli-X Gaussians Q and D3, so their first three control expressions stay in that declared free algebra. The independently reviewed full Clifford obstruction still excludes their fourth endpoint from the required monomial span. + +Specifically, the source-bound proof `outputs/released-operator-intersection-20261023/PROOF.md`, SHA256 `c20b2f13e875d9e1fb6d0743693f9deccc4f7d68dcdc55ec241324ed3d7de7bc`, has independent PASS review at `outputs/released-operator-independent-review-20261023/REVIEW.md`, SHA256 `23b5aa5df049cb4a8fe3349c054e27d84ca151e29e263c658670639bad082428`. It forces the entire endpoint to vanish if it belongs to that fixed columnwise monomial-Clifford class, contradicting invertibility. S and T are therefore examples of a failed proof extension, not candidate t0 solutions. + +The always-valid ambient factorization inequality rank(S-I)<=dim(one gathered field) does not restore the scalar owner bound: extracting the identity group coefficient can increase scalar owner rank. To derive a useful paid-query lower bound for a changed gather model, one needs an additional whole-endpoint or information constraint tied to the actual queried field, width, source access and restoration. To claim a constructive saving, one needs an actual native/query observation and a complete mixed sandwich and inverse bill. Neither follows from naming an operator-valued gather. diff --git a/research/nonlinear-monomial-control-ceiling/evidence/paid-control/captures/04-PROOF.md b/research/nonlinear-monomial-control-ceiling/evidence/paid-control/captures/04-PROOF.md new file mode 100644 index 0000000000..6ddce7b794 --- /dev/null +++ b/research/nonlinear-monomial-control-ceiling/evidence/paid-control/captures/04-PROOF.md @@ -0,0 +1,395 @@ +# An active local-query obstruction for the actual 112-port code + +Prepared for Dillon, 2026-10-23, with substantial OpenAI Codex assistance. +This is a scoped exact local theorem with finite source checks, not a +complete-controller lower bound, a multiplication algorithm or an admission +claim. Apache-2.0. + +## Statement + +Use the actual physical native code from `research/coded-native-caller`: + +\[ + C_{ij}=a_{ij}K_i^{-1}K_j,\qquad J=Q^{-1}CQ, +\] + +where there are 112 owners, Q is the one-central-axis Gaussian, each K_i is +the retained signed orthogonal permutation normalizer, and the scalar matrix +a has zero diagonal and 52 nonzero off-diagonal entries in every column. +The 112 vectors q_i=G_i e_0 are distinct. The source factorization +C=I-14P has an eight-mode projector P. + +Fix any finite projective group H of invertible one-column monomial maps on the +declared fixed ambient address space, containing the source normalizers and +all admitted free address maps. For example, H may be the full group of +one-column permutations with phases in a fixed finite root-of-unity group; +the free permutations need not be Clifford. Every address map is lifted as +the SAME one-column map h on all f columns, h^tensor(f). Independent choices +on individual columns are outside this grammar. Free operations are +arbitrary linear owner programs over the +monomial subgroup algebra, including scalar operations, every +actual C or C inverse, owned COPY/CLEAR, arbitrary clean buffers, saved +banks, and fanout. These are charged as ordinary finite work; "free" here +means no recursive Gaussian call. Allow every paid unary call to apply an +arbitrary fixed one-field linear operator, with chronological linear +feedback from all available fields. In particular its input may use all +earlier paid responses. All coefficients are data-independent, although +they may depend on width. Every counted field has the same address width as +one source owner; a wider packed channel is not counted as one field. + +**Theorem.** A standalone exact implementation of J, or of J-I, on 112 +arbitrary fresh owner fields requires at least 112 paid unary field calls. +The statement is valid for an identity at all positive tensor widths, and +allows the scalar owner coefficients to depend on width. The same conclusion +holds after adding any free owner operator to J. +More precisely, the bound holds at every sufficiently large tensor width, +so no all-width implementation uses fewer than 112 calls. It is not a bound +at every small width: at width one, +Q=((1+i)/2)I+((1-i)/2)X is in the scalar span of free monomials. + +**Sharp one-sided corollary.** Put P=(I-C)/14. Each of QP and PQ, as a +112-input/112-output owner operator, requires at least 112 paid unary field +calls. Exactly 112 H1 calls suffice: apply P for free and then Q to every +output, or apply Q to every input and then P. Thus the eight scalar code +modes do not yield an eight-call one-sided Q repair. The statement also +holds with Q replaced by Q inverse. + +**Canonical eight-mode corollary.** For the actual source mode matrix M, +let U_mode=diag(K_i^-1)M and V_mode=M^T diag(K_i)/56. Then +P=U_mode V_mode and V_mode U_mode=I_8. Each of Q U_mode (eight fresh inputs, +112 outputs) and V_mode Q (112 fresh inputs, eight outputs) also has exact +paid-field-call complexity 112, attained by 112 H1 calls. Thus the canonical +rectangular mode ABI does not avoid the one-sided obstruction. + +It does not cover receiving Gaussian-dependent saved data, absorbing the +correction into the paid native FULLs or their prefixes, recycling an existing +paid IDLE, or a direct coded consumer with a different endpoint. Nonlinear +controls, noncolumnwise address operations, approximate endpoints, and +unequal-width packing are outside the stated model. + +## 1. Phase-correct finite algebra and all-width evaluation + +This uses the finite-phase algebra and sufficiently-large-width tensor +faithfulness already proved in `coded-port-adapter-ceiling/PROOF.md`, section +3, and clarified by the passive saved-query package's uniform-alphabet +addendum. Here only the FREE group H must be finite projectively. An ambient +group generated by H and Q need not be finite or Clifford. + +Choose one invertible matrix representative for every projective element +of H. At width f its nonzero scalar multiplication cocycle is raised to +the f-th power; the scalar factors need not have modulus one. +Let B_f be the resulting finite-dimensional twisted group algebra over +k=C, with one basis element per projective element of H. Its evaluation +is the span of the literal tensor lifts h^tensor(f). Scalar factors are +retained through the cocycle; owner coefficients are applied only once. + +Besides H itself, we will need only a fixed finite union of its projective +cosets: the sets g_ij H used for J, the analogous right cosets for PQ and +V_mode Q, and the analogous left cosets for QP and Q U_mode. Take S to be +their union with H, with repeated projective elements identified. This set +is finite because H and the source owner/mode sets are finite. Closure of +S under multiplication is not needed: all free products remain in H, and +every target-times-free expression used below stays in its declared coset +set. A different fixed finite number of source stages merely enlarges S. + +For projectively distinct nonzero representatives v_1,...,v_N, individually +normalize each by its Hilbert-Schmidt norm. Their tensor-power +Gram matrix has diagonal one and off-diagonal entries +^f, whose magnitudes are at most rho^f for some rho<1. For +(N-1)rho^f<1 the Gram matrix is strictly diagonally dominant and positive +definite. Apply this to the projective representatives in S. Their literal +tensor lifts are linearly independent for every sufficiently large f. +Thus B_f and all the required coset subspaces evaluate faithfully, with +distinct cosets in direct sum, at those widths. Scalar factors remain raised +to the f-th power; owner coefficients are never raised to that power. + +We work at any one such width. The threshold is uniform over all free +coefficients, control choices, and control lengths because H and S are +fixed. Width-dependent scalar coefficients are allowed. A width-dependent +or noncolumnwise group of arbitrary payload maps is not allowed; neither +is an infinite projective phase grammar. No assumption that arbitrary +permutations together with Q generate a finite group is made. + +## 2. Actual-source right-coset separation + +Write g_ij=Q^-1 K_i^-1 K_j Q. For a fixed i, + +\[ + g_{ij}^{-1}g_{ik}=Q^{-1}K_j^{-1}K_kQ. (1) +\] + +For the signed orthogonal permutation M=K_j^-1 K_k, let G=G_j^-1 G_k be +its binary orthogonal coordinate matrix and e=e_0. Pauli signs affect only +tableau phases. The Z-to-X block of Q^-1 M Q is + +\[ + B=G e e^T+e e^T G. (2) +\] + +If G e=e, orthogonality gives e^T G=e^T, and B=0. If G e differs from e, +then both pairs (G e,e) and (e,G^T e) are independent; the two outer products +in (2) have rank two. Thus Q^-1 M Q is monomial exactly when G e=e. +Equivalently, G_j e=G_k e, or q_j=q_k. The actual 112 source vectors are +distinct, so (1) belongs to H exactly when j=k. + +Therefore the 112 g_ij lie in distinct right H-cosets for each fixed i. +The j=i coset is H itself, since g_ii=I. The scalar phase quotient does not +alter this test: multiplying by a scalar never changes monomiality. +The companion checker verifies all 112^2 ordered pairs by both formula +(2) and the retained exact tableau implementation. It separately checks +the actual code's zero diagonal and nonempty column supports. + +## 3. No nonzero free input survives modulo a free output + +Let F_0 be ANY 112-by-112 matrix over B_f. Suppose w in B_f^112 obeys + +\[ + (J-F_0)w=0. (3) +\] + +Row i of Jw is sum_j a_ij g_ij w_j. Each summand lies in the right-coset +subspace g_ij B_f. These subspaces are in direct sum. Every component of +F_0 w lies in B_f, the H coset. The diagonal source coefficient is zero. +For each off-diagonal supported pair (i,j), projecting (3) onto g_ij B_f +therefore gives a_ij g_ij w_j=0. A nonzero scalar and a group unit are +invertible, so w_j=0. Each column j has a supported off-diagonal entry, +and hence w=0. + +This is stronger than an eigenvector statement: J-F_0 has zero kernel on +B_f^112 for every free F_0. In particular the Q-conjugated eigenspaces of +C contain no nonzero free input embedding, despite the eight-mode +factorization of C-I. A cheap compression of J-F_0 through fewer than 112 +owner fields is impossible. + +### One-sided projector operators + +Write p=(I-a)/14, so P_ij=p_ij K_i^-1 K_j and p_ii=1/14. For PQ, fix row +i. Its blocks are p_ij h_ij with h_ij=K_i^-1 K_j Q. Every h_ij has Fourier +grade one and lies outside H. Further, + + h_ij^-1 h_ik = Q^-1 K_j^-1 K_k Q. + +Thus the h_ij occupy distinct right H-cosets as j varies. The same argument +proves ker_{B_f}(PQ-F_0)=0 for every free F_0; the nonzero diagonal of p +already supplies a supported entry in each column. + +For QP fix column j instead, and put t_ij=Q K_i^-1 K_j. These grade-one +words lie in distinct LEFT H-cosets as i varies, because + + t_ij t_kj^-1 = Q K_i^-1 K_k Q^-1 + +is nonmonomial for i different from k. Consequently a free row w satisfying +w(QP-F_0)=0 must vanish: project each fixed output column onto the distinct +left-coset subspaces B_f t_ij, and use p_ii different from zero. This is the +left-kernel version of the same source-specific separation. + +### Canonical rectangular eight-mode maps and free rebasing + +The retained 112-by-8 scalar mode matrix M obeys M^T M=56I_8 and +a=I-MM^T/4; each row has four nonzero entries. The checker verifies every +entry of these identities from the pinned source receipt. Define + + U_mode = diag(K_i^-1) M, shape 112 by 8, + V_mode = M^T diag(K_i)/56, shape 8 by 112. + +These maps are free, with P=U_mode V_mode and V_mode U_mode=I_8. +For Q U_mode the fixed mode-column a has entries M_ia Q K_i^-1. The words +t_i=Q K_i^-1 lie in distinct left H-cosets, since +t_i t_k^-1=Q K_i^-1 K_k Q^-1 is nonmonomial when i differs from k. +Each t_i itself is nonmonomial. Every row of M has a nonzero entry, so +the same columnwise projection proves that + + w (Q U_mode-F_0)=0, w in B_f^(1 by 112), + +forces w=0 for every free 112-by-8 F_0. Thus the no-left-kernel conclusion +holds even though there are only eight fresh inputs. + +Dually, V_mode Q has entries M_ja K_j Q/56. The words K_j Q lie in +distinct right H-cosets because their pairwise relative transports are +Q^-1 K_j^-1 K_k Q. Every source column is represented in at least one mode +row. Hence (V_mode Q-F_0)w=0, with w in B_f^112, forces w=0 for every free +8-by-112 F_0, even though there are only eight outputs. + +These conclusions cover all free changes of retraction basis. If another +free pair U',V' has P=U'V' and V'U'=I_8, then + + W=V_mode U', W^-1=V' U_mode, + U'=U_mode W, V'=W^-1 V_mode. + +The two inverse identities follow by substituting U'V'=P and +U_mode V_mode=P. Thus Q U' and V'Q differ from the canonical operators by +invertible free input or output boundaries. No cheaper implementation is +obtained by any such free eight-mode basis choice. + +## 4. Arbitrary chronological paid calls + +Normalize any finite acyclic linear implementation by retaining all paid +responses y_1,...,y_r. COPY, saves, fanout, overwrite and CLEAR are included +in this normalization: enlarging owned storage can preserve the history +without introducing another recursive call. At paid call t its input is + +\[ + b_t x+\sum_{s B_f^r. As a k-linear map its +domain has dimension 112 dim_k B_f and its codomain has dimension +r dim_k B_f. If r<112, choose a nonzero w in its kernel. Substitute the +formal input x=w z, with z one arbitrary field. Every direct term b_t w is +zero. Induction on t in (4) gives every paid response y_t=0, regardless of +T_t and regardless of the active feedback coefficients. Thus the claimed +endpoint J must satisfy Jw=F_0w. Section 3 contradicts this. + +This proves r>=112. Replacing J by J-I or J+F_free merely changes the free +matrix F_0 and gives the same result. + +The input-kernel proof applies verbatim to PQ. For QP use the dual output +cut: concatenate the free output columns in (5) into L in B_f^(112 by r). +If r<112, the k-linear map w -> wL from B_f^(1 by 112) to B_f^(1 by r) +has a nonzero kernel row w. Multiplying the endpoint by that row annihilates +every paid-response term, so w(QP-F_0)=0, contrary to the left-kernel result. +No assumption about the paid calls' chronology is needed beyond the +existence of the finite linear circuit normal form. + +For Q U_mode the same output cut has 112 rows and r paid-response columns, +although F_0 has only eight input columns. It again gives r>=112. For +V_mode Q the same input cut has 112 input columns, although F_0 has only +eight output rows, and also gives r>=112. Both bounds are attained by +112 H1 calls: expand eight fields through U_mode for free and apply Q to +all 112 results; or apply Q to the 112 inputs and contract through V_mode +for free. Invertible free retraction basis changes preserve these counts. +The same arguments apply with Q inverse throughout. + +## 5. Direct sums and local amortization + +For any fixed positive m, consider m fresh physical code blocks, embedded +in one fixed ambient address space. Each block may use its own eight fresh +axes and physical normalizers, with the same 112 distinct-direction +property. Let N=112m. The same proof gives + +* diag(J_1,...,J_m), or its free perturbation: at least N paid field calls; +* diag(P_1 Q,...,P_m Q): exactly N H1 calls are necessary and sufficient; +* diag(QP_1,...,QP_m): exactly N H1 calls are necessary and sufficient. +* diag(Q U_mode,1,...,Q U_mode,m), and its free retraction rebases: + exactly N H1 calls are necessary and sufficient; +* diag(V_mode,1 Q,...,V_mode,m Q), and its free retraction rebases: + exactly N H1 calls are necessary and sufficient. + +Free controls may mix different blocks arbitrarily. To see why this does +not change the proof, write a candidate free kernel column or row in m +blocks. Project each target block's rows or columns onto its nontrivial +cosets. The free comparison operator still contributes only the H coset, +even when it mixes all N owners. The corresponding single-block argument +forces every block of the candidate kernel to vanish. Apply the input or +output dimension cut with B_f^N and r=1 and price at least one, so the local correction costs at least 112. +Its added recursive price exceeds the removed-center saving by at least +48 per FULL. The direct-sum theorem gives a gap of at least 48m for m such +standalone local repairs, even when their free controls mix all inputs. + +Thus this isolated interface-restoration strategy cannot turn the removed +eight H8 calls into a recursive saving anywhere in 0<=c<1. Extra ordinary +work can only increase the cost. The conclusion does not price or rule out +absorption into correlated FULL, IDLE, or saved-query data in a complete +caller, nor a different direct coded endpoint. + +### Relationship to predecessor and active caller work + +The predecessor `coded-port-adapter-ceiling` (#4932) proves that one parallel +bank needs 112 grade-2 channels for its physical recovery adapter, using a +grade projection of full rank. It expressly excludes multiple chronological +banks. The argument here uses source-specific distinct right cosets and an +input-kernel cut to cover arbitrary chronological paid unary calls and +active feedback. Its conclusion is 112 calls of any grade; it does not +claim 112 grade-2 calls in this enlarged model. + +The conclusion excludes an independently implemented 16-, 32-, or 42-H1 +fresh-input correction of J, J-I, QP, or PQ through the native code's eight +modes. For J and J-I this is a lower bound, with the elementary 224-H1 +conjugation implementation as an upper bound; 112-call attainability is +not asserted. For QP and PQ the 112-H1 bound is attained. +It does not prove that p2 needs four thin banks, nor that a complete active +three-bank caller is impossible. Such a caller can already hold nonfree +data from its FULLs, queries, and IDLE. A useful construction must exploit +that shared paid access, change the endpoint, or leave this local grammar. +The particular eight-mode ABIs Q U_mode and V_mode Q, including their free +retraction rebases, are covered above. An arbitrary different eight-mode +endpoint, the middle compression V_mode Q U_mode, and the two-sided +compression PQP remain outside the claims. + +The actual #4932 entrance adapter is also covered by free equivalence. Its +physical monomial X wrapper is M_i=U_i K_i Q^-1. Therefore +U a U*=M (Q C Q^-1) M*, and Q C Q^-1=Q^2 J Q^-2. Since Q^2 and M are free, +and a^-1=(a+12I)/13, the physical U a^-1 U* differs from J by invertible +free boundaries, a nonzero scalar, and a free bypass. These operations +preserve the local paid-call lower bound. This correspondence uses the +retained exact physical wrapper identities; it does not replace U_i with +a bare Fourier along q_i. + +## Source and predecessor credit + +The source code and native producer are Dugongue's #4929. The actual signed +orthogonal normalizers and physical C formulation come from the retained +`coded-native-caller` package. The finite-phase group algebra, tensor +faithfulness, adapter scope and warning that rank-eight scalar modes are not +eight query heads come from #4932. The uniform finite-alphabet clarification +and related passive-controller limitations come from the retained passive +saved-query package. Credit also Cinna Davis/1337isnot1337's #4694/#4698, +Cao Yifan/intqwq's #4610 source, zxv's retained exact Clifford/compiler work, +and all source notices. This active local input-kernel argument was derived +in the current research session with substantial OpenAI Codex assistance. + +No independent human or formal-kernel acceptance is claimed. The source +native all-input theorem, numerical outer admission, and complete caller +endpoint are inherited or open, not replayed by this scratch result. + diff --git a/research/nonlinear-monomial-control-ceiling/evidence/paid-control/captures/05-REPORT.md b/research/nonlinear-monomial-control-ceiling/evidence/paid-control/captures/05-REPORT.md new file mode 100644 index 0000000000..8a200e2535 --- /dev/null +++ b/research/nonlinear-monomial-control-ceiling/evidence/paid-control/captures/05-REPORT.md @@ -0,0 +1,230 @@ +# Released two-FULL/one-query problem: source inventory + +Source inventory only, 2026-10-23. No t0 solver, search, native replay, compiler, +factory or external post was performed. The monitor authenticated Dillon's +release of original #4930 comment 19141406 and the new handoff at comment +19143357. Its exact local receipt is preserved as `source/RELEASE-MONITOR.json`. +The public theorem is `0e8a13c251a9be1029114cda8b4194f05f3a23ec`, +`research/coded-active-query-bound`; the actual code/normalizer scientific pin +is `4c0a359d88f22f05189d51f48387750b801a45d4`. The publication base is +`20a1e31bc6009b9b3f4d084bbaadc6350e34fcba`. Eight exact source files and their +hashes are listed in `SOURCES.json`. + +## Stock and actual operators + +There are two main banks of 112 arbitrary independent owner fields each. +Each field has the fixed 17-axis address stock: one shared central axis q, +eight stage-1 axes and eight disjoint stage-3 axes, lifted to f columns, plus +all retained external row/spectator fields. There is no new anchor or extra +address axis. Two fresh source FULLs add two disjoint banks of 408 arbitrary +helper inputs: 816 helper owners, 1040 original main-plus-helper owners total. +An owned query bank is dependent workspace, not productive input credit. + +Q is the actual central Gaussian, D1 and D3 the products on their respective +eight axes. They are invertible symmetric actual Gaussian operators; the +different axis factors commute. Q is not order two: its exact square is a +Pauli translation. Put G=D1 D3 Q, a rank-17 Gaussian on the complete stock. + +The physical code is not the scalar matrix A. The source gives + + Cs[i,j] = A[i,j] Ki,s^-1 Kj,s, + Fs = diag(Ds,Ds) L(Cs Q^-1). + +The exact local K words, G_rows, Z_mask and compiled monomial wrappers are in +`NORMALIZERS.json`. Embed their nine local coordinates as q plus the selected +stage's eight coordinates, with the other stage as identity. These are stored +matrix ROWS; replacing them by columns changes the operator. The K maps are +real signed orthogonal permutations, commuting with Ds Q but not generally +with Q or Ds separately. Their all-column lift retains the accumulated phase; +owner coefficients are applied once, not tensor-powered with the address map. + +## Released sufficient template and all four conditions + +All controls are 112-by-112 owner-operator matrices on this same field width. +They are not necessarily single Clifford operators. Seek an invertible t0 +such that each of the following has an explicit physically free program AND +an explicit program for its inverse: + +| Control | Definition | Mathematical inverse, in exact order | +|---|---|---| +| R1 | t0 | t0^-1 | +| R2 | S0=Q^-1 t0 Q | Q^-1 t0^-1 Q | +| R3 | t=D3 t0 D3^-1 | D3 t0^-1 D3^-1 | +| R4 | K=Q^-1 C3 t C1 Q^-1 | Q C1^-1 t^-1 C3^-1 Q | + +The released middle operator is + + J(t0) = [[-t0 C1, t0 Q], [0, -C3 S0]]. + +Under the source identities, the handoff states + + M = F3 J(t0) F1 + = [[0,t],[-K,0]] diag(G,G). + +The cancellation uses the actual source relations [C1,D1 Q]=0 and +[C3,D3 Q]=0, plus [C1,D3]=0 from the disjoint fresh-axis embedding. +For example the upper-left block of J F1 is +t0(-C1 D1+D1 Q C1 Q^-1)=0; the lower-right block of F3 J F1 cancels +by S0=Q^-1 t0 Q. No commutation of t0 with either Gaussian is assumed. + +Writing S=[[0,t],[-K,0]], its full main endpoint is + + Xout=t G Yin, Yout=-K G Xin. + +There is no main/helper leakage in the inherited FULL endpoint. The helpers +retain their autonomous source transformations. If the four explicit free +program/inverse requirements hold, the post-map + + S^-1 = [[0,-K^-1],[t^-1,0]] + +would return the main outputs to diag(G,G) on the 224 original inputs, with +the two banks routed back to their declared owners. Invertibility alone is +not a physical implementation of this post-map. + +The mathematical raw inverse transpose is, using transpose rather than +Hermitian adjoint, + + M^-T = S^-T diag(G^-1,G^-1), + S^-T = [[0,t^-T],[-K^-T,0]]. + +An actual reversed/transpose tape must preserve the source raw-dual convention, +all phases and owned-buffer lifecycles. For a source backward FULL the retained +normalization is X-only Q^2 conjugation, with D_s^2 output corrections; the +older Y-only convention cannot be substituted. The formulas here do not emit +that tape or automatically provide physical transpose programs for a generic +algebraic unit. Returning inverse-Gaussian jobs to positive jobs likewise +requires the actual Pauli maps, not an order-two assumption. + +## One paid query and chart contacts + +A direct conditional implementation of J first obtains QY in 112 owned query +fields, then forms -t0 C1 X+t0(QY) and -C3 S0 Y using the required free programs. +All reads of the old Y must finish before Y is overwritten. This explains the +one bank of 112 H1 queries. The query bank is cleared/reset only under the +retained owned-stream rules. A concrete safe COPY/query/use/reset route has +112 query fields; additional scratch for t0 or the free recodings must be +separately allocated and charged. No helper is scratch and no copied stream +earns independent output credit. The handoff does not publish a full emitter. + +Each FULL uses its actual stage-local normalizers. The source forward wrappers +at each port i, rightmost acting first, are + + input X: Ei Ki Q^-1, input Y: Ki, + output X: Ki^-1, output Y: Ds Ki^-1 Si^-1, + +where Ei,Si are the retained actual entrance-line/exit-hyperplane operators. +The complete products shown by the source are monomial; their factors cannot +be individually dropped or declared free solely from a frame label. The +middle J and final controls act on the normalized physical main fields over +all 17 axes. They must match those exact contact coordinates and preserve the +unused stage block as a spectator where required. + +## Promised recursive invoice and the helper complement + +The inherited center-free native price and helper outputs per FULL are + + P0 = 2201H1+412H2+510H3+127H4+40H5, + H = 28H4+7H7+47H8+326H9. + +The released promised complete target/paid census is + + T = 224H17 + 816H9, + P = 4608H1+838H2+1020H3+254H4+136H5. + +Its accounting decomposes exactly as + + P = 2P0 + 112H1 + (94H1+14H2+56H5). + +The last bracket is the complement price for the two native helper banks: +94 rank-eight jobs need rank-one complements; 14 rank-seven jobs need rank-two +complements; 56 rank-four jobs need rank-five complements. The remaining 652 +are already rank nine. Thus all 816 helper original inputs are credited once, +and no original rank-four/seven/eight reward is also retained after counting +its completed rank-nine job. The price has rank mass 11040 and the target +11152, giving the stated raw surplus 112. + +This arithmetic is consistent with the handoff, not an independent native +certificate for its unpublished construction. The actual complement words +must meet the helper's physical entrance/exit charts and retain the other +eight-stage coordinates as spectators, in both orientations. A rank difference +alone does not prove those physical contacts or their monomial adapters. The +source FULL/autonomous-helper premise is inherited; this review performs no +new 816-owner phase campaign. Nor does the positive raw surplus certify a +moment, finite compiler reserve, precision or outer transfer. + +Known ordinary source charges remain nonzero: 448 port-wrapper field maps per +FULL, up to 448 further maps per backward normalization, and 6596 dyadic +precompensation ADDs per native FULL in addition to the original native work. +C_s or C_s^-1 has a source program with 224 K maps, 896 scalar ADDs and 16 +owned CLEARs, using eight reusable owned mode fields. The inverse coefficient +is -1/52; its odd-denominator precision work is not free. Counts of these +macros in J/output decoders depend on the proposed t0 programs and cannot be +invented here. Query COPY/reset, inverse-query Paulis, routing, all buffers, +heights and full fixed-tape cost remain charged obligations. + +## Actual free dictionary versus a theorem's enlarged algebra + +The public active-query theorem deliberately grants arbitrary finite linear +owner programs over a fixed finite projective group H of one-column monomial +maps, possibly non-Clifford, to strengthen its LOWER BOUND. Each map is lifted +as the same h in all f columns. It permits width-dependent scalar coefficients +within that mathematical model and proves its obstruction at sufficiently +large f. It does not present an executable tariff for every such H map, every +element of its span, or every invertible matrix over that span. + +The construction handoff expressly asks for EXPLICIT free implementations. +Its actual known source dictionary is narrower and operational: + +* elementary scalar owner ADDs/mixing, signs and owner routing, with fixed + coefficients computed and rounded under the caller's precision rules; +* owned COPY/reset/clear with original independent fields preserved; +* the exact K_i,s and inverses, their supplied phase-correct P/D/X words, + ordinary phases/Paulis and any explicitly compiled fixed coordinate map; +* C1,C3 and their inverses through the concrete eight-mode scalar programs; +* Q^2 and Ds^2 Pauli maps with their actual finite passes. + +This dictionary is closed under concatenating actual finite programs. A unit +must have a returned inverse program; mathematical nonsingularity supplies +neither its finite word nor its cost. There is no admitted closure rule saying +that conjugation by Q or D3 preserves this dictionary. In particular a word +with an even Gaussian count can still be nonmonomial/nonfree. Generic finite-H +or full Clifford-span hypotheses from a different theorem must not be silently +installed here. Arbitrary width-dependent coefficients also need a uniform +descriptor, bit-height and runtime prescription, not a unit-cost oracle. + +There is also a general exact monomial-Clifford constructor in the retained +symbolic source, not just a table of K_i: `normalizers.py` lines 123–127 calls +`ce.monomial(op)`, checks exact tableau and phase, and emits only P/D/X/phase +letters. A specified rank-zero Clifford candidate can therefore be represented +by a literal fixed-axis word and its inverse. It still needs its 17-axis +embedding, all-column phase lift and finite indexed-movement/precision bill +under the retained monomial implementation. This is distinct from granting +every arbitrary non-Clifford permutation in the theorem's enlarged H. No new +letter was synthesized or tested in this review. + +### Smallest concrete units subclass for the design lead + +A useful source-grounded first class is t0=B tensor I_address, where B is a +fixed scalar owner matrix accompanied by a finite elementary program and its +inverse. A particularly strict finite generator alphabet is owner ADD with +coefficients +1 or -1, owner swaps and signs. Any fixed word in these generators +has an explicit inverse by reversing and inverting its gates, uses no Gaussian +call or new axis, and has a finite ordinary cost. B is then unimodular. Other +fixed rational B can be admitted only with their explicit scaling/division and +precision programs. This is a candidate subclass, not a claim of a solution. + +For such scalar B, the first three conditions reduce to B and B^-1 because +they commute with Q and D3. The remaining test is the actual ordered operator +Q^-1 C3 B C1 Q^-1 and its inverse. The source normalizers fix its phase-correct +17-axis terms. Astra owns the creative intersection derivation; no linear +system, determinant test, t0 candidate enumeration or operator solver is run +by this inventory review. Proving a cancellation in a one-column matrix alone +would not justify every width: an all-column word identity and its physical +finite program are still required. + +Credit Dillon for the released template and active local theorem, Dugongue for +#4929, the #4933 normalizers/coded ABI authors, Cinna Davis, Cao Yifan, zxv, +Jacob Sussman and all retained native/helper/compiler predecessors. Prepared +with substantial OpenAI Codex assistance; Apache-2.0. The joint #5124 mechanism, +finite-packing work and the other agents' operator-intersection design remain +separate ownership scopes. diff --git a/research/nonlinear-monomial-control-ceiling/evidence/paid-control/captures/06-REVIEW.md b/research/nonlinear-monomial-control-ceiling/evidence/paid-control/captures/06-REVIEW.md new file mode 100644 index 0000000000..f06f34d333 --- /dev/null +++ b/research/nonlinear-monomial-control-ceiling/evidence/paid-control/captures/06-REVIEW.md @@ -0,0 +1,190 @@ +# Independent review of the released t0 operator-intersection argument + +**PASS for the revised, explicitly scoped theorem.** No all-width invertible +t0 in the fixed finite columnwise monomial-Clifford/scalar-owner control class +satisfies the four declared free-membership conditions. The direct scalar-owner +subclass proof also passes. This is not a classification of physical FREE, a +universal obstruction to the released caller, or an application of the enlarged +query-lower-bound alphabet as an executable free dictionary. + +Reviewed proof SHA256: +`c20b2f13e875d9e1fb6d0743693f9deccc4f7d68dcdc55ec241324ed3d7de7bc`, +copied unchanged as `frozen/REVIEWED-PROOF.md`. Twelve source/proof/inventory +captures authenticate in SOURCES.json. The scientific source is +`4c0a359d88f22f05189d51f48387750b801a45d4`, with the actual CODE.json and +NORMALIZERS.json; the released handoff is Discussion4930 comment19143357. +Public #5126 `0e8a13c251a9be1029114cda8b4194f05f3a23ec` supplies separate +local-query context, not the constructive physical-free alphabet assumed here. + +## 1. Source and model discipline + +The fixed address space is q+U+W, with one central and two disjoint eight-axis +blocks:17 axes, lifted as the same one-column map on every f column. The owner +matrix A has zero diagonal, distinct source port directions q_i=G_i e_q, and +nonempty rows/columns. The source polynomial +`A^2+12A-13I=0` gives the exact inverse `(A+12I)/13`, hence each physical code +`C_s=diag(K_s)^-1 A diag(K_s)` is invertible at every width. These are the +authenticated source premises/static source-check results; no new all-port +operator evaluation or source native replay was performed for this review. + +The declared H is the projective group of **monomial Clifford** maps on those +17 one-column axes. It is finite projectively. Fixed finite scalar owner +programs, COPY/history and owned clear/reset yield matrices over its linear +span; coefficients may depend on width. Such a span is a mathematical +relaxation of explicit programs. Membership is necessary for a program in this +class; it is not by itself an executable tariff or an inverse program. + +The proof does not assume an arbitrary nonsingular algebra element has a free +inverse. H, its conjugates and their intersection are groups and are inverse +closed, but that abstract fact supplies no physical implementation for a unit +of their matrix algebra. The forward membership obstruction already excludes +the handoff's invertible solution; proving inverse-program sufficiency is +unnecessary for this negative result. + +## 2. Support and width separation are valid + +For pairwise nonproportional nonzero operator vectors v_i, choose, for each +j!=i, a functional annihilating v_j but not v_i. Their tensor product isolates +v_i^(tensor(m-1)); extra factors nonzero on v_i extend the argument to larger +widths. Thus any fixed finite union of projective classes has independent +literal tensor lifts for every sufficiently large f. Equal projective classes +are grouped first and their scalar phases are raised to f. This handles +width-dependent owner coefficients without tensoring an owner-space sum. + +The union must include the candidate control letters, their Q/D3 conjugates, +the terminal endpoint products and the letters of a proposed free endpoint +program. It is finite in this declared fixed-columnwise Clifford class. The +threshold is not claimed small or practical. No assertion is made at every +small width: a family supported only at finitely many small widths can evade +the nonzero-family wording, but cannot meet all-width invertibility. + +The precise conclusion is pointwise at sufficiently large width: any t0_f +satisfying all four memberships must be zero there. It excludes every fixed +nonzero t0 and every all-width invertible family, including width-dependent +scalar coefficients. It does not assert that no exceptional small-width +identity exists. + +## 3. D3 transport and the necessary quotient flag + +The first three controls put surviving t0 support in + +`H0=H intersect Q H Q^-1 intersect D3^-1 H D3`. + +For t=D3 t0 D3^-1 the correct transported support is +`Ht=D3 H0 D3^-1`. It must not be identified with H0. In particular +`Q^-1 t Q=D3(Q^-1 t0 Q)D3^-1` is not automatically monomial. The revised +proof correctly retains this failure of the tempting closure inference. + +On binary symplectic space Q and D3 are involutions, while their operator +squares remain nontrivial Pauli/phase maps. If h lies in H0, it stabilizes +L0,Lq,Lw. A transported k in Ht stabilizes L0,Lw,Lqw. Because k is monomial, +its X block R_k is invertible and preserves both W and span(e_q)+W. It fixes +the nonzero central class in the quotient by W; it need not fix e_q itself. +The allowed behavior e_q -> e_q+w is precisely why the stronger false +transport inference cannot be reused. + +For **nonzero** v in span(e_q)+U and nonzero w in span(e_q)+W, if +k L(v)=L(w), with `L(v)={(v v^T z,z)}`, X projection gives R_k v=w. +The induced invertible quotient map has the central class as a fixed point, +so the preimage of its span is that span. Therefore v=e_q, while w may be +e_q plus a W vector. This proves exactly the one-sided implication needed; +it does not establish w=e_q. + +## 4. The endpoint has no monomial support: cancellation does not rescue it + +A terminal endpoint block has terms + +`Q^-1 K_(3,i)^-1 K_(3,r) k K_(1,s)^-1 K_(1,j) Q^-1`, + +with coefficient `A[i,r] b[r,s,k] A[s,j]` and k in Ht. For such a word to be +monomial, its central Gaussian chart must be carried to the central chart. +Using the actual orthogonal source normalizers, this requires k to carry + +`L(G_(1,s)^-1 q_j)` to `L(G_(3,r)^-1 q_i)`. + +The quotient lemma forces `G_(1,s)^-1 q_j=e_q`, hence s=j by distinct source +directions. Its coefficient then contains A[j,j]=0. Thus **every term with +nonzero coefficient is outside H**. + +The corrected proof groups any equal nonmonomial projective classes. Internal +cancellation can remove such classes, but cannot create a class in H. Uniform +tensor separation makes their remaining span disjoint from the monomial span. +If the endpoint E also has a free representation in the declared class, it +must therefore be zero. But + +`E=Q^-1 C3 t C1 Q^-1=0` + +implies t=0 and then t0=0 because the source Q,C1,C3 are invertible. This closes +the wider-class argument without assuming unique intermediate products or a +free inverse for t0. Invertibility of the handoff's requested t0 makes the +contradiction immediate. + +This direct-sum argument was the useful correction during review: an initial +draft left an unnecessary cancellation gap. The reviewed version resolves it +without reintroducing the invalid claim that Ht=H0 or R_k e_q=e_q. + +## 5. Independent scalar-owner crosscheck + +For t0=B tensor I, t=t0 and the first three controls are scalar owner maps. +Each endpoint word has an orthogonal binary map + +`M=G_(3,i)^-1 G_(3,r) G_(1,s)^-1 G_(1,j)`. + +Writing `Q^-1=aI+bX_q`, both a,b are nonzero. If M e_q differs from e_q, +the four translations0,e_q,M e_q,e_q+M e_q are distinct and have nonzero +coefficients; the word is nonmonomial. If the directions agree, the actual +Pauli sign yields either a single translation or a scalar times M. Thus the +stated monomial criterion is correct, including sign choices and the two +Q^-1 factors. Q is never treated as order two. + +Returning the direction to e_q forces s=j and r=i by the disjoint stage +supports and distinct source q_i, annihilating the source diagonal factors. +For fixed i,j, equality of two remaining binary products would give + +`G_(3,r')^-1 G_(3,r)=G_(1,s')^-1 G_(1,s)`. + +Their stage-local intersection is the identity binary map: both sides fix U +and W, preserve the respective local spaces, and the only invertible map of +the single common binary bit fixes it. Distinct q_i then force r=r',s=s'. +So a nonzero B[r,s], together with a nonzero source entry in column r and row +s, gives a unique nonmonomial term. This independently confirms the scalar +subclass obstruction, with the same sufficiently-large-width qualification. + +## 6. Scope, corrections and constructive continuation + +The following draft issues were corrected before the reviewed hash was frozen: +the prematurely broad theorem claim while cancellation was open; an incorrect +sentence placing transported t support back in H0; missing nonzero-vector +quantifiers in the chart lemma; and the width-dependent coefficient wording. +The revised cancellation argument now legitimately proves the broader +**declared monomial-Clifford** class, rather than a universal FREE theorem. + +This result does not import #5126's enlarged arbitrary finite monomial H as +an actual physical dictionary. It does not exclude non-Clifford monomial +controls, noncolumnwise/growing alphabets, extra paid corrections, a changed +controller or productive endpoint, extra axes, a different code, or reuse of +nonfree native information outside the four-free-controls template. + +The primary constructive direction remains a **priced** coupled operator using +already available native information, or another explicit changed hypothesis. +For the handoff's promised T={9:816,17:224} and +P={1:4608,2:838,3:1020,4:254,5:136}, a separate source-reused exact point +calculation gives raw allowance52.133114229301600459209653..654 at +C=.0030217758 and44.676959553589208855078743..744 at C=.00339999. +These are conditional point budgets only, before compiler reserve/tails and +all finite costs. They do not price any proposed t0 or prove a uniform interval. +An additional112H1 or224H1 already exceeds either point allowance, so a useful +paid continuation must exploit actual shared nonfree access or a cheaper +complete endpoint rather than a standalone full-bank correction. + +No t0 solver, candidate enumeration, native/factory/emitter/compiler/Lean run, +publication or κ claim occurred in this review. Only source text/byte reads, +mathematical derivation and the labelled scalar moment calculation occurred. + +Credit Dillon's released equations and source interface; Astra's corrected +operator-intersection derivation; Dugongue/mdumitrean's source code; Cao Yifan, +Cinna Davis, zxv, Jacob Alan Sussman and the retained native/Clifford/compiler +lineage. The tensor-separation technique is retained predecessor mathematics. +Substantial OpenAI Codex assistance. Apache-2.0. Root owns research and +publication decisions; rejection of this control class does not end the search +for a constructive paid-profile gain. diff --git a/research/nonlinear-monomial-control-ceiling/evidence/paid-control/captures/07-REPORT.md b/research/nonlinear-monomial-control-ceiling/evidence/paid-control/captures/07-REPORT.md new file mode 100644 index 0000000000..ddc6e82c07 --- /dev/null +++ b/research/nonlinear-monomial-control-ceiling/evidence/paid-control/captures/07-REPORT.md @@ -0,0 +1,82 @@ +# Released two-FULL invoice: optimistic additional recursive-control budget + +**Exact arithmetic result under an unproved complete operator/inventory premise:** at most **44 additional H1 calls** can be allowed while retaining strict raw moment positivity uniformly for `C in [0.0030217758,0.00339999]`. Forty-four succeeds uniformly; forty-five fails at the upper endpoint. This is an optimistic arithmetic grant for that declared invoice, not actual caller funding, compiler reserve or κ admission. + +The stock is Dillon's released [Discussion4930 comment19143357](https://github.com/CrocSwap/integer-mult-bounds/discussions/4930#discussioncomment-19143357), as authenticated in `coded-port-adapter-ceiling-20261023/outputs/released-operator-inventory-20261023`. Its scientific source pin is `4c0a359d88f22f05189d51f48387750b801a45d4`; public scoped theorem pin is `0e8a13c251a9be1029114cda8b4194f05f3a23ec`. The checker authenticates all eleven files listed by that frozen inventory's FILES.json, records their hashes and manifest hash, and checks its release receipt's exact T/P. It imports no source/native/operator/compiler code. + +## Complete promised invoice, including every helper + +``` +T = 224 H17 + 816 H9 +P = 4608 H1 + 838 H2 + 1020 H3 + 254 H4 + 136 H5 +``` + +There are 224 original main inputs and exactly 816 original helper inputs, for 1,040 productive owners. Each of the two FULLs has native profile `P0={1:2201,2:412,3:510,4:127,5:40}` and helper profile `H={4:28,7:7,8:47,9:326}`. The accounting is + +``` +P = 2 P0 + 112 H1 + (94 H1 + 14 H2 + 56 H5). +``` + +The query bank costs 112 H1. The final bracket pays complements for the 94 rank-eight, 14 rank-seven and 56 rank-four helper jobs across both FULLs, completing each to rank nine; 652 helpers already have rank nine. All 816 original helper owners are credited once. Their old lower-rank outputs are not also retained as extra targets. The complement rank mass is 402. Total target and paid rank masses are respectively **11,152** and **11,040**, a raw rank-mass difference of **112**. That difference is not the extra-control budget at positive C. + +## Declared comparison interval and exact bounds + +Write `a=0.0030217758`, `b=0.00339999` and + +``` +M(C) = 224*17^(1-C) + 816*9^(1-C) + - 4608 - 838*2^(1-C) - 1020*3^(1-C) + - 254*4^(1-C) - 136*5^(1-C). +``` + +Exact rational outward enclosures give: + +| Quantity | Lower bound | Upper bound | +|---|---:|---:| +| M(a) | 52.133114229301600459209653 | 52.133114229301600459209654 | +| M(b) | 44.676959553589208855078743 | 44.676959553589208855078744 | +| M'(C), uniformly on [a,b] | −19728.040142366829130975258373 | −19700.174693469327002916426638 | + +The exact Fraction endpoints are retained in RESULT.json. For the derivative, define `L_T(C)=sum T_r*r^(1-C)*log r` and likewise `L_P`. Both are nonincreasing positive sums. Therefore + +``` +L_P(b)-L_T(a) <= M'(C) <= L_P(a)-L_T(b) < 0. +``` + +This proves strict decrease of M throughout the interval. Its minimum occurs at b; the endpoint enclosure proves `44 0.676959553589208855078743` uniformly on [a,b]. +* At b, `M(b)-45 < -0.323040446410791144921256`, so no integer grant of 45 or more is uniformly compatible with strict positivity. +* One **additional complete 112-query bank** gives a negative margin everywhere in the interval: its uniform upper bound is below `-59.866885770698399540790346`. This prices that fixed extra-query policy; it is not a universal query lower bound on other operator implementations. + +Thus **44 is maximal**, rather than merely a sufficient rounded allowance, for additional H1 jobs under this exact unchanged T/P and the requested uniform interval. The reported raw surplus 112 cannot be substituted for it. + +For a different finite nonnegative additional paid profile E, the relevant condition is `sum E_r*r^(1-C) B_f^r`. Finite-dimensional vector-space counting +gives a nonzero column `w` in its kernel. Substitution `x=w z` makes every +paid response vanish by induction, so `(T-F0)w=0`. This proof does not require +the algebra to be a division ring. The dual output cut gives a nonzero free +left relation when there are fewer paid responses than output fields. + +#5126 closes this condition for its listed endpoints by actual source coset +separation: no such free right or left relation exists for any free bypass. +It is the source-specific no-relation fact that creates the numerical bound. + +The changed four-control problem has an unknown `t0` and an unknown legal +endpoint `E=Q^-1 C3 (D3 t0 D3^-1) C1 Q^-1`. No reduction of every admissible +candidate to one of the endpoints in the table has been supplied here. +Applying the 112 bound to this entire family would therefore be unjustified. +A useful obstruction must first bind the actual endpoint or prove its own +no-free-relation statement. A proposed one/two-child control must instead +exhibit its actual words and all required inverse programs. + +## Published Schur identities supply conditional inverses + +Gelfand, Gelfand, Retakh and Wilson, *Quasideterminants*, Advances in Mathematics +193 (2005), DOI [10.1016/j.aim.2004.03.018](https://doi.org/10.1016/j.aim.2004.03.018), +give the ordered Schur expression over a unital ring in Proposition 1.2.6, +p.10 of the [inspected author preprint](https://arxiv.org/pdf/math/0208146). +The actual inverses in the expression must exist. This is useful for +constructing an inverse identity in the concrete operator algebra, rather +than transferring a generic rank lower bound. + +For an operator-valued gather row `V`, scatter column `U` and one paid child +`H`, take `S=I+U H V`. When the middle inverse exists, ordered elimination gives + +`S^-1 = I - U (I + H VU)^-1 H V`. + +In particular the actual operator identity `VU=0` implies +`S^-1=I-U H V`. The inverse uses the same `H` with the opposite scatter sign. +No commutation assumption and no `H^-1` call are required. If `VU` is nonzero, +writing the one-field middle inverse does not implement or price it. Its +source access, precision, temporary ownership and all recursive calls remain +obligations. + +For two chronological children, the normal form already preserves their +ordered interaction: + +`T = F0 + l1 H1 b1 + l2 H2 b2 + l2 H2 e21 H1 b1`. + +It cannot be replaced by the sum of two independent rank-one updates unless +the cross term is actually absent. Likewise, a product of two invertible +nilpotent-update shears has its inverse in reverse order; each use and every +control conjugate still needs its own physical cost binding. + +Porat and Vinnikov, *Realizations of non-commutative rational functions around +a matrix centre, I*, JLMS 104 (2021), DOI +[10.1112/jlms.12459](https://doi.org/10.1112/jlms.12459), supply realization +synthesis over a unital algebra in Theorem 2.4, p.10 of the +[inspected preprint](https://arxiv.org/pdf/1905.11304). Its ordered product and +inverse constructions are (2.7) and (2.8), pp.12–13. These preserve algebraic +identities when the stated domain and invertibility conditions hold. +Their algebraic state dimension is not a theorem about native H1 call count. + +## Why generic noncommutative rank is insufficient + +The actual free algebra has relations and can have zero divisors. For a +nontrivial involutive Pauli translation `X`, `(I-X)(I+X)=0` although both +factors are nonzero. A universal free-field calculation does not retain this +specialized algebra unchanged. Generic inversion height, generic minimal +realization size or generic inner rank can therefore neither exclude nor +construct the required physical control without an exact specialization +argument. The source module cut avoids that problem by using ordinary +finite-dimensional linear algebra and an explicitly verified coset property. + +No new paper was needed for these implications. No search over controls, +group enumeration, solver, native/history experiment or external write was +performed. The next useful input is the construction owner's actual candidate +and invoice, or an exact endpoint reduction missing from the present source. + +## Construction-owner follow-up: a narrower translation module cut + +The construction owner subsequently proposed a more specific family: every +internal free coefficient is in the Pauli-X translation algebra `B`, while +paid X-Gaussian children and translations generate a fixed finite projective +group algebra `A`. The target `T=t0` is invertible, commutes with Q and D3, +and its fourth endpoint is required to be monomial-Clifford free. This excludes +intervening free K/C programs outside B, arbitrary monomial gathers, +noncommuting children and a paid fourth endpoint. + +The proposed source step is `projection_B(T)=0`. The author is deriving it +by isolating every translation coefficient in the actual fourth endpoint. +Matching such a term with a different owner pair would force the relative +source linear part to be identity; after the owner pair is fixed, the +translation itself must match. A translation does not change the symplectic +nonmonomial witness furnished by the source's zero diagonal. The full written +source-isolation and module argument have now been read in the construction +owner's report, with no blocking mathematical error found under its retained +source facts. Normalizer data, macro counts and native execution were not +independently replayed by this lane. + +Conditional on that step, the resulting module argument is valid. At one +sufficiently large faithful width, a circuit with fewer than 112 paid children +has a nonzero `w in B^112` annihilating all direct paid-input rows. The circuit +then gives `T w=F0 w in B^112`. But right multiplication by B preserves the +nontrivial right cosets, so `projection_B(T)=0` puts `T w` in their direct sum. +The intersection is zero. Literal invertibility of T forces w to be zero, +a contradiction. This uses no scalar coefficient-rank bound and no +division-ring or semisimplicity hypothesis. + +The conceptual check requires a fixed projectively finite A and free endpoint +alphabet, the same column lift at every width, exact cocycle phases and owner +coefficients applied once. It requires fresh equal-width input fields and all +bypass/gather/scatter/feedback coefficients in B. It covers the translation +operator-valued counterexample under those premises. It does not exclude the +full arbitrary-monomial family originally assigned to the construction owner. + +The reviewed construction-owner report is `outputs/operator-valued-paid-control-gate-20261023/REPORT.md`, SHA256 `19932a49cda97acb2e3f4933f95d4bed5598e441d7593132bd07d15a63c23731`. Its explicit `U_i=X_(q_i)`, `V_i=c_i X_(q_i)^-1`, `H=Q` shear has `VU=0`; its direct fourth-endpoint implementation produces an additional `228 H1` invoice, which is an upper construction cost, not a lower bound for all paid endpoints. The report retains the incomplete ordinary bill and unexecuted caller. One minor clarity suggestion was to state explicitly that the commuting X-Gaussian generators have finite projective order, proving the fixed algebra is finite. diff --git a/research/nonlinear-monomial-control-ceiling/evidence/paid-control/captures/RAW-GIT-5126-PROOF.md b/research/nonlinear-monomial-control-ceiling/evidence/paid-control/captures/RAW-GIT-5126-PROOF.md new file mode 100644 index 0000000000..1ebf18af5b --- /dev/null +++ b/research/nonlinear-monomial-control-ceiling/evidence/paid-control/captures/RAW-GIT-5126-PROOF.md @@ -0,0 +1,394 @@ +# An active local-query obstruction for the actual 112-port code + +Prepared for Dillon, 2026-10-23, with substantial OpenAI Codex assistance. +This is a scoped exact local theorem with finite source checks, not a +complete-controller lower bound, a multiplication algorithm or an admission +claim. Apache-2.0. + +## Statement + +Use the actual physical native code from `research/coded-native-caller`: + +\[ + C_{ij}=a_{ij}K_i^{-1}K_j,\qquad J=Q^{-1}CQ, +\] + +where there are 112 owners, Q is the one-central-axis Gaussian, each K_i is +the retained signed orthogonal permutation normalizer, and the scalar matrix +a has zero diagonal and 52 nonzero off-diagonal entries in every column. +The 112 vectors q_i=G_i e_0 are distinct. The source factorization +C=I-14P has an eight-mode projector P. + +Fix any finite projective group H of invertible one-column monomial maps on the +declared fixed ambient address space, containing the source normalizers and +all admitted free address maps. For example, H may be the full group of +one-column permutations with phases in a fixed finite root-of-unity group; +the free permutations need not be Clifford. Every address map is lifted as +the SAME one-column map h on all f columns, h^tensor(f). Independent choices +on individual columns are outside this grammar. Free operations are +arbitrary linear owner programs over the +monomial subgroup algebra, including scalar operations, every +actual C or C inverse, owned COPY/CLEAR, arbitrary clean buffers, saved +banks, and fanout. These are charged as ordinary finite work; "free" here +means no recursive Gaussian call. Allow every paid unary call to apply an +arbitrary fixed one-field linear operator, with chronological linear +feedback from all available fields. In particular its input may use all +earlier paid responses. All coefficients are data-independent, although +they may depend on width. Every counted field has the same address width as +one source owner; a wider packed channel is not counted as one field. + +**Theorem.** A standalone exact implementation of J, or of J-I, on 112 +arbitrary fresh owner fields requires at least 112 paid unary field calls. +The statement is valid for an identity at all positive tensor widths, and +allows the scalar owner coefficients to depend on width. The same conclusion +holds after adding any free owner operator to J. +More precisely, the bound holds at every sufficiently large tensor width, +so no all-width implementation uses fewer than 112 calls. It is not a bound +at every small width: at width one, +Q=((1+i)/2)I+((1-i)/2)X is in the scalar span of free monomials. + +**Sharp one-sided corollary.** Put P=(I-C)/14. Each of QP and PQ, as a +112-input/112-output owner operator, requires at least 112 paid unary field +calls. Exactly 112 H1 calls suffice: apply P for free and then Q to every +output, or apply Q to every input and then P. Thus the eight scalar code +modes do not yield an eight-call one-sided Q repair. The statement also +holds with Q replaced by Q inverse. + +**Canonical eight-mode corollary.** For the actual source mode matrix M, +let U_mode=diag(K_i^-1)M and V_mode=M^T diag(K_i)/56. Then +P=U_mode V_mode and V_mode U_mode=I_8. Each of Q U_mode (eight fresh inputs, +112 outputs) and V_mode Q (112 fresh inputs, eight outputs) also has exact +paid-field-call complexity 112, attained by 112 H1 calls. Thus the canonical +rectangular mode ABI does not avoid the one-sided obstruction. + +It does not cover receiving Gaussian-dependent saved data, absorbing the +correction into the paid native FULLs or their prefixes, recycling an existing +paid IDLE, or a direct coded consumer with a different endpoint. Nonlinear +controls, noncolumnwise address operations, approximate endpoints, and +unequal-width packing are outside the stated model. + +## 1. Phase-correct finite algebra and all-width evaluation + +This uses the finite-phase algebra and sufficiently-large-width tensor +faithfulness already proved in `coded-port-adapter-ceiling/PROOF.md`, section +3, and clarified by the passive saved-query package's uniform-alphabet +addendum. Here only the FREE group H must be finite projectively. An ambient +group generated by H and Q need not be finite or Clifford. + +Choose one invertible matrix representative for every projective element +of H. At width f its nonzero scalar multiplication cocycle is raised to +the f-th power; the scalar factors need not have modulus one. +Let B_f be the resulting finite-dimensional twisted group algebra over +k=C, with one basis element per projective element of H. Its evaluation +is the span of the literal tensor lifts h^tensor(f). Scalar factors are +retained through the cocycle; owner coefficients are applied only once. + +Besides H itself, we will need only a fixed finite union of its projective +cosets: the sets g_ij H used for J, the analogous right cosets for PQ and +V_mode Q, and the analogous left cosets for QP and Q U_mode. Take S to be +their union with H, with repeated projective elements identified. This set +is finite because H and the source owner/mode sets are finite. Closure of +S under multiplication is not needed: all free products remain in H, and +every target-times-free expression used below stays in its declared coset +set. A different fixed finite number of source stages merely enlarges S. + +For projectively distinct nonzero representatives v_1,...,v_N, individually +normalize each by its Hilbert-Schmidt norm. Their tensor-power +Gram matrix has diagonal one and off-diagonal entries +^f, whose magnitudes are at most rho^f for some rho<1. For +(N-1)rho^f<1 the Gram matrix is strictly diagonally dominant and positive +definite. Apply this to the projective representatives in S. Their literal +tensor lifts are linearly independent for every sufficiently large f. +Thus B_f and all the required coset subspaces evaluate faithfully, with +distinct cosets in direct sum, at those widths. Scalar factors remain raised +to the f-th power; owner coefficients are never raised to that power. + +We work at any one such width. The threshold is uniform over all free +coefficients, control choices, and control lengths because H and S are +fixed. Width-dependent scalar coefficients are allowed. A width-dependent +or noncolumnwise group of arbitrary payload maps is not allowed; neither +is an infinite projective phase grammar. No assumption that arbitrary +permutations together with Q generate a finite group is made. + +## 2. Actual-source right-coset separation + +Write g_ij=Q^-1 K_i^-1 K_j Q. For a fixed i, + +\[ + g_{ij}^{-1}g_{ik}=Q^{-1}K_j^{-1}K_kQ. (1) +\] + +For the signed orthogonal permutation M=K_j^-1 K_k, let G=G_j^-1 G_k be +its binary orthogonal coordinate matrix and e=e_0. Pauli signs affect only +tableau phases. The Z-to-X block of Q^-1 M Q is + +\[ + B=G e e^T+e e^T G. (2) +\] + +If G e=e, orthogonality gives e^T G=e^T, and B=0. If G e differs from e, +then both pairs (G e,e) and (e,G^T e) are independent; the two outer products +in (2) have rank two. Thus Q^-1 M Q is monomial exactly when G e=e. +Equivalently, G_j e=G_k e, or q_j=q_k. The actual 112 source vectors are +distinct, so (1) belongs to H exactly when j=k. + +Therefore the 112 g_ij lie in distinct right H-cosets for each fixed i. +The j=i coset is H itself, since g_ii=I. The scalar phase quotient does not +alter this test: multiplying by a scalar never changes monomiality. +The companion checker verifies all 112^2 ordered pairs by both formula +(2) and the retained exact tableau implementation. It separately checks +the actual code's zero diagonal and nonempty column supports. + +## 3. No nonzero free input survives modulo a free output + +Let F_0 be ANY 112-by-112 matrix over B_f. Suppose w in B_f^112 obeys + +\[ + (J-F_0)w=0. (3) +\] + +Row i of Jw is sum_j a_ij g_ij w_j. Each summand lies in the right-coset +subspace g_ij B_f. These subspaces are in direct sum. Every component of +F_0 w lies in B_f, the H coset. The diagonal source coefficient is zero. +For each off-diagonal supported pair (i,j), projecting (3) onto g_ij B_f +therefore gives a_ij g_ij w_j=0. A nonzero scalar and a group unit are +invertible, so w_j=0. Each column j has a supported off-diagonal entry, +and hence w=0. + +This is stronger than an eigenvector statement: J-F_0 has zero kernel on +B_f^112 for every free F_0. In particular the Q-conjugated eigenspaces of +C contain no nonzero free input embedding, despite the eight-mode +factorization of C-I. A cheap compression of J-F_0 through fewer than 112 +owner fields is impossible. + +### One-sided projector operators + +Write p=(I-a)/14, so P_ij=p_ij K_i^-1 K_j and p_ii=1/14. For PQ, fix row +i. Its blocks are p_ij h_ij with h_ij=K_i^-1 K_j Q. Every h_ij has Fourier +grade one and lies outside H. Further, + + h_ij^-1 h_ik = Q^-1 K_j^-1 K_k Q. + +Thus the h_ij occupy distinct right H-cosets as j varies. The same argument +proves ker_{B_f}(PQ-F_0)=0 for every free F_0; the nonzero diagonal of p +already supplies a supported entry in each column. + +For QP fix column j instead, and put t_ij=Q K_i^-1 K_j. These grade-one +words lie in distinct LEFT H-cosets as i varies, because + + t_ij t_kj^-1 = Q K_i^-1 K_k Q^-1 + +is nonmonomial for i different from k. Consequently a free row w satisfying +w(QP-F_0)=0 must vanish: project each fixed output column onto the distinct +left-coset subspaces B_f t_ij, and use p_ii different from zero. This is the +left-kernel version of the same source-specific separation. + +### Canonical rectangular eight-mode maps and free rebasing + +The retained 112-by-8 scalar mode matrix M obeys M^T M=56I_8 and +a=I-MM^T/4; each row has four nonzero entries. The checker verifies every +entry of these identities from the pinned source receipt. Define + + U_mode = diag(K_i^-1) M, shape 112 by 8, + V_mode = M^T diag(K_i)/56, shape 8 by 112. + +These maps are free, with P=U_mode V_mode and V_mode U_mode=I_8. +For Q U_mode the fixed mode-column a has entries M_ia Q K_i^-1. The words +t_i=Q K_i^-1 lie in distinct left H-cosets, since +t_i t_k^-1=Q K_i^-1 K_k Q^-1 is nonmonomial when i differs from k. +Each t_i itself is nonmonomial. Every row of M has a nonzero entry, so +the same columnwise projection proves that + + w (Q U_mode-F_0)=0, w in B_f^(1 by 112), + +forces w=0 for every free 112-by-8 F_0. Thus the no-left-kernel conclusion +holds even though there are only eight fresh inputs. + +Dually, V_mode Q has entries M_ja K_j Q/56. The words K_j Q lie in +distinct right H-cosets because their pairwise relative transports are +Q^-1 K_j^-1 K_k Q. Every source column is represented in at least one mode +row. Hence (V_mode Q-F_0)w=0, with w in B_f^112, forces w=0 for every free +8-by-112 F_0, even though there are only eight outputs. + +These conclusions cover all free changes of retraction basis. If another +free pair U',V' has P=U'V' and V'U'=I_8, then + + W=V_mode U', W^-1=V' U_mode, + U'=U_mode W, V'=W^-1 V_mode. + +The two inverse identities follow by substituting U'V'=P and +U_mode V_mode=P. Thus Q U' and V'Q differ from the canonical operators by +invertible free input or output boundaries. No cheaper implementation is +obtained by any such free eight-mode basis choice. + +## 4. Arbitrary chronological paid calls + +Normalize any finite acyclic linear implementation by retaining all paid +responses y_1,...,y_r. COPY, saves, fanout, overwrite and CLEAR are included +in this normalization: enlarging owned storage can preserve the history +without introducing another recursive call. At paid call t its input is + +\[ + b_t x+\sum_{s B_f^r. As a k-linear map its +domain has dimension 112 dim_k B_f and its codomain has dimension +r dim_k B_f. If r<112, choose a nonzero w in its kernel. Substitute the +formal input x=w z, with z one arbitrary field. Every direct term b_t w is +zero. Induction on t in (4) gives every paid response y_t=0, regardless of +T_t and regardless of the active feedback coefficients. Thus the claimed +endpoint J must satisfy Jw=F_0w. Section 3 contradicts this. + +This proves r>=112. Replacing J by J-I or J+F_free merely changes the free +matrix F_0 and gives the same result. + +The input-kernel proof applies verbatim to PQ. For QP use the dual output +cut: concatenate the free output columns in (5) into L in B_f^(112 by r). +If r<112, the k-linear map w -> wL from B_f^(1 by 112) to B_f^(1 by r) +has a nonzero kernel row w. Multiplying the endpoint by that row annihilates +every paid-response term, so w(QP-F_0)=0, contrary to the left-kernel result. +No assumption about the paid calls' chronology is needed beyond the +existence of the finite linear circuit normal form. + +For Q U_mode the same output cut has 112 rows and r paid-response columns, +although F_0 has only eight input columns. It again gives r>=112. For +V_mode Q the same input cut has 112 input columns, although F_0 has only +eight output rows, and also gives r>=112. Both bounds are attained by +112 H1 calls: expand eight fields through U_mode for free and apply Q to +all 112 results; or apply Q to the 112 inputs and contract through V_mode +for free. Invertible free retraction basis changes preserve these counts. +The same arguments apply with Q inverse throughout. + +## 5. Direct sums and local amortization + +For any fixed positive m, consider m fresh physical code blocks, embedded +in one fixed ambient address space. Each block may use its own eight fresh +axes and physical normalizers, with the same 112 distinct-direction +property. Let N=112m. The same proof gives + +* diag(J_1,...,J_m), or its free perturbation: at least N paid field calls; +* diag(P_1 Q,...,P_m Q): exactly N H1 calls are necessary and sufficient; +* diag(QP_1,...,QP_m): exactly N H1 calls are necessary and sufficient. +* diag(Q U_mode,1,...,Q U_mode,m), and its free retraction rebases: + exactly N H1 calls are necessary and sufficient; +* diag(V_mode,1 Q,...,V_mode,m Q), and its free retraction rebases: + exactly N H1 calls are necessary and sufficient. + +Free controls may mix different blocks arbitrarily. To see why this does +not change the proof, write a candidate free kernel column or row in m +blocks. Project each target block's rows or columns onto its nontrivial +cosets. The free comparison operator still contributes only the H coset, +even when it mixes all N owners. The corresponding single-block argument +forces every block of the candidate kernel to vanish. Apply the input or +output dimension cut with B_f^N and r=1 and price at least one, so the local correction costs at least 112. +Its added recursive price exceeds the removed-center saving by at least +48 per FULL. The direct-sum theorem gives a gap of at least 48m for m such +standalone local repairs, even when their free controls mix all inputs. + +Thus this isolated interface-restoration strategy cannot turn the removed +eight H8 calls into a recursive saving anywhere in 0<=c<1. Extra ordinary +work can only increase the cost. The conclusion does not price or rule out +absorption into correlated FULL, IDLE, or saved-query data in a complete +caller, nor a different direct coded endpoint. + +### Relationship to predecessor and active caller work + +The predecessor `coded-port-adapter-ceiling` (#4932) proves that one parallel +bank needs 112 grade-2 channels for its physical recovery adapter, using a +grade projection of full rank. It expressly excludes multiple chronological +banks. The argument here uses source-specific distinct right cosets and an +input-kernel cut to cover arbitrary chronological paid unary calls and +active feedback. Its conclusion is 112 calls of any grade; it does not +claim 112 grade-2 calls in this enlarged model. + +The conclusion excludes an independently implemented 16-, 32-, or 42-H1 +fresh-input correction of J, J-I, QP, or PQ through the native code's eight +modes. For J and J-I this is a lower bound, with the elementary 224-H1 +conjugation implementation as an upper bound; 112-call attainability is +not asserted. For QP and PQ the 112-H1 bound is attained. +It does not prove that p2 needs four thin banks, nor that a complete active +three-bank caller is impossible. Such a caller can already hold nonfree +data from its FULLs, queries, and IDLE. A useful construction must exploit +that shared paid access, change the endpoint, or leave this local grammar. +The particular eight-mode ABIs Q U_mode and V_mode Q, including their free +retraction rebases, are covered above. An arbitrary different eight-mode +endpoint, the middle compression V_mode Q U_mode, and the two-sided +compression PQP remain outside the claims. + +The actual #4932 entrance adapter is also covered by free equivalence. Its +physical monomial X wrapper is M_i=U_i K_i Q^-1. Therefore +U a U*=M (Q C Q^-1) M*, and Q C Q^-1=Q^2 J Q^-2. Since Q^2 and M are free, +and a^-1=(a+12I)/13, the physical U a^-1 U* differs from J by invertible +free boundaries, a nonzero scalar, and a free bypass. These operations +preserve the local paid-call lower bound. This correspondence uses the +retained exact physical wrapper identities; it does not replace U_i with +a bare Fourier along q_i. + +## Source and predecessor credit + +The source code and native producer are Dugongue's #4929. The actual signed +orthogonal normalizers and physical C formulation come from the retained +`coded-native-caller` package. The finite-phase group algebra, tensor +faithfulness, adapter scope and warning that rank-eight scalar modes are not +eight query heads come from #4932. The uniform finite-alphabet clarification +and related passive-controller limitations come from the retained passive +saved-query package. Credit also Cinna Davis/1337isnot1337's #4694/#4698, +Cao Yifan/intqwq's #4610 source, zxv's retained exact Clifford/compiler work, +and all source notices. This active local input-kernel argument was derived +in the current research session with substantial OpenAI Codex assistance. + +No independent human or formal-kernel acceptance is claimed. The source +native all-input theorem, numerical outer admission, and complete caller +endpoint are inherited or open, not replayed by this scratch result. From 942775f86ef11bbfa2f03e98476edccd8c3a41ca Mon Sep 17 00:00:00 2001 From: d3lerium Date: Fri, 23 Oct 2026 23:22:36 +1100 Subject: [PATCH 3/4] Polish spacing in the scoped followup guide --- .../README.md | 20 +++++++++---------- 1 file changed, 10 insertions(+), 10 deletions(-) diff --git a/research/nonlinear-monomial-control-ceiling/README.md b/research/nonlinear-monomial-control-ceiling/README.md index 8a579276ca..9ce87dc40a 100644 --- a/research/nonlinear-monomial-control-ceiling/README.md +++ b/research/nonlinear-monomial-control-ceiling/README.md @@ -34,7 +34,7 @@ supplies the sufficiently-large-width conclusion. Cancellation cannot create a monomial class absent from every surviving source term. The scientific source remains `4c0a359d88f22f05189d51f48387750b801a45d4`, -`research/coded-native-caller`, on q+U8+W8 (17 axes,112 owners). +`research/coded-native-caller`, on q+U8+W8 (17 axes, 112 owners). The released endpoint is `E=Q^-1 C3 t C1 Q^-1`, `t=D3 t0 D3^-1`. The predecessor is #5160 head `7800589ce99d932bdb063aafca1f501f1a38626d`; #5126 head `0e8a13c251a9be1029114cda8b4194f05f3a23ec` is context, not @@ -52,8 +52,8 @@ this package does not reexecute its native source or all-port evaluator. ## What the finite checker establishes The unchanged [checker](evidence/finite-check/check_support.py) uses exact -Gaussian-rational16x16 matrices on q,u1,u2,w. It checks three examples: -weighted nonlinear Toffoli on q or w, and CCZ(q,w,u1). Its108 terminal +Gaussian-rational 16x16 matrices on q,u1,u2,w. It checks three examples: +weighted nonlinear Toffoli on q or w, and CCZ(q,w,u1). Its 108 terminal sandwiches are nonmonomial. The exact controls demonstrate that dropping join preservation or the nonzero-U condition changes the answer; transported CCZ need not preserve central q-cosets; reversing D conjugation or omitting @@ -70,16 +70,16 @@ No t0 search, native/emitter/compiler/Lean replay or complete κ claim is made. The unchanged [author report](evidence/paid-control/AUTHOR-REPORT.md) and [separate independent review](evidence/paid-control/INDEPENDENT-REVIEW.md) record a narrower paid-control test. The explicit operator-valued shear -`T=I+U Q V`, with `U_i=X_(q_i)`, `V_i=c_i X_(q_i)^-1` and56 signs of each +`T=I+U Q V`, with `U_i=X_(q_i)`, `V_i=c_i X_(q_i)^-1` and 56 signs of each kind, has `VU=0`, inverse `I-U Q V` and raw dual `T^-T=I-V^T Q U^T`. All allowed translations and X-Gaussian children commute with Q and D3, so the declared T circuit also commutes with them. The one-child algebraic word does not supply a physical tape/precision emitter. -Its direct whole-control policy adds228 H1, including the paid terminal +Its direct whole-control policy adds 228 H1, including the paid terminal inverse. That is an upper construction bill, not a minimum for every paid E decoder, and it loses against the inherited conditional raw44H1 allowance -before ordinary costs or reserves. Original224 main and816 independent +before ordinary costs or reserves. Original 224 main and 816 independent helper jobs remain counted once; the complete physical helper/caller remains unrealized. Source recoding, /52, raw-dual corrections and finite work remain charged obligations. @@ -88,8 +88,8 @@ A separate written kernel argument proves `r>=112` only for a **fresh standalone T circuit with no initially supplied nonfree observations**, translation-algebra free coefficients, commuting same-width X-Gaussian children, and a FREE terminal E in the declared fixed finite endpoint class, -at sufficiently large faithful widths. Under exactly these conditions112 -exceeds the conditional44 allowance. This is **not** a whole-caller amortized +at sufficiently large faithful widths. Under exactly these conditions 112 +exceeds the conditional 44 allowance. This is **not** a whole-caller amortized barrier: already-paid QY/native/IDLE/centre observations violate that initial- information premise. Arbitrary nontranslation/nonlinear weighted gathers, noncommuting children, wider channels, a paid E or changed targets are outside @@ -98,7 +98,7 @@ it. The primary nonlinear-monomial theorem is not enlarged by this appendix. No new proof code, source evaluator or native/compiler replay checks this written appendix; the hook only authenticates its immutable bytes. Captures reuse primary inputs by digest. [CAPTURES.json](evidence/paid-control/CAPTURES.json) -records the monitor5126 proof's extra LF relative to raw Git; both byte versions +records the monitor #5126 proof's extra LF relative to raw Git; both byte versions are retained without replacing the raw scientific pin. The separate p19 compiler-contracts reviewer supplied the final scoped PASS. @@ -112,7 +112,7 @@ python3 -B scripts/research_package.py --package research/nonlinear-monomial-con python3 scripts/check_research_commit.py --package research/nonlinear-monomial-control-ceiling --ref HEAD --check-source-history ``` -CPython3.12, standard library only; direct runs take about1second. The wrapper +CPython 3.12, standard library only; direct runs take about 1 second. The wrapper authenticates every original packet manifest and runs an unchanged checker copy in private scratch with the original four source paths. The author checker writes RESULT/MATRICES next to itself, so direct execution against its frozen From bddbdfab1cdba27ce69ce4a83d9c92a5d9e66bd8 Mon Sep 17 00:00:00 2001 From: d3lerium Date: Fri, 23 Oct 2026 23:24:18 +1100 Subject: [PATCH 4/4] Specify the paid appendix terminal alphabet explicitly --- research/nonlinear-monomial-control-ceiling/README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/research/nonlinear-monomial-control-ceiling/README.md b/research/nonlinear-monomial-control-ceiling/README.md index 9ce87dc40a..d586643c28 100644 --- a/research/nonlinear-monomial-control-ceiling/README.md +++ b/research/nonlinear-monomial-control-ceiling/README.md @@ -87,7 +87,7 @@ charged obligations. A separate written kernel argument proves `r>=112` only for a **fresh standalone T circuit with no initially supplied nonfree observations**, translation-algebra free coefficients, commuting same-width X-Gaussian -children, and a FREE terminal E in the declared fixed finite endpoint class, +children, and a FREE terminal E in the declared fixed finite monomial-Clifford endpoint class, at sufficiently large faithful widths. Under exactly these conditions 112 exceeds the conditional 44 allowance. This is **not** a whole-caller amortized barrier: already-paid QY/native/IDLE/centre observations violate that initial-