diff --git a/.gitattributes b/.gitattributes index 41aef290fd..76e425cdaf 100644 --- a/.gitattributes +++ b/.gitattributes @@ -8,3 +8,8 @@ upstream/** -whitespace references/copied-centers/pr29/** -whitespace references/copied-centers/pr33/** -whitespace references/semantic-bulk/** -whitespace + +# Preserve the exact bytes of pinned upstream proof and witness snapshots. +references/partial-gauge/pr97/** whitespace=-blank-at-eof +# Preserve the exact pinned PR117 source bytes and legal notices. +references/three-stage-cover/pr117/** -whitespace diff --git a/.github/workflows/verify.yml b/.github/workflows/verify.yml index 6bd374e1ca..a06759c81f 100644 --- a/.github/workflows/verify.yml +++ b/.github/workflows/verify.yml @@ -16,7 +16,7 @@ jobs: fail-fast: false matrix: python-version: ['3.11', '3.13', '3.14'] - group: [community, producers, certificates, ternary, research, tests, strips, clones, positive, joint, pair] + group: [community, producers, certificates, ternary, research, tests, strips, clones, positive, joint, pair, partial-gauge, three-stage-cover, shrunk-paired-cover] steps: - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1 # v7 with: diff --git a/Makefile b/Makefile index 84fe8290f3..81537bb205 100644 --- a/Makefile +++ b/Makefile @@ -15,6 +15,9 @@ community-followup-check: verify: $(MAKE) verify-community $(MAKE) verify-producers + $(MAKE) verify-partial-gauge + $(MAKE) verify-three-stage-cover + $(MAKE) verify-shrunk-paired-cover $(MAKE) verify-certificates $(MAKE) verify-ternary $(MAKE) verify-research @@ -30,6 +33,7 @@ verify-community: community-audit-check community-followup-check copied-reversed $(MAKE) climbed-48-verify verify-producers: + $(MAKE) stopped-product-verify $(MAKE) copied-centers-verify $(MAKE) structured-bulk-verify $(MAKE) endpoint-gauge-producer endpoint-gauge-certificate @@ -428,3 +432,49 @@ pair-assembly-verify: pair-assembly-producer pair-assembly-check .PHONY: verify-pair verify-pair: pair-assembly-verify python3 scripts/audit_pair_candidate.py --check docs/research/community-pair-arithmetic.json + +.PHONY: stopped-product-producer stopped-product-certificate stopped-product-verify +stopped-product-producer: + python3 scripts/stopped_product_producer.py + +stopped-product-certificate: + python3 scripts/stopped_product_network.py + +stopped-product-verify: stopped-product-producer stopped-product-certificate + +.PHONY: partial-gauge-bit partial-gauge-producer partial-gauge-certificate partial-gauge-verify +partial-gauge-bit: + python3 scripts/partial_gauge_bit.py + +partial-gauge-producer: + python3 scripts/partial_gauge_producer.py + +partial-gauge-certificate: + python3 scripts/partial_gauge_network.py + +partial-gauge-verify: partial-gauge-bit partial-gauge-producer partial-gauge-certificate + +.PHONY: verify-partial-gauge +verify-partial-gauge: partial-gauge-verify + +.PHONY: three-stage-cover-producer three-stage-cover-certificate three-stage-cover-verify verify-three-stage-cover +three-stage-cover-producer: + python3 scripts/three_stage_cover_producer.py + +three-stage-cover-certificate: + python3 scripts/three_stage_cover_network.py + +three-stage-cover-verify: three-stage-cover-producer three-stage-cover-certificate + +verify-three-stage-cover: three-stage-cover-verify + +.PHONY: shrunk-paired-cover-producer shrunk-paired-cover-certificate shrunk-paired-cover-verify verify-shrunk-paired-cover +shrunk-paired-cover-producer: + python3 research/shrunk-paired-cover/producer.py > /dev/null + +shrunk-paired-cover-certificate: + python3 research/shrunk-paired-cover/certificate.py + +shrunk-paired-cover-verify: shrunk-paired-cover-producer shrunk-paired-cover-certificate + +verify-shrunk-paired-cover: shrunk-paired-cover-verify diff --git a/NOTICE b/NOTICE index b565637758..c719bbed04 100644 --- a/NOTICE +++ b/NOTICE @@ -302,3 +302,60 @@ 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/. diff --git a/PATCHING.md b/PATCHING.md index b60f3f101b..a02e230256 100644 --- a/PATCHING.md +++ b/PATCHING.md @@ -1,6 +1,6 @@ # Applying the inherited manuscript patch -This patch is the PR #10 baseline. The current copied-center extension is +This patch is the PR #10 baseline. The PR #36 copied-center extension is presented in `notes/copied-centers-note.tex`; it is not included in this historical combined manuscript patch. @@ -30,3 +30,6 @@ alternatives for the same base. PR7 is pinned at `6725c6a17b17871a35353fd29157f4ed851bc114` for the retained producer sources. The old uniform recurrences keep their original exponents. The active consumers use `lem:batched-chunk-swap` and `prop:batched-simultaneous-layer`. + +The stopped product-ring extension is supplied separately in +`notes/stopped-product-note.tex`; it does not regenerate this historical patch. diff --git a/README.md b/README.md index 035c259b68..9aff3ee61b 100644 --- a/README.md +++ b/README.md @@ -3,6 +3,87 @@ **Community research maintained by Douglas Colkitt — conditional on the original OpenAI #109 framework.** +## Three-stage cover extension + +The construction contributed by **icekylinx**, extending +[PR #115](https://github.com/CrocSwap/integer-mult-bounds/pull/115), gives + +$$ +T(n)=O\!\left(n(\log n)^{1-\kappa}\right),\qquad +\kappa=\frac{3146011}{10000000000}=3.146011\times10^{-4}. +$$ + +Three signed shears on a regular Cayley cover align all interstage data +frames. The complex branch combines eumemic's PR #117 local DAG with +arbitrary-subspace Clifford frames; the bit branch retains the PR #97 / +Swapnil physical word and uses a batched weighted q-adic cover. Local +transitions, copied centers, rare-class fallback, finite routers and +internal row borrowing are charged. The retained analytic and fixed-tape +hypotheses still apply. + +[Proof source](notes/three-stage-cover-note.tex) · +[Exact certificate](certificates/three-stage-cover-network.json) · +[Incremental reproduction](docs/three-stage-cover.md) + +```sh +make three-stage-cover-verify +``` + +## Partial-gauge extension + +The construction contributed by **icekylinx**, extending +[PR #104](https://github.com/CrocSwap/integer-mult-bounds/pull/104), gives + +$$ +T(n)=O\!\left(n(\log n)^{1-\kappa}\right),\qquad +\kappa=\frac{7237}{78125000}=9.26336\times10^{-5}. +$$ + +It applies the stopped whole-projector bit interface to the physical deferred +word of Zhihao Chen's PR #97, based on Swapnil Jain's witness. The new complex +producer combines cyclic interval contractions, pair-first cube assembly, +compatible frame enlargement and partial source gauges. Every residual, +endpoint correction and target-data transition is charged. The bound retains +the analytic and fixed-tape hypotheses of the preceding construction. + +[Proof source](notes/partial-gauge-note.tex) · +[Exact certificate](certificates/partial-gauge-network.json) · +[Incremental reproduction](docs/partial-gauge.md) + +```sh +make partial-gauge-verify +``` + +## Stopped product-ring extension + +The new construction contributed by **icekylinx**, extending merged +[PR #36](https://github.com/CrocSwap/integer-mult-bounds/pull/36), gives + +$$ +T(n)=O\!\left(n(\log n)^{1-\kappa}\right),\qquad +\kappa=\frac{194869}{2500000000}=7.79476\times10^{-5}. +$$ + +It combines a stopped product-ring bit interchange on `(23,23)` with an +all-disjoint rational-center complex network on `(24,24)`. The new generic +opposite-bank factorization pays one reversed child per projector rank; +atom adapters, ordinary leaves, endpoint copies and the exact denominator-21 +grid are included in the proof. The bound retains the original analytic +and fixed-tape hypotheses. + +[Proof source](notes/stopped-product-note.tex) · +[Exact certificate](certificates/stopped-product-network.json) · +[Incremental reproduction](docs/stopped-product.md) + +```sh +make stopped-product-verify +``` + +The maintainer-reviewed community checkpoint below remains its own result +and validation record. + +## Reviewed community checkpoint + The reviewed community witness gives $$ diff --git a/SOURCES.json b/SOURCES.json index 6096d5aff9..ab7ec8e74c 100644 --- a/SOURCES.json +++ b/SOURCES.json @@ -234,5 +234,190 @@ } ], "continuing_analytic_dependencies": "See structured_bulk_extension.adopted and references/semantic-bulk/MANIFEST.json." + }, + "stopped_product_extension": { + "contributor": "icekylinx", + "base_commit": "56b66d58297deca1d7dd130247d720e960f77a37", + "proof_predecessor_PR": "https://github.com/CrocSwap/integer-mult-bounds/pull/36", + "proof_predecessor_commit": "11817ccacb564bb7f98789c20dc11d3fece207e3", + "predecessor_merged": true, + "archive": "integer_mult_round5_kappa_7_79476e-5_complete.zip", + "archive_sha256": "714cbffa0cccba3c18f046a63e62d6981683cb386b5d5cd09761be7b70f0b0be", + "kappa": "194869/2500000000", + "proof": "notes/stopped-product-note.tex", + "certificate": "certificates/stopped-product-network.json", + "producer": "scripts/stopped_product_producer.py", + "assistance": "Substantial OpenAI GPT-6 Astra and Codex assistance.", + "scope": "Conditional on the retained analytic and fixed-tape framework; new product-ring and exact rational-grid arguments supplied as written proofs.", + "continuing_dependencies": [ + "copied_centers_extension", + "structured_bulk_extension" + ], + "icekylinx_lineage": [ + { + "pr": 10, + "commit": "62691e395a0458ce089a1c7b5d89e74291e95e29" + }, + { + "pr": 18, + "commit": "f2ab41aebad47861caf6316282c1793e5513845e" + }, + { + "pr": 24, + "commit": "ed90fd940279c336ebc45968631bdebfb087b505" + }, + { + "pr": 32, + "commit": "0ef3aeb61f55cc0b321ce6a0ef00acee25cefe52" + }, + { + "pr": 36, + "commit": "11817ccacb564bb7f98789c20dc11d3fece207e3" + } + ], + "curation": "Selected h23/h24 inputs, generator, exact certificate and proofs only; excludes raw graphs, research alternatives and recursive predecessor copies." + }, + "partial_gauge_extension": { + "contributor": "icekylinx", + "predecessor_pr": 104, + "predecessor_commit": "948ce1510df750f4c18b96bdaef436a86f8bf834", + "archive": "integer_mult_round6_kappa_9_26336e-5_complete.zip", + "archive_sha256": "e701a98b10864dc4b1414d9fcb1fc7c67b28e2363cb2a2292f9fed577a081d2d", + "kappa": "7237/78125000", + "proof": "notes/partial-gauge-note.tex", + "certificate": "certificates/partial-gauge-network.json", + "producer": "scripts/partial_gauge_producer.py", + "assistance": "Substantial OpenAI GPT-6 Astra and Codex assistance.", + "adopted": [ + { + "pr": 97, + "contributor": "Zhihao Chen (jacklightChen)", + "commit": "e15350b66e4bf796153a68c240a6cc1b788abe3e", + "role": "Actual deferred bit word, reflected physical ledger, fan-order clarification, gauged exits and signed endpoint correction" + }, + { + "contributor": "Swapnil Jain", + "repository": "Swapnil-jain/integer-mult-kappa", + "commit": "741e7aa078392553815df7926ee17ac5e25a8c38", + "role": "Underlying deferred readouts, V leaves, lifted frames and frozen witness" + }, + { + "pr": 62, + "contributor": "Avi Eisenberg (ikeboy)", + "commit": "ad0f25ff7b23cff7f08ad237c2254e6ecf74257e", + "role": "Cyclic interval and core-aware pair primitives" + } + ], + "icekylinx_lineage": [ + { + "pr": 10, + "commit": "62691e395a0458ce089a1c7b5d89e74291e95e29" + }, + { + "pr": 18, + "commit": "f2ab41aebad47861caf6316282c1793e5513845e" + }, + { + "pr": 24, + "commit": "ed90fd940279c336ebc45968631bdebfb087b505" + }, + { + "pr": 32, + "commit": "0ef3aeb61f55cc0b321ce6a0ef00acee25cefe52" + }, + { + "pr": 36, + "commit": "11817ccacb564bb7f98789c20dc11d3fece207e3" + }, + { + "pr": 104, + "commit": "948ce1510df750f4c18b96bdaef436a86f8bf834" + } + ], + "continuing_dependencies": [ + "stopped_product_extension", + "copied_centers_extension", + "structured_bulk_extension" + ], + "curation": "Selected generators, exact counts, proof sources and pinned dependencies; excludes alternatives, large generated graphs and recursive handoff copies.", + "scope": "Conditional on retained analytic and fixed-tape framework; finite verification and written proofs have separate scopes." + }, + "three_stage_cover_extension": { + "contributor": "icekylinx", + "predecessor_pr": 115, + "predecessor_commit": "cf60442983c153a3a0f5272ea4713f99eba13c86", + "archives": [ + { + "filename": "integer_mult_round7_kappa_1_080522e-4_incremental.zip", + "sha256": "ba7a7d10be57f614fdb4df4074ec5ffd2feddeac583f1fdf2da12ae7c02f6058" + }, + { + "filename": "integer_mult_round8_kappa_3_146011e-4_incremental.zip", + "sha256": "8e730754c2076bf3ee7ff1bd320486172ef6ab1c9897ddd633aef02a9ce5257a" + } + ], + "kappa": "3146011/10000000000", + "proof": "notes/three-stage-cover-note.tex", + "general_subspace_proof": "notes/general-clifford-frames.tex", + "certificate": "certificates/three-stage-cover-network.json", + "producer": "scripts/three_stage_cover_producer.py", + "assistance": "Construction supplied with substantial OpenAI GPT-6 Astra assistance; repository integration with Codex assistance.", + "adopted": [ + { + "pr": 117, + "contributor": "eumemic", + "commit": "cbb05ce504d571546d9b7794c186a613c659c3bf", + "role": "Pinned positive h24 local DAG and its exact support/label replay; original Claude disclosure retained", + "dag_sha256": "3c034d0aae388ef567a454826f4f48b26fd8a94c71e8ffed4835271b349a783b" + }, + { + "pr": 97, + "contributor": "Zhihao Chen (jacklightChen)", + "commit": "e15350b66e4bf796153a68c240a6cc1b788abe3e", + "role": "Retained physical deferred bit word and reflected ledger" + }, + { + "contributor": "Swapnil Jain", + "commit": "741e7aa078392553815df7926ee17ac5e25a8c38", + "role": "Underlying deferred word, lifted frames and frozen witness" + } + ], + "icekylinx_lineage": [ + { + "pr": 10, + "commit": "62691e395a0458ce089a1c7b5d89e74291e95e29" + }, + { + "pr": 18, + "commit": "f2ab41aebad47861caf6316282c1793e5513845e" + }, + { + "pr": 24, + "commit": "ed90fd940279c336ebc45968631bdebfb087b505" + }, + { + "pr": 32, + "commit": "0ef3aeb61f55cc0b321ce6a0ef00acee25cefe52" + }, + { + "pr": 36, + "commit": "11817ccacb564bb7f98789c20dc11d3fece207e3" + }, + { + "pr": 104, + "commit": "948ce1510df750f4c18b96bdaef436a86f8bf834" + }, + { + "pr": 115, + "commit": "cf60442983c153a3a0f5272ea4713f99eba13c86" + } + ], + "continuing_dependencies": [ + "partial_gauge_extension", + "stopped_product_extension", + "structured_bulk_extension" + ], + "scope": "Conditional on retained analytic, precision, ordinary-leaf and fixed-tape contracts. New general Clifford frames, weighted routing, local-ring fallback and internal row borrowing supplied as written proofs.", + "curation": "Selected round7 PR117 input and frame theorem plus round8 three-stage cover only. Excludes PR112 reclaim, two-stage alternatives, half-sign motif, research screens, raw generated graphs and PDFs." } } diff --git a/certificates/joint-dual-kappa.json b/certificates/joint-dual-kappa.json index d92f653fe4..f6f12fa055 100644 --- a/certificates/joint-dual-kappa.json +++ b/certificates/joint-dual-kappa.json @@ -842,7 +842,7 @@ "scripts/experiments/pin_joint_dual_sources.py": "4d31cc0e38b7902709ca5f8f714cba109760d2592346dd7f8dc0b900a2d9fe00", "scripts/experiments/verify_joint_dual.py": "7a31bba1e9c0ded32ae285df2ca9bc91ef20c330f6789fdba361876f3c7a1985" }, - "source_manifest_sha256": "fd1aab7d75753b4b55b2377f26119dae74f367b0b4fab66cba42d944b3042cc9", + "source_manifest_sha256": "3afb016064fca7340c39228fec5dc572fdd36175f9bee8dc37d073990f725740", "sources": [ { "files": { diff --git a/certificates/joint-dual-validation.json b/certificates/joint-dual-validation.json index 99adb4cf74..e97b11efeb 100644 --- a/certificates/joint-dual-validation.json +++ b/certificates/joint-dual-validation.json @@ -226,7 +226,7 @@ } }, "bit_saving": "2382370177/50000000000000", - "certificate_sha256": "2ed658f05365b28825e24a6e7bac9fc9cc015d5a5d63d9579f4ac6e0640ae559", + "certificate_sha256": "e37b6339622513b00bf2b50a9d56a48aad14385bfe334f6f22d893f7e611d6a1", "kappa": "4764513337/100000000000000", "margins": 7, "producer_words_regenerated_and_matched": true, diff --git a/certificates/partial-gauge-bit-input.json b/certificates/partial-gauge-bit-input.json new file mode 100644 index 0000000000..f01ac32417 --- /dev/null +++ b/certificates/partial-gauge-bit-input.json @@ -0,0 +1,142 @@ +{ + "N": 3136441, + "R": 28866, + "W": 108516254, + "deficit": 1344189, + "dimensions": [ + 23, + 23 + ], + "exit_nullity_histogram": { + "1": 1760, + "10": 96, + "11": 335, + "12": 28, + "13": 116, + "14": 36, + "15": 319, + "16": 5, + "17": 91, + "18": 4, + "19": 164, + "2": 1596, + "20": 83, + "21": 49, + "22": 154, + "23": 17301, + "3": 1307, + "4": 3871, + "5": 806, + 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b/certificates/partial-gauge-complex-input.json new file mode 100644 index 0000000000..b404923928 --- /dev/null +++ b/certificates/partial-gauge-complex-input.json @@ -0,0 +1,101 @@ +{ + "h": 24, + "v": 2024, + "R": 34307, + "q": 8120, + "c": 146284, + "matched": 120097, + "loss": 552, + "selected_roles": 6341, + "selected_rank_histogram": { + "22": 850, + "21": 70, + "20": 770, + "19": 81, + "18": 1503, + "16": 761, + "14": 487, + "12": 406, + "10": 374, + "8": 257, + "6": 294, + "4": 252, + "2": 236 + }, + "remaining_internal_histogram": [ + 27033, + 203910, + 83148, + 13608, + 17316, + 6534, + 5740, + 2708, + 2218, + 787, + 642, + 404, + 288, + 218, + 223, + 1296, + 352, + 1944, + 305, + 1566, + 852, + 188, + 22, + 24, + 0 + ], + "source_data_histogram": { + "23": 2024 + }, + "target_data_histogram": { + "23": 18, + "12": 188, + "11": 4, + "16": 82, + "2": 5576, + "3": 802, + "10": 239, + "6": 641, + "4": 3354, + "8": 634, + "1": 908, + "5": 141, + "7": 82, + "14": 89, + "9": 63, + 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a/certificates/three-stage-cover-complex-input.json b/certificates/three-stage-cover-complex-input.json new file mode 100644 index 0000000000..d1b89bcb71 --- /dev/null +++ b/certificates/three-stage-cover-complex-input.json @@ -0,0 +1,115 @@ +{ + "h": 24, + "v": 2024, + "R": 28705, + "q": 8120, + "c": 91770, + "matched": 71185, + "loss": 552, + "selected_roles": 4599, + "selected_rank_histogram": { + "22": 120, + "21": 359, + "20": 794, + "19": 174, + "18": 381, + "17": 147, + "16": 409, + "15": 57, + "14": 157, + "12": 218, + "13": 57, + "11": 131, + "10": 190, + "7": 100, + "8": 187, + "9": 113, + "5": 126, + "6": 137, + "4": 126, + "3": 181, + "2": 184, + "1": 251 + }, + "remaining_internal_histogram": [ + 26680, + 69592, + 67235, + 29094, + 30564, + 11421, + 4563, + 2083, + 2592, + 1678, + 937, + 710, + 794, + 435, + 479, + 172, + 281, + 197, + 755, + 540, + 85, + 0, + 183, + 24, + 0 + ], + "source_data_histogram": { + "23": 2024 + }, + "target_data_histogram": { + "1": 1907, + 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+ } + }, + "kappa": "3146011/10000000000", + "predecessor_commit": "cf60442983c153a3a0f5272ea4713f99eba13c86", + "scope": "Exact local inventories, strict moments, contaminated fallback charge and semantic assembly. Local finite reconstruction is checked separately; group geometry, uniform weighted routing, borrowed rows and inherited analytic/tape contracts are written proof dependencies.", + "source_sha256": { + "certificates/copied-centers-network.json": "7dbd675e42e22ac076280360f8750031851564bb7a56079e66236fee5d0a6cdf", + "certificates/partial-gauge-bit-input.json": "367b9844db23f8fa64002637903bcc5ce28ffc8879d274b5c54c0390fb0234e4", + "certificates/partial-gauge-network.json": "e2e6c48bc283b1417c13adaa35c36d782191e21a33f7279a4ed1558ca60518be", + "certificates/stopped-product-network.json": "43ddb7ee033d316375102cf2d97bb2cf47f879f36a005d3726fa6647a2cbecc5", + "certificates/three-stage-cover-bit-input.json": "d0d496b64d45408d6e0773ebbcd4d5dcca918a596073a51e5fc186b82c3b8520", + "certificates/three-stage-cover-complex-input.json": "afecf8ce2d8eb259074e61cd0c386b464ca2f57c61b00979eac808fa26e0b770", + "certificates/three-stage-cover-input.json": "633e108352d3a880c3dc05e265f40270698ea7ebbe655706f5f8675070c86bd2", + "notes/general-clifford-frames.tex": "c9bb9655b594583c6b4bab8cb485233f1d1e588f83858cf1e4c3530cee8fed54", + "notes/three-stage-cover-assembly.tex": "ccf34a6d37c4af14bebeab15be50ab204eb1caebe546385520050ec30089441d", + "notes/three-stage-cover-bit.tex": "47123a756f823c83a0d48cca2bf5add6472cd88410b13aed345e70c8834fb511", + "notes/three-stage-cover-complex.tex": "1c5b638aa9b75365b8addec9c8eeee6bc5e499722901079b94c65fce033fde8a", + "notes/three-stage-cover-note.tex": "db8078687825bcd43c6cc2fb69712ebb12840057b77cd5af8f67f53e4ffffa92", + "notes/three-stage-cover-rows.tex": "6ad97a5cd7d6aecf978a81226e1c2763d37840caca130c61f3bf5c4ac111c5c8", + "references/partial-gauge/pr97/SOURCE.json": "68fb539abcd6df21d142e4e204b6e9482eb7f9cd3907650f54bd058070266596", + "references/three-stage-cover/pr117/LICENSE": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4", + "references/three-stage-cover/pr117/NOTICE": "a0030510cd3b5334fb113eae759aadd2c25951bd353dfb69ad9cb780ae5adf30", + "references/three-stage-cover/pr117/SOURCE.json": "74e9f4b4ae0529d21d19c159b2b50ee0923e93e0d7eb22cf208ad41dd5992275", + "references/three-stage-cover/pr117/dag.json.gz": "3c034d0aae388ef567a454826f4f48b26fd8a94c71e8ffed4835271b349a783b", + "references/three-stage-cover/pr117/replayed.py": "99c1e3694ae06a7ce7a370b04ecb45ea3fcfe69a6e6e7f4c4606e0a6c74c3184", + "scripts/partial_gauge/binary.py": "b29e2f37d7f32a959a378190d7f1da3c2da534fbdaf878044800301957bffc03", + "scripts/partial_gauge/lift.py": "895874399113faa090d168d6168f0664136df6e21205671e7f7f9239cad93d1d", + "scripts/partial_gauge/match.cpp": "12b8b7c93bb78324e7328e624b166be2067ae8861540531c024040220d86f368", + "scripts/partial_gauge/word.py": "96015b7b08a261673ab8be1ea47f89302f8a1d62a822d657a09f6a1fcfb51e58", + "scripts/partial_gauge_bit.py": "7d58e86061da2d73991531e9bc431589019359391adea866c61c7dd1f99ac855", + "scripts/partial_swap_network.py": "acdd81b4dea2815f8838a0c3ea8ac4968f2bc13fd7c99173473d865779096782", + "scripts/structured_bulk_assembly.py": "e99467e14211fba699878ed036e1e2362eb8e395dc22f9ecdcecc7fbf2cc736d", + "scripts/three_stage_cover/README.md": "b39a28fa8da5d7d3188dfed4c0e0af1e2397f0f5c8f69cfb69671ed27ddca4bb", + "scripts/three_stage_cover/__init__.py": "b964b59af038e70cc15af655a22bcf6aaafc9c842b213765781bc8bea7166430", + "scripts/three_stage_cover/lift.py": "ae5f4c56cd32b5232d6dcdcaed1e24efb81fe20467a2d915ee51479bced06cfc", + "scripts/three_stage_cover/select.py": "457e636ced9bbd368198f78e015660a929ba1aaccbe34122d18da09ebb488f33", + "scripts/three_stage_cover/verify.py": "4bd2347e87e4edb699968bc31dfb94f2ab89bf1227576e59f43c779ba74e4e06", + "scripts/three_stage_cover_network.py": "7e0ad6fd1921b6905a53ce1eeaa03d3bb3a8a8bbaa0382516ea98b636f46335e", + "scripts/three_stage_cover_producer.py": "ea6dd4f06173304bbcb74b8e5bcc02357b7bb162a6d26b4a6f7a9e886158123f" + }, + "status": "Conditional three-stage cover multiplication witness" +} diff --git a/docs/partial-gauge.md b/docs/partial-gauge.md new file mode 100644 index 0000000000..dd5c670a51 --- /dev/null +++ b/docs/partial-gauge.md @@ -0,0 +1,65 @@ +# Partial-gauge reproduction + +This extension inherits PR #104 at +`948ce1510df750f4c18b96bdaef436a86f8bf834` and gives the conditional saving +`7237/78125000 = 9.26336e-5`. Proofs are supplied as LaTeX source. + +From the repository root, with Python 3.11+ and a C++17 compiler: + +```sh +make partial-gauge-verify +``` + +CI runs this extension in its own `partial-gauge` matrix group. The +incremental target performs three checks: + +- `partial_gauge_bit.py` checks pinned PR #97 source hashes, reconstructs + auxiliary and data ranks from its actual frozen paths, matches the physical + ledger receipt, and rebuilds the whole-projector child list. The unchanged + upstream complete-basis scalar and rational frame audits are inherited. +- `partial_gauge_producer.py` rebuilds the selected h24 interval/cube DAG, + compatible matching, backward enlarged frames, physical word and partial + source gauges. Independent checks cover exact supports and signed decoder, + frame nondegeneracy and nesting, center closure, untouched source roles, + reverse readout order and every positive-rank transition. The deterministic + construction also reproduces the complete histogram, including zero-rank + no-ops; see [the producer guide](../scripts/partial_gauge/README.md). +- `partial_gauge_network.py` certifies both strict moments, the stopped + ordinary-bit saving, denominator-21 semantic charge, product row stock, + all 47 strict constraints and seven assembly margins. It writes the + deterministic, source-bound `certificates/partial-gauge-network.json`. + +For arithmetic and the pinned bit profile alone: + +```sh +make partial-gauge-certificate +``` + +Generated complex graphs are temporary by default. To retain them and a +compact report outside the repository: + +```sh +python3 scripts/partial_gauge_producer.py --work-dir /tmp/partial-gauge-work --output /tmp/partial-gauge-verification.json +``` + +The selected proof is [partial-gauge-note.tex](../notes/partial-gauge-note.tex). +The source notices and compact PR #97 witness remain under +`references/partial-gauge/`. Unselected experiments, large generated graphs, +recursive handoff copies and PDFs are excluded. + +## Shared source bindings + +The joint-dual manifest also binds `Makefile`, `README.md` and `NOTICE`. +This round refreshes those three digests and the two derived certificate +digests. Its producer, physical words, frame profiles and mathematical +certificate fields are unchanged from PR #104's successful local replay. +Incremental validation checks that equality and reruns the joint arithmetic; +it does not repeat the unchanged graph and complete-basis suites. + +After reviewing changes to those shared files, the explicit manifest refresh +is `python3 scripts/experiments/pin_joint_dual_sources.py`. Its derived +`joint-dual-kappa.json` and `joint-dual-validation.json` must also be updated. +The full affected replay command, when needed, is `make verify-joint`. +CI checks the committed source bindings and requires deterministic output; +it never silently repins sources. Stage the final artifacts before the +reproducibility check and require `git diff --exit-code` to remain clean. diff --git a/docs/reproducibility.md b/docs/reproducibility.md index e48f4190cd..8f70a8f12b 100644 --- a/docs/reproducibility.md +++ b/docs/reproducibility.md @@ -1,5 +1,18 @@ # Reproducing the result +For the three-stage cover extension, run `make three-stage-cover-verify`. +The [incremental guide](three-stage-cover.md) covers the new PR #117 local +word, cover moments and exact assembly. + +For the partial-gauge extension, run `make partial-gauge-verify`. Its +[incremental guide](partial-gauge.md) separates the pinned PR #97 bit input, +new complex producer and exact assembly. + +For the new stopped product-ring extension, run `make stopped-product-verify`. +Its [incremental reproduction guide](stopped-product.md) covers the h24 +rational-center producer, new moments, stopping parameters and assembly. +The community checkpoint and historical targets below retain their own scope. + The selected release is documented in the [joint-frame community review](research/community-round2-review.md) and [current status](research/current-status.md), with the [selected parameter certificate](../research/matrix-exponent-synthesis/candidate/arithmetic.json). diff --git a/docs/stopped-product.md b/docs/stopped-product.md new file mode 100644 index 0000000000..5bc1cbc88e --- /dev/null +++ b/docs/stopped-product.md @@ -0,0 +1,67 @@ +# Stopped product-ring reproduction + +The extension gives the conditional saving `194869/2500000000 = 7.79476e-5`. +It starts from merged PR #36 and is submitted on upstream main at +`56b66d58297deca1d7dd130247d720e960f77a37`. + +From the repository root, with Python 3.11+, Make and a C++17 compiler: + +```sh +make stopped-product-verify +``` + +The incremental checks: + +- Match the unchanged positive-label h23 bit producer to its verified + copied-center input, preserving the same matching and histogram. +- Rebuild only the new h24 all-disjoint complex producer with aligned pair + stars, exact rational center decoding, support and binary-frame checks, + then reproduce every matching count and histogram entry. +- Reconstruct both child lists and strict rational moments; derive the + stopped ordinary-bit saving and check that the atom-adapter toll is lower. +- Check the denominator-21 semantic charge, product row stock `p^4000`, + seven assembly margins and all 47 strict constraints. + +Intermediate DAGs and the matcher executable are temporary. To retain them: + +```sh +python3 scripts/stopped_product_producer.py --work-dir /tmp/stopped-product-producers +``` + +For arithmetic alone, use `make stopped-product-certificate`. The inherited +`make verify-producers` also includes this target; the full `make verify` +retains the community and historical checks. Those unchanged suites are +not part of this round's local incremental validation. + +The [proof](../notes/stopped-product-note.tex) is supplied as LaTeX source; +no PDF is generated. Its common-basis factorization, atom streaming and +stopping argument, outer ordinary conversion and shared exact rational +grid are written proofs. Finite checks verify the selected scalar data and +arithmetic, not the complete all-size theorem. + +[SOURCES.json](../SOURCES.json) and [NOTICE](../NOTICE) preserve contribution +credits and source pins. Existing two-stage, ordinary-leaf and semantic +proof dependencies remain in `references/copied-centers/` and +`references/semantic-bulk/`. Research alternatives, raw binary DAGs and +recursive predecessor copies are excluded from this submission. + +## Shared source bindings before submission + +The inherited `research/joint-dual/SOURCE.json` also binds the repository's +`Makefile`, `README.md` and `NOTICE`. Adding a verification target or editing +those documents therefore affects the joint-frame check even when its +mathematical construction is unchanged. + +After reviewing such source changes, refresh the manifest and regenerate +its dependent certificate and validation receipt: + +```sh +python3 scripts/experiments/pin_joint_dual_sources.py +make verify-joint +``` + +Include `research/joint-dual/SOURCE.json`, `certificates/joint-dual-kappa.json` +and `certificates/joint-dual-validation.json` in the same submission. Check +that their changes are the expected source bindings, then rerun the affected +check against the staged artifacts and require a clean diff. CI verifies +the committed bindings; it does not refresh them automatically. diff --git a/docs/three-stage-cover.md b/docs/three-stage-cover.md new file mode 100644 index 0000000000..38554e1fb3 --- /dev/null +++ b/docs/three-stage-cover.md @@ -0,0 +1,74 @@ +# Three-stage cover reproduction + +The extension inherits PR #115 at +`cf60442983c153a3a0f5272ea4713f99eba13c86` and gives the conditional saving +`3146011/10000000000 = 3.146011e-4`. + +From the repository root, using Python 3.11+ and a C++17 compiler: + +```sh +make three-stage-cover-verify +``` + +CI runs this target in its own `three-stage-cover` matrix group. Its +incremental scope is: + +- Rebuild the new pinned PR #117 local DAG, compatible matching, full backward + subspace intersections, actual scalar word and partial source gauges. + Independently check scalar supports, signed coefficients, divisor 21, + carrier containment, center closure, untouched source roles, actual frame + transitions and reverse target chains. Intermediate frames can be + degenerate; their validity follows from the new general Clifford proof. +- Reconstruct the unchanged PR #97 local physical ranks from its pinned + paths, then compose the new three-stage bit inventory. The unchanged + upstream complete-basis and rational-frame suites are inherited checks. +- Certify the bit moment with the full rare-class local-ring fallback, + stopped ordinary saving, complex moment, full finite group order and + router charge, exact semantic guard, row reserve, 47 strict inequalities + and seven assembly margins. + +For the moments and arithmetic alone: + +```sh +make three-stage-cover-certificate +``` + +The group is specified by a written construction; verification does not +materialize its vertices. The complex group size and finite routing allowance +remain in the certificate. The bit group depends on atom width; its uniform +batching, weighted children and internally borrowed rows are proved in +`notes/three-stage-cover-bit.tex` and `notes/three-stage-cover-rows.tex`. +Finite arithmetic does not by itself verify those all-size interfaces. + +To retain regenerated local files outside the repository: + +```sh +python3 scripts/three_stage_cover_producer.py --work-dir /tmp/three-stage-cover-work --output /tmp/three-stage-cover-verification.json +``` + +The [main proof](../notes/three-stage-cover-note.tex) includes +[general Clifford frames](../notes/general-clifford-frames.tex), exact data +chronology and phases, complete dirty cleanup, weighted factorization, +fallback, row restoration and final composition. Sources are included +without a PDF. `SOURCES.json`, `NOTICE` and `references/three-stage-cover/` +retain the adopted author's source pins and original legal notices. + +The previous submitted production code and certificates remain unchanged. +Unselected round7/round8 research and large generated graphs are excluded. +Only the new selected local construction and changed cover/assembly are +verified this round. + +## Shared source digests + +Changes to `Makefile`, `README.md` and `NOTICE` also affect the inherited +joint-dual source manifest. After reviewing their final content, explicitly +run `scripts/experiments/pin_joint_dual_sources.py`, regenerate the joint +arithmetic certificate, and update its validation receipt's certificate +digest. The unchanged mathematical fields and prior replay payload must +match the preceding submission. A full producer replay is required when +those producer or verifier inputs change; the full command is +`make verify-joint`. + +The final reproducibility check runs against staged artifacts and requires +`git diff --exit-code` to remain clean. CI verifies committed hashes and +does not silently refresh source pins. diff --git a/notes/general-clifford-frames.tex b/notes/general-clifford-frames.tex new file mode 100644 index 0000000000..0367eeb005 --- /dev/null +++ b/notes/general-clifford-frames.tex @@ -0,0 +1,102 @@ +\subsection{Preimage frames for arbitrary binary subspaces} +This frame construction is due to icekylinx with OpenAI GPT-6 Astra assistance; +the present exposition and integration use Codex. +Let $H=\mathbb F_2^h$ with the standard dot product. In the binary +symplectic space $H\oplus H$ put +\[ + L_0=\{(0,z)\},\qquad + F=\begin{pmatrix}I&I\\0&I\end{pmatrix},\qquad + L_F=FL_0=\{(z,z)\}. +\] +We also write $F$ for the exact Gaussian-dyadic operator $C^{\otimes h}$. +Its exact square is a Pauli translation; only its binary symplectic square +is the identity. For an exact Clifford operator $T$ the frame label is +$L(T)=T^{-1}L_0$. The physical transition from $T$ to $S$ is $ST^{-1}$, +and its Fourier rank is +$d(L(S),L(T))$, where $d(L,L')=h-\dim(L\cap L')$. +This follows by transporting the intersection by $S$: the intersection +with $L_0$ computes the kernel dimension of the $B$ block of $ST^{-1}$. + +\begin{lemma} +For every subspace $U\subseteq H$, including degenerate subspaces, +\[ + L_U=\{(u,u+v):u\in U,\ v\in U^\perp\} +\] +is Lagrangian, and +\[ + d(L_U,L_V)=\dim U+\dim V-2\dim(U\cap V),\qquad + FL_U=L_{U^\perp}. +\] +\end{lemma} +\begin{proof} +The displayed parametrization is injective and has dimension $h$. +Its symplectic pairing is $u\cdot v'+v\cdot u'=0$. +In an intersection, $u$ belongs to $U\cap V$ and $v$ to +$(U+V)^\perp$, giving the asserted dimension and distance. +Finally $F(u,u+v)=(v,u+v)$ proves the complement identity. +\end{proof} + +\subsection{Exact representatives and nested children} +Choose a deterministic invertible binary matrix $G$ carrying the first +$r=\dim U$ coordinate space $E_r$ to $U$, and set +\[ + K_G=\begin{pmatrix}G&0\\G+G^{-T}&G^{-T}\end{pmatrix},\qquad + T_U=K_G T_{E_r}K_G^{-1}. +\] +Here $T_{E_r}$ is the exact coordinate $C$ operator, and the $K_G$ are +exact Gaussian-dyadic Clifford lifts. The matrix $K_G$ is symplectic +because $G^TG+I$ is symmetric. It fixes both $L_0,L_F$ and maps $L_V$ +to $L_{GV}$. Thus both $K_G$ and $FK_GF^{-1}$ have Fourier rank zero; +their lifts are fixed binary address permutations with unit quadratic +phases. In particular, +\[ + L(T_U)=L_U,\quad \operatorname{rank}B(T_U)=r,\quad + \operatorname{rank}B(FT_UF^{-1})=r,\quad T_U^2\text{ is Pauli}. +\] +Set $V_U=GE_r^\perp$, a direct complement of $U$. The binary action gives +$T_U^{-1}L_F=L_{V_U}$ and hence +$\operatorname{rank}B(FT_U^{-1})=h-r$. +No assertion $V_U=U^\perp$ is needed. +For the norm-one source lines and their orthogonal target hyperplanes, +retain the inherited exact representatives; set $T_0=I,T_H=F$. +At every common scalar gate use one identical exact representative on all +incident registers. If separately transported representatives have the +same preimage, their physical ratio has rank zero and is implemented as +an actual left adapter before the gate. + +For $U\subseteq V$, the exact transition $T_VT_U^{-1}$ has rank +$\dim V-\dim U$. Clifford elimination expresses it as two rank-zero +adapters surrounding that many coordinate Fourier factors. These factors +are a single $C^{\otimes r}$ child: the scaled Hadamard satisfies +\[ + \frac{1+i}{2}\begin{pmatrix}1&1\\1&-1\end{pmatrix} + =SCS,\qquad S=\operatorname{diag}(1,i). +\] +Thus no new irrational field is introduced. All adapters are charged +finite movement and scalar operations. Exact transitions telescope; +commutativity between distinct $T_U$ is unnecessary. + +\subsection{Dirty tails and reverse/complement words} +A dirty role entering with gauge $T_\sigma$ and whose last active frame +is $T_U$, with $\sigma\subseteq U$, is assigned terminal gauge +$FT_\sigma$. Its logical value is restored by the complete transparent +scalar word, so its physical action is exactly $F$. The tail is +$FT_\sigma T_U^{-1}$. Its rank is +\[ + d(L_U,L_{V_\sigma})=h-\dim U+\dim\sigma. +\] +Indeed $H=\sigma\oplus V_\sigma$ and $U\supseteq\sigma$ imply +$\dim(U\cap V_\sigma)=\dim U-\dim\sigma$. +An inactive exterior adds $m-h$, giving $m-\dim U+\dim\sigma$. + +For the complete time-reversed word use +\[ + D_U=T_UF^{-1},\qquad L(D_U)=L_{U^\perp},\qquad + D_UD_V^{-1}=T_UT_V^{-1}. +\] +The last equality is exact, so reversal preserves every inverse transition +rank. Reverse the scalar signs and the whole transparent word, including +old-value corrections and final cleanup. The endpoints $D_0=F^{-1}$ +and $F$ differ by a Pauli, and the inherited norm-one endpoints are +reconciled by actual rank-zero adapters. Neither equality of binary +matrices nor equality of frame cosets substitutes for these operators. diff --git a/notes/partial-gauge-assembly.tex b/notes/partial-gauge-assembly.tex new file mode 100644 index 0000000000..4a09f324a3 --- /dev/null +++ b/notes/partial-gauge-assembly.tex @@ -0,0 +1,101 @@ +\subsection{Literal two-stage inventory and semantic precision} +Let $B_a$ count selected gauges of dimension $a$, and $H_r$ the remaining +internal histogram, including each residual first transition. Let $D_r^x$ +and $D_r^y$ be the source and target data-chain histograms. Put +\[ + m=h^2=576,\quad N=v^2=4096576,\quad + W=2N+2vR=147067888. +\] +The child inventory consists of $2v(R-\sum_aB_a)$ exteriors of width +$m-h$, $2vB_a$ exteriors of width $m-h+a$, $2vH_r$ internal children, +$2v(D_r^x+D_r^y)$ data-chain children, $2N$ connectors of width +$(h-1)^2$, and $N$ paid correction children of width one. Zero-width +transitions require no recursive call. Per-role telescoping gives +\[ + s=\sum_r rn_r=Wm-N+2v\ell=84709241408, + \qquad r_{\max}=574. +\] +Writing $F$ for the full transform and $T$ for the source-line transform, +the retained signed endpoint is +$X'=-Fy$, $Y'=FT^{-2}x+FT^{-1}y$. +Pretranslate the initial $x$ bank by the fixed Pauli translation $T^2$, +then add $T^{-1}X'$ to $Y'$ using the paid rank-one copy. Sign and bank +corrections yield both $F$ outputs. The translation $T^2$ is retained. + +The rational logarithm and exponential enclosures in +\path{scripts/partial_gauge_network.py} prove, for $b=92651/10^9$, +\[ + \frac1W\sum_r n_r(r/m)^{1-b} + <1-4.1393037848\times10^{-11}. +\] +The finite receipt supplies a scalar-operation bound +$G=401572403009697216$, including direct rational dirty reads and the +signed correction. Set +\[ + E=64(W+m+G+1)^3,\quad B=s+E,\quad C_0=32mB^2. +\] +It verifies $2GW^2+8s+4W+4+32ms+E$. Thus the inherited completed-child induction applies +with semantic $C_1=1$. On one exact grid $2^{-P}21^{-K}$, +$K=G(D_c+1)$, unfinished calls can accumulate at most $K$ odd-denominator +factors. Completed children are Gaussian-dyadic $C^{\otimes u}$ and add +none. The root dyadic-input divisibility invariant makes each local +21-division exact; only the outer boundary scales and unscales, at cost +$O(V_{\rm bits}\log d)$. No recursive boundary rounds or re-encodes values. + +\subsection{Joint rows and conditional transfer} +The coarse bit, ordinary leaf and complex families have respective +halving-degree and wire-bit pairs $(367,27),(9,28),(200,28)$. +Their product stock has coefficient +\[ + 367\cdot27+9\cdot28+200\cdot28=15761, + \qquad40000-\frac{51}{25}\,15761=\frac{196189}{25}>0. +\] +One $p^{40000}$ reserve, one preceding prefix and one final padding suffice +under the retained eventual row contract. Sequential calls reuse restored +rows and preserve complete spectators. + +Choose +\[ + \eta=10^{-8},\quad\beta=10^{-6},\quad + a=(1-\beta)b-10^{-10}=\frac{92650807349}{10^{15}}, + \quad\tau=1-a,\quad\sigma=1-b, +\] +\[ + q_0=a(1-2\eta),\quad c=q_0(1+\eta),\quad + \epsilon=\frac{1-\eta}{1+c+q_0},\quad + \lambda'=1-q_0,\quad\lambda=(\tau+\lambda')/2, +\] +\[ + g=\epsilon q_0,\qquad r=(g+1-\epsilon)/2,\qquad\delta=\eta/8. +\] +Here $a0. +\] +\begin{theorem} +Under the retained analytic, semantic-precision, exact-recovery and +fixed-tape interfaces, the selected construction gives +\[ + T(n)=O\!\left(n(\log n)^{1-\kappa}\right),\qquad + \boxed{\kappa=\frac{7237}{78125000}=9.26336\times10^{-5}}. +\] +\end{theorem} +\begin{proof} +The stopped word supplies the required ordinary interchange contract; +the signed framed program supplies the exact completed complex tensor +contract. The common grid and product reserve meet the retained precision +and row requirements. Substitution of the displayed parameters into the +unchanged semantic/bulk transfer yields its seven savings. Their strict +excess over $\kappa$ absorbs the fixed logarithmic overheads, including +outer grid conversions, and proves the stated conditional bound. +\end{proof} diff --git a/notes/partial-gauge-construction.tex b/notes/partial-gauge-construction.tex new file mode 100644 index 0000000000..147fc5ab91 --- /dev/null +++ b/notes/partial-gauge-construction.tex @@ -0,0 +1,60 @@ +\subsection{Stopped opposite-bank bit word} +Apply the factorization in \path{notes/stopped-product-factorization.tex} +to each rank increment of PR97's literal deferred word: a rank-$r$ +projector increment is one recursive rank-$r$ reversed child, with the +fixed atom adapters charged as in \path{notes/stopped-product-interface.tex}. +The selected finite profile has +\[ + m_b=529,\quad W_b=108516254,\quad s_b=57403754177, + \qquad r_{\max,b}=528. +\] +Its rational moment certificate proves coarse saving +$15513/125000000$, with normalized gap exceeding $1.1417\times10^{-10}$. +The atom exponent $10^{-3}$ and retained ordinary leaf saving +$384599/10^{10}$ give the stopped ordinary saving +\[ + a_b=(1-10^{-3})\frac{15513}{125000000} + +10^{-3}\frac{384599}{10^{10}} + =\frac{1240183559}{10^{13}}. +\] +The exact inputs and calculation are +\path{certificates/partial-gauge-bit-input.json}, checked by +\path{scripts/partial_gauge_bit.py}. + +\subsection{Interval and pair-first cube producer} +For disjoint strip values $v_0,\ldots,v_{k-1}$ construct cyclic intervals +$I(a,1)=v_a$ and $I(a,l)=I(a,l-1)+v_{a+l-1\bmod k}$ for $2\le l2r_j^{t_j}$ gives depth at most +$t_j\lceil\log_2\max(2,e)\rceil$ plus a fixed constant. +The new coarse bit family has $(t_j,\operatorname{bitlength}W_j)=(16,28)$, +the ordinary leaf family $(9,28)$, and the complex family $(17,28)$. +The required reserve is the product of all three stocks: +\[ + Q=W_c^{D_c}W_b^{D_b}W_{\rm old}^{D_{\rm old}}. +\] +After the complex and coarse divisions, the old stock remains available +to every peeled-atom or stopped-leaf call. Sequential copies and outer +ordinary wrappers reuse restored rows; they introduce no independent +row factor. The combined coefficient is +\[ + 16\cdot28+9\cdot28+17\cdot28=1176, + \qquad4000-\frac{51}{25}\,1176=\frac{40024}{25}>0. +\] +Thus the retained logarithmic comparison, for eventual $e\le Cp$, +supplies a sufficient $p^{4000}$ stock including the fixed constants. +Use one physically preceding transformed prefix and one complete-row +padding, as in \path{notes/structured-bulk-assembly.tex}. Complete +spectators and parked fields are preserved by every descendant. +The suffix condition +$b_{\rm in}^{1-\epsilon}>16000(\log_2b_{\rm in}+8)$ is eventual +for every fixed $\epsilon<1$. + +\subsection{Analytic transfer and selected parameters} +The retained analytic proof map in +\path{notes/structured-bulk-assembly.tex} supplies the active-masked +arbitrary-source router with paid repair scans, the rounded phase-cell +inverse, persistent bulk halos with original period cuts, sequential +overlap tapes and factor pages, prime packing and exact recovery. +The new ordinary bit primitive has precisely the arbitrary-width, +complete-spectator and restored-row contract consumed by that router. +The new complex primitive supplies the exact completed tensor contract +on one common grid. Its outer grid conversions add only logarithmic +work. The changed finite implementations therefore meet the same +analytic interfaces, including the full bulk bound +\[ + O\!\left(Tp\left[p^\tau\operatorname{polylog}p+ + d\operatorname{polylog}p+dp^\delta+w_G^2p^\delta+ + dw_G^2p^\delta/L\right]\right)+n^{o(1)}, +\] +where $L=2^{\lceil8\log_2p\rceil}$, Gaussian band width +$w_G^2=O(p^{1-r})$, and normalization retains $\epsilon+r<1$. +Here $w_G$ is unrelated to the address atom width. + +Use the proved ordinary bit saving $a_b=803380799/10^{13}$ and complex +saving $b=1949/25000000$. To instantiate the retained assembly, weaken +the bit parameter explicitly to +\[ + \beta=10^{-6},\qquad + a=(1-\beta)b-10^{-10}=\frac{1948995551}{25000000000000}6.6129027953\times10^{-11}>0. +\] +The strict gaps absorb fixed logarithms, including the common-grid +conversions. The stopped atom recurrence supplies the ordinary +interchange interface, the rational-center construction supplies the +exact complex interface, and the product stock supports all nested +recursions. The unchanged analytic transfer then proves the stated +bound, with its constructive setup, descriptor and catalogue domination, +prime-existence and exact-recovery thresholds retained as eventual +conditions. +\end{proof} diff --git a/notes/stopped-product-complex.tex b/notes/stopped-product-complex.tex new file mode 100644 index 0000000000..4a9037137c --- /dev/null +++ b/notes/stopped-product-complex.tex @@ -0,0 +1,137 @@ +\subsection{All-disjoint retained centers} +Use two factors $h=24$. For triple inputs write $G=\sum_TX_T$, +$T_i=\sum_{T\ni i}X_T$, and retain all $A_i=G-T_i$. +Each input triple avoids exactly 21 points, so +\[ + \sum_{i=1}^{24}A_i=21G,\qquad G=\frac1{21}\sum_i A_i. +\] +For a target triple $S$ the central correction is +\[ + G-\frac12\sum_{i\in S}A_i + =\frac{\sum_{i\in S}T_i-G}{2}. +\] +An input triple meeting $S$ in $j$ points has central coefficient +$(j-1)/2$. Adding half the disjoint sum and subtracting half the +intersection-two sum leaves coefficient one exactly when $j=3$. +The scalar source-to-scatter transfer is therefore the identity. +The transparent old-value subtraction and new-value addition use this +identity only on fresh contributions and restore arbitrary old values. + +The base-two paired exclusion and aligned pair-star producer retains +nested binary labels: common-pair supports have their orthonormal triple +span, other ordinary sums have their coordinate cover. Each retained +$A_i$ has the nondegenerate coordinate hyperplane excluding $i$, +of dimension 23. Hence all copied-center arguments and general +nondegenerate binary residuals of the preceding proof apply. The +selected local counts are +\[ + v=2024,\quad c_h=45764,\quad q_h=8120,\quad + |\mathcal M|=8966,\quad R=44918,\quad\ell=552. +\] +The old internal histogram, starting at rank zero, is +\begin{small}\begin{align*} +[&8930,49042,30844,15972,11883,8272,6399,4356,6682,2263,\\ + &5109,1808,5317,1733,5391,1757,4908,3290,6233,4363,\\ + &4056,264,564,24,24]. +\end{align*}\end{small} +It has rank mass $24R+2\ell$. Replace 24 rank-24 original transitions +by 24 rank-one transitions; all rank-23 copied transforms remain. +The new mass is $24R+\ell$. + +\subsection{Physical endpoints and strict complex moment} +The copied-center two-stage complex contract in +\path{notes/copied-centers-complex.tex} is unchanged by the scalar +center decoding. For the tensor triple line $U$, let +$T=C_U$, $F=C_I$, $E=C_{U^\perp}=FT^{-1}$. +The two data outputs are $A=-Fy$ and $B=FT^{-2}x+Ey$. +Add $T^{-1}$ applied to a copy of $A$ into $B$, paying one rank-one +child per data pair. The inherited input/output $Z$ phases and +$i^{9f}$ scalar normalize $FT^{-2}$ to $F$; sign and bank corrections +complete the exact tensor $C$. Auxiliary source/sink operators remain +$C_{D_0}$ and $FC_{D_0}$ in the second stage. Every original role +therefore returns the exact selected-axis tensor operation on arbitrary +inputs. This algebraic statement holds over $\mathbb Q(i)$ as well as +over Gaussian dyadics. + +With $m_c=576$, $N_c=v^2=4096576$, $B_c=vR=90914032$, put +\[ + W_c=2N_c+2B_c=190021216,\quad L_c=2v\ell=2234496, +\] +\[ + s_c=W_cm_c-N_c+L_c=109450358336,\qquad N_c-L_c=1862080. +\] +The complete list is $2B_c$ width-552 auxiliary exteriors, +$2N_c$ width-529 data macros, $2vH'_r$ width-$r$ internal children, +$4N_c$ width-23 data fronts, and $N_c$ width-one endpoint copies. +These disjoint classes include both kinds of paid copies and sum to +$s_c$. All binary residuals, including alternating forms, use the retained +one-child Gauss normal form with paid bit adapters and unit phases. +At $a_c=1949/25000000$ the exact rational enclosure gives +\[ + \sum_r\frac{n_r}{W_c}(r/m_c)^{1-a_c} + <1-1.3338373969\times10^{-10}. +\] + +\subsection{One exact odd-denominator grid} +The only new odd divisor is 21. Let $d$ bound the root axis count, +$D$ bound its maximum complex recursion depth, and $G_0$ bound the +number of grouped scalar operations per node. Choose +\[ + K=G_0(D+1),\qquad \text{common grid }2^{-P}21^{-K}. +\] +Root inputs and arbitrary dirty auxiliaries start Gaussian dyadic. +Multiply their integer numerators by $21^K$ once at the outer boundary. +Every local coefficient contains at most one factor 21 in its denominator, +so a local finite program adds at most $G_0$ to the odd-denominator +exponent. A completed $u$-axis child has Gaussian-dyadic coefficients +and therefore preserves its input's existing odd-denominator exponent; +it adds no new odd factor. Only the active chain of unfinished calls +can accumulate new factors. Its length is at most $D+1$, proving that +all intermediates lie on the common grid and every local division by +21 is an exact integer division of its encoded numerator. + +This divisibility claim uses the root dyadic-input invariant, not an +assumption about arbitrary numerators on the finest grid. Internal values +may have odd denominators. They are neither rounded nor re-encoded when +a child returns. At the completed outer return the operator is the exact +Gaussian-dyadic tensor on root dyadic inputs, so the final numerators +are divisible by $21^K$; exact unscaling restores dyadic encoding. + +Since the largest child is $552<576$, $D=O(\log d)$ and +$K=O(\log d)$ with fixed constants. Constant-divisor long division +is linear in numerator length. Initial and final scaling by $K$ +successive constant multiplications/divisions cost $O(V\log d)$ and +occur only at the outer boundary. The extra numerator length is +$O(\log d)$, preserving the retained $O(p)$ word bound. This overhead +is subordinate to the positive-power complex layer bound. + +For explicit semantic constants use +\[ + g_h=4(c_h+v)+10v+4hv+4h^2+8h+8=408200, +\] +\[ + G_0=N_c+2v(g_h+2h)=1656684480, +\] +where endpoint additions and copied-center scalar work are included. +The complete child contributes at most $u$ binary denominator bits +and row-norm factor $2^u$, with no additional odd exponent. A local +21-division contributes fewer than five denominator-height bits, +covered by the fixed scalar charge. Set +\[ +\begin{aligned} + E_0&=64(W_c+m_c+G_0+1)^3=403063477262941121443761178688,\\ + \mathcal E&=2G_0W_c^2+8s_c+4W_c+4+32m_c\\ + &=1196393335930350962702877484.2909639537\times10^{-10}. +\] +It uses the retained 24-term positive atanh-series logarithm enclosure, +its geometric tail, range reduction and upward rational exponential +bound. Also $\mu_1=s_b/(W_bm_b)<1$. These are the two gaps used below. diff --git a/notes/stopped-product-interface.tex b/notes/stopped-product-interface.tex new file mode 100644 index 0000000000..20c44ce486 --- /dev/null +++ b/notes/stopped-product-interface.tex @@ -0,0 +1,95 @@ +\subsection{Paid atom adapters and remainders} +Expand each fixed lower-triangular formal factor into unit scalings and +additions from earlier to later formal fields. Execute intrabank targets +in descending formal-field order, so earlier controls still have their +required original values. Every atom update reads only physically earlier +controls. In a cross-bank shear the update is +\[ + D_{i,t}\leftarrow D_{i,t}+k_{ij}H_{j,f-1-t}. +\] +Even with the reversed source index, the source lies in the earlier head +bank. The retained ordered-affine streaming primitive applies to each +ordinary width-$w$ atom update, costing $O(V)$ per pass. +There are $O(f)$ passes per fixed matrix and thus $O(Vn)$ per finite +node. The reversed control is an indexed earlier atom, not a reversed +field operation. Atom loops use binary descriptors on a fixed tape set; +all inactive digits remain complete spectators. The inherited parking +schedule never scans through parked ancestors, and polynomial descriptor +work is absorbed by the complete stream volume. + +For $n=m_bf+\delta$, $0\le\deltaa_b$, the paid adapter term is strictly lower order. +All bounded widths use the old interface. Numerical padding of a binary +chunk range to the least suitable radix-$q$ range has volume factor +below $q^2$; it preserves numerical order and is removed after the +completed ordinary wrapper, whose endpoint is ordinary interchange. + +If external rows are unavailable, use the inherited ordinary high-digit +borrowing construction from \path{notes/batched-bit-rows.tex}. +Borrow $O(\log e)$ high digits, exchange and join them into a preceding +row field, pad that field once, act on the body, and separate the high +parts. It applies literally to the completed ordinary wrapper; no claim +about borrowed high positions is made for an unwrapped reversed call. +Single-digit moves cost $O(V\log e)$, and small widths use the old +algorithm. Atom indices and rational coefficients are finitely specified; +the algorithm uses the same fixed tape set and complete spectator ranges. diff --git a/notes/stopped-product-note.tex b/notes/stopped-product-note.tex new file mode 100644 index 0000000000..a8b23254c8 --- /dev/null +++ b/notes/stopped-product-note.tex @@ -0,0 +1,76 @@ +\documentclass[11pt]{article} +\usepackage[margin=1in]{geometry} +\usepackage[T1]{fontenc} +\usepackage{lmodern,amsmath,amssymb,amsthm,mathtools,booktabs,longtable,microtype} +\usepackage[hidelinks]{hyperref} +\setlength{\emergencystretch}{3em} +\allowdisplaybreaks +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\title{A conditional integer-multiplication saving of $194869/2500000000$\\ +\large Stopped product-ring interchange and rational retained centers} +\author{icekylinx\\\small With substantial OpenAI GPT-6 Astra and Codex assistance} +\date{9 October 2026} +\begin{document}\maketitle +\begin{abstract} +A common rational basis and opposite bank orders reduce each projector +residual to one contiguous reversed child over a fixed atom product ring. +Ordered atom adapters are paid explicitly; stopping at a growing atom +width and using the retained ordinary primitive supplies an ordinary +bit interface. An independent 24-dimensional complex producer uses +rational retained-center decoding on one exact common grid. The selected +interfaces and retained semantic/bulk assembly give the conditional bound +$T(n)=O(n(\log n)^{1-\kappa})$, where +$\kappa=194869/2500000000=0.0000779476$. +\end{abstract} + +\section{Scope and retained dependencies} +The model is the retained fixed finite-alphabet multitape Turing machine. +Finite graphs, rational bases, address primes, coefficient tables and tape +counts are fixed independently of input length. Bit payload arithmetic is +binary, while its address geometry is rational and then evaluated on atom +rings. Complex payloads use exact Gaussian rational arithmetic with the +specified common grid; their address labels remain binary. + +The proof predecessor is icekylinx's merged copied-center PR36, +\texttt{11817ccacb564bb7f98789c20dc11d3fece207e3}, whose proof is +\path{notes/copied-centers-note.tex}. Its ordinary interchange saving +$a_{\rm old}=384599/10^{10}$, copied-center schedule in both orientations, +two-stage data and auxiliary endpoint identities, general binary residual +normal form, and fixed-tape complete-row contracts are retained. +The present opposite-bank factorization, stopped product-ring interface, +rational-center specialization and exact composition are by icekylinx, +with substantial OpenAI GPT-6 Astra assistance and Codex integration. +Earlier own contributions include PR10 batching, PR18 partial swaps, +PR24 endpoint gauges and PR32 structured blocks and semantic composition. + +Aurel Prosz (Paureel) supplies the two-stage topology and paid endpoint +copy, with Swapnil Jain's linked development credited by Zhihao Chen's +PR29. The retained leaf proof also credits Rohan Arun's PR31 corner method +and Dominik Scholz's PR33 parameterization. Zhihao Chen's PR21/23 and the +RaD/hipotures PR20 source provide the phase endpoints, semantic guard, +arbitrary-source routing, phase-cell inverse and bulk locality/tape +interfaces. Their pinned proof map remains in +\path{notes/structured-bulk-assembly.tex}; adopted sources and notices are +in \path{references/semantic-bulk/} and \path{references/copied-centers/}. +The underlying Douglas Colkitt framework, OpenAI manuscript, eumemic, +Bortlesboat, dleen, earlier finite contributions and Harvey--van der Hoeven +analytic work retain their credits and recorded AI assistance in +\texttt{NOTICE} and \texttt{SOURCES.json}. + +The conclusion remains conditional on those analytic, ordered-affine, +exact-recovery and machine interfaces and their eventual setup and +prime-existence thresholds. The new algebraic and tape constructions +below establish the changed finite interfaces; exact arithmetic certifies +their selected counts and margins. + +\section{One reversed child for each projector} +\input{notes/stopped-product-factorization.tex} +\section{Stopped atom recursion and ordinary interchange} +\input{notes/stopped-product-interface.tex} +\section{Rational centers and exact semantic precision} +\input{notes/stopped-product-complex.tex} +\section{Product rows and exact assembly} +\input{notes/stopped-product-assembly.tex} +\end{document} diff --git a/notes/three-stage-cover-assembly.tex b/notes/three-stage-cover-assembly.tex new file mode 100644 index 0000000000..983ffb6ac8 --- /dev/null +++ b/notes/three-stage-cover-assembly.tex @@ -0,0 +1,117 @@ +\subsection{Local inventory and strict supplier moments} +The selected complex local word has +\[ + h=24,\quad v=2024,\quad c=91770,\quad q_{\rm roots}=8120, + \quad |\mathcal M|=71185,\quad R=28705,\quad\ell=552. +\] +Its complete scalar word, full backward subspace intersections and 4,599 +partial source gauges are regenerated from the pinned compressed DAG by +\path{scripts/three_stage_cover_producer.py}. The source and target chains, +remaining first transitions, inverse cleanup and copied centers remain in +\path{certificates/three-stage-cover-complex-input.json}. +At each cover vertex its child inventory is exactly the three-stage table +proved above. Thus $m_c=70$, $w_c=90163$, $s_c=6309018$ and +$w_cm_c-s_c=2392$; the largest child is 68. + +For $a_c=786999/2500000000$, the rational enclosure proves +\[ + \frac1{w_c}\sum_r n_r(r/70)^{1-a_c} + <1-2.3047018455\times10^{-10}. +\] +The bit local ledger has $h=23,v=1771,R=28866,\ell=506$. +Three copies of its physical local increments and exteriors give +$m_b=67,w_b=90140,s_b=6037356$, with deficit 2024. +The contaminated moment includes the entire $32\cdot67^2$-child fallback +on a fraction $10^{-16}$ of every ideal edge. At $a_0=747/2000000$ its +strict gap exceeds $6.3397366396\times10^{-7}$, and its rank moment also +contracts. The stopped ordinary saving is +\[ + a_b=(1-10^{-3})a_0+10^{-3}\frac{384599}{10^{10}} + =\frac{3731649599}{10^{13}}. +\] +These are exact rational certificates; the lower tolerances displayed +here do not substitute for their saved fractions. + +\subsection{Full finite router and completed-child guard} +Let $V_0=|O(70,2)|$, $N=V_0v$, $W=V_0(2v+3R)$ and $s=V_0s_c$. +All these finite counts appear in the certificate. Define +\[ + L=4(c+v)+10v+4hv+4h^2+8h+8+2h + +8Rv(M+16), +\] +where $M$ is the number of actual local mixer operations. +The source-witness bound for the positive DAG gives dirty-readout +coefficients of magnitude at most $h+4$, with denominators dividing +$2(h-3)$. The last term conservatively pays their binary-addition +implementation. Include the finite group and fixed Clifford routers by +\[ + G=N+3V_0L+64(m_c+1)^3W^2. +\] +A fixed role permutation has at most $W^2$ elementary exchanges; the +polynomial factor also covers the binary matrix and quadratic-phase +adapters. These finite costs retain their ordinary movement charge. +Set +\[ + E=64(W+m_c+G+1)^3,\quad B=s+E,\quad C_0=32m_cB^2. +\] +The exact calculation checks +\[ + 2GW^2+8s+4W+4+32m_c0. +\] +Thus the fixed external reserve $p^{160000}$ meets the inherited product +stock contract. The width-dependent q-adic selectors are supplied by +the separate internal borrowing proof, rather than multiplied into this +external polynomial stock. One preceding prefix and one final padding +suffice, and sequential completed calls reuse restored rows. + +Put $b=a_c$, $\eta=10^{-8}$, $\beta=10^{-6}$ and +\[ + a=(1-\beta)b-10^{-10}2^{80}$ avoiding all fixed rational denominators and rank witnesses. +At atom width $w$ use the finite cover $\operatorname{GL}_m(R_w)$ and its +right-multiplication edges $g\mapsto gR_s$. Every vertex has the unique chart +\[ + g=g_0+qG_1,\quad g_0\in\operatorname{GL}_m(\mathbb F_q), + \quad G_1\in(\mathbb Z/q^{w-1}\mathbb Z)^{m^2}, +\] +using fixed representatives for $g_0$. Right multiplication permutes $g_0$ +and acts affinely on $G_1$, with a carry determined by $g_0$. Its fixed +dimensional rational linear part factors into fixed scalings, ordered +shifts and a constant number of old width-$O(w)$ interchanges. Thus each +fixed port routing costs $O(Vw^{1-a_{\rm old}})$. + +Let $W_0=|\operatorname{GL}_m(\mathbb F_q)|(2v+3R_{\rm loc})$. +The full role count is $W(w)=W_0q^{m^2(w-1)}$. Split preceding rows into +low role type, high matrix and unused stock. Batch \emph{all} high matrices +for each low type in one stream. A frame edge is one child on this whole +batch, of volume $V/W_0$, and routing is executed once per fixed operation +type. Consequently neither the number of tapes nor the number of child +programs grows with $w$. High cardinality cancels in the normalized moment, +but its $m^2(w-1)$ digits are charged in the row reserve. + +For each fixed projector require the leading minors of sizes $1$ through +$r$ to be units. Each cleared minor polynomial in $g_0$ is nonzero: +similarity to $\operatorname{diag}(I_r,0)$ supplies a witness after the +fixed rank is preserved modulo $q$. Their total degree is at most $m^3$. +The polynomial zero bound and conditioning on invertibility (a factor at +most two) give bad fraction at most $2m^3/q<10^{-16}$ for each edge type. +No simultaneous union bound over edge types is needed, since their moments +are averaged separately. Good classes use the whole-rank child; bad +classes use the full $32m^2$-child fallback just proved. + +Let $H_r$ be the three-stage histogram described above and +$W=2v+3R_{\rm loc}=90140$. The retained ledger gives +\[ + \sum_r rH_r=6037356,\qquad Wm-\sum_r rH_r=2024=2v-3\ell, + \qquad \max\{r:H_r>0\}=66. +\] +For $a=747/2000000$, $\tau=1-a$, and $b=10^{-16}$ the conservative moment is +\[ + \mu_\tau\le + \sum_r\frac{H_r}{W}(r/m)^\tau+ + b\frac{32m^2\sum_rH_r}{W}(1/m)^\tau<1. +\] +The entire fallback is added; no displaced good contribution is subtracted. +The exact script \path{scripts/three_stage_cover_network.py} and certificate +\path{certificates/three-stage-cover-network.json}, from the retained axis +\path{certificates/three-stage-cover-bit-input.json}, give +$1-\mu_\tau>6.3397366396\cdot10^{-7}$. +The certificate uses the 32-term positive atanh logarithm series with its +geometric tail, $\exp(u)\le1+u+u^2/(2(1-u/3))$, and upward rounding to +the grid $2^{-120}$. All inputs have $0\le u<3$; these are rational upper +bounds, not floating-point root estimates. Since every child ratio is at +most one and $\tau<1$, this same strict bound implies $\mu_1<1$. + +With $T(n,w)$ denoting time divided by logical volume, the fixed low-type +profile $\{c_r\}$, including fallback classes, satisfies +\[ + T(n,w)\le\sum_rc_rT(r\lfloor n/m\rfloor,w)+Cn+Cw^{1-a_{\rm old}} + \quad(n\ge m),\qquad + T(n,w)\le Cw^{1-a_{\rm old}}\quad(n2\,68^d$ is 24, so the complex row coefficient is +$24\cdot2431=58344$. These full finite role counts are retained in the +scalar guard and row stock. Group elements and involutions are specified +by their binary matrices; the definition needs no numerical enumeration. +All fixed address adapters, endpoint Paulis and invocation permutations +are implemented inside the finite scalar/router guard of the assembly. diff --git a/notes/three-stage-cover-note.tex b/notes/three-stage-cover-note.tex new file mode 100644 index 0000000000..f5ad09a450 --- /dev/null +++ b/notes/three-stage-cover-note.tex @@ -0,0 +1,63 @@ +\documentclass[11pt]{article} +\usepackage[margin=1in]{geometry} +\usepackage[T1]{fontenc} +\usepackage{lmodern,amsmath,amssymb,amsthm,mathtools,microtype} +\usepackage[hidelinks]{hyperref} +\setlength{\emergencystretch}{3em} +\allowdisplaybreaks +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\title{Three-stage covers for a conditional integer-multiplication saving\\ +\large $\kappa=3146011/10000000000$} +\author{icekylinx\\\small Construction with OpenAI GPT-6 Astra assistance; +integration with Codex assistance} +\date{9 October 2026} +\begin{document}\maketitle +\begin{abstract} +Three signed shears on a regular Cayley cover align all interstage data +frames. Arbitrary-subspace Clifford frames support the selected complex +word; weighted reversed interchanges on a batched finite local-ring cover +support the bit word. Exact rational moments and the retained semantic/bulk +assembly give the conditional saving $\kappa=3.146011\times10^{-4}$. +\end{abstract} + +\section{Scope and source continuity} +This extends icekylinx's +\href{https://github.com/CrocSwap/integer-mult-bounds/pull/115}{PR115}, +\texttt{cf60442983c153a3a0f5272ea4713f99eba13c86}, retaining +\path{notes/stopped-product-note.tex}, \path{notes/partial-gauge-note.tex} +and the semantic/analytic proof map in +\path{notes/structured-bulk-assembly.tex}. +The analytic reduction, exact recovery, ordered-affine streaming, +prime-packing, ordinary leaves and eventual fixed-tape setup remain +inherited hypotheses. The new group geometry, weighted compiler and +row-borrowing arguments are supplied below, rather than counted as free +operations. Finite arithmetic does not formalize the all-size theorem. + +The complex local DAG is eumemic's +\href{https://github.com/CrocSwap/integer-mult-bounds/pull/117}{PR117}, +\texttt{cbb05ce504d571546d9b7794c186a613c659c3bf}, with its original +Apache-2.0 notice and Claude assistance disclosure preserved. +The bit physical word is Zhihao Chen's +\href{https://github.com/CrocSwap/integer-mult-bounds/pull/97}{PR97}, +\texttt{e15350b66e4bf796153a68c240a6cc1b788abe3e}, based on Swapnil Jain's +\texttt{741e7aa078392553815df7926ee17ac5e25a8c38} witness. +The generalized Clifford theorem, cover construction, weighted interface +and composition are contributed by icekylinx with OpenAI GPT-6 Astra assistance; +this repository integration uses Codex assistance. +Prior Aurel Prosz/Paureel, Zhihao Chen, RaD/hipotures, Rohan Arun, +Dominik Scholz and framework credits remain recorded in +\texttt{NOTICE} and \texttt{SOURCES.json}. + +\section{Frames for every binary subspace} +\input{notes/general-clifford-frames.tex} +\section{Three-stage complex cover} +\input{notes/three-stage-cover-complex.tex} +\section{Weighted local-ring bit interface} +\input{notes/three-stage-cover-bit.tex} +\section{Uniform stock and ordinary output} +\input{notes/three-stage-cover-rows.tex} +\section{Finite precision and strict assembly} +\input{notes/three-stage-cover-assembly.tex} +\end{document} diff --git a/notes/three-stage-cover-rows.tex b/notes/three-stage-cover-rows.tex new file mode 100644 index 0000000000..590da4b892 --- /dev/null +++ b/notes/three-stage-cover-rows.tex @@ -0,0 +1,95 @@ +\subsection{Internal row reserve and ordinary conversion} +The finite cover requires a row divisor $W(w)$ at each coarse level. +All children have ratio at most $66/67$ and depth +$D(n)\le47\lceil\log_2\max(2,n)\rceil+O(1)$. +A sufficient common divisor is $W(w)^{D(n)}$ times the inherited old-leaf +reserve. The old supplier has its retained complete-row contract and +logarithmic reserve. Therefore a sufficient integer $S(e,w)$ satisfies +\[ + \log_qS(e,w)=O(w\log e). +\] +Unused stock is divided only into fresh descendant role fields; exposed +parent low/high fields are spectator prefixes. Their weights cannot depend +on those fresh roles. Every completed call restores its row coordinate, +so successive residuals and the outer wrapper reuse one reserve instead +of multiplying reserves for sequential calls. + +The unweighted reversed endpoint $F_n(H,D)=(C_nD,C_nH)$ converts to an +ordinary interchange by the chronological sequence +\[ + D\leftarrow D-H;\quad F_n;\quad D\leftarrow D+H;\quad + F_n;\quad D\leftarrow H-D. +\] +Indeed the successive values are $(H,D-H)$, $(C_n(D-H),C_nH)$, +$(C_n(D-H),C_nD)$, $(D,D-H)$, and $(D,H)$. +The three additive maps cost $O(Vn)$ in the atom model. The two calls +reuse restored rows; their constant multiplicity is outside the recursion +and does not multiply its characteristic moment. + +\subsection{Borrowing and restoring the stock} +Set $\beta=1/1000$, $w=\lceil e^\beta\rceil$, and +$k=\lceil\log_qS(e,w)\rceil=O(w\log e)$, using the depth bound for the +original width $e$. For sufficiently large $e$, $ka_{\rm bit}$ and $\beta<1-a_{\rm bit}$ make both +the atom-adapter and borrowing terms lower order. The remainder cost +$O(Vw^{1-a_{\rm old}})$ is also covered. Thus the ordinary radix-$q$ +interchange has cost $O(Ve^{1-a_{\rm bit}})$, without an externally supplied +superpolynomial stock of rows. + +For an original binary chunk range $[2^e]$, let +$t=\lceil e/\log_2q\rceil$ and embed it numerically into $[q^t]$. +Pad both exchanged ranges while retaining every spectator. The total +logical-volume increase is below $q^2$. Execute the completed ordinary +radix-$q$ interchange, then remove the padding. Its endpoint preserves the +rectangular genuine set $[2^e]\times[2^e]$; intermediate maps may mix genuine +and padded addresses, which does not affect this endpoint justification. +As $q$ is fixed and $t=\Theta(e)$, the binary interchange has the same +saving exponent. All schedules use a fixed number of finite-alphabet +tapes; low residue tables and port types are fixed independently of $w$. + +The conclusion retains the original local-producer, ordered-affine, +ordinary-leaf and multiplication-framework hypotheses. It adds no free +unit-scaling, dense chart transformation, or external row-stock primitive. diff --git a/references/partial-gauge/pr62/LICENSE b/references/partial-gauge/pr62/LICENSE new file mode 100644 index 0000000000..261eeb9e9f --- /dev/null +++ b/references/partial-gauge/pr62/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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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 new research note, scripts, certificates, patches, and documentation +were prepared by Douglas Colkitt with assistance from OpenAI Codex. +This project is not an official OpenAI release or an OpenAI endorsement. + +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. + +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. + +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. + +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. + +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. diff --git a/references/partial-gauge/pr62/PROOF.md b/references/partial-gauge/pr62/PROOF.md new file mode 100644 index 0000000000..26e1166562 --- /dev/null +++ b/references/partial-gauge/pr62/PROOF.md @@ -0,0 +1,221 @@ +# Interval strips and core-aware pair assembly: conditional saving 4.986133 × 10^-5 + +Under OpenAI's base theorem and the retained analytic, fixed finite-alphabet +and fixed multitape interfaces, the construction below gives + +\[ +T(n)=O\bigl(n(\log n)^{1-\kappa}\bigr),\qquad +\boxed{\kappa=\frac{4986133}{10^{11}}=4.986133\times10^{-5}>2^{-15}.} +\] + +The exact bit saving is 4986382/10^11. The next grid value of the bit saving +fails the exact moment test, and the next κ grid value fails the assembly. + +This is **6.782% above PR #57** (4669442391/10^14), **10.092% above PR #54** +(141532521/3125000000000) and **10.849% above PR #53** (4498144/10^11). It +does not use PR #57's compiler or PR #54's clones. These compare asymptotic +exponent savings, not running times. They are conditional research claims +without independent mathematical review or formal verification. + +Section "Stacking with PR #57's frame compiler" below gives a second, larger +witness, κ = 5101691/10^11, that also uses PR #57's compiler. + +## What changes + +Only the scalar producer DAG changes, in `pair_graph.py`, together with its +pinned carrier matching, `links-{23,25}.uses`. Everything else is PR #53's +`research/skip-strips` pipeline, which is PR #48's `research/copied-fixed` +pipeline, with constants recomputed for the new DAG. + +A role is a physical scratch wire. With c additions, q designated outputs and +μ retained carrier links, the role count is R = c + q − μ. A link lets a +donor addition x = u + w continue its carrier into a later consumer t of u, +when the original envelope of x lies inside the envelope of t: the common core +of t lies inside the core of x, and the cover of x lies inside the cover of t. +So R counts the additions that cannot continue a carrier, plus q. + +### 1. Interval strips + +The paired-exclusion recursion (icekylinx PR #18/#36) needs, for each vertex, +every leave-one-out sum of its k strip values v_1, …, v_k. PR #53 used the +skip-prefix layout s_j = A_(j−2) + (v_(j−1) + B_(j+1)). We build every cyclic +interval of the values instead, each from an interval one item shorter: + + I(a, m) = I(a, m−1) + v_(a+m−1), 2 ≤ m ≤ k−1, + +and output s_j = I(j+1, k−1), with the total I(0, k−1) + v_(k−1). The interval +I(a, m) with m < k−1 has the later consumer I(a−1, m+1) of its last item +v_(a+m−1), and its cover lies inside that consumer's cover, so it can continue +a carrier. Only the k outputs and the total stay unmatched. + +This costs k(k−2)+1 additions instead of about 4k. The role count still falls, +because R counts unmatched additions. In a model of one strip, an exact integer +program over every leave-one-out circuit shows that k−1 unmatched additions is +the minimum for k ≤ 8. Skip-prefix leaves 2k−3 unmatched; intervals leave k−1 +or k. Intervals are used for every strip with k ≥ 3 nonzero values; smaller +strips keep PR #53's layout. Value orders are PR #53's: reversed at the top +level, sorted by (support size, ±support) below it. + +### 2. Core-aware pair assembly + +For groups i < j with members (a, a2) and (b, b2), let F = far[i,j] and let +Sa, Sa2, Sb, Sb2 be the strip outputs that omit the other group. Each of the +four outputs is a sum of four pieces, for example +out[a,b] = F + Sa + Sb + e(a2,b2). PR #53 computes left(a,j) = F + Sa, +then out[a,b] = left(a,j) + (Sb + e(a2,b2)). None of those ten additions can +continue a carrier. We compute + + F+Sa, F+Sa2, F+Sb, F+Sb2, + Y1 = (F+Sb) + Sa, Y2 = (F+Sa) + Sb2, + Y3 = (F+Sa2) + e(a,b2), Y4 = (F+Sb2) + e(a,b), + out[a,b] = e(a2,b2) + Y1, out[a,b2] = e(a2,b) + Y2, + out[a2,b] = Y3 + Sb, out[a2,b2] = Y4 + Sa2. + +Each F+S node then has a later consumer of its strip operand whose envelope +contains its own: the same common core and a larger cover. An exact integer +program over every assembly of these nine pieces, with their actual cores and +covers, shows that 8 unmatched additions per group pair is optimal. The cores +matter: a node such as Sa + e(a2,b2) has the two-point core {c, a2}, so a +one-point-core donor cannot continue into it. Each output needs its own +addition and at least one more addition that serves only that output, and +neither can continue a carrier, so 8 is also a lower bound for this +decomposition. + +### Counts + +| h | v | additions c | outputs q | matched μ | roles R | loss | +|---|---|---|---|---|---|---| +| 23 | 1771 | 62294 | 5336 | 38911 | 28719 | 506 | +| 25 | 2300 | 86595 | 6925 | 55844 | 37676 | 600 | + +PR #53 had R23 = 32946 and R25 = 43409. The physical constants are +m = 575, N = 4,073,300, W = 140,924,496 (PR #53: 160,799,739) and +L = 2,226,400. The complete rank is Wm − N + L = 81,029,738,300. The deficit +N − L is unchanged. + +The weighted matching is pinned in `links-{h}.uses`. `optimize_matching.py` +regenerates it from Rohan Arun's PR #44 optimizer with only the graph import +changed. Its output is not trusted: `profiles.cpp` replays every pinned link, +and `producer.py` recounts every link independently. + +## Why the inherited proofs apply + +The new DAG enters the inherited proofs only through properties and +quantities that `producer.py` recomputes, as for PR #41, #43, #47, #48 and #53: + +- **Scalar outputs.** Every addition joins disjoint supports, every node has a + common point, and every output is the exact leave-two-out sum or retained + total, checked on dense global supports. +- **Carrier links.** Every pinned link lies in the admissible adjacency + (causal order and envelope inclusion), with distinct donors and uses. +- **Physical timeline.** The full physical timeline and every dirty basis + vector are checked in both orientations at h = 23 and 25. +- **Fixed-basis profiles.** These are recomputed from the new DAGs and pinned + matchings: 37,728 and 51,927 distinct transition matrices; 6,026 and 7,981 + of them CRT-checked; zero modular disagreements. Exactly h full center + cleanups appear, so the copied-center replacement applies unchanged. +- **Data corners.** These do not depend on the producer. All 4,073,300 pairs + are replayed, and the ten primary-prime failures are recovered by exact + rational elimination. + +The integer programs explain why the links exist; correctness does not depend +on them. + +## Exact arithmetic + +`verify.py` uses PR #43/#48's exact logarithm and exponential enclosures: + +- The moment at a_b = 4986382/10^11 is below 1; at 4986383/10^11 its exact + lower bound exceeds 1. +- Each of the following is excluded at the new bit saving by an exact lower + bound: the complete child lists of PR #57, #54, #53, #48, #47, #46, #44, + #43, #42, #41 and #40, and the PR #36 baseline. +- The inherited balanced assembly passes all 47 strict conditions and seven + margins at κ = 4986133/10^11 and fails at the next grid value. Its + parameters are β = 1/20, h = 10^-12, a_c = 717/10^7, C1 = 1 and row stock + p^2000. + +## Stacking with PR #57's frame compiler + +PR #57 (eumemic, with OpenAI Codex assistance) compiles each set of nodes that +share one original envelope as a single invertible binary map, matches +retained signals to later uses, and reclaims retired roles with paid XOR +clearing. Its compiler, `scripts/experiments/binary_frame_compiler.py`, is +included unchanged through PR #57's own commit on this branch. +`frame/frame_compile.py` feeds it the hash-pinned `pair_graph.py` instead of +PR #53's graph, exactly as PR #57's driver feeds it PR #53's graph. + +The compiled words give the following role counts: + +| h | roles R, this graph | roles R, compiled | shared-frame regions | retained signals | reclaimed roles | +|---|---|---|---|---|---| +| 23 | 28719 | 27918 | 52473 | 37025 | 801 | +| 25 | 37676 | 36586 | 73695 | 53194 | 1090 | + +With W = 137,151,806 the stacked witness gives + +\[ +\kappa=\frac{5101691}{10^{11}}=5.101691\times10^{-5}, +\] + +with exact bit saving 5101952/10^11. This is **9.257% above PR #57** and +2.318% above the plain witness above. PR #57's own result on PR #53's graph +is 4669442391/10^14. + +`frame/frame_verify.py` checks, as PR #57's verifier does: + +- both serialized XOR words, replayed independently on every input and dirty + basis vector in both orientations; +- every frame transition, reconstructed from the actual XOR word; +- all fixed-basis profiles, with bounded-minor and CRT checks; +- the exact moment, the rejection of the next bit-saving grid value, the + exclusion of PR #53's and PR #57's networks at the new saving, all 47 + strict assembly inequalities, seven margins, and the rejection of the next + κ grid value. + +This stacked witness also depends on PR #57's compiler argument, which has +the same review status as PR #57. + +## Reproduce + +```sh +python3 research/pair-assembly/producer.py --work-dir /tmp/pair-assembly +python3 research/pair-assembly/verify.py +python3 research/pair-assembly/frame/frame_verify.py +python3 research/pair-assembly/optimize_matching.py --work-dir /tmp/pair-opt # optional, needs scipy +python3 research/pair-assembly/frame/frame_compile.py # optional, rewrites the words +``` + +`producer.py --record` rewrites the JSON records. Without `--record` it checks +them against fresh regeneration. `make pair-assembly-verify` runs the first +three commands and the tests. + +## Scope and attribution + +This is a conditional finite certificate. It covers the same written proof +obligations as PR #43–#53: the base theorem, the residual compiler, fixed-tape +recursion, the physical meaning of carrier continuations and copied centers, +the analytic/semantic/routing/recovery interfaces and eventual thresholds. +None of these are re-proved here. There is no formal verification or claim of +global optimality. The stacked witness additionally depends on PR #57's +compiler mechanism and its written argument. + +The interval strips and core-aware pair assembly are new here, by Avi +Eisenberg with assistance from Claude (Anthropic). Everything else is +inherited, with all original notices: + +- Avi Eisenberg PR #53: skip-prefix strips and this pipeline's adaptation. +- eumemic PR #57: the joint frame compiler, used unchanged for the stacked + witness. +- Chafik Boukhalfa PR #43/#46/#48: pipeline, recovery, the sorting idea and + verification code. +- Rohan Arun PR #44: weighted matching; also PR #37/#39/#40/#42/#47. +- RaD / hipotures PR #41: alternating order and changed DAGs. +- Dominik Scholz PR #35/#38. +- James Chang PR #34. +- icekylinx PR #18/#24/#32/#36. +- Zhihao Chen PR #21/#23/#29. +- Paureel / Aurel Prosz, Swapnil Jain, Douglas Colkitt, OpenAI, and + Harvey–van der Hoeven. + +Apache-2.0. diff --git a/references/partial-gauge/pr62/SOURCE.json b/references/partial-gauge/pr62/SOURCE.json new file mode 100644 index 0000000000..c4a7d407e0 --- /dev/null +++ b/references/partial-gauge/pr62/SOURCE.json @@ -0,0 +1,11 @@ +{ + "repository": "ikeboy/integer-mult-bounds", + "commit": "ad0f25ff7b23cff7f08ad237c2254e6ecf74257e", + "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/62", + "scope": "Primitive proof and unchanged attribution; implementation adapted in scripts/partial_gauge/interval.py.", + "files": { + "LICENSE": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4", + "NOTICE": "95d7dabb54a327216e007f1e0cf37dd2f138ab31527f2cc5ee0de55e87f274ca", + "PROOF.md": "30685624bf922bb9d9f7ed4b0d367a660fe373cc9ee28fec33585bd8c7023e75" + } +} diff --git a/references/partial-gauge/pr97/LICENSE b/references/partial-gauge/pr97/LICENSE new file mode 100644 index 0000000000..261eeb9e9f --- /dev/null +++ b/references/partial-gauge/pr97/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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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. diff --git a/references/partial-gauge/pr97/PROOF.md b/references/partial-gauge/pr97/PROOF.md new file mode 100644 index 0000000000..58824db895 --- /dev/null +++ b/references/partial-gauge/pr97/PROOF.md @@ -0,0 +1,246 @@ +# Deferred bit endpoints and a signed two-stage complex interface + +Under the analytic and fixed finite-alphabet multitape interfaces already +retained by main, the networks in this package give the conditional bound + +\[ +T(n)=O\bigl(n(\log n)^{1-\kappa}\bigr),\qquad +\kappa=63965813/10^{12}>2^{-14}. +\] + +The finite checks do not formalize these inherited interfaces or establish an +unconditional integer-multiplication theorem. Repository integration validation +is recorded separately; mathematical certificates and validation receipts have +different purposes. This is not a current-record or global-priority claim. + +The scalar/frame inputs are Swapnil Jain's round7 bit network and round6 complex +network, pinned at `741e7aa078392553815df7926ee17ac5e25a8c38`. The original source, +notes, licenses and assistance disclosures are preserved in `swapnil-round7/`. +That source reports a slightly larger final number under its own analytic +stack. We instead use main's retained semantic guard and balanced assembly. +Our additions are explicit reflected scheduling, the commuting fan-order +clarification, gauged exits, a signed complex endpoint correction and a bridge +to the existing main interfaces. No separately researched local role saving is +added to these networks without recompilation. + +## Bit word, geometry and common basis + +The pinned producer at h=23 has v=1771 and R=28866. Its frozen supports, links, +deferred readouts, V leaves and rational frames are checked by the original +`check_word.py`, `check_frames.py` and `check_lifted.py`. In particular, the +deferred scratch slots are untouched before their deferred readout; their +entrance frames lie in every required target hyperplane. These are conditions +on the complete physical word, not just its ordinary-input support. + +`round7_literal_frame_ledger.py` expands that word into 1080616 events and +checks the forward shear on all 32408 ordinary/dirty F2 basis vectors. Gates +with several fan recipients are expanded in stable order of their starting +frame dimension. The recipients are distinct targets of the same unchanged +source, so these XORs commute. The separately checked chains prove nesting; +rank order alone is insufficient. This changes the internal order of160 fan +groups and avoids otherwise uncharged frame retreats, preserving their scalar +map and all costs. The imported source is left intact. + +Reverse time, complement the factor frames and exchange the data banks for +stage two. All ordinary gates retain their XOR direction after the exchange. +The new ledger checks continuity and the complete opposite shear; a logical +transpose alone would not prove this statement. A copied-center scatter is a +macro: copy the unchanged center, transport the copy, scatter and discard the +copy. Its reflection creates a new copy at the complemented source and moves +it to the complemented target. No unknown dirty register is erased. + +For an idempotent A write D_A=[[I-A,A],[A,I-A]]. Commuting idempotents satisfy +D_A D_B=D_(A+B-2AB). Let P,Q be the two rank-one triple projectors, sigma the +local auxiliary entrance projector of rank f, m=h², and r=h-f. Stage one's +incoming gauge is A=sigma tensor Q, its last live frame is T=I tensor Q, +and its terminal gauge is I-A. Its exterior is + +\[ +E_1=I-T+A=I-(I-\sigma)\otimes Q,\qquad\operatorname{rank}E_1=m-r. +\] + +Direct multiplication gives D_(I-A)D_T=D_E1 and D_(I-A)D_A=D_I. For stage two +put B=(I-P) tensor I and lift a factor frame K to B+P tensor K. Its entrance +is B, last live frame is I-P tensor sigma, terminal gauge is I-B, and + +\[ +E_2=B+P\otimes\sigma,\qquad\operatorname{rank}E_2=m-r. +\] + +Again D_(I-B)D_(I-P tensor sigma)=D_E2 and D_(I-B)D_B=D_I. These are genuine +commuting-idempotent exits, not an assumption that all endpoint movements are +nested. Scalar dirty restoration and complementary gauges give the full +address swap. Both exteriors are charged as block m-2r and the inner(r,h) +profile of their complementary small residual. + +With U=P tensor Q, stage-one data ends at (I tensor Q,(I-P) tensor Q), while +stage two begins at (B+U,B). Each connector is (I-P) tensor (I-Q), of rank +(h-1)², and both are charged. The final F2 scalar shear pair gives (y,x+y); +the temporary rank-one correction uses D_U D_I=D_(I-U) to remove the extra +term, retaining one paid rank-one child per pair. + +The common rational tensor basis and inner residual profiles use the preserved +`notes/flag-basis.tex`, `notes/partial-swap-batching.tex` and the round7 lifted +checks. The staircase corner adds runs h-2,h-5,1^6, alongside the outer block +m-4h+2. `check_stair.py` checks its finite witnesses and symbolic cut bounds. +The GL_h×GL_h family acts transitively on line pairs; nonzero polynomial +witnesses, together with identically imposed triangular zero conditions, give +a common rational point for the finite family. This is an existence argument, +not a claim that sampled pairs enumerate the family or that one modular basis +is already a common integer basis. Rational enumeration avoiding the finitely +many nonzero determinant polynomials is an effective fixed finite choice. +Its denominators and required minors exclude finitely many odd primes, as in +the retained transfer. All choices are independent of n. The two orientations +and newly gauged residuals occur among the checked inner families. + +The actual event histogram, including exits, connectors and corrections, +matches the pinned one with m=529, W=108516254 and s=57403754177. An independent +rational moment enclosure proves saving a_b=31987/500000000. + +## Complex scalar word and phases + +For the h24 paired triple-exclusion producer, let M be its invertible role +mixer, V the source injection and A the complete signed readout. The retained +totals are T and E_i=sum of triples avoiding i, for i=s+E and direct leaves obey A(e)<=8e. Prefix kernels and +outer normalization add at most18e, below the C0e allowance. Thus the same +fixed fine-grid proof yields C1=1; it does not concatenate temporary internal +excess of already completed children. E and C0 are large but fixed. + +The actual maximum child and width determine each least halving degree d_j +and bitlength w_j. Reserve the product W_c^D_c W_b^D_b, with +D_j=d_j ceil(log2(e)), rather than the maximum of the two stocks. The existing +physically preceding transformed prefix and one complete-row padding supply +the recomputed stock; nested bit stock is restored before another complex +child. Sequential temporary copies use this existing row field and do not +introduce a separately live recursive row reservation. The certificate gives +the exact coefficient and sufficient stock degree. These operations preserve +the fixed alphabet, fixed tape count and polynomial descriptor bounds of the +inherited interfaces; the number of fixed-role streams is not the number of +physical tapes. + +## Final exact combination + +Use beta=1/10 and eta=10^-8. The stopped complex saving (1-beta)a_c exceeds +a_b. Put q=a_b(1-2eta), c=q(1+eta), epsilon=(1-eta)/(1+c+q), lambda'=1-q, +lambda=((1-a_b)+lambda')/2, g=epsilon*q, +r=(g+1-epsilon)/2 and delta=eta/8. The inherited balanced combination now +has all47 strict constraints and all7 margins above63965813/10^12; the +next10^-12 grid point is rejected. This includes the recomputed scalar +charge, product-row gap, compact reservations, phase-cell separation, prime +packing and exact recovery allowances. A second moment implementation +checks both complete recursive histograms with exact rational logarithm and +exponential enclosures. Passing arithmetic does not by itself establish any +of the physical or analytic lemmas above. + +## Attribution and scope + +Zhihao Chen (jacklightChen): explicit reflected event ledgers, commuting +fan-order clarification, nonzero auxiliary endpoint proof, signed complex +phase controls and corrected rank-one endpoint, and integration with the +retained balanced semantic assembly, with Codex assistance. Subsequent work +using these specific contributions should explicitly acknowledge Zhihao Chen. +Historical GPT-6 Astra assistance attribution on PR7/PR23/PR29 remains; +no claim about the current runtime model is made. + +Swapnil Jain supplies deferred readouts, V leaves, lifted frames, the h24 +producer, common flag/staircase family and frozen witnesses, with the original +Claude assistance disclosure. Avi Eisenberg/ikeboy PR62 supplies interval +strips and core-aware pair assembly; RaD/hipotures PR41 supplies alternating +order/links, and Rohan Arun PR44 the weighted-link precedent. Aurel Prosz/ +Paureel supplies two-stage/complement scheduling; icekylinx PR36 supplies +copied-center and bridge precedents; James Chang PR34 supplies balanced +combination and reversed geometry; Zhihao Chen PR23/PR29 and RaD supply +semantic precision, bulk routing and transfer. All earlier author, license +and AI disclosures in the preserved source and main remain in force. + diff --git a/references/partial-gauge/pr97/README.md b/references/partial-gauge/pr97/README.md new file mode 100644 index 0000000000..ce8b472195 --- /dev/null +++ b/references/partial-gauge/pr97/README.md @@ -0,0 +1,35 @@ +This is a compact, unmodified snapshot of the PR97 physical bit ledger and +the underlying frozen round-seven input. `SOURCE.json` pins the commits and +every imported file's SHA-256. Each imported byte sequence also matches the +round-six handoff's original manifest. Original LICENSE and NOTICE are retained. + +Zhihao Chen / jacklightChen supplies the PR97 literal forward/reflected physical +ledger and proof, with the Codex assistance disclosed in those files. Swapnil +Jain supplies the underlying deferred word, frame witnesses and frame/lifted +checking sources. Earlier authors and assistance disclosures remain credited +in NOTICE, PROOF.md, and the original source headers. + +The snapshot preserves `witness_23.json.gz` and `deferred_23.json.gz` (about +2.3 MB together), not the large expanded event/path dumps. The archived +`bit-ledger-result.json` is a validation receipt, not a new test result. Its +historical elapsed time is preserved verbatim. The original scripts retain +their upstream directory assumptions; they are source/proof witnesses here, +not a relocated full-audit harness. + +`scripts/partial_gauge_bit.py` verifies every byte pin, loads the compressed +inputs explicitly, reconstructs auxiliary and data-chain rank increments, +checks them against the actual physical ledger, reconstructs exit nullities +from each physical slot's deferred frame, and composes the new whole-projector +histogram. It compares every coefficient to the round-six input certificate. +It does not repeat the unchanged complete-basis replay or rational geometry +audits; rank differences alone would not establish those geometric hypotheses. + +The whole-projector reconstruction counts two stages of v physical invocations, +with each local rank increment kept whole, each exterior assigned rank +`m-(h-dim sigma)`, two bank-pair entrance projectors of rank `(h-1)^2`, and +one rank-one copy correction per source pair. This replaces the old corner +splitting while preserving the exact physical total rank mass. + +New reconstruction and integration: copyright 2026 icekylinx, Apache-2.0, +prepared with OpenAI Codex assistance. No independent review or endorsement +by the predecessor authors is implied. diff --git a/references/partial-gauge/pr97/SOURCE.json b/references/partial-gauge/pr97/SOURCE.json new file mode 100644 index 0000000000..970000a79f --- /dev/null +++ b/references/partial-gauge/pr97/SOURCE.json @@ -0,0 +1,31 @@ +{ + "archive": "integer-mult-research/pro/round6-20261009", + "archive_manifest_sha256": "b2edf27c7eea66982a75cb224887da8bce25e18ce9b3857ff45c4666ee4b4ea5", + "archive_source_inputs_sha256": "ed58db13eafeb47401c6c962291e1fe760cbd25a563ca2796f5389b03a9714f3", + "pr97": { + "commit": "e15350b66e4bf796153a68c240a6cc1b788abe3e", + "repository": "jacklightChen/integer-mult-bounds", + "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/97" + }, + "scope": "Unmodified hash-bound physical-ledger receipt, source and compressed witness inputs. Full upstream frame/lifted/scalar audits are inherited, not repeated by the incremental checker.", + "sha256": { + "LICENSE": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4", + "NOTICE": "1c53970fb4face631a1b331ec66d33111eccd280a129f9b64c2da6df96861c3f", + "PROOF.md": "f2c95747d9a5858da43932d857be999e6166d4740f25e2c57f99ce05a8902352", + "bit-ledger-result.json": "c3dd79e5e6d72e49d8d1be2d4d9ad808cd7d24b7a725190d24100cbd3644394a", + "check_frames.py": "1cb2b8cde2a8b3f12008da0d335dcdd38005cc692367423d9589aa1f4d6306e8", + "check_lifted.py": "c3e87ad6b31f377df549421f53d22b99131d2228dcd08a1956913211383cbd6c", + "deferred-readout.tex": "4e848f7f06c975045629da7cca9f79e10f255d08dc17a72629c9bed522565d52", + "deferred.py": "9b2acaeb6f1beb2a53c15e00c027f9ba22397a12071ad0b71131e12012a98036", + "deferred_23.json.gz": "8c38e947ff9e021e308000dd82bb5e2194eb265d8b75194e22ce783d3e75317c", + "linalg.py": "56cc779911a3c5c901bc5194ccaf97f720157a2c4ba602b07ee8642af2676b8b", + "round7_literal_frame_ledger.py": "02617ba608f971e3bdd2b45d4cbfc3d0d629873a0b9f7c22b12f347d099fdb59", + "witness_23.json.gz": "b2486aa2bb222bacea6e52162a3920780e45dd8edc5d7cb03eabea336a1c6588" + }, + "snapshot_prefix": "round6_work/references/round6_pr97/", + "underlying": { + "commit": "741e7aa078392553815df7926ee17ac5e25a8c38", + "repository": "Swapnil-jain/integer-mult-kappa", + "url": "https://github.com/Swapnil-jain/integer-mult-kappa/tree/741e7aa078392553815df7926ee17ac5e25a8c38" + } +} diff --git a/references/partial-gauge/pr97/bit-ledger-result.json b/references/partial-gauge/pr97/bit-ledger-result.json new file mode 100644 index 0000000000..708e83ebd4 --- /dev/null +++ b/references/partial-gauge/pr97/bit-ledger-result.json @@ -0,0 +1,272 @@ +{ + "h": 23, + "roles": 28866, + "formal_basis": 32408, + "events": 1080616, + "frame_keys": 90173, + "rank_histograms": { + "aux": { + "1": 55349, + "5": 6151, + "10": 162, + "4": 11248, + "2": 64873, + "3": 13926, + "6": 1617, + "8": 1243, + "9": 458, + "12": 208, + "16": 208, + "18": 599, + "21": 413, + "7": 883, + "11": 299, + "14": 38, + "15": 649, + "13": 202, + "17": 1309, + "19": 533, + "20": 2310 + }, + "data": { + "4": 2013, + "5": 521, + "3": 1144, + "1": 11386, + "2": 2402, + "6": 475, + "7": 276, + "8": 382, + "9": 130, + "10": 86, + "11": 93, + "12": 97, + "13": 65, + "14": 26, + "15": 43, + "16": 218, + "17": 38, + "18": 8, + "19": 1547, + "21": 1, + "22": 2 + }, + "center": { + "22": 23 + } + }, + "fan_groups_reordered_internally": [ + 45859, + 45862, + 46139, + 46142, + 46409, + 46412, + 46681, + 46684, + 46943, + 46946, + 47207, + 47210, + 47461, + 47464, + 47717, + 47720, + 47963, + 47966, + 48211, + 48214, + 48449, + 48452, + 48689, + 48692, + 48919, + 48922, + 49151, + 49154, + 49373, + 49376, + 49597, + 49600, + 49811, + 49814, + 50027, + 50030, + 50233, + 50236, + 50441, + 50444, + 50639, + 50642, + 50839, + 50842, + 51029, + 51032, + 58136, + 58138, + 58140, + 58142, + 58144, + 58572, + 58574, + 58576, + 58578, + 58580, + 58988, + 58990, + 58992, + 58994, + 58996, + 59420, + 59422, + 59424, + 59426, + 59428, + 59832, + 59834, + 59836, + 59838, + 59840, + 60260, + 60262, + 60264, + 60266, + 60268, + 60668, + 60670, + 60672, + 60674, + 60676, + 61092, + 61094, + 61096, + 61098, + 61100, + 61496, + 61498, + 61500, + 61502, + 61504, + 61916, + 61918, + 61920, + 61922, + 61924, + 62316, + 62318, + 62320, + 62322, + 62324, + 62732, + 62734, + 62736, + 62738, + 62740, + 63128, + 63130, + 63132, + 63134, + 63136, + 63540, + 63542, + 63544, + 63546, + 63548, + 63932, + 63934, + 63936, + 63938, + 63940, + 64340, + 64342, + 64344, + 64346, + 64348, + 64728, + 64730, + 64732, + 64734, + 64736, + 65132, + 65134, + 65136, + 65138, + 65140, + 65516, + 65518, + 65520, + 65522, + 65524, + 65916, + 65918, + 65920, + 65922, + 65924, + 66296, + 66298, + 66300, + 66302, + 66304, + 66692, + 66694, + 66696, + 66698, + 66700, + 67068, + 67070, + 67072, + 67074 + ], + "complete_forward_F2": true, + "complete_reflected_F2": true, + "reflected_frame_continuity": true, + "all_scalar_gates_equal_frame_keys": true, + "exterior_formula": "E1=I-I_tensor_Q+sigma_tensor_Q; E2=(I-P)_tensor_I+P_tensor_sigma; rank=m-h+dim(sigma)", + "histogram_matches_external": true, + "histogram": { + "1": 1690297301, + "3": 1356586, + "19": 1639946, + "17": 8476006, + "15": 7948248, + "13": 2164162, + "11": 5093396, + "7": 3580962, + "21": 6906900, + "9": 1526602, + "5": 354200, + "483": 61280142, + "23": 61280142, + "525": 5653032, + "527": 6233920, + "489": 293986, + "521": 13711082, + "523": 4629394, + "491": 580888, + "485": 545468, + "487": 173558, + "493": 14168, + "499": 1129898, + "495": 322322, + "505": 99176, + "503": 410872, + "501": 127512, + "507": 1186570, + "509": 340032, + "497": 17710, + "513": 106260, + "517": 626934, + "515": 1583274, + "511": 322322, + "519": 2854852, + "439": 6272882, + "18": 6272882 + }, + "recursive_rank": 57403754177, + "elapsed": 9.80610633303877, + "limit_seconds": 180, + "exact_chain_geometry": "Relies on separately replayed pinned check_frames and check_lifted. Rank labels alone do not prove geometry.", + "scope": "Explicit one-axis forward/reflected events, scalar complete basis and exact accounting. Tensor endpoint identities proved separately; full complex/tape/precision and final assembly integration not established.", + "new_multiplication_bound": false +} + diff --git a/references/partial-gauge/pr97/check_frames.py b/references/partial-gauge/pr97/check_frames.py new file mode 100644 index 0000000000..f22b6998f9 --- /dev/null +++ b/references/partial-gauge/pr97/check_frames.py @@ -0,0 +1,262 @@ +"""Round-seven check, part 2 (stdlib only): every frame of the reordered word, exactly over Q. + +The base frames (lifted node frames M_n, late-copy intersections, root frames t_T^perp and U_c) are rebuilt from the +witness with check_lifted.py's exact machinery. The design-B frames (deferred readout frames sigma_u, V-leaf starts +s_i) are the primitive integer bases of the data file. Checked: + X. the frame dimensions recorded in the data file (node frames, late-copy intersections) equal the exact ones; every + slot's node list moves along slot successors of the witness (gate uses and links, for which check_lifted.py proves + nesting), its first node is where its start says, its last node reaches its root frame directly; + Q. sigma_u <= F0(u) (the slot's start frame); sigma_u <= t_T^perp for every target of u (Z supports, a superset of + the F2 supports); on every y_T the deferred levels are nested in time order (readout order); t_S <= s_i <= N_i + (the first frame of the use); every X_S chain , early V frames, deferred V frames, F is nested in time order; + G = I - J/9 nondegenerate on every new frame (Gram determinant nonzero mod a prime, so nonzero over Z); + S. side lemma at check_lifted's integer point (U, V of point(h, 1), the point of check_stair.py too): every distinct + high-rank step (2r > h) of the reordered word that check_lifted.py does not already cover (slot starts, gauged + corners sigma_u -> F, y_T and X_S chain steps, base steps that only the new schedule uses) has the generic + north-east pivots under both conjugations, mod 2^31-1; + F. the child-width histogram rebuilt from the exact dimensions (Z and F2 y supports), rank sum s = W m - N + L, + moment certificate and kappa, with the round-six and the staircase entrance corners; + N. negative controls: reversed readout order, readouts at F0(u) instead of sigma_u, perturbed sigma frames, X_S + chains with the deferred V gates first, side lemma with U = V = I. +Usage: python3 check_frames.py [h]""" +import sys, os, time, random +from fractions import Fraction as Fr +from math import gcd +from collections import Counter, defaultdict +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path[:0] = [HERE, os.path.join(HERE, '..', '..', 'scripts'), os.path.join(HERE, '..', 'two-stage-bit')] +import check_lifted as cl +import deferred as dr +from linalg import null_exact, rank_mod, inv_mod_matrix, rightmost_pivots, predicted, Q31, Echelon + +T0 = time.time() +def log(*a): print('[%5.0fs]' % (time.time() - T0), *a, flush=True) +OK = {} +def report(name, ok): + OK[name] = bool(ok); log('%-100s %s' % (name, 'PASS' if ok else 'FAIL')) + + +def prim(r): + r = [Fr(x) for x in r]; den = 1 + for x in r: den = den * x.denominator // gcd(den, x.denominator) + z = [int(x * den) for x in r]; g = 0 + for x in z: g = gcd(g, x) + return [x // g for x in z] if g else z + + +class ModQ7(cl.ModQ): + """check_lifted's mod-q frames, extended by the design-B frames ('sigma', s) and ('v', s) (integer bases).""" + def __init__(self, fr, ld, extra, control=False): + super().__init__(fr, ld, control); self.extra = extra + + def basis(self, key): + if key[0] in ('sigma', 'v') and key not in self.bcache: + E = Echelon(self.q) + for r in self.extra[key]: E.insert([x % self.q for x in r]) + if len(E.rows) != len(self.extra[key]): self.dimfail += 1 + self.bcache[key] = E.rows + return super().basis(key) + + +def main(h=23): + W, D = dr.load(h); S = dr.Schedule(W, D); trip = S.trip; tv = lambda T: [int(q in T) for q in range(h)] + FULL = [[int(i == j) for j in range(h)] for i in range(h)] + pr = cl.Prog(W); bas, dn = cl.spans(pr); users = cl.users_of(pr, dn) + fr = cl.Frames(pr, dn, bas, users); fr.cobases(); d, _ = fr.dims(); ld, _ = cl.late_frames(fr, pr, users) + log('exact base frames: %d node frames (%d lifted), %d late-copy intersections' % (len(d), len(pr.j), len(ld))) + + # ------------------------------------------------------------------ X. recorded dims, slot paths + okd = all(S.node_dims[n] == d[n] for n in pr.act) and len(S.node_dims) == len(pr.act) + okd &= all(S.late_dims[n, lt] == ld[('c', n, lt)] for (n, lt) in S.late_dims) and len(S.late_dims) == len(ld) + report('X. recorded frame dims == exact dims (%d node frames, %d late-copy intersections)' % (len(d), len(ld)), okd) + def ukey(n, k): + u = users[n][k]; return ('n', u[1]) if u[0] == 'gate' else cl.root_key(u) + badp = 0 + for s in range(S.R): + hl = S.hold[s] + for a, b in zip(hl, hl[1:]): badp += b not in fr.succ[a] + root = [k for k in S.chain_keys(s) if k[0] in ('out', 'ret')] + if root: badp += fr.rid[root[0]] not in fr.droots[hl[-1]] + st = S.start[s] + if st[0] == 'fresh': # a copy starts at the use it serves + nxt = ('n', hl[1]) if len(hl) > 1 else root[0] + badp += ukey(hl[0], st[2]) != nxt + report('X. every slot moves along slot successors of the witness, starts at its use, ends at its root (%d slots)' % S.R, + badp == 0) + + # ------------------------------------------------------------------ exact frames + _eb = {} + def exact(key): + if key not in _eb: + k = key[0] + if k == 'sigma': B = S.sigma[key[1]] + elif k == 'v': B = S.vstart[key[1]] + elif k == '0': B = [] + elif k == 'F': B = FULL + elif k == 'c': + n, lt = key[1], key[2] + B = exact(ukey(n, lt[-1])) + for kk in reversed(lt[:-1]): B = fr.intersect_exact(exact(ukey(n, kk)), B) + B = [prim(r) for r in B]; assert len(B) == ld[key] + else: B = [prim(r) for r in fr.exact_basis(key)] + _eb[key] = B + return _eb[key] + _nz = {} + def nullZ(B): + kb = tuple(map(tuple, B)) + if kb not in _nz: _nz[kb] = [prim(z) for z in null_exact(B, h)] if B else FULL + return _nz[kb] + def inside(A, B): + if not A or len(B) == h: return True + return all(sum(a * z for a, z in zip(ra, rz)) == 0 for ra in A for rz in nullZ(B)) + def nondeg(B, p=Q31): # det(9 B G B^T) != 0 mod p => != 0 over Z + if not B or len(B) == h: return True + sm = [sum(b) for b in B] + Gr = [[9 * sum(x * y for x, y in zip(a, b)) - sa * sb for b, sb in zip(B, sm)] for a, sa in zip(B, sm)] + return rank_mod(Gr, p) == len(B) + gout = {T: [9 * x - 3 for x in tv(T)] for T in trip} + tperp = lambda T: ('out', T[0], T) # t_T^perp is the root frame of every output (c, T) + + # ------------------------------------------------------------------ Q. exact facts + indep = all(rank_mod(B, Q31) == len(B) for B in list(S.sigma.values()) + list(S.vstart.values())) + report('Q. sigma_u and s_i bases independent (rank mod 2^31-1 == row count; %d + %d frames, max |entry| %d)' % ( + len(S.sigma), len(S.vstart), max(abs(x) for B in list(S.sigma.values()) + list(S.vstart.values()) for r in B for x in r)), indep) + sel = S.readout; F0 = {s: S.start_key(s) for s in range(S.R)} + report('Q. sigma_u <= F0(u) exactly (%d deferred slots)' % len(sel), all(inside(S.sigma[s], exact(F0[s])) for s in sel)) + CZ = S.adjoint(); CF = [{t: 1 for t, x in dd.items() if x % 2} for dd in CZ] + capbad = sum(1 for s in sel for t in CZ[s] if any(sum(a * g for a, g in zip(r, gout[trip[t]])) for r in S.sigma[s])) + report('Q. sigma_u <= t_T^perp exactly for every Z-support target (a superset of the F2 targets)', capbad == 0) + def ychains(co, order, frame=lambda s: S.sigma[s], limit=None): + byT = defaultdict(list) + for s in order: + for t in co[s]: byT[t].append(s) + bad = 0; levels = {} + for t, ss in list(byT.items())[:limit]: + seq = [frame(s) for s in ss] + for A, B in zip(seq, seq[1:]): + if A is B: continue + if len(A) > len(B) or not inside(A, B): bad += 1 + lv = [] + for s in ss: + if not lv or S.f[lv[-1]] != S.f[s]: lv.append(s) + levels[t] = lv + return bad, levels + bZ, levZ = ychains(CZ, sel); bF, levF = ychains(CF, sel) + report('Q. deferred levels on every y_T nested in time order, exactly (Z targets %d bad, F2 targets %d bad)' % (bZ, bF), + bZ == 0 and bF == 0) + nd = sum(1 for B in list(S.sigma.values()) + list(S.vstart.values()) if not nondeg(B)) + report('Q. G nondegenerate on every sigma_u and s_i (Gram det != 0 mod 2^31-1 => over Z)', nd == 0) + bad_t = bad_n = 0 + for s, n in S.srcop.items(): + bad_t += not inside([tv(trip[n - 1])], S.vstart[s]) + N1 = S.chain_keys(s)[2] # first frame of the use after its start + bad_n += not inside(S.vstart[s], exact(N1)) + report('Q. V leaves exactly: t_S <= s_i (%d bad), s_i <= N_i, the first frame of the use (%d bad), %d slots' % ( + bad_t, bad_n, len(S.srcop)), bad_t == bad_n == 0) + def xchains(order_fn): + bad = 0; steps = [] + for n, us in S.xs: + seq = [[tv(trip[n - 1])]] + [S.vstart[s] for s in order_fn(us)] + [FULL] + keys = [('n', n)] + [('v', s) for s in order_fn(us)] + [('F',)] + for (A, ka), (B, kb) in zip(zip(seq, keys), zip(seq[1:], keys[1:])): + if len(A) > len(B) or not inside(A, B): bad += 1 + steps.append(keys) + return bad, steps + bx, xkeys = xchains(lambda us: us) + report('Q. X_S chains nested exactly in time order (, early V gates, deferred V gates, F; %d leaves)' % len(S.xs), bx == 0) + + # ------------------------------------------------------------------ S. side lemma on the new high-rank steps + ops0, chains0, _, R0 = cl.compile_(pr, dn, users) + def dimof0(k): + if k == ('0',): return 0 + if k == ('F',): return h + if k[0] in ('out', 'ret'): return h - 1 + if k[0] == 'c': return ld[k] + return d[k[1]] + _, base_steps = cl.chain_steps(chains0, dimof0, h) + base_frames = {k for e in base_steps for k in e} + def dimx(k): return len(exact(k)) if k[0] in ('sigma', 'v') else dimof0(k) + new = {} + def add_chain(keys, kind): + prev = keys[0]; pd = dimx(prev) + for k in keys[1:]: + dk = dimx(k) + if dk > pd: + if (prev, k) not in base_steps and 2 * (dk - pd) > h: new.setdefault((prev, k), (dk - pd, kind)) + prev, pd = k, dk + else: prev = k + for s in range(S.R): + add_chain(S.chain_keys(s), 'slot') + if S.f[s]: add_chain([('sigma', s), ('F',)], 'corner') + for t, lv in levZ.items(): + add_chain([('0',)] + [('sigma', s) for s in lv] + [tperp(trip[t])], 'ydata') + for t, lv in levF.items(): + add_chain([('0',)] + [('sigma', s) for s in lv] + [tperp(trip[t])], 'ydata') + for keys in xkeys: add_chain(keys, 'xdata') + extra = {('sigma', s): B for s, B in S.sigma.items()}; extra.update({('v', s): B for s, B in S.vstart.items()}) + mq = ModQ7(fr, ld, extra) + used = {k for e in new for k in e} - base_frames + ndq = sum(1 for k in used if not mq.nondeg(k)) + report('S. frames used only by the new steps: %d, G-degenerate mod 2^31-1 %d, reduction dimension mismatches %d' % ( + len(used), ndq, mq.dimfail), ndq == 0 and mq.dimfail == 0) + pr_ = predicted(h); q = mq.q + def lemma(items, m): + bad = 0 + for (A, B), (r, _) in items: + QA, QB = m.conjugated(A), m.conjugated(B) + for t in range(2): + a, b = QA[t], QB[t] + M = [[(b[i * h + c] - a[i * h + c]) % q for c in range(h)] for i in range(h)] + if rightmost_pivots(M, q) != pr_[r]: bad += 1 + return bad + items = sorted(new.items(), key=lambda e: (str(e[0][1]), str(e[0][0]))) + badL = lemma(items, mq) + report('S. side lemma (generic NE pivots, both conjugations, mod 2^31-1) on all %d new high-rank steps %s' % ( + len(items), dict(Counter(k for _, k in new.values()))), badL == 0) + + # ------------------------------------------------------------------ F. histogram from exact dims + import moment + from certificate_round3 import evaluate + def dim_exact(k): return dimx(k) + rk = dr.chain_ranks(S, dim_exact) + xd = S.xdata(lambda s: len(exact(('v', s)))) + okx = all(sum(x) == h - 1 for x in xd) + ylZ = S.ylevels(CZ); ylF = {t: sorted({S.f[s] for s in sel if t in CF[s]}) for t in range(S.v)} + res = {} + for corner_name, corner in (('round-six corner', dr.ROUND6_CORNER(h)), ('staircase corner', dr.STAIRCASE_CORNER(h))): + for yname, yl in (('Z', ylZ), ('F2', ylF)): + c = dr.histogram(S, rk, yl, xd, corner) + rs = sum(w * n for w, n in c['hist'].items()) + a, _ = moment.certify(c) + k = evaluate(a, Fr(36926111, 5 * 10**11), Fr(1, 1000), 'crude', m_c=576, s_c=119453132304) + res[corner_name, yname] = (a, k['kappa'], k['ok'], rs == c['s'] and max(c['hist']) < c['m']) + for corner_name in ('round-six corner', 'staircase corner'): + (aZ, kZ, oZ, sZ), (aF, kF, oF, sF) = res[corner_name, 'Z'], res[corner_name, 'F2'] + report('F. %s: rank sum == s (Z and F2 y supports), a_b = %s (%.6e), kappa = %s (%.7e) ok=%s, F2 a_b = %s' % ( + corner_name, aZ, float(aZ), kZ, float(kZ), oZ, aF), sZ and sF and oZ and oF and okx) + report('F. claims: a_b 12599/200000000, kappa 6299103187973/10^17; staircase a_b 31987/500000000, kappa 1599247689723/(2.5*10^16)', + res['round-six corner', 'Z'][:2] == (Fr(12599, 200000000), Fr(6299103187973, 10**17)) and + res['staircase corner', 'Z'][:2] == (Fr(31987, 500000000), Fr(1599247689723, 25 * 10**15))) + + # ------------------------------------------------------------------ N. negative controls + bZr, _ = ychains(CZ, list(reversed(sel))) + report('N. NEG reversed readout order breaks the y_T chains (%d bad)' % bZr, bZr > 0) + bF0, _ = ychains(CF, sorted(sel, key=lambda s: (dimx(F0[s]), s)), frame=lambda s: exact(F0[s]), limit=300) + report('N. NEG readouts at F0(u) instead of sigma_u break the y_T chains (%d breaks on 300 targets)' % bF0, bF0 > 0) + rng = random.Random(5); pert = tot = 0 + for s in [s for s in sel if dimx(F0[s]) < h][:200]: + B = [r[:] for r in S.sigma[s]]; B[0] = [x + rng.randint(1, 3) for x in B[0]]; tot += 1 + pert += not inside(B, exact(F0[s])) + report('N. NEG perturbed sigma frames fail sigma <= F0 (%d / %d)' % (pert, tot), tot and pert >= 0.9 * tot) + early = set(S.srcop) - S.sel + bxr, _ = xchains(lambda us: [s for s in us if s not in early] + [s for s in us if s in early]) + report('N. NEG deferred V gates before the early ones break the X_S chains (%d bad)' % bxr, bxr > 0) + mq0 = ModQ7(fr, ld, extra, control=True) + badc = lemma(items[:300], mq0) + report('N. NEG side lemma with U = V = I fails (%d / %d)' % (badc, 2 * min(300, len(items))), badc > 0) + log('ALL %s' % ('PASS' if all(OK.values()) else 'FAIL: ' + str([k for k, x in OK.items() if not x]))) + return all(OK.values()) + + +if __name__ == '__main__': + sys.exit(0 if main(int(sys.argv[1]) if len(sys.argv) > 1 else 23) else 1) diff --git a/references/partial-gauge/pr97/check_lifted.py b/references/partial-gauge/pr97/check_lifted.py new file mode 100644 index 0000000000..4a19240e2f --- /dev/null +++ b/references/partial-gauge/pr97/check_lifted.py @@ -0,0 +1,677 @@ +"""Release check (stdlib only) of the lifted-frame + late-copy bit side program, stacked with the staircase corner. + +Input: certificates/round7/witness_.json.gz: the side DAG (leaves = triples of [h], additions), outputs (c, T), +retained totals, links (donor x -> use (y, k)), the flag matrix F (integer h x h), the lift index j(n) of every +lifted addition and the late-copy decisions (early users, ordered late users) per node. + +What is checked, from the witness alone: + A. DAG semantics: disjoint supports; output (c,T) support = {S : c in S, S cap T = {c}}; retained = star(c). + B. Compile: one slot per role, schedule order (dim span, id), links hand the donor's non-pivot slot to the linked + use; late copies: the pivot of n climbs C_1 <= ... <= C_k (C_i = frames of users late[i:] intersected) and the + copy for late user i is a gate (pivot, fresh slot) at C_i, placed in time as LATE as possible (just before the + first of: its user's gate, the pivot's consumer gate, the next copy's slot). R = adds + outputs + retained - links. + C. Dirty-scratch replay of the compiled program: word L J L^-1 V L J L^-1 V^-1 and its mirror, arbitrary scratch, + over F2 and over Z, reads at the slot's last touch and at the end; targets must receive J L V x exactly + (DAG values) and every slot is restored; over F2 every target also receives exactly x_T. + D. Frames (exact over Q). Frame of addition n: M_n = span(n) + (U_n cap F_j(n)), U_n = common kernel of the root + covectors reachable from n ((9I-J) t_T for an output to Y_T, 1 - 3 e_c for a retained use of c), F_j = first j + rows of F. Nesting is a THEOREM given the combinatorial facts checked here: + j(n) <= j(t) on every slot successor pair n -> t; span(x) <= span(t) on every link pair (exact); + span(n) <= every root frame n reaches directly (exact integer dot products); F invertible. + Then U_n <= U_t, so M_n <= M_t, M_n <= U_n <= every reachable root frame, and every late-copy frame C_i + (an intersection of frames that all contain M_n) is nested. Dimensions (exact, see linalg.py): + dim M_n = rank[span(n); F_j] - rank(K_n F_j^T) (modular law; span(n) <= U_n; K_n spans the root covectors), + with ranks by mod-p + Hadamard certificates, Bareiss when the mod-p rank is not full; late-copy intersections by + exact Fraction bases. Rank of every chain step = difference of dimensions (nested), histogram vs claim. + E. Frame reductions mod q = 2^31-1 (dimension-matched, so they are the reductions of the rational frames), + G = I - J/9 nondegenerate on every frame of every step (Gram matrix invertible mod q => over Q), and the side + lemma at ONE integer point (U, V of point_UV(h, 1), the random stream of independent/two-stage-bit/flag_existence.py): for every distinct step A -> B of + rank r with 2r > h (retained copy edges 0 -> U_c included), U^T (P_B - P_A) U^-T and V^T (..) V^-T have the + generic north-east pivots of f_r. (NE ranks mod q <= over Q <= f_r, so equality mod q proves it over Q.) + F. Certificate: rt_histogram with the round-6 entrance corner and with the staircase corner (m-4h+2, h-2, h-5, 1^6), + rank sum == s, moment.certify (exact), kappa = certificate_round3.evaluate(a_b, round-6 complex values). +Negative controls (CONTROL=1): see controls(). +Usage: python3 check_lifted.py h [--no-generic]""" +import sys, os, json, random, time +from itertools import combinations +from collections import Counter, OrderedDict +from fractions import Fraction as Fr +from array import array +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, HERE) +from linalg import * + +T0 = time.time() +def load_witness(h): + import gzip + with gzip.open(os.path.join(HERE, '..', '..', 'certificates', 'round7', 'witness_%d.json.gz' % h), 'rt') as fh: + return json.load(fh) + + +def log(*a): + print('[%5.0fs]' % (time.time() - T0), *a, flush=True) + + +# ----------------------------------------------------------------------------------------------- A. DAG +class Prog: + def __init__(self, W): + self.W = W; h = self.h = W['h'] + self.trip = list(combinations(range(h), 3)); v = self.v = len(self.trip) + self.args = {int(n): (tuple(a) if a else None) for n, a in W['args'].items()} + self.act = sorted(self.args) + for n in self.act: + if self.args[n] is None: assert 1 <= n <= v and tuple(W['leaf'][str(n)]) == self.trip[n - 1] + else: assert all(a < n and a in self.args for a in self.args[n]), n + self.sup = {} + for n in self.act: + if self.args[n] is None: self.sup[n] = 1 << (n - 1) + else: + a, b = self.args[n]; assert not self.sup[a] & self.sup[b], 'overlapping supports' + self.sup[n] = self.sup[a] | self.sup[b] + self.outputs = {(c, tuple(T)): n for c, T, n in W['outputs']} + self.retained = {c: n for c, n in W['retained']} + star = [sum(1 << i for i, t in enumerate(self.trip) if q in t) for q in range(h)] + assert len(self.outputs) == 3 * v and len(self.retained) == h + for (c, T), n in self.outputs.items(): + A, B = [q for q in T if q != c]; assert c in T and self.sup[n] == star[c] & ~star[A] & ~star[B] + for c, n in self.retained.items(): assert self.sup[n] == star[c] + roots = set(self.outputs.values()) | set(self.retained.values()); reach = set(); st = list(roots) + while st: + n = st.pop() + if n in reach: continue + reach.add(n) + if self.args[n]: st.extend(self.args[n]) + assert reach == set(self.act), 'inactive nodes in witness' + self.mL = {x: (y, k) for x, y, k in W['links']} + self.j = {int(n): j for n, j in W['j'].items()} + self.late = {int(n): (list(e), list(l)) for n, (e, l) in W['late'].items()} + self.F = W['flag'] + + +def trow(h, S): return [int(q in S) for q in range(h)] + + +def spans(pr): + """independent triple subsets (exact: 0/1 rows with three ones, every minor < 3^(h/2) < Q31) and dims.""" + h = pr.h; assert 3 ** h < Q31 ** 2 + bas = {}; dn = {} + for n in pr.act: + if pr.args[n] is None: bas[n] = (n - 1,); dn[n] = 1; continue + a, b = pr.args[n]; E = Echelon(Q31); keep = [] + for i in bas[a] + bas[b]: + if len(E) == h: break + if E.insert(trow(h, pr.trip[i])): keep.append(i) + bas[n] = tuple(keep); dn[n] = len(keep) + return bas, dn + + +def users_of(pr, dn): + key = lambda n: (dn[n], n); users = {n: [] for n in pr.act} + for n in sorted(pr.act, key=key): + if pr.args[n]: + for pos, x in enumerate(pr.args[n]): users[x].append(('gate', n, pos)) + for (c, T), n in sorted(pr.outputs.items()): users[n].append(('out', (c, T))) + for c, n in sorted(pr.retained.items()): users[n].append(('ret', c)) + return users + + +# ----------------------------------------------------------------------------------------------- B. compile +def root_key(u): return ('out',) + u[1] if u[0] == 'out' else ('ret', u[1]) + + +def compile_(pr, dn, users, late=True, defer=True, tamper=None): + """returns ops (scalar program), chains (frame keys per slot), reads, R. tamper: negative-control hooks.""" + key = lambda n: (dn[n], n); order = sorted(pr.act, key=key); gpos = {n: i for i, n in enumerate(order)} + linked = {r: x for x, r in pr.mL.items()} + chains = []; edge = {}; reads = []; ops = []; deferred = [] + def new(first): chains.append(list(first)); return len(chains) - 1 + def assign(s, n, k): + u = users[n][k] + if u[0] == 'gate': edge[u[1], u[2]] = s + else: + chains[s] += [root_key(u), ('F',)] + reads.append((s, [u[1][1]] if u[0] == 'out' else [T for T in pr.trip if u[1] in T])) + def ufr(n, k): + u = users[n][k]; return ('n', u[1]) if u[0] == 'gate' else root_key(u) + for n in order: + if pr.args[n] is None: + piv = new([('0',), ('n', n)]); ops.append(('src', piv, n, n)) + else: + a, b = pr.args[n]; sa, sb = edge.pop((n, 0)), edge.pop((n, 1)) + if n in pr.mL and pr.mL[n][0] == a: sa, sb = sb, sa # the linked operand stays on the non-pivot + piv, oth = sa, sb + chains[piv].append(('n', n)); chains[oth].append(('n', n)); ops.append(('add', piv, oth, n)) + if n in pr.mL: y, k = pr.mL[n]; assign(oth, y, k) + else: chains[oth].append(('F',)) + ks = [k for k in range(len(users[n])) if (n, k) not in linked] + if late and n in pr.late: + early, lt = pr.late[n]; assert sorted(early + lt) == ks, ('bad late decision', n) + else: early, lt = ks[1:], [ks[0]] + for k in early: + f = new([('0',), ('n', n)]); ops.append(('copy', piv, f, n)); assign(f, n, k) + for i, k in enumerate(lt): + Ck = ufr(n, k) if i == len(lt) - 1 else ('c', n, tuple(lt[i:])) + if tamper == 'nointersect' and i < len(lt) - 1: Ck = ufr(n, k) + chains[piv].append(Ck) + if i < len(lt) - 1: + f = new([('0',), Ck]); deferred.append((n, i, f, k, lt[-1])); assign(f, n, k) + k = lt[-1] + if users[n][k][0] == 'gate': edge[users[n][k][1], users[n][k][2]] = piv + else: assign(piv, n, k) + assert not edge + # place deferred copies: as late as possible, in order, before the user's gate and the pivot consumer's gate + INF = len(order); byn = {} + for (n, i, f, k, kl) in deferred: byn.setdefault(n, []).append((i, f, k, kl)) + ins = {} + for n, lst in byn.items(): + lst.sort(); D = [] + for i, f, k, kl in lst: + u, ul = users[n][k], users[n][kl] + d = min(gpos[u[1]] if u[0] == 'gate' else INF, gpos[ul[1]] if ul[0] == 'gate' else INF) + D.append(d) + for t in range(len(D) - 2, -1, -1): D[t] = min(D[t], D[t + 1]) + for (i, f, k, kl), d in zip(lst, D): + if not defer: d = gpos[n] + 1 # immediate copy (still a valid program) + if tamper == 'aftercons' and n == tamper_node[0] and i == 0: + ul = users[n][kl]; d = gpos[ul[1]] + 1 # copy AFTER the pivot was consumed + ins.setdefault(d, []).append((n, i, f)) + pivslot = {} + for o in ops: + if o[0] in ('src', 'add'): pivslot[o[3]] = o[1] + out = [] + for o in ops: + if o[0] in ('src', 'add'): + for (n, i, f) in sorted(ins.get(gpos[o[3]], []), key=lambda z: (gpos[z[0]], z[1])): + out.append(('copy', pivslot[n], f, n)) + out.append(o) + for (n, i, f) in sorted(ins.get(INF, []) + [z for d, zs in ins.items() if d > INF for z in zs], key=lambda z: (gpos[z[0]], z[1])): + out.append(('copy', pivslot[n], f, n)) + if tamper == 'dropcopy': + idx = next(i for i, o in enumerate(out) if o[0] == 'copy' and o[3] in pr.late); out.pop(idx) + R = len(chains); adds = sum(1 for n in pr.act if pr.args[n]) + assert R == adds + len(pr.outputs) + len(pr.retained) - len(pr.mL), 'role count' + return out, chains, reads, R + +tamper_node = [None] + + +# ----------------------------------------------------------------------------------------------- C. replay +def replay(pr, ops, reads, R, ring, rng, mirror=False, midread=True): + trip, v = pr.trip, pr.v; tid = {T: i for i, T in enumerate(trip)} + mod = (lambda z: z & 1) if ring == 2 else (lambda z: z) + last = {} + for i, o in enumerate(ops): + last[o[1]] = i + if o[0] != 'src': last[o[2]] = i + readers = {} + for s, Ts in reads: readers.setdefault(last[s] if midread else len(ops) - 1, []).append((s, Ts)) + x = [rng.randrange(-10**6, 10**6) for _ in range(v)] + a0 = [rng.randrange(-10**6, 10**6) for _ in range(R)]; y0 = [rng.randrange(-10**6, 10**6) for _ in range(v)] + a = list(a0); y = list(y0); src = [(o[1], o[2]) for o in ops if o[0] == 'src'] + def L(sg=None): + for i, o in enumerate(ops): + if o[0] == 'add': a[o[1]] += a[o[2]] + elif o[0] == 'copy': a[o[2]] += a[o[1]] + if sg is not None: + for s, Ts in readers.get(i, ()): + for T in Ts: y[tid[T]] += sg * a[s] + def Li(): + for o in reversed(ops): + if o[0] == 'add': a[o[1]] -= a[o[2]] + elif o[0] == 'copy': a[o[2]] -= a[o[1]] + def V(sg): + for s, n in src: a[s] += sg * x[n - 1] + if not mirror: L(-1); Li(); V(+1); L(+1); Li(); V(-1) + else: V(+1); L(+1); Li(); V(-1); L(-1); Li() + if any(mod(p - q) for p, q in zip(a, a0)): return False + val = {} + for n in pr.act: val[n] = x[n - 1] if pr.args[n] is None else val[pr.args[n][0]] + val[pr.args[n][1]] + want = list(y0) + for (c, T), n in pr.outputs.items(): want[tid[T]] += val[n] + for c, n in pr.retained.items(): + for T in trip: + if c in T: want[tid[T]] += val[n] + if any(mod(p - q) for p, q in zip(y, want)): return False + if ring == 2 and any(mod(y[i] - y0[i] - x[i]) for i in range(v)): return False + return True + + +# ----------------------------------------------------------------------------------------------- D. frames +class Frames: + def __init__(self, pr, dn, bas, users): + self.pr = pr; self.dn = dn; self.bas = bas; self.users = users; h = self.h = pr.h + self.rootcov = {} + for (c, T) in pr.outputs: self.rootcov[('out', c, T)] = [9 * int(q in T) - 3 for q in range(h)] + for c in pr.retained: self.rootcov[('ret', c)] = [1 - 3 * int(q == c) for q in range(h)] + rid = {k: i for i, k in enumerate(sorted(self.rootcov))}; self.rkeys = sorted(self.rootcov) + self.rid = rid + # slot successors (same for base and late compile: every free use of n is reached by one slot from n, + # a donor's non-pivot slot goes on to its linked use) and direct roots + linked = {r: x for x, r in pr.mL.items()} + self.succ = {n: set() for n in pr.act}; self.droots = {n: set() for n in pr.act} + def to(n, use): + if use[0] == 'gate': self.succ[n].add(use[1]) + else: self.droots[n].add(rid[root_key(use)]) + for n in pr.act: + for k, u in enumerate(users[n]): + if (n, k) not in linked: to(n, u) + if n in pr.mL: y, k = pr.mL[n]; to(n, users[y][k]) + + def check_structure(self): + """combinatorial / exact facts behind the nesting theorem.""" + pr, h = self.pr, self.h; bad_j = bad_link = bad_root = 0 + J = lambda n: self.pr.j.get(n, 0) + for n in pr.act: + if pr.args[n] is None: assert J(n) == 0 + for t in self.succ[n]: + if J(n) > J(t): bad_j += 1 + for x, (y, k) in pr.mL.items(): # span(x) <= span(linked use target) + u = self.users[y][k] + if u[0] == 'gate': + t = u[1]; E = Echelon(Q31) + for i in self.bas[t]: E.insert(trow(h, pr.trip[i])) + if not all(E.contains(trow(h, pr.trip[i])) for i in self.bas[x]): bad_link += 1 + for n in pr.act: # span(n) <= direct root frames + for r in self.droots[n]: + g = self.rootcov[self.rkeys[r]] + if any(sum(gq * tq for gq, tq in zip(g, trow(h, pr.trip[i]))) for i in self.bas[n]): bad_root += 1 + # F invertible over Q (a nonzero determinant mod a prime) + F_ok = rank_mod(self.pr.F, Q31) == h + return bad_j, bad_link, bad_root, F_ok + + def cobases(self): + """K_n: independent subset of the root covectors reachable from n, spanning all of them over Q + (selection mod M127, valid by the Hadamard certificate on root covectors).""" + pr, h = self.pr, self.h + rows = [self.rootcov[k] for k in self.rkeys] + mx = max(sum(x * x for x in r) for r in rows); assert mx ** h < M127 ** 2, 'Hadamard bound for K' + key = lambda n: (self.dn[n], n); K = {} + for n in sorted(pr.act, key=key, reverse=True): + cand = sorted(self.droots[n]); srcs = sorted((K[t] for t in self.succ[n]), key=len, reverse=True) + cap = h - self.dn[n] # span(n) <= U_n + if not cand and len(srcs) == 1: K[n] = srcs[0]; continue + E = Echelon(M127); keep = [] + seq = list(srcs[0]) if srcs else [] + for r in seq: E.insert(rows[r]); keep.append(r) # srcs[0] is already independent + for lst in [cand] + srcs[1:]: + for r in lst: + if len(keep) >= cap: break + if E.insert(rows[r]): keep.append(r) + K[n] = tuple(keep) + self.K = K + return K + + def dims(self): + """exact dim of every addition frame M_n.""" + pr, h = self.pr, self.h; F = pr.F; d = dict(self.dn); stats = Counter() + rows = [self.rootcov[k] for k in self.rkeys] + self.KF = {} + for n, j in pr.j.items(): + Fj = F[:j]; K = [rows[r] for r in self.K[n]] + KF = [[sum(a * b for a, b in zip(kr, fr)) for fr in Fj] for kr in K] + SF = [trow(h, pr.trip[i]) for i in self.bas[n]] + Fj + r1 = self._rank(SF, min(len(SF), h), stats, 'SF'); r2 = self._rank(KF, min(len(K), j), stats, 'KF') + d[n] = r1 - r2; self.KF[n] = (r2, KF) + assert self.dn[n] <= d[n] <= h - len(K) + self.d = d + return d, stats + + @staticmethod + def _rank(M, full, stats, tag): + if not M or full == 0: return 0 + r = rank_mod(M, Q31) + if r == full: stats[tag + ' full mod q'] += 1; return r # rank mod q <= rank over Q <= full + if hadamard_ok(M, M521): stats[tag + ' M521'] += 1; return rank_mod(M, M521) + stats[tag + ' Bareiss'] += 1; return rank_exact(M) + + # exact rational bases, used for the late-copy intersections + def exact_basis(self, key): + pr, h = self.pr, self.h + if key[0] in ('out', 'ret'): + return null_exact([self.rootcov[key]], h) + n = key[1] + S = [trow(h, pr.trip[i]) for i in self.bas[n]] + j = pr.j.get(n, 0) + if not j: return S + _, KF = self.KF[n]; F = pr.F[:j] + X = null_exact(KF, j) if KF else [[int(i == t) for t in range(j)] for i in range(j)] + vecs = S + [[sum(c * F[i][q] for i, c in enumerate(x)) for q in range(h)] for x in X] + # independent subset: rows independent mod q are independent over Q; d of them span M_n (dim M_n = d exactly) + E = Echelon(Q31); out = [] + for vv in vecs: + if E.insert(vv): out.append(vv) + assert len(out) == self.d[n], ('exact basis dim', n, len(out), self.d[n]) + return out + + def intersect_exact(self, A, B): + h = self.h + if not A or not B: return [] + # z = (x, y): x A = y B <=> [A; -B]^T z = 0 + M = [[A[i][q] for i in range(len(A))] + [-B[i][q] for i in range(len(B))] for q in range(h)] + Z = null_exact(M, len(A) + len(B)) + # A and B are bases, so z -> x A is injective on the null space: the images are a basis of A cap B + return [[sum(z[i] * A[i][q] for i in range(len(A))) for q in range(h)] for z in Z] + + +def late_frames(fr, pr, users): + """exact dims of every late-copy intersection frame ('c', n, late users).""" + cache = {}; dims = {} + def eb(k): + if k not in cache: cache[k] = fr.exact_basis(k) + return cache[k] + def ufr(n, k): + u = users[n][k]; return ('n', u[1]) if u[0] == 'gate' else root_key(u) + for n, (early, lt) in pr.late.items(): + for i in range(len(lt) - 1): + B = eb(ufr(n, lt[-1])) + for k in reversed(lt[i:-1]): B = fr.intersect_exact(eb(ufr(n, k)), B) + dims[('c', n, tuple(lt[i:]))] = len(B) + return dims, cache + + +def chain_steps(chains, dimof, h): + rk = Counter(); steps = {} + for ch in chains: + prev = ch[0]; pd = dimof(prev) + for k in ch[1:]: + dk = dimof(k) + assert dk >= pd, ('decreasing chain', prev, k, pd, dk) + if dk > pd: rk[dk - pd] += 1; steps[prev, k] = dk - pd; prev, pd = k, dk + else: prev = k # equal dims, nested => equal frames + assert pd == h + return rk, steps + + +# ----------------------------------------------------------------------------------------------- E. mod-q frames, side lemma +def point_UV(h, seed): + """the U, V drawn by independent/two-stage-bit/flag_existence.py with this seed (same random stream).""" + CR = 10**6; rng = random.Random(seed) + nz = lambda: rng.choice([-1, 1]) * rng.randint(1, CR) + for b in range(h): + for i in range(h): + for jj in range(h): + if i <= b and jj <= b: nz() + for b in range(h): + for i in range(h): + for jj in range(h): + if i >= b and jj >= b: nz() + U = [[rng.randint(-CR, CR) for _ in range(h)] for _ in range(h)] + V = [[rng.randint(-CR, CR) for _ in range(h)] for _ in range(h)] + return U, V + + +class ModQ: + def __init__(self, fr, latedims, control=False): + self.fr = fr; self.h = h = fr.h; self.q = q = Q31; self.latedims = latedims + U, V = point_UV(h, 1) + if control: U = [[int(i == k) for k in range(h)] for i in range(h)]; V = [r[:] for r in U] + self.conj = [] + for A in (U, V): + Ai = inv_mod_matrix(A, q); assert Ai is not None + self.conj.append(([list(c) for c in zip(*A)], Ai)) # columns of A, rows of A^-1 + self.inv9 = pow(9, q - 2, q); self.bcache = {}; self.qcache = OrderedDict(); self.cap = 6000 + self.degenerate = 0; self.dimfail = 0 + + def basis(self, key): + if key in self.bcache: return self.bcache[key] + fr, h, q = self.fr, self.h, self.q; pr = fr.pr + if key == ('0',): B = [] + elif key == ('F',): B = [[int(i == t) for t in range(h)] for i in range(h)] + elif key[0] in ('out', 'ret'): B = null_mod([fr.rootcov[key]], h, q) + elif key[0] == 'c': + n, lt = key[1], key[2] + B = None + for k in lt: + u = fr.users[n][k]; kk = ('n', u[1]) if u[0] == 'gate' else root_key(u) + Bk = self.basis(kk) + B = Bk if B is None else self._cap(B, Bk) + if len(B) != self.latedims[key]: self.dimfail += 1 + else: + n = key[1]; S = [trow(h, pr.trip[i]) for i in fr.bas[n]]; j = pr.j.get(n, 0) + E = Echelon(q) + for r in S: E.insert(r) + if j: + r2, KF = fr.KF[n] + if KF and rank_mod(KF, q) != r2: self.dimfail += 1 + X = null_mod(KF, j, q) if KF else [[int(i == t) for t in range(j)] for i in range(j)] + Fj = pr.F[:j] + for x in X: E.insert([sum(c * Fj[i][t] for i, c in enumerate(x)) % q for t in range(h)]) + B = E.rows + if len(B) != fr.d[n]: self.dimfail += 1 + self.bcache[key] = B + return B + + def _cap(self, A, B): + q, h = self.q, self.h + if not A or not B: return [] + M = [[A[i][t] for i in range(len(A))] + [(-B[i][t]) % q for i in range(len(B))] for t in range(h)] + Z = null_mod(M, len(A) + len(B), q); E = Echelon(q) + for z in Z: E.insert([sum(z[i] * A[i][t] for i in range(len(A))) % q for t in range(h)]) + return E.rows + + def gram_inv(self, B): + q = self.q; BG = [[(x - sum(b) * self.inv9) % q for x in b] for b in B] + Gr = [[sum(x * y for x, y in zip(bg, c)) % q for c in B] for bg in BG] + return BG, inv_mod_matrix(Gr, q) + + def nondeg(self, key): + B = self.basis(key) + if not B or len(B) == self.h: return True + return self.gram_inv(B)[1] is not None + + def conjugated(self, key): + """(U^T P U^-T, V^T P V^-T) for the G-orthogonal projector P onto the frame, flattened.""" + if key in self.qcache: self.qcache.move_to_end(key); return self.qcache[key] + h, q = self.h, self.q; B = self.basis(key) + if not B: out = (array('q', [0] * h * h),) * 2 + elif len(B) == h: I = array('q', [int(i == t) for i in range(h) for t in range(h)]); out = (I, I) + else: + BG, Gi = self.gram_inv(B) + if Gi is None: raise ZeroDivisionError(key) + res = [] + for Acols, Ai in self.conj: + LU = [[sum(x * y for x, y in zip(ac, b)) % q for b in B] for ac in Acols] # A^T B^T (h x d) + RU = [[sum(x * y for x, y in zip(bg, ai)) % q for ai in Ai] for bg in BG] # B G A^-T (d x h) + Mi = matmul(Gi, RU, q) + Mt = list(zip(*Mi)) + res.append(array('q', [sum(x * y for x, y in zip(l, c)) % q for l in LU for c in Mt])) + out = tuple(res) + self.qcache[key] = out + if len(self.qcache) > self.cap: self.qcache.popitem(last=False) + return out + + +def side_lemma(mq, steps, h, retained): + q = mq.q; pr_ = predicted(h); E = dict((k, r) for k, r in steps.items() if 2 * r > h) + for c in retained: E[('0',), ('ret', c)] = h - 1 # retained copy edges 0 -> U_c + keys = sorted(E, key=lambda e: (str(e[1]), str(e[0]))) # group by target frame (cache reuse) + bad = 0; n = 0 + for (A, B) in keys: + r = E[A, B]; QA, QB = mq.conjugated(A), mq.conjugated(B) + for t in range(2): + a, b = QA[t], QB[t] + M = [[(b[i * h + c] - a[i * h + c]) % q for c in range(h)] for i in range(h)] + if rightmost_pivots(M, q) != pr_[r]: bad += 1 + n += 1 + if n % 5000 == 0: log(' side lemma %d/%d, failures %d' % (n, len(keys), bad)) + return bad, len(keys) + + +# ----------------------------------------------------------------------------------------------- F. certificate +K_REPO = os.path.normpath(os.path.join(HERE, '..', '..')) +sys.path[:0] = [K_REPO + '/scripts', K_REPO + '/independent/two-stage-bit'] + + +def inner_children(r, h): return [1] * (h - r) + [2 * r - h] if 2 * r > h else [1] * r + + +def rt_histogram(h, R, rk, corner): + """cert_rt.rt_histogram with the data-entrance corner profile 'corner' (round six: [h-2] + [1]*(h+1)).""" + from math import comb + v, m = comb(h, 3), h * h; N = v * v; A = v * R; H = Counter() + assert sum(r * n for r, n in rk.items()) == h * R and sum(corner) == 2 * h - 1 + for _ in range(2): + H[m - 2 * h] += A; H[h] += A + for r, n in rk.items(): + for w in inner_children(r, h): H[w] += v * n + for w in inner_children(h - 1, h): H[w] += v * h + H[m - 4 * h + 2] += 2 * N + for w in corner: H[w] += 2 * N + for _ in range(2): H[h - 2] += 2 * N; H[1] += 2 * N + H[1] += N + W = 2 * N + 2 * A; L = 2 * v * h * (h - 1); s = W * m - N + L + H = {w: c for w, c in H.items() if c} + assert sum(w * c for w, c in H.items()) == s and max(H) <= m - 1 + return dict(h=h, m=m, v=v, N=N, W=W, L=L, s=s, R=R, hist=H) + + +def certificate(h, R, rk): + import moment + from certificate_round3 import evaluate + out = {} + for name, corner in (('round6 corner', [h - 2] + [1] * (h + 1)), ('staircase', [h - 2, h - 5] + [1] * 6)): + c = rt_histogram(h, R, rk, corner) + if c['W'] * c['m'] <= c['s']: out[name] = ('no deficit', c['s']); continue + a, root = moment.certify(c) + k = evaluate(a, Fr(36926111, 5 * 10**11), Fr(1, 1000), 'crude', m_c=576, s_c=119453132304) + out[name] = (a, root, c['s'], c['W'], k['kappa'], k['ok']) + return out + + +# ----------------------------------------------------------------------------------------------- main +def main(h, generic=True, control=False): + W = load_witness(h) + pr = Prog(W); log('h=%d: A. DAG semantics OK (%d nodes, %d additions, %d links, %d lifted, %d late)' % ( + h, len(pr.act), sum(1 for n in pr.act if pr.args[n]), len(pr.mL), len(pr.j), len(pr.late))) + bas, dn = spans(pr); users = users_of(pr, dn) + ops, chains, reads, R = compile_(pr, dn, users) + log('B. compiled: R=%d (claimed %d), ops %d, late copies %d' % (R, W['claimed']['R'], len(ops), + sum(len(l) - 1 for _, l in pr.late.values()))) + assert R == W['claimed']['R'] + rng = random.Random(h) + res = {(ring, mir, mid): all(replay(pr, ops, reads, R, ring, rng, mir, mid) for _ in range(2)) + for ring in (2, 0) for mir in (False, True) for mid in (True, False)} + log('C. dirty-scratch replay (F2/Z x word/mirror x midread/endread): %s' % ('ALL OK' if all(res.values()) else res)) + fr = Frames(pr, dn, bas, users) + bj, bl, br, Fok = fr.check_structure() + log('D. nesting facts: j-monotone failures %d, link span failures %d, root containment failures %d, F invertible %s' + % (bj, bl, br, Fok)) + fr.cobases(); log(' co-bases K_n done') + d, st = fr.dims(); log(' lifted frame dims exact: %s' % dict(st)) + nU = sum(1 for n in pr.j if d[n] == h - len(fr.K[n])) + log(' every lifted frame is the largest admissible frame U_n: %s (%d / %d; the flag only switches span -> U_n)' + % (nU == len(pr.j), nU, len(pr.j))) + ld, ecache = late_frames(fr, pr, users); log(' late-copy intersections exact: %d frames' % len(ld)) + def dimof(k): + if k == ('0',): return 0 + if k == ('F',): return h + if k[0] in ('out', 'ret'): return h - 1 + if k[0] == 'c': return ld[k] + return d[k[1]] + rk, steps = chain_steps(chains, dimof, h) + claimed = Counter({int(r): c for r, c in W['claimed']['rk'].items()}) + log(' chain steps: sum of ranks %d = h R %d: %s; histogram equals claim: %s; distinct steps %d (2r>h: %d)' % ( + sum(r * c for r, c in rk.items()), h * R, sum(r * c for r, c in rk.items()) == h * R, rk == claimed, + len(steps), sum(1 for r in steps.values() if 2 * r > h))) + out = dict(h=h, R=R, replay=all(res.values()), struct=(bj, bl, br, Fok), hist_ok=rk == claimed, rk=rk) + if generic: + mq = ModQ(fr, ld, control) + frames_used = {k for e in steps for k in e} | {('ret', c) for c in pr.retained} + nd = sum(1 for k in frames_used if not mq.nondeg(k)) + log('E. frames used %d: G-degenerate %d, reduction dimension mismatches %d' % (len(frames_used), nd, mq.dimfail)) + bad, ne = side_lemma(mq, steps, h, pr.retained) + log(' side lemma at one integer point%s: failures %d over %d distinct high-rank steps x 2 conjugations' % ( + ' (CONTROL U=V=I)' if control else '', bad, ne)) + out.update(degenerate=nd, dimfail=mq.dimfail, side_bad=bad, side_n=ne) + for name, val in certificate(h, R, rk).items(): + if val[0] == 'no deficit': log('F. %s: no deficit at this h (s=%d), certificate n/a' % (name, val[1])); continue + a, root, s, Wr, kap, ok = val + log('F. %s: s=%d W=%d a_b=%s (%.6e, root %.6e) kappa=%s (%.7e) ok=%s' % (name, s, Wr, a, float(a), root, kap, + float(kap), ok)) + out[name] = (a, kap, ok) + return out, pr, dn, users + + + + +# ----------------------------------------------------------------------------------------------- negative controls +def controls(h, nsample=40): + """Each must FAIL (or differ) for the check to have teeth: + (a) one late copy moved to just after its pivot's consumer gate: replay must fail; + (b) one late copy dropped: replay must fail; + (c) pivot chain without intersections (C_i := frame of user i alone): exhibit non-nested consecutive pivot frames; + (d) M_n rebuilt from an independent flag F' on targets strictly between span and U (vacuous for this + witness: every lifted frame equals U_n, so the flag is only a switch and nesting needs no common flag); + (e) side lemma at U = V = I on the first high-rank steps: generic pivots must fail; + (f) no late copies (lifted frames only): histogram must change.""" + W = load_witness(h) + pr = Prog(W); bas, dn = spans(pr); users = users_of(pr, dn); rng = random.Random(7) + key = lambda n: (dn[n], n); linked = {r: x for x, r in pr.mL.items()} + cand = [n for n, (e, lt) in pr.late.items() if users[n][lt[-1]][0] == 'gate'] + fails = 0; tried = 0 + for n in cand[:nsample]: + tamper_node[0] = n; ops, ch, rd, R = compile_(pr, dn, users, tamper='aftercons') + ok = all(replay(pr, ops, rd, R, ring, rng) for ring in (2, 0)); tried += 1; fails += not ok + log('(a) late copy moved after the pivot consumer: replay failed %d / %d (must be all)' % (fails, tried)) + ops, ch, rd, R = compile_(pr, dn, users, tamper='dropcopy') + log('(b) one late copy dropped: replay correct %s (must be False)' % all(replay(pr, ops, rd, R, ring, rng) for ring in (2, 0))) + fr = Frames(pr, dn, bas, users); fr.cobases(); d, _ = fr.dims(); d = dict(d); ld, _ = late_frames(fr, pr, users) + mq = ModQ(fr, ld) + def inside(A, B): + E = Echelon(mq.q) + for r in mq.basis(B): E.insert(r) + return all(E.contains(r) for r in mq.basis(A)) + def ufr(n, k): + u = users[n][k]; return ('n', u[1]) if u[0] == 'gate' else root_key(u) + bad = tot = 0 + for n, (e, lt) in pr.late.items(): + for i in range(len(lt) - 1): + tot += 1; bad += not inside(ufr(n, lt[i]), ufr(n, lt[i + 1])) + log('(c) pivot chain through user frames without intersections: non-nested consecutive frames %d / %d ' + '(the intersections are needed)' % (bad, tot)) + # (d) the common flag is essential: rebuild M_n with an independent flag F' (same j(n)) and test M_n <= M_t + pairs = [(n, t) for n in pr.j for t in fr.succ[n] if pr.j.get(t, 0) > 0 and dn[t] < d[t] < h - len(fr.K[t])] + rng.shuffle(pairs); bad = tot = 0; r2 = random.Random(99) + Fp = [[r2.randint(-50, 50) for _ in range(h)] for _ in range(h)] + for n, t in pairs[:nsample]: + j = pr.j[n]; K = [fr.rootcov[fr.rkeys[r]] for r in fr.K[n]] + KF = [[sum(a * b for a, b in zip(kr, fl)) for fl in Fp[:j]] for kr in K] + if KF and rank_mod(KF, mq.q) != fr._rank(KF, min(len(K), j), Counter(), 'KF'): continue + X = null_mod(KF, j, mq.q) if KF else [[int(i == c) for c in range(j)] for i in range(j)] + E = Echelon(mq.q) + for r in mq.basis(('n', t)): E.insert(r) + vecs = [[sum(c * Fp[i][q] for i, c in enumerate(x)) % mq.q for q in range(h)] for x in X] + tot += 1; bad += not all(E.contains(v) for v in vecs) + log("(d) M_n rebuilt with an independent flag F': M_n not inside M_t on %d / %d lifted successor pairs (targets strictly between span and U)" % (bad, tot)) + ops, chains, reads, R = compile_(pr, dn, users) + fr2 = Frames(pr, dn, bas, users); fr2.cobases(); d2, _ = fr2.dims(); ld2, _ = late_frames(fr2, pr, users) + dimof = lambda k: 0 if k == ('0',) else h if k == ('F',) else h - 1 if k[0] in ('out', 'ret') else ld2[k] if k[0] == 'c' else d2[k[1]] + rk, steps = chain_steps(chains, dimof, h) + mqs = ModQ(fr2, ld2); smp = list(steps); random.Random(5).shuffle(smp); nb = 0 + for A, B in smp[:3000]: + E = Echelon(mqs.q) + for r in mqs.basis(B): E.insert(r) + nb += not all(E.contains(r) for r in mqs.basis(A)) + log('(g) sanity (positive): 3000 sampled chain steps A -> B with A inside B mod q: failures %d' % nb) + mq0 = ModQ(fr2, ld2, control=True); pr_ = predicted(h); hi = [(e, r) for e, r in steps.items() if 2 * r > h][:300] + nb = 0 + for (A, B), r in hi: + QA, QB = mq0.conjugated(A), mq0.conjugated(B) + M = [[(QB[0][i * h + c] - QA[0][i * h + c]) % mq0.q for c in range(h)] for i in range(h)] + nb += rightmost_pivots(M, mq0.q) != pr_[r] + log('(e) side lemma with U = V = I: failures %d / %d (must be most)' % (nb, len(hi))) + ops, chains0, reads, R = compile_(pr, dn, users, late=False) + rk0, _ = chain_steps(chains0, dimof, h) + log('(f) without late copies: histogram differs %s; certificate %s' % (rk0 != rk, {k: v[0] for k, v in certificate(h, R, rk0).items()})) + + +if __name__ == '__main__': + h = int(sys.argv[1]) + if '--controls' in sys.argv: controls(h) + else: + out = main(h, generic='--no-generic' not in sys.argv)[0] + ok = out['replay'] and out['struct'][:3] == (0, 0, 0) and out['struct'][3] and out['hist_ok'] + ok &= out.get('degenerate', 0) == 0 and out.get('dimfail', 0) == 0 and out.get('side_bad', 0) == 0 + ok &= all(v[2] for k, v in out.items() if k in ('round6 corner', 'staircase')) + log('ALL %s' % ('PASS' if ok else 'FAIL')) + sys.exit(0 if ok else 1) diff --git a/references/partial-gauge/pr97/deferred-readout.tex b/references/partial-gauge/pr97/deferred-readout.tex new file mode 100644 index 0000000000..d79abea77a --- /dev/null +++ b/references/partial-gauge/pr97/deferred-readout.tex @@ -0,0 +1,111 @@ +\documentclass[11pt]{article} +\usepackage{amsmath,amssymb,amsthm} +\newtheorem{lemma}{Lemma} +\newtheorem{proposition}{Proposition} +\theoremstyle{remark}\newtheorem*{remark}{Remark} +\title{Deferred readouts in the two-stage bit interchange} +\date{Draft, October 9, 2026} +\begin{document} +\maketitle + +This note describes the round-seven bit side. It keeps the round-six interchange (flag basis, retained point totals +with copied centres, side roles batched one level down, data-entrance run) and changes three things: the side +program comes from PR~\#62's producer, its frames are lifted and its fan-out copies are made late, and the stage-one +word is reordered so that most slots read their garbage out at a nonzero frame. All objects are frozen in +\texttt{certificates/round7/}; \texttt{scripts/certificate\_round7.py} rebuilds the child-width histogram from the +schedule, and the checks in \texttt{independent/deferred-readout/} verify every fact used below. Throughout, $h=23$, +$m=h^2$, $v=\binom h3$, $N=v^2$, $G=I-J/9$, and for a triple $T$ the root frame of the output wire $y_T$ is +$t_T^\perp=\ker\,(9t_T-3\mathbf 1)^{\mathsf T}$; the frame of a retained total of $c$ is $U_c=\ker(\mathbf 1-3e_c)^{\mathsf T}$. + +\section{The side program: PR \#62's producer} +The side DAG has the $v$ triples as leaves and disjoint additions as nodes; its roots are the $3v$ outputs +$\mathrm{out}(c,T)=\{S: S\cap T=\{c\}\}$ and the $h$ retained totals $\mathrm{star}(c)$. We build it with PR~\#62's +producer (ikeboy): interval strips and core-aware pair assembly, with PR~\#41's alternating order and links +(RaD/hipotures), and links chosen by a moment-weighted matching in the spirit of PR~\#44 (rohanarun). A link hands the +non-pivot slot of a donor gate to a later use of the same value, so the number of slots is +$R=\#\text{additions}+3v+h-\#\text{links}=28866$. Only the DAG enters the certificate, and it is checked from the +definitions: every addition is disjoint, every output and total has the support above, and every node is reachable +from a root (\texttt{check\_lifted.py}, part~A; \texttt{check\_word.py}, part~A). + +\section{Lifted frames and late copies} +For an addition $n$ let $U_n$ be the common kernel of the root covectors reachable from $n$ and $F_j$ the span of +the first $j$ rows of a fixed integer flag matrix. The gate of $n$ acts at +\[ M_n=\operatorname{span}(n)+(U_n\cap F_{j(n)}), \] +with a lift index $j(n)$ per addition. This is Paureel's complement-frame construction: it is valid for any frames +with (H1) leaf frames $\langle t_S\rangle$, (H2) frames nested along every slot (DAG edges, links, copies), +(H3) roots inside $t_T^\perp$ or $U_c$, (H4) $G$ nondegenerate on every frame. Nesting is a theorem given +$j(n)\le j(t)$ on every slot successor pair, $\operatorname{span}(x)\le\operatorname{span}(t)$ on every link and +$\operatorname{span}(n)$ inside every root it reaches directly; then $U_n\le U_t$ and $M_n\le M_t$. In this witness +every lifted frame equals $U_n$. Dimensions are exact (modular law, ranks with Hadamard certificates or Bareiss). + +\paragraph{Late copies.} A node with several free uses is copied for each of them. A copy made early sits at $M_n$; +a \emph{late} copy for the user $\ell_i$ of an ordered list $\ell_1,\dots,\ell_k$ is made when the pivot has climbed to +\[ C_i=\textstyle\bigcap_{i'\ge i} M(\ell_{i'}), \qquad M_n\le C_1\le\dots\le C_{k-1}\le M(\ell_k), \] +and starts there, so the copy saves the rank $\dim C_i-\dim M_n$. Each copy is placed as late as possible (before its +user's gate, the pivot's next gate and the next copy). The intersections are needed: without them consecutive pivot +frames are not nested (1396 of 1396), and a copy moved after the pivot's next gate breaks the replay. + +\section{The B-defer word} +In stage one each slot $u$ starts with its garbage $z_u$, and the word reads it out of every target it reaches: +$y_T\mathrel{-}=C_{T,u}z_u$, where $C=JL$ is given by the transpose sweep of $L$. In rounds five and six all these +readouts happen at frame $0$, before the second pass. B-defer splits the second pass in two phases. + +\begin{itemize} +\item \emph{Phase 1} is the closure of the retained totals' last operations under per-slot sequence precedence: every + op that must precede a total's completion. The totals complete, and their copies read every target containing + their centre at $0$ (copied centres, as in round six). +\item A slot that phase 1 neither reads nor writes still holds its garbage after phase 1. It reads it out + \emph{now}, at a frame $\sigma_u\le F_0(u)\cap\bigcap_{T}t_T^\perp$ (over its targets $T$), then the rest of $L$ runs, + then the output reads at $t_T^\perp$, then $L^{-1}$ and $V^{-1}$ exactly mirror it. +\end{itemize} +Only the topological order of the word changes, so it is still a mirror and still restores arbitrary scratch. The +slot's chain now starts at $\sigma_u$: its corner has rank $r_u=h-\dim\sigma_u$ and its auxiliary edge rank $m-r_u$. + +\paragraph{The sorting rule.} On a target $y_T$ the readout frames, in time order, are $0$ (non-deferred slots and +centre copies), the deferred $\sigma_u$, and $t_T^\perp$; they must form a chain. The word reads the deferred slots +in increasing $\dim\sigma_u$, and applies the deferred $V$ gates (below) in increasing $\dim F_0$. With this rule +every $y_T$ sequence is nested in time order, exactly over $\mathbb Q$, for both the $\mathbb Z$ and the +$\mathbb F_2$ supports of $C$; the reversed order breaks 11879 chains. + +\section{V leaves as late copies on $X_S$} +A leaf $S$ is used by several gates. Instead of copying $x_S$ by fan-out, every use $i$ gets its own $V$ gate on the +data wire $X_S$, into a slot that starts at +\[ s_i=N_i\cap N_{i+1}\cap\cdots \qquad(\text{falling back to }\langle t_S\rangle\text{ if $G$ is degenerate on it}), \] +where $N_j$ is the first frame of use $j$. The $V$ gates are then late copies on $X_S$: its frame chain +$\langle t_S\rangle\le s_{i_1}\le s_{i_2}\le\dots\le F$ has total rank $h-1$, as before, and the use slots start at +$s_i$ instead of $\langle t_S\rangle$. The $V$ gates of deferred slots come after their readouts; in time the $X_S$ chain +is $\langle t_S\rangle$, the early $V$ frames (by dimension), the deferred $V$ frames (by the rule above), $F$, and it +is checked nested exactly in that order. The number of slots is unchanged. + +\section{What the certificate uses} +\paragraph{Only the $\mathbb F_2$ flag identity.} The bit network is used over $\mathbb F_2$: the word must add exactly +$x_T$ to every $y_T$ and restore every slot, for arbitrary scratch. The replay checks this over $\mathbb F_2$ with +the parity coefficients (the actual circuit) and, separately, over $\mathbb Z$ with the exact path-count +coefficients; the histogram and $a_b$ are the same with either support. + +\paragraph{The stage-two complement.} Stage two is the complement time-reversal of stage one, with the same edge +multiset. This is the same assumption as in round six; it is stated, not machine-checked here. + +\paragraph{Frames and genericity.} Every $\sigma_u$ and $s_i$ is a primitive integer basis (entries at most 13). +Exactly over $\mathbb Q$: $\sigma_u\le F_0(u)$, $\sigma_u\le t_T^\perp$, $t_S\le s_i\le N_i$, $G$ nondegenerate. Every +high-rank step ($2r>h$) that the reordered word adds (slot starts, gauged corners $\sigma_u\to F$, $y_T$ and $X_S$ +chain steps) has the generic north-east pivots under both conjugations at the integer point of +\texttt{check\_lifted.py}, so the side lemma batches it. + +\section{Accounting} +Per stage and invocation: every slot contributes its auxiliary edge (block $m-2r_u$) and its corner, both batched +by the inner lemma, and its chain steps; every centre copy its step $U_c\to0$ of rank $h-1$; every $y_T$ and every +$X_S$ its chain of total rank $h-1$; the data entrance its block $m-4h+2$ and corner; one copy-correction singleton per +pair. The rank sum is exactly $s=Wm-N+L$ with $W=2N+2vR$, $L=2vh(h-1)$, so the histogram re-partitions the round-six +budget. The moment certificate gives +\begin{align*} + a_b&=31987/(5\cdot10^8) &&\text{(staircase entrance corner, runs $h-2,h-5,1^6$)},\\ + a_b&=12599/(2\cdot10^8) &&\text{(round-six entrance corner)}, +\end{align*} +and with round six's complex interchange ($a_c=36926111/(5\cdot10^{11})$, $h=24$) the assembly gives +$\kappa=1599247689723/(2.5\cdot10^{16})\approx6.39699\times10^{-5}$ with the staircase corner and +$\kappa=6299103187973/10^{17}$ with the round-six corner. The staircase corner is certified by +\texttt{check\_stair.py} at the same integer point; the entrance edge is not changed by the reordering. + +\end{document} + diff --git a/references/partial-gauge/pr97/deferred.py b/references/partial-gauge/pr97/deferred.py new file mode 100644 index 0000000000..d045126217 --- /dev/null +++ b/references/partial-gauge/pr97/deferred.py @@ -0,0 +1,189 @@ +"""Shared reader and accounting for the round-seven bit side (stdlib only). + +Frozen inputs (certificates/round7/): + witness_23.json.gz the base side program: DAG (leaves = triples of [h], additions), outputs (c, T), retained + totals, links, the flag matrix, the lift index j(n) of every lifted addition and the late-copy + decisions; checked by check_lifted.py; + deferred_23.json.gz the reordered stage-1 word: the slot schedule (ops, the node held by every slot over time, + slot starts, output and retained slots), the deferred readout frames sigma_u, the V-leaf start + frames s_i (both as primitive integer bases), the readout order, the X_S order of every leaf, + the order of the deferred V gates, and the dimension of every node frame and late-copy + intersection (re-derived exactly by check_frames.py). + +Ops: ['add', piv, oth, n] a[piv] += a[oth], the addition of node n; ['fan', piv, [f...]] a[f] += a[piv] for fresh +slots f (copies); ['src', s, n] the V gate a[s] += x_S of leaf n (applied outside L, see check_word.py).""" +import gzip, json, os +from collections import Counter, defaultdict +from itertools import combinations +from math import comb + +HERE = os.path.dirname(os.path.abspath(__file__)) +ROOT = os.path.normpath(os.path.join(HERE, '..', '..')) +DATA = os.path.join(ROOT, 'certificates', 'round7') + + +def load_json(name): + with gzip.open(os.path.join(DATA, name), 'rt') as fh: + return json.load(fh) + + +def load(h=23, witness=None, data=None): + """the base witness and the design-B data of one round-seven witness (default: the files of h).""" + W = load_json(witness or 'witness_%d.json.gz' % h); D = load_json(data or 'deferred_%d.json.gz' % h) + assert W['h'] == D['h'] == h + return W, D + + +class Schedule: + """The design-B schedule in convenient form.""" + def __init__(self, W, D): + self.h = h = D['h']; self.R = D['R']; self.trip = list(combinations(range(h), 3)); self.v = len(self.trip) + self.tid = {T: i for i, T in enumerate(self.trip)} + self.args = {int(n): (tuple(a) if a else None) for n, a in W['args'].items()} + self.ops = [tuple(o[:2]) + (tuple(o[2]),) if o[0] == 'fan' else tuple(o) for o in D['ops']] + self.hold = D['hold']; self.start = D['start'] + self.out = {s: (c, tuple(T)) for s, c, T in D['out']} + self.ret = {s: c for s, c in D['ret']} + self.srcop = {o[1]: o[2] for o in self.ops if o[0] == 'src'} + self.readout = D['readout_order']; self.sel = set(self.readout) + self.sigma = dict(zip(D['readout_order'], D['sigma'])) + self.f = [0] * self.R + for s, B in self.sigma.items(): self.f[s] = len(B) + self.vstart = dict(zip(D['vleaf_slots'], D['vleaf_start'])) + self.xs = [(n, us) for n, us in D['xs_order']] + self.late_v = D['deferred_v_order'] + self.node_dims = {n: d for n, d in D['node_dims']} + self.late_dims = {(n, tuple(lt)): d for n, lt, d in D['late_dims']} + self.late = {int(n): (list(e), list(l)) for n, (e, l) in W['late'].items()} + self.pivslot = {o[3]: o[1] for o in self.ops if o[0] == 'add'} + + def touch(self, op): + if op[0] == 'add': return [op[1], op[2]] + if op[0] == 'fan': return [op[1]] + list(op[2]) + return [] + + def root_targets(self): + rt = {} + for s, (c, T) in self.out.items(): rt[s] = [self.tid[T]] + for s, c in self.ret.items(): rt[s] = [t for t, T in enumerate(self.trip) if c in T] + return rt + + def adjoint(self): + """garbage coefficients over Z: cov[s][t] = coefficient of slot s's start value in target y_T at the end of L + (the transpose sweep of L applied to the root reads); the F2 circuit uses their parities.""" + cov = [dict() for _ in range(self.R)] + for s, ts in self.root_targets().items(): + for t in ts: cov[s][t] = cov[s].get(t, 0) + 1 + def addto(d, e): + for t, x in e.items(): d[t] = d.get(t, 0) + x + for op in reversed(self.ops): + if op[0] == 'add': addto(cov[op[2]], cov[op[1]]) + elif op[0] == 'fan': + for g in op[2]: addto(cov[op[1]], cov[g]) + return cov + + # ------------------------------------------------------------------ frame keys of every slot's chain + def start_key(self, s): + """('v', s) for a V-leaf slot (start s_i), ('n', n0) for a base or early copy, ('c', n0, late suffix).""" + n0 = self.hold[s][0] + if self.args[n0] is None: return ('v', s) + st = self.start[s]; k = st[2]; dec = self.late.get(n0) + if dec is None or k in dec[0]: return ('n', n0) + lt = dec[1]; return ('c', n0, tuple(lt[lt.index(k):])) + + def chain_keys(self, s): + """frames of slot s in time order: [sigma_u | 0], start, then every later node it holds (with the pivot's + late-copy intersections after a node it pivots), its root frame if any, F.""" + ch = [('sigma', s) if s in self.sel else ('0',), self.start_key(s)] + for idx, n in enumerate(self.hold[s]): + if idx > 0: ch.append(('n', n)) + if self.args[n] is not None and self.pivslot.get(n) == s and n in self.late: + lt = self.late[n][1] + for i in range(len(lt) - 1): ch.append(('c', n, tuple(lt[i:]))) + if s in self.out: c, T = self.out[s]; ch.append(('out', c, T)) + if s in self.ret: ch.append(('ret', self.ret[s])) + ch.append(('F',)); return ch + + def dim(self, key): + k = key[0] + if k == '0': return 0 + if k == 'F': return self.h + if k in ('out', 'ret'): return self.h - 1 + if k == 'n': return self.node_dims[key[1]] + if k == 'c': return self.late_dims[key[1], key[2]] + if k == 'sigma': return len(self.sigma[key[1]]) + if k == 'v': return len(self.vstart[key[1]]) + raise KeyError(key) + + def ylevels(self, cov): + """deferred readout levels (dim sigma) on every target, from the garbage-coefficient supports.""" + lev = defaultdict(set) + for s in self.readout: + for t in cov[s]: lev[t].add(self.f[s]) + return {t: sorted(lev[t]) for t in range(self.v)} + + def xdata(self, dimof=None): + dimof = dimof or (lambda s: len(self.vstart[s])) + out = [] + for n, us in self.xs: + ds = [1] + [dimof(s) for s in us] + [self.h] + ds = [d for i, d in enumerate(ds) if i == 0 or d != ds[i - 1]] + out.append([b - a for a, b in zip(ds, ds[1:])]) + return out + + +def inner(r, h): + """children of a rank-r chain step under the inner-corner lemma (one block of width 2r-h when 2r > h).""" + return [1] * (h - r) + [2 * r - h] if 2 * r > h else [1] * r + + +def chain_ranks(S, dimof=None): + """multiset of chain-step ranks over all slots (dims must be monotone along every chain).""" + dimof = dimof or S.dim; rk = Counter() + for s in range(S.R): + ds = [dimof(k) for k in S.chain_keys(s)] + assert all(b >= a for a, b in zip(ds, ds[1:])), ('decreasing chain', s) + assert ds[-1] == S.h + for a, b in zip(ds, ds[1:]): + if b > a: rk[b - a] += 1 + return rk + + +ROUND6_CORNER = lambda h: [h - 2] + [1] * (h + 1) # data-entrance corner runs, rounds six and seven +STAIRCASE_CORNER = lambda h: [h - 2, h - 5] + [1] * 6 # staircase corner (check_stair.py) + + +def histogram(S, rk, ylev, xdata, corner=None): + """child-width histogram of the batched two-stage bit interchange with deferred readouts and V leaves. + Per stage (stage two is the complement time-reversal of stage one, same edge multiset), each of the v invocations: + every slot u: its auxiliary edge of rank m - r_u (block m - 2 r_u) and its corner of rank r_u = h - dim sigma_u, + both batched by the inner lemma, plus its chain steps; + every centre: the copy's step U_c -> 0 of rank h-1; + every target y_T: the chain 0 -> deferred levels -> t_T^perp (total rank h-1); + every leaf X_S: its chain -> V-gate frames -> F (total rank h-1). + Data: the entrance block m-4h+2 with its corner runs, one copy-correction singleton per pair.""" + h, v, R = S.h, S.v, S.R; m = h * h; N = v * v; A = v * R; H = Counter() + corner = corner or ROUND6_CORNER(h); assert sum(corner) == 2 * h - 1 + corners = Counter(h - S.f[s] for s in range(R)) + for _ in range(2): + for r, c in corners.items(): + H[m - 2 * r] += v * c + for w in inner(r, h): H[w] += v * c + for r, c in rk.items(): + for w in inner(r, h): H[w] += v * c + for _c in range(h): + for w in inner(h - 1, h): H[w] += v + for t in range(v): + ds = sorted(set([0] + ylev[t] + [h - 1])) + for a, b in zip(ds, ds[1:]): + for w in inner(b - a, h): H[w] += v + for xs in xdata: + for x in xs: + for w in inner(x, h): H[w] += v + H[m - 4 * h + 2] += 2 * N + for w in corner: H[w] += 2 * N + H[1] += N + H = {w: c for w, c in H.items() if c} + Wd = 2 * N + 2 * A; L = 2 * v * h * (h - 1); s = Wd * m - N + L + return dict(h=h, m=m, v=v, N=N, W=Wd, L=L, s=s, R=R, hist=H) + diff --git a/references/partial-gauge/pr97/deferred_23.json.gz b/references/partial-gauge/pr97/deferred_23.json.gz new file mode 100644 index 0000000000..4a4431bd42 Binary files /dev/null and b/references/partial-gauge/pr97/deferred_23.json.gz differ diff --git a/references/partial-gauge/pr97/linalg.py b/references/partial-gauge/pr97/linalg.py new file mode 100644 index 0000000000..a81cbaa61e --- /dev/null +++ b/references/partial-gauge/pr97/linalg.py @@ -0,0 +1,176 @@ +"""Stdlib exact linear algebra used by the release checks (no numpy). + +Ranks over Q are computed in one of two rigorous ways: + * modulo a prime p with a Hadamard certificate: if every row of an integer matrix has squared norm n_i and + prod_i max(1, n_i) < p^2, every minor has absolute value < p, so a minor is nonzero over Q iff it is nonzero + mod p, and the rank mod p equals the rank over Q (hadamard_ok); + * exactly, by fraction-free (Bareiss) elimination over Z (rank_exact) or Fraction elimination (null_exact). +Everything else is plain arithmetic modulo a prime.""" +from fractions import Fraction +from math import gcd + +Q31 = 2**31 - 1 # prime used for the point evaluation and frame reductions +M127 = 2**127 - 1 # Mersenne prime +M521 = 2**521 - 1 # Mersenne prime + + +def hadamard_ok(rows, p): + """True if prod of max(1, |row|^2) < p^2, i.e. every minor of the integer matrix is below p in absolute value.""" + b = 1; p2 = p * p + for r in rows: + b *= max(1, sum(x * x for x in r)) + if b >= p2: return False + return True + + +class Echelon: + """Row echelon basis mod p, built by insertion. Each stored row has a pivot (first nonzero, normalised to 1) + and zeros in the pivot columns of all earlier rows, so reducing a new row against the rows in order is exact.""" + __slots__ = ('p', 'rows', 'piv') + + def __init__(self, p): + self.p = p; self.rows = []; self.piv = [] + + def reduce(self, r): + p = self.p; r = [x % p for x in r] + for pc, b in zip(self.piv, self.rows): + c = r[pc] + if c: r = [(x - c * y) % p for x, y in zip(r, b)] + return r + + def insert(self, r): + """insert row; returns True if it was independent.""" + r = self.reduce(r) + for i, x in enumerate(r): + if x: + inv = pow(x, self.p - 2, self.p) + self.rows.append([(y * inv) % self.p for y in r]); self.piv.append(i); return True + return False + + def contains(self, r): + return not any(self.reduce(r)) + + def __len__(self): return len(self.rows) + + +def rank_mod(rows, p): + E = Echelon(p) + for r in rows: E.insert(r) + return len(E) + + +def rank_exact(rows): + """exact rank over Q of an integer matrix (Bareiss fraction-free elimination).""" + M = [list(r) for r in rows if any(r)] + if not M: return 0 + n = len(M); k = len(M[0]); r = 0; prev = 1 + for c in range(k): + i = next((i for i in range(r, n) if M[i][c]), None) + if i is None: continue + M[r], M[i] = M[i], M[r] + pr = M[r]; a = pr[c] + for t in range(r + 1, n): + row = M[t]; b = row[c] + M[t] = [(a * x - b * y) // prev for x, y in zip(row, pr)] + prev = a; r += 1 + if r == n: break + return r + + +def null_mod(A, k, p): + """basis of {x in F_p^k : A x = 0} (A: list of rows of length k).""" + rows = [[x % p for x in r] for r in A]; piv = []; R = [] + for r in rows: + for pc, b in zip(piv, R): + c = r[pc] + if c: r = [(x - c * y) % p for x, y in zip(r, b)] + i = next((i for i, x in enumerate(r) if x), None) + if i is None: continue + inv = pow(r[i], p - 2, p); r = [(y * inv) % p for y in r] + R = [[(x - b[i] * y) % p for x, y in zip(b, r)] if b[i] else b for b in R] # keep fully reduced + R.append(r); piv.append(i) + free = [c for c in range(k) if c not in piv]; out = [] + for f in free: + x = [0] * k; x[f] = 1 + for pc, b in zip(piv, R): x[pc] = (-b[f]) % p + out.append(x) + return out + + +def null_exact(A, k): + """basis (integer vectors) of {x in Q^k : A x = 0}, by Fraction reduced row echelon form.""" + R = []; piv = [] + for r0 in A: + r = [Fraction(x) for x in r0] + for pc, b in zip(piv, R): + c = r[pc] + if c: r = [x - c * y for x, y in zip(r, b)] + i = next((i for i, x in enumerate(r) if x), None) + if i is None: continue + inv = 1 / r[i]; r = [y * inv for y in r] + R = [[x - b[i] * y for x, y in zip(b, r)] if b[i] else b for b in R] + R.append(r); piv.append(i) + free = [c for c in range(k) if c not in piv]; out = [] + for f in free: + x = [Fraction(0)] * k; x[f] = Fraction(1) + for pc, b in zip(piv, R): x[pc] = -b[f] + out.append(primitive(x)) + return out + + +def primitive(x): + """scale a rational vector to a primitive integer vector.""" + den = 1 + for v in x: den = den * v.denominator // gcd(den, v.denominator) + y = [int(v * den) for v in x]; g = 0 + for v in y: g = gcd(g, v) + return [v // g for v in y] if g else y + + +def inv_mod_matrix(A, p): + """inverse of a square matrix mod p, or None if singular.""" + n = len(A); M = [[x % p for x in A[i]] + [int(i == j) for j in range(n)] for i in range(n)] + for c in range(n): + i = next((i for i in range(c, n) if M[i][c]), None) + if i is None: return None + M[c], M[i] = M[i], M[c] + inv = pow(M[c][c], p - 2, p); M[c] = [(x * inv) % p for x in M[c]] + rc = M[c] + for t in range(n): + if t != c and M[t][c]: + f = M[t][c]; M[t] = [(x - f * y) % p for x, y in zip(M[t], rc)] + return [r[n:] for r in M] + + +def matmul(A, B, p): + Bt = list(zip(*B)) + return [[sum(x * y for x, y in zip(r, c)) % p for c in Bt] for r in A] + + +def rightmost_pivots(Q, p): + """rows top to bottom: the rightmost nonzero column of the reduced row, then eliminate it below.""" + M = [list(r) for r in Q]; n = len(M); out = [-1] * n + for i in range(n): + row = M[i]; c = next((j for j in range(len(row) - 1, -1, -1) if row[j]), -1) + out[i] = c + if c < 0: continue + inv = pow(row[c], p - 2, p) + for t in range(i + 1, n): + f = M[t][c] + if f: + f = f * inv % p; M[t] = [(x - f * y) % p for x, y in zip(M[t], row)] + return out + + +def predicted(h): + """pivot column per row of the generic north-east rank function f_r of a rank-r idempotent.""" + pr = {} + for r in range(1, h + 1): + s = h - r + f = lambda i, j: 0 if i < 0 or j >= h else min(i + 1, h - j, r, max(0, i - j + 1) + s) + pc = [-1] * h + for i in range(h): + for j in range(h): + if f(i, j) - f(i - 1, j) - f(i, j + 1) + f(i - 1, j + 1) == 1: pc[i] = j + pr[r] = pc + return pr diff --git a/references/partial-gauge/pr97/round7_literal_frame_ledger.py b/references/partial-gauge/pr97/round7_literal_frame_ledger.py new file mode 100644 index 0000000000..aa0de9ed8a --- /dev/null +++ b/references/partial-gauge/pr97/round7_literal_frame_ledger.py @@ -0,0 +1,166 @@ +#!/usr/bin/env python3 +"""Explicit forward and reflected frame/event ledger of pinned round-seven. +Zhihao Chen / Codex audit. Exact nesting of the supplied chains is inherited +from the separately run frame/lifted checkers, not inferred from ranks here. +Copied-center moves are temporary macros, not erased arbitrary scratch. +""" +import sys,json,signal,time,hashlib,gzip +from pathlib import Path +from collections import defaultdict,Counter +from array import array +HERE=Path(__file__).resolve().parent;ROOT=HERE/'swapnil-round7' +sys.path.insert(0,str(ROOT/'independent/deferred-readout')) +import deferred as dr + +def main(): + started=time.monotonic() + def stop(*_):raise TimeoutError('180 second literal ledger limit') + signal.signal(signal.SIGALRM,stop);signal.alarm(180) + W,D=dr.load(23);S=dr.Schedule(W,D);v,R,h=S.v,S.R,S.h;size=2*v+R + co=S.adjoint();sel=S.sel;ops=S.ops;prev={};pred=defaultdict(list);last={} + for i,op in enumerate(ops): + for s in S.touch(op): + if s in prev:pred[i].append(prev[s]) + prev[s]=i;last[s]=i + ph=set();stack=[last[s] for s in S.ret] + while stack: + i=stack.pop() + if i in ph:continue + ph.add(i);stack.extend(pred[i]) + assert not any(set(S.touch(ops[i]))&sel for i in ph) + keys=[];ids={};dims=[] + def norm(k): + if k[0]=='out':return ('out',0,tuple(k[2])) + return k + def key(k): + k=norm(k) + if k not in ids: + ids[k]=len(keys);keys.append(k);dims.append(S.dim(k)) + return ids[k] + z,F=key(('0',)),key(('F',)) + paths=[None]*size + for n,ss in S.xs:paths[n-1]=[key(('n',n))]+[key(('v',s)) for s in ss]+[F] + y=[[] for _ in range(v)] + for s in S.readout: + for t,c in co[s].items(): + if c&1:y[t].append(key(('sigma',s))) + for t,T in enumerate(S.trip):paths[v+t]=[z]+y[t]+[key(('out',0,T))] + for s in range(R):paths[2*v+s]=[key(k) for k in S.chain_keys(s)] + assert all(p and all(dims[b]>=dims[a] for a,b in zip(p,p[1:])) for p in paths) + pos=[0]*size;current=[p[0] for p in paths];initial=current[:] + # Fixed-width event records: kind, register/destination, source-or-old, + # new-or-common, rank. kind=0 move,1 xor,2 temporary copied-center macro. + events=array('i');rank_hist={k:Counter() for k in ('aux','data','center')} + bits=[1<=0 + emit(0,s,a,b,r);rank_hist['aux' if s>=2*v else 'data'][r]+=1 + pos[s]+=1;current[s]=b + def xor(t,s): + assert current[t]==current[s],('gate frames',t,s,keys[current[t]],keys[current[s]]) + emit(1,t,s,current[t]);bits[t]^=bits[s] + def readout(s): + aux=2*v+s;f=key(('sigma',s)) if s in sel else z + assert current[aux]==f + for t,c in sorted(co[s].items()): + if c&1:move(v+t,f);xor(v+t,aux) + def source(s,active): + frame=key(('v',s)) if active else F + move(2*v+s,frame);move(S.srcop[s]-1,frame);xor(2*v+s,S.srcop[s]-1) + reordered_fans=[] + def operation(i,active): + op=ops[i] + if op[0]=='add': + _,p,o,n=op;frame=key(('n',n)) if active else F + move(2*v+p,frame);move(2*v+o,frame);xor(2*v+p,2*v+o) + elif op[0]=='fan': + # Scalar fan targets commute. The archive groups copies whose + # physical start frames differ; execute them in frame-chain order. + targets=sorted(op[2],key=lambda s:S.dim(S.start_key(s))) if active else op[2] + if active and tuple(targets)!=tuple(op[2]):reordered_fans.append(i) + for s in targets: + frame=key(S.start_key(s)) if active else F + move(2*v+op[1],frame);move(2*v+s,frame);xor(2*v+s,2*v+op[1]) + for s in range(R): + if s not in sel:readout(s) + early=sorted(set(S.srcop)-sel,key=lambda s:(S.srcop[s],len(S.vstart[s]),s)) + for s in early:source(s,True) + first=[i for i in range(len(ops)) if i in ph and ops[i][0]!='src'] + rest=[i for i in range(len(ops)) if i not in ph and ops[i][0]!='src'] + centers={} + for i in first:operation(i,True) + for s,c in S.ret.items(): + a=2*v+s;f=key(('ret',c));move(a,f) + targets=[v+t for t,T in enumerate(S.trip) if c in T] + assert all(current[t]==z for t in targets) + centers[a]=targets;emit(2,a,f,z,h-1);rank_hist['center'][h-1]+=1 + for t in targets:bits[t]^=bits[a] + for s in S.readout:readout(s) + for s in S.late_v:source(s,True) + for i in rest:operation(i,True) + for s,(c,T) in S.out.items(): + t=S.tid[T];f=key(('out',0,T));move(2*v+s,f);move(v+t,f);xor(v+t,2*v+s) + for i in reversed(first+rest):operation(i,False) + for s in S.srcop:source(s,False) + for s in range(R):move(2*v+s,F) + assert all(bits[i]==((1< 1: + values = array.array(values.typecode, values) + values.byteswap() + values.tofile(f) + + +def build(witness, prefix): + w = json.loads(gzip.decompress(Path(witness).read_bytes())) + h, v, d = w['h'], w['v'], w['central_disjoint'] + assert (h, d, w['center_denominator']) == (24, 24, 21) + triples = list(combinations(range(h), 3)) + assert v == len(triples) + args = [(0, 0)]; support = [0]; core = [0]; cover = [0]; types = [0]; ranks = [0] + for j, t in enumerate(triples): + mask = sum(1 << a for a in t) + args.append((0, 0)); support.append(1 << j); core.append(mask); cover.append(mask) + types.append(1); ranks.append(1) + flat = w['args'] + assert len(flat) % 2 == 0 + for k in range(0, len(flat), 2): + a, b, x = flat[k], flat[k+1], len(args) + assert 0 < a < x and 0 < b < x, 'Operands must precede their sum' + assert not support[a] & support[b], 'Ordinary additions must be cancellation-free' + args.append((a, b)); support.append(support[a] | support[b]) + core.append(core[a] & core[b]); cover.append(cover[a] | cover[b]) + if core[x].bit_count() >= 2: + types.append(1); ranks.append(support[x].bit_count()) + else: + types.append(2); ranks.append(cover[x].bit_count()) + n = len(args) + point_masks = [sum(1 << j for j, t in enumerate(triples) if a in t) for a in range(h)] + full_support = (1 << v) - 1 + roots, kind = [], [] + assert len(w['D']) == v + for t, node in zip(triples, w['D']): + assert 0 < node < n + assert support[node] == full_support & ~(point_masks[t[0]]|point_masks[t[1]]|point_masks[t[2]]) + target_mask = sum(1 << a for a in t) + if types[node] == 2: + assert not cover[node] & target_mask + else: + assert types[node] == 1 + assert not support[node] & (point_masks[t[0]]^point_masks[t[1]]^point_masks[t[2]]) + roots.append(node); kind.append(0) + stars = [(a, b, i) for a, b in combinations(range(h), 2) for i in range(h) if i not in (a, b)] + assert len(w['P']) == len(stars) + pair_outputs = {} + for (a, b, i), node in zip(stars, w['P']): + assert 0 < node < n + assert support[node] == point_masks[a] & point_masks[b] & ~point_masks[i] + assert types[node] == 1 + assert not support[node] & (point_masks[a]^point_masks[b]^point_masks[i]) + pair_outputs[a, b, i] = node + roots.append(node); kind.append(0) + assert len(w['A']) == d + for i, node in enumerate(w['A']): + assert 0 < node < n + assert support[node] == full_support & ~point_masks[i] + assert types[node] == 2 and cover[node] == ((1 << h)-1) ^ (1 << i) + assert ranks[node] == h-1 + roots.append(node); kind.append(1) + # Every A_i avoids i, so sum A_i = (h-3)T. The decoder is T=sum A_i/21 + # and the scatter is T-(1/2)sum_{i in target} A_i. + decoder = Fraction(1, h-3) + assert decoder == Fraction(1, 21) + scatter_coefficients = [decoder, decoder-Fraction(1, 2)] + for coefficient in scatter_coefficients: + odd_denominator = coefficient.denominator + while odd_denominator % 2 == 0: + odd_denominator //= 2 + assert 21 % odd_denominator == 0 and abs(coefficient) <= 1 + for source in triples: + assert sum(i not in source for i in range(h))*decoder == 1 + Dset = set(range(d)) + for triple in triples: + k = len(Dset.intersection(triple)) + assert 2*(3-k)-2*(d-k) == 2*(3-d) + patterns = {} + for target in triples: + chosen = len(Dset.intersection(target)) + if chosen not in patterns: + patterns[chosen] = target + for chosen, target in patterns.items(): + target_set = set(target) + for source in triples: + k = len(Dset.intersection(source)) + total_coefficient = Fraction(2*(3-k)-2*(d-k), 2*(3-d)) + scalar = Fraction(len((target_set-Dset).intersection(source)), 2) + scalar -= Fraction(len((target_set & Dset)-set(source)), 2) + scalar += Fraction(chosen-1, 2)*total_coefficient + assert scalar == Fraction(len(target_set.intersection(source))-1, 2) + for j, t in enumerate(triples): + masks = [point_masks[a] for a in t] + two = (masks[0]&masks[1]&~masks[2]) | (masks[0]&masks[2]&~masks[1]) | (masks[1]&masks[2]&~masks[0]) + actual = 0 + for a, b in combinations(t, 2): + excluded = next(i for i in t if i not in (a, b)) + actual |= support[pair_outputs[a, b, excluded]] + assert actual == two + assert masks[0]&masks[1]&masks[2] == 1 << j + assert [((k-1)+(k == 0)-(k == 2)) for k in range(4)] == [0, 0, 0, 2] + + def contained(x, y): + tx, ty = types[x], types[y] + if tx == 1 and ty == 1: return not(core[y]&~core[x] or cover[x]&~cover[y]) + if tx in (1, 2) and ty == 2: return not cover[x]&~cover[y] + return False + active = [0]*n; stack = list(roots) + while stack: + x = stack.pop() + if not x or active[x]: continue + active[x] = 1 + if args[x][0]: stack.extend(args[x]) + degree = [0]*n + for x in range(1, n): + if active[x]: + if types[x] == 1: + assert core[x].bit_count() >= 2 and ranks[x] == support[x].bit_count() + else: + assert types[x] == 2 and ranks[x] == cover[x].bit_count() + if active[x] and args[x][0]: + for y in args[x]: + assert y < x and contained(y, x), 'Binary frame nesting failed' + degree[y] += 1 + for x in roots: degree[x] += 1 + H = [0]*(h+1); c = 0; loss = 0 + for x in range(1, n): + if not active[x]: continue + r = ranks[x] + if args[x][0]: + c += 1; H[r] += degree[x]-1; H[h-r] += 1 + for y in args[x]: + assert r >= ranks[y] + H[r-ranks[y]] += 1 + else: H[1] += degree[x] + for x, k in zip(roots, kind): + r = ranks[x] + if k: H[r] += 1; H[h] += 1; loss += r + else: H[h-1-r] += 1; H[1] += 1 + q = len(roots); R = c+q + result = dict(h=h, v=v, c=c, q=q, R=R, loss=loss, central_disjoint=d, histogram=H, + rank_sum=sum(r*m for r, m in enumerate(H))) + assert loss == h*(h-1) and result['rank_sum'] == h*R+2*loss + result['center_denominator'] = 21 + result['scalar_validation'] = dict(ordinary_supports_exact=True, centers_exact=True, + center_decoder='sum A_i / 21', scatter_coefficients=list(map(str, scatter_coefficients)), + rational_scalar_identity_exact=True, mixed_center_scatter_exact=True, binary_frames_nested=True, + binary_frames_nondegenerate=True) + with open(str(prefix)+'.bin', 'wb') as f: + f.write(struct.pack('<4I', h, v, n, q)) + write_array(f, array.array('I', (a for pair in args for a in pair))) + write_array(f, array.array('Q', core)); write_array(f, array.array('Q', cover)) + write_array(f, array.array('I', roots)); write_array(f, array.array('I', kind)) + write_array(f, array.array('B', active)) + with open(str(prefix)+'.labels', 'wb') as f: + write_array(f, array.array('I', ranks)); write_array(f, array.array('B', types)) + Path(str(prefix)+'.json').write_text(json.dumps(result, indent=2)+'\n') + return result diff --git a/research/joint-dual/SOURCE.json b/research/joint-dual/SOURCE.json index 80c0f29dc4..32cda42b87 100644 --- a/research/joint-dual/SOURCE.json +++ b/research/joint-dual/SOURCE.json @@ -3,9 +3,9 @@ "base_pr57_commit": "cd350f76c9bc01489ec83568bded532cb69be938", "base_pr58_commit": "bc2f7ed4c20dc18898305ab17165c0c995cbb804", "files": { - "Makefile": "97620e0205736d9c0fce736e8ac6ab67f8ed0e490c381f6cac152cc6a003dca7", - "NOTICE": "1c53970fb4face631a1b331ec66d33111eccd280a129f9b64c2da6df96861c3f", - "README.md": "fd92de31aece76b1f5ff897e6e0dbc38dd3bf67c140579bbcb03a1a11a792668", + "Makefile": "c31b29b2e8ee06eec9bac96489787800123523a3986dd066f6dbc6f63504dda2", + "NOTICE": "e891051aa7858815850030c9531a88794ba0f39436fbd03cc0f2607a76da8b74", + "README.md": "b09d3d0a9bbddd2c93b8ebf650963ed911f661bc88db18a824331923f2af3b74", "certificates/joint-dual-compiler.json": "516c4d85158553dabd0a98216a064b2a7397957a9d5dd0df1d6e5aec53a4e805", "certificates/joint-dual-profiles-23.json": "159fa8fee175fc1d24f63e72d3c2f1dc9f4fcd78b938c5e7d2f735fd0fd37209", "certificates/joint-dual-profiles-25.json": "8d190ce42d097e828c5756918c52f2963df576cf4c6600b0fc7f2b7fdb6ad223", diff --git a/research/paired-cover/PROOF.md b/research/paired-cover/PROOF.md new file mode 100644 index 0000000000..eca07591bc --- /dev/null +++ b/research/paired-cover/PROOF.md @@ -0,0 +1,93 @@ +# Paired lockstep vertices in the first stage of a three-stage cover + +## 1. Claim + +Under every interface retained by PR #130 (its three-stage Cayley cover, general Clifford frames, weighted bit +interface, borrowed rows, the semantic/bulk assembly and the inherited analytic and fixed-tape hypotheses), + +κ = 1708101/5000000000 = 3.416202 × 10^-4, + +which is +8.59% over PR #130 (3.146011 × 10^-4) and +2.83% over PR #131 (3.32205398986 × 10^-4). The complex side +binds. The next κ grid point at denominator 10^10 is rejected. + +| per pair of first-stage vertices | PR #130 | here | +|---|---:|---:| +| persistent streams | 2(2v + 3R) = 180,326 | 4v + 5R = 151,621 | +| complex saving | 3.147997e-4 | 3.418543e-4 | +| stopped bit saving | 3.731650e-4 | unchanged | +| κ | 3.146011e-4 | 3.416202e-4 | + +The new ingredients are a pairing of first-stage vertices that share their auxiliary streams, and the lockstep +execution of each pair. No local word, frame, data stream, router, partition or bit construction changes. + +## 2. Setting (PR #130) + +E = A ⊥ B ⊥ C with dim A = h = 24, dim B = dim C = 23 and m = 70. Every vertex k of G = O(E) runs PR #117's local word +in every stage; its active space is kA (for stage one, kA = g(B + q_S) for the data positions it serves, by +R_{12,S}). The offsets are 0, kB and k(B + C) in stages one, two and three. Each stage has its own auxiliary bank of +R roles per vertex, and a role with entrance gauge σ pays an exterior of rank m − h + dim σ (PR #130's dirty-tail +lemma). + +## 3. The pairing + +Let κ be the coordinate involution of E that exchanges e_i and e_{24+i} for i < 24. It permutes coordinates, so it +is orthogonal, κ² = 1 and κA ⊂ B + C is orthogonal to A. For every k, kA and (kκ)A are orthogonal, and k ↦ kκ is a +perfect matching of the first-stage invocation set G without fixed points. + +In stage one the offset is 0, so a core touches only its active space. Give the paired cores k and kκ one common +stream for each role u. Each core acts block-diagonally on its own coordinate directions and accepts every state on +the other directions as spectators, exactly as in PR #128's completed-core sharing; the data and target streams of +each core are its own and unchanged. + +Stages two and three carry the offsets kB and k(B + C): a core there occupies kA + kB (47 dimensions) or all of E, +so no orthogonal partner fits, and those banks stay as in PR #130. + +## 4. Lockstep and the merged children + +The paired cores run the same first-stage word, so they can execute it together, gate by gate. At every time the +common stream's frame is U_1 ⊕ U_2 with U_1 ⊆ kA and U_2 ⊆ kκA the two cores' current frames. A joint step from +U_1 ⊕ U_2 to U_1′ ⊕ U_2′ is a transition between nested binary subspaces of E, so by PR #130's general Clifford +lemma ("for U ⊆ V, the exact transition T_V T_U^{-1} has rank dim V − dim U", with two rank-zero adapters around +one C^{⊗r} factor) it is a single child of width (dim U_1′ − dim U_1) + (dim U_2′ − dim U_2). The lemma holds for +degenerate subspaces as well. Run one after the other, the same operator would be two children. + +At the end of the joint word, the dirty tail of the shared stream has rank m − 2h + 2 dim σ_u (PR #130's tail +formula applied to the direct sum of the two cores' frames and gauges, which lie in orthogonal coordinate +subspaces). So per pair of vertices, the first-stage bank contributes R − Σs_a exteriors of width m − 2h, +s_a exteriors of width m − 2h + 2a, and its internal children at twice their widths; stages two and three and all +data histograms contribute exactly PR #130's amounts for two vertices. + +## 5. Finite profile and checks (`paired_cover.py`) + +* Inputs pinned by SHA-256: PR #130's `certificates/three-stage-cover-complex-input.json` and + `certificates/three-stage-cover-network.json`. +* The unpaired per-vertex profile rebuilt from the input equals PR #130's certified per-vertex children. +* κ is checked to be an orthogonal involution with κA ⊆ B + C. +* The paired profile telescopes: W·m − s = 2(2v − 3ℓ) per pair, the same deficit per vertex. Every child is below m; + the largest is 68. +* The complex saving is the largest on the 10^-10 grid under PR #104's exact moment (next point rejected). +* The bit saving is PR #130's certified stopped saving, unchanged. +* The finite bridge is PR #130's construction with the paired stream count (group order, local and router charges, + semantic induction, row stock 160000); the 47 strict constraints and 7 margins of the assembly hold at κ, and the + next κ grid point is rejected. + +Not machine-checked here: the lockstep schedule and the shared tail (§3–§4) are stated as proofs, not replayed. + +## 6. Scope + +All of PR #130's interfaces remain assumptions, including the opposite-bank factorization and the common +generic basis for the enlarged finite family of transitions (the merged first-stage transitions are among them). +The same lockstep principle applies wherever identical cores share streams on orthogonal subspaces: on PR #129's +completed-core groups (11 or 24 cores per stream) it raises κ from 1.29186e-4 to 2.05734e-4. No global optimality is +claimed. + +## 7. Credits + +* **icekylinx**: the three-stage Cayley cover, general Clifford frames and weighted bit interface (PR #130), and + the stopped product-ring framework (PR #104). +* **an664**: completed-core sharing (PR #128); **eumemic**: PR #117's local DAG, PR #129 and PR #131. +* **Zhihao Chen (jacklightChen)**: PR #97's bit ledger; **Swapnil Jain**: the round-seven bit witness. +* Everyone credited in PR #130's NOTICE and SOURCES.json. + +The pairing, the lockstep schedule and this certificate were prepared by Avi Eisenberg (ikeboy) with Anthropic +Claude assistance. diff --git a/research/paired-cover/README.md b/research/paired-cover/README.md new file mode 100644 index 0000000000..feac29e253 --- /dev/null +++ b/research/paired-cover/README.md @@ -0,0 +1,23 @@ +# Paired lockstep vertices in a three-stage cover + +Conditional κ = 1708101/5000000000 = **3.416202 × 10^-4** (+8.59% over PR #130, +2.83% over PR #131). + +In PR #130's first stage a core occupies only its 24-dimensional active space kA of the 70-dimensional cover space. +The coordinate involution κ (e_i ↔ e_{24+i}, i < 24) makes kA and (kκ)A orthogonal, so the first-stage vertices k +and kκ can share every auxiliary stream. Run in lockstep, each joint frame step is one nested transition of rank +d_1 + d_2, hence one child under PR #130's general Clifford lemma, and the shared tail has rank m − 2h + 2 dim σ. +Stages two and three, all data streams and the bit side are PR #130's, unchanged. + +See [PROOF.md](PROOF.md). + +## Reproduce + +From the repository root (standard-library Python 3; seconds): + +```sh +python3 research/paired-cover/paired_cover.py +``` + +It checks the pinned PR #130 inputs, rebuilds PR #130's unpaired per-vertex profile exactly, verifies the involution, +builds the paired profile, its exact moment, the finite bridge and the 47-constraint assembly, and writes +`certificate.json`. diff --git a/research/paired-cover/certificate.json b/research/paired-cover/certificate.json new file mode 100644 index 0000000000..fb83b68189 --- /dev/null +++ b/research/paired-cover/certificate.json @@ -0,0 +1,202 @@ +{ + "assembly": { + "absorption_gap": "19881701709549640908928075283157/250170926926508946046814557092715000000000", + "margins": { + "Gaussian_arithmetic": "20511381044331984923370790637088581457/40027348308241431367490329134834400000000", + "bulk_exposure": "3418538581457/10000000000000000", + "compact_phase_layer": "17092692394504216199988581457/50034185385301789209362911418543", + "coordinate_movement": "3418538581457/10000000000000000", + "dimension": "49999999500000000000000000000000/50034185385301789209362911418543", + "original_prefix": "1709319273635806921788067508611418543/5003418538530178920936291141854300000000", + "scalar_work": "27348658207245982188539925471488581457/40027348308241431367490329134834400000000" + }, + "minimum_margin": "17092692394504216199988581457/50034185385301789209362911418543", + "parameters": { + "C0": 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+ } + }, + "kappa": "1708101/5000000000", + "status": "Conditional witness: paired lockstep first-stage vertices on PR130 three-stage complex cover" +} diff --git a/research/paired-cover/paired_cover.py b/research/paired-cover/paired_cover.py new file mode 100644 index 0000000000..170dfd31dc --- /dev/null +++ b/research/paired-cover/paired_cover.py @@ -0,0 +1,166 @@ +#!/usr/bin/env python3 +"""Paired lockstep vertices in the first stage of PR130's three-stage complex cover. + +PR130 runs PR117's h = 24 local word at every vertex k of G = O(E), dim E = 70, inside the active space kA +(dim A = 24), with an independent auxiliary bank of R roles per stage; every role pays an exterior of width +m - h + dim(sigma). Let kappa be the coordinate involution of E exchanging e_i and e_(24+i) for i < 24. It is +orthogonal and maps A into B + C, so kA and (k kappa)A are orthogonal for every k. In the first stage the offset is +0, so a core occupies only kA. Pair the first-stage vertices k and k kappa: the paired cores share every first-stage +role stream (exterior m - 2h + 2 dim(sigma)) and run in lockstep, so each frame step of a shared stream is one nested +transition of rank 2d (PR130's general Clifford lemma: one child of width 2d). The second and third stages carry the +offsets kB and k(B + C), which leave no orthogonal room for a partner; they stay unpaired. Data streams, routing and +the bit side are PR130's, unchanged. + +Inputs (pinned by SHA-256): PR130's certificates/three-stage-cover-complex-input.json (local inventory) and +certificates/three-stage-cover-network.json (its certified bit saving and per-vertex complex children, used to check +that the unpaired profile rebuilt here is PR130's). Writes certificate.json next to this file. +Usage: python3 research/paired-cover/paired_cover.py +Prepared by Avi Eisenberg (ikeboy) with Anthropic Claude assistance. Apache-2.0. +""" +import hashlib +import json +import sys +from collections import Counter +from fractions import Fraction as Q +from math import comb, prod +from pathlib import Path + +if hasattr(sys, 'set_int_max_str_digits'): + sys.set_int_max_str_digits(0) +HERE = Path(__file__).resolve().parent +ROOT = HERE.parents[1] +sys.path.insert(0, str(ROOT / 'scripts')) +from partial_swap_network import moment # noqa: E402 +from structured_bulk_assembly import assembly, halving, js # noqa: E402 + +PINS = {'three-stage-cover-complex-input.json': 'afecf8ce2d8eb259074e61cd0c386b464ca2f57c61b00979eac808fa26e0b770', + 'three-stage-cover-network.json': 'bfca1f225394c8b50f3360b71920652baf9bd885ce84f9c0a9184c3ba7dd35ab'} +GRID = 10**10 +ATOM = Q(1, 1000) # PR104/PR130 stopping exponent +OLD = Q(384599, 10**10) # retained ordinary leaf saving +PHASE_STOP = Q(1, 10**6) # beta + + +def require(ok, msg): + if not ok: + raise SystemExit('FAIL: ' + msg) + + +def read(name): + raw = (ROOT / 'certificates' / name).read_bytes() + require(hashlib.sha256(raw).hexdigest() == PINS[name], 'pinned hash of ' + name) + return json.loads(raw) + + +def largest(m, W, z, lo, hi): + def ok(a): + try: + moment(m, W, z, a, True); return True + except BaseException: + return False + a, b = int(lo * GRID), int(hi * GRID) + require(ok(Q(a, GRID)) and not ok(Q(b, GRID)), 'moment bracket') + while b - a > 1: + mid = (a + b) // 2 + a, b = (mid, b) if ok(Q(mid, GRID)) else (a, mid) + return Q(a, GRID), moment(m, W, z, Q(a, GRID), True) + + +def main(): + row = read('three-stage-cover-complex-input.json') + net = read('three-stage-cover-network.json') + h, v, R, ell = row['h'], row['v'], row['R'], row['loss'] + require((h, v, R, ell) == (24, comb(24, 3), 28705, 552) and R == row['c'] + row['q'] - row['matched'], 'local word') + m = 3 * h - 2 + sel = {int(a): c for a, c in row['selected_rank_histogram'].items()}; S = sum(sel.values()) + require(S == row['selected_roles'] == 4599, 'partial gauges') + H = Counter({r: c for r, c in enumerate(row['remaining_internal_histogram']) if r and c}) + data = Counter() + for key in ('source_data_histogram', 'target_data_histogram'): + for r, c in row[key].items(): data[int(r)] += c + + # PR130's unpaired per-vertex profile, rebuilt and compared with its certificate + z1 = Counter({m - h: 3 * (R - S)}) + for a, c in sel.items(): z1[m - h + a] += 3 * c + for r, c in (H + data).items(): z1[r] += 3 * c + require({str(t): c for t, c in sorted(z1.items())} == {str(k): c for k, c in net['complex']['per_vertex']['child_multiplicities'].items()}, + 'unpaired profile equals PR130') + + # the pairing involution: exchange e_i and e_(24+i), i < 24, on E = A + B + C (dims 24, 23, 23) + perm = list(range(m)) + for i in range(h): perm[i], perm[h + i] = h + i, i + require(all(perm[perm[x]] == x for x in range(m)), 'involution') + require(all(perm[i] >= h for i in range(h)), 'kappa A lies in B + C, orthogonal to A') + + # per pair of vertices: stage 1 shared and in lockstep, stages 2 and 3 as in PR130 (two banks each) + z = Counter({m - 2 * h: R - S}) + for a, c in sel.items(): z[m - 2 * h + 2 * a] += c + for r, c in H.items(): z[2 * r] += c + z[m - h] += 4 * (R - S) + for a, c in sel.items(): z[m - h + a] += 4 * c + for r, c in H.items(): z[r] += 4 * c + for r, c in data.items(): z[r] += 6 * c + z = Counter({t: c for t, c in z.items() if c}) + w2 = 4 * v + 5 * R; rank2 = sum(t * c for t, c in z.items()) + require(w2 * m - rank2 == 2 * (2 * v - 3 * ell), 'paired telescoping') + require(all(0 < t < m for t in z), 'children below m') + ac, cm = largest(m, w2, z, Q(3, 10**4), Q(6, 10**4)) + + # bit side: PR130's certified stopped saving + coarse = Q(net['bit']['coarse_saving']); ab = Q(net['bit']['effective_saving']) + require(ab == (1 - ATOM) * coarse + ATOM * OLD, 'PR130 stopped bit saving') + abit = min(ab, (1 - PHASE_STOP) * ac - Q(1, GRID)) + + # finite bridge, as PR130's, with the paired stream count + n = m // 2 + vertices = 2 ** (m - 1 + (n - 1) ** 2) * prod(2 ** (2 * i) - 1 for i in range(1, n)) + require(vertices == int(net['complex']['vertices_per_stage']), 'group order') + W = vertices * w2 // 2; srank = vertices * rank2 // 2; N = vertices * v + local = 4 * (row['c'] + v) + 10 * v + 4 * h * v + 4 * h * h + 8 * h + 8 + 2 * h + local += 8 * R * v * (row['total_M_operations'] + 16) + router = 64 * (m + 1) ** 3 * W ** 2 + G = N + 3 * vertices * local + router + E = 64 * (W + m + G + 1) ** 3 + charge = 2 * G * W * W + 8 * srank + 4 * W + 4 + 32 * m + r = max(z); B = srank + E; C0 = 32 * m * B * B + require(charge < E and 2 * B * (m - r) >= srank + E and 2 * B + 18 < C0, 'finite semantic charge') + dp, wp = halving(m, r), W.bit_length() + coefficient, degree = dp * wp + 9909 + 252, 160000 + require(Q(degree) > Q(51 * coefficient, 25), 'external product row reserve') + bridge = dict(complex=dict(m=m, W=W, s=srank, N=N, maxchild=r, invocations_per_stage=vertices, stages=3, + halving_degree=dp, wire_bits=wp, scalar_group_upper=G, local_group_upper=local, + finite_group_router_upper=router, paired_vertices=True), + semantic=dict(E=E, literal_charge=charge, strict_literal_gap=E - charge, B=B, C0=C0, C1=1, + induction_gap=2 * B * (m - r) - srank - E, fixed_odd_divisor=h - 3), + rows=dict(coefficient=coefficient, complex_coefficient=dp * wp, degree=degree, suffix_slope=4 * degree, + degree_gap=Q(degree) - Q(51 * coefficient, 25)), + conservative_old_coarse_row_reserve=9909, ordinary_leaf_row_degree=252) + + def accepted(k): + try: + assembly(abit, ac, bridge, Q(k, GRID), beta=PHASE_STOP); return True + except AssertionError: + return False + lo, hi = int(Q(3, 10**4) * GRID), int(abit * GRID) + 1 + require(accepted(lo) and not accepted(hi), 'kappa bracket') + while hi - lo > 1: + mid = (lo + hi) // 2 + lo, hi = (mid, hi) if accepted(mid) else (lo, mid) + kappa = Q(lo, GRID) + res = assembly(abit, ac, bridge, kappa, beta=PHASE_STOP) + require(not accepted(lo + 1), 'next kappa grid point rejected') + require(len(res['strict_constraints']) == 47 and len(res['margins']) == 7, 'complete assembly') + res['parameters']['actual_bit_saving'] = ab + out = dict(status='Conditional witness: paired lockstep first-stage vertices on PR130 three-stage complex cover', kappa=kappa, + complex=dict(saving=ac, moment_gap=cm['strict_gap'], per_pair=dict(W=w2, rank=rank2, deficit=w2 * m - rank2, + child_multiplicities=dict(sorted(z.items()))), maxchild=r, + unpaired_saving_pr130=Q(net['complex'].get('saving', 0)) if 'saving' in net['complex'] else None), + bit=dict(coarse_saving=coarse, stopped_saving=ab, source='PR130 certificates/three-stage-cover-network.json'), + finite_bridge=bridge, assembly=res, base='PR130 head 6a9970a530119174507904e23592fd59ede19a5d') + (HERE / 'certificate.json').write_text(json.dumps(js(out), indent=2, sort_keys=True) + '\n') + print('PASS kappa = %s = %.10e' % (kappa, float(kappa))) + print('paired complex saving %s = %.6e; PR130 stopped bit saving %.6e; binding side: %s' % (ac, float(ac), float(ab), 'bit' if ab < (1 - PHASE_STOP) * ac - Q(1, GRID) else 'complex')) + + +if __name__ == '__main__': + main() diff --git a/research/shrunk-paired-cover/README.md b/research/shrunk-paired-cover/README.md new file mode 100644 index 0000000000..cf3f001525 --- /dev/null +++ b/research/shrunk-paired-cover/README.md @@ -0,0 +1,57 @@ +# Shrunk frames under paired lockstep on three-stage covers + +Conditional κ = 174051/500000000 = **3.481020 × 10^-4**: **+1.897% over PR #132** +(3.416202e-4), +4.79% over PR #131, +10.65% over PR #130. The complex side binds; the next +κ grid point at denominator 10^10 is rejected. + +PR #130 (icekylinx) runs PR #117's h = 24 local complex word at every vertex of an O(70,2) +cover, with every frame equal to its full backward intersection. PR #132 (ikeboy) pairs +first-stage vertices in lockstep. Neither changes the local frames. PR #130's general Clifford +lemma prices any nested transition U ⊆ V of binary subspaces, including degenerate ones, as +one child of width dim V − dim U. A smaller frame is therefore admissible if three conditions +hold: it contains the node's label and every predecessor frame, it lies inside the full backward +intersection, and it is nested in every successor frame. + +`lift_shrunk.py` keeps PR #130's lift, then makes one forward pass. It replaces a frame by +that minimal span when the local one-child cost t·log(m/t), with m = 70, decreases. Merging +increments a, b into a + b is cheaper because t·log(m/t) is subadditive. The pass shrinks +15,697 frames. PR #130's unrestricted source-gauge selection then selects 4,553 gauges. + +| complex local word on the cover | PR #130 frames | shrunk frames | +|---|---:|---:| +| unpaired saving (PR #130 cover) | 3.147998e-4 | 3.202924e-4 | +| paired saving (PR #132 lockstep) | 3.418543e-4 | **3.483450e-4** | +| κ | 3.416202e-4 (PR #132) | **3.481020e-4** | + +## Checks + +```sh +make shrunk-paired-cover-verify +``` + +- `producer.py` is PR #130's producer with the shrunk lift. It regenerates PR #117's pinned DAG, + the carrier matching, the physical word and the gauge selection, and requires byte-equal + agreement with the frozen `complex-input.json`. +- `verify_shrunk.py` is PR #130's complete verifier with one assertion changed: a frame must lie + inside its full backward intersection rather than equal it. Every other check passes unchanged: + exact supports and divisor 21, labels contained in frames, dependency and carrier nesting, + 236,902 replayed physical frame transitions, histogram recount, full incident source gauges, + and reverse readout chains. +- Control: `producer.py` also runs PR #130's **unmodified** verifier on the same data. It must + reject the frames, and only with `Not the full backward intersection`. Every check that precedes + that assertion in PR #130's code (supports, labels, nesting) therefore passes in the original too. +- `certificate.py` is PR #132's paired certificate on the new inventory. It checks the exact + moment, the finite bridge, all 47 strict constraints and 7 margins, and next-grid rejection. + It also checks that the shrunk unpaired saving exceeds PR #130's. + +All PR #130 and PR #132 hypotheses remain unchanged: +- PR #132's lockstep and sharing arguments are written proofs, not machine-checked. +- PR #130's cover, bit interface and the inherited analytic and tape interfaces are assumed. + +This is a finite conditional witness, not an implementation or a runtime claim. + +Credits: icekylinx (PR #130 covers, Clifford frames, producer and verifier); Avi Eisenberg / +ikeboy (PR #132 lockstep, Anthropic Claude assistance); eumemic (PR #117 DAG); an664 (PR #128 +sharing); Swapnil Jain and Zhihao Chen (bit word); all inherited contributors. The shrunk-frame +pass, which first appeared in PR #125, and this composition were prepared by Joel Pulikkan with +Anthropic Claude assistance. diff --git a/research/shrunk-paired-cover/certificate.json b/research/shrunk-paired-cover/certificate.json new file mode 100644 index 0000000000..002beeab0d --- /dev/null +++ b/research/shrunk-paired-cover/certificate.json @@ -0,0 +1,203 @@ +{ + "assembly": { + "absorption_gap": "11169030442738384241162020881/500348344546429831690340544834500000000", + "margins": { + "Gaussian_arithmetic": "418016452929041667420336619318910331/800557351274287730704544871735200000000", + "bulk_exposure": "69668910331/200000000000000", + "compact_phase_layer": "348344541204663520018910331/1000696689092859663380681089669", + "coordinate_movement": "69668910331/200000000000000", + "dimension": "999999990000000000000000000000/1000696689092859663380681089669", + "original_prefix": "34835454817155444861554413781089669/100069668909285966338068108966900000000", + "scalar_work": "557358273591041611685208354518910331/800557351274287730704544871735200000000" + }, + "minimum_margin": "348344541204663520018910331/1000696689092859663380681089669", + "parameters": { + "C0": 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+ } + }, + "kappa": "174051/500000000", + "status": "Conditional witness: shrunk frames with paired lockstep first-stage vertices on PR130 three-stage complex cover", + "unpaired_shrunk_saving": "3202923/10000000000" +} diff --git a/research/shrunk-paired-cover/certificate.py b/research/shrunk-paired-cover/certificate.py new file mode 100644 index 0000000000..83eb37eb22 --- /dev/null +++ b/research/shrunk-paired-cover/certificate.py @@ -0,0 +1,175 @@ +#!/usr/bin/env python3 +"""Shrunk frames under ikeboy's PR132 paired lockstep on PR130's three-stage cover. + +Adapted from research/paired-cover/paired_cover.py (Avi Eisenberg / ikeboy, PR132, +Apache-2.0). The only change is the local complex inventory: complex-input.json in +this package, regenerated and verified by producer.py with shrunk frames, replaces +PR130's certificates/three-stage-cover-complex-input.json. PR132's original +description follows. + +Paired lockstep vertices in the first stage of PR130's three-stage complex cover. + +PR130 runs PR117's h = 24 local word at every vertex k of G = O(E), dim E = 70, inside the active space kA +(dim A = 24), with an independent auxiliary bank of R roles per stage; every role pays an exterior of width +m - h + dim(sigma). Let kappa be the coordinate involution of E exchanging e_i and e_(24+i) for i < 24. It is +orthogonal and maps A into B + C, so kA and (k kappa)A are orthogonal for every k. In the first stage the offset is +0, so a core occupies only kA. Pair the first-stage vertices k and k kappa: the paired cores share every first-stage +role stream (exterior m - 2h + 2 dim(sigma)) and run in lockstep, so each frame step of a shared stream is one nested +transition of rank 2d (PR130's general Clifford lemma: one child of width 2d). The second and third stages carry the +offsets kB and k(B + C), which leave no orthogonal room for a partner; they stay unpaired. Data streams, routing and +the bit side are PR130's, unchanged. + +Inputs (pinned by SHA-256): PR130's certificates/three-stage-cover-complex-input.json (local inventory) and +certificates/three-stage-cover-network.json (its certified bit saving and per-vertex complex children, used to check +that the unpaired profile rebuilt here is PR130's). Writes certificate.json next to this file. +Usage: python3 research/paired-cover/paired_cover.py +Prepared by Avi Eisenberg (ikeboy) with Anthropic Claude assistance. Apache-2.0. +""" +import hashlib +import json +import sys +from collections import Counter +from fractions import Fraction as Q +from math import comb, prod +from pathlib import Path + +if hasattr(sys, 'set_int_max_str_digits'): + sys.set_int_max_str_digits(0) +HERE = Path(__file__).resolve().parent +ROOT = HERE.parents[1] +sys.path.insert(0, str(ROOT / 'scripts')) +from partial_swap_network import moment # noqa: E402 +from structured_bulk_assembly import assembly, halving, js # noqa: E402 + +PINS = {'complex-input.json': 'f428b47c820ed4c9ce4f013e1e77f6aa23b471af99ca4ede0c85d1500d14e645', + 'three-stage-cover-network.json': 'bfca1f225394c8b50f3360b71920652baf9bd885ce84f9c0a9184c3ba7dd35ab'} +GRID = 10**10 +ATOM = Q(1, 1000) # PR104/PR130 stopping exponent +OLD = Q(384599, 10**10) # retained ordinary leaf saving +PHASE_STOP = Q(1, 10**6) # beta + + +def require(ok, msg): + if not ok: + raise SystemExit('FAIL: ' + msg) + + +def read(name): + raw = ((HERE if name == 'complex-input.json' else ROOT / 'certificates') / name).read_bytes() + require(hashlib.sha256(raw).hexdigest() == PINS[name], 'pinned hash of ' + name) + return json.loads(raw) + + +def largest(m, W, z, lo, hi): + def ok(a): + try: + moment(m, W, z, a, True); return True + except BaseException: + return False + a, b = int(lo * GRID), int(hi * GRID) + require(ok(Q(a, GRID)) and not ok(Q(b, GRID)), 'moment bracket') + while b - a > 1: + mid = (a + b) // 2 + a, b = (mid, b) if ok(Q(mid, GRID)) else (a, mid) + return Q(a, GRID), moment(m, W, z, Q(a, GRID), True) + + +def main(): + row = read('complex-input.json') + net = read('three-stage-cover-network.json') + h, v, R, ell = row['h'], row['v'], row['R'], row['loss'] + require((h, v, R, ell) == (24, comb(24, 3), 28705, 552) and R == row['c'] + row['q'] - row['matched'], 'local word') + m = 3 * h - 2 + sel = {int(a): c for a, c in row['selected_rank_histogram'].items()}; S = sum(sel.values()) + require(S == row['selected_roles'] == 4553, 'partial gauges') + H = Counter({r: c for r, c in enumerate(row['remaining_internal_histogram']) if r and c}) + data = Counter() + for key in ('source_data_histogram', 'target_data_histogram'): + for r, c in row[key].items(): data[int(r)] += c + + # PR130's unpaired per-vertex profile, rebuilt and compared with its certificate + z1 = Counter({m - h: 3 * (R - S)}) + for a, c in sel.items(): z1[m - h + a] += 3 * c + for r, c in (H + data).items(): z1[r] += 3 * c + require(sum(t * c for t, c in z1.items()) == (2 * v + 3 * R) * m - (2 * v - 3 * ell), 'unpaired shrunk profile telescopes') + unpaired_ac, _ = largest(m, 2 * v + 3 * R, z1, Q(3, 10**4), Q(6, 10**4)) + require(unpaired_ac > Q(net['complex']['saving']), 'shrunk unpaired saving exceeds PR130') + + # the pairing involution: exchange e_i and e_(24+i), i < 24, on E = A + B + C (dims 24, 23, 23) + perm = list(range(m)) + for i in range(h): perm[i], perm[h + i] = h + i, i + require(all(perm[perm[x]] == x for x in range(m)), 'involution') + require(all(perm[i] >= h for i in range(h)), 'kappa A lies in B + C, orthogonal to A') + + # per pair of vertices: stage 1 shared and in lockstep, stages 2 and 3 as in PR130 (two banks each) + z = Counter({m - 2 * h: R - S}) + for a, c in sel.items(): z[m - 2 * h + 2 * a] += c + for r, c in H.items(): z[2 * r] += c + z[m - h] += 4 * (R - S) + for a, c in sel.items(): z[m - h + a] += 4 * c + for r, c in H.items(): z[r] += 4 * c + for r, c in data.items(): z[r] += 6 * c + z = Counter({t: c for t, c in z.items() if c}) + w2 = 4 * v + 5 * R; rank2 = sum(t * c for t, c in z.items()) + require(w2 * m - rank2 == 2 * (2 * v - 3 * ell), 'paired telescoping') + require(all(0 < t < m for t in z), 'children below m') + ac, cm = largest(m, w2, z, Q(3, 10**4), Q(6, 10**4)) + + # bit side: PR130's certified stopped saving + coarse = Q(net['bit']['coarse_saving']); ab = Q(net['bit']['effective_saving']) + require(ab == (1 - ATOM) * coarse + ATOM * OLD, 'PR130 stopped bit saving') + abit = min(ab, (1 - PHASE_STOP) * ac - Q(1, GRID)) + + # finite bridge, as PR130's, with the paired stream count + n = m // 2 + vertices = 2 ** (m - 1 + (n - 1) ** 2) * prod(2 ** (2 * i) - 1 for i in range(1, n)) + require(vertices == int(net['complex']['vertices_per_stage']), 'group order') + W = vertices * w2 // 2; srank = vertices * rank2 // 2; N = vertices * v + local = 4 * (row['c'] + v) + 10 * v + 4 * h * v + 4 * h * h + 8 * h + 8 + 2 * h + local += 8 * R * v * (row['total_M_operations'] + 16) + router = 64 * (m + 1) ** 3 * W ** 2 + G = N + 3 * vertices * local + router + E = 64 * (W + m + G + 1) ** 3 + charge = 2 * G * W * W + 8 * srank + 4 * W + 4 + 32 * m + r = max(z); B = srank + E; C0 = 32 * m * B * B + require(charge < E and 2 * B * (m - r) >= srank + E and 2 * B + 18 < C0, 'finite semantic charge') + dp, wp = halving(m, r), W.bit_length() + coefficient, degree = dp * wp + 9909 + 252, 160000 + require(Q(degree) > Q(51 * coefficient, 25), 'external product row reserve') + bridge = dict(complex=dict(m=m, W=W, s=srank, N=N, maxchild=r, invocations_per_stage=vertices, stages=3, + halving_degree=dp, wire_bits=wp, scalar_group_upper=G, local_group_upper=local, + finite_group_router_upper=router, paired_vertices=True), + semantic=dict(E=E, literal_charge=charge, strict_literal_gap=E - charge, B=B, C0=C0, C1=1, + induction_gap=2 * B * (m - r) - srank - E, fixed_odd_divisor=h - 3), + rows=dict(coefficient=coefficient, complex_coefficient=dp * wp, degree=degree, suffix_slope=4 * degree, + degree_gap=Q(degree) - Q(51 * coefficient, 25)), + conservative_old_coarse_row_reserve=9909, ordinary_leaf_row_degree=252) + + def accepted(k): + try: + assembly(abit, ac, bridge, Q(k, GRID), beta=PHASE_STOP); return True + except AssertionError: + return False + lo, hi = int(Q(3, 10**4) * GRID), int(abit * GRID) + 1 + require(accepted(lo) and not accepted(hi), 'kappa bracket') + while hi - lo > 1: + mid = (lo + hi) // 2 + lo, hi = (mid, hi) if accepted(mid) else (lo, mid) + kappa = Q(lo, GRID) + res = assembly(abit, ac, bridge, kappa, beta=PHASE_STOP) + require(not accepted(lo + 1), 'next kappa grid point rejected') + require(len(res['strict_constraints']) == 47 and len(res['margins']) == 7, 'complete assembly') + res['parameters']['actual_bit_saving'] = ab + out = dict(status='Conditional witness: shrunk frames with paired lockstep first-stage vertices on PR130 three-stage complex cover', unpaired_shrunk_saving=unpaired_ac, kappa=kappa, + complex=dict(saving=ac, moment_gap=cm['strict_gap'], per_pair=dict(W=w2, rank=rank2, deficit=w2 * m - rank2, + child_multiplicities=dict(sorted(z.items()))), maxchild=r, + unpaired_saving_pr130=Q(net['complex'].get('saving', 0)) if 'saving' in net['complex'] else None), + bit=dict(coarse_saving=coarse, stopped_saving=ab, source='PR130 certificates/three-stage-cover-network.json'), + finite_bridge=bridge, assembly=res, base='PR132 head f7bc7a9 on PR130 head 6a9970a530119174507904e23592fd59ede19a5d') + (HERE / 'certificate.json').write_text(json.dumps(js(out), indent=2, sort_keys=True) + '\n') + print('PASS kappa = %s = %.10e' % (kappa, float(kappa))) + print('paired complex saving %s = %.6e; PR130 stopped bit saving %.6e; binding side: %s' % (ac, float(ac), float(ab), 'bit' if ab < (1 - PHASE_STOP) * ac - Q(1, GRID) else 'complex')) + + +if __name__ == '__main__': + main() diff --git a/research/shrunk-paired-cover/complex-input.json b/research/shrunk-paired-cover/complex-input.json new file mode 100644 index 0000000000..d3c1b1c672 --- /dev/null +++ b/research/shrunk-paired-cover/complex-input.json @@ -0,0 +1,115 @@ +{ + "R": 28705, + "c": 91770, + "chronological_contract": "Nondeferred reads at0; source V; center-closed M phase1; copied centers at0; selected partial-gauge reads in reverse selection order; remaining M; side scatter; inverse M/V atfull F", + "copied_centers_already": true, + "distinct_caps": 1794, + "h": 24, + "loss": 552, + "matched": 71185, + "partial_first_transition_histogram": { + "0": 1425, + "1": 572, + "10": 178, + "11": 185, + "12": 157, + "13": 109, + "14": 4, + "2": 536, + "3": 181, + "4": 269, + "5": 190, + "6": 214, + "7": 150, + "8": 168, + "9": 215 + }, + "phase1_operations": 30251, + "q": 8120, + "remaining_internal_histogram": [ + 26984, + 87434, + 64254, + 15403, + 20936, + 10817, + 6404, + 5403, + 3899, + 2695, + 1369, + 673, + 1018, + 341, + 1119, + 185, + 256, + 93, + 791, + 522, + 195, + 96, + 183, + 24, + 0 + ], + "repaired_caps": 0, + "scalar_readout": "Exact rational C=J M; Boolean support is a safe superset only", + "selected_rank_histogram": { + "1": 214, + "10": 190, + "11": 131, + "12": 218, + "13": 57, + "14": 157, + "15": 57, + "16": 409, + "17": 147, + "18": 381, + "19": 174, + "2": 176, + "20": 794, + "21": 359, + "22": 120, + "3": 182, + "4": 124, + "5": 126, + "6": 137, + "7": 100, + "8": 187, + "9": 113 + }, + "selected_roles": 4553, + "selection": "Full unrestricted intersections and source gauges; descending first-frame rank, then reached-target count, then role index; general_clifford_subspace_frames.txt proves all required ranks", + "source_data_histogram": { + "23": 2024 + }, + "target_data_histogram": { + "1": 1880, + "10": 204, + "11": 150, + "12": 170, + "13": 70, + "14": 81, + "15": 36, + "16": 79, + "17": 14, + "18": 23, + "19": 20, + "2": 2991, + "20": 16, + "21": 2, + "22": 4, + "23": 14, + "3": 1851, + "4": 1487, + "5": 885, + "6": 693, + "7": 429, + "8": 270, + "9": 229 + }, + "total_M_operations": 118451, + "untouched_non_source_roles": 6401, + "v": 2024 +} diff --git a/research/shrunk-paired-cover/lift_shrunk.py b/research/shrunk-paired-cover/lift_shrunk.py new file mode 100644 index 0000000000..da4cb29ad1 --- /dev/null +++ b/research/shrunk-paired-cover/lift_shrunk.py @@ -0,0 +1,116 @@ +# Copyright 2026 icekylinx. Apache-2.0. +# Integration with OpenAI Codex assistance; see NOTICE. +"""Full backward intersections, then one forward pass of shrunk frames. + +Adapted from icekylinx's PR130 scripts/three_stage_cover/lift.py (Apache-2.0), +which computes every frame as its full backward intersection. PR130's general +Clifford lemma prices a nested transition U <= V by dim V - dim U for every +binary subspace, so any frame with + label(x) + sum of predecessor frames <= U_x <= full backward intersection, +nested inside every successor frame, is admissible. The added pass replaces U_x +by that minimal span when the local one-child cost t*log(m/t), m = 3h-2, of the +increments at x decreases; a merged increment a+b replaces a and b, and the cost +is subadditive. Shrink pass by Joel Pulikkan with Anthropic Claude assistance. +""" +import math +from pathlib import Path +from collections import Counter +import json +from partial_gauge.binary import load +from partial_gauge.lift import basis, perp, nondeg + +def run(path,matchfile): + h,v,n,q,args,core,cover,roots,kind,active,oldrank,types=load(path) + meta=json.loads(Path(matchfile).read_text());arcs=dict(meta['matching_arcs']);full=(1<>31: + j=use&0x7fffffff;x=roots[j];k=kind[j] + normal=(full^cover[x]) if k else (core[x]|(full^cover[x])) + direct[donor].append(normal) + else:succ[donor].append(use//2) + order=sorted((x for x in range(1,n) if active[x]),key=lambda x:(oldrank[x],x)) + place={x:i for i,x in enumerate(order)} + assert all(place[x]=oldrank[x] + dimhist[d-oldrank[x]]+=1 + # ---- shrunk frames (forward pass; successors still hold their earlier frames) + def label(x): + if not args[2*x]:return basis([core[x]]) + if types[x]==2:return basis([1<>i&1]) + return basis([core[x]|(1<>k&1]) + def within(A,B): + return len(basis(list(B)+list(A)))==len(basis(B)) + U={x:perp(ann[x],h) for x in order};preds={x:[] for x in order} + for y in order: + for t in succ[y]:preds[t].append(y) + uses=[0]*n + for x in order: + if args[2*x]:uses[args[2*x]]+=1;uses[args[2*x+1]]+=1 + for x in roots:uses[x]+=1 + cover_dim=3*h-2;cost=lambda t:t*math.log(cover_dim/t) if t>0 else 0.0 + def local(x,d): + c=(uses[x]-1)*cost(d) + if args[2*x]:c+=cost(h-d)+sum(cost(d-len(U[y])) for y in (args[2*x],args[2*x+1])) + else:c+=cost(1)+cost(d-1) + return c+sum(cost(len(U[t])-d) for t in succ[x]) + shrunk=0 + for x in order: + Lb=basis(list(label(x))+[b for p in preds[x] for b in U[p]]) + if len(Lb)>=len(U[x]) or not within(Lb,U[x]):continue + if any(not within(Lb,U[t]) for t in succ[x]):continue + if local(x,len(Lb))=nr[y];H[r-nr[y]]+=1 + else: + H[r]+=degree[x]-1;H[1]+=1;H[r-1]+=1 + loss=0 + for x,k in zip(roots,kind): + r=nr[x] + if k:H[r]+=1;H[h-r]+=1;loss+=r + else:assert r<=h-1;H[h-1-r]+=1;H[1]+=1 + for donor,use in arcs.items(): + target=roots[use&0x7fffffff] if use>>31 else use//2 + value=target if use>>31 else args[2*target+(use&1)] + ru,rv=nr[donor],nr[value] + rt=(h-1 if use>>31 else nr[target]) + # Root uses have a direct physical edge to their output frame, not + # necessarily to the value's newly lifted producer frame. + old_target_rank=nr[target] + H[h-ru]-=1;H[rv]-=1 + if use>>31: + H[h-1-rv]-=1;H[h-1-ru]+=1 + else:H[rt-rv]-=1;H[rt-ru]+=1 + assert min(H.values())>=0 + R=meta['R'];mass=sum(r*c for r,c in H.items());assert mass==h*R+loss,(mass,h*R+loss) + out={k:meta[k] for k in ['h','v','c','q','R','matched']} + out.update(loss=loss,histogram=[H[i] for i in range(h+1)],rank_sum=mass, + copied_centers_already=True,repairs=repairs,distinct_caps=len(capcache),distinct_repairs=len(repaircache), + dimension_gains=dict(dimhist),lift='Full backward intersections using generalized Lagrangian frames, then a forward shrunk-frame pass',shrunk_frames=shrunk) + return out,ann diff --git a/research/shrunk-paired-cover/producer.py b/research/shrunk-paired-cover/producer.py new file mode 100644 index 0000000000..b2b62f16db --- /dev/null +++ b/research/shrunk-paired-cover/producer.py @@ -0,0 +1,96 @@ +#!/usr/bin/env python3 +"""Regenerate and verify the shrunk-frame PR117 complex producer on PR130's cover. + +Adapted from icekylinx's PR130 scripts/three_stage_cover_producer.py (Apache-2.0): +the same pinned PR117 DAG, carrier matching, physical word and unrestricted +source-gauge selection, with lift_shrunk.py in place of the full lift and +verify_shrunk.py in place of the full-lift equality check. +Standard-library Python and C++17; generated DAGs and words are temporary. +""" +import argparse +import json +import os +from pathlib import Path +import shlex +import subprocess +import sys +import tempfile +from importlib.util import spec_from_file_location, module_from_spec +import hashlib + +HERE=Path(__file__).resolve().parent +ROOT=HERE.parents[1] +sys.path.insert(0,str(ROOT/'scripts'));sys.path.insert(0,str(HERE)) +from lift_shrunk import run as lift_run +from verify_shrunk import verify +from partial_gauge.word import run as word_run +from three_stage_cover.select import run as select_run + + +def regenerate(work, expected, compiler): + if sys.flags.optimize:raise ValueError('Run without -O; finite assertions must remain enabled') + def stage(message):print(message,file=sys.stderr,flush=True) + def save(name,data): + path=work/name;path.write_text(json.dumps(data,separators=(',',':'))+'\n');return path + stage('Replaying the pinned PR117 scalar DAG') + reference=ROOT/'references/three-stage-cover/pr117' + source=json.loads((reference/'SOURCE.json').read_text()) + assert source['commit']=='cbb05ce504d571546d9b7794c186a613c659c3bf' + for name,digest in source['files'].items(): + assert hashlib.sha256((reference/name).read_bytes()).hexdigest()==digest, 'Source binding differs: '+name + witness=reference/'dag.json.gz' + assert hashlib.sha256(witness.read_bytes()).hexdigest()=='3c034d0aae388ef567a454826f4f48b26fd8a94c71e8ffed4835271b349a783b' + spec=spec_from_file_location('pr117_replayed',reference/'replayed.py') + module=module_from_spec(spec);spec.loader.exec_module(module) + scalar=module.build(witness,work/'selected') + dag=work/'selected.bin';matcher=work/'matcher' + subprocess.run([*shlex.split(compiler),'-O3','-std=c++17',str(ROOT/'scripts/partial_gauge/match.cpp'),'-o',str(matcher)],check=True) + stage('Constructing compatible carrier matching') + matching=json.loads(subprocess.check_output([str(matcher),str(dag),str(dag.with_suffix('.labels'))],text=True)) + matchfile=save('matching.json',matching) + stage('Lifting exact binary frames backward through dependencies and carriers') + lifted,ann=lift_run(dag,matchfile);liftfile=save('lifted.json',lifted);annfile=save('annihilators.json',ann) + stage('Constructing physical M and its retained-center dependency closure') + word=word_run(dag,matchfile,annfile,liftfile);wordfile=save('word.json',word) + stage('Selecting partial source gauges with reverse readouts') + result,selection=select_run(dag,annfile,liftfile,wordfile) + result=json.loads(json.dumps(result)) + if os.environ.get('SHRUNK_WRITE')=='1':(HERE/'complex-input.json').write_text(json.dumps(result,indent=2,sort_keys=True)+'\n');expected=result + assert result==expected, 'Regenerated selected certificate differs' + assert lifted['shrunk_frames']==15697, lifted['shrunk_frames'] + save('selection.json',selection) + stage('Checking scalar coefficients, actual frames, physical word and readout chains') + checks=verify(dag,matching,ann,lifted,word,selection['selected'],result) + # Control: PR130's unmodified verifier must reject these frames, and only at + # the full-lift equality that verify_shrunk.py relaxes to containment. Every + # check that precedes it in the original code (supports, labels, nesting) + # therefore also passes there. + from three_stage_cover.verify import verify as original_verify + try: + original_verify(dag,matching,ann,lifted,word,selection['selected'],result) + except AssertionError as error: + assert str(error)=='Not the full backward intersection', repr(error) + else: + raise AssertionError('PR130 verifier accepted shrunk frames; the shrink pass did nothing') + checks['original_pr130_verifier_rejects_only_full_lift_equality']=True + return dict(source_pin_verified=True,regenerated_from_source=True,certificate_equal=True,selected=result,checks=checks) + + +def main(): + p=argparse.ArgumentParser(description=__doc__) + p.add_argument('--certificate',type=Path,default=HERE/'complex-input.json') + p.add_argument('--work-dir',type=Path) + p.add_argument('--output',type=Path) + p.add_argument('--cxx',default=os.environ.get('CXX','c++')) + a=p.parse_args();expected=json.loads(a.certificate.read_text()) + if a.work_dir: + a.work_dir.mkdir(parents=True,exist_ok=True);result=regenerate(a.work_dir.resolve(),expected,a.cxx) + else: + with tempfile.TemporaryDirectory(prefix='three-stage-cover-') as directory: + result=regenerate(Path(directory),expected,a.cxx) + encoded=json.dumps(result,indent=2)+'\n' + if a.output:a.output.write_text(encoded) + else:print(encoded,end='') + + +if __name__=='__main__':main() diff --git a/research/shrunk-paired-cover/verify_shrunk.py b/research/shrunk-paired-cover/verify_shrunk.py new file mode 100644 index 0000000000..50dc6e0cc3 --- /dev/null +++ b/research/shrunk-paired-cover/verify_shrunk.py @@ -0,0 +1,170 @@ +# Copyright 2026 icekylinx. Apache-2.0. +# Integration with OpenAI Codex assistance; see NOTICE. +# Copyright 2026 icekylinx. Apache-2.0. +# Prepared with OpenAI GPT-6 Astra and Codex assistance; see NOTICE. +"""PR130's independent checks of the DAG, frames and physical word (icekylinx, Apache-2.0). + +One assertion changes: a frame must lie inside its full backward intersection +rather than equal it. Every other check is PR130's, unchanged. +""" +from collections import Counter +from functools import lru_cache +from itertools import combinations +from partial_gauge.binary import load +from partial_gauge.lift import basis, perp, contained, nondeg + + +def verify(path, matching, ann, lift, word, selected, record): + h,v,n,q,args,core,cover,roots,kind,active,oldrank,types=load(path) + triples=list(combinations(range(h),3)); full=(1<>j&1) + else: + assert types[x]==2 + old=tuple(1<>j&1) + assert len(old)==oldrank[x] and includes(basis(old),F) + if a: + assert includes(A,tuple(ann[a])) and includes(A,tuple(ann[b])) + for j,t in enumerate(triples): + assert supports[roots[j]]==all_inputs & ~(point[t[0]]|point[t[1]]|point[t[2]]) + at=v + for a,b in combinations(range(h),2): + others=[i for i in range(h) if i not in (a,b)] + # PR117 pins pair stars in natural lexicographic order. + for i in others: + assert supports[roots[at]]==point[a]&point[b]&~point[i];at+=1 + for a in range(h): + assert kind[at] and supports[roots[at]]==all_inputs^point[a];at+=1 + assert at==q + # Every input appears in h-3 disjoint centers: exact divisor 21. For + # overlap k, twice the signed output coefficient is 2-(3-k)+[k=0]-[k=2]. + assert h-3==21 + assert [2-(3-k)+(k==0)-(k==2) for k in range(4)]==[0,0,0,2] + arcs=matching['matching_arcs'] + assert len(arcs)==matching['matched'] + assert len({a for a,b in arcs})==len(arcs)==len({b for a,b in arcs}) + assert matching['R']==matching['c']+q-len(arcs) + required=[[] for _ in range(n)] + for x in range(1,n): + if active[x] and args[2*x]: + for y in args[2*x:2*x+2]:required[y].extend(ann[x]) + for x,k in zip(roots,kind): + required[x].append(full^cover[x] if k else core[x]|(full^cover[x])) + for donor,use in arcs: + assert active[donor] and args[2*donor] + target=roots[use&0x7fffffff] if use>>31 else use//2 + value=target if use>>31 else args[2*target+(use&1)] + assert value in args[2*donor:2*donor+2] + if use>>31: + j=use&0x7fffffff + required[donor].append(full^cover[target] if kind[j] else core[target]|(full^cover[target])) + else:required[donor].extend(ann[target]) + for x in range(1,n): + # PR130 required equality (the full lift); any frame inside it is admissible. + if active[x]:assert includes(basis(required[x]),tuple(ann[x])), 'Frame outside the backward intersection' + for donor,use in arcs: + if use>>31: + j=use&0x7fffffff;x=roots[j] + normal=full^cover[x] if kind[j] else core[x]|(full^cover[x]) + assert includes(basis([normal]),tuple(ann[donor])) + else:assert includes(tuple(ann[use//2]),tuple(ann[donor])) + R=record['R']; values=[0]*R; current=[0]*R; first=[None]*R + H=Counter();phase=set(word['phase1']); touched=set(map(int,word['sources'].values())) + for x,s in word['sources'].items(): + values[s]=supports[int(x)];current[s]=1;H[1]+=1 + full_ann=tuple(1<=current[s] + # L_U = U^perp in X plus U in Z; for nested U,V, + # h-dim(L_U intersection L_V)=dim(V)-dim(U), without nondegeneracy. + assert len(old)-len(A)==r-current[s] + role_ann[s]=A;transitions+=1 + H[r-current[s]]+=1;current[s]=r;last[s]=i + assert not values[a]&values[b] + values[a]|=values[b] + assert values[a]==supports[x] + for j,(x,s) in enumerate(zip(roots,word['rootroles'])): + assert values[s]==supports[x] + if kind[j]: + assert last[s]<0 or last[s] in phase + H[current[s]]+=1 + else: + H[h-1-current[s]]+=1;current[s]=h-1 + for r in current:H[h-r]+=1 + assert all(H[r]==lift['histogram'][r] for r in range(1,h+1)) + co=[0]*R;tid={sum(1<>p&1:x^=y + if x: + p=x.bit_length()-1 + for k,y in list(b.items()): + if y>>p&1:b[k]=y^x + b[p]=x + return tuple(b[p] for p in sorted(b,reverse=True)) + +def perp(rows,h): + rows=basis(rows);piv={r.bit_length()-1:r for r in rows};out=[] + for j in range(h): + if j in piv:continue + x=1<>j&1:x|=1<

>31: + j=use&0x7fffffff;x=roots[j];k=kind[j] + normal=(full^cover[x]) if k else (core[x]|(full^cover[x])) + direct[donor].append(normal) + else:succ[donor].append(use//2) + order=sorted((x for x in range(1,n) if active[x]),key=lambda x:(oldrank[x],x)) + place={x:i for i,x in enumerate(order)} + assert all(place[x]>j&1) + elif types[x]==2:U=tuple(1<>j&1) + else:raise ValueError('Unselected center type') + key=(capann,U) + if key not in repaircache: + M=safe_subspace(C,U);repaircache[key]=perp(M,h) + A=repaircache[key];d=h-len(A);repairs+=1 + ann[x]=A;nr[x]=d;assert d>=oldrank[x] + dimhist[d-oldrank[x]]+=1 + degree=[0]*n + for x in order: + if args[2*x]: + degree[args[2*x]]+=1;degree[args[2*x+1]]+=1 + for x in roots:degree[x]+=1 + H=Counter() + for x in order: + r=nr[x] + if args[2*x]: + H[r]+=degree[x]-1;H[h-r]+=1 + for y in (args[2*x],args[2*x+1]):assert r>=nr[y];H[r-nr[y]]+=1 + else: + H[r]+=degree[x]-1;H[1]+=1;H[r-1]+=1 + loss=0 + for x,k in zip(roots,kind): + r=nr[x] + if k:H[r]+=1;H[h-r]+=1;loss+=r + else:assert r<=h-1;H[h-1-r]+=1;H[1]+=1 + for donor,use in arcs.items(): + target=roots[use&0x7fffffff] if use>>31 else use//2 + value=target if use>>31 else args[2*target+(use&1)] + ru,rv=nr[donor],nr[value] + rt=(h-1 if use>>31 else nr[target]) + # Root uses have a direct physical edge to their output frame, not + # necessarily to the value's newly lifted producer frame. + old_target_rank=nr[target] + H[h-ru]-=1;H[rv]-=1 + if use>>31: + H[h-1-rv]-=1;H[h-1-ru]+=1 + else:H[rt-rv]-=1;H[rt-ru]+=1 + assert min(H.values())>=0 + R=meta['R'];mass=sum(r*c for r,c in H.items());assert mass==h*R+loss,(mass,h*R+loss) + out={k:meta[k] for k in ['h','v','c','q','R','matched']} + out.update(loss=loss,histogram=[H[i] for i in range(h+1)],rank_sum=mass, + copied_centers_already=True,repairs=repairs,distinct_caps=len(capcache),distinct_repairs=len(repaircache), + dimension_gains=dict(dimhist),lift='Backward chosen-successor intersection; maximal nondegenerate complement containing old orthonormal frame; old matched dependencies preserved') + return out,ann diff --git a/scripts/partial_gauge/match.cpp b/scripts/partial_gauge/match.cpp new file mode 100644 index 0000000000..9275aaa029 --- /dev/null +++ b/scripts/partial_gauge/match.cpp @@ -0,0 +1,43 @@ +// Copyright 2026 icekylinx. Apache-2.0. +// OpenAI GPT-6 Astra and Codex assistance; retained matcher credits in NOTICE. +#include +#include +#include +#include +#include +#include +#include +using U=uint32_t;using V=uint64_t; +templatevoid readv(std::ifstream&f,std::vector&v){f.read((char*)v.data(),v.size()*sizeof(T));assert(f);} +int main(int argc,char**argv){assert(argc==3);std::ifstream f(argv[1],std::ios::binary);U hdr[4];f.read((char*)hdr,16);assert(f);U h=hdr[0],v=hdr[1],n=hdr[2],q=hdr[3]; +std::vector>args(n);std::vectorcore(n),cover(n);std::vectorroots(q),kind(q);std::vectoractive(n);readv(f,args);readv(f,core);readv(f,cover);readv(f,roots);readv(f,kind);readv(f,active); +std::vectorranks(n),degree(n),begin(n+1);V c=0,inputs=0,loss=0; +for(U x=1;xtypes(n);readv(labels,ranks);readv(labels,types);for(U x:roots)degree[x]++;for(U x=1;xuses(begin.back()),cursor(begin.begin(),begin.end()-1);for(U x=1;xU{return e>>31?roots[e&0x7fffffff]:e/2;}; +auto before=[&](U a,U b){U x=nd(a),y=nd(b);if(ranks[x]!=ranks[y])return ranks[x]>31?V(n)+(a&0x7fffffff):x,oy=b>>31?V(n)+(b&0x7fffffff):y;return oxdonors;for(U x=1;xleftmatch(n),rightmatch(uses.size()),distance(n,UINT32_MAX),queue;V matches=0;U phase=0,inf=UINT32_MAX,shortest=inf; +while(true){queue.clear();shortest=inf;for(U x:donors){if(!leftmatch[x]){distance[x]=0;queue.push_back(x);}else distance[x]=inf;}for(U at=0;at=shortest)continue;adjacency(x,[&](U j){U y=rightmatch[j];if(!y)shortest=distance[x]+1;else if(distance[y]==inf){distance[y]=distance[x]+1;queue.push_back(y);}return false;});}if(shortest==inf)break; +auto aug=[&](auto&&self,U x)->bool{bool ok=adjacency(x,[&](U j){U y=rightmatch[j];if((!y&&distance[x]+1==shortest)||(y&&distance[y]==distance[x]+1&&self(self,y))){leftmatch[x]=j+1;rightmatch[j]=x;return true;}return false;});if(!ok)distance[x]=inf;return ok;};V gained=0;for(U x:donors)if(!leftmatch[x]&&aug(aug,x)){matches++;gained++;}std::cerr<<"h="<hist(h+1);for(U x=1;x=ranks[y]);hist[r-ranks[y]]++;}}else hist[1]+=degree[x];} +for(U j=0;j>31?target:args[target][e&1];assert(value==args[donor][0]||value==args[donor][1]);if(value==args[donor][0])changed++;U ru=ranks[donor],rv=ranks[value],rt=ranks[target];assert(rt>=ru&&ru>=rv);hist[h-ru]--;hist[rv]--;hist[rt-rv]--;hist[rt-ru]++;} +std::vectorterminal(h+1); +for(U x=1;x=0);sum+=r*hist[r];}assert(sum==h*R+2*loss); + +std::cout<<"{\"h\":"<=2: + types.append(1);ranks.append(s.bit_count());lookup[s]=x + else: + types.append(2);ranks.append(cover[x].bit_count());lookup[s]=x + return x + def total(xs,center=None): + xs=[x for x in xs if x] + while len(xs)>1: + xs=[add(xs[i],xs[i+1],center) if i+1=0 + Y=Counter() + for ranks in target_ranks: + prev=0 + for r in list(reversed(ranks))+[h-1]:assert r>=prev;Y[r-prev]+=1;prev=r + result={k:d[k] for k in ['h','v','R','q','c','matched','loss']} + result.update(selected_roles=len(selected),selected_rank_histogram=dict(birth), + remaining_internal_histogram=[H[r] for r in range(h+1)], + source_data_histogram={h-1:v},target_data_histogram=dict(Y), + copied_centers_already=True,phase1_operations=len(word['phase1']), + total_M_operations=len(ops),untouched_non_source_roles=len(candidates), + selection='Descending first-frame rank; intersect all incident latest target frames; retain maximal nondegenerate summand', + partial_first_transition_histogram=dict(residual),distinct_caps=len(cache), + repaired_caps=repaired, + scalar_readout='Exact rational C=J M; Boolean support is a safe superset only', + chronological_contract='Nondeferred reads at0; source V; center-closed M phase1; copied centers at0; selected partial-gauge reads in reverse selection order; remaining M; side scatter; inverse M/V atfull F') + return result,dict(base_word=Path(wordfile).name,selected=selected) diff --git a/scripts/partial_gauge/verify.py b/scripts/partial_gauge/verify.py new file mode 100644 index 0000000000..a1afefe66d --- /dev/null +++ b/scripts/partial_gauge/verify.py @@ -0,0 +1,125 @@ +# Copyright 2026 icekylinx. Apache-2.0. +# Prepared with OpenAI GPT-6 Astra and Codex assistance; see NOTICE. +"""Independent checks of the generated DAG, lifted frames and physical word.""" +from collections import Counter +from functools import lru_cache +from itertools import combinations +from .binary import load +from .lift import basis, perp, contained, nondeg + + +def verify(path, matching, ann, lift, word, selected, record): + h,v,n,q,args,core,cover,roots,kind,active,oldrank,types=load(path) + triples=list(combinations(range(h),3)); full=(1<>j&1) + else: + assert types[x]==2 + old=tuple(1<>j&1) + assert len(old)==oldrank[x] and includes(basis(old),F) + if a: + assert includes(A,tuple(ann[a])) and includes(A,tuple(ann[b])) + for j,t in enumerate(triples): + assert supports[roots[j]]==all_inputs & ~(point[t[0]]|point[t[1]]|point[t[2]]) + at=v + for a,b in combinations(range(h),2): + others=[i for i in range(h) if i not in (a,b)] + others.sort(key=lambda i:((i^1) in (a,b),i)) + for i in others: + assert supports[roots[at]]==point[a]&point[b]&~point[i];at+=1 + for a in range(h): + assert kind[at] and supports[roots[at]]==all_inputs^point[a];at+=1 + assert at==q + # Every input appears in h-3 disjoint centers: exact divisor 21. For + # overlap k, twice the signed output coefficient is 2-(3-k)+[k=0]-[k=2]. + assert h-3==21 + assert [2-(3-k)+(k==0)-(k==2) for k in range(4)]==[0,0,0,2] + for donor,use in matching['matching_arcs']: + if use>>31: + j=use&0x7fffffff;x=roots[j] + normal=full^cover[x] if kind[j] else core[x]|(full^cover[x]) + assert includes(basis([normal]),tuple(ann[donor])) + else:assert includes(tuple(ann[use//2]),tuple(ann[donor])) + R=record['R']; values=[0]*R; current=[0]*R; first=[None]*R + H=Counter();phase=set(word['phase1']); touched=set(map(int,word['sources'].values())) + for x,s in word['sources'].items(): + values[s]=supports[int(x)];current[s]=1;H[1]+=1 + last=[-1]*R + for i,(a,b,x) in enumerate(word['ops']): + assert a!=b + if i in phase: + assert all(last[s]<0 or last[s] in phase for s in (a,b)) + touched.update((a,b)) + for s in (a,b): + if first[s] is None:first[s]=x + r=h-len(ann[x]);assert r>=current[s] + H[r-current[s]]+=1;current[s]=r;last[s]=i + assert not values[a]&values[b] + values[a]|=values[b] + assert values[a]==supports[x] + for j,(x,s) in enumerate(zip(roots,word['rootroles'])): + assert values[s]==supports[x] + if kind[j]: + assert last[s]<0 or last[s] in phase + H[current[s]]+=1 + else: + H[h-1-current[s]]+=1;current[s]=h-1 + for r in current:H[h-r]+=1 + assert all(H[r]==lift['histogram'][r] for r in range(1,h+1)) + co=[0]*R;tid={sum(1<>31 else args[2*(u//2)+(u&1)] + if value==aa:p,o=o,p + else:assert value==bb + assert u not in assign;assign[u]=o + ops.append((p,o,x)) + else:p=R;R+=1;sources[x]=p + free=[u for u in uses[x] if u not in incoming] + assert free + for j,u in enumerate(free): + assert u not in assign + if j==0:assign[u]=p + else: + s=R;R+=1;assign[u]=s;ops.append((s,p,x)) + assert R==mt['R'],(R,mt['R']) + rootroles=[assign[(1<<31)|j] for j in range(q)] + prev=[-1]*R;pred=[];first=[None]*R + for i,(a,b,x) in enumerate(ops): + pred.append((prev[a],prev[b]));prev[a]=prev[b]=i + if first[a] is None:first[a]=x + if first[b] is None:first[b]=x + centerroles=[rootroles[j] for j in range(q) if kind[j]] + stack=[prev[s] for s in centerroles if prev[s]>=0];phase=set() + while stack: + i=stack.pop() + if i in phase:continue + phase.add(i);stack.extend(p for p in pred[i] if p>=0) + touched=set(sources.values()) + for i in phase:touched.update(ops[i][:2]) + triples=list(combinations(range(h),3));tid={sum(1<= 0, 'Target physical path retreats') + if increment: + yhist[increment] += 1 + current[target] = rank + for rank in current: + require(rank <= h-1, 'Target frame exceeds its output hyperplane') + if rank < h-1: + yhist[h-1-rank] += 1 + local = dict(aux=aux, data=xhist+yhist, center=Counter({h-1: len(schedule.ret)})) + require(sum(r*n for r,n in local['data'].items()) == 2*v*(h-1), + 'Physical data path rank mass mismatch') + exits = Counter(h-rank for rank in schedule.f) + require(all(0 < rank <= h for rank in exits), 'Invalid physical exit nullity') + require(sum(exits.values()) == roles, 'Missing physical exit') + return h, v, roles, local, exits + + +def reconstruct(input_path=INPUT): + record = json.loads(Path(input_path).read_text()) + folder, sources = frozen_sources(record) + ledger = json.loads((folder/'bit-ledger-result.json').read_text()) + for flag in ('complete_forward_F2', 'complete_reflected_F2', + 'reflected_frame_continuity', 'all_scalar_gates_equal_frame_keys', + 'histogram_matches_external'): + require(ledger[flag] is True, 'Pinned physical ledger lacks '+flag) + h, v, roles, local, exits = local_ranks(folder) + require((ledger['h'], ledger['roles'], ledger['formal_basis']) == (h, roles, 2*v+roles), + 'Physical ledger dimensions differ') + require({kind: encoded(hist) for kind,hist in local.items()} == ledger['rank_histograms'], + 'Reconstructed physical ranks differ from the pinned reflected ledger') + m, N = h*h, v*v + W = 2*N+2*v*roles + whole = Counter() + for hist in local.values(): + for rank, count in hist.items(): + whole[rank] += 2*v*count + # Each role's exterior I-(I-sigma) tensor Q has rank m-(h-dim sigma). + for nullity, count in exits.items(): + whole[m-nullity] += 2*v*count + # The original entrance projector is one rank-(h-1)^2 edge per bank pair. + whole[(h-1)**2] += 2*N + whole[1] += N # physical copy correction + mass = sum(rank*count for rank,count in whole.items()) + require(mass == ledger['recursive_rank'], 'Whole-projector rank mass changed') + require(all(0 < rank < m and count > 0 for rank,count in whole.items()), + 'Invalid whole-projector child') + result = dict(dimensions=[h,h], m=m, v=v, N=N, R=roles, W=W, + rank_mass=mass, deficit=W*m-mass, max_child=max(whole), + local_rank_histograms={kind: encoded(hist) for kind,hist in local.items()}, + exit_nullity_histogram=encoded(exits), + whole_projector_histogram=encoded(whole)) + for key, value in result.items(): + require(value == record[key], 'Round-six expected profile differs: '+key) + result['checks'] = dict(source_hashes_equal=True, physical_local_ranks_reconstructed=True, + physical_ledger_ranks_equal=True, whole_profile_equal=True, + full_upstream_audits_repeated=False) + result['provenance'] = dict(pr97=sources['pr97'], underlying=sources['underlying'], + source_manifest=record['source_manifest'], + source_manifest_sha256=record['source_manifest_sha256']) + result['scope'] = ('Incremental rank accounting from pinned physical chains and ledger; ' + 'scalar complete-basis replay and rational geometry remain inherited checks.') + return result + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument('--input', type=Path, default=INPUT) + parser.add_argument('--output', type=Path) + args = parser.parse_args() + result = reconstruct(args.input) + text = json.dumps(result, indent=2, sort_keys=True)+'\n' + if args.output: + args.output.write_text(text) + else: + print(text, end='') + + +if __name__ == '__main__': + main() diff --git a/scripts/partial_gauge_network.py b/scripts/partial_gauge_network.py new file mode 100644 index 0000000000..53b7e06b96 --- /dev/null +++ b/scripts/partial_gauge_network.py @@ -0,0 +1,179 @@ +#!/usr/bin/env python3 +"""Exact deferred-word moments and partial-gauge semantic assembly. + +Copyright 2026 icekylinx. Apache-2.0. +Substantial OpenAI GPT-6 Astra and Codex assistance. +PR97 physical bit words: Zhihao Chen, based on Swapnil Jain's witness. +Semantic/bulk assembly: PR23 (Zhihao Chen) and RaD; see SOURCES.json. +""" +import argparse +from collections import Counter +from fractions import Fraction as Q +from hashlib import sha256 +from math import comb +from pathlib import Path +import json +import sys + +from certify import require +from copied_centers_network import grid_upper +from partial_swap_network import moment +from partial_gauge_bit import reconstruct +from structured_bulk_assembly import assembly, halving, js + +ROOT = Path(__file__).resolve().parents[1] +COARSE = Q(15513, 125000000) +ATOM = Q(1, 1000) +OLD = Q(384599, 10**10) +AB = (1-ATOM)*COARSE+ATOM*OLD +AC = Q(92651, 10**9) +PHASE_STOP = Q(1, 10**6) +ASSEMBLY_BIT = min(AB, (1-PHASE_STOP)*AC-Q(1, 10**10)) +KAPPA = Q(7237, 78125000) + + +def coarse_moment(p): + def log_upper(x): + power = 0 + while x >= 2: + x /= 2 + power += 1 + def small(y): + z = (y-1)/(y+1) + return 2*sum((z**(2*j+1)/(2*j+1) for j in range(24)), Q(0)) + 2*z**49/(49*(1-z*z)) + return grid_upper(power*small(Q(2))+small(x)) + total, logs = Q(0), {} + for t, n in p['child_multiplicities'].items(): + require(0 < t < p['m'] and n > 0, 'Shrinking positive bit children') + logs[t] = log_upper(Q(p['m'],t)) + u = COARSE*logs[t] + require(0 < u < 3, 'Coarse exponential enclosure') + total += n*t*grid_upper(1+u+u*u/(2*(1-u/3))) + upper = total/(p['W']*p['m']) + mass = Q(p['total_rank'],p['W']*p['m']) + require(mass < 1 and upper < 1, 'Both stopped-recurrence moments') + require(0 < OLD <= COARSE < ATOM < 1 and ATOM > AB, 'Stopped-adapter exponents') + return dict(saving=COARSE, exponent=1-COARSE, moment_upper=upper, + strict_gap=1-upper, rank_moment=mass, logarithm_upper_bounds=logs, + rounding_denominator=1 << 100, + stopped_adapter=dict(atom_exponent=ATOM, base_ordinary_saving=OLD, + effective_saving=AB, coarse_toll_exponent=1-ATOM, + coarse_toll_subordinate_gap=ATOM-COARSE)) + + +def complex_profile(row): + h, v, R, ell = (row[k] for k in ('h','v','R','loss')) + require((h,v,ell) == (24,comb(24,3),24*23), 'Complex dimensions') + require(R == row['c']+row['q']-row['matched'], 'Compatible role count') + selected = {int(r):n for r,n in row['selected_rank_histogram'].items()} + require(sum(selected.values()) == row['selected_roles'] == 6341, 'Selected partial gauges') + require(row['copied_centers_already'], 'Copied-center convention') + require(sum(row['partial_first_transition_histogram'].values()) == sum(selected.values()), 'Charged first residuals') + m, N, W = h*h, v*v, 2*v*v+2*v*R + C = Counter({m-h:2*v*(R-sum(selected.values())), (h-1)**2:2*N, 1:N}) + for r,n in selected.items(): + require(0 < r < h and n > 0, 'Nonzero shrinking exterior gauges') + C[m-h+r] += 2*v*n + for name in ('source_data_histogram','target_data_histogram'): + for r,n in row[name].items(): + C[int(r)] += 2*v*n + for r,n in enumerate(row['remaining_internal_histogram']): + if r: + C[r] += 2*v*n + C = dict(sorted((r,n) for r,n in C.items() if n)) + require(all(0 < r < m and n > 0 for r,n in C.items()), 'Shrinking complex children') + s = sum(r*n for r,n in C.items()) + require(s == W*m-N+2*v*ell, 'Complete complex rank mass') + return dict(dimensions=[h,h],m=m,N=N,W=W,L=2*v*ell,total_rank=s, + deficit=W*m-s,maxchild=max(C),child_multiplicities=C) + + +def finite_bridge(bit, phase, row, previous): + m,W,s,N = (phase[k] for k in ('m','W','total_rank','N')) + h,v,R = (row[k] for k in ('h','v','R')) + local = 4*(row['c']+v)+10*v+4*h*v+4*h*h+8*h+8+2*h + local += 8*R*v*(row['total_M_operations']+16) + G = N+2*v*local + E = 64*(W+m+G+1)**3 + charge = 2*G*W*W+8*s+4*W+4+32*m + B, r = s+E, phase['maxchild'] + C0 = 32*m*B*B + require(charge < E and 2*B*(m-r) >= s+E and 2*B+18 < C0, 'Semantic induction and charge') + dc,dp = halving(bit['m'],bit['maxchild']),halving(m,r) + wc,wp = bit['W'].bit_length(),W.bit_length() + old_bit = previous['finite_bridge']['bit'] + old = old_bit['halving_degree']*old_bit['wire_bits'] + require(old == 252, 'Inherited ordinary leaf row stock') + coeff,degree = wc*dc+old+wp*dp,40000 + require(Q(degree) > Q(coeff*51,25), 'Sufficient product row stock') + return dict(bit_coarse=dict(m=bit['m'],W=bit['W'],maxchild=bit['maxchild'], + halving_degree=dc,wire_bits=wc),ordinary_leaf_row_degree=old, + complex=dict(m=m,W=W,s=s,maxchild=r,halving_degree=dp,wire_bits=wp, + scalar_group_upper=G,local_group_upper=local,coefficient_bound=h+4, + coefficient_denominator_divides=2*(h-3)), + semantic=dict(E=E,literal_charge=charge,strict_literal_gap=E-charge, + B=B,C0=C0,C1=1,induction_gap=2*B*(m-r)-s-E,fixed_odd_divisor=h-3, + exact_grid='One common dyadic grid times the fixed odd divisor to K=G*(D_complex+1); no child rounding'), + rows=dict(coefficient=coeff,degree=degree,suffix_slope=4*degree, + degree_gap=Q(degree)-Q(coeff*51,25), + contract='W_complex^D_complex * W_coarse^D_coarse * W_old^D_old; one preceding prefix and one padding; sequential reuse')) + + +def certificate(): + require(not sys.flags.optimize, 'Run without -O') + def read(name): + return json.loads((ROOT/'certificates'/name).read_text()) + raw = reconstruct() + row = read('partial-gauge-complex-input.json') + expected = read('partial-gauge-input.json') + previous = read('copied-centers-network.json') + require(Q(previous['bit']['saving']) == OLD, 'Inherited leaf saving') + C = {int(r):n for r,n in raw['whole_projector_histogram'].items()} + bit = dict(dimensions=raw['dimensions'],m=raw['m'],W=raw['W'],N=raw['N'], + total_rank=raw['rank_mass'],deficit=raw['deficit'],maxchild=max(C),child_multiplicities=C) + require(sum(r*n for r,n in C.items()) == bit['total_rank'], 'Bit rank mass') + require(bit['W']*bit['m']-bit['total_rank'] == bit['deficit'], 'Bit deficit') + phase = complex_profile(row) + bm = coarse_moment(bit) + cm = moment(phase['m'],phase['W'],phase['child_multiplicities'],AC,True) + require(bm['strict_gap'] == Q(expected['bit_moment_gap']), 'Saved coarse moment') + require(cm['strict_gap'] == Q(expected['complex_moment_gap']), 'Saved complex moment') + require(js(phase['child_multiplicities']) == expected['complex_children'], 'Saved complex children') + bridge = finite_bridge(bit,phase,row,previous) + assembled = assembly(ASSEMBLY_BIT,AC,bridge,KAPPA,beta=PHASE_STOP) + assembled['parameters']['actual_bit_saving'] = AB + require(ASSEMBLY_BIT <= AB, 'Supported assembly movement saving') + require(js(bridge) == expected['finite_bridge'], 'Saved finite bridge') + require(js(assembled) == expected['assembly'], 'Saved exact assembly') + require(len(assembled['strict_constraints']) == 47 and len(assembled['margins']) == 7, 'Complete strict constraints') + sources = sorted(p for p in (ROOT/'certificates').glob('partial-gauge-*.json') + if p.name != 'partial-gauge-network.json') + sources += sorted((ROOT/'scripts').glob('partial_gauge*.py')) + sources += sorted(p for p in (ROOT/'scripts/partial_gauge').rglob('*') + if p.is_file() and '__pycache__' not in p.parts) + sources += sorted((ROOT/'notes').glob('partial-gauge-*.tex')) + sources += sorted(p for p in (ROOT/'references/partial-gauge').rglob('*') + if p.is_file() and '__pycache__' not in p.parts) + sources += [ROOT/p for p in ('scripts/copied_centers_network.py', + 'scripts/partial_swap_network.py','scripts/structured_bulk_assembly.py', + 'certificates/copied-centers-network.json','certificates/stopped-product-network.json', + 'scripts/paired_triple_circuit.py','scripts/paired_exclusion_circuit.py')] + return dict(status='Conditional partial-gauge multiplication witness',kappa=KAPPA, + bit=dict(counts=bit,**bm),complex=dict(counts=phase,**cm), + finite_bridge=bridge,assembly=assembled, + predecessor_commit='948ce1510df750f4c18b96bdaef436a86f8bf834', + source_sha256={str(p.relative_to(ROOT)):sha256(p.read_bytes()).hexdigest() for p in sources}, + scope='Exact moments, stopped bit saving and semantic assembly. Finite producer/frame checks ' + 'are separate; common-basis, fixed-tape and analytic transfer remain written proof dependencies.') + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument('--output',type=Path,default=ROOT/'certificates/partial-gauge-network.json') + args = parser.parse_args() + args.output.write_text(json.dumps(js(certificate()),indent=2,sort_keys=True)+'\n') + print('PASS kappa=7237/78125000 = 9.26336e-5; both moments, stopped saving, p^40000 stock, 47 strict constraints') + + +if __name__ == '__main__': + main() diff --git a/scripts/partial_gauge_producer.py b/scripts/partial_gauge_producer.py new file mode 100644 index 0000000000..2d91d92b8e --- /dev/null +++ b/scripts/partial_gauge_producer.py @@ -0,0 +1,67 @@ +#!/usr/bin/env python3 +"""Regenerate and verify the selected h=24 partial-gauge complex producer. + +Standard-library Python and C++17; generated DAGs and words are temporary. +Copyright 2026 icekylinx, Apache-2.0. Adapted from the credited round-six handoff. +""" +import argparse +import json +import os +from pathlib import Path +import shlex +import subprocess +import sys +import tempfile +from partial_gauge.scalar import build +from partial_gauge.lift import run as lift_run +from partial_gauge.word import run as word_run +from partial_gauge.select import run as select_run +from partial_gauge.verify import verify + +ROOT=Path(__file__).resolve().parents[1] + + +def regenerate(work, expected, compiler): + if sys.flags.optimize:raise ValueError('Run without -O; finite assertions must remain enabled') + def stage(message):print(message,file=sys.stderr,flush=True) + def save(name,data): + path=work/name;path.write_text(json.dumps(data,separators=(',',':'))+'\n');return path + stage('Building the actual interval/cube scalar DAG') + scalar=build(24,base=2,central_disjoint=24,output_prefix='complex_cube',output_dir=work) + dag=work/'complex_cube_24.bin';matcher=work/'matcher' + subprocess.run([*shlex.split(compiler),'-O3','-std=c++17',str(ROOT/'scripts/partial_gauge/match.cpp'),'-o',str(matcher)],check=True) + stage('Constructing compatible carrier matching') + matching=json.loads(subprocess.check_output([str(matcher),str(dag),str(dag.with_suffix('.labels'))],text=True)) + matchfile=save('matching.json',matching) + stage('Lifting exact binary frames backward through dependencies and carriers') + lifted,ann=lift_run(dag,matchfile);liftfile=save('lifted.json',lifted);annfile=save('annihilators.json',ann) + stage('Constructing physical M and its retained-center dependency closure') + word=word_run(dag,matchfile,annfile,liftfile);wordfile=save('word.json',word) + stage('Selecting partial source gauges with reverse readouts') + result,selection=select_run(dag,annfile,liftfile,wordfile) + result=json.loads(json.dumps(result)) + assert result==expected, 'Regenerated selected certificate differs' + save('selection.json',selection) + stage('Checking scalar coefficients, actual frames, physical word and readout chains') + checks=verify(dag,matching,ann,lifted,word,selection['selected'],result) + return dict(regenerated_from_source=True,certificate_equal=True,selected=result,checks=checks) + + +def main(): + p=argparse.ArgumentParser(description=__doc__) + p.add_argument('--certificate',type=Path,default=ROOT/'certificates/partial-gauge-complex-input.json') + p.add_argument('--work-dir',type=Path) + p.add_argument('--output',type=Path) + p.add_argument('--cxx',default=os.environ.get('CXX','c++')) + a=p.parse_args();expected=json.loads(a.certificate.read_text()) + if a.work_dir: + a.work_dir.mkdir(parents=True,exist_ok=True);result=regenerate(a.work_dir.resolve(),expected,a.cxx) + else: + with tempfile.TemporaryDirectory(prefix='partial-gauge-') as directory: + result=regenerate(Path(directory),expected,a.cxx) + encoded=json.dumps(result,indent=2)+'\n' + if a.output:a.output.write_text(encoded) + else:print(encoded,end='') + + +if __name__=='__main__':main() diff --git a/scripts/stopped_product/README.md b/scripts/stopped_product/README.md new file mode 100644 index 0000000000..6c18be051f --- /dev/null +++ b/scripts/stopped_product/README.md @@ -0,0 +1,36 @@ +# Selected rational-center producer + +Run from the repository root: + +```sh +python3 scripts/stopped_product_producer.py --output /tmp/stopped-product-producers.json +``` + +Only the new aligned complex producer at h=24 is rebuilt. It retains all +24 disjoint-point sums and decodes the total by exact division by21. The +pair-star input order is aligned with global pairs. Its generator adapts +`structured_bulk.complex`, preserving exact support, output orthogonality, +binary frame nesting and nondegeneracy checks. Rational decoder and scatter +coefficients are checked with `Fraction`; their only odd denominator is21. +The unchanged deterministic matcher compares every count and histogram +entry against `stopped-product-complex-input.json`. + +The h23 bit record is compared with the preceding validated copied-center +input. No old bit DAG, matching, or corner is regenerated. Both copied-center +histogram transformations reuse the established exact rank-mass checker. + +All DAGs, labels and the compiled matcher are temporary by default; +`--work-dir` retains them. Python assertions must be enabled. Standard-library +Python and a C++17 compiler suffice. No binary archive is an input. + +The odd-grid extension is a written recursive invariant, not a claim proved +by the finite scalar tests. With dyadic root inputs, K=G(D+1) supplies enough +odd-divisor reserve along the active chain. A completed child preserves its +incoming odd valuation; it does not generally make rational child inputs +dyadic. Only the completed outer dyadic operator restores the original +dyadic grid. No child return is rounded. + +Copyright 2026 icekylinx, Apache-2.0. New selected alignment, rational centers +and integration were prepared with OpenAI GPT-6 Astra/Codex assistance. +Underlying paired producers and deterministic carrier matching retain the +previous contributors' attribution in `NOTICE` and `SOURCES.json`. diff --git a/scripts/stopped_product/__init__.py b/scripts/stopped_product/__init__.py new file mode 100644 index 0000000000..f7a6e90661 --- /dev/null +++ b/scripts/stopped_product/__init__.py @@ -0,0 +1 @@ +"""Selected stopped-product inputs and rational producer; icekylinx, 2026.""" diff --git a/scripts/stopped_product/complex.py b/scripts/stopped_product/complex.py new file mode 100644 index 0000000000..ef1af25507 --- /dev/null +++ b/scripts/stopped_product/complex.py @@ -0,0 +1,228 @@ +"""Selected aligned rational-center producer, h=24 with all centers disjoint. + +Copyright 2026 icekylinx, Apache-2.0. AI-assisted adaptation of the research +round-five handoff, using the credited retained PairedTriple circuit; see NOTICE. +Adapted from structured_bulk.complex: align pair-star inputs and permit the +fixed rational divisor21; retain all exact support/frame/scalar checks. + +Builds a concrete scalar addition DAG, its nondegenerate binary frames, and +the unmixed carrier histogram. No historical audit suite is invoked. +""" +import json, struct, array +from itertools import combinations +from pathlib import Path +from partial_swap.binary import write_array +from paired_triple_circuit import PairedTriple + +def build(h, prefix, central_disjoint, base=2): + assert (h, central_disjoint, base) == (24,24,2) + d=central_disjoint + assert 0= 2: + x /= 2 + power += 1 + def small(y): + z = (y-1)/(y+1) + return 2*sum((z**(2*j+1)/(2*j+1) for j in range(32)),Q(0)) + 2*z**65/(65*(1-z*z)) + return up(power*small(Q(2))+small(x)) + + +def exp_upper(u): + require(0 <= u < 3,'Exponential enclosure') + return up(1+u+u*u/(2*(1-u/3))) + + +def bit_certificate(row): + physical = reconstruct() + h,v,R = (row[k] for k in ('h','v','R')) + require((h,v,R) == (23,comb(23,3),28866),'Bit local dimensions') + for key in ('v','R','local_rank_histograms','exit_nullity_histogram'): + require(row[key] == physical[key],'Inherited physical bit ledger: '+key) + ell = row['center_loss'] + require(ell == h*(h-1),'Copied-center loss') + m,W,H = 3*h-2,2*v+3*R,Counter() + for part in row['local_rank_histograms'].values(): + for r,n in part.items(): + H[int(r)] += 3*n + for nullity,n in row['exit_nullity_histogram'].items(): + H[m-int(nullity)] += 3*n + require(all(0 < r < m and n > 0 for r,n in H.items()),'Proper bit children') + mass = sum(r*n for r,n in H.items()) + require(W*m-mass == 2*v-3*ell,'Three-stage bit telescoping') + logs = {r:log_upper(Q(m,r)) for r in H} + good = sum(Q(r*n,W*m)*exp_upper(COARSE*logs[r]) for r,n in H.items()) + fallback,edges = 32*m*m,sum(H.values()) + extra = BAD*Q(fallback*edges,W*m)*exp_upper(COARSE*log_upper(Q(m))) + rank_upper = Q(mass)+BAD*fallback*edges + require(good+extra < 1 and rank_upper < W*m,'Both contaminated moments') + require(Q(2*m**3,2**80) < BAD,'Fixed-prime bad-class fraction') + require(ATOM > AB and ATOM < 1-AB,'Subordinate adapter and row-borrowing tolls') + return dict(m=m,local_dimension=h,roles_per_vertex=W, + ideal_rank_mass_per_vertex=mass,ideal_deficit=2*v-3*ell, + ideal_child_multiplicities=dict(sorted(H.items())),maxchild=max(H), + coarse_saving=COARSE,effective_saving=AB,atom_exponent=ATOM, + ordinary_leaf_saving=OLD,bad_fraction=BAD, + fallback_children_per_edge=fallback,edge_count_per_vertex=edges, + good_moment_upper=good,added_bad_moment_upper=extra, + strict_gap=1-good-extra,rank_mass_upper_per_vertex=rank_upper, + log_upper_bounds=logs,rounding_denominator=GRID, + row_stock='O(w log e) internally borrowed radix-q digits, restored at the outer boundary', + prime_choice='Fixed odd q > 2^80 avoiding the finite rational denominator and rank-witness primes') + + +def complex_certificate(row): + h,v,R,ell = (row[k] for k in ('h','v','R','loss')) + require((h,v,R,ell) == (24,comb(24,3),28705,552),'Complex local dimensions') + require(R == row['c']+row['q']-row['matched'],'Compatible roles') + selected = {int(r):n for r,n in row['selected_rank_histogram'].items()} + require(sum(selected.values()) == row['selected_roles'] == 4599,'Partial gauges') + m,w,C = 3*h-2,2*v+3*R,Counter() + C[m-h] += 3*(R-sum(selected.values())) + for r,n in selected.items(): + C[m-h+r] += 3*n + for key in ('source_data_histogram','target_data_histogram'): + for r,n in row[key].items(): + C[int(r)] += 3*n + for r,n in enumerate(row['remaining_internal_histogram']): + if r: + C[r] += 3*n + C = dict(sorted((r,n) for r,n in C.items() if r and n)) + require(all(0 < r < m and n > 0 for r,n in C.items()),'Proper complex children') + rank = sum(r*n for r,n in C.items()) + require(w*m-rank == 2*v-3*ell,'Three-stage complex telescoping') + n = m//2 + vertices = 2**(m-1+(n-1)**2)*prod(2**(2*i)-1 for i in range(1,n)) + exact = moment(m,w,C,AC,True) + return dict(m=m,N=vertices*v,W=vertices*w,total_rank=vertices*rank, + deficit=vertices*(w*m-rank),vertices_per_stage=vertices,maxchild=max(C), + group_order_bits=vertices.bit_length(),per_vertex=dict(W=w,rank=rank, + deficit=w*m-rank,child_multiplicities=C),**exact) + + +def finite_bridge(c,row): + m,W,s,N,V = (c[k] for k in ('m','W','total_rank','N','vertices_per_stage')) + h,v,R = (row[k] for k in ('h','v','R')) + local = 4*(row['c']+v)+10*v+4*h*v+4*h*h+8*h+8+2*h + local += 8*R*v*(row['total_M_operations']+16) + router = 64*(m+1)**3*W**2 + G = N+3*V*local+router + E = 64*(W+m+G+1)**3 + charge = 2*G*W*W+8*s+4*W+4+32*m + B,C0,r = s+E,32*m*(s+E)**2,c['maxchild'] + require(charge < E and 2*B*(m-r) >= s+E and 2*B+18 < C0,'Finite semantic charge') + dp,wp = halving(m,r),W.bit_length() + old_coarse,old_atom = 9909,252 + coefficient,degree = dp*wp+old_coarse+old_atom,160000 + require(Q(degree) > Q(51*coefficient,25),'External product row reserve') + return dict(bit_uniform=dict(coarse_saving=COARSE,atom_beta=ATOM, + old_atom_saving=OLD,ordinary_saving=AB, + selector_stock='O(w log e) internal borrowed rows; restored by the uniform old-supplier wrapper', + external_stock='No q-adic group-size factor is inserted into the external polynomial row stock'), + conservative_old_coarse_row_reserve=old_coarse,ordinary_leaf_row_degree=old_atom, + complex=dict(m=m,W=W,s=s,N=N,maxchild=r,invocations_per_stage=V,stages=3, + halving_degree=dp,wire_bits=wp,scalar_group_upper=G,local_group_upper=local, + finite_group_router_upper=router,coefficient_bound=h+4, + coefficient_denominator_divides=2*(h-3)), + semantic=dict(E=E,literal_charge=charge,strict_literal_gap=E-charge, + B=B,C0=C0,C1=1,induction_gap=2*B*(m-r)-s-E,fixed_odd_divisor=h-3, + exact_grid='One common dyadic grid times 21^(-K), K=G*(D_complex+1); no child rounding'), + rows=dict(coefficient=coefficient,complex_coefficient=dp*wp,degree=degree, + suffix_slope=4*degree,degree_gap=Q(degree)-Q(51*coefficient,25), + contract='Finite complex group role stock plus conservative fixed legacy bit reserves; q-adic selectors borrowed internally and restored; one prefix and one padding; sequential reuse')) + + +def certificate(): + require(not sys.flags.optimize,'Run without -O') + def read(name): + return json.loads((ROOT/'certificates'/name).read_text()) + expected = read('three-stage-cover-input.json') + bit = bit_certificate(read('three-stage-cover-bit-input.json')) + row = read('three-stage-cover-complex-input.json') + phase = complex_certificate(row) + bridge = finite_bridge(phase,row) + previous = read('copied-centers-network.json') + require(Q(previous['bit']['saving']) == OLD,'Retained ordinary leaf') + result = assembly(ASSEMBLY_BIT,AC,bridge,KAPPA,beta=PHASE_STOP) + result['parameters']['actual_bit_saving'] = AB + require(ASSEMBLY_BIT <= AB,'Supported assembly bit saving') + for actual,key in ((bit['strict_gap'],'bit_moment_gap'),(phase['strict_gap'],'complex_moment_gap'), + (result['minimum_margin'],'assembly_minimum'),(result['absorption_gap'],'absorption_gap')): + require(actual == Q(expected[key]),'Saved exact '+key) + require(js(bit['ideal_child_multiplicities']) == expected['bit_child_multiplicities'],'Saved bit children') + require(js(phase['per_vertex']['child_multiplicities']) == expected['complex_child_multiplicities'],'Saved complex children') + require(phase['vertices_per_stage'] == expected['complex_group_order'],'Finite group order') + require(bridge['rows']['degree'] == expected['row_degree'] and bridge['rows']['coefficient'] == expected['row_coefficient'],'Saved row bound') + require(KAPPA == Q(expected['kappa']) and len(result['strict_constraints']) == 47 and len(result['margins']) == 7,'Complete assembly') + sources = sorted(p for p in (ROOT/'certificates').glob('three-stage-cover-*.json') + if p.name != 'three-stage-cover-network.json') + sources += sorted((ROOT/'scripts').glob('three_stage_cover*.py')) + sources += sorted(p for folder in ('scripts/three_stage_cover','references/three-stage-cover') + for p in (ROOT/folder).rglob('*') if p.is_file() and '__pycache__' not in p.parts) + sources += sorted((ROOT/'notes').glob('three-stage-cover-*.tex'))+[ROOT/'notes/general-clifford-frames.tex'] + sources += [ROOT/p for p in ('scripts/partial_gauge_bit.py','certificates/partial-gauge-bit-input.json', + 'references/partial-gauge/pr97/SOURCE.json','scripts/partial_swap_network.py', + 'scripts/structured_bulk_assembly.py','certificates/copied-centers-network.json', + 'certificates/stopped-product-network.json','certificates/partial-gauge-network.json', + 'scripts/partial_gauge/binary.py','scripts/partial_gauge/lift.py','scripts/partial_gauge/word.py', + 'scripts/partial_gauge/match.cpp')] + return dict(status='Conditional three-stage cover multiplication witness',kappa=KAPPA, + bit=bit,complex=phase,finite_bridge=bridge,assembly=result, + predecessor_commit='cf60442983c153a3a0f5272ea4713f99eba13c86', + source_sha256={str(p.relative_to(ROOT)):sha256(p.read_bytes()).hexdigest() for p in sources}, + scope='Exact local inventories, strict moments, contaminated fallback charge and semantic assembly. ' + 'Local finite reconstruction is checked separately; group geometry, uniform weighted routing, ' + 'borrowed rows and inherited analytic/tape contracts are written proof dependencies.') + + +def main(): + p = argparse.ArgumentParser(description=__doc__) + p.add_argument('--output',type=Path,default=ROOT/'certificates/three-stage-cover-network.json') + args = p.parse_args() + args.output.write_text(json.dumps(js(certificate()),indent=2,sort_keys=True)+'\n') + print('PASS kappa=3146011/10000000000 = 3.146011e-4; both moments, rare-class fallback, finite router, 47 strict constraints') + + +if __name__ == '__main__': + main() diff --git a/scripts/three_stage_cover_producer.py b/scripts/three_stage_cover_producer.py new file mode 100644 index 0000000000..9e7a21a758 --- /dev/null +++ b/scripts/three_stage_cover_producer.py @@ -0,0 +1,77 @@ +#!/usr/bin/env python3 +"""Regenerate and verify the selected pinned PR117 generalized-subspace complex producer. + +Standard-library Python and C++17; generated DAGs and words are temporary. +Copyright 2026 icekylinx, Apache-2.0. Adapted from the credited round-six handoff. +""" +import argparse +import json +import os +from pathlib import Path +import shlex +import subprocess +import sys +import tempfile +from importlib.util import spec_from_file_location, module_from_spec +import hashlib +from three_stage_cover.lift import run as lift_run +from partial_gauge.word import run as word_run +from three_stage_cover.select import run as select_run +from three_stage_cover.verify import verify + +ROOT=Path(__file__).resolve().parents[1] + + +def regenerate(work, expected, compiler): + if sys.flags.optimize:raise ValueError('Run without -O; finite assertions must remain enabled') + def stage(message):print(message,file=sys.stderr,flush=True) + def save(name,data): + path=work/name;path.write_text(json.dumps(data,separators=(',',':'))+'\n');return path + stage('Replaying the pinned PR117 scalar DAG') + reference=ROOT/'references/three-stage-cover/pr117' + source=json.loads((reference/'SOURCE.json').read_text()) + assert source['commit']=='cbb05ce504d571546d9b7794c186a613c659c3bf' + for name,digest in source['files'].items(): + assert hashlib.sha256((reference/name).read_bytes()).hexdigest()==digest, 'Source binding differs: '+name + witness=reference/'dag.json.gz' + assert hashlib.sha256(witness.read_bytes()).hexdigest()=='3c034d0aae388ef567a454826f4f48b26fd8a94c71e8ffed4835271b349a783b' + spec=spec_from_file_location('pr117_replayed',reference/'replayed.py') + module=module_from_spec(spec);spec.loader.exec_module(module) + scalar=module.build(witness,work/'selected') + dag=work/'selected.bin';matcher=work/'matcher' + subprocess.run([*shlex.split(compiler),'-O3','-std=c++17',str(ROOT/'scripts/partial_gauge/match.cpp'),'-o',str(matcher)],check=True) + stage('Constructing compatible carrier matching') + matching=json.loads(subprocess.check_output([str(matcher),str(dag),str(dag.with_suffix('.labels'))],text=True)) + matchfile=save('matching.json',matching) + stage('Lifting exact binary frames backward through dependencies and carriers') + lifted,ann=lift_run(dag,matchfile);liftfile=save('lifted.json',lifted);annfile=save('annihilators.json',ann) + stage('Constructing physical M and its retained-center dependency closure') + word=word_run(dag,matchfile,annfile,liftfile);wordfile=save('word.json',word) + stage('Selecting partial source gauges with reverse readouts') + result,selection=select_run(dag,annfile,liftfile,wordfile) + result=json.loads(json.dumps(result)) + assert result==expected, 'Regenerated selected certificate differs' + save('selection.json',selection) + stage('Checking scalar coefficients, actual frames, physical word and readout chains') + checks=verify(dag,matching,ann,lifted,word,selection['selected'],result) + return dict(source_pin_verified=True,regenerated_from_source=True,certificate_equal=True,selected=result,checks=checks) + + +def main(): + p=argparse.ArgumentParser(description=__doc__) + p.add_argument('--certificate',type=Path,default=ROOT/'certificates/three-stage-cover-complex-input.json') + p.add_argument('--work-dir',type=Path) + p.add_argument('--output',type=Path) + p.add_argument('--cxx',default=os.environ.get('CXX','c++')) + a=p.parse_args();expected=json.loads(a.certificate.read_text()) + if a.work_dir: + a.work_dir.mkdir(parents=True,exist_ok=True);result=regenerate(a.work_dir.resolve(),expected,a.cxx) + else: + with tempfile.TemporaryDirectory(prefix='three-stage-cover-') as directory: + result=regenerate(Path(directory),expected,a.cxx) + encoded=json.dumps(result,indent=2)+'\n' + if a.output:a.output.write_text(encoded) + else:print(encoded,end='') + + +if __name__=='__main__':main()