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interface result; no new circuit, supplier or kappa.** On the +pinned 11,500-ADD PR1062 R439 word, omit one terminal helper-star restoration +`a ^= b`, and toggle one donor in each of two existing phase-A writes of `a` +before its first read. There are **314 exact arbitrary-dirty completions**, +including **52 with one fewer scalar ADD**, but **every completion violates the +unchanged monotone rational-frame law**. Each rejection has an explicit old-frame +source label outside the proposed receiving frame's rational span. + +This answers only a bounded part of Sagar Thakrar's +[PR1868](https://github.com/CrocSwap/integer-mult-bounds/pull/1868) open question +about multi-gate splits. That predecessor already supplies the twisted-helper +contract and three legal **single-writer** absorptions. Its result is credited, +not rediscovered here. One of its actual legal moves is a positive control. + +## Boundary contract and complete domain + +Use zero-based physical register and A-gate indices in +`research/r439-bsection-stars-seed01/candidate.json.gz` at main +`0b1cccf86e2f5132ed286a26ef0a681a495740ea`. Sources are 0–110, dirty targets +111–221, and dirty helpers 222–660. The source word has 236 terminal star +restorations. Every leaf is distinct and is never a centre. + +For **each one** of those 236 edges `(a,b)` separately: + +1. Remove only its final D restoration `a ^= b`. Keep the other D gates. +2. Choose two distinct original A gates `i < j`, both writing `a`, strictly + before the first original A gate reading `a`. If A never reads it, the + deadline is the end of A. +3. Toggle exactly one donor `s` in gate i's source set, and exactly one donor + `t` in gate j's source set. A toggle removes an existing donor or adds a new + one. The donor alphabet is all original source/helper registers except a: + **549 choices at each cut**. Targets and new dirty auxiliaries are excluded. +4. Keep every other A gate, all scatter/COPY/ERASE/B/C operations, original gate + frames, starts and final frames. The donor lists are canonically sorted; + donor order within a common-frame XOR gate does not change the boundary. + No new gate, frame transport, free reset, cleanup rewrite or source synthesis + is permitted. A donorless centre gate still retains its original frame move. + +The certificate lists all 236 `(centre,leaf,first-read,writers)` records. Their +writer-pair counts per edge are: + +| pairs per edge | edges | +|---:|---:| +| 0 | 81 | +| 1 | 118 | +| 3 | 10 | +| 6 | 1 | +| 10 | 26 | + +Thus there are **414 edge/writer pairs**, not an independently sampled set. +There are 414 × 549 = **227,286 first-donor trials**. The second donor is solved +uniquely, rather than sampled; the next argument proves this covers all +414 × 549² = 124,780,014 possible donor pairs in the declared domain. + +## Exact replacement lemma + +Let `r_k(u)` be the row vector giving register u's current value immediately +before original A gate k, in the 661 original dirty-input coordinates. Because +the D star suffix restores every helper, deleting `a ^= b` leaves precisely the +extra original leaf value `h_b` in centre a at the D/A cut, and no other defect. +The disjoint-star checks are material to this statement. + +Before a's first read, no other register acquires the defect. Writes to a add +values independent of a, so toggling s at i adds `r_i(s)` to that defect and +toggling t at j adds `r_j(t)`. The candidate and original states agree on every +register immediately after j **if and only if** + +``` +r_i(s) XOR r_j(t) = e_b. (1) +``` + +Necessity follows by choosing a unit dirty-input column where the residual row +is nonzero; sufficiency follows by linearity. This is an identity over every +F2-module, not merely clean helper inputs. After that cut the unchanged +continuation receives identical scalar values. In particular, the copied totals +and their chronological ERASE obligations are unchanged. The full-word replay +checks them independently in both orientations. + +Conversely, a nonzero residual at this cut cannot disappear in a legal, +unchanged continuation: ADD is invertible, COPY is injective into a clean +temporary, and ERASE is invertible on its required matching-value domain. +Consequently equality at this cut is also necessary for the complete legal +dirty boundary, rather than merely a convenient sufficient filter. + +The original prefix is an invertible XOR word. Its 661 state rows are distinct +and nonzero, so for a fixed first donor s at most one register t has row +`e_b XOR r_i(s)`. The verifier builds that exact row-to-register dictionary over +the allowed second-donor alphabet. It tries every s at every eligible writer +pair. A missing row excludes every second donor; a present row gives the unique +candidate. This establishes complete coverage without a quadratic search. + +Equation (1) is only a **scalar boundary criterion**. It does not imply that +its reads are legal at their proposed frames. That missing implication is the +one falsified here. It is not a defect in the existing checker or in PR1868's +explicitly frame-checked single-writer construction. + +## Exact source-bound counterexample + +The lexicographically first saving completion is: + +``` +omit D restoration: 231 ^= 459 +A[9]: [frame 6, dest 231, donors [4]] -> donors [4,109] +A[527]: [frame 234, dest 231, donors [459]] -> donors [] +``` + +The first read of 231 is A[528]. Before A[9], register 109 holds `x_109`. +Before A[527], register 459 holds `h_459 XOR x_109`. Their XOR is exactly +`h_459`, so both corrections together eliminate the omitted-restoration defect +for **arbitrary dirty inputs**. Literal forward/reverse execution on all 661 +columns succeeds, and the complete scalar count is 11,499 instead of 11,500. + +But source 109 starts at frame 111, whose label support is `{109}`. A[9] asks +it to visit frame 6, support `{4}`. Exact rational elimination gives + +``` +rank_Q(line[4]) = 1 +rank_Q(line[4], line[109]) = 2. +``` + +The required old frame is therefore not contained in the receiving frame. +There is no legal monotone climb, regardless of the perfect scalar endpoint. +All other 313 completions similarly have a checked rational noncontainment +witness, not merely a copied checker failure flag. + +## Accounting + +Each completion changes the complete ADD invoice by + +``` +-1 + (1 - 2*[s was an old donor at i]) + + (1 - 2*[t was an old donor at j]). +``` + +The `-1` is the omitted D restoration; C is unchanged. Exactly 52 completions +have delta -1 and 262 have delta +1. No delta -3 completion exists. None has an +admissible paid histogram under the retained frame contract; the verifier +refuses to bill an illegal path. Scalar savings cannot be substituted for a +legal recursive profile or multiplication-exponent improvement. + +The source control's **complete** profile is independently recomputed from +all rational frame-rank increments plus all nine COPY ranks: +`{1:1323,2:1342,3:79,4:44,5:102,6:11,7:19}`, rank mass **5,129**. +All nine COPY/ERASE pairs are retained. As a nontrivial positive control, +PR1868's `506 ^= 374` omission with A[288] donors changed from `[374]` to `[69]` +passes the full dirty boundary and has identical ordered frame paths and paid +profile, while reducing the scalar count to 11,499. This saving belongs to +PR1868. Both source controls' complete positive envelopes pass the inherited +strict 104/192-bit inequalities; exact integers are in `CERTIFICATE.json`. + +## Reproduce + +Standard-library Python 3.12, from the repository root: + +```sh +python3.12 scripts/research_package.py --package research/r439-two-writer-absorption +``` + +Direct commands: + +```sh +python3.12 -I -O -B research/r439-two-writer-absorption/verify.py +python3.12 -I -O -B research/r439-two-writer-absorption/test_verify.py +``` + +The verifier prints: + +``` +PASS_R439_TWO_WRITER_ABSORPTION pairs=414 trials=227286 exact=314 scalar_saving=52 legal=0 columns=661 controls=7 no_new_kappa +``` + +`--write` explicitly regenerates only this package's certificate. Default +replay is read-only, compares every certificate field and checks input hashes. +Seven failure controls cover omitted corrections, retaining the supposedly +omitted restoration, a wrong donor, billing an illegal path and source +corruption. Seven focused tests include an independent all-eight-input Boolean +identity, exact source rows, the rational frame obstruction, and clean-helper +false acceptance of an omitted restoration. The borrowed working example +`r439-bsection-stars-seed01/verify.py` passed on CPython 3.12.12; the system +Python 3.9 attempt failed at `int.bit_count`, before the supported replay. + +The registered command passed on macOS arm64 with CPython **3.12.12**, including +all seven tests and the optimized direct verifier, in **4.66 seconds** wall +time. Certificate generation took 4.32 seconds. The environment's equivalent +command is `uv run --python 3.12 --no-project python scripts/research_package.py +--package research/r439-two-writer-absorption`. An initial comparison exposed an +integer-versus-string histogram-key ordering mismatch; JSON normalization fixed +that receipt-only issue before the passing replay. No mathematical census or +source input was changed. + +## Scope, overlap and provenance + +The selected complex supplier binds; this audit does not change the selected +record or any frozen evidence. It neither constructs a legal two-writer word +nor rules out multi-donor splits, three or more correction gates, post-read +propagation and compensation, changed gate frames/itineraries, rebuilt A/B/C, +additional registers, or other source words. Native multistage compilation, +whole-call hosting, changed precision/tape realization, outer assembly and a +Lean multiplication theorem are not supplied or run. + +The source word and unchanged checker-as-specification have exact Git/SHA256 +pins in `SOURCES.json`. No predecessor executable is imported. The selected +record and catalogue are historical provenance, not moving runtime inputs. +The initial cached overlap search covered 540 open PRs including drafts at +2026-10-14 13:15:09Z and 524 discussion records at 13:10:59Z, with complete +pagination. A pre-publication offline search used the parent's newer complete +snapshot of 543 PRs at 13:23:53Z and 529 discussion records at 13:25:07Z. +Closed work is covered only by local README/TASK/ledger records; arbitrary +diffs and nested files are not searched. The precise split-absorption terms +found PR1868's open question, not a completed two-toggle census. Full bodies of +PR1868/1580/1592 and discussion1743 were read. A targeted live query before +publication confirmed PR1868 remains open at the recorded head +`6b222d01fee072bcd3a680e6a96ae7d22e13bcac`. Later claims remain possible; +these searches do not establish novelty. Exact heads and terms are in `TASK.json`. +[Discussion1884](https://github.com/CrocSwap/integer-mult-bounds/discussions/1884) +records the parent-coordinated advisory scope; this package makes no priority +guarantee or public reservation. + +## Open next questions + +Can two writers absorb one star restoration when each may toggle a **set** of +donors? First solve the constrained row-sum equation at one actual pair of +cuts, admitting only donors whose entire subsequent frame itinerary remains +legal and pricing every ADD. No favorable legal completion or hosted invoice +is supplied here. `TASK.json` records the structured question and blockers. + +## Credit and disclosure + +Sagar Thakrar / **PR1868** supplies the twisted contract, single-writer census, +positive fixture and multi-gate-split question. **woklerx / PR1194** supplies +semantic-D substitution; **Arham Tahir / PR1311, PR1580, PR1592** supplies its +application and distinct terminal-deferral/compensation obstructions. The source +word is **Sagar Thakrar / PR1062**, using **Sricharan Vagvala / PR967** helper +stars, **Gareth Davidson / PR915/926** bases, **Aaron Wang / PR1015** and +**Ross Richardson / PR923** blocks, on **DreamingOfClouds / PR589** and +**eumemic / PR555/543**. The source checker and inherited frame law are eumemic's; +Zhihao Chen's PR425 and the retained scalar-interface lineage remain credited. +No earlier construction's scalar savings are claimed as new here. + +New material prepared for **Adam Gall** with substantial **OpenAI Codex** +assistance in mathematics, implementation, investigation, verification and +writing. Apache-2.0. No independent human review or formal proof-assistant +verification is claimed. diff --git a/research/r439-two-writer-absorption/SOURCES.json b/research/r439-two-writer-absorption/SOURCES.json new file mode 100644 index 0000000000..9868f0c265 --- /dev/null +++ b/research/r439-two-writer-absorption/SOURCES.json @@ -0,0 +1,30 @@ +{ + "schema_version": 1, + "base_commit": "0b1cccf86e2f5132ed286a26ef0a681a495740ea", + "files": [ + { + "path": "certificates/selected-result.json", + "usage": "provenance", + "git_blob": "ae6a99f28aa468649eb66a361e764d769363bf11", + "sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32" + }, + { + "path": "docs/research/agent-task-catalog.json", + "usage": "provenance", + "git_blob": "06c96f5579bb67ed1b86d334e78fbed7da6945f1", + "sha256": "6686ad58f1fbdbc87134a78901b60fd53a20d3984355e91defbbebc9e9fa8075" + }, + { + "path": "research/e8-bit-p111-tuned/code/xcheck_p.py", + "usage": "input", + "git_blob": "184f5dcf94a486cc7db91ca4db5865916bf26009", + "sha256": "af3decc877e27c00001cf56f50c6c09566e8e101297ab34c7860aed96a06d80e" + }, + { + "path": "research/r439-bsection-stars-seed01/candidate.json.gz", + "usage": "input", + "git_blob": "9a17b4920334878d6c949191f6931bd97126fe3c", + "sha256": "b452d2c2b0b2a546d1105b1da9769aa78e5c54d6f00e97b30c77e3f54ce445ee" + } + ] +} diff --git a/research/r439-two-writer-absorption/TASK.json b/research/r439-two-writer-absorption/TASK.json new file mode 100644 index 0000000000..8675b44699 --- /dev/null +++ b/research/r439-two-writer-absorption/TASK.json @@ -0,0 +1,80 @@ +{ + "schema_version": 1, + "task_id": "bit-helper-contract", + "priority_revision": 3, + "base_commit": "0b1cccf86e2f5132ed286a26ef0a681a495740ea", + "original_card": { + "id": "bit-helper-contract", + "question": "Can one precisely specified helper-mixing word satisfy a replacement for the inherited H2/K interface?", + "starting_points": [ + "research/r439-bsection-stars-seed01/README.md", + "research/r439-copy-path-sharing-barrier/README.md" + ], + "methods": [ + "Derive a replacement interface lemma and price its implementation.", + "Seek an exact dirty-input or chronological-cut counterexample to the proposed replacement." + ], + "acceptance_gate": "A full boundary contract and paid profile; unchanged scalar counts alone are insufficient.", + "first_step": { + "command": [ + "python3", + "-B", + "research/r439-bsection-stars-seed01/verify.py" + ], + "expected": "PASS_R439_BSECTION_STARS", + "inspect": "Compare the two false H2/K flags with the unchanged paid profile in its README.", + "scope": "Positive control / source-bound orientation only; this is not a new construction or full cold replay." + }, + "first_experiment": "Write one replacement H2/K implication for a single helper-star operation, then try to falsify it on dirty inputs.", + "bottleneck_link": "A legitimate replacement could admit a cheaper bit circuit; scalar savings alone do not alter the selected moment.", + "allowed_variations": [ + "Choose one helper-star or quartet instance; narrow to one boundary implication." + ], + "scope_change_triggers": [ + "Changing only source count or scalar addition count without an interface argument." + ], + "overlap_terms": [ + "H2_D_frame0_targets_from_helpers", + "K_equals_D", + "helper-star" + ] + }, + "scope_status": "refined", + "actual_scope": "Exact arbitrary-dirty iff condition for absorbing one PR1062 terminal helper-star restoration into exactly two existing A writes of its centre before the first centre read, toggling one non-target donor per writer. Exhaust all 236 source edges and 414 eligible writer pairs.", + "scope_change_reason": "PR1194/1311 already cover fully restored semantic D. PR1868 supplies single-writer absorption and explicitly leaves multi-gate splits open. This investigates only the two-writer, one-toggle-per-writer subclass, retaining the original frame contract.", + "bottleneck_link": "A scalar restoration saving is not a paid interface improvement. Every exact two-toggle completion here violates a rational monotone-frame containment, so its tempting scalar saving cannot be used as a legal supplier bill.", + "budget_minutes": 60, + "stop_rule": "Complete the 414 writer pairs and 549 first donors per pair using the unique exact second-donor lookup; replay every exact completion, then stop without widening the gate alphabet.", + "outcome": "completed", + "evidence": "CERTIFICATE.json: 227286 first-donor trials, 314 exact dirty completions, 52 reducing ADD count by one, zero frame-legal completions. Every exclusion has an exact rational source-label noncontainment witness. Source-bound legal single-writer positive and seven failure controls.", + "author": "Adam Gall with substantial OpenAI Codex assistance", + "ai_assistance": "Substantial OpenAI Codex assistance in mathematics, implementation, investigation, verification and writing; no independent human review or proof-assistant verification.", + "coordination": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1884", + "predecessors": [ + {"pr": 1868, "head": "6b222d01fee072bcd3a680e6a96ae7d22e13bcac", "author": "Sagar Thakrar", "credit": "twisted helper contract and single-writer absorption"}, + {"pr": 1194, "head": "2e39c98238f543d988f9efc1b076590a4b780dca", "author": "woklerx", "credit": "semantic-D substitution"}, + {"pr": 1580, "head": "2490dac98eacc294bac29f136f0715343611d55e", "author": "Arham Tahir", "credit": "unchanged terminal-restoration obstruction"}, + {"pr": 1592, "head": "20588690897133e9a811f9ba434b0cce8d156d04", "author": "Arham Tahir", "credit": "terminal repair obstruction"} + ], + "overlap": { + "open_prs": {"fetched_at": "2026-10-14T13:15:09.525042+00:00", "records": 540, "complete_pagination": true}, + "discussions": {"fetched_at": "2026-10-14T13:10:59.319576+00:00", "records": 524, "complete_pagination": true}, + "terms": ["multi-gate splits", "two-writer", "multi-writer absorption", "split absorption"], + "result": "Only PR1868's explicit open next question matched the refined terms. Full bodies of PR1868/1580/1592 and discussion1743 read. Provisional overlap evidence, not novelty certification.", + "limits": "Open PR bodies including drafts and complete Discussion snapshot; closed work via local README/TASK/ledger only. No arbitrary code diffs or nested files; snapshots may miss later work.", + "prepublication_cached_snapshot": { + "open_prs": {"fetched_at": "2026-10-14T13:23:53.012964+00:00", "records": 543, "complete_pagination": true}, + "discussions": {"fetched_at": "2026-10-14T13:25:07.056183+00:00", "records": 529, "complete_pagination": true}, + "mode": "offline; cached data may miss current changes", + "targeted_live_check": "PR1868 remains OPEN at 6b222d01fee072bcd3a680e6a96ae7d22e13bcac before publication" + } + }, + "open_questions": [ + { + "question": "Can two centre writes absorb a star restoration when each may toggle a multi-donor set?", + "why_it_matters": "The present exact exclusion is for one donor toggle at each of two writers, not the full split-restoration problem.", + "first_step": "At a chosen writer pair, solve the row-sum compensation equation constrained to donors that preserve both current and subsequent rational frames, charging every added and removed ADD.", + "blocked_by": "No favorable legal multi-donor completion or complete hosted invoice is supplied. Native common-frame realization and supplier admission remain separate obligations." + } + ] +} diff --git a/research/r439-two-writer-absorption/VERIFY.json b/research/r439-two-writer-absorption/VERIFY.json new file mode 100644 index 0000000000..d9f09395e0 --- /dev/null +++ b/research/r439-two-writer-absorption/VERIFY.json @@ -0,0 +1,28 @@ +{ + "schema_version": 1, + "status": "ready", + "python": "3.12", + "requirements": [], + "inputs": [ + "research/e8-bit-p111-tuned/code/xcheck_p.py", + "research/r439-bsection-stars-seed01/candidate.json.gz" + ], + "commands": [ + { + "argv": [ + "{python}", + "-I", + "-O", + "-B", + "{package}/verify.py" + ], + "timeout_seconds": 120, + "checks": "Exhaustive exact two-writer compensation domain, all 661 columns in both orientations for all 314 completions, rational frame witnesses, full source-control invoices and seven failure controls" + }, + { + "argv": ["{python}", "-I", "-O", "-B", "{package}/test_verify.py"], + "timeout_seconds": 120, + "checks": "Seven focused tests for the universal dirty identity, source rows, rational noncontainment, clean-only false acceptance, second-donor uniqueness, pins and star-domain mutations" + } + ] +} diff --git a/research/r439-two-writer-absorption/test_verify.py b/research/r439-two-writer-absorption/test_verify.py new file mode 100755 index 0000000000..1772846da7 --- /dev/null +++ b/research/r439-two-writer-absorption/test_verify.py @@ -0,0 +1,78 @@ +#!/usr/bin/env python3 +# SPDX-License-Identifier: Apache-2.0 +# Adam Gall, with substantial OpenAI Codex assistance. +"""Focused tests; no whole-repository replay.""" +import copy +import importlib.util +import itertools +import unittest +from pathlib import Path + +spec = importlib.util.spec_from_file_location('two_writer', Path(__file__).with_name('verify.py')) +v = importlib.util.module_from_spec(spec) +spec.loader.exec_module(v) + + +class BoundaryTests(unittest.TestCase): + @classmethod + def setUpClass(cls): + cls.word = v.load_word() + cls.count, cls.edges, cls.first, cls.writers, cls.snapshots = v.star_domain(cls.word) + + def test_two_writer_boolean_identity(self): + # Three arbitrary dirty values; the middle gate changes the later donor. + # Independent scalar Boolean interpreter, all eight input assignments. + for x, a, b in itertools.product((0, 1), repeat=3): + pending = a ^ b + pending ^= x + later_leaf = b ^ x + pending ^= later_leaf + self.assertEqual(pending, a) + # Assuming the second donor is still the original b is false. + if x: + self.assertNotEqual((a ^ b ^ x) ^ b, a) + + def test_source_pair_and_rows(self): + self.assertIn((231, 459), self.edges) + self.assertEqual(self.writers[231], [9, 527]) + self.assertEqual(self.first[231], 528) + self.assertEqual(self.snapshots[9][109], 1 << 109) + self.assertEqual(self.snapshots[527][459], (1 << 109) ^ (1 << 459)) + + def test_rational_frame_failure(self): + candidate = v.candidate(self.word, (-1, 231, 459, 9, 527, 109, 459), self.count) + paths, error = v.Geometry(self.word).paths(candidate) + self.assertIsNone(paths) + self.assertEqual(error, {'section': 'A', 'gate': 9, 'register': 109, + 'old_frame': 111, 'new_frame': 6, 'outside_label': 109, + 'receiving_rank': 1, 'extended_rank': 2}) + self.assertEqual(v.literal(candidate), v.wanted(candidate)) + + def test_clean_only_false_acceptance(self): + candidate = v.omit_restoration(self.word, 231, 459, self.count) + actual, target = v.literal(candidate), v.wanted(candidate) + self.assertNotEqual(actual, target) + # Projecting away every dirty-helper input hides the missing restoration. + clean_mask = (1 << (2 * self.word['v'])) - 1 + self.assertEqual([r & clean_mask for r in actual], [r & clean_mask for r in target]) + + def test_second_donor_uniqueness(self): + rows = self.snapshots[527] + required = self.snapshots[9][109] ^ (1 << 459) + self.assertEqual([r for r, row in enumerate(rows) if row == required], [459]) + + def test_source_pin_corruption(self): + v.check_pin(b'unchanged', v.sha(b'unchanged')) + with self.assertRaisesRegex(ValueError, 'source pin mismatch'): + v.check_pin(b'changed', v.sha(b'unchanged')) + + def test_domain_rejects_overlapping_stars(self): + candidate = copy.deepcopy(self.word) + candidate['D'][0][2] = [candidate['D'][0][1]] + candidate['D'][-1][2] = [candidate['D'][-1][1]] + with self.assertRaisesRegex(ValueError, 'disjoint star leaves'): + v.star_domain(candidate) + + +if __name__ == '__main__': + unittest.main() diff --git a/research/r439-two-writer-absorption/verify.py b/research/r439-two-writer-absorption/verify.py new file mode 100755 index 0000000000..3d900e2162 --- /dev/null +++ b/research/r439-two-writer-absorption/verify.py @@ -0,0 +1,369 @@ +#!/usr/bin/env python3 +# SPDX-License-Identifier: Apache-2.0 +# Adam Gall, with substantial OpenAI Codex assistance. +"""Exact two-writer dirty compensation census; no supplier admission or new kappa. + +No predecessor executable is imported. Source geometry implements the unchanged +nested-frame law specified in the pinned eumemic xcheck_p.py, lines 18--23. +The single-writer positive fixture is credited to Sagar Thakrar / PR1868. +""" +import argparse +import copy +import gzip +import hashlib +import itertools +import json +from collections import Counter +from fractions import Fraction +from pathlib import Path + +HERE = Path(__file__).resolve().parent +ROOT = HERE.parents[1] +WORD = 'research/r439-bsection-stars-seed01/candidate.json.gz' +SPEC = 'research/e8-bit-p111-tuned/code/xcheck_p.py' +INPUTS = [SPEC, WORD] +PROFILE = {1: 1323, 2: 1342, 3: 79, 4: 44, 5: 102, 6: 11, 7: 19} + + +def need(ok, message): + if not ok: + raise ValueError(message) + + +def canonical(value): + return json.dumps(value, sort_keys=True, separators=(',', ':')).encode() + + +def sha(data): + return hashlib.sha256(data).hexdigest() + + +def check_pin(data, expected): + need(sha(data) == expected, 'source pin mismatch') + + +def load_word(): + manifest = json.loads((HERE / 'SOURCES.json').read_text(encoding='utf-8')) + pins = {x['path']: x['sha256'] for x in manifest['files'] if x['usage'] == 'input'} + need(sorted(pins) == INPUTS, 'complete source inventory') + need(json.loads((HERE / 'VERIFY.json').read_text(encoding='utf-8'))['inputs'] == INPUTS, + 'complete replay inventory') + data = {} + for path in INPUTS: + data[path] = (ROOT / path).read_bytes() + check_pin(data[path], pins[path]) + return json.loads(gzip.decompress(data[WORD])) + + +def xor_rows(rows, gates): + for _, dest, donors in gates: + for donor in donors: + rows[dest] ^= rows[donor] + return rows + + +def stream(word): + n = 2 * word['v'] + word['R'] + ops = [('add', d, s) for section in 'DA' for _, d, ss in word[section] for s in ss] + for j, centre in enumerate(word['ret']): + ops.append(('copy', n, centre)) + ops.extend(('add', word['v'] + t, n) for t, row in enumerate(word['scatter']) if j in row) + ops.append(('erase', n, centre)) + ops.extend(('add', d, s) for section in 'BC' for _, d, ss in word[section] for s in ss) + return ops + + +def literal(word, reverse=False): + n = 2 * word['v'] + word['R'] + rows = [1 << k for k in range(n)] + [0] + ops = stream(word) + for kind, d, s in reversed(ops) if reverse else ops: + if reverse: + kind = {'copy': 'erase', 'erase': 'copy'}.get(kind, kind) + if kind == 'add': + rows[d] ^= rows[s] + elif kind == 'copy': + need(rows[d] == 0, 'COPY requires a clean temporary') + rows[d] = rows[s] + else: + need(rows[d] == rows[s], 'ERASE requires matching current value') + rows[d] = 0 + need(rows[-1] == 0, 'unrestored temporary') + return rows[:-1] + + +def wanted(word): + v = word['v'] + return [(1 << k) ^ ((1 << (k - v)) if v <= k < 2 * v else 0) + for k in range(2 * v + word['R'])] + + +def bill(word): + ops = stream(word) + maxima = [] + for reverse in (False, True): + vals = [1] * (2 * word['v'] + word['R'] + 1) + peak = 1 + for kind, d, s in reversed(ops) if reverse else ops: + if reverse: + kind = {'copy': 'erase', 'erase': 'copy'}.get(kind, kind) + vals[d] = vals[d] + vals[s] if kind == 'add' else vals[s] if kind == 'copy' else 0 + peak = max(peak, vals[d]) + maxima.append(peak) + f, b = maxima + return {'ADDs': sum(k == 'add' for k, _, _ in ops), 'F': f, 'B': b, + 'local_bound': str(64 * f**3 * b**2), + 'nine_stage_bound': str((1 << 18) * f**5 * b**4)} + + +def rref(rows): + rows = [[Fraction(x) for x in row] for row in rows] + pivot = 0 + for col in range(8): + found = next((k for k in range(pivot, len(rows)) if rows[k][col]), None) + if found is None: + continue + rows[pivot], rows[found] = rows[found], rows[pivot] + divisor = rows[pivot][col] + rows[pivot] = [x / divisor for x in rows[pivot]] + for k in range(len(rows)): + if k != pivot and rows[k][col]: + multiplier = rows[k][col] + rows[k] = [x - multiplier * y for x, y in zip(rows[k], rows[pivot])] + pivot += 1 + return rows[:pivot] + + +class Geometry: + def __init__(self, word): + self.lines = word['lines'] + self.frames = [frozenset(x) for x in word['frames']] + self.bases = {} + + def basis(self, frame): + if frame not in self.bases: + self.bases[frame] = rref([self.lines[k] for k in sorted(self.frames[frame])]) + return self.bases[frame] + + def failure_witness(self, section, gate, register, old, new): + basis = self.basis(new) + # The set test alone is not the certificate: require an actual rational + # source-label vector outside the proposed receiving frame's span. + for label in sorted(self.frames[old] - self.frames[new]): + extended = rref(basis + [self.lines[label]]) + if len(extended) > len(basis): + return {'section': section, 'gate': gate, 'register': register, + 'old_frame': old, 'new_frame': new, 'outside_label': label, + 'receiving_rank': len(basis), 'extended_rank': len(extended)} + raise ValueError('frame-set rejection lacks a rational noncontainment witness') + + def paths(self, word): + paths = [[f] for f in word['start']] + for section in 'DABC': + for index, (frame, dest, donors) in enumerate(word[section]): + for register in (dest, *donors): + old = paths[register][-1] + if old == frame: + continue + if not self.frames[old] < self.frames[frame]: + return None, self.failure_witness(section, index, register, old, frame) + paths[register].append(frame) + for register, frame in enumerate(word['final']): + old = paths[register][-1] + if old != frame: + if not self.frames[old] < self.frames[frame]: + return None, self.failure_witness('final', -1, register, old, frame) + paths[register].append(frame) + return paths, None + + def paid_profile(self, word): + paths, error = self.paths(word) + need(error is None, 'no paid profile admitted for an illegal frame path') + counts = Counter(len(self.basis(b)) - len(self.basis(a)) + for path in paths for a, b in zip(path, path[1:])) + current = list(word['start']) + for section in 'DA': + for frame, dest, donors in word[section]: + for register in (dest, *donors): + current[register] = frame + counts.update(len(self.basis(current[r])) for r in word['ret']) + return dict(sorted(counts.items())) + + +def star_domain(word): + v, d = word['v'], word['D'] + count = sum(dest >= 2 * v for _, dest, _ in d) + need(count % 2 == 0, 'paired helper preparation/restoration') + count //= 2 + prep, rest = d[:count], d[-count:] + need(rest == prep[::-1], 'literal star restoration suffix') + need(all(f == 0 and a >= 2*v and len(ss) == 1 and ss[0] >= 2*v for f,a,ss in prep), + 'source-free zero-frame helper stars') + need(all(f == 0 and v <= a < 2*v for f,a,ss in d[count:-count]), 'target-only middle') + edges = [(a, ss[0]) for _, a, ss in rest] + centres = {a for a, _ in edges} + leaves = [b for _, b in edges] + need(len(set(leaves)) == len(leaves) and centres.isdisjoint(leaves), 'disjoint star leaves') + first = {a: next((i for i, (_, _, ss) in enumerate(word['A']) if a in ss), len(word['A'])) + for a in centres} + writers = {a: [i for i, (_, d, _) in enumerate(word['A'][:first[a]]) if d == a] for a in centres} + n = 2*v + word['R'] + rows = xor_rows([1 << k for k in range(n)], word['D']) + need(all(rows[k] == 1 << k for k in list(range(v)) + list(range(2*v, n))), + 'D restores every source/helper') + need(all(d not in ss and len(ss) == len(set(ss)) and all(not v <= s < 2*v for s in ss) + for _, d, ss in word['A']), 'target-free A donor sets without self reads') + cuts = {i for ws in writers.values() for i in ws} + snapshots = {} + for i, gate in enumerate(word['A']): + if i in cuts: + need(len(set(rows)) == n and all(rows), 'invertible state has distinct nonzero rows') + snapshots[i] = tuple(rows) + xor_rows(rows, [gate]) + return count, edges, first, writers, snapshots + + +def omit_restoration(word, centre, leaf, count): + out = copy.deepcopy(word) + matches = [k for k in range(len(out['D'])-count, len(out['D'])) + if out['D'][k] == [0, centre, [leaf]]] + need(len(matches) == 1, 'unique specified terminal restoration') + out['D'].pop(matches[0]) + return out + + +def candidate(word, solution, count): + _, centre, leaf, i, j, s, t = solution + out = omit_restoration(word, centre, leaf, count) + for gate, donor in ((i, s), (j, t)): + out['A'][gate][2] = sorted(set(out['A'][gate][2]) ^ {donor}) + return out + + +def solve(word): + n, v = 2 * word['v'] + word['R'], word['v'] + count, edges, first, writers, snapshots = star_domain(word) + solutions, trials, pairs = [], 0, 0 + for centre, leaf in sorted(edges): + donors = [r for r in range(n) if r != centre and not v <= r < 2*v] + for i, j in itertools.combinations(writers[centre], 2): + pairs += 1 + lookup = {snapshots[j][r]: r for r in donors} + need(len(lookup) == len(donors), 'unique second donor lookup') + for s in donors: + trials += 1 + # r_i(s) XOR r_j(t) = e_leaf is necessary and sufficient. + t = lookup.get(snapshots[i][s] ^ (1 << leaf)) + if t is not None: + delta = -1 + (1 - 2*(s in word['A'][i][2])) + (1 - 2*(t in word['A'][j][2])) + solutions.append((delta, centre, leaf, i, j, s, t)) + return count, pairs, trials, sorted(solutions), first, writers, snapshots + + +def expect_rejection(function): + try: + function() + except ValueError: + return + raise ValueError('negative control was accepted') + + +def run(word): + need((word['v'], word['R']) == (111, 439), 'fixed source dimensions') + target = wanted(word) + need(literal(word) == target and literal(word, True) == target, 'baseline dirty boundary') + geo = Geometry(word) + paths, error = geo.paths(word) + need(error is None, 'baseline frame legality') + profile = geo.paid_profile(word) + need(profile == PROFILE, 'baseline complete paid histogram') + base_bill = bill(word) + need(base_bill['ADDs'] == 11500, 'baseline scalar invoice') + count, pairs, trials, solutions, first, writers, snapshots = solve(word) + rows = [] + for solution in solutions: + change, centre, leaf, i, j, s, t = solution + out = candidate(word, solution, count) + need(literal(out) == target and literal(out, True) == target, 'full dirty completion') + need(sum(len(ss) for sec in 'DABC' for _, _, ss in out[sec]) - + sum(len(ss) for sec in 'DABC' for _, _, ss in word[sec]) == change, 'complete ADD delta') + _, error = geo.paths(out) + need(error is not None, 'a frame-legal two-toggle completion exists') + rows.append({'solution': list(solution), 'failure': error}) + need((count, pairs, trials, len(solutions)) == (236, 414, 227286, 314), 'complete census counts') + need(Counter(s[0] for s in solutions) == {-1: 52, 1: 262}, 'scalar saving census') + + # Source-bound legal single-writer positive: PR1868, not a new construction. + positive = omit_restoration(word, 506, 374, count) + need(positive['A'][288] == [71, 506, [374]], 'positive fixture source gate') + positive['A'][288][2] = [69] + need(literal(positive) == target and literal(positive, True) == target, 'single-writer positive boundary') + need(geo.paths(positive)[0] == paths and geo.paid_profile(positive) == profile, + 'single-writer positive frame paths and full paid profile') + positive_bill = bill(positive) + need(positive_bill['ADDs'] == 11499, 'single-writer positive invoice') + for priced in (base_bill, positive_bill): + need(int(priced['local_bound']) < 1 << 104 and int(priced['nine_stage_bound']) < 1 << 192, + 'source control precision guards') + + fixture = solutions[0] + need(fixture == (-1, 231, 459, 9, 527, 109, 459), 'canonical counterexample') + _, centre, leaf, i, j, s, t = fixture + need(snapshots[i][s] == 1 << 109 and snapshots[j][t] == (1 << 109) ^ (1 << 459), + 'explicit arbitrary-dirty correction rows') + full = candidate(word, fixture, count) + controls = [] + for name, altered in [('omit_both_corrections', omit_restoration(word, centre, leaf, count)), + ('keep_restoration', copy.deepcopy(full)), + ('omit_first_correction', copy.deepcopy(full)), + ('omit_second_correction', copy.deepcopy(full)), + ('wrong_first_donor', copy.deepcopy(full))]: + if name == 'keep_restoration': + altered['D'] = copy.deepcopy(word['D']) + elif name == 'omit_first_correction': + altered['A'][i] = copy.deepcopy(word['A'][i]) + elif name == 'omit_second_correction': + altered['A'][j] = copy.deepcopy(word['A'][j]) + elif name == 'wrong_first_donor': + altered['A'][i][2] = sorted(set(altered['A'][i][2]) ^ {109, 108}) + actual = literal(altered) + need(actual != target, 'boundary control: ' + name) + register = next(k for k, (a, b) in enumerate(zip(actual, target)) if a != b) + defect = actual[register] ^ target[register] + column = (defect & -defect).bit_length() - 1 + controls.append({'name': name, 'first_bad_register': register, 'witness_input_column': column}) + # Both exact frame failure and integrity checks must remain active with -O. + expect_rejection(lambda: geo.paid_profile(full)) + expect_rejection(lambda: check_pin(b'changed source', sha(b'source'))) + controls.extend([{'name': 'illegal_frame_cannot_be_billed'}, {'name': 'source_corruption'}]) + return {'format': 'r439-two-writer-absorption/1', 'columns': 661, + 'star_edges': count, 'edge_writer_pair_counts': dict(sorted(Counter( + len(list(itertools.combinations(writers[a], 2))) for a, _ in star_domain(word)[1]).items())), + 'writer_pairs': pairs, 'donors_per_first_writer': 549, 'first_donor_trials': trials, + 'exact_completions': len(solutions), 'scalar_improving': 52, 'frame_legal': 0, + 'domain': [{'centre': a, 'leaf': b, 'first_read_A': first[a], 'writers_A': writers[a]} + for a, b in sorted(star_domain(word)[1])], + 'solutions': rows, 'baseline_bill': base_bill, 'paid_profile': profile, + 'rank_mass': sum(r * c for r, c in profile.items()), + 'single_writer_positive': {'credit': 'Sagar Thakrar / PR1868', 'centre': 506, 'leaf': 374, + 'A_gate': 288, 'new_donors': [69], 'bill': positive_bill, + 'same_frame_paths_and_profile': True}, + 'controls': controls, 'no_new_kappa': True} + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument('--write', action='store_true', help='regenerate only this package certificate') + args = parser.parse_args() + # Normalize integer histogram keys before canonical comparison with JSON. + report = json.loads(canonical(run(load_word()))) + path = HERE / 'CERTIFICATE.json' + if args.write: + path.write_text(json.dumps(report, sort_keys=True, indent=2) + '\n', encoding='utf-8') + else: + need(canonical(report) == canonical(json.loads(path.read_text(encoding='utf-8'))), 'certificate mismatch') + print('PASS_R439_TWO_WRITER_ABSORPTION pairs=414 trials=227286 exact=314 scalar_saving=52 legal=0 columns=661 controls=7 no_new_kappa') + + +if __name__ == '__main__': + main()