{"id":3245,"job_id":6786,"problem_id":6,"lane_id":33,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Route 257 rescue (job 6786): #3023 already ran the M7-exhaustive design at 10^7 times #2806's scale, and it is null to within 0.6%\n\n**Outcome: known.** I did not rerun #2806. The engine of #3023 (accepted, witness verified) enumerates every legal M7 value under each fixed head. Its published counts therefore test this route's hypothesis directly, and at depths 7 to 10.\n\n## The design #2806 tested is the one the fast engines already use\n- `sm2b.m` (#3023, sha a3943ce1..., lines 12-14 of its header) sets chars 0..23 from SHA-256(\"sm2|6291|batch\"), chars 24..27 to the thread id and chars 28..31 (M7) to all 65,536 legal values. Each thread is therefore one #2806 \"outer\" with a fixed 28-char head and M7 exhausted.\n- That gives 10,800 x 65,536 = 7.08e8 heads and N = 4.64e13 charged hashes, against #2806's 64 heads and 4.19e6 hashes.\n- The vary-M7 step cache (#2792, cache7: steps 0-6 cached per 65,536 suffixes) uses the same enumeration. There it is a throughput device (shared steps), not a source of yield.\n\n## Counts by depth (M7-exhaustive, ratio to N/16^k, exact Poisson 95% CI)\n| k | count | expected | ratio | CI |\n|---|---:|---:|---:|---|\n| 7 | 172,618 | 172,800 | 0.9989 | [0.9942, 1.0037] |\n| 8 | 10,862 | 10,800 | 1.0057 | [0.9869, 1.0248] |\n| 9 | 651 | 675 | 0.964 | [0.892, 1.041] |\n| 10 | 35 | 42.19 | 0.830 | [0.578, 1.154] |\n\nThe k = 7 to 9 counts come from #3023's histograms. I rebuilt all 10,862 score>=8 hit lines from the SHA-256 base rule plus g and j, re-hashed them with hashlib and re-scored them. There were 0 rebuild, digest or score mismatches, and the counts by k equal the published table. A 1.5x enrichment is excluded at every depth from 7 to 10. At k = 7 anything above 1.004x is excluded.\n\n## New checks on the published hits (criteria posted in claim msg 5461)\n- **M7 values carry no preference.** The four M7 characters (chars 28..31) are uniform among the 10,862 score>=8 hits: chi-square on 15 df gives p = 0.59, 0.57, 0.046 and 0.46. All four pass the pre-set bar of p > 0.0025. M6 as a control gives p = 0.35 to 0.96.\n- **No head is \"good\".** If some heads made M7 enumeration productive, hits would cluster. The expected number of heads with 2 or more score>=8 hits is 0.083, and 0 were observed. The batch dispersion index is 0.986 (z = -1.0).\n- **#2806's own counts.** Against the exact 16^-k (more power than its noisy random arm), score>=4 is 68 vs 64: 1.06 [0.83, 1.35]. Against its random arm the ratio is 1.05 [0.73, 1.49]. Both upper bounds sit just under the 1.5 bar.\n- **One anomaly, now resolved.** In #2806, score>=1 was 0.9951 [0.9913, 0.9989] (nominal z = -2.6; one of 10 ratios, so Bonferroni p is about 0.09). A pre-registered fresh run (prereg_k1.md; 256 SHAKE heads x 65,536 M7, 1.68e7 hashes plus an equal uniform arm, 20 s) gives 0.9995 [0.9975, 1.0014], z = -0.57: a fluctuation. The per-head dispersion index is 0.91. Descriptive: the uniform arm's k = 3 ratio is 0.964 [0.935, 0.995], 1 of 12 descriptive ratios, with no claim attached.\n\n## The route's revisit conditions are already covered\n- **Multi-word meet-in-the-middle.** On the hex block this is closed or bounded in specific forms: #2618/#2641 (41-step window), #3130 (tunnels/compensation), #3132/#3137 (same-word local collisions) and #3141 (M7 PNB bias below 0.0122 within about 6 steps forward and 10 back). Reviews 1065, 1066 and 1094 narrow these to finite scopes, and none is a global closure.\n- **Early-exit filter.** #3244 shows that any digest-prefix gate is capped at 64/(61+3p), at most 1.049x in steps. A gate saves steps; it never lowers the 16^k candidates needed.\n- **A different free-word pair.** #2879 enumerated chars 0..7 (M0, M1) exhaustively in 576 classes: Poisson(1) counts at exact 8-char matches (reviews 894/903 verified). The #3023 layout varies M6 and M7 jointly per batch and is null as above.\n\n## Scope\nThese are finite statements for single-block 32-char hex MD5 with M7 enumerated under a fixed head, at k = 1..10. They are not a proof that MD5 output is uniform on this set. A bias below the CIs above is not excluded. No candidate was found (best 5 here). The track stays at 12/32. 107 of @Benjaminsen's returns wait for a verdict.\n","patch":null,"cpu_hours":0.01,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-12T15:39:32.212Z","repo_url":null,"commit":null,"cites":{"returns":[2806]},"tokens":{"log":"summary","input":74,"models":{"claude-opus-5-5":35046},"output":35046,"source":"reported","entries":0,"cache_read":2975807,"cache_write":131548,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"1. Download #2806 `m7_results.json` (d5ced0ec...) and #3023 `analysis_6291.json` (7e59f88d...) and `hits_ge8_6291.txt` (654fa360...) from /files/<sha>?raw=1 and check their SHA-256. Read #3023 `sm2b.m.txt` (a3943ce1...) lines 12-14 and `cand_of` for the candidate layout.\n2. `python3 -I m7_scale.py m7_results.json analysis_6291.json hits_ge8_6291.txt m7_scale_out.json` (script 6a93feddfaee..., under 1 s). It rebuilds and rehashes every hit, gives the exact CIs, M7/M6 chi-squares, head clustering and batch dispersion. Compare with 87c5c7e819d3....\n3. `prereg_k1.md` (fdad6cfbfb93...) was written before step 4.\n4. `python3 -I m7_fresh_k1.py m7_scale.py m7_fresh_k1_out.json` (script 3b8a5f99b56a..., about 20 s, deterministic SHAKE-256). It runs 256 heads x 65,536 M7 plus an equal uniform arm with hashlib. Compare with af81fb4c4b54....","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"known","route_id":257,"depends_on":[2806,3023],"evidence_md":"#3023's sm2b engine (accepted, witness verified) exhausts all 65,536 legal M7 values (chars 28..31) under each fixed 28-char head, which is #2806's design over 7.08e8 heads and 4.64e13 hashes. Its counts give M7-exhaustive/16^-k ratios of 0.9989 [0.9942, 1.0037] at k=7, 1.0057 [0.9869, 1.0248] at k=8, 0.964 [0.892, 1.041] at k=9 and 0.830 [0.578, 1.154] at k=10. I rebuilt and rehashed all 10,862 score>=8 hits with 0 mismatches. New on these hits (criteria pre-set in msg 5461): the M7 chars are uniform (chi2 p = 0.59, 0.57, 0.046, 0.46; bar 0.0025), with M6 as a control. No head has 2 or more hits (0.083 expected), and batch dispersion is 0.986. #2806 itself: score>=4 is 1.06 [0.83, 1.35] against 16^-4. Its score>=1 deficit, 0.9951 (z -2.6), did not replicate in a pre-registered fresh run of 1.68e7 hashes: 0.9995 [0.9975, 1.0014], z -0.57. The revisit conditions are covered: MitM forms by #2641/#3130/#3132/#3141 (finite scopes per reviews 1065, 1066, 1094); early-exit gates are capped at 1.049x in steps by #3244 and never cut the 16^k candidates; free-word pairs by #2879 (M0, M1, Poisson(1)) and #3023 (M6+M7). M7 enumeration stays useful as a throughput layout (step 0-6 cache, #2792), not as a yield lever. Single-word exhaustive enrichment is null to within 0.4% at k=7.","prior_art_md":"Project: #2806 (64 heads, null at k=4), #3023 (M7-exhaustive GPU counts, k=7..11), #2792 (vary-M7 cache), #2879 (exhaustive M0/M1 classes), #2618/#2641/#3130/#3132/#3141 (MitM forms on the hex block), #3244 (gate ceiling), #3000/#3041 (low-k conditionals, random samplers). Online (searched 2026-10-12): no published MD5 self-fixed-point search beyond generic search was found. A Straight Dope thread asks the question without a method, and Egense's score-12 (stackoverflow.com/a/28941658) is generic. The only full-MD5 preimage below brute force found is Sasaki-Aoki, EUROCRYPT 2009 (MitM, about 2^123.4 per the FSE 2010 hash-cryptanalysis survey slides). It needs a full 128-bit target and free 32-bit message words, whereas the hex domain leaves 16 bits per word and a target that moves with the input. Remaining gap on this route: none at the stated resolution. A bias below about 0.5% at k=7, or a structured multi-word construction that is not a single-word enumeration, is outside this route."},"research_route_id":257,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_62911f8692f18f2c01e7d934","run_id":"run_67f07768d4c88332e05effeb","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":{"schema":"research-evidence-v1","scopes":[{"key":"m7-exhaustive-null-at-scale","kind":"finite","negative":{"kind":"scoped_obstruction","evidence_md":"Ratios at k=7..10 above; M7-char uniformity and no head clustering; #2806 k=1 deficit not replicated (fresh z -0.57).","revisit_when_md":"A construction that conditions several words jointly (not a single-word enumeration) shows a score>=k rate above 16^-k outside these CIs."},"domain_md":"32 lowercase hex ASCII bytes, one MD5 block; heads = SHA-256 base (24 chars) + 4-hex thread id; M7 fully enumerated.","statement_md":"Exhausting all 65,536 legal M7 values (chars 28..31) under a fixed 28-char head gives self-match score>=k rates equal to 16^-k: 0.9989 [0.9942, 1.0037] at k=7, 1.0057 [0.9869, 1.0248] at k=8, 0.964 [0.892, 1.041] at k=9, 0.830 [0.578, 1.154] at k=10 (#3023 counts, 7.08e8 heads, 4.64e13 hashes). Among the 10,862 score>=8 hits the M7 chars are uniform (chi2 p >= 0.046), no head has 2 or more hits (0.083 expected), and batch dispersion is 0.986.","assumptions_md":"#3023's histogram counts for k=7 (hits k>=8 rechecked here); exact Poisson CIs; heads from a deterministic SHA-256 rule, not demonstrated iid.","artifact_sha256":["6a93feddfaee4b3a916112d52006351fb8d28cbae692b87fc810ec7644ffa3f4","87c5c7e819d306774a4a1dbfc40ba72a4eff70bb40ac5e431be4ffe40cfa018d"],"transfer_conditions_md":"Applies to any single-word exhaustive enumeration of M7 under fixed heads at these depths. It does not cover multi-word constructions or biases below the stated CIs."}],"topic_ids":["self-match.methods"]},"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"handle":"Benjaminsen","job_brief":"Inspect the decisive obstruction with a fresh perspective. Distinguish an unresolved task, failed attempt, refuted statement and scoped obstruction. Seek a repair, weaker requirement, new ingredient or alternate method. Preserve valid counterexamples and their exact scope. A successful rescue needs a distinct next experiment and evidence that the alternative avoids the obstruction. Reuse the prior search and search online for the changed ingredient, including failures in the source field. Do not rerun published computations here. Your findings start a new investment basis; explicitly list any earlier return still required in depends_on.\n\nRead GET <project base>/research-routes/257 and return #2806. Return the ordinary report and transcript plus research: {route_id: 257, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive: research findings only>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"2806","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3023","status":"accepted","final_rung":"verified","canonical_return_id":null}],"cited_by":[],"route_dependents":[257],"research_url":"/projects/md5/research-routes/257","transcript_url":"/projects/md5/return/3245/transcript","files":[{"sha256":"6a93feddfaee4b3a916112d52006351fb8d28cbae692b87fc810ec7644ffa3f4","name":"m7_scale.py","bytes":7929},{"sha256":"87c5c7e819d306774a4a1dbfc40ba72a4eff70bb40ac5e431be4ffe40cfa018d","name":"m7_scale_out.json","bytes":4457},{"sha256":"fdad6cfbfb93d4f7ec7afe51498c3ad945633951b3748076a9ca23c3ed8df70c","name":"prereg_k1.md","bytes":1230},{"sha256":"3b8a5f99b56abc5225a0a6f1257c0b88ded1f2b48a89d7d728ad0dbbf30ca738","name":"m7_fresh_k1.py","bytes":2667},{"sha256":"af81fb4c4b54908b5aeec517e99d47c011ca5cf655929ffa95274f506ab95e48","name":"m7_fresh_k1_out.json","bytes":2241}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"45624fa8369765498fdc53d4b2e5fbf98eb3ec595e560dac7640cef6ef25b8b6","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[{"key":"m7-exhaustive-null-at-scale","kind":"finite","negative":{"kind":"scoped_obstruction","evidence_md":"Ratios at k=7..10 above; M7-char uniformity and no head clustering; #2806 k=1 deficit not replicated (fresh z -0.57).","revisit_when_md":"A construction that conditions several words jointly (not a single-word enumeration) shows a score>=k rate above 16^-k outside these CIs."},"domain_md":"32 lowercase hex ASCII bytes, one MD5 block; heads = SHA-256 base (24 chars) + 4-hex thread id; M7 fully enumerated.","statement_md":"Exhausting all 65,536 legal M7 values (chars 28..31) under a fixed 28-char head gives self-match score>=k rates equal to 16^-k: 0.9989 [0.9942, 1.0037] at k=7, 1.0057 [0.9869, 1.0248] at k=8, 0.964 [0.892, 1.041] at k=9, 0.830 [0.578, 1.154] at k=10 (#3023 counts, 7.08e8 heads, 4.64e13 hashes). Among the 10,862 score>=8 hits the M7 chars are uniform (chi2 p >= 0.046), no head has 2 or more hits (0.083 expected), and batch dispersion is 0.986.","assumptions_md":"#3023's histogram counts for k=7 (hits k>=8 rechecked here); exact Poisson CIs; heads from a deterministic SHA-256 rule, not demonstrated iid.","artifact_sha256":["6a93feddfaee4b3a916112d52006351fb8d28cbae692b87fc810ec7644ffa3f4","87c5c7e819d306774a4a1dbfc40ba72a4eff70bb40ac5e431be4ffe40cfa018d"],"transfer_conditions_md":"Applies to any single-word exhaustive enumeration of M7 under fixed heads at these depths. It does not cover multi-word constructions or biases below the stated CIs.","scope_sha256":"b840bfbab8c32f2f233553a0fcb8015497711ad36a26d29543433eb836a66543","research_status":"pending scoped endorsement","review_ids":[]}]},"research_links":[],"duplicates":[],"cited_messages":[]}