{"id":4574,"job_id":9388,"problem_id":6,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Tiny-input MD5 (<10 B/side): a validated signed-difference trail estimator finds no sparse 32-bit E13 trail for the padding differences\nDirection: MD5 collisions with both inputs under 10 bytes. Step-reduced metric: largest s with a produced collision stays **s = 2**; next is s = 15 (8 vs 9 bytes), 2^65 generic.\n\n## Question\nThe s = 15 route needs, for pairs that share m0, m1, the round-1 transition from (0,0,0,delta) after step 2 to (d,0,0,0) after step 13 (Q12..Q14 equal, dQ11 = d = -8*dlen).\nEarlier toys (#4567) showed single-d favoring that does not fade with word size n. Hypothesis tested here: sparse signed-difference trails have a weight that\ndoes not grow with n, which could put the 32-bit probability far above the generic 2^-128. Pre-registered (prereg.md): gain < 6 bits at 32 bits.\n\n## Method\n- Markov trail model of steps 3..13: state (additive dQ[i-3], BSD of Q[i-2], Q[i-1], Q[i]); F output digits exact per bit; rotation carries exact (4-case formula);\n  BSD of the new word 2^-weight. Threshold meet-in-the-middle sum over trails (forward from delta, backward from (d,0,0,0), join after step 7). Python and C agree exactly.\n- Bug caught by a C sanity check and fixed: an F bit with only x active has three outcomes (y - z: -1, 0, +1 with 1/4, 1/2, 1/4).\n\n## Validation\n- V1 (pre-registered 10%): parts exact by full enumeration (BSD reps, rotation outcomes); the composed one-step kernel is within 17.5% on sparse states but up to 17x off on dense ones. FAILED as stated.\n- V2 vs #4567 exhaustive toy counts (ratios vs generic): of 10 favored cells (>= 8x), 5 within 2x, 8 within 3.5x, 2 missed (G d=512, H d=508); the generic set B is predicted 0.\n- V3, new exhaustive fixed-delta count (n = 8, rotations 7,4,1,6, delta = +1, 8 constant sets): d = -16 observed 92x generic, predicted 29.7x.\n- Conclusion: a conservative lower bound (usually 2-4x under) that does find strongly favored sparse cells.\n\n## Results\n- 32 bits, real rotations, delta in {+1, 2^24, -2^24} (the padding families), d in {-8..-72}: **no trail found**. Excluded: trails whose forward half (steps 3..7) and backward half (steps 8..13) are both >= 2^-32 (delta = +1, d = -8) or 2^-30 (the others).\n- Scaling with real rotations mod n (delta = +1, d = -8k, budgets 30/30): n = 8 +4.9 bits (d = -16, 2^-27.1), n = 9 -6.0, n = 10 -1.5, n = 13, 14, 20, 24 none.\n  n = 16 with rotations (7,12,1,6) gives d = -64 at 2^-46.9, **+17.2 bits** over generic 2^-64.\n- Reading: favoring through sparse trails depends on the rotation pattern (small rotations 1 and 6), not on n alone. The real pattern (7,12,17,22) shows none at n = 20, 24, 32 within budget.\n  H5 (gain < 6 bits) is not decided: unbalanced trails with one half below 2^-32 are not excluded. The s = 15 cost stays 2^65 generic (best case 2^60.7), and the metric stays s = 2.\n\n## Dead ends\n- Backward-only top-K beam: myopic (2.8x vs 34.8x on the best toy cell). Replaced by the threshold MITM.\n\n## Next\nLean 32-bit search for unbalanced trails (shift the join point, asymmetric budgets, best-trail DFS), plus a sampled positive control of the n = 16 trail.\n","patch":null,"cpu_hours":2.7,"hashes":{"job9388-dcount.c":"72a37b93202c17d219d6c9cebe42501e72a3db7733bc1edc691a3d6d6a0a8c46","job9388-trail.py":"4cda423ac189d2857bc398d461b4d22764d501402f3fedee7d02e610d74b496a","job9388-trailc.c":"d3edb9c61dedde159ac15ba6098876242a04eaa0844901333e05e01dbb1c66a3","job9388-prereg.md":"6fbef20f20aaa091e918066cd51fdc00a9132222c6b219fdb6287549f845f052","job9388-summary.json":"41237658e7a124f1a4703e59a3deab1e094a47df7ae0b312762a6f1e98e93ed9"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-18T15:00:36.446Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[4567,4566],"messages":[7489]},"tokens":{"log":"summary","input":138,"models":{"claude-opus-5-5":108181},"output":108181,"source":"reported","entries":0,"cache_read":6985165,"cache_write":155067,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe (job 9388)\ncc -O3 -march=native -o trailc trailc.c -lm ; cc -O3 -march=native -o dcount dcount.c\nCross-check Python vs C: ./xcheck_c.sh (identical state counts and log2p).\nV1: python3 v1.py ; python3 v1s.py   (needs trail.py)\nV2: ./v2batch.sh then python3 v2score.py (needs meas_ratios.json = #4567 per-d ratios vs generic)\nV3: ./dcount 8 7 4 1 6 1 SEED for SEED = 1001..1008 (2^32 pairs each, about 5 min per seed)\n32-bit: ./trailc 32 7 12 17 22 1 12 Wf Wb 8 DELTA d1 d2 ...  (args: n r0..r3 slack wmax Wf Wb mid delta d...; delta = -1 means all deltas)\nScaling: ./scale.sh 30\nHardware: Apple M1 Mac mini, Apple clang -O3. trailc 32/32 run: 4.6 s, 2.0 GB.","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":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_62911f8692f18f2c01e7d934","run_id":"run_50f4a14778404b59fc809c9b","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":{"schema":"research-evidence-v1","scopes":[{"key":"md5-tiny-e13-bsd-trail-estimator","kind":"finite","domain_md":"MD5 round 1 (F) steps 3..13 with zero message difference; pairs that differ only in Q3 after step 2; target (d,0,0,0) after step 13; n-bit toys and 32-bit.","statement_md":"A signed-difference Markov trail sum (state: additive dQ[i-3] and BSDs of Q[i-2..i]; exact F digits, exact rotation carries, 2^-w BSD cost; threshold meet-in-the-middle) is a conservative lower bound for the E13 transition: of 10 strongly favored toy cells (>= 8x, exhaustive counts of #4567) 5 are within 2x, 8 within 3.5x, 2 missed (G d=512, H d=508), and a generic set is predicted 0. A new exhaustive fixed-delta count (n = 8, rotations 7,4,1,6, delta = +1) gives 92x at d = -16 against 29.7x predicted. The one-step kernel is within 17.5% on sparse states and up to 17x off on dense ones.","assumptions_md":"Bits of different words are treated as independent (Markov); constants K are averaged. Budgets 18/28 and 20/30 bits per half for toys.","artifact_sha256":["4cda423ac189d2857bc398d461b4d22764d501402f3fedee7d02e610d74b496a","b59104466d34368a2f573eb92acd41207c9f542a2e5490207d66c7aecefeef4c","d3edb9c61dedde159ac15ba6098876242a04eaa0844901333e05e01dbb1c66a3","72a37b93202c17d219d6c9cebe42501e72a3db7733bc1edc691a3d6d6a0a8c46","d6e1eb30415254671da507a972af4a808754e7afe55b327bc585d2a4a054bcc7","9437be4457d83a1a67ec2bd8a8e3661684d2d966c16ef84505c94e7c6f4369d6","5b488601fd18786b5284c6e68b0a977745a45cab864354f7611ac54f0f35f44c","6500baf2eb099d5d697fa2a1b6fdfd6da49bbc61eb18573a3577fc7db1e6dcaa","b80f87fd8ee4cc147eed5544adeed25008ca2a6c9b2cff3074f5ac43ecb41595","ab6cb9e62c2d0d3b0ebc5081efe1528a981d555eee1bf25a1f098786a4c33738","30bce87b758ca6b4d6719e3a7b15b0cf2a412a0d47c0113e40676c8d7f8a5161","3eab685e498f61e75a4c108f98632185d4646ad49b25980d6b5dfe47f613caba","1ea1819f48e4d21b0dcb7022a90758636d0030e521208b7c6a3f68614e0052b8","20eb4c93231449ee93044068f7c38593df335ecf08233af8c76c40248eff7cd1","632689c5c8f0b8e1d0d96bf6a0cd332901114007a0e5d8227404fe2e19a0c9c6","50ecab90cc9644c44d08940f4aa1c13654d6e2ad8bc7e02f374455d96a6a5c00","7919a3d5e0fdb9aa1a4f165689ad7fae28d2d84de5dfd9da60ac582523a8343f","6fbef20f20aaa091e918066cd51fdc00a9132222c6b219fdb6287549f845f052"],"transfer_conditions_md":"Any MD4-family round-1 step Q[t+1] = Q[t] + rotl(Q[t-3] + f + K, r) with a bitwise f; dense differences need a better kernel."},{"key":"md5-tiny-e13-32bit-sparse-trail-null","kind":"finite","domain_md":"MD5 round 1 (F) steps 3..13 with zero message difference; pairs that differ only in Q3 after step 2; target (d,0,0,0) after step 13; n-bit toys and 32-bit.","statement_md":"For 32-bit MD5 with the real rotations and the padding differences of the s = 15 tiny-input route (delta = +1, 2^24, -2^24 in Q3; d = -8..-72), no E13 trail exists whose forward half (steps 3..7) and backward half (steps 8..13) both have probability >= 2^-32 (delta = +1, d = -8) or >= 2^-30 (the others). With real rotations mod n, sparse favoring appears at n = 8 (+4.9 bits) and with the (7,12,1,6) pattern at n = 16 (+17.2 bits, 2^-46.9), but none at n = 13, 14, 20, 24 within 2^-30 per half. Favoring depends on the rotation pattern, not on n alone. The step-reduced metric stays s = 2.","assumptions_md":"Markov trail model of the first scope (a lower bound); BSD slack 1, at most 12 digits per word at 32 bits; unbalanced trails with one half below the budget are not excluded.","artifact_sha256":["d3edb9c61dedde159ac15ba6098876242a04eaa0844901333e05e01dbb1c66a3","45e7f815126c726a22930adb78468064543a30699d347cb10037889e0711d04f","4736272f4acf1cc475549b01fc64764261a542967efa42d9716c6c6dd43011f5","0a7fc66ca499a858690414bf8cf55ea5b1e86c632ad886af35cad9720cfb0e7f","db0387d6df589bb2e68e32ad4c1ffccf35f77e87de6cded5defb088743c7a9d5","41237658e7a124f1a4703e59a3deab1e094a47df7ae0b312762a6f1e98e93ed9","6fbef20f20aaa091e918066cd51fdc00a9132222c6b219fdb6287549f845f052","a2f3eae0531edb5153ef98a7e0ed6fea500430c55d793bd5b3f7aa4f1f6be407"],"transfer_conditions_md":"Changes if a lean search finds an unbalanced 32-bit trail above 2^-122 (a >= 6-bit gain) for these deltas."}],"topic_ids":[]},"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"Benjaminsen","job_brief":"Consult the relevant shared local evidence. 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Structural measurement only, no search.","created_at":"2026-10-18T14:59:08.194Z","url":"/projects/md5/chat/messages/7489"}]}