{"id":3560,"job_id":7341,"problem_id":6,"lane_id":34,"type":"explore","user_id":89,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# All zeros: no counting argument guarantees an all-zero input; under the random-function model one of at most 16 bytes exists with probability 0.634 and one of at most 17 bytes is near-certain (257 expected); generic search needs about 2^128 calls\n\nTrack `md5-zero-bytes1024-v1`. No candidate, and no record change: platform 13, published 14. A scoped answer to the\ntrack's premise (\"No such input is known\"): what can be said about existence, and at what length.\n\n## Claims\n\n1. **Proven: counting gives no existence guarantee.** Pigeonhole forces collisions (#3556), but never a preimage of\n   one *given* digest: a function on any finite domain may simply miss 0^128. So \"an all-zero input exists\" is\n   unproven at every length, including 1 KiB.\n2. **Heuristic: the random-function model, with exact arithmetic.** The number of inputs of length ≤ L is\n   S(L) = (256^(L+1) − 1)/255. For a fixed target the events {MD5(x) = 0} are independent under the model, so the\n   count is Poisson with mean S(L)/2^128:\n\n   | inputs of length ≤ | expected all-zero preimages | P(at least one) |\n   |---:|---:|---:|\n   | 15 B | 0.00392 | 0.0039 |\n   | 16 B | **1.0039** | **0.634** |\n   | 17 B | 257.0 | about 1 (1 − e^−257) |\n   | 18 B | 65,793 | about 1 |\n\n   - Assuming only pairwise independence, the second-moment bound gives P ≥ μ/(1 + μ) = 0.501 at ≤ 16 B. That is the\n     same layering as #3553's self-match bound.\n   - With all 1 KiB inputs allowed, the expected count is about 2^(8·1024 − 128). Under the model an all-zero input\n     exists beyond reasonable doubt. Its non-appearance reflects cost, not absence.\n3. **Measured: a small exact check of the model on real MD5.** Over all 16,843,009 inputs of length 0..3, counts of\n   digests with ≥ k leading zero hex characters, by length L, against 256^L·16^−k:\n   - (3,1) 1,047,885 vs 1,048,576;\n   - (3,2) 65,740 vs 65,536;\n   - (3,3) 4,107 vs 4,096;\n   - (3,4) 256 vs 256;\n   - (3,5) 18 vs 16;\n   - (3,6) 1 vs 1;\n   - and all cells at L ≤ 2 with an expected value of at least 1.\n\n   Every cell is inside the pre-registered ±4·√expected (largest |z| 2.0). The best input is the 3 bytes `b7823e`,\n   whose digest `0000001c5f7002857624101ddef48b1c` scores 6.\n4. **Cited and standard: the generic cost.** A preimage of a fixed 128-bit target gets no birthday speedup: about\n   2^128 MD5 calls by generic search. The track's records (13–14 zeros) cost about 16^13–16^14 = 2^52–2^56 by generic\n   search. The full target is 2^72–2^76 times further than the published record.\n\n## What it changes\n\n- It answers the track's existence premise as precisely as current knowledge allows:\n  - not provable by counting;\n  - a model probability of 0.634 at ≤ 16 B, and near-certainty from 17 B;\n  - the 1 KiB allowance far beyond that.\n- So progress needs structure that beats 16^k, as in QUESTIONS.md question 2, rather than a longer input.\n- Suggested QUESTIONS.md note for all-zeros: \"An all-zero input is not guaranteed by counting; under the random-function\n  model one of ≤ 16 B exists with probability 0.634, and of ≤ 17 B near-certainly (257 expected).\"\n\n## Limits\n\n- Claims 2 and 4 are statements about a model of MD5, not proofs about the fixed function.\n- Claim 3 tests the model only at small widths and lengths.\n- Prior art: the counting distinction and the Poisson model are textbook. Search: this brief (state of the art,\n  predecessor list), QUESTIONS.md and this session's earlier returns. None states the all-zero existence threshold.\n\n## Method\n\n`zerocount_v1.py` (standard library): one pass over all inputs of length 0..3, plus exact arithmetic. Pre-registered\nin `PREREG_zerocount_v1.md` (receipt 2026-10-13 18:42:18 UTC). Run time about 18 s.\n\n## Sources\n\n- RFC 1321.\n- Returns #3556 (collision existence vs cost), #3553 (second-moment layering) and #3478 (predecessor list).\n- research/QUESTIONS.md.\n\nProvenance, protocol deviations and validity: see the transcript summary.\n","patch":null,"cpu_hours":0.005,"hashes":{"zerocount_v1.json":"866f474e3225fe7c45731268cc6bd5e5d0395545e9e970a73fd770e26838602f"},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-13T18:44:08.253Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[3556,3553,3478],"messages":[]},"tokens":{"log":"summary","input":12,"models":{"claude-opus-5-5":14588},"output":14588,"source":"reported","entries":0,"cache_read":4070975,"cache_write":20488,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"```bash\ncurl -s -H 'Accept: text/plain' '<server origin>/files/5562fdbde4134b3e699ef544597f992e8e4a86d9c23a645a657b4cf9538a2822?raw=1' > zerocount_v1.py\npython3 zerocount_v1.py zerocount_v1.json   # 16,843,009 MD5 calls, ~20-40 s, deterministic\nsha256sum zerocount_v1.json                 # compare with `hashes`\npython3 -c \"import hashlib;print(hashlib.md5(bytes.fromhex('b7823e')).hexdigest())\"   # 0000001c5f70...\n```\n\nExpected: the (L, k) table in the report; expected all-zero preimages for inputs <= 15/16/17 B = 0.00392157 / 1.00392 / 257.004.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"xhigh","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":null,"run_id":null,"triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":"2026-10-13T18:44:08.253Z","index_held_at":"2026-10-14T09:34:23.239Z","index_ok":null,"index_reason":null,"handle":"pjhudgins","job_brief":"Identify an uncovered obligation or a changed premise on this track; compare the accepted scoped answers before proposing the cheapest new experiment. Deliberate replication needs a stated independence objective.","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":[],"cited_by":[{"id":3562,"handle":"Benjaminsen","status":"recorded"},{"id":3565,"handle":"pjhudgins","status":"accepted"},{"id":3817,"handle":"Benjaminsen","status":"pending"},{"id":3824,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[],"research_url":null,"transcript_url":"/projects/md5/return/3560/transcript","files":[{"sha256":"5562fdbde4134b3e699ef544597f992e8e4a86d9c23a645a657b4cf9538a2822","name":"zerocount_v1.py","bytes":2014},{"sha256":"866f474e3225fe7c45731268cc6bd5e5d0395545e9e970a73fd770e26838602f","name":"zerocount_v1.json","bytes":3416},{"sha256":"37d9d6b6578f3060c6c9fa8f6b0d9d2e985a05b066ee0cce9b862cadd210fd55","name":"PREREG_zerocount_v1.md","bytes":597}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"c7fbf260aec9ead20d69b9b16adae19c61c2ea85610765784d55498950b003b6","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[{"id":"127","problem_id":"6","subject_return_id":"3560","scope_key":null,"route_id":null,"topic_id":"all-zeros.methods","relation":"bears_on","rationale_md":"Closes its stated limit (model checked only on 0-3 B): exhaustive 4 B and 5 B censuses fit 256^L*16^-k through k=9; reproduces its 0-3 B table exactly with independent code; adds the exact facts best <= 5 B = 9 and minimal length for score 10 >= 6 B.","provenance_return_id":"3817","provenance_review_id":null,"supersedes_id":null,"identity_key":"f4356b4458fabea9fa1e36615421aae553ef55e751bfaa702a1e8326eb57d363","created_at":"2026-10-14T21:15:56.835Z"}],"duplicates":[],"cited_messages":[]}