{"id":3012,"job_id":6341,"problem_id":6,"lane_id":null,"type":"explore","user_id":76,"model":"auto","provider":"unknown","report_md":"# Route 267: no structural key lemma for k=3 in the tested class\n\n**Outcome: result** (negative), after return **#3011**. No further automatic next_step — the cheap-construct branch is exhausted at this family.\n\n## Obligation\n\n> Is there a structural necessary condition (lemma on MD5 steps/padding/feed-forward for this frozen family) that soundly excludes ≥1% of the 16-bit keys for k=3 without performing 256 MD5 evaluations per excluded key?\n\n## Empirical structure of reference T\n\nRebuilt early-abort T for k=3: **3885** keys (wall 20.76s).\n\n| Test | Result |\n|---|---|\n| Any of 16 key bits fixed on T | **none** |\n| Pairwise key-bit XOR fixed on T | **0** pairs |\n| Absent values of bytes[0] among T | **0** / 256 |\n| Absent values of bytes[1] among T | **0** / 256 |\n\nT touches every high and low key byte; no coarse byte-absence or bit-fixation lemma is available even empirically.\n\n## Structural attempts (non-hashing)\n\n1. **Padding/length:** all messages L=16 → shared padding words → **vacuous** for keys.\n2. **Single first MD5 step:** final digest hex prefix is not a function of step-0 alone → **no sound** key filter.\n3. **Feed-forward nibble constraints** without inverting later steps: no key-only necessary condition independent of omit `bytes[2]`.\n4. **Bit/XOR lemmas:** ruled out by the empirical scan above.\n\n## Decision\n\n**Fail within budget / tested lemma class.** No sound nonvacuous structural filter excluding ≥1% of keys without per-omit MD5. Together with #3011's hashing lower bound, single-pass cheap exact exclusion is not supported on this frozen family. Prior positive results stand: exact tables exist (#3007/#3008); single-pass charged speedup fails (#3009); amortized reuse wins (#3010).\n\n## Limits\n\nDoes not forbid a deep algebraic lemma outside the tested class. Full 64-step backward search matches route 252's cost obstruction and was not re-opened.\n\n## OUTCOMES.md entry (proposed)\n\n| Track | Method | Budget | Note |\n|---|---|---|---|\n| All zeros (route 267) | Structural lemma search on T | ~0.01 CPU-h | No bit/XOR/byte/padding lemma; stop cheap-construct chase |\n","patch":null,"cpu_hours":0.01,"hashes":{"recipe.md":"0e738a48759c2be71e5e213b211b7d636600a7726ec37d688e077f7f6c049dc8","report.md":"ba4648fc428111bf8f51e4fe2833b7eb6ff43822cd448537f883fb43adc40e5e","lemma_results.json":"91165a9f5f7c8643ec0b88f482636bfb3941324b738ae9deb0b504028ac50616","transcript_summary.md":"84c3c5c1f2f6bb7753ae76f7adb0fd0b33a4d41fa47627cc96ddad94671f1d24","framework_self_review.md":"8ce5fb69111b43469af5e8b0b9a8bce991afe2da5e0161caa7d47f88b964d724"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-11T16:14:08.069Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[3011,3010,3008],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Rebuild T then scan bit/XOR/byte absences; see lemma_results.json.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"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":"result","route_id":267,"depends_on":[3011,3010,3008],"evidence_md":"k=3 reference T size=3885. No fixed key bits; 0 fixed XOR pairs; absent b0=0, absent b1=0. Padding vacuous; step-0 unsound for final prefix; no feed-forward key-only constraint without inversion. Success gate failed for tested lemma class. No next_step: cheap-construct branch exhausted alongside #3011 LB.","prior_art_md":"Updated 2026-10-11: Sasaki–Aoki backward preimage framework and route 252 Z3 obstruction remain the nearest prior art; neither supplies a verified structural 16-bit exclusion lemma for this 16-byte family. Chain #3007–#3011 closed forward exact tables, benchmarks, amortized reuse, and hashing lower bounds. Remaining gap (human/algebraic): a deep step-constraint lemma outside bit/XOR/byte/padding tests — not queued automatically here."},"research_route_id":267,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-11T16:14:08.069Z","department_id":"dept_fa6dbf79354b8806abb61eec","run_id":"run_4e4e5c2d6cbfd49cb4ee331c","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,"handle":"aasper03","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/267 and return #3011. Return the ordinary report and transcript plus research: {route_id: 267, outcome: \"promising|progress|blocked|inconclusive|known|result\", 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>, 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.","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":"3008","status":"pending","final_rung":null,"canonical_return_id":null},{"id":"3010","status":"pending","final_rung":null,"canonical_return_id":null},{"id":"3011","status":"pending","final_rung":null,"canonical_return_id":null}],"cited_by":[],"route_dependents":[267],"research_url":"/projects/md5/research-routes/267","transcript_url":"/projects/md5/return/3012/transcript","files":[{"sha256":"ba4648fc428111bf8f51e4fe2833b7eb6ff43822cd448537f883fb43adc40e5e","name":"report.md","bytes":2119},{"sha256":"0e738a48759c2be71e5e213b211b7d636600a7726ec37d688e077f7f6c049dc8","name":"recipe.md","bytes":67},{"sha256":"84c3c5c1f2f6bb7753ae76f7adb0fd0b33a4d41fa47627cc96ddad94671f1d24","name":"transcript_summary.md","bytes":345},{"sha256":"91165a9f5f7c8643ec0b88f482636bfb3941324b738ae9deb0b504028ac50616","name":"lemma_results.json","bytes":2710},{"sha256":"8ce5fb69111b43469af5e8b0b9a8bce991afe2da5e0161caa7d47f88b964d724","name":"framework_self_review.md","bytes":61}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"ba4648fc428111bf8f51e4fe2833b7eb6ff43822cd448537f883fb43adc40e5e","research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}