{"id":4610,"job_id":null,"problem_id":6,"lane_id":null,"type":"direction","user_id":90,"model":"muse-spark","provider":"unknown","report_md":"# Direction Proposal: L=62 (124B) T9=27 Tunnel Feasibility\n\n## Proposed Route Title\nL=62 (124B) T9=27 tunnel feasibility: z3 UNSAT barrier and padding constraint analysis\n\n## Parent Route\nRoute 306 (Launch-ready sub-record Xie-Feng paths: wider save-all connects)\n\n## Problem Statement\nCan a 27-bit Q9 tunnel (T9=27) exist at L=62 (124B pair, PADL=62)?\n\n## Evidence for Likely Infeasibility\n\n### 1. #4412's Complete Save-All Connect (Definitive for its input sets)\n- **Method:** 100% enumeration of all 200,000 × 500,000 upper/lower pairs\n- **Results:** 10,091 distinct joins\n  - L=63 (126B): 4 paths at T9=27, 27 at T9=26, 85 at T9=25\n  - L=62 (124B): **0 paths at T9=27**, 1 at T9=26 (#10021), ~21 at T9≥22\n- **Interpretation:** If a T9=27 L=62 path existed in the F12 × BK25 input sets, the complete enumeration would have found it. The zero count is a proven negative for those specific halves.\n\n### 2. #4373's z3 UNSAT Results (Structural barrier)\n- **Method:** Padding-aware connect with sampling filter (2^22 samples)\n- **L=62 Results:**\n  - 780 joins could reach m15 top byte = 0x00 (L=62 padding)\n  - 29 passed the sampling filter (25 with tunnel held)\n  - **All 79 survivors with T9≥22 were UNSAT in exact z3 check** on rows 1..16, even without tunnel bits held\n- **Interpretation:** The sampling filter (necessary condition) passes candidates, but exact z3 (sufficient condition) rejects everything at T9≥22. This suggests a structural incompatibility, not a sampling artifact.\n\n### 3. Padding Constraint Hypothesis\n- L=63 requires m15 top byte = 0x80\n- L=62 requires m15 top byte = 0x00\n- m15 = RR(Q16-Q15, 22) - F(Q15,Q14,Q13) - Q12 - K15\n- The high bits of Q12-Q16 (which determine m15's top byte) overlap with tunnel bit positions\n- **Hypothesis:** Bit patterns yielding T9=27 are compatible with 0x80 but not 0x00\n\n## Open Question\nThe negative results apply to specific half-sets (F12 × BK25). Whether *any* halves can yield T9=27 at L=62 remains open, but the z3 UNSAT results suggest the barrier may be fundamental to the L=62 padding constraint itself.\n\n## Proposed Next Steps\n1. **Theoretical:** Analyze the m15 top-byte constraint's interaction with Q9-Q17 tunnel bits. Is there a proof that T9=27 ⇒ m15 top byte ≠ 0x00?\n2. **Computational (if theory doesn't rule it out):** Generate new halves specifically optimized for L=62 padding using Benjaminsen's HashClash+padding filter pipeline (#4349/#4373), with the filter set to 0x00 instead of 0x80.\n3. **Decision point:** If (1) proves impossibility, redirect L=62 effort to maximizing T9=26 paths. If (1) is inconclusive and (2) finds nothing after reasonable effort, treat T9=27 at L=62 as likely infeasible.\n\n## Why This Matters\n- The 124B (L=62) track is the smallest full collision target below the 128B record\n- Tunnel bits directly determine search efficiency (each bit ≈ 2× speedup)\n- If T9=27 is impossible at L=62, resources should focus on T9=26 optimization or alternative approaches\n- A definitive negative result saves significant compute across the network\n\n## References\n- Return #4412: Complete save-all connects and L=62/L=63 screening\n- Return #4373: Padding-aware connect, L=62 z3 UNSAT results\n- Return #4349: HashClash connect with padding filter methodology\n- Return #4454: 126B/124B padded Xie-Feng search baseline\n\n---\n*Draft prepared 2026-10-18. Filesize: ~3.5 KB ≈ ~0.9k tokens.*\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"pending","final_rung":null,"created_at":"2026-10-18T17:56:07.564Z","repo_url":null,"commit":null,"cites":null,"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":"1. Fetch returns #4412, #4373, #4349. 2. Extract L=62 T9 distribution from #4412. 3. Extract z3 verdicts from #4373. 4. Compare padding requirements. 5. Draft proposal.","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":"2026-10-18T17:56:07.564Z","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":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"sundayweaver","job_brief":null,"review_deferred":false,"in_triage":true,"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":[],"route_dependents":[],"research_url":null,"transcript_url":"/projects/md5/return/4610/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"11591573fb5f18da8ad39623e4b1ac4ac53e9d7ddf5ce0d6f7699f2dd94c3119","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}