{"id":4256,"job_id":8641,"problem_id":6,"lane_id":null,"type":"explore","user_id":90,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job 8641: middle path for the 4-word-shifted Xie-Feng difference (m1 +2^10, m6 +2^31). Connecting paths exist, but every one found is over the L = 55 budget, and none is satisfiable under padding\n\n**Outcome: STOP (scoped negative).** HashClash finds signed paths that join the shifted XF round-1 end to XF rows 33..36 at shifted Q37..Q40. The best one carries 473 bit conditions on Q1..Q40. The budget is 8L - 12, which is 428 at L = 55 and 244 at L = 32. Even with a free connection, the forward and backward halves alone already cost 350-367 for every connection step tried, which rules out L = 32. A further finding: z3 reports UNSAT for every connected path checked under L = 55 padding (also for every round-1 prefix and for padding-aware starts), and also for 4 of the 5 without padding that finished (the fifth timed out). The contract's stop rule applies: dM' is dropped for 32+32 too. The next candidates are the Stevens shifts m0/m5 (L >= 24) and m4/m9 (L >= 40).\n\nPre-registration: `prereg.md`, written before any path search. This extends return #4213 (job 8631). Served inputs (sha256 checked): `job8372-xf2010_path.txt` 35d21fde..., `job8372-xf2010_pair.txt` 8005288a..., `job8631-results.json` d1ced71a..., `job8631-shsat2.py` 1a0f6499... (read only, not run).\n\n## What was run\n- **Tools.** HashClash (Marc Stevens, MIT), GitHub master tarball. md5_diffpathforward, md5_diffpathbackward, md5_diffpathconnect and md5_diffpathhelper were built by hand (`build_hc.sh`) against a local static Boost 1.88 (`build_boost.sh`). There were no autotools and no zlib, so the gzip filters were removed and path files are plain boost archives. Path viewer and selector: `showpaths.cpp`, `bestn.cpp`. z3 5.3.1.\n- **My own code** (stdlib, plus z3 where named):\n  - `md5steps.py`: integer MD5 steps and a condition checker.\n  - `condfill.py`: derives sufficient bit conditions for a signed path from one conforming pair.\n  - `make_start_paths.py`, `make_padstart.py`: build the start paths.\n  - `sweep.sh`, `connect_only.sh`, `backward_chain.sh`: run the searches.\n  - `summarize_sweep.py`, `make_results.py`: collect results.\n  - `solve_path.py` (z3): conforming pair through row R, with optional padding, a row range, and an XOR-weight bound.\n  - `verify_pair.py`: integer recheck of z3 witnesses.\n- **Start rows Q1..Q12** (shifted row t = XF row t+4):\n  - B: Q1..Q11 are served XF rows 5..15, 127 conditions. Q12 keeps XF's modular difference, and the forward step re-signs it.\n  - A: #4213's L = 32 witness signs on Q8..Q12 plus derived conditions, 148 conditions on Q1..Q12. Its forward was 34 conditions worse to Q18, so it was not continued.\n  - Padding-aware starts padA / padB (see below).\n- **Upper rows: Q37..Q48 = XF rows 33..44.** That is a valid characteristic for dM' through step 47, because shifted steps 40..47 are XF's H-steps 36..43 with the same rotations and dm6 at 43. Everything beyond Q40 is covered by #4213's 2^-12 tail; counts stop at Q40.\n- **Search parameters.**\n  - forward `-t 12.. -a POOL -b 1 -q 64`\n  - backward `-t 39.. -a POOL -b 40 -q 64`\n  - connect at T (lower rows <= Q_T, upper rows >= Q_{T+1}), Qcondstart T+4, `--noenhancepath`\n  - Pools of 50k (T = 18..24) and 200k (T = 17..24, 26).\n- **Two connect fixes (diff in `connect_j8641.diff`):**\n  - Qcondstart <= T+3 makes stock connect index an empty tunnel table and segfault.\n  - The stock ranking (round-1 tunnels) and its round-1 `check_path_collfind` were replaced by ranking on total conditions only.\n- **Upper-path fix.** A first upper path carried XF rows to Q64. That is invalid for dM' at steps 48..51 (I versus H), so test_path failed all 2.5M candidate connections. Truncating it at Q48 fixed this.\n- **Threads.** Until about 14:33Z everything ran on CPUs 16-17, with 1-2 threads per tool. After that it ran on CPUs 0-7, 10-15 and 18: forward with 8-10 threads, backward with 3, 4-6 parallel connects with 2-3 threads each, and z3 single-thread solves, one per CPU, up to 13 at once. Some of these overlapped, so for a while those CPUs were oversubscribed.\n\n## Paths found and condition counts\nThe floor is best forward (Q1..Q_T) plus best backward (Q_{T+1}..Q40), i.e. the cost if the connection were free. \"Best\" is the best connected path, Q1..Q40.\n\n| start / pool | T=17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 26 |\n|---|---|---|---|---|---|---|---|---|---|\n| B 50k: floor / best | - | 364 / 480 | - | 354 / 483 | - | 355 / none | - | 358 / 482 | - |\n| B 200k: floor / best | 367 / 476 | 362 / 488 | 357 / 474 | 350 / 486 | 352 / **473** | 351 / none | 352 / 479 | 350 / 484 | 359 / 488 |\n| padA 200k (L=55-aware to Q16) | | | 409 / 527 | 408 / 523 | 407 / 516 | | | | |\n| padB 200k (L=55-aware to Q17) | | 431 / none | 427 / 547 | 427 / 548 | 428 / 543 | | | | |\n\n- **Budget comparison.** The total C(Q1..Q40) must satisfy C <= 8L - 12 (that is, 2^(8L - C) pairs through Q40 times the 2^-12 tail). That is 244 at L = 32 and 428 at L = 55.\n  - The best found, 473, is 45 over at L = 55.\n  - As #4213's middle count (C - 152), it is 321 against the 276 middle budget.\n  - Raising the pool from 50k to 200k gained only 7 conditions (480 to 473).\n- **The connection is what costs.**\n  - It adds about 110-125 conditions on the six rows Q_{T-2}..Q_{T+3}, which reach 25-31 conditions each.\n  - Example, the best path (T = 21): Q1..Q11 = 127, Q12..Q28 = 309, Q29..Q40 = 37.\n  - The backward side is cheap. Q33..Q36 are nearly forced: H-steps 36..39 must produce XF rows 33..36, which pins their modular differences close to XF's Q29..Q32. Q29..Q40 costs only 37 conditions.\n  - The forward side is expensive. The dense round-1 end (XF Q13..Q16) costs 13-30 conditions per row through Q21.\n- **L = 32 is out with margin.** Every floor (350 or more) exceeds 244 before any connection cost, and the start rows Q1..Q11 already use 127 of it.\n- **Padding-aware starts cost more, and still meet the budget only with a free connection.** A z3 pair at L = 55 that conforms to served Q1..Q11 with XOR weight <= 10 on Q12..Q16 was found in 418 s, and its integer recheck is clean. Built from that pair, the padA/padB starts raise the floor to 407-431, which equals the 428 budget only if the connection were free.\n\n## z3 checks (conforming pair through row R; all SAT witnesses rechecked with integer steps, 0 failing conditions)\n- **Encoding controls.**\n  - Integer check: the real 2010 XF pair against the served path breaks only (9, 30, '!'), the same as #4213, and it collides.\n  - z3 on the XF path with XF's dM: SAT for rows 1..20 (15 s) and 1..24 (30 s); unknown for rows 1..40 at 900 s.\n- **B 50k, best T = 18 (480 conditions), no padding:** rows 1..16, 1..17 and 1..18 are SAT. Rows 1..20 (182 s), 1..24 and 1..40 (139 s) are UNSAT. The contradiction sits in the connection steps 18..19: HashClash's per-step checks pass, but the conditions are not jointly satisfiable.\n- **B 200k:**\n  - T = 17 is SAT through its connection (rows 1..21, 29 s). Rows 1..40: unknown at 2700 s for two seeds. A 15-cube split (Q1 bits 0..3, incremental from row 21) reaches SAT through row 28 at most, with row 29 still open after ~25 min.\n  - T = 21 (rows 1..25 and 1..40) and T = 23 (rows 1..27) are UNSAT.\n- **L = 55 padding:**\n  - The T = 17 path is UNSAT from row 14 on: rows 1..13 are SAT, 1..14 UNSAT in 2 s.\n  - All 13 distinct Q1..Q16 prefixes of the start-B connected paths are UNSAT at L = 55, while 3 of 3 are SAT without padding.\n  - The cause: m13's top byte (byte 55 = 0x80), m14 and m15 are fixed. That makes Q14..Q16 functions of earlier rows, and the forward search put conditions on them.\n- **Padding-aware starts at L = 55:**\n  - The padB start is SAT through row 17.\n  - Every padA/padB connected path is still UNSAT: padA from row 16 on, padB from row 18 on (0.6 s). Forward steps 16-17 add conditions on the padding-determined rows Q15/Q16.\n  - padA T = 21 without padding is UNSAT (rows 1..40, 22 s).\n\nSo, under padding, no connected path from this search is a satisfiable characteristic, and its count exceeds the budget even before that.\n\n## Limits of the claim\n- **What the negative covers.** It is scoped to: HashClash with these parameters (pools of up to 200k, maxsdrs 64, the default maxweight, connection at T = 17..26 in round 2); starts B/A/padA/padB; and the fixed entry at XF rows 33..36 = Q37..Q40.\n- **What it is not.** It is not a lower bound. Not tried:\n  - other round-1 signings;\n  - connecting inside round 1 (T <= 15);\n  - a search that knows m13..m15 are fixed;\n  - other tail entries.\n- **How firm the counts are.** They are counts against a 2^(8L - C) heuristic, not exact probabilities. A 1M-pool run was started and then stopped, because its intermediate files exceeded the memory-disk cap. The pool trend (480 to 473 for 4x) makes closing a gap of 45 unlikely.\n\n## Next step\nPer the contract, drop dM' (m1, m6) for 32+32 and test the Stevens shift m0 +2^25, m5 +2^31 (L >= 24) the same way. The work done here makes that cheap:\n- The upper path is #4213's shifted tail.\n- The lower start is the shifted Stevens round 1.\n- The shifted Stevens round 1 does not start from the IV (extension below), so it needs a new round-1 path from step 0.\n- The search should be padding-aware from the start: a z3 L-padded seed (`make_padstart.py`) and no forward conditions on padding-determined rows.\n\n## Extension (separate from the contract's question): Stevens m0/m5 shift from the IV\nAn exact negative, no padding. Stevens' 2012 pair with a k = 8 shift (dM' = m0 +2^25, m5 +2^31; shifted row t = his row t+8, so his rows Q5..Q8 become the IV):\n- Shifted rows 1..2 with only his modular differences (dQ1 = dQ2 = +1, any signs) are UNSAT in 0.01 s. Step 1 needs dF = 0, but bit 0 of Q1 always flips, and the IV words Q0 and Q-1 differ at that bit (Q0 xor Q-1 = 0x77777777). His pair has Q7 = Q8 at bits 0..1.\n- Controls: unshifted from the IV, rows 1..16 signed and rows 1..24 are SAT.\n- So the m0/m5 shift needs a new round-1 path from step 0, not only a new middle. Script `stevens_pairpath.py`; paths in `paths/`.\n\n## Files (this folder)\n- `prereg.md`, `report.md`, `results.json`\n- Scripts: `md5steps.py`, `condfill.py`, `make_start_paths.py`, `make_padstart.py`, `solve_path.py`, `verify_pair.py`, `stevens_pairpath.py`, `sweep.sh`, `connect_only.sh`, `backward_chain.sh`, `summarize_sweep.py`, `make_results.py`\n- Build and HashClash tools: `build_boost.sh`, `build_hc.sh`, `connect_j8641.diff`, `showpaths.cpp`, `bestn.cpp`\n- Start and upper paths: `lowerA.txt`, `lowerB.txt`, `padA.txt`, `upper48.txt`\n- `paths/`: best connected paths, plus `padB.txt`\n- `padded_pair_L55_rows1_11.json`: the z3 L = 55 seed pair\n\nCPU time: 8.35 CPU-h measured (from 15:30Z on), plus about 12 CPU-h estimated before that (not timed); see results.json (`cpu`). Work run for sundayweaver.\n\nFiles are uploaded with the prefix `job8641-` (`paths/x` as `job8641-paths-x`).\n","patch":null,"cpu_hours":20.35,"hashes":{"job8641-padA.txt":"ed9f6a10339081b8da3c0116bca0c9e540b359eb04757e62443e8b86eeb162c5","job8641-sweep.sh":"1d5354dda5344583d7c93f85aaeadf304b27273f243cdcfc21e689911a4e2aca","job8641-bestn.cpp":"2cd8117be629bed131469ffd3760f2f8d865c4c67763bef89aeae52b1e14f5af","job8641-prereg.md":"90397046dd77033c8266d870f673fc9ae9a522429388af47b1366b246bfc1708","job8641-lowerA.txt":"f7739c40bac3bf2f81e2612476857ffed5bc920c47834151e4fe15bf87fe7fc1","job8641-lowerB.txt":"a5396362f1c9f8559b9d7a82cfb007abc28b45d2dd2db6beecc387a27b13dada","job8641-build_hc.sh":"814ddf84d64cf3125a6360e091886b613f61ae5fde7423f7d3f5115b182775a9","job8641-condfill.py":"fed3988500732bfd943edd05e1c254299fe29d696c975dcc5db315b95a473961","job8641-md5steps.py":"dee5d27028edfce466236c007dae685fd018c03a6d463b72126d60a2dd8ae747","job8641-upper48.txt":"52e34db334697e04de8dd4e46ff105ea029f56d12cd6972ff6387ebf343af883","job8641-results.json":"129329ff62b4c4444e0b0eadd4d9bd9aaf95f2beb553bb605699d7c50949501d","job8641-showpaths.cpp":"f622cdc5c792b8ba7b973c1852bf0d8d918104a54a0ba81dfd033a1de120ce9e","job8641-solve_path.py":"ece11fa5126c9b46f63dccdbe908d834457625c9268a8281ae17ec6409913428","job8641-build_boost.sh":"30d15746c3efc644652923f4d240b017105d7e24561d9c0f7a466a903f9f2f66","job8641-paths-padB.txt":"0a3be33db4ebdd75c4ce373d64306cb8c9ddf165474b8c3c881642070017cafc","job8641-verify_pair.py":"f2ebbbce82e6ade626b7b0a84876d40f02af3f0cec1fda6f130059ab6a8579a0","job8641-connect_only.sh":"414b412b01e9ec1459d6667e4cbf6342a28f3e820a7a9d76c9e9ca27dcc0b225","job8641-make_results.py":"b71ff935825af36aa3a656fe8645d84ffacf5c42f430a330c8ad2217adb98404","job8641-make_padstart.py":"8140b93b323b705e5851c7c7a13bcacbc3f44988f30f3433fb8a93e93f70893c","job8641-backward_chain.sh":"e2ef2f4b839f8841a177083a2cb5e70c6ceab5993a49fdafefafb484aa5ce0a2","job8641-connect_j8641.diff":"642c05810feefe955a714b10f7520288691c08d917bf62d98d0d338c6dc00366","job8641-summarize_sweep.py":"770140bb6584586b0756803b3232930a9f08368a8e072f9f08cfaefaf92512f4","job8641-make_start_paths.py":"a89ea21a77b749328595626e84fd581977dca199f09061eb46d1c21c01ba719d","job8641-stevens_pairpath.py":"9bdbe73007b57ff3144fb31a51a8cf112b6acdfdcc6e999c82483f7dafcc9949","job8641-paths-best_P_c21.txt":"db49c17c81c7977036727a45115faebb65bc64fb2b3e0abe8a712c1e943e8663","job8641-paths-best_Q_c21.txt":"4248ea2e848ef88ef8a0c0f7708ac00499e8684a68602f646cb0b71a4d08e1de","job8641-paths-best_B2_c18.txt":"925a577168ecddc737ce6ba6f0c89c0be9ba04d63172822d06812ca2224b3913","job8641-paths-best_B3_c17.txt":"6c96c41a8c89ac5a7bc7b5af45b1ea72bb9b210cbf8972aaa5f71d77446c0603","job8641-paths-best_B3_c19.txt":"b890ba9a0d607d62ff91ad719edc205964b2a49991da1a9ec244a3663d3bc158","job8641-paths-best_B3_c21.txt":"9e2c44fddabab399dd135ba591b7f4f8e3d0a507804636c9782420fbc012a47f","job8641-paths-best_B3_c23.txt":"57cba65824805b83f66a15e9c5335faf9ce176ca62ec288ea40e8c922f97eff8","job8641-paths-best_B3_c24.txt":"3b08dff783347af297c4be22b89ec3879847af6c3d635190ec92749c0dd354f2","job8641-paths-stevens_pairpath.txt":"167ecd3d31e53dec67d69b982a44e3f9adcebab6824a45d0de301cb9d62d77c2","job8641-paths-stevens_k8_shifted.txt":"c15735c73bacec9b028444748163738ee42a44fc99a4db931df10abb906dfdf5","job8641-padded_pair_L55_rows1_11.json":"d48f8fd9eceb9b4f0951bc3e32037ff31264d04e3c3e344774c12de36eebeac9"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-16T16:08:03.921Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[4213],"messages":[]},"tokens":{"log":"summary","input":646,"models":{"claude-opus-5-5":81251},"output":81251,"source":"reported","entries":0,"cache_read":50000000,"cache_write":2827441,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Build HashClash's md5_diffpath{forward,backward,connect,helper} with `job8641-build_boost.sh` and `job8641-build_hc.sh`\n(apply `job8641-connect_j8641.diff`); python 3 with z3-solver.\n- `job8641-sweep.sh LOWER UPPER OUTDIR POOL \"TLIST\"` runs forward, backward and connect for one start.\n- `job8641-summarize_sweep.py OUTDIR \"TLIST\" showpaths` counts conditions on Q1..Q40 (floor and best per T).\n- `python3 -I job8641-solve_path.py PATH --rows R --L 55` (or `--L none`) asks z3 for a conforming pair; witnesses are\n  rechecked with `job8641-verify_pair.py PATH witness.json`.\n`job8641-results.json` holds every count, z3 verdict, time and the CPU accounting.","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":"blocked","obstacle":{"kind":"attempt_failed","evidence":"job8641-results.json (all counts and z3 verdicts), job8641-paths-* (best paths), job8641-solve_path.py and job8641-verify_pair.py. Limitation: a search result, not a lower bound; other round-1 signings, connecting inside round 1, a search aware of the fixed m13..m15 and other tail entries were not tried.","statement":"No middle path for dM' = (m1 +2^10, m6 +2^31) from the shifted XF round-1 end to XF rows 33..36 at Q37..Q40 fits 8L - 12 conditions at L = 55 (best 473 > 428) or L = 32 (floor 350 > 244), and none found is satisfiable under L = 55 padding.","assumptions":"HashClash forward/backward/connect with pools up to 200k, maxsdrs 64, default maxweight, connection at T = 17..26 (round 2), starts B/A/padA/padB, fixed entry at XF rows 33..36; conditions counted on Q1..Q40 against the 2^(8L - C) heuristic with #4213's 2^-12 tail.","revisit_when":"A padding-aware forward search (no conditions on Q14..Q16 rows fixed by padding), a connection inside round 1, or another tail entry. Per the contract, the next difference to try is the Stevens shift m0/m5 (needs a new round-1 path from step 0, see the extension) or m4/m9 (L >= 40)."},"route_id":296,"depends_on":[4213],"evidence_md":"HashClash connects the shifted XF round-1 end (dM' = m1 +2^10, m6 +2^31) to XF rows 33..36 at Q37..Q40, but the best connected path carries 473 bit conditions on Q1..Q40 (start B, pool 200k, T = 21) against the 8L - 12 budget of 428 at L = 55 and 244 at L = 32. The floor (best forward + best backward, free connection) is 350-367 for every T in 17..26, so L = 32 is out; the connection adds ~110-125. 4x the pool gained 7 conditions (480 -> 473). Padding-aware starts (z3 L = 55 seed pair, integer recheck clean) raise the floor to 407-431 and the best path to 516-548. z3: every connected path is UNSAT under L = 55 padding (all 13 distinct Q1..Q16 prefixes; padding fixes Q14..Q16 via m13's top byte, m14, m15); without padding 4 of 5 checked are UNSAT and the best (T = 17, 476) stays unknown at row 29 (two 2700 s solves, 15-cube split to row 28). Controls: the real XF pair breaks only (9, 30) as in #4213; z3 on XF's own path is SAT through rows 1..24. Extension (separate claim): the Stevens 2012 pair shifted by 8 (m0 +2^25, m5 +2^31) is UNSAT at rows 1..2 from the MD5 IV (bit 0 of Q1 flips while Q0 xor Q-1 = 0x77777777), so that shift needs a new round-1 path from step 0."},"research_route_id":296,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_2a239d5b5a891d355de5bb8e","run_id":"run_f5ac2afae6968f964e2138ad","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":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in a first look. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/296 and return #4213. Return the ordinary report and transcript plus research: {route_id: 296, 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 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