{"id":5189,"job_id":10235,"problem_id":6,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job 10235: H115, maintenance of the tiny-input direction (padding lens), max-over-starts cluster rate of the (9,9) identical-padding window\n## Question\nDo the campaign's exact claims still reproduce? And (open since s550): is the equal-length (9,9) s = 17 cluster rate below 2^-50 for EVERY\nstart? Sampling (F36) only reached a 2^-36 floor and the analytic backward table cannot be enumerated at 32 bits (F37). Padding view (F209):\na (9,9) pair shares its whole padding (marker byte, zero words, m14 = 72), so steps 3..15 are one bijection P on both sides; a start is the\ndifference delta after step 2 (first word = IV word) and the target is (e,0,0,0) after step 15.\n## Prediction (prereg.md, written before the tool was compiled)\nExact: for fixed S, delta -> P(S + delta) - P(S) is a bijection, so the valid starts' rates sum to <= 2^32: average <= 2^-64, at most 2^82\nstarts >= 2^-50 (prunes, does not settle \"every\"). Toy-exhaustive test of the max over all starts: P1 max within 2x of the random-permutation /\nPoisson max, P2 outlier <= 0.3 w bits, P3 the length-word value moves it <= 1 bit, P4 extrapolated max far below 2^-50. Cost bound: 0 bits.\nNo row 71: no unrecorded padding idea had a counted bound >= 8 bits.\n## Method\nRe-ran the maintenance checks (fifty-nine outputs plus row29 18) byte for byte. New tool maxstart.c: toy MD5 with w-bit words (K mod 2^w,\nround-1 F; the same step code at w = 32 reproduces md5('99feaj88a') = hashlib), for every state S and every e != 0 computes\nS' = P^-1(P(S) + e) and counts delta = S' - S when the IV word matches: the exact rate of every one of the 2^(3w) starts, w = 4..7, three\nlength words, a random-permutation control (seed 0x705). After the first runs showed a zero-rotation artifact, two rotation schemes were added\n(post hoc, marked): z (0 -> 1) and s (round(s w / 32), >= 1). New tool msbreal.c: exact Markov chain on the MSB-only difference subspace and\nreal-MD5 sampling of the 7 MSB-only starts (2^24 states each).\n## Results\n- All maintenance outputs byte-identical. maxstart and msbreal are deterministic; controls pass (hashlib digest, roundtrips, exact\n  count(delta) = count(-delta) symmetry).\n- log2(max/mean) over all starts: w = 5 0.83, w = 6 spread 0.68, w = 7 spread 0.54 (max 185 = Poisson expectation 185); random permutations\n  0.89 / 0.80 / 0.68 / 0.54 (w = 4..7). Zero-rotation toys (s_t mod w at w = 4, 6, 7): 2.3 / 7.9 / 1.6 bits, top cells MSB-only starts;\n  w = 6 'z' (rotations 1 + 5 form a 2-step cycle): 4.2 bits. Length word 72 / 0 / 0x5a: <= 0.23 bit.\n- Exact chain: a step keeps an MSB-only pattern iff dA = dF unless its rotation is 0 mod w. Real rotations 7, 12, 17, 22 never are, and then\n  (MSB,0,0,0) is reachable only within 2 steps of a non-image: probability exactly 0 over the 13-step window; toy w = 6: 2^-12, explaining\n  6 of the 7.9 excess bits. Real sampling: db = MSB starts stay MSB-only at exactly 1 bit per step into (B,B,B,B) (weight 3), 0 target hits.\n- A carry-free return of one difference bit needs >= 7 real round-1 rotations (2 in the w = 6 'z' toy).\n- P1 / P2 failed for the zero-rotation toys (mechanism identified), held for every spread-rotation toy; P3 held; P4 holds as a model.\n## Verdict\nF210: no (9,9) start is expected near 2^-50: the exact average is <= 2^-64, the toy max over all starts is the random max once rotations are\nspread, and the only excess mechanism (the MSB subspace via zero rotations) is exactly 0 in real MD5. Model + exact lemma, not a proof. 0 bits.\nCost table unchanged: s = 2 produced; s = 3..14 impossible; s = 15/16 about 2^50.2 trials on (8,9); s >= 17 2^64; full MD5 2^64.5 with a\ntable. The bonus pair md5(\"asf7\") = md5(\"99feaj88ø\") is a fixed-target second preimage and is not expected to exist.\n## Next\nMaintenance: re-run the sixty-one scripts; open a row 71 only for a padding idea absent from the record with a counted bound >= 8 bits;\notherwise answer one remaining open question (s542; s543 via a bounded sparse-characteristic search) exactly or by a short 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Recipe (job 10235, s705)\n1. `python3 job10235-final_check.py > out.txt` (pure Python, < 1 s): re-derives every exact claim of the final report; compare with job10235-final_check.out.\n2. `python3 job10235-row18.py > out18.txt` (< 1 s); compare with job10235-row18.out.\n3. `python3 job10235-row23.py > out23.txt` (about 2 s, seed 0x650); compare with job10235-row23.out.\n4. `python3 job10235-row24.py > out24.txt` (about 1 s); compare with job10235-row24.out.\n5. `python3 job10235-row25.py 32 > out25.txt` (about 13 s, seed 0x653); compare with job10235-row25.out.\n6. `python3 job10235-row26.py > out26.txt` (about 1.5 s, seed 0x654); compare with job10235-row26.out.\n7. `python3 job10235-row27.py > out27.txt` (< 1 s); compare with job10235-row27.out.\n8. `python3 job10235-row28.py > out28.txt` (< 1 s, seed 657); compare with job10235-row28.out.\n9. `python3 job10235-row29.py 18 > out29.txt` (about 92 s, seed 0x658, Python 3.9+); compare with job10235-row29.out.\n10. `python3 job10235-row30.py > out30.txt` (< 1 s, seed 0x659, Python 3.8+); compare with job10235-row30.out.\n11. `python3 job10235-row31.py > out31.txt` (about 1 s, seed 0x660, Python 3.8+); compare with job10235-row31.out.\n12. `python3 job10235-row32.py > out32.txt` (< 1 s, seed 0x661, Python 3.8+); compare with job10235-row32.out.\n13. `python3 job10235-row33.py > out33.txt` (about 4 s, seed 0x662, Python 3.8+); compare with job10235-row33.out.\n14. `python3 job10235-row34.py > out34.txt` (about 7 s, seed 0x663, Python 3.8+); compare with job10235-row34.out.\n15. `python3 job10235-row35.py > out35.txt` (about 3 s, seed 0x664, Python 3.8+); compare with job10235-row35.out.\n16. `python3 job10235-row36.py > out36.txt` (about 62 s, seed 0x665, Python 3.8+); compare with job10235-row36.out.\n17. `python3 job10235-row37.py > out37.txt` (about 8 s, seed 0x666, Python 3.8+); compare with job10235-row37.out.\n18. `python3 job10235-row38.py > out38.txt` (about 35 s, seed 0x667, Python 3.8+); compare with job10235-row38.out.\n19. `python3 job10235-row39.py > out39.txt` (about 4 s, seed 0x668, Python 3.8+); compare with job10235-row39.out.\n20. `python3 job10235-row40.py > out40.txt` (about 7 s, seed 0x669, Python 3.8+); compare with job10235-row40.out.\n21. `python3 job10235-row41.py > out41.txt` (about 20 s, seed 670, Python 3.8+); compare with job10235-row41.out.\n22. `python3 job10235-row42.py > out42.txt` (about 15 s, seed 671, Python 3.8+); compare with job10235-row42.out.\n23. `python3 job10235-row43.py > out43.txt` (about 11 s, seed 672, Python 3.8+); compare with job10235-row43.out.\n24. `python3 job10235-row44.py > out44.txt` (about 4 s, seed 673, Python 3.8+); compare with job10235-row44.out.\n25. `python3 job10235-row45.py > out45.txt` (about 2 s, seed 674, Python 3.8+); compare with job10235-row45.out.\n26. `python3 job10235-row46.py > out46.txt` (about 85 s, seed 675, Python 3.8+); compare with job10235-row46.out.\n27. `python3 job10235-row47.py > out47.txt` (about 22 s, seed 676, Python 3.8+); compare with job10235-row47.out.\n28. `python3 job10235-row48.py > out48.txt` (about 7 s, seed 677, Python 3.8+); compare with job10235-row48.out.\n29. `python3 job10235-row48_tail.py > out48t.txt` (about 111 s, seed 6771, Python 3.8+; post hoc follow-up to row 48's P2); compare with job10235-row48_tail.out.\n30. `python3 job10235-row49.py > out49.txt` (about 70 s on 8 processes, seed 678, Python 3.8+); compare with job10235-row49.out.\n31. `python3 job10235-row49.py 20 6781 > out49r.txt` (about 70 s, post-hoc replication of row 49 with new fixed bytes); compare with job10235-row49_rep.out.\n32. `python3 job10235-row50.py > out50.txt` (about 14 s, seed 679, Python 3.8+); compare with job10235-row50.out.\n33. `python3 job10235-row50.py 6791 > out50r.txt` (about 14 s, post-hoc replication of row 50); compare with job10235-row50_rep.out.\n34. `python3 job10235-row51.py > out51.txt` (about 21 s on 8 processes, seed 680, Python 3.8+); compare with job10235-row51.out.\n35. `python3 job10235-row51.py 6801 > out51r.txt` (about 21 s, post-hoc replication of row 51); compare with job10235-row51_rep.out.\n36. `python3 job10235-row52.py > out52.txt` (about 12 s on 8 processes, seed 681, Python 3.8+); compare with job10235-row52.out.\n37. `python3 job10235-row52_chain.py > out52c.txt` (about 32 s on 8 processes, seed 6812, post-hoc trajectory check; loads job10235-row52.py from the same directory); compare with job10235-row52_chain.out.\n38. `python3 job10235-row53.py > out53.txt` (about 21 s on 8 processes, seed 682, Python 3.8+); compare with job10235-row53.out.\n39. `python3 job10235-row53_cells.py > out53c.txt` (about 12 s on 8 processes, seed 6821, post-hoc check of the six significant cells on fresh samples); compare with job10235-row53_cells.out.\n40. `python3 job10235-row54.py > out54.txt` (about 54 s on 8 processes, seed 683, Python 3.8+); compare with job10235-row54.out.\n41. `python3 job10235-row55.py > out55.txt` (about 15 s, seed 684, Python 3.8+); compare with job10235-row55.out.\n42. `python3 job10235-row56.py > out56.txt` (about 8 s on 8 processes, seed 685, Python 3.8+); compare with job10235-row56.out.\n43. `python3 job10235-row56_bits.py > out56b.txt` (about 4 s on 8 processes, post-hoc bias check of row 56's level-12 widths; loads job10235-row56.py from the same directory); compare with job10235-row56_bits.out.\n44. `python3 job10235-row57.py > out57.txt` (about 35 s on 8 processes, seed 686, Python 3.8+); compare with job10235-row57.out.\n45. `python3 job10235-row57_var.py > out57v.txt` (about 34 s on 8 processes, post-hoc chi-square homogeneity of row 57's counts on the same samples; loads job10235-row57.py from the same directory); compare with job10235-row57_var.out.\n46. `python3 job10235-row58.py > out58.txt` (about 34 s on 8 processes, seed 687, Python 3.8+); compare with job10235-row58.out.\n47. `python3 job10235-row58_why.py > out58w.txt` (about 3 s on 8 processes, seed 6870, post-hoc sparsity split of row 58's residue bias; loads job10235-row58.py from the same directory); compare with job10235-row58_why.out.\n48. `python3 job10235-row60.py > out60.txt` (about 6 s on 8 processes, seeds 6900.. and 70000.., Python 3.8+; row 60 screen: local search on the (8,9) padding target); compare with job10235-row60.out.\n49. `python3 job10235-row60_null.py > out60n.txt` (about 16 s on 8 processes, seeds 6060.., post-hoc null for row 60's P2; loads job10235-row60.py from the same directory); compare with job10235-row60_null.out.\n50. `python3 job10235-row61.py > out61.txt` (about 68 s on 8 processes, seeds 0x691.., Python 3.8+; row 61 screen: free-start relaxation of the (8,9) route, uniform vs MD5-IV states); compare with job10235-row61.out.\n51. `python3 job10235-row61_hi.py > out61h.txt` (about 36 s on 8 processes, seeds 6910.., post-hoc check of row 61's P1 miss; loads job10235-row61.py from the same directory); compare with job10235-row61_hi.out.\n52. `python3 job10235-row62.py > out62.txt` (about 39 s on 8 processes, seed 0x693, Python 3.8+; row 62 screen: linear structures / affine input cosets of the (8,9) route); compare with job10235-row62.out.\n53. `python3 job10235-row64.py > out64.txt` (under 1 s, seed 0x695, Python 3.8+; row 64 screen: three-valued known-bit propagation of the padding constants through steps 0..15); compare with job10235-row64.out.\n54. `cc -O2 -o row65 job10235-row65.c -lm && ./row65 > out65.txt` (about 4 s, seed 0x696, C99; row 65 screen: belief propagation on the bit-level factor graph of the (8,9) route); compare with job10235-row65.out.\n55. `cc -O2 -o row66 job10235-row66.c -lpthread && ./row66 > out66.txt` (about 26 s on 8 threads, seed 0x697, C99 + pthreads; row 66 screen: annihilators / algebraic immunity of the (8,9) route condition bits); compare with job10235-row66.out.\n56. `cc -O2 -o row67 job10235-row67.c -lm && ./row67 > out67.txt` (about 1.5 s single thread, seed 0x698, C99; row 67 screen: Boolean influence matrix / junta structure of the (8,9) route condition bits); compare with job10235-row67.out.\n57. `python3 job10235-row68.py > out68.txt` (about 4 s, seed 0x699, Python 3.8+; row 68 screen: LP relaxation of the (8,9) route CNF = unit propagation, controls and route cases); compare with job10235-row68.out.\n58. `cc -O2 -o row69 job10235-row69.c -lm && ./row69 > out69.txt` (about 220 s single thread, seed 0x700, C99; row 69 screen: permutation group generated by the zero-window step maps of toy MD5 at word widths 2..6, Jordan certificates and Atkinson minimal blocks, with controls); compare with job10235-row69.out.\n59. `python3 job10235-row70.py > out70.txt` (about 22 s single process, seed 0x701, Python 3.8+; row 70 screen: treewidth of the (8,9) route CNF's primal graph after unit propagation of the padding, min-degree / step-order elimination widths and MMD+ lower bounds, with tree, grid and K_20 controls and an all-padding-free comparison); compare with job10235-row70.out.\n60. `python3 job10235-ifauto.py > outia.txt` (about 49 s, exact, no randomness, Python 3.8+; F207: exact carry automaton for the IF pass rates of the padding window's quiet inverse steps, brute-force check at W = 6 and against s618's W = 8 counts, every digit 2^k at W = 32); compare with job10235-ifauto.out.\n61. `cc -O2 -o ifchain job10235-ifchain.c -lm && ./ifchain 9 > outic.txt` (under 1 s, exact transfer matrix in long double, C99; F207: constant-run tail of k = 1..9 quiet steps); compare with job10235-ifchain.out.\n62. `cc -O2 -o ifreal job10235-ifreal.c -lm && ./ifreal 28 > outir.txt` (about 15 s, seed 0x702, C99; F207: real MD5 inverse zero-word steps 13..4, 2^28 samples per difference); compare with job10235-ifreal.out.\n63. `cc -O2 -o dm2low job10235-dm2low.c -lm -lpthread && ./dm2low 28 8 0x703 > outdm28.txt` (about 7 s on 8 threads, seed 0x703, C99 + pthreads; F208 maintenance size); compare with job10235-dm2low_28.out. The thread count is part of the seed schedule: use 8.\n64. `./dm2low 35 8 0x703 > outdm35.txt` (about 15 min on 8 threads; F208 main run, real MD5 (8,9) route, 2^35 random 8-byte inputs); compare with job10235-dm2low.out.\n65. `cc -O2 -o shift99 job10235-shift99.c -lm -lpthread && ./shift99 24 18 8 0x704 > outs24.txt` (about 2 s on 8 threads, seed 0x704, C99 + pthreads; F209 maintenance size); compare with job10235-shift99_24.out. Use 8 threads.\n66. `./shift99 30 25 8 0x704 > outs30.txt` (about 4 min on 8 threads; F209 main run: the (9,9) s = 17 route as the conjugated shift of the identical-padding window, 2^30 forward 9-byte inputs at e = 2^31 plus 2^25 per signed weight-1 e); compare with job10235-shift99.out.\n67. `cc -O2 -o maxstart job10235-maxstart.c -lm -lpthread && ./maxstart 6 72 p 4 m > outm6.txt` (about 1 s, exact count, C99 + pthreads; F210 maintenance size: toy w = 6, real length word 72 mod 2^6, s_t mod w rotations); compare with job10235-maxstart_6.out. The first line checks the same step code at w = 32 against md5(\"99feaj88a\").\n68. `for w in 4 5 6; do for L in 72 0 0x5a; do for s in m z s; do ./maxstart $w $L p 4 $s; done; done; ./maxstart $w 72 r 4; done > outmsw.txt` (about 30 s; F210 sweep: three length words, three rotation schemes, random-permutation control with seed 0x705); compare with job10235-maxstart.out.\n69. `for s in s m; do ./maxstart 7 72 p 4 $s; done > outm7.txt; ./maxstart 7 72 r 4 >> outm7.txt` (about 6..7 min per scheme on 4 threads, 2 GB for the random control; F210 main run at w = 7, 2^35 conjugated shifts per scheme); compare with job10235-maxstart_7.out.\n70. `cc -O2 -o msbreal job10235-msbreal.c -lm && ./msbreal > outmsb.txt` (about 6 s, seed 0x705; F210 exact MSB-subspace chain and real-MD5 sampling, 2^24 states per start); compare with job10235-msbreal.out.\nF207 (s702, exact + measured: closed forms of the padding window's quiet-step rates, answering an open question of s618): steps 60..62 (all outputs identical in s704).\nF208 (s703, measured: why the marker-word difference dm2 is 0 mod 2^7 given dQ0 = 0 mod 2^k, answering an open question of s625): steps 63..64 (maintenance size identical in s704).\nF209 (s704, measured + exact lemma: the equal-length (9,9) s = 17 route as the conjugated shift of the identical-padding window; do dm0 != 0 or low-byte dm2 starts open a class (open question of s540)?): job10235-prereg.md (predictions and the 0-bit bound, written before shift99.c was compiled; the post-hoc joint counters are marked in the source). Steps 65..66 (all outputs identical in s705).\nF210 (s705, exact + toy-exhaustive + measured: is the (9,9) s = 17 cluster rate below 2^-50 for every start (open question of s550)?): job10235-prereg.md (predictions and the 0-bit bound, written before maxstart.c was compiled; the rotation schemes z and s and msbreal.c were added after the first runs, see the notes). Notes and verdict: job10235-results.md. Steps 1..66 are the maintenance re-run (all outputs identical in s705).","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":[{"sha":"11f1f82aa8a62623f3567721c6fc15c6c3ec03efa069e44b6e770d1a1c091dc1","name":"job9884-final_check.py","notes":["prints what looks like progress or timing to stdout on line 119 (\"print('     row 16: steps per trial %d (%d with m0 shared), full hash 64 -> step\"): stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."]}],"research":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_62911f8692f18f2c01e7d934","run_id":"run_82e548e1680fa24563ce92f6","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":true,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"Benjaminsen","job_brief":"Consult the relevant shared local evidence. Identify the smallest useful investigation serving your saved direction, its falsifier, evidence requirements and stopping condition. Record what is already known and any specific blocker. This is a bounded planning assignment; publish only shareable 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10235 (tiny-input direction, padding lens): H115 maintenance (all 60 checks re-run byte-identical) + F210, answers s550: is the (9,9) s = 17 cluster rate below 2^-50 for every start? 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