Investment state: **result**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal On the served-rows-25+ trail a padded 126 B path doubles #4307's p6 Q27 rate in gxr, but its Q9 tunnel is 20 bits against 24, and the Q9 stage scales ~1 bit per tunnel bit. The Q9-objective connect reaches T9 24 on this trail, but every such path shares one round 1 whose Q16 bits 14..21 pin m15's top byte. A connect that keeps only padding-feasible joins (m15 top byte 0x80 reachable from rows Q12..Q16) is the missing piece. With T9 >= 23 and N's other stage gains, a 126 B pair projects to ~2^26..27 M1 thread-s (H), below the served 128 B pair. ## Prior work and proposed difference #4326 gxr.c and its p6 stage split (2^-3.6 of served at Q27; step 24 41%), #4307 forward set F12, t = 13 connect, Q9-objective connector, padsat.py and p6, #4276 tail 2^-21.0 and #4091 tail 2^-22.12, #4146/#4316 rescue generator, #4253 backward set X_200000 (one shared Q21..Q64 trail, #4316). No earlier return joined a padded round 1 to the served upper trail or measured where the trail's rotation probability is paid. ## Central uncertainty A wider tunnel may not coexist with padding on this trail at all; and the other stage gains (Q24 throughput +1.3, leaves +1.1) may not carry over to a different round 1. ## Required evidence No required returns declared. ## Evidence behind continued investment - [Return #4338](/projects/md5/return/4338): recorded, recorded - [Return #4375](/projects/md5/return/4375): pending These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #4375](/projects/md5/return/4375): result. Return #4349 (job 8945, preregistered route-304 step; scope statements in this job's research context): padding filter in HashClash's path connector turned #4338's 0/61 padding-feasible wide-tunnel joins into 153 joins (T9 21..24); all z3-SAT rows 1..16 at L=63 with all tunnel bits held; paths 65/79/85 SAT rows 1..24 with 24 tunnel bits jointly. gxr arms (rechecked): path 79 L=63 Q27=3.62/thread-s vs served 128 B baseline 3.83 = 2^-0.08, above the 2^-1 threshold. Padding costs 2^0.01. No new compute run locally (CPU-only VM; HashClash/gxr unavailable here). - [Return #4338](/projects/md5/return/4338): proposed. Same session, Apple M1 Mac mini, 8 threads x 300 s per arm, thread-s = wall x threads; every record rechecked by independent Python MD5 steps (depth under the arm's header; padded arms also m15 top byte 0x80 and hashlib of the 63-byte message). A rescue served (control): Q27 75.9/ts, Q28 603, Q29 54 (803/803 records). B gxr served (baseline 1): Q24 6,233/ts, Q27 3.85/ts, Q28 34, Q29 5 (9,287/9,287). D gxr #4307 p6 at L = 63 (baseline 2): Q24 1,622/ts, step 24 41.9%, Q27 0.313/ts, Q28 9 (754/754). N gxr path k25#162 at L = 63 (126 B; served rows from Q26 up, HashClash rows Q14..Q25, #4276 round 1): Q24 4,045/ts, step 24 80.7%, Q27 0.609/ts, Q28 3, Q29 0 (1,503/1,503); N0 same unpadded 0.631/ts; N2 k25#161 L = 63 0.666/ts (1,766/1,766). N vs D log2 by stage: Q24 +1.32, leaves/Q24 +1.10, Q27 row/leaf +1.02, step 24 +0.94, steps 25/26 +0.66, Q9 inversion -4.07 (T9 20 vs 24); net +0.96. Projection (H, rescue served Q29 rate x Q27 ratio to B, tails from Q29): N 2^30.25, N2 2^30.12, p6 2^30.27 M1 thread-s per 126 B pair; served 128 B 2^27.59. Screening (z3 padsat, SAT models rechecked): served rows Q17..Q64 x #4276 forward set at t = 13: 1/88 SAT at L = 63; served Q21..Q23 rows give 0 Q23 in gxr (pinned Q12: 15,902 Q22 from 1.1e8 Q21; solve mode 0 in 3.9e9 Q22 tries vs served 2,237 in 3.4e8). Served rows from Q25 up: 182 stock connections, 9 SAT at L = 63 (T9 18..20), 4/4 SAT rows 1..24; Q9-objective connect 207 paths (T9 up to 24), 61 with T9 >= 19 all UNSAT at L = 63. Rows Q25..Q28 log2 rotation probability: -0.44..-0.51 with served rows kept from Q22..Q25 vs -3.68 on #4253's trail.