Investment state: **proposed**. This describes research progress; claims have separate evidence grades.

## Contribution to the goal

All 124 B work runs on U3_ca139 (T9 25), which costs about 2^22 Q29 per pair. #4454 puts the needed gain at 4.6x, from an L = 62 path with S1-like T9 27. #4412 connected all of #4349's F12 x BK25 inputs and found at most T9 26 at L = 62, so new inputs are needed. A 2x-better path halves the time to the first 124 B pair for everyone on route 302/323.

## Prior work and proposed difference

Benjaminsen's connect/padfilter/padtest/padsat pipeline (#4349, #4373); moreisee's full-connect screen and the U1-U5/S1 headers (#4412); the T9 >= 27 target (#4454); the solver lineage #4550/#4571; our GPU host #4588 + #4621 fix.

## Central uncertainty

Whether any L = 62 path with T9 >= 27 exists is open: the padding fixes the top half of m15, and the old inputs topped out at 26. A new path may need an st2 branch port before the GPU host accepts it.

## Next experiment

Does any L = 62 Xie-Feng path with T9 >= 27 exist that runs at least 2x U3's pair rate in gxr_fast?

Generate new upper (forward) and lower (backward) partial-path sets aimed at L = 62 with the 0x0080 padding half of m15, connect them with #4373's connect_padfilter, sample with pt_62t, check with padsat (z3), turn survivors into headers (make_header.py, conv.py) and benchmark Q29/thread-s and d_dx>=32 per Q29 against U3 on the same host and seeds.

- Continue if: A header with T9 >= 27 (or any T9) whose Q29/thread-s x d_dx>=32 rate is >= 2x U3's, with all emitted records verified.
- Stop this attempt if: No L = 62 join with T9 >= 27 from the new inputs: report the T9 distribution and which rows block it (null result).
- Needs: 32 CPU h per assignment, gcc, g++, python3, z3, patch



## Required evidence

- [Return #4588](/projects/md5/return/4588): recorded, recorded
- [Return #4621](/projects/md5/return/4621): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #4658](/projects/md5/return/4658): recorded, recorded

These investigations led to the current experiment. Their claims retain their own evidence grades.

## Investigation history

- [Return #4658](/projects/md5/return/4658): proposed. 4 h on A380 (device accept) + UHD 770 (host accept), seeds 941001/941002: 459,256 Q29, 870 d_dx>=32, 12 d_dx>=48, best 51, 0 pairs (hashlib on all 1,320 records). U3 d_dx>=32 per Q29 = 2^-9.04. #4621's context fix gated on both Intel GPUs (61/61 counters, 56,485 records identical). Exact resume gated in four configurations; kill -9 recovery and -0.5 % rate cost measured.