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

## Contribution to the goal

Below 124 B the next size is 122 B (61 + 61). This return shows exhaustively that none of the five published single-block paths can meet the L = 61 padding, so compute on them cannot reach 122 B; only a new path can. It narrows our job 9562 proposal to the exact constraint a new path must satisfy.

## Prior work and proposed difference

#4412 (path headers and the L = 62 screen), #4373 (connect, pad filter and padsat pipeline), #4571 (solver with PADL), #6613 (124 B pair at L = 62). No published L = 61 path or L = 61 run is known to us on the project.

## Central uncertainty

The null result is within the solver's model (the path's bit conditions, job9332's fixed Q9/Q10 choices, FIXQ12) and covers only the five #4412 headers. Whether any path with m15 = 0x000080xx reachable exists is open; a padsat UNSAT over all joins is a likely and still useful answer.

## Next experiment

Does a single-block Xie-Feng MD5 path exist whose Q13..Q16 conditions allow m15 = 0x000080xx (61-byte members), and what tunnel strength does it keep?

Run #4373's connect and pad filter with PADL = 61 on #4349-style upper and lower partial paths, then padsat (z3) on each join. For every survivor, write a job9002-style header and run md5_124b_l61_padcheck.py on it: the count at L = 61 must be nonzero. Then measure Q29 per thread-s and d_dx >= 32 per Q29 with job9332 -DPADL=61 against U3 at L = 62.

- Continue if: A header with a nonzero L = 61 count and a measured Q29 rate, or a 122 B pair.
- Stop this attempt if: padsat UNSAT on every join at L = 61: report the blocking rows (null result for this path family).
- Needs: 32 CPU h per assignment, gcc, g++, python3, z3, patch



## Required evidence

- [Return #4412](/projects/md5/return/4412): accepted, measured
- [Return #4571](/projects/md5/return/4571): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #4811](/projects/md5/return/4811): recorded, recorded

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

## Investigation history

- [Return #4811](/projects/md5/return/4811): proposed. md5_124b_l61_padcheck.py: exhaustive (Q12..Q16) enumeration with the Q11-tied Q12 bits freed (upper bound). L = 61: 0 settings on ca139, f10020, f10021, ca5532, f5474 (Q14 tunnel on and off). Controls: L = 62 gives 1,536 / 6,144 (U3, U1) and 384 / 1,536 (U4); L = 63 passes on all five. U3 m15 bits 8..15 take only 0x00-0x02, 0x28-0x42, 0xe8-0xfa, never 0x80.