Research · Papers · The SHA-256 record and exact synthesis · ML-039
Conditional arithmetic for nonlinear Maj gauge closure
After ML-036’s affine closure, the register states that a score would require one of two conjectured mechanisms.
Published 2026-08-29
For everyone
Plain summary
ML-039 studies a proposed way to finish the Maj part of the route, where Maj is the majority operation named in the register. The remaining obstacle is a set of residual quantities called gauges. The proposed closure could use carry-row transvections, special row operations whose projected gauge rank, the number of independent residual directions visible after projection, reaches at least four for each round pair, or a chained packer that combines two gauges in one counted row. The packer uses g_i + g_j + g_i g_j, an expression that includes the two gauges and their product. If the first mechanism works, the register gives conditional totals of 128, then 256, 512, 768, and 992. The second mechanism has an abstract 512-row calculation in the simplified model associated with MF-065. These figures describe route arithmetic. They do not validate the SHA-256 instance. No validating test masks, replay, recount, or machine-checkable Lean proof artifact exists. The prescribed first test is a projection study on one ordinary two-round block from the pinned Boolean-logic file. The register records no prior-art comparison or novelty claim.
Result
After ML-036’s affine closure, the register states that a score would require one of two conjectured mechanisms. The first is a set of already-paid carry-row transvections with gauge-projection rank at least four per round pair. Its listed conditional arithmetic is 128, followed by 256/512/768/992.
The second is a chained two-gauge packer that represents
g_i + g_j + g_i g_j
by one counted row. The register gives this route an abstract 512 through MF-065. The word “abstract” matters here because the MF-065 entry places that figure in a named model and records that it is not a SHA construction; the current Maj residuals remain outside that model’s completed validation.
ML-039 remains a CONJECTURE. No masks, replay, recount, or Lean artifact exists. The register’s first instrument is to export one ordinary two-round block from the pinned BLIF and run the gauge-projection atlas. It directs the experiment away from mitring the unmodified edge relation.
Setting and definitions
The setting begins after ML-036’s affine closure of the Maj route. A gauge is the residual quantity that the proposed closure must constrain after that phase. The register indexes two such quantities as g_i and g_j.
A carry-row transvection is the proposed row operation whose usefulness is measured by gauge-projection rank. Gauge-projection rank is the rank visible after the relevant row data is projected onto the gauge coordinates. The required threshold is at least four per round pair.
A two-gauge packer is the proposed single counted row for the quadratic expression g_i + g_j + g_i g_j. “Already-paid” carry rows are rows treated as available in the conditional arithmetic. BLIF is the pinned Boolean-logic representation from which the first instrument is to export an ordinary two-round block. The gauge-projection atlas is the named measurement instrument for that block.
Method
The source record combines an experiment specification, exact export requests, and a verification report, with the underlying specifications and verification reports preserved in the downloadable evidence pack.
Those records establish the proposed alternatives and their conditional arithmetic. The transvection route is stated with a minimum gauge-projection rank of four per round pair and the sequence 128, then 256/512/768/992. The packer route is recorded as the expression g_i + g_j + g_i g_j in one counted row, with an abstract 512 linked to MF-065.
The ledger prescribes the first measurement rather than reporting its completion: export one ordinary two-round block from the pinned BLIF and run the gauge-projection atlas. It also directs the operator not to mitre the unmodified edge relation. No masks, replay, recount, or Lean artifact is recorded.
Discussion
ML-039 is a CONJECTURE about non-flat gauge closure after the affine phase described through ML-036. The entry identifies a concrete threshold for the carry-row route and a concrete algebraic form for the two-gauge packer. Those conditions describe what a successful closure would need to expose. They do not show that the pinned SHA block has the required rank or that the packer can be implemented as one counted row there.
The 512 attached to MF-065 belongs to an abstract model. MF-065 records that its 16-equation pairing is sufficient in that model and that the SHA instance is UNKNOWN because current Maj residuals are quadratic. ML-039 therefore carries the abstract figure as route arithmetic, with no completed SHA validation.
The curation notes record no correction for ML-039. They record no prior-art comparison or novelty position. The register does not support a stronger claim about a working non-flat gauge closure or a novelty result.
For everyone — the takeaway
What this means
ML-039 marks a precise bottleneck in the Maj route. The earlier affine step leaves residual quantities that may need a product term to close. The proposed next step either uses special row operations with enough independent projected directions or compresses two residual quantities into one counted row. The listed numbers show how the bookkeeping would look if one of those mechanisms works. The 512 packer number comes from an abstract calculation, so it does not demonstrate a working SHA-256 circuit. The register’s next move is a small measurement on one ordinary two-round block. Until masks and replayable evidence exist, the entry remains a conjecture. No prior-art claim is part of the record.
Register references
- ML-039
- Package 05-NEXT-EXPERIMENT-SPEC.json
- Package 04-EXACT-EXPORT-REQUESTS.md
- CONT lens_r2b_majgauge_verification_report.md
- Named register entry: MF-065
- Prior art: the register does not record a prior-art work.
Every artifact named above is bundled in, or hashed by, this paper's evidence pack below.
Evidence pack
Everything needed to check this entry against its receipts: the register text, a manifest with a SHA-256 hash for every named receipt, and 3 of 3 receipt files bundled (8 KB). Anything not bundled is still hashed in the manifest and lives in the compute-box working trees.
Changelog
Last reviewed 2026-08-29
- 2026-08-29Published on this site.