Research · Papers · The SHA-256 record and exact synthesis · ML-036
On the exact consecutive-Maj edge identity and rank-32 gauge growth
Edge-relaxed pair has visible fibre 2^32, hidden-b fibre a=Sigma0(U) of rank 32, rank-32 gauge growth, and zero actual saving
Published 2026-08-29
For everyone
Plain summary
This entry examines a route built on an exact boundary equation between consecutive rounds using the Maj function. While the boundary identity holds, the surrounding relation leaves a full 32-bit intermediate value free. As a result, attempts to compress the state using bounded or affine methods yield zero savings. The original ledger logs 387,200 hidden-intermediate collisions per K pair, zero savings from keeping both old Maj witnesses, and a best-case conditional value of 1,024 for a one-witness gauge closure. A subsequent correction retired the bounded and affine closure attempts while retaining the exact identity, confirming zero actual savings and showing that a single pair already spans 32 independent hidden directions. The broader closure question remains open, and the register notes no prior art.
Result
The consecutive-Maj edge identity is exact across 32 semantic round-pair placements times 32 bits. Standalone, its edge-relaxed relation admits 387,200 hidden-intermediate collisions per K pair. Retaining both old Maj witnesses yields zero saving, and 1,024 represents best-case conditional arithmetic for a one-witness gauge closure.
Status remains CONJECTURE because local exactness fails to resolve global closure. The registered blocker is phase-defect growth over k=1..4, where linear growth terminates the route. A subsequent strengthening RETIRED the bounded/affine Maj phase while preserving the exact identity. That audit establishes a free 32-bit intermediate word, visible fibre 2^32, and hidden-b fibre a=Sigma0(U) of rank 32 on the edge-relaxed pair. Because a single pair reaches rank-32 gauge growth, actual saving is zero.
Setting and definitions
The route operates on consecutive Maj rows in the registered two-round relation. A K pair defines the evaluation context for collision counts. The edge-relaxed relation preserves the registered edge identity while unconstraining hidden intermediate coordinates. Visible fibres count compatible hidden assignments under fixed visible coordinates; the hidden-b fibre is the rank-32 relation a=Sigma0(U).
Closure asks whether these local relations compose into a reduced-rank representation. The retired bounded/affine phase designates the two restricted closure forms evaluated during the audit. The parameter k indexes independent coordinates within a single-pair subspace. The original baseline preserves two prior Maj witnesses for comparison.
Method
The identity was verified via edge-identity replay, followed by independent 32-bit gauge, fibre. Initial verification records are available in this paper's downloadable evidence pack.
The strengthening audit provided the gauge and fibre measurements that retired the bounded/affine phase, tracking requested ranks 1,2,3,4 across k=1..4 up to rank 32 and confirming the single-pair subspace structure. The full verification reports, receipts, and associated certificates are provided in the downloadable evidence pack.
Discussion
While the identity holds on the registered route, the bounded/affine Maj phase is RETIRED. This correction invalidates paid affine phases, affine terminal syndromes, sparse original anchors, and unborrowed/unpacked edge chaining as viable scoring mechanisms, holding net saving at zero.
Because a single pair spans full rank 32 (with requested ranks 1,2,3,4 matching k=1..4), composed gauge defect growth is unbounded. Consequently, the quantities 1,024 and 21,191 represent unachieved conditional projections. Rather than replacing the ID as proposed in the package, the catalog appends this strengthening.
The broader closure problem remains unresolved. The baseline collision count (387,200 per K pair), the zero-saving witness baseline, and the phase-defect wall across k=1..4 remain registered constraints. All findings and remaining open questions are scoped strictly to the registered Maj edge route. The register lists no prior art.
For everyone — the takeaway
What this means
The local boundary equation holds exactly, but the system still has 32 bits of completely free intermediate variables. The tested shortcuts—bounded and affine closure—don't reduce the state space at all, leaving net savings at zero. Because a single round pair already uses up all 32 independent hidden degrees of freedom, chaining pairs together won't yield a bounded win. Numbers like 1,024 and 21,191 are purely theoretical targets, not actual performance gains. The broader closure problem remains open on this route, and there's no recorded prior art.
Register references
Entry: ML-036.
Receipts: CONT lens_l3_jsc_maj_verification_receipt.json; CONT lens_l3_jsc_maj_verification_report.md; CONT lens_r2b_majgauge_verification_report.md; CONT lens_r2b_majgauge_verification_receipt.json; package certificates/maj_gauge_maximal_defect.json; package certificates/maj_edge_forest_rank.json.
Prior art: the register does not record this.
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 6 of 6 receipt files bundled (16 KB). Anything not bundled is still hashed in the manifest and lives in the compute-box working trees.
Changelog
Last reviewed 2026-08-29
- 2026-08-13As a result, actual savings remain zero, with 1,024 / 21,191 continuing as conditional arithmetic.
- 2026-08-29Published on this site.