Research · Papers · The SHA-256 record and exact synthesis · MF-048
Quadratic-hull screens of two-column SHA-256 seed allocations
All four K-pairs: 92,274,688 honest rows, degree-two rank 837 of 862, ideal dimension 25; JSC-13, JSC-12, JSC-11 fail instrument gate
Published 2026-08-29
For everyone
Plain summary
This entry tests whether a two-column constraint in SHA-256 can double up the work of four core Ch/Maj products, using them to catch carry defects between adjacent columns. The concept remains an unproven conjecture. Three candidate seed allocations—JSC-13, JSC-12, and JSC-11—fail the initial quadratic-hull screen. In each case, degree-two combinations of the available coordinates still admit an invalid internal point that shares the same external input as a valid point. The associated row counts of 17,841, 16,881, and 15,921 (compared against the 22,215-row baseline leader) are hypothetical accounting targets rather than verified constructions. Alternative joint configurations remain untested, and the register notes no prior art.
Result
The register records the conjecture that a genuine two-column SHA-256 relation can repurpose the four mandatory Ch/Maj products to reject carry defects. Relative to the 22,215-row leader, the conditional row ledgers evaluate to 17,841 for JSC-13, 16,881 for JSC-12, and 15,921 for JSC-11, with no verified score claimed.
Naive one-column factor deformations fail completely. Across the exported full-raw two-column graph, all four K-pairs exhibit 92,274,688 honest rows, degree-two rank 837 of 862, and ideal dimension 25. Seed allocations JSC-13, JSC-12, and JSC-11 each admit an explicit same-input wrong point within their exact quadratic hulls, failing the instrument gate. Chained coordinates, cyclic lifts, nonidentity C-sides, and other joint allocations remain UNKNOWN. The cheapest decision procedure is to test candidate joint allocations for hull separation prior to rank-one interpolation.
Setting and definitions
The allocations JSC-13, JSC-12, and JSC-11 designate three specific seed configurations evaluated on the exported full-raw two-column graph across four K-pairs. The exact quadratic hull is the degree-two linear span generated by the graph coordinates and their pairwise products. A same-input wrong point is an explicit non-honest assignment that matches an honest assignment on all input coordinates. The instrument gate denotes the quadratic-hull separation check, which precedes rank-one interpolation. The entry provides no additional implementation details for K-pairs or C-sides.
Method
The evaluation paired conditional row accounting with rank computations on the exported full-raw two-column graph across all four K-pairs. Passing candidate allocations through the quadratic-hull screen revealed explicit same-input wrong points, eliminating JSC-13, JSC-12, and JSC-11 along with naive one-column factor deformations.
Verification relies on three CONT priority-7 replay certificates in this paper's downloadable evidence pack.
Discussion
Status remains CONJECTURE. The ledger values reflect hypothetical row counts under an admissible relation, not realized designs. While the screen disproves the three JSC seeds and one-column factor deformations, it leaves chained coordinates, cyclic lifts, nonidentity C-sides, and broader joint layouts UNKNOWN. Future allocations must demonstrate quadratic-hull separation before advancing to rank-one interpolation. Curation records list no amendments for MF-048, and the register records no prior art.
For everyone — the takeaway
What this means
The proposal tries to make four existing SHA-256 multiplications pull double duty: handle their normal operations while simultaneously catching broken carries passed between columns. The three tested starting layouts don't work because their degree-two combinations still accept false internal data on valid inputs. Their row savings remain purely theoretical. Other joint layouts might still succeed; any new candidate should simply pass the pairwise hull test before anyone invests time in the heavier interpolation step.
Register references
Entry: MF-048.
Receipts: CONT jsc_two_column_screen.priority7_replay.json (SHA-256 52dfedaf95b5b779eae2cd59de8284aae5179de75ee49d7132524ac06e5eff64); CONT jsc_pinning_factorized_certificate.priority7_replay.json (SHA-256 f81fedc845fdf48f25d00c1d0b14a4f49a05ed29039e4d50d782bd51190f6e44); CONT jsc_delta2_exact.priority7_replay.json (SHA-256 1c16927bdfd256564ddc53dd7616ae5bbfb6517c843db04dfb42cbe37f4eb042).
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 3 of 3 receipt files bundled (24 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.