Research · Papers · The SHA-256 record and exact synthesis · ML-031
Closure of flat Joint Semantic-Carry systems at every row budget
Every system in C_flat affine in the existing 41 coordinates is closed at every row budget
Published 2026-08-29
For everyone
Plain summary
The flat version of the Joint Semantic-Carry problem is closed. In this setting, flat means using only linear combinations plus constants of the 41 listed quantities. The row budget counts product operations (AND-like steps). Exhaustive tests show that no flat system survives at any row budget.
The underlying ideal yields exactly 51 factorable equations spanning 21 dimensions, and none mixes the semantic and state roles needed for the construction. A candidate attempting this splice also fails the full test suite.
This obstruction applies strictly to the flat class. It leaves open systems with hidden or chained helper variables, higher-degree equations, larger blocks, modified internal state, cyclic lifts, or alternative relations. Those avenues remain UNKNOWN, and the broader search stays a CONJECTURE. The register records no prior art.
Result
Let C_flat be the class of Joint Semantic-Carry systems affine in the existing 41 coordinates. Every system in C_flat is closed at every row budget: the flat-JSC wall is PROVED.
The exact 25-dimensional ideal yields exactly 51 factorable equations spanning 21 dimensions, with no mixed equations. A false semantic/state splice satisfies the complete atlas, leaving zero surviving flat systems at any row budget.
The conditional targets attached to ML-031 are JSC-13 17,841, JSC-12 16,881, and JSC-11 15,921. This closure rules out the flat route toward those targets without settling non-flat constructions or establishing any construction at these three targets.
The supplied full-raw seed allocations for JSC-13, JSC-12, and JSC-11 each contain an explicit same-input wrong point in the exact degree-two hull, excluding them from SAT. Chained coordinates, cyclic lifts, nonidentity C-sides, and alternative relational allocations remain UNKNOWN. The non-flat route remains CONJECTURE.
Setting and definitions
The entry targets Joint Semantic-Carry conditional bounds JSC-13 17,841, JSC-12 16,881, and JSC-11 15,921.
- C_flat: affine systems restricted to linear combinations and constants of the existing 41 coordinates.
- Factorable equation: an equation matching the bilinear product structure required by the JSC construction.
- Mixed equation: an equation coupling semantic and state roles.
- Exact degree-two hull: the algebraic screen evaluating quadratic relations among seed allocations.
- Same-input wrong point: an evaluation where a proposed relation matches the degree-two hull but produces incorrect output.
- Non-flat class: extensions introducing hidden or chained witnesses, cyclic lifts, modified state presentations, or higher-degree elimination.
Method
Initial screening proceeded in two stages. Naive one-column deformations yielded exact-negative results. A genuine two-column raw graph was exported and exactly ranked across all four K-pairs. Degree-two-hull and rank checks identified an explicit same-input wrong point for each raw seed allocation in JSC-13, JSC-12, and JSC-11, barring these seeds from SAT input.
Strengthening analyzed the 25-dimensional ideal, extracting all 51 factorable equations spanning 21 dimensions. None were mixed. Testing a false semantic/state splice against the complete atlas and executing an independent factorable-equation replay confirmed flat closure across all row budgets.
Verification relies on base receipts alongside strengthening receipts and reports, all included in this paper's downloadable evidence pack.
Discussion
The result establishes an obstruction for the 41-coordinate affine class rather than resolving the full Joint Semantic-Carry program. C_flat is PROVED closed at every row budget, and supplied full-raw seeds fail via degree-two hull counterexamples.
The broader non-flat formulation remains CONJECTURE. Unresolved structures—hidden or chained witnesses, higher-degree elimination, fibres, larger blocks, alternative state choices, cyclic lifts, nonidentity C-sides, and other relational allocations—remain UNKNOWN. The flat obstruction isolates the exact boundary where affine coordinate reuse terminates, leaving non-affine formulations open. The register records no prior art.
For everyone — the takeaway
What this means
The 41-coordinate flat approach cannot produce a valid circuit at any row count. To find working solutions for JSC-13, JSC-12, or JSC-11, future attempts must break the flat assumption. Viable paths include adding chained helper variables, using higher-degree algebra, expanding the block size, or restructuring the state relations. Because these non-flat routes remain unverified, the overarching problem remains a CONJECTURE. The register records no prior art.
Register references
ML-031; receipts: CONT jsc_two_column_screen.priority7_replay.json, jsc_pinning_factorized_certificate.priority7_replay.json, jsc_delta2_exact.priority7_replay.json, lens_l3_jsc_maj_verification_report.md, lens_l3_jsc_maj_verification_receipt.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 2 of 2 receipt files bundled (5 KB). Anything not bundled is still hashed in the manifest and lives in the compute-box working trees.
Changelog
Last reviewed 2026-08-29
- 2026-08-13This result is proved for flat systems, where the exact 25-dimensional ideal has 51 factorable equations spanning 21 dimensions, closing all systems in the 41 coordinates at every row budget. Non-flat cases—including higher-degree elimination, larger blocks, and changed state—remain an open conjecture.
- 2026-08-29Published on this site.