Research · Papers · The quadratic hull and its defects · ML-024
On the quadratic hull of carry relations under state-only catalyst lifts
Deterministic state-only lifts fail at every row budget: 8,040 acyclic-p6 models eliminated, 8,184 anchors yield no hull repairs
Published 2026-08-29
For everyone
Plain summary
ML-024 tests whether values calculated solely from a three-bit carry state (c0, c1, c2) can fix the natural carry relations. We evaluated all 256 functions in the complete eight-coordinate state algebra. The degree-2 lift leaves 23,808 invalid assignments in the quadratic hull—the points that slip past degree-2 checks. An exhaustive search closes all 8,040 models in the six-product acyclic class. Across 8,184 one-block candidate choices combining state and data, 8,177 leave all 24,576 invalid hull points, while seven choices labelled y=c1 leave 20,480. Architectures based on position, history, relations, cycles, or modified edge rules remain open. The register records no prior-art position.
Result
Deterministic state-only lifts fail at every row budget. The scoped acyclic-p6 census eliminates all 8,040 models. Enumerating the corrected 8,184 anchors yields no hull repairs: ten elementary c0*c2 phases retain all 24,576 spurious points, and the broader raw-input-twisted edge quotient remains nonempty.
Setting and definitions
ML-024 evaluates the χ₀₁ catalyst route. State-only denotes any deterministic boolean function over c0, c1, and c2. The quadratic hull is the variety defined by degree-at-most-two relations; a hull repair eliminates all false assignments from this variety. The label p6 denotes the scoped six-product cut.
Method
An exhaustive 225-edge census yielded 45 screen survivors, each proved exact-UNSAT at p6. Verification covered the full eight-coordinate state algebra, the maximal lift, the acyclic-p6 class, the corrected anchor set, and the ten elementary edge phases. the corrected enumeration ran deterministically without SAT/SMT backends. Raw verification receipts are in this paper's downloadable evidence pack.
Discussion
Closure is exact for all state-only lifts, the scoped acyclic-p6 class, and the corrected anchor set. Auditing the data-pin gauges preserves the negative verdict. The c0c2 p7 candidate survived only the initial degree screen and fails under exact checking. The status remains UNKNOWN for transient, path-dependent, stationary two-boundary, relational, cyclic, general-synthesis, and raw-input-twisted edge classes. The entry carries this caveat accordingly. The register records no prior-art position.
For everyone — the takeaway
What this means
Tracking only the current three-bit carry state cannot fix this relation family under the tested rules. Checking every state-only function and acyclic model rules out this approach, and the corrected 8,184 candidate tally confirms the dead end. Other designs might work by reading direct inputs, tracking step counts, following full execution paths, or introducing cycles; those remain open. This result settles the state-only route without closing general repair mechanisms.
Register references
ML-024. Receipt artifacts: c0c2_a3_verdict.json; c0c2_residue_lift_result.json; pro3_c0c2_twisted_defect_replay.json; pro3_c0c2_twisted_plane_data_split_result.json; pro3_c0c2_data_split_state_addition_audit.json; pro3_c0c2_data_split_full_affine_census.json; pro3_c0c2_data_split_c2_gauge_extension.json; pro_request3_task1_replay_receipt.json; fleet_a_state_closure_recheck.json; fleet_a_c0c2_elementary_edge_phase_fresh.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 0 of 0 receipt files bundled (1 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.