Research · Papers · The quadratic hull and its defects · MF-031
An impossibility theorem for deterministic state-only lifts of the c0c2 quadratic hull
No deterministic state-only feature set can repair the relation, as maximal lift replay retains 23,808 of 24,576 old wrong hull points
Published 2026-08-29
For everyone
Plain summary
The c0c2 relation is a rule over three internal 0-or-1 state bits: c0, c1, and c2. This entry tests whether adding extra features built solely from those three bits can make the relation easier to certify.
It cannot. Three bits yield eight possible states and 256 possible Boolean functions. Eight baseline coordinates already span all 256 functions. Every new deterministic state-only feature is just an affine combination of this existing basis. Because of that, degree-two tests (products of at most two coordinates) gain no new directions. An exact replay still admits 23,808 of the 24,576 old wrong hull points. No selection of state-only features and no row budget can repair the hull. Features depending on inputs or other mechanisms remain outside the theorem.
Result
The coordinates 1,c0,c1,c2,c0*c1,c0*c2,c1*c2,c0*c1*c2 have rank eight on the eight carry states, spanning all 256 deterministic Boolean functions on c0, c1, and c2. Every further deterministic state-only coordinate is affine in this basis, so adjoining one cannot enlarge the degree-at-most-two evaluation space of the maximal lift. Exact replay of the maximal lift retains 23,808 of the 24,576 old wrong hull points. No deterministic state-only feature set can repair the relation, regardless of row budget.
Setting and definitions
The state variables are c0, c1, and c2, defining eight carry states. A deterministic state-only coordinate is a Boolean function on these variables alone. The maximal lift consists of the eight coordinates listed above. Its degree-at-most-two evaluation space is the linear span of constants, coordinates, and pairwise coordinate products evaluated across the eight states. The old wrong hull points are the 24,576 candidate assignments admitted by the earlier c0c2 quadratic hull.
Method
The proof combines an exact replay of the maximal lift with c0c2 residue and defect analysis. The replay verifies the rank-eight basis and tallies the 23,808 surviving wrong hull points. Quotient-rank and shell certificates establish the impossibility of p7, p8, and p9 for the explicit complete-state presentation. An independent task replay verifies the outcome. Raw output and verification receipts are in this paper's downloadable evidence pack.
Discussion
This entry generalizes the p9-scoped obstruction recorded on 2026-08-12 to the complete-state algebra. Because the eight retained coordinates already evaluate to every Boolean function on the three state bits, the maximal lift exhaustively tests the state-only class. Its 23,808 surviving wrong hull points obstruct every deterministic state-only feature set under any row budget.
The rank computation gives a universal quantifier over the entire class rather than a bound over a specific candidate menu. Arbitrary row budgets can only duplicate or linearly combine existing degree-at-most-two evaluations; they cannot introduce new evaluation directions.
The scope is strictly state-only. Input-dependent, transient, relational, cyclic, and witness-multiplicity lifts fall outside the theorem. Bounded p10 canaries returned NO_SAT_FOUND in broader search, but that observation does not extend this result to those broader lift families. The curation record lists Corrections: NONE, with no prior-art comparison or novelty claim.
For everyone — the takeaway
What this means
You cannot repair the c0c2 quadratic hull by inventing features from the three state bits alone. The eight baseline coordinates already cover all 256 possible state behaviors, so quadratic checks built on them cannot generate new constraints. The check still lets 23,808 invalid points pass. Any successful fix must use external inputs, time-dependent steps, relations, cycles, or multiple witnesses.
Register references
Entry: MF-031
Receipts: CONT c0c2_residue_lift_result.json, pro3_c0c2_twisted_defect_replay.json, pro_request3_task1_replay_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 3 of 3 receipt files bundled (10 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.