Research · Papers · The quadratic hull and its defects · ML-000
Pinning obstructions and the one-missing-direction empirical wall
State-affine pure products lack 2-point zeros and AND₃-vanishing quadratics pin at positive wrong point; global explanation is CONJECTURE
Published 2026-09-04
For everyone
Plain summary
When automated search tools look for minimal logic circuits or exact equations, candidates across many mathematical families run into the same wall. A candidate clears all initial tests, adds one valid direction, and stalls exactly one step short of a full solution at rank n−1.
Two underlying algebraic roadblocks are proved: certain products of state variables cannot equal zero on exactly two points, and every degree-two equation vanishing on a three-input AND truth table also zeroes out an invalid coordinate. But whether these two proved roadblocks fully explain why every family stalls remains an unproved conjecture. The formal audit therefore rates this entry DEFICIENT, leaving its status as an empirical wall.
Result
Across all candidate families tested—trimmed states, schedule prefixes, Pascal candidates, PCT orbits, all cyclic systems, and all 45 transport-census edges—candidates pass every admissibility screen, extend along exactly one legal direction, and admit zero subsequent extensions, terminating at rank n−1 of n.
Two algebraic mechanisms governing this behavior are proved:
- State-{0,1} wall: Pure products of state-affine forms cannot have a two-point zero set.
- Fixed-coordinate positive-pin obstruction: Every quadratic form vanishing on the AND₃ graph also vanishes at its positive wrong point.
The global identification—that these two proved mechanisms fully account for the one-missing-direction carry signature across every listed family—is unproved and holds CONJECTURE status.
Setting and definitions
The signature occurs in classifying affine linear and quadratic representations over Boolean spaces:
- State-affine pure products: Products of affine linear forms evaluated on state variables, with vanishing loci constrained on binary domains.
- AND₃ graph: The evaluation graph of three-input Boolean conjunction.
- Positive wrong point: The deterministic non-solution coordinate forced to zero by all quadratic relations vanishing on the target evaluation graph.
- One-missing-direction signature: Structural termination where an intermediate candidate of rank k reaches rank k+1 = n−1 but permits no rank-n extension.
- Rank-tight coefficient/divisor hypothesis: The conjecture that transfers the two-point and AND₃ pinning obstructions to the specific coefficient spaces of each candidate family.
Method
Direct mathematical proofs established the two foundational algebraic lemmas (upgraded 2026-08-12). Candidate evaluations across all 45 transport-census edges, cyclic systems, trimmed states, schedule prefixes, Pascal candidates, and PCT orbits ran through automated certificate generation and exhaustive search replays.
Package 3 incorporated a complementary quadratic pinning profile and a scoped DP10 missing-plane obstruction certificate. Register verification artifacts comprise:
- brief §A.1, §B.1, §B.3
- handoff §3.4
- CONT batch_a_replay_receipt.json
- CONT batch_b_replay_receipt.json
- CONT quadratic_pinning_profile_live_results.json
- CONT dp10_phase_ab_certificate.json
Discussion
The register audit assigned this entry a verdict of DEFICIENT. Proving that state-affine pure products reject two-point zero sets and that AND₃ quadratic forms force positive-pin vanishing does not establish that these mechanisms account for the rank n−1 wall across every candidate family.
Exporting the obstruction requires proving the rank-tight coefficient/divisor hypothesis for each family individually. While Package 3's live quadratic pinning profile and scoped DP10 certificate confirm local missing-plane obstructions, they do not supply the global identification theorem. ML-000 remains classified as EMPIRICAL-WALL. No historical route verdict is modified or deleted, and the register records no prior-art position.
For everyone — the takeaway
What this means
Exact synthesis searches often stall one dimension before the target rank n. We know why certain local equations fail: two proved algebraic rules block them from closing. What we do not know is whether those two rules explain the failure in every family tested. The proved pieces give concrete diagnostic tools, but the general wall remains an open problem.
Register references
- Entry ML-000
- brief §A.1, §B.1, §B.3
- handoff §3.4
- CONT batch_a_replay_receipt.json
- CONT batch_b_replay_receipt.json
- CONT quadratic_pinning_profile_live_results.json
- CONT dp10_phase_ab_certificate.json
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 4 of 4 receipt files bundled (17 KB). Anything not bundled is still hashed in the manifest and lives in the compute-box working trees.
Changelog
Last reviewed 2026-09-04
- 2026-09-04Published on this site.