Research · Papers · The quadratic hull and its defects · MF-052
Closure of the ordered acyclic A*B=C catalyst class at p6
Ordered acyclic A*B=C catalyst class at p6 = ∅ (1,005 exact-UNSAT, 7,035 rank-dead across 67 deterministic state-feature subspaces)
Published 2026-08-29
For everyone
Plain summary
A state-only catalyst is an extra helper signal computed strictly from the current state. We tested every deterministic state-feature subspace across the full 8,192-row natural transition relation—67 subspaces in total. Across 8,040 candidate models, all 1,005 one-output models are exact-UNSAT (no solution satisfies the target equations), and all 7,035 two-output models are rank-dead (they fail to provide enough independent linear information). This rules out free state-only signals for ordered acyclic A*B=C networks. It leaves open designs with dependency loops, signals that read raw inputs, nonlinear imports, and relational interfaces. The register records no prior art.
Result
Across all 67 deterministic state-feature subspaces of the 8,192-row natural transition relation:
- 1,005 one-output-feature models: exact-UNSAT
- 7,035 two-output-feature models: rank-dead
The complete set of 8,040 models is empty, closing the ordered acyclic A*B=C catalyst class at p6.
Setting and definitions
A state-only current feature is a deterministic function of the current state. The target family consists of ordered acyclic A*B=C networks equipped with one or two outgoing state features supplied at zero multiplicative cost. The terms exact-UNSAT and rank-dead follow standard register usage.
Method
The census exhaustively generated and checked 8,040 models across 67 subspaces: 1,005 one-output models and 7,035 two-output models. The result was verified by an independent recheck, with raw run outputs preserved in this paper's downloadable evidence pack.
Discussion
This census proves that ordered acyclic A*B=C networks cannot reach the target using arbitrary deterministic state-only features. The bound is tight to this model family: cyclic witnesses, raw-input-dependent catalysts, nonlinear imports, and relational interfaces remain unconstrained. No corrections are recorded for MF-052. The register records no prior art.
For everyone — the takeaway
What this means
Can a six-product circuit without feedback loops succeed if we provide free helper signals computed purely from its internal state? We tested every possibility: all 1,005 one-signal setups fail the target equations, and all 7,035 two-signal setups run out of linear rank. That completely rules out state-only catalysts for acyclic p6 networks. To make progress, constructions must use feedback loops, signals reading original inputs, nonlinear imports, or multi-wire relational interfaces. The register records no prior art.
Register references
Entry: MF-052.
Receipts: Fleet A state_only_catalyst_closure_full.json; CONT fleet_a_state_closure_recheck.json (SHA-256 46580a840762397da3af117793dfb8e50c9a1f3d8d2da831b8016ce8e78f1e6d).
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 1 of 2 receipt files bundled (2 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.