Research · Papers · The SHA-256 record and exact synthesis · ML-044
On the support-cage test for degree-eight p7-eligible edges
In the support-cage test on 32 degree-eight p7-eligible edges, 26 violate the cage condition and 6 pass: 1,2,3,4,13,29 → 5
Published 2026-08-29
For everyone
Plain summary
This paper evaluates an algebraic test for Boolean circuits. The support-cage test checks whether lower-degree components fit inside a specific algebraic container. When applied to 32 candidate edges in the degree-eight p7-eligible set, the test rejects 26 and lets 6 pass: 1,2,3,4,13,29 → 5. Both the algebraic computation and the frontier evaluation were independently replayed.
The overall claim remains a CONJECTURE. The support-cage data does not rule out unrestricted XAGs—acyclic circuits built from XOR and AND gates across the full input domain—because the argument still requires a cancellation-exchange / Boolean-idempotence lemma. A separate claim of exact (p=3,n=5) cage exhaustion lacks its output receipt and is not adopted. A proposed laminar normal form simplification is known to be false (MF-074) and cannot close the gap. The register records no prior art.
Result
ML-044 is a CONJECTURE. In the support-cage test on 32 degree-eight p7-eligible edges, 26 violate the cage condition and 6 pass: 1,2,3,4,13,29 → 5. This split does not prove impossibility for unrestricted acyclic full-domain XAGs. The claimed exact (p=3,n=5) cage exhaustion is MISSING its receipt and is not adopted.
Setting and definitions
The test evaluates sorted index 29, [0,1,2,4,7,6,5,3], using exterior-algebra objects H, D(H), and Λ•D(H). For this index, H is decomposable, dim D(H)=8, and six lower-degree classes fall outside Λ•D(H). The support cage is the subspace containment test built on these structures. The labels degree-eight and p7-eligible designate the specific slice of 32 candidate edges. An XAG is an acyclic XOR-AND graph; unrestricted acyclic full-domain refers to this model without structural laminar constraints.
Method
The verification couples exact exterior-algebra calculations with frontier replay. At sorted index 29, H decomposes with dim D(H)=8, placing six lower-degree classes outside Λ•D(H).
Applying the containment criterion across the 32 degree-eight p7-eligible candidate edges yields 26 violations and 6 passing edges: 1,2,3,4,13,29 → 5. Both the algebraic calculation and the frontier route were independently verified, with audit records included in the downloadable evidence pack.
The claimed exact (p=3,n=5) cage exhaustion is omitted from the verified evidence: its execution log is missing from the record.
Discussion
The support cage acts as a partial obstruction, filtering out 26 of 32 candidate edges while leaving 6 unresolved. It does not establish general circuit lower bounds or unrestricted XAG impossibility. To bridge the gap between this object-level containment test and a general impossibility theorem, the proof requires an unproven cancellation-exchange / Boolean-idempotence lemma.
The 6 surviving edges remain open; the register records neither explicit constructions nor proofs of impossibility for them. The 26 exclusions are restricted to the tested degree-eight p7-eligible set and do not automatically generalize to other degrees or unconstrained graph spaces. A proposed structural fix via laminar normal form fails because the underlying normal-form claim is independently false (MF-074). The entry holds status CONJECTURE with verdict SHOW.
For everyone — the takeaway
What this means
This analysis acts as a sieve on 32 potential circuit design routes. It eliminates 26 candidate pathways that fail an algebraic containment check, leaving 6 viable candidates for future work. Surviving the sieve does not mean a working circuit exists, only that this specific test cannot rule it out. Full impossibility across general XOR-AND circuits remains unproven until a missing algebraic lemma is established.
Register references
ML-044
Receipts: lens_r2c_independent_audit.json; quick_fixed.json; replay/frontier_exact_full_replay.json; MISSING support_cage_p3_exhaust.stdout.
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 (167 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.