Research · Papers · Formal verification and machine-checked proof · MF-027
Verifier-level audit of finite phase-trellis relations
Subset construction gives exact completeness/soundness checks for finite phase relations; natural sample & ITER19 K0 adapter fail soundness
Published 2026-08-29
For everyone
Plain summary
MF-027 presents a verifier-level audit for finite phase-trellis relations. A phase-trellis relation is a finite table of allowed local phase transitions. The audit checks completeness, meaning that required cases are represented, and soundness, meaning that every represented case is valid. It also checks Court Zero, the verifier's legality rule for globally allocated cyclic paths, where local steps are connected around a complete cycle. This matters because a table can follow the expected format while still admitting a bad transition. The supplied natural sample failed the soundness check. An exact adapter for the ITER19 K0 case also failed on every real SHA round-0 case involving the lowest three bits. The record says the candidate labelled p5 still needs five singleton-C pins per column, pins that fix a single C value, affine output masks, output adjustments made from XORs and constants, boundary semantics, and a deterministic lower-index-only witness selector, which chooses a witness using only earlier indices. The register states no prior-art position and records no correction for this entry.
Result
For a finite phase relation, the subset construction gives exact checks for completeness and soundness. Court Zero licenses globally allocated cyclic paths for the relation. The verifier-shaped p5 candidate recorded by the audit still requires five singleton-C pins per column, affine output masks, boundary semantics, and a deterministic lower-index-only witness selector.
The supplied natural sample fails soundness. An exact ITER19 K0 adapter also fails soundness on every real SHA round-0 low-three-bit case tested in the record. The result is therefore a method for making these checks and a negative outcome for the two supplied samples. It is not a positive soundness result for either sample.
Setting and definitions
The object under audit is a finite phase relation together with its subset construction. The subset construction is the representation used to test the relation as a collection of admissible phase subsets. Completeness asks whether the required cases are covered. Soundness asks whether the cases admitted by the construction belong to the intended relation.
Court Zero is the verifier-legality layer for globally allocated cyclic paths. The audit keeps this legality question separate from semantic soundness. The notation p5 names the verifier-shaped phase-trellis candidate in the entry. ITER19 K0 names the fixed relation for which an exact adapter was replayed. A singleton-C pin, an affine output mask, boundary semantics, and a lower-index-only witness selector are the additional pieces named by the register for a verifier-shaped p5 candidate.
Method
The audit constructed the subset representation of the finite phase relation and used it to check completeness and soundness exactly. It then applied the Court Zero legality conditions to the globally allocated cyclic paths. This makes the legality test explicit at verifier level instead of treating a syntactically available path as sufficient evidence.
The work recorded two concrete checks. First, the supplied natural sample was tested against the subset construction and failed soundness. Second, the fixed ITER19 K0 relation was connected to an exact adapter and replayed on the real SHA round-0 low-three-bit cases. Every case in that set failed soundness. The register also records the remaining verifier-shaped p5 requirements: five singleton-C pins per column, affine output masks, boundary semantics, and a deterministic lower-index-only witness selector.
The raw evidence receipts for the verifier-legality audit and the exact adapter replay are available in this paper's downloadable evidence pack.
Discussion
Court Zero legality and semantic soundness are separate obligations. The audit can therefore report a path as legal while still reporting the relation as unsound. The natural sample and the exact ITER19 K0 adapter illustrate that distinction in the recorded tests.
The entry records the pieces that a verifier-shaped p5 candidate still needs: five singleton-C pins per column, affine output masks, boundary semantics, and a deterministic lower-index-only witness selector. The two failed samples cannot support a claim that the tested relations are sound. The result is limited to those recorded cases and requirements; it does not establish a positive conclusion for every phase relation.
No corrections are recorded for MF-027, and no prior-art position is stated. The register therefore supports a method paper about verifier-level checking and the recorded negative tests, with no novelty position attached to it.
For everyone — the takeaway
What this means
MF-027 is a test instrument for proposals that use finite phase rules. It separates two questions: whether a path follows the verifier's format, and whether every case accepted by the rules is valid. The natural example and the case labelled ITER19 K0 both enter the audit and fail the second question. The result gives later work a precise test to rerun. A candidate labelled p5 still needs the listed pins, output masks, boundary rules, and a rule for choosing a witness from earlier indices before it can be assessed as complete. This paper records a checking method and two failed tests. It does not record a successful construction.
Register references
- Entry: MF-027.
- Receipt artifacts:
phase_trellis_p5_verifier_legality_audit.json,ghost_iter19_k0_phase_trellis_adapter_receipt.json. - Prior-art work: 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 2 of 2 receipt files bundled (9 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.