Research · Papers · Adders, counters and the heap law · MF-053
Refutation of stationary additive raw/state phases by a three-column divergent carry path
Natural edge quotient has 491,040 candidates across 245,520 classes with span rank 74; 40 elementary q_i(c)*u_j are linearly independent
Published 2026-08-29
For everyone
Plain summary
One proposed way to fix errors in a natural Boolean relation attaches an adjustment to each step, letting the adjustment depend on the raw input and current state. A census cataloged 491,040 nonzero candidate adjustments, which group into 245,520 classes across 74 independent directions. All 40 basic q_i(c)*u_j directions are independent. An 88-dimensional audit covering every candidate found a three-column detour that leaves the honest carry path and returns to it with zero net difference across every candidate adjustment. If the only check at the end is the accumulated total, no fixed additive raw-and-state rule can spot this detour. This closes the stationary additive raw/state repair route. Other approaches—such as output-sensitive, position-dependent, non-additive, relational-phase, or separate constraint methods—remain open. The register records no prior art.
Result
The natural edge quotient contains 491,040 nonzero raw-input-twisted one-product candidates across 245,520 quotient classes with span rank 74. The 40 elementary q_i(c)*u_j directions are linearly independent.
It exposes a three-column divergent carry path that returns to the honest carry with zero relative moment across every candidate phase. Every stationary additive raw/state phase fails when evaluated solely by accumulated terminal syndrome.
Deterministic state-only closure leaves the edge/path-dependent perimeter nonempty. The result does not construct a sound phase repair.
Setting and definitions
An edge phase is an increment attached to a transition in the natural relation. Raw-input-twisted candidates depend on both raw inputs and state. The natural edge quotient partitions candidates under the register quotient; its span rank measures the dimension of the resulting function space.
The elementary directions are the 40 products q_i(c)*u_j. The audit basis consists of the functions indicator(c=s)*{1,u0,...,u9}. A divergent carry path is any trajectory deviating from the honest carry sequence; its relative moment is its accumulated phase difference relative to the honest path. A stationary additive raw/state phase sums position-invariant input/state contributions along the path and decides acceptance strictly on the final accumulated syndrome.
Method
The result derives from three computational verifications: an edge-phase census, an independent rank-and-holonomy audit, and a complete phase-span holonomy screen. Raw verification outputs and checksums are provided in this paper's downloadable evidence pack.
Discussion
The 88-dimensional audit shows that a three-column divergent carry path achieves zero relative moment across all 491,040 candidate increments, independent of honest-graph quotient gauges. Stationary additive raw/state phases with accumulated terminal syndromes cannot repair the relation, and deterministic state-only closure fails to empty the edge/path-dependent perimeter.
The result does not prove a sound phase repair. It leaves semantic-output-sensitive syndromes, position-dependent or non-additive automata, relational phase choices, and separate QCS constraints as UNKNOWN. Curation limits this refutation strictly to stationary additive raw-state phases tested by an accumulated syndrome. There are no corrections, and the register records no prior art.
For everyone — the takeaway
What this means
This repair strategy adds a number at each step and checks the total at the end. It doesn't work. A wrong path can drift away from the correct carry for three columns and merge right back with a net difference of zero, no matter which of the 491,040 rules you choose. That rules out this whole family of stationary additive checks. Other designs might still work: tracking output meanings, changing rules by position, using nonlinear updates, altering relational phases, or enforcing separate QCS constraints.
Register references
- Entry: MF-053
- Receipt: Fleet A
edge_phase_escape_census.json - Receipt: CONT
fleet_a_independent_rank_and_holonomy_audit.json(SHA-25610d5a9e7848b0ce67461bd10f1f9dea1bf0ccfeba83d975c8bd88052eda7b354) - Receipt: CONT
fleet_a_full_phase_span_holonomy_screen.json(SHA-25611a5119469e2df168f1214ba6c1e7fde027281133090b8f072e896c475058666) - 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 2 of 3 receipt files bundled (4 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.