R12 Relation-Complete Transport Review Result

Decision: finite S_3 identification mechanics GO; uniform neural reasoning mechanism, resource advantage, preregistration, fitting, and H100 allocation NO-GO.

Reviewed claim

The candidate proposed globally enforced Coxeter relations as a way to recover missing transitions with fewer labeled endpoints than an unconstrained atlas. The finite S_3 falsifier correctly derives:

• 76 involutions on six labels;
• 120 globally relation-valid transitive actions;
• equality of those actions with labeled regular-action relabelings;
• unique completion of one erased canonical edge;
• a target-specific four-edge identifying set for the canonical table.

Those are valid finite statements. They do not establish a uniform neural sample-efficiency or reasoning advantage.

Uniform theorem

Let N = m! and let the adjacent transpositions of S_m act transitively on an N-state carrier.

1. Orbit-stabilizer gives a trivial stabilizer, so every such action is regular.
2. Up to conjugacy, the regular action is unique. If semantic carrier labels do not matter, zero transition anchors are required to identify the action.
3. On a fixed labeled carrier there are (N - 1)! distinct regular action tables, because the centralizer of the regular action has size N.
4. Exact semantic labeling therefore remains the unresolved resource. Direct state labels require N - 1 labels; transition anchors have a target- specific lower bound ceil((N - 1) / 2) and a spanning-tree upper bound N - 2.
5. A uniform learner that identifies every labeled action requires at least ceil(log_(N-1)((N-1)!)) = N - Theta(N / log N) transition queries in the worst case.

The semantic identification cost is therefore Theta(m!). The exact coefficient is not needed to decide the neural lane.

Scaling ledger

There are m(m-1)/2 Coxeter relation schemas. Exhaustively enforcing them from every state costs m! * (2m(m-1) - 2) transition applications.

Matched-control collapse

The apparent target-bit reduction survives only against an untied atlas.

• An untied atlas stores m!(m-1) successors and pays factorial state alignment.
• A relation-aware tied recurrence on an atomic carrier has the same (N - 1)! gauge ambiguity.
• A recurrence with permutation coordinates needs only O(m log m) state bits and one shared adjacent-swap rule.
• A hard-coded coordinate update swaps positions i and i+1 and requires no learned transition atlas or relation oracle.

The favorable recurrence and hard-coded controls remove the claimed advantage. Relation consistency may still be a useful regularizer, but it is not a new reasoning primitive.

Omitted resources in the candidate ledger

The current 36-target-bit versus 12-target-bit comparison does not charge:

• the selected anchor indices;
• the semantic carrier-to-permutation decoder;
• supplied carrier size and transitivity;
• generator-token and presentation semantics;
• factorial relation-oracle applications;
• query decoding from arbitrary state labels;
• the group operation used to generate supervision.

If relation consistency is architectural, the favorable tied recurrence must receive it. If it is supervised, relation-oracle generation and optimization must be counted.

Gate table

Preservation boundary

Preserve the finite S_3 artifact as an exact identifiability certificate and possible relation-consistency regularizer. An optional CPU closure may solve the exact S_4 anchor coefficient, but it cannot overturn the factorial scaling result and has no capability priority.

The highest-leverage Shohin frontier remains natural-language compilation, common-mode operation-selection errors, internal state actuation, recurrent consumption, and termination.