# R12 Relation-Complete Transport Hypothesis **Status:** theory and finite CPU collapse-test candidate only. No neural implementation, data generation, fit, score, accelerator allocation, reasoning claim, or novelty claim is authorized. ## 1. Motivation Shohin's direct DRS probes show a characteristic asymmetry: many local digit transitions are correct, while operation order, long-range state transport, and terminal serialization fail. Generic replication is a poor match for this failure because three lanes can agree on the same wrong semantic action. Invertible transport is also insufficient because a bijection preserves an error rather than correcting it. The remaining cross-domain analogy is more specific. Physical gauge systems, error-correcting constraint complexes, and biological proofreading all exploit closed consistency relations. A missing or corrupted local transition can be identified when it creates nonzero defect around enough independent closed loops. The proposed object is therefore not another hidden scratchpad. It is a globally relation-constrained transition law whose local errors create algebraic syndromes. ## 2. Capability object Use the R12 late-query permutation witness. For `m` objects, the event alphabet contains adjacent transpositions `tau_i`, a history is a word in those generators, and a query revealed after the history asks where one object moved. The exact causal state is the resulting permutation. The smallest noncommutative instance is `S_3` with generators `s=(01)` and `t=(12)`. Its Coxeter presentation is ```text s^2 = e t^2 = e sts = tst ``` A six-state action table assigns one successor to every one of the twelve `(state, generator)` pairs. A relation-complete table must satisfy all three relations from every state and its generated orbit must contain all six states. Checking the relators only at the identity is explicitly insufficient. ## 3. Finite identification theorem For labeled six-state deterministic actions of `s` and `t`: 1. exactly 120 transitive action tables satisfy `s^2=t^2=e` and `sts=tst` globally; 2. these 120 tables are precisely the labeled regular actions of `S_3`; 3. if one directed edge is erased from the canonical table while the other eleven remain fixed, only the canonical successor completes all global relations; 4. among the 120 globally valid tables, four suitably chosen labeled edges are sufficient and necessary to identify the canonical table. The CPU falsifier must derive all four statements by enumeration. It may not encode the counts as acceptance literals. This is a finite identification result, not an asymptotic reasoning theorem. In particular, it assumes a fixed six-state carrier and exact global relation enforcement. A neural hidden state does not arrive with those semantic labels. ## 4. Resource hypothesis An unconstrained six-state, two-generator atlas has `6^12 = 2,176,782,336` possible tables and needs twelve successor labels. Using a fixed-width three-bit state identifier, that is 36 labeled target bits. The global presentation reduces the candidate set to 120 tables. Four exact edge labels can then identify the canonical table, using 12 labeled target bits. This apparent threefold reduction is not free: - the generator and relation presentation is retained side information; - all relations are checked from every state, costing 60 transition applications for one full six-state check; - the carrier size and transitivity requirement are supplied; - exact state labels or an equivalent observation decoder are supplied; - a hard-coded group action can realize the same behavior with no learned atlas at all. The surviving conjecture is therefore narrow: > At matched trainable parameters and total optimization compute, global > relation-syndrome supervision can reduce the number of labeled transition > examples needed to learn a uniform action, compared with endpoint-only > training, while generalizing to withheld edges and longer equivalent words. The claim is about learnability and data allocation. It is not a new state ontology or a separation from finite-state recurrence. ## 5. Equivalence dossier ### Tied recurrence A deterministic relation-aware recurrent updater over the same six states has exactly the same feasible action tables. This is the mandatory favorable control. Any neural treatment must beat an equally parameterized recurrence given the same relation checks, labeled edges, updates, and inference depth. ### Finite atlas An untied atlas can represent every candidate but receives no structural generalization unless the presentation is imposed. It is the endpoint-only control, not the strongest comparator. ### Hard-coded execution Swapping permutation coordinates implements the witness exactly. That is a known symbolic algorithm and a capability ceiling. It cannot be described as a learned reasoning primitive. ### SFT and recurrence Relation losses are structured supervision. Unrolling their optimizer yields ordinary training computation, and the learned updater is still a recurrent finite-state transducer. The only reopenable question is whether the structural constraints improve sample efficiency or scale extrapolation at matched resources. ### Retrieval, fast weights, and external execution The finite CPU board uses explicit tables only to test identifiability. A future neural experiment may not retrieve a gold table, generate weights from the source, or execute the group action on the host at inference. Oracle relations may supervise training, but no oracle state, query answer, repair loop, or verifier may enter autonomous evaluation. ## 6. Exact collapse test The CPU artifact must: 1. enumerate all 76 involutions on six labels and every pair; 2. retain only globally relation-valid, transitive pairs and derive the count 120; 3. independently enumerate every completion of one erased edge and prove the unique valid completion is exact; 4. enumerate all 4,096 edge-observation subsets and derive the minimum exact identifying size and version-space profile; 5. exhibit a wrong patch that passes identity-only relators but fails global relators; 6. show that the relation-complete atlas candidate set equals the matched tied relation-aware recurrence candidate set; 7. emit target-bit, relation-check, candidate-count, and presentation-byte ledgers. A CPU pass permits only independent review and, if that review accepts the resource accounting and prior-art boundary, drafting a neural preregistration. It does not authorize neural source or fitting. ## 7. Future neural falsifier boundary Any later preregistration must use increasing unseen scales and lengths, not only `S_3`. It must freeze: - relation treatment, endpoint-only control, shuffled-relation control, and a favorable relation-aware tied recurrence; - identical trainable parameter ceilings and optimizer-update budgets; - exact labeled-edge and oracle-relation counts; - held-out transitions, unseen equivalent words, non-equivalent order twins, and every late query; - state bits, source bytes, training examples, training/inference FLOPs, sequential depth, and external execution; - an autonomous evaluation with no oracle cursor, state, schedule, repair, or host executor. The direct Shohin relevance would remain conditional. A permutation witness pass would establish relation-guided state transport, not natural-language compilation, decimal arithmetic, or terminal serialization. ## 8. Prior-art and claim boundary The mathematical ingredients are known: Coxeter presentations, Cayley graphs, constraint syndromes, cycle consistency, group-equivariant learning, and finite-state recurrence. No primitive-novelty claim is allowed. The potentially new contribution is only a rigorously controlled training protocol for a tiny language model whose supervision is allocated through global algebraic defect. That delta requires a dedicated prior-art review after the CPU object is frozen.