R12 Finite State-versus-Motor No-Go

Status: exact identifiability boundary. No finite challenge board, by itself, can distinguish a reusable state from every finite motor table.

1. Behavioral object

Let a deterministic task be

M = (S, G, Q, A, delta, output),

where G contains update generators and Q contains late consumers. Define the residual behavior of state s by

rho_s(w,q) = output(delta_w(s), q),  w in G*.

Two states are behaviorally equivalent exactly when

s ~ t iff rho_s(w,q) = rho_t(w,q) for every w,q.

A source-deleted realization has writer W:S->Z, update maps T_g, and reader D. It is reusable over the declared system exactly when

D(T_w(W(s)), q, pi) = pi(rho_s(w,q))

for every reachable state, generated update word, consumer, and supplied output recoding pi.

2. Finite-protocol table theorem

For any finite evaluation protocol P, including adaptively chosen but finite-support challenges, construct a machine with states

Z_P = {(source_id, tested_update_prefix)}.

The writer stores source_id; each tested updater appends its label; the reader returns a table entry for (source_id,prefix,consumer) and then applies the supplied output recoding. Every untested transition enters a failure sink.

This machine can pass all of the following on P:

physical source deletion;
multi-step continuation;
consumers and updates hidden from the scorer until after commitment;
arbitrary supplied output recodings;
packet swaps and complement ablations.

It has no behavior beyond the finite protocol tree. Secret scoring prevents manual board tuning but does not turn a finite board into a universal theorem. Any valid claim must therefore bound description length, retained state, hypothesis class, or scale dependence.

3. Four exact counterexamples

Consumer insufficiency: (a,b) and fitted parity consumers admit the one-bit packet a XOR b, which cannot answer a held-out query for a.
Unseen operator: two worlds may agree on all observations while a new symbol denotes identity in one and bit-flip in the other. Its semantics require a declared grammar, examples, or an oracle.
Finite horizon: a machine may match every continuation through depth L and deliberately fail at L+1.
Output recoding: withholding pi makes two recodings jointly impossible; supplying pi lets a motor table recode too. Recoding rejects token-specific actuators, not arbitrary answer bundles.

4. Weakest conditional sufficiency

The weakest non-circular exact criterion is generator-complete, separator-complete bisimulation:

declared generators span every admitted update;
a consumer core separates all residual states;
source packets are complete before late challenge disclosure;
every generator update commutes with the packet realization;
every separating consumer reads the correct answer;
source deletion and output recoding are process-enforced.

Induction proves every generated continuation. The resulting minimal reachable realization is isomorphic to the Myhill-Nerode residual quotient. A complete answer bundle closed under every generator is behaviorally a state; its internal ontology is not separately identifiable.

5. What finite experiments may establish

A score-blind experiment can establish a resource-bounded result:

One uniform mechanism generalizes across post-commit generated interfaces and unseen scale using fewer retained bits, parameters, examples, or compute than a named motor-table/control family.

It must freeze the complete resource vector, test increasing scales, include a favorable table/horizon control, and state exactly which hypothesis class was rejected. A finite pass never excludes unlimited tables or proves universal reasoning.

6. Prior-art boundary

residual equivalence and minimal deterministic realization are the Myhill-Nerode/Moore-machine construction;
local transition closure is bisimulation;
predictive-state representations intentionally treat a sufficient vector of future-test predictions as state;
minimal observable/controllable state is unique only up to coordinates in classical realization theory;
IIT and DAS test or install a supplied causal abstraction; they do not prove that the abstraction is the unique reusable state.

Primary references:

Myhill-Nerode: https://doi.org/10.1090/S0002-9939-1958-0135681-9
Predictive state representations: https://papers.neurips.cc/paper/1983-predictive-representations-of-state.pdf
Interchange Intervention Training: https://proceedings.mlr.press/v162/geiger22a.html
Distributed Alignment Search: https://proceedings.mlr.press/v236/geiger24a.html

7. Shohin decision

No new Shohin fit may advance from a finite consumer suite, MCBS projection, J-lens basis, or output recoding alone. The active bounded target is a uniform post-commit interface protocol with explicit packet and model-description limits. R12_POST_COMMIT_INTERFACE_FALSIFIER_PREREG.md tests only whether its CPU scorer correctly separates the declared complete-state and motor controls.