R12 Local Reversible Rule Control

Status: retained as the strongest finite nonlinear structured-action control found so far. It gives a real polynomial description and sample advantage over global tables and defeats low-rank linear comparators, but the advantage is purchased by handed local coordinates and rule sharing.

1. Family

Let the exact state be x in {0,1}^n. Each event supplies a rule label and a labeled tuple of at most k wires. The label selects one of L unknown local reversible maps

g_l : {0,1}^k -> {0,1}^k.

The event replaces only the selected coordinates by g_l of their current values. Training traces expose pre-state, post-state, label, and affected wire tuple; each observed output bit is independently flipped with probability eta < 1/2.

This is nonlinear when the rule set contains Toffoli-type gates. NOT and Toffoli generate universal reversible Boolean computation, so the family is not a disguised linear automaton.

2. Learnability theorem under handed locality

For each rule, input pattern, and output bit, majority vote has error at most exp(-Theta(m(1-2 eta)^2)) after m occurrences. With balanced coverage of all rule-pattern cells, a union bound gives exact recovery with probability at least 1-delta after

T = O(
  L 2^k / (1-2 eta)^2
  * log(L k 2^k / delta)
)

examples, up to the coverage constant. The learned presentation uses O(L k 2^k + log n) bits plus the wire labels, retains exactly n state bits, and applies an event in O(k) work. A global transition table instead has 2^n rows per event.

This is a real structured resource separation. It is also an ordinary local rule learner once the coordinates and sharing map are supplied.

3. Separation from low-rank predictive-state controls

Let N=2^n and let the reachable action be transitive on the N states. Use all balanced Boolean readouts as late queries. The state-by-readout incidence matrix has full row rank N. More strongly, every rank-d approximation has normalized mean-square error at least

(N-d) / (4(N-1)).

Thus a low-dimensional WFA/OOM/PSR/Hankel model cannot uniformly approximate the balanced readout family unless d is exponential. The local nonlinear presentation remains polynomial.

The separation is useful because it rules out the project's easiest linear collapse. It does not rule out locality-aware neural cellular automata, program learners, sparse circuits, or equivariant models.

4. Fatal hidden-coordinate assumption

Conjugate the global state by an arbitrary bijection

phi : {0,1}^n -> {0,1}^n.

The transformed dynamics phi g phi^-1 have identical abstract transition behavior but generally destroy every visible locality and sparsity property. Ordinary input/output traces identify the action only up to such a conjugacy unless observations anchor the coordinates. Therefore the sample theorem does not establish that a learner can discover the local presentation.

The same problem survives softer versions:

• supplying wire tuples is already a program trace;
• supplying shared rule labels is meta-data about the factorization;
• runtime state noise is not corrected merely because training labels are denoised;
• reversible dynamics cannot erase accumulated state corruption without extra redundancy and irreversible correction;
• known locality-aware, kernel, equivariant, and program-induction controls can exploit the same gift.

5. Verdict

This family should be used as a hard control for any later R12 mechanism. A candidate must learn a robust local or modular presentation from ordinary partial observations, without receiving wire coordinates, rule labels, or the sharing map, and must survive runtime noise. Until a theorem supplies those missing steps, no Shohin fit or H100 job is authorized.