R12 Polynomial-Coded Action No-Go

Status: exact positive control; reject as a novel mechanism or fair recurrent separation.

Candidate

Represent each event action as a bounded-degree polynomial over a finite field, learn it by interpolation, encode the latent state with an error-correcting code, and apply decode-compute-reencode at every reasoning step. This gives an exact, nonlinear, length-extrapolating and noise-robust recurrence.

Identification and robustness theorem

Let each action be

U_a : F_q^d -> F_q^d

with coordinate degree at most k < q. The scalar polynomial space has

M = binomial(d+k, k)

coefficients. For sampled input states S_a = (x_1,...,x_m), let V_a be the multivariate evaluation matrix and define the evaluation code

C_{S_a} = {(p(x_1),...,p(x_m)) : degree(p) <= k}.

For every output coordinate:

1. noiseless exact identification is possible exactly when rank(V_a)=M;
2. exact recovery from at most e adversarially corrupted transition labels is possible exactly when d_min(C_{S_a}) >= 2e+1;
3. once every generator U_a is identified, every finite composition is exact by induction.

If E is a code correcting t physical errors and D its decoder, then

Phi_a(z) = E(U_a(D(z)))

is an exact t-robust recurrent action under the stated fault boundary. Learning and runtime repair are therefore interpolation and decoding problems.

Smallest nonlinear witnesses

Over F_3, U(x)=x^2 with ternary repetition E(x)=(x,x,x) is nonlinear and corrects one symbol error. Traces confined to {0,1} cannot distinguish it from the identity, exposing the excitation requirement.

For a reversible Boolean witness, Toffoli

U(x,y,z) = (x,y,z xor (x and y))

is the smallest nonlinear reversible action: every permutation of two Boolean bits is affine because AGL(2,2) already has order 24 = |S_4|. Combining Toffoli with a binary [6,3,3] code corrects one physical bit error; the Hamming bound excludes a binary one-error-correcting encoding of three data bits with length at most five.

Fatal matched-comparator theorem

Give a universal recurrent comparator the same physical state bits, polynomial degree promise, samples, and enough update computation to implement D, U_a, and E. It can execute the identical interpolation and the identical decode-compute-reencode recurrence. It therefore has the same sample complexity, length extrapolation, and robustness.

The apparent description gap

binomial(d+k,k)  versus  q^d

is only a low-degree learner versus an arbitrary transition table. It vanishes as an architecture-wide separation once the comparator receives the same target promise. Removing the promise reintroduces ordinary finite-system nonidentifiability: any finite transition map has a finite-field polynomial representation, and off-support patches survive.

Collapse audit

• bounded-degree interpolation is classical polynomial learning/coding;
• state repair is error-correcting or fault-tolerant computation;
• bijective maps are permutation-group actions and arbitrary maps form a transformation semigroup;
• unknown coordinate changes destroy the presentation-dependent degree and locality unless a field basis is externally anchored;
• unknown noisy linear structure already contains parity with noise;
• explicitly supplied arithmetic operators are an arithmetic inductive bias, not discovered reasoning.

This family remains an excellent exact control for any future proposal claiming nonlinear extrapolation plus runtime noise stability. It is not a novel R12 primitive and does not justify a CPU falsifier, Shohin fit, or H100 experiment.

Reopening condition

Reconsider only if a task-native observable identifies the field basis and code from ordinary traces, and the proposed resource restriction excludes simulation by a fairly matched universal recurrent circuit without simply denying it the same computation.

Primary references:

• Z. Dvir and A. Shpilka, noisy interpolation sets and punctured Reed-Muller constructions.
• D. Spielman, reliable computation with efficient error-correcting codes.