R12 Commutator Factorization No-Go

Status: rejected as an R12 invention. Observable event commutators can recover a conditional central/direct product of transition groups, but do not identify a product residual state. Complete positive cases reduce to known group, automata, trace-monoid, or Cartesian-graph decomposition.

1. Observable commutator support

Let X be N finite residual states, A be m labeled events with permutation transitions T_a, and let joint late-query signatures separate states. Define

S(a,b) = {x in X : T_a T_b x != T_b T_a x}.

Separating queries make S(a,b) experimentally observable when states can be prepared reproducibly. State-uniform event independence means S(a,b) is empty.

2. Commutator-to-central-product theorem

Let G=<T_a:a in A>. Connect event labels that do not commute and let the connected components be A_1,...,A_k, with

G_i = <T_a : a in A_i>.

Then the G_i commute pairwise and multiplication gives a surjection

mu : product_i G_i -> G.

Every coordinate of ker(mu) lies in

G_i intersect <G_j:j!=i> subset Z(G_i).

Commutators therefore identify at most a central product. The group is the direct product of the G_i exactly when all cross-intersections are trivial; centerless factors are sufficient.

This algebraic condition still does not factor the state.

3. Stabilizer obstruction

Assume the action is transitive and fix state x. Let

H = stabilizer_G(x),
H_i = H intersect G_i.

Even when G is a direct product, a factorwise state decomposition

X isomorphic to product_i G_i/H_i

exists iff

H = product_i H_i,

equivalently |Gx|=product_i |G_i x|. A diagonal or subdirect stabilizer couples the apparent modules.

The smallest useful counterexamples are:

1. on Z_3, a(x)=x+1 and b(x)=x-1 commute but generate the same C_3, not two modules;
2. S_3 x S_3 acts faithfully on six states X=S_3 by (g,h).x=g x h^-1; its factors commute and form a true direct product, but the stabilizer is diagonal S_3, so 6 != 6*6 and no product state exists;
3. the regular C_2 x C_2 action has three inequivalent choices of two C_2 factors, so abelian centralizers do not select one decomposition.

4. Probe and information cost

With reproducible access to every state, recovering all event tables costs mNq readouts for q separating query bits. Directly testing every pair costs

2 binomial(m,2) N q

readouts. Independent bit noise eta<1/2 adds the repetition factor

O((1-2eta)^-2 log(Pq/delta)).

Without full-support state preparation or a positive minimum state probability, no finite exact guarantee exists. An adversary can change one unobserved state-event transition and destroy commutation or factorization. Thus unrestricted state-uniform locality needs Omega(mN) transition probes.

If a product chart and event assignment are already known, local tables use

sum_i m_i n_i log n_i

bits instead of mN log N, for N=product_i n_i. But an opaque chart can cost log(N!)=Theta(N log N) bits and exact discovery already pays the extensional probe cost. Retained state does not shrink:

sum_i log n_i = log N.

Query savings additionally require low-arity factorized readouts.

5. Prior-art collapse

• finite machine decomposition is covered more generally by Krohn-Rhodes cascade/wreath products;
• direct-product decomposition of permutation groups is polynomial-time from generators (Wilson);
• Cartesian transition-graph factorization and permutation-automata decomposition are established algorithms;
• trace monoids encode commuting event words but do not prove separate storage coordinates;
• subdirect and diagonal couplings are the standard Goursat obstruction.

A structure-aware group, automata, graph-factorization, or message-passing control receives the same conditional resource gain.

6. Verdict

No CPU experiment is authorized. A survivor must expose an ordinary-trace invariant that defeats central and subdirect coupling, identifies modules from sub-extensional coverage, and yields a resource advantage over established decomposition algorithms. Pairwise commutators do not.