R12 Query-Kernel Factorization No-Go

Status: exact diagnostic theorem, rejected as a new reasoning mechanism. Task-native queries canonically identify predictive quotients, but the result is Moore-machine output projection plus universal-algebra factor congruences.

1. Future-stable query kernels

For event maps T_a:X->X, query family F, and outputs o_q, define

kappa_F = {(x,y): o_q(T_w x)=o_q(T_w y) for every q in F and event word w}.

kappa_F is the greatest event congruence contained in the immediate query kernel. It is the exact behavioral quotient relevant to that query family.

For query families F_1,...,F_k, the map

x -> ([x]_(kappa_1),..., [x]_(kappa_k))

embeds the minimal joint residual machine into the product of its query quotients. Every projection is surjective, so the image is always a subdirect product. It is a full direct product only when every tuple of quotient classes is jointly realizable.

A sufficient Chinese-remainder certificate is:

1. the intersection of the kernels is equality;
2. the kernels are pairwise comaximal;
3. their generated congruence lattice is distributive;
4. the congruences permute.

If the query-generated factor congruences form a finite Boolean algebra, its co-atoms give canonical factors up to permutation. A query depends only on coordinate set S exactly when the intersection of those coordinate kernels is contained in the query's output kernel.

2. Exact counterexamples

2.1 Smallest subdirect obstruction

Use three states, identity dynamics, and two query signatures

00, 01, 11.

The two kernels meet at equality and join universally, but signature 10 is missing. The state space is a proper subdirect image rather than a product.

2.2 Pairwise tests are insufficient

On F_2^2, expose queries x, y, and x xor y. Every pair supplies valid coordinates, but the three-bit image contains only four parity-consistent tuples rather than eight. The generated congruence lattice is the nondistributive M_3. Symmetric exposure of all three queries does not select a canonical basis.

2.3 Coupled dynamics consume factors

Under CNOT, the future-stable kernel of the target-bit query collapses to equality because a later target readout can reveal the control. Query-kernel CRT therefore finds genuinely independent predictive modules, not interacting reasoning modules.

2.4 Finite traces do not certify the kernels

Any unobserved state-event transition can be changed to violate a proposed congruence while preserving the finite transcript. Unrestricted exact certification requires extensional transition coverage.

3. Resource ledger

Let N=|X|, m=|Sigma|, p be total query-output bits, and s be separating signature bits. Complete deterministic reconstruction uses

C = N p + N m s

readout bits. With independent flip noise eta<1/2, repeat each cell on the order of

2/(1-2 eta)^2 * log(2C/delta).

Passive data additionally needs every required cell to have positive mass. If some cell has probability zero, exact identification is impossible.

For true factor sizes n_i, a supplied decomposition can reduce transition description from roughly m N log N to m sum_i n_i log n_i. It does not beat the residual-state information lower bound log N.

4. Prior-art boundary

The CRT conditions are standard congruence decomposition. Output-projected Moore-machine learning and product-automata learning already exploit the same component reduction. Proper subdirect images are the same missing-combination phenomenon as lossless-join theory; future-query coordinates also sit inside predictive-state, observable-operator, and weighted-automata representations.

5. Decision

Use future-stable query congruences only as a control that diagnoses whether a task really contains independent predictive modules. No CPU falsifier or Shohin mechanism is authorized. Reconsider only with a theorem that learns interacting modules from ordinary traces and beats product automata, PSRs, tensor-factor models, and congruence decomposition under matched information.