R12 Active Witness Allocation No-Go

Status: exact partial separation and exact collapse. Adaptive target queries can beat passive/random allocation, but residual-witness supervision has no oracle-complexity advantage over a fair active answer-only learner when every derived label is computed from the same counted transcript.

1. Active versus passive theorem

Let a target threshold be theta in {1,...,N} and let an ordinary answer query at x in {1,...,N-1} return

O_theta(x) = 1[x >= theta].

Adaptive binary search identifies theta with ceil(log2 N) one-bit answers, and this is optimal because a depth-m binary decision tree has at most 2^m leaves.

Any nonadaptive schedule of m locations partitions the N possible thresholds into at most m+1 answer transcripts. Under the uniform target prior, even the optimal decoder therefore has

P(theta_hat = theta) <= (m + 1) / N.

Success at least 1-delta requires m >= (1-delta)N - 1, and worst-case exact identification requires N-1 calls. This is a real Theta(log N) versus Theta(N) active/passive separation.

2. Active answer-only simulation theorem

Suppose a WGRQ policy chooses query x_t from the public transcript

T_(t-1) = (x_1,y_1,...,x_(t-1),y_(t-1))

and receives the ordinary answer y_t=O_theta(x_t). If every merge, separation, collision, or witness label is computed from those public queries and counted answers, an active answer-only learner can:

1. run the identical query-selection policy;
2. submit the identical ordinary answer query;
3. receive the identical answer;
4. compute the identical derived labels;
5. perform the identical model update.

Induction on t gives identical transcripts, parameters, and outputs for every target and random seed. Thus WGRQ has no strict oracle or sample advantage over the fair active answer-only class.

If WGRQ instead receives exact residual-equivalence labels, target-selected counterexamples, hidden state IDs, or simulator-produced witness identities, it has a stronger oracle. Equal call counts do not restore fairness. The ledger must count oracle semantics, returned information bits, query-description bits, target-dependent witness-search work, and adaptive rounds.

3. Smallest exhaustive audit

N=4 is minimal. Adaptive binary search and active answer-only both identify all four targets in two calls. Every nonadaptive two-call schedule induces at most three transcripts, so uniform-prior exact success is at most 3/4. Enumerating all depth-two adaptive trees and all nonadaptive schedules can only verify this identity; it cannot rescue a WGRQ oracle advantage.

4. Decision

Reject WGRQ as an oracle-complexity or finite-sample invention relative to active answer-only supervision. Preserve adaptive allocation as a known data acquisition control. A remaining CPU board may test only a narrower neural optimization claim under frozen oracle transcripts, favorable active controls, and a complete information ledger.