Prompt-Conditioned Determining Law

Status: one bounded representation falsifier; no novelty or reasoning claim.

Capability hypothesis

CEER can lower source-consequence energy but its learned consequence head fails when applied to the held-out query. The latent system fits observed evidence without forming one reusable law. Prompt-Conditioned Determining Law (PCDL) replaces the opaque state/readout split with an explicit episode law.

A shared learned basis maps every source probe and the late query into the same rank-R coordinates. Source outcomes determine class-valued law coefficients by a differentiable regularized solve:

Phi_j = phi_theta(source, probe_j)
W = (Phi^T M Phi + lambda I)^-1 Phi^T M one_hot(outcome)
answer_logits = phi_theta(source, query)^T W

The query is absent from law formation. It can only evaluate the law determined by source witnesses. The treatment cannot use a separate query reader.

Matched control

The DENSE arm is a standard outcome-embedded set-attention reader from all source witnesses to the query. Every model instantiates and executes both PCDL and DENSE paths; the arm changes only which answer logits receive the answer loss and are evaluated. Encoder, data, parameter count, module execution, training updates, and frozen cohorts are identical. Both arms receive the same PCDL witness-reconstruction auxiliary loss, so the shared encoder has the same extra supervision.

This mechanism is related to deep kernel learning, conditional neural processes, differentiable least squares, and system identification. Those are prior art. This pilot tests the representation diagnosis; it does not claim that a differentiable law solve is itself novel.

Frozen pilot

• seed 41;
• 1,000 updates, 256 examples/update;
• training depths 2--4;
• rank 8, width 64, ridge 0.1;
• the same six frozen depth-5/7 cohorts used by FCPT, CGSGR, QVESR, and CEER;
• one H100 per arm, estimated under five minutes total GPU time.

PCDL advances only if it beats DENSE by at least five absolute macro points, improves the mean of both depths in every family, exceeds 15% mean induction accuracy, loses at least five points when witness outcomes are shuffled, and loses at least five points when solved law coefficients are exchanged across episodes. A miss closes this exact learned-basis law solver; it does not permit rank, ridge, seed, width, duration, or basis-network sweeps.

Result

Jobs 739305/739306 complete cleanly in 27/24 seconds. Both arms contain exactly 51,038 parameters and see 256,000 examples.

The learned law reconstructs approximately 85.5--95.2% of observed binding and induction witnesses and 87.6--94.4% of noncommuting witnesses, while scoring zero on unseen noncommuting queries. Its controls invert the intended causal prediction: shuffling witness outcomes raises macro to 10.254%, and exchanging solved law coefficients across episodes raises it to 10.677%.

Report SHA-256 values are:

• PCDL: fc1d4810a09f07af7f7e87fb14009bd76faa53dbbb0d0fb959c43c7f9933a894
• DENSE: 6aa1442f4d83cb4987408032d688f3c80388b40e2d4098eaddf6b42ddd563021

PCDL is therefore closed. A low-rank learned feature map plus differentiable coefficient solve is not a determining representation: it creates a compact interpolation surface with no enforced algebra between probes. The next candidate must select a restricted hypothesis family and compose a learned generator or operation law inside that family. This is a different mechanism, not a rank, ridge, seed, width, or duration repair.