19.10 Calibration of the mixed-order diagnostic

Before asking a system to extend an artistic theory, the mixed differential mechanism must be calibrated independently of that stronger claim. The following sequence tests whether a local bracket predicts the useful order of two registered updates, whether the estimate survives changes of observer and sample size, and when the correct response is abstention. It deliberately stops before theory extension.

DIAL-CAN–0.

In a smooth two-dimensional CAN proxy, the finite order-difference converged to the exact Lie bracket with first-order slope approximately one. The bracket selected the better order in 97.7 percent of states, while an exactly commuting control remained null.

DIAL-CAN–1.

A trained neural CAN with a LeakyReLU observer failed the registered differential audit. Rare activation-stratum crossings produced large finite-witness outliers; first-order convergence, norm correlation, matched-order accuracy, and the shuffled-control comparison all missed their frozen gates.

DIAL-CAN–1B.

A separately registered Softplus ablation restored first-order convergence, near-perfect norm correlation, and perfect order prediction. It nevertheless failed admission because one order dominated every sampled probe, leaving the bracket controller no adaptive advantage over the best fixed order.

DIAL-CAN–2.

Sampling three paired training checkpoints produced real decision switching. The smooth bracket predicted the better order on every registered probe and exceeded both fixed-order and shuffled controls. The registered study nevertheless missed its preregistered aggregate diversity threshold because one order occurred on 16.7 rather than 20 percent of probes.

DIAL-CAN–3.

A first-order finite-difference estimator was trained on checkpoints 200 and 600 and transported to the switching checkpoint 400. The supervised calibrator failed because its training regions contained no examples of one order. In a retrospective diagnostic, the theory-supplied zero boundary improved from 77.1 to 87.5 percent held-out accuracy as the latent sample increased from 16 to 128; the diagnostic does not override the failed registered study.

Taken together, the calibrations license one narrow conclusion: in the smooth registered proxies, mixed geometry can carry useful local order information. They do not establish a successful learned finite-sample controller, a nonzero quotient residual, a Weil-persistent missing direction, or the construction of a visual generator. The final calibration instead supplies an epistemic constraint for the full architecture: when training data have no support on one side of a structurally defined boundary, a learned controller should abstain or retain the declared boundary rather than invent an exterior threshold.