What the Prepared Return Proves

11. Discussion

The static folds produce delayed switching through a mixed amplitude and memory direction. Complete left and right normalization fixes both load and quadratic coefficients, and therefore the prefactor of the two-thirds-power law. Preparation selects the Airy solution. A long stable approach and an exact-fold start remain distinct contracts even in the slow limit.

The local warrant is the invariant graph for seven complementary directions, with a finite-chart contraction estimate and parameter motion. The global warrant is strict loss at the fixed fold, supplying a complete connection, followed by continuity and regular tracking on the ramp. Neither substitutes stationary memory or assumes constant-input monotonicity during a sweep.

The prepared legs select different branches across coexistence, converge away from folds, and have the proved local input offsets and positive area limit. They can overshoot and do not prescribe an exact vertical trace. Fixed-rate attraction to a periodic response remains a separate question.

All declared rates and both numerical levels support the leading prediction, with preparation, symmetry and measurement controls. The levels have no certified chart radius. Explicit full-source finite-rate constants, higher-order terms, calibrated amplitude events, shrinking-neighborhood arrival, periodic-orbit attraction, noise and physical implementation remain open. A fitted log slope would not resolve them.

Equal stationary branch power fixes the leading work coefficient while memory changes the state and passage time. Slower traversal reduces switching-input displacement while increasing accumulated supplied work. Geometric hysteresis does not measure this work. Conditional coherent and dissipative linkage remains tied to one carrier; complete response retains source state and history. This finite constitutive mechanism extends maintained-state dynamics without claiming a common microscopic origin of unrelated spectral profiles.

Appendix A. Finite-preparation residual audit

For exact-fold preparation define

$$E_{\rm quad}=\Delta a_{\rm actual}-h_\nu T_d(k=1/\sqrt3).\tag{69}$$

The four original derived fields instead subtracted $h_\nu T_d(k=0)$, the tracked-branch comparison, even though their stored predictions had been updated to the correct preparation. The checked output preserves the event locations, preparation-specific predictions, leading ratios and frozen comparison results. It changes a reporting difference rather than an experiment or tolerance.

Table 9. The four preserved raw differences and their checked counterparts at $\nu=3\times10^{-6}$. The public correction record retains the raw hash and all old and new values. Its executable stage computes (69) from the actual run's events and predictions. Scientific source and comparison criteria are unchanged.

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