What the Threshold and Readout Obstruction Establish

10. Discussion and limits

The sharp threshold compares baseline stiffness with the most negative stationary edge curvature. Excitation weakens an edge; the quadratic memory cost bounds the curvature loss. At and above $\mu=\chi\lVert L\rVert$ strict convexity selects one full stationary source at every constant drive. Below the path threshold two stable states can persist at the same drive, separated by a saddle in the counted regime. The full Hessian governs their survival under arbitrary perturbations.

Highest-mode forcing aligns explicit scalar reduction with global exclusion of reflection-broken equilibria when $\mu>\chi$. The memory lift and inertia lemma restore all coordinates. The low-mode failure shows why the return matters. At $\mu\le\chi$ the two outer high-mode roots still disprove uniqueness, but additional roots have not been completely counted. Individual networks below the universal bound need separate geometry.

Complete saddle-nodes rely on the stated nondegeneracy conditions. Reactive mobility preserves inertia at fixed Hessian yet changes stationary load at fixed physical input. A comparison must specify which it holds fixed. Memory clamping also changes the experiment: conditional linked generators survive while free memory adds a causal kernel and changes static gain.

The barrier regions are sufficient, not a complete description of the saddle stable set. The numerical square pulses switch at finite time and return to identical input with explicit energy ledgers. They do not prove a quasistatic limit, minimum work, noisy retention, or experimental realization. A physical implementation needs calibrated reciprocal constitutive behavior, controlled drive, phase-sensitive access and a justified free-versus-clamped probe protocol.

Identical coarse present data can fail to determine the admitted future response; richer calibrated stationary amplitude can recover memory. Continuous external power sustains both alternatives. This is a complete constitutive mechanism in the counted regime, not a common microscopic explanation of different spectral domains. For slowly swept inputs, continue to Slow Passage and Prepared Hysteresis.

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