Parabolic Switching and Terminal Margins

What the note studies

Consider a state with two changing coordinates and a threshold. The baseline mode contracts the first coordinate; an intervention omits that contraction. Both modes share the second-coordinate update and freeze the first coordinate after threshold crossing. The paper asks how an intervention schedule affects the eventual first-coordinate value and the smallest margin along the trajectory.

The main estimate compares that terminal value with the sum of contractions omitted before crossing, uniformly over schedules. From an exact critical start, it yields a delayed-intervention margin proportional to the delay raised to −5/2. With at most K−1 consecutive baseline updates, an interchange argument identifies the minimum-margin schedule, and a complete-tail bound gives the K−5/2 asymptotic with a zeta(5/2) factor.

Contribution and limits

The prototype’s local critical geometry has established antecedents. The note’s claimed contribution is terminal transfer and constrained switching within its declared class. A leakage example shows why leading weighted local terms alone do not determine the terminal observable when the continuation changes.

The margin is a functional of an exact trajectory, not a proof of robustness to disturbances fed into the dynamics. The result does not cover arbitrary monotone maps, linear-loss models, approximate critical initialization, or cognition. Historical priority beyond the bounded literature review and the work’s significance for a journal remain for human specialists to assess.

Place in B-Theory

Evolutionary Logic asks how continuous change relates to an observable logical description. This note supplies a narrow switching-and-margin model relevant to that question. It does not construct an observation procedure or prove that a system acquires a concept or predictive model; those tasks remain separate in The MIND programme.

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The final source and PDF passed compilation and page-by-page visual production checks. The reviews were AI-assisted and are not human peer review or external endorsement. The source has no named author line; none has been inferred for this reading copy.