To unify continuity, causality, and information

To unify continuity, causality, and information into a single admissibility rule, the sources propose the Marginal Admissibility Principle. This rule acts as a zeroth-order admissibility axiom or filter, stating that a model is only physically and computationally valid if vanishingly small input perturbations produce vanishingly small output responses. This unified rule is operationalized through the following three integrated components: 1. Continuity as the Operational Foundation The rule reformulates continuity from a static topological property into a local response constraint. Using the marginal (increment) operator: Delta_f(x₀; h) = f(x₀ + h) − f(x₀) continuity is satisfied if and only if this operator vanishes as the perturbation h approaches zero. This provides a minimal local form of realizability, ensuring that the system does not undergo unearned local ruptures or finite jumps without infinite marginal cost. 2. Causality as the “No-Free-Response” Principle Within this unified rule, causality is expressed through a local causality inequality: finite effects cannot be generated by vanishing causes at a point. Admissible models maintain causal consistency by requiring that any finite state change be earned through a corresponding finite input. Discontinuities are viewed as inadmissible ruptures unless they are completed by additional structures, such as impulse terms (Dirac measures), guard–reset maps, or conservation-consistent weak formulations. Without these, a model fails the admissibility filter because it generates finite structure out of nothing. 3. Information as Bounded Gain and Accounting The rule incorporates information flow by treating continuity as a finite-gain constraint. Finite gain: Continuity implies the existence of a noise envelope in which shrinking input uncertainty maps to a bounded output response. Infinite information extraction: A jump discontinuity corresponds to infinite small-signal gain, representing an inadmissible scenario where infinite information is extracted from an infinitesimal signal. Resource accounting: By enforcing the admissibility rule, the framework ensures that information and energy accounting remain well-defined, preventing a breakdown in structured reasoning. Summary of the Unified Admissibility Rule The sources synthesize these concepts into a single admissibility floor: Component Role in the Rule Failure State (Inadmissible) Continuity Zeroth-order stability axiom Numerical refinement collapses Causality No-free-response safeguard Finite effects from vanishing causes Information Bounded small-signal gain Infinite information gain at zero scale Algorithmic Verification This unified rule can be tested via the 100-Tick Lattice Protocol, which uses binary refinement (hₙ₊₁ = hₙ / 2) to record the marginal response (Omegaₙ). An Admissibility Filter—an algorithmic gate—then determines whether the model is valid based on whether the marginal response decays under refinement (absolute value of Omegaₙ₊₁ less than c times absolute value of Omegaₙ, for c less than 1). If the response fails to decay, the model is flagged as INADMISSIBLE due to a detected rupture.