Technical Standard

The Marginal Admissibility Principle (MAP) for Physical and Computational Models

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1. Strategic Context and Admissibility Mandate

This standard establishes the Marginal Admissibility Principle (MAP) as the primary zeroth-order admissibility filter for all physical and computational models.

Scientific validity depends on the preservation of structured reasoning under refinement. MAP therefore serves as the admissibility floor that prevents marginal rupture—the emergence of finite structure from vanishing causes.

Higher-order properties (derivatives, Lipschitz continuity, Lyapunov stability, conservation laws) are non-referencable until MAP compliance is verified. If MAP is violated, derivatives are undefined in any physically meaningful sense, and all subsequent stability or conservation analysis is mathematically incoherent.

Mandate
Vanishingly small input perturbations shall produce vanishingly small output responses.

Any model that permits a finite response to arise from a vanishing cause represents an unearned rupture of the causal chain, making conservation, stability, and information accounting impossible. Such models are inadmissible by definition.

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2. Theoretical Foundation: Marginal Operator and Continuity–Marginality Equivalence

MAP reframes continuity from a topological neighborhood condition into a dynamic marginal response constraint. Continuity is treated not as a static property of sets, but as a computable requirement for refinement stability and resource accounting.

2.1 Marginal Framework

Definition (Marginal Operator)
For a function $f$ and evaluation point $x_0$, the marginal response to a perturbation $h$ is defined as:

$$\Delta_f(x_0;h) = f(x_0 + h) - f(x_0)$$

This operator is the primitive object of admissibility analysis.

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2.2 Specification Block: Continuity–Marginality Equivalence

Theorem 2.1 (Continuity–Marginality Equivalence)
A model $f$ is admissible at evaluation point $x_0$ if and only if:

$$\lim_{h \to 0} \Delta_f(x_0;h) = 0$$

This condition is logically equivalent to the classical $\epsilon\!-\!\delta$ definition of continuity, but is operationally superior because it is:

• scale-explicit

• locally testable

• independent of derivatives

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2.3 The Infinite-Bill Principle

Admissibility is governed by the Infinite-Bill Principle:

No finite jump in state may occur without infinite marginal cost.

Continuity is therefore the refusal of marginal rupture. A discontinuity without an explicit physical completion mechanism constitutes a violation of realizability and is prohibited in safety-critical and high-integrity systems.

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3. Continuity as the Zeroth-Order Axiom Across Domains

MAP is a structural requirement across scientific disciplines. Without it, the mathematical tools used to assess stability, conservation, convergence, and information flow lose meaning.

Domain	Role of Continuity	Inadmissibility Result
Stability Theory	Guarantees one-step stability; prevents instantaneous divergence	Arbitrarily small perturbations cause finite deviations
Conservation & Causality	Enforces local causality; no finite effects from vanishing causes	Free-response rupture; resource budgets undefined
Numerical Convergence	Prerequisite for consistency and refinement	Step-size reduction fails; convergence collapses
Information Flow	Bounds small-signal gain via noise envelopes	Infinite information extraction at zero scale

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3.1 The No-Free-Response Principle

Continuity implies the existence of a noise envelope $\eta(\delta) \to 0$ such that:

$$|x-x_0| < \delta \;\Rightarrow\; |f(x)-f(x_0)| < \eta(\delta)$$

This bounds small-signal amplification. Failure of this envelope implies infinite information extraction from an infinitesimal signal—violating causal and informational accounting.

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4. Doctrine of Earned Discontinuity

Discontinuities occur in physical reality (shocks, switches, phase transitions). MAP distinguishes between unearned rupture and completed discontinuity.

Discontinuities are not forbidden. They are burdens.

4.1 Admissible Completion Mechanisms

A discontinuity is admissible only if paired with at least one explicit completion mechanism:

1. Impulse / measure-valued terms (e.g., Dirac distributions)

2. Guard–reset maps in hybrid dynamical systems

3. Conservation-consistent weak formulations

4. Limiting procedures (zero-viscosity, singular perturbation, multiscale collapse)

Absent completion, a discontinuity is not “non-smooth”—it is incoherent.

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4.2 Derivation: Divergent Empirical Gain

Proof Sketch
If $f$ has an unearned jump at $x_0$, then there exists $\epsilon > 0$ such that:

$$|\Delta_f(x_0;h)| \ge \epsilon \quad \text{as } h \to 0$$

Define empirical gain:

$$G(h) = \frac{|\Delta_f(x_0;h)|}{|h|}$$

Then:

$$G(h) \ge \frac{\epsilon}{|h|} \;\to\; \infty \quad \text{as } h \to 0$$

This proves finite effect from vanishing cause—violating local causality and zeroth-order stability.

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5. Marginal Admissibility Protocol (MAP-P)

MAP compliance shall be verified algorithmically using the MAP-P v2 protocol.

5.1 100-Tick Lattice Protocol

• Initialize perturbation: $h_0 = 1.0$

• Binary refinement: $h_{n+1} = h_n / 2$

• Record at each tick:

  • Marginal response $\Omega_n = |\Delta_f(x_0;h_n)|$

  • Empirical gain

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5.2 Hardened Admissibility Filter v2

Mandatory parameters:

• Sustained-Decay Window: 3 consecutive decay-violating steps required to flag rupture

• Contraction Bound: $|\Omega_{n+1}| < c\,|\Omega_n|$, with $c < 0.9$

• Minimum Depth: ≥ 5 ticks before admissibility may be asserted

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5.3 Outcome Classification

• ADMISSIBLE
  Sustained marginal decay observed; zeroth-order stability satisfied.

• INADMISSIBLE
  Sustained marginal rupture detected; model is incoherent and prohibited.

• INCONCLUSIVE
  Insufficient refinement or persistent noise; model may not proceed to certification.

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6. Compliance and Enforcement

MAP compliance is mandatory for all models claiming physical or computational validity.

Final Compliance Checklist

• Marginal response vanishes as $h \to 0$ at all critical points

• MAP-P v2 passes with no sustained rupture

• All discontinuities explicitly completed and documented

• Small-signal gain remains finite as noise envelope shrinks

Non-compliance classification:
Any model failing these requirements is hereby designated Mathematically Incoherent and is strictly prohibited from use in safety-critical, conservation-dependent, or high-integrity simulations.