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MAG-1.0 / MAG-1.1

Marginal Admissibility Governance

Formal Theorem–Proof Constitution

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I. Axiomatic Foundation

Let ( f : \mathbb{R} \to \mathbb{R} ).
Fix ( x_0 \in \mathbb{R} ).

Axiom 1 — Marginal Operator

[
\Delta_f(x_0;h) := f(x_0 + h) - f(x_0)
]

The marginal operator is the canonical unit of structural change.

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Axiom 2 — Empirical Gain

[
G_f(x_0;h) := \frac{|\Delta_f(x_0;h)|}{|h|}
]

This is the sole authorized proxy for local amplification.

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Axiom 3 — Admissibility Floor

( f ) is admissible at ( x_0 ) iff

[
\lim_{h \to 0} |\Delta_f(x_0;h)| = 0
]

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Axiom 4 — Rupture Condition

If there exists a sequence ( h_k \to 0 ) such that

[
|\Delta_f(x_0;h_k)| \ge \varepsilon_0 > 0
]

then

[
G_f(x_0;h_k) \to \infty
]

and the model is marginally inadmissible.

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Axiom 5 — Earned Discontinuity

A discontinuity is permitted only if accompanied by a completion mechanism restoring bounded gain.

Otherwise, it constitutes an unearned rupture.

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II. Core Theorems

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Theorem 2.1 — Continuity–Marginality Equivalence

A function ( f ) is continuous at ( x_0 )
iff

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

Proof

By definition of continuity:

[
\lim_{x \to x_0} f(x) = f(x_0)
]

Substitute ( x = x_0 + h ):

[
\lim_{h \to 0} f(x_0 + h) = f(x_0)
]

Rewriting:

[
\lim_{h \to 0} (f(x_0 + h) - f(x_0)) = 0
]

Thus:

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

∎

This establishes continuity as a zeroth-order marginal stability condition.

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Proposition 2.2 — One-Step Stability

If ( f ) is continuous at ( x_0 ), then arbitrarily small perturbations produce arbitrarily small one-step deviations.

Conversely, if ( f ) is discontinuous at ( x_0 ), one-step local stability cannot be formulated.

Proof follows directly from Axiom 3 and the contrapositive of Theorem 2.1.

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III. Rupture Theorem

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Theorem 3.1 — Jump Discontinuity Implies Gain Divergence

If ( f ) has a jump discontinuity at ( x_0 ), then empirical gain diverges along some refinement path.

Proof

By jump discontinuity:

There exists ( \varepsilon > 0 ) such that
for every ( \delta > 0 ),
there exists ( h ) with ( 0 < |h| < \delta ) and

[
|\Delta_f(x_0;h)| \ge \varepsilon
]

Thus:

[
G_f(x_0;h) = \frac{|\Delta_f|}{|h|}
\ge \frac{\varepsilon}{|h|}
]

As ( |h| \to 0 ):

[
\frac{\varepsilon}{|h|} \to \infty
]

Hence gain diverges.

∎

This formally encodes the No-Free-Response Principle.

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IV. Operational Verification

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Definition 4.1 — Binary Lattice Refinement

[
h_{n+1} = \frac12 h_n
]

Define

[
\Omega_n := |\Delta_f(x_0;h_n)|
]

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Definition 4.2 — Admissibility Filter (v2)

A model is algorithmically admissible if

[
\exists c < 1 \text{ such that } \Omega_{n+1} \le c \Omega_n
]

for sufficiently large ( n ).

This enforces sustained contraction.

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Theorem 4.1 — Refinement Convergence Implies Admissibility

If ( \Omega_n \to 0 ) under binary refinement, then ( f ) satisfies the Admissibility Floor.

Proof:

Binary refinement implies ( h_n \to 0 ).
If ( \Omega_n \to 0 ), then

[
\lim_{n\to\infty} \Delta_f(x_0;h_n) = 0
]

Thus Axiom 3 holds.

∎

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V. Order Spectrum & Directional Curvature (MAG-1.1)

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Definition 5.1 — Local Order Exponent

If

[
\Delta_f(x_0;h) \sim C |h|^p
]

then ( p ) is the local order exponent.

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Definition 5.2 — Refinement Ratio

[
\rho = \frac{\Omega_{n+1}}{\Omega_n}
]

Under binary refinement:

[
\rho = 2^{-p}
]

Hence:

[
p = -\log_2(\rho)
]

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Definition 5.3 — Directional Curvature Index

[
\rho_+,\rho_- \Rightarrow
p_+ = -\log_2(\rho_+),
\quad
p_- = -\log_2(\rho_-)
]

Define:

[
\mathrm{DCI}(x_0) = (p_+, p_-)
]

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VI. Order-Admissibility Hierarchy

Let ( p_{\min} = \min(p_+, p_-) ).

Level	Condition	Interpretation
0	( p_{\min} > 0 )	Continuous
1	( p_{\min} \ge 1 )	Gain-Bounded (Lipschitz)
2	( p_+ = p_- \ge 2 )	Curvature-Coherent
k	( p_{\min} \ge k )	k-Order Stable

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VII. Structural Principle

Local geometry is encoded in refinement decay.

Differentiability corresponds to:

[
p_+ = p_- \ge 1
]

Second-order smoothness corresponds to:

[
p_+ = p_- \ge 2
]

Higher-order structural coherence is determined solely by scale contraction behavior.

No symbolic differentiation required.

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