# MAG-1.0 GOVERNANCE REPORT: THEOREM-PROOF STRUCTURE
# ============================================================
# MAG-1.0 GOVERNANCE REPORT: THEOREM-PROOF STRUCTURE
# ============================================================
"""
This report formalizes the theorem-proof architecture of Marginal Admissibility
Governance (MAG), establishing the logical chain from zeroth-order axioms
to operational verification.
"""
# ============================================================
# I. THE AXIOMATIC FOUNDATION (Constitutional Layer)
# ============================================================
The MAG framework rejects traditional topological assumptions in favor of a
strict local response constraint. The theorem-proof structure begins with
five canonical axioms:
1. Axiom 1 (Marginal Response): Defines Δf(x0; h) := f(x0 + h) - f(x0) as the
canonical unit of structural change.
2. Axiom 2 (Empirical Gain): Establishes G_f(x0; h) := ||Δf(x0; h)|| / ||h||
as the sole authorized proxy for local amplification analysis.
3. Axiom 3 (Admissibility Floor): Mandates that a model is admissible iff
lim_{||h||→0} ||Δf(x0; h)|| = 0.
4. Axiom 4 (Rupture Condition): Declares marginal inadmissibility if a
sequence h_k exists where response remains above ε0 as h_k vanishes,
forcing gain to diverge.
5. Axiom 5 (Earned Discontinuity): Prohibits discontinuities unless paired
with completion mechanisms (e.g., impulse terms or guard-reset maps) that
restore bounded gain.
# ============================================================
# II. CORE THEOREMS: EQUIVALENCE AND STABILITY
# ============================================================
The primary logical derivation in this framework is the reframing of continuity
as an operational response metric.
Theorem 2.1 — Continuity–Marginality Equivalence
A function f is continuous at x0 if and only if the marginal operator
vanishes as h approaches zero (lim_{h→0} Δf(x0; h) = 0). This theorem
establishes continuity as a zeroth-order stability axiom, prior to derivatives
or Lyapunov functions.
Proposition 3.1 — Instantaneous Stability
Continuity guarantees one-step stability. Conversely, if f is discontinuous,
arbitrarily small perturbations produce finite deviations in a single
iteration, rendering local stability theory unformulated at that point.
# ============================================================
# III. PROOF OF INADMISSIBILITY (Rupture Logic)
# ============================================================
The "Proof of Rupture" serves as the mechanism for model rejection. The logic
follows a causal and resource-based constraint known as the
"No-Free-Response Principle".
Proof Sketch: The Case of Jump Discontinuity
1. Assume f has a jump discontinuity at x0.
2. There exists ε > 0 such that for every δ > 0, there is a perturbation h
where 0 < |h| < δ and |Δf(x0; h)| ≥ ε.
3. As |h| approaches zero, the output change stays bounded away from zero.
4. Consequently, the empirical gain |Δf(x0; h)| / |h| ≥ ε / |h| diverges to
infinity.
5. This violation of local causality (finite effect from vanishing cause)
triggers an "Unearned Local Rupture" declaration.
# ============================================================
# IV. OPERATIONAL VERIFICATION (Refinement Protocol)
# ============================================================
Logical proofs are operationalized via "Refinement Sovereignty," requiring
that admissibility survive algorithmic testing.
The 100-Tick Lattice Protocol
Refinement is executed via the binary sequence h_{n+1} = (1/2)h_n. The
marginal magnitude sequence Ω_n := ||Δf(x0; h_n)|| must decay toward zero
.
The Admissibility Filter (v2)
A model is algorithmically proven admissible if it satisfies:
|Ω_{n+1}| ≤ c * |Ω_n| for some c < 1.
The "Hardened" v2 filter allows for occasional non-decay (numerical noise) but
requires sustained decay over a window of refinement ticks to verify structural
legitimacy.
# ============================================================
# V. HIERARCHICAL PROOF (Order Spectrum Test)
# ============================================================
Beyond the zeroth-order floor, the MAG-1.1 doctrine establishes a proof
structure for higher-order structural behavior.
Directional Curvature Index (DCI)
By measuring the decay rates (ρ+ and ρ-) along positive and negative refinement
paths, the framework derives Local Order Exponents (p+ and p-).
The Order-Admissibility Hierarchy:
- Level 0: p_min > 0 (Zeroth-Order Admissible/Continuous).
- Level 1: p_min ≥ 1 (First-Order Gain-Bounded/Lipschitz).
- Level 2: p+ = p- ≥ 2 (Second-Order Curvature-Coherent/Smooth).
This hierarchy allows structural inference—such as detecting curvature
asymmetry—using only refinement behavior rather than symbolic
differentiation.
# ============================================================
# MAG-1.0 GOVERNANCE REPORT: THEOREM-PROOF STRUCTURE
# ============================================================
"""
This report formalizes the theorem-proof architecture of Marginal Admissibility
Governance (MAG), establishing the logical chain from zeroth-order axioms
to operational verification.
"""
# ============================================================
# I. THE AXIOMATIC FOUNDATION (Constitutional Layer)
# ============================================================
The MAG framework rejects traditional topological assumptions in favor of a
strict local response constraint. The theorem-proof structure begins with
five canonical axioms:
1. Axiom 1 (Marginal Response): Defines Δf(x0; h) := f(x0 + h) - f(x0) as the
canonical unit of structural change.
2. Axiom 2 (Empirical Gain): Establishes G_f(x0; h) := ||Δf(x0; h)|| / ||h||
as the sole authorized proxy for local amplification analysis.
3. Axiom 3 (Admissibility Floor): Mandates that a model is admissible iff
lim_{||h||→0} ||Δf(x0; h)|| = 0.
4. Axiom 4 (Rupture Condition): Declares marginal inadmissibility if a
sequence h_k exists where response remains above ε0 as h_k vanishes,
forcing gain to diverge.
5. Axiom 5 (Earned Discontinuity): Prohibits discontinuities unless paired
with completion mechanisms (e.g., impulse terms or guard-reset maps) that
restore bounded gain.
# ============================================================
# II. CORE THEOREMS: EQUIVALENCE AND STABILITY
# ============================================================
The primary logical derivation in this framework is the reframing of continuity
as an operational response metric.
Theorem 2.1 — Continuity–Marginality Equivalence
A function f is continuous at x0 if and only if the marginal operator
vanishes as h approaches zero (lim_{h→0} Δf(x0; h) = 0). This theorem
establishes continuity as a zeroth-order stability axiom, prior to derivatives
or Lyapunov functions.
Proposition 3.1 — Instantaneous Stability
Continuity guarantees one-step stability. Conversely, if f is discontinuous,
arbitrarily small perturbations produce finite deviations in a single
iteration, rendering local stability theory unformulated at that point.
# ============================================================
# III. PROOF OF INADMISSIBILITY (Rupture Logic)
# ============================================================
The "Proof of Rupture" serves as the mechanism for model rejection. The logic
follows a causal and resource-based constraint known as the
"No-Free-Response Principle".
Proof Sketch: The Case of Jump Discontinuity
1. Assume f has a jump discontinuity at x0.
2. There exists ε > 0 such that for every δ > 0, there is a perturbation h
where 0 < |h| < δ and |Δf(x0; h)| ≥ ε.
3. As |h| approaches zero, the output change stays bounded away from zero.
4. Consequently, the empirical gain |Δf(x0; h)| / |h| ≥ ε / |h| diverges to
infinity.
5. This violation of local causality (finite effect from vanishing cause)
triggers an "Unearned Local Rupture" declaration.
# ============================================================
# IV. OPERATIONAL VERIFICATION (Refinement Protocol)
# ============================================================
Logical proofs are operationalized via "Refinement Sovereignty," requiring
that admissibility survive algorithmic testing.
The 100-Tick Lattice Protocol
Refinement is executed via the binary sequence h_{n+1} = (1/2)h_n. The
marginal magnitude sequence Ω_n := ||Δf(x0; h_n)|| must decay toward zero
.
The Admissibility Filter (v2)
A model is algorithmically proven admissible if it satisfies:
|Ω_{n+1}| ≤ c * |Ω_n| for some c < 1.
The "Hardened" v2 filter allows for occasional non-decay (numerical noise) but
requires sustained decay over a window of refinement ticks to verify structural
legitimacy.
# ============================================================
# V. HIERARCHICAL PROOF (Order Spectrum Test)
# ============================================================
Beyond the zeroth-order floor, the MAG-1.1 doctrine establishes a proof
structure for higher-order structural behavior.
Directional Curvature Index (DCI)
By measuring the decay rates (ρ+ and ρ-) along positive and negative refinement
paths, the framework derives Local Order Exponents (p+ and p-).
The Order-Admissibility Hierarchy:
- Level 0: p_min > 0 (Zeroth-Order Admissible/Continuous).
- Level 1: p_min ≥ 1 (First-Order Gain-Bounded/Lipschitz).
- Level 2: p+ = p- ≥ 2 (Second-Order Curvature-Coherent/Smooth).
This hierarchy allows structural inference—such as detecting curvature
asymmetry—using only refinement behavior rather than symbolic
differentiation.
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