Regime Geography Theorem Registry

Recursion Geometry Lab

Version 1.0

Preamble

This registry formalizes the structural foundations of Regime Geography and the governed Δ-metric framework implemented in Q4 (version q4-v0) .

These statements define the mathematical properties of atlas comparison under a strict comparability contract.

This document is versioned and serves as a stable citation anchor for subsequent publications and articles.

Definitions

Definition 1 — Parameter Grid

Let $G \subset \mathbb{R}^n$ be a fixed finite parameter grid with shape held constant under a governance contract.

Definition 2 — Regime Codebook

Let $\mathcal{R}$ be a finite set of regime labels.

Definition 3 — Atlas (Regime Geography)

An atlas is a mapping $\Omega : G \rightarrow \mathcal{R}$ together with optional scalar diagnostic fields $X_\ell : G \rightarrow \mathbb{R}, \quad \ell \in \mathcal{L}$.

Definition 4 — Governed Comparability Contract

A contract $C$ fixes:

• grid shape,
• categorical layers $\mathcal{L}_{cat}$,
• scalar layers $\mathcal{L}_{sc}$,
• measurement policy,
• grid policy (reject on mismatch),
• version constraints.

Only atlases satisfying $C$ are comparable .

Definition 5 — Signature Functional

Under contract $C$, define a deterministic signature mapping $S_C(\Omega) \in \mathbb{R}^d$ constructed from:

• categorical mass fractions,
• scalar summary statistics (mean, quantiles, etc.),
• boundary fractions (e.g. seam indicators).

Definition 6 — Δ Distance

Let $W \succ 0$ be a positive definite weight matrix.

Define the governed deformation metric

$$\Delta_C(\Omega_1, \Omega_2)$$

II. Structural Theorems

Theorem 1 — Well-Definedness Under Governance

Let $C$ be a fixed comparability contract.

For any atlas $\Omega$ satisfying $C$, the signature $S_C(\Omega)$ exists, is unique, and depends only on atlas data compliant with $C$.

Therefore, $\Delta_C$ is well-defined on the governed atlas set.

Theorem 2 — Pseudometric Structure

For any atlases $\Omega_1, \Omega_2, \Omega_3$ satisfying $C$:

(1) Non-negativity

$$\Delta_C(\Omega_1,\Omega_2) \ge 0$$

(2) Symmetry

$$\Delta_C(\Omega_1,\Omega_2)=\Delta_C(\Omega_2,\Omega_1)$$

(3) Triangle Inequality

$$\Delta_C(\Omega_1,\Omega_3)$$

Hence, $\Delta_C$ is a pseudometric.

Corollary — Metric on Quotient Space

Define an equivalence relation

$$\Omega_1 \sim \Omega_2 \quad \text{iff} \quad S_C(\Omega_1) = S_C(\Omega_2)$$

Then $\Delta_C$ is a metric on $\mathcal{A}_C / \sim$

Theorem 3 — Layer Decomposition

Let the signature decompose as

$$ S_C(\Omega) = \big( S_C^{cat}(\Omega), S_C^{sc}(\Omega) \big) $$

Under compatible weighting,

$$\Delta_C^2 = \Delta_{cat}^2 + \Delta_{sc}^2$$

Categorical and scalar deformation contributions are independently detectable.

Theorem 4 — Stability Under Bounded Perturbation

Suppose:

• scalar layers are perturbed by at most $\varepsilon_\ell$ in sup norm,
• categorical labels change on at most an $\eta$ fraction of grid points.

Then there exist constants $K_{sc}, K_{cat}$ such that

$$\Delta_C(\Omega,\tilde{\Omega}) \le K_{sc}(\{\varepsilon_\ell\}) + K_{cat} \eta$$

Thus, small perturbations induce bounded Δ variation.

Theorem 5 — Boundary Overlap Metric

Let $B(\Omega) \subseteq G$ denote a boundary indicator set (e.g. seam_hard).

Define Jaccard similarity

$$J(\Omega_1,\Omega_2) = \frac{ |B(\Omega_1)\cap B(\Omega_2)| }{ |B(\Omega_1)\cup B(\Omega_2)| }$$

Define $d_J = 1 - J$

Then $d_J$ is a metric on boundary indicator sets modulo equality on null unions.

Theorem 6 — Signature Invariance Under Observable-Preserving Transformations

Let $g$ be a transformation that preserves all contract-observed quantities used by $S_C$.

Then

$$S_C(g \cdot \Omega) = S_C(\Omega) \Rightarrow \Delta_C(\Omega, g \cdot \Omega) = 0$$

III. Interpretation

The above theorems establish:

1. Regime atlases are first-class geometric objects.
2. Governed signature extraction embeds atlases into a metric space.
3. Structural deformation between systems is quantifiable.
4. Boundary geometry admits complementary set-based metrics.
5. Comparability is contract-enforced, not assumed.

This registry defines the structural foundation of comparative regime geometry.

Governance Reference

All statements above assume compliance with the Q4 governed comparability contract (q4-v0) as implemented in the Recursion Geometry Lab engine .

End of Registry

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Recursion Geometry Lab

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