Regime Geography as a First-Class Object in Dynamical and Learned Systems

The global behavioral structure of a dynamical system forms a measurable geometric object. We formalize this object as a Regime Geography and define reproducible metrics for constructing and comparing such geographies across systems and model versions.

Motivation

Modern dynamical systems and trained machine learning models exhibit:

• Sharp regime shifts
• Hidden instability basins
• Training-phase bifurcations
• Silent structural drift between model versions

Most evaluation tools are local:

• Lyapunov exponents
• Spectral norms
• Loss landscapes
• Validation accuracy

These metrics describe pointwise properties.

They do not describe the global organization of behavioral regimes across parameter and state space.

We propose treating that global organization itself as a measurable geometric object.

Parameterized Systems and Behavioral Classification

Let a parameterized system be defined by a state update operator

$S(\theta)(x) = F_\theta(x)$

where:

• $\theta \in \Theta$ denotes model parameters
• $x \in X$ denotes state
• $F_\theta : X \to X$ is the transition function induced by the model architecture

In the linear operator case:

$F_\theta(x) = M(\theta)x,$

where $M(\theta)$ is a parameterized matrix.

In a nonlinear recurrent system:

$F_\theta(x) = \phi(W(\theta)x),$

where $\phi$ is an activation function and $W(\theta)$ parameterizes the recurrent weights.

Given an initial condition $x_0$, we define a behavioral classification map:

$$ B(\theta, x_0) = \mathrm{classify} \left( \{ S(\theta)^t(x_0) \}_{t=0}^{T} \right) $$

where classification is determined by diagnostic criteria such as:

• Convergence to fixed point
• Periodicity
• Divergence
• Chaotic indicators
• Variance thresholds

This classification map is computable under deterministic instrumentation.

Definition: Regime Geography

Definition (Regime Geography)

The Regime Geography of system $S$ is the measurable partition

$$ \mathcal{G}_S = \{ R_i \subset \Theta \times X \} $$

induced by the classification map $B$, such that:

• Each region $R_i$ corresponds to a stable behavioral class
• Boundaries between regions represent qualitative transitions
• The partition is defined by invariant diagnostics

This treats regime structure not as incidental output, but as a primary object of study.

Atlas Construction Protocol

A regime geography is constructed via deterministic sweeps over:

• Parameter space $\Theta$
• Initial condition space $X$

For each grid point $(\theta, x_0)$, we compute:

• Lyapunov estimates
• Divergence rates
• Periodicity tests
• Fixed-point convergence checks
• Variance measures

Each evaluation yields:

• A regime label
• A diagnostic vector
• A hash-anchored signature

Reproducibility Contract

Two independent implementations following the protocol must produce equivalent atlases up to floating-point tolerance.

Without reproducibility, geography is metaphor.
With it, geography becomes measurable structure.

Boundaries and Seam Detection

Let $R_i$ and $R_j$ be distinct regime regions.

Their boundary is defined as:

$$ \partial R_i = \overline{R_i} \cap \overline{R_j} $$

To localize transition regions, we define seam density:

$$ \sigma(\theta) = \left\| \nabla B(\theta) \right\| $$

High seam density indicates structural instability regions.

Seam detection provides:

• Transition localization
• Boundary complexity measurement
• Early warning indicators

Tail Blindness Phenomenon

Local stability metrics do not determine global regime topology.

A system may satisfy:

• Stable Lyapunov exponent
• Subcritical spectral radius

yet possess disconnected instability basins elsewhere in parameter space.

Local metrics are therefore tail-blind to global regime structure.

This motivates mapping the full regime geography rather than relying solely on pointwise diagnostics.

Dimensional $\Delta$: Comparative Regime Geometry

Given two systems $S_1$ and $S_2$, we compare their geographies using:

• Regime overlap (Jaccard / Dice indices)
• Boundary distance measures
• Volume distribution divergence
• Seam density shifts

We define a structural divergence vector

$$ \Delta(S_1, S_2) = \left( \Delta_{\text{volume}}, \Delta_{\text{boundary}}, \Delta_{\text{seam}}, \Delta_{\text{regime}} \right) $$

This enables:

• Model version comparison
• Training drift detection
• Cross-architecture structural analysis

Regime comparison becomes geometric rather than task-specific.

Empirical Illustration (Minimal Case)

In a recurrent system where spectral radius is varied:

• A chaotic basin emerges beyond a critical threshold
• Boundary folding occurs during training
• Later epochs contract unstable regions

The geography evolves during training.

Not merely the parameter values.

This demonstrates that learned systems possess measurable regime structure.

Limitations

• Sweep resolution scaling
• Curse of dimensionality
• Diagnostic sensitivity
• Classification threshold dependence

Future work includes adaptive atlas refinement and cross-scale invariance analysis.

Conclusion

Regime Geography is a measurable object.

Systems should be compared structurally, not merely functionally.

Treating behavioral organization as geometry reframes stability research for both classical dynamical systems and learned models.

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