A Taxonomy of Recursive Instability

Recursion Geometry Lab — Research Note

Recursive systems fail in structurally distinct ways. Recursion Geometry classifies instability at the operator level, independent of domain semantics.

1. The Structural Problem

Feedback systems appear across disciplines: dynamical systems, machine learning, ecological models, economic systems, and logical constructions.

Despite domain differences, instability consistently arises from the interaction between:

• A transition operator
• A state space
• A boundary or classification rule

Recursion Geometry studies this interaction formally.

A feedback system is modeled as:

S = (X, T, B)

where:

• X is a state space
• T : X → X is a transition operator
• B : X → {0,1} is a boundary predicate defining collapse, termination, or classification

Instability is defined structurally — not semantically — through properties of T and its interaction with B.

2. Spectral Instability

In spectral instability, failure emerges from operator growth.

Let ρ(T) denote the spectral radius of the relevant strongly connected component.

As ρ(T) → 1 from below, transient dwell time diverges. Under generic linear approach:

τ ~ (1 − ρ)⁻¹

This class admits perturbation analysis and spectral characterization.

Examples include:

• Edge-of-criticality dynamics
• Recurrent network crack onset
• Slow escape from metastable regions

This instability is measurable through operator invariants.

3. Elimination Geometry

Instability can emerge even when spectral growth is absent.

In elimination geometry:

• The operator may be contractive
• Instability arises from progressive state elimination under boundary refinement

Here, collapse is driven by B rather than ρ(T).

This class cannot be detected by spectral radius alone.

4. Infinite-Tail Geometry

Finite truncations may remain stable while the limit object collapses.

For a sequence of truncated systems Sₙ:

• All finite Sₙ satisfy stability
• The limit Sω exhibits failure

No finite spectral invariant predicts this instability.

This class is not reducible to spectral criticality.

5. Boundary Variance Collapse

Instability may emerge under reduction or coarse-graining.

A reduced system S’ may preserve:

• Eigenvalues
• Operator norms

Yet fail to preserve boundary behavior.

This produces instability under scale transformation rather than intrinsic operator growth.

6. Structural Separation

These instability classes are not semantic metaphors.

They are defined by measurable structural properties:

• Spectral invariants
• Boundary predicates
• Reduction maps
• Limit behavior

Two systems are structurally equivalent only if:

• Operator dynamics correspond under reduction
• Boundary classifications are preserved

Recursion Geometry investigates whether instability classes form equivalence strata in operator space.

7. Program Direction

Ongoing work includes:

• Formalization of the Spectral Criticality Principle
• Non-reducibility results for infinite-tail systems
• Boundary-preserving reduction theory
• Cross-domain applications in learning systems and multi-agent feedback models

Recursion Geometry aims to establish a structural classification theory for recursive instability.