Recursion Geometry: A Structural Program for Feedback Systems

Recursion Geometry Lab — Research Overview Recursion Geometry is a research program dedicated to the operator-level analysis of feedback systems. Rather than classifying systems by application domain or semantic interpretation, it studies the structural behavior of recursive dynamics through measurable operator invariants and boundary interactions. Across dynamical systems, learning architectures, and logical constructions, instability arises from the interaction between state evolution and classification or termination rules. Recursion Geometry formalizes this interaction using a minimal structural model.

Coherence Objects

A feedback system is modeled as a coherence object:

S = (X, T, B)

where:

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

The geometry of recursion is determined by:

• Spectral properties of the operator T,
• Invariant subspaces and strongly connected components,
• Limit behavior of iterates Tⁿ,
• Interaction between operator structure and boundary predicate B.

All classification proceeds through operator-theoretic quantities that are measurable or structurally defined. Classes of Recursive Instability

Recursion Geometry identifies structurally distinct classes of instability.

1. Spectral Instability

Instability occurs when a transient strongly connected component has spectral radius approaching unity. In such systems, dwell time diverges as ρ(T) → 1.

Under generic linear approach to criticality:

τ ~ (1 − ρ)⁻¹

This class admits operator perturbation analysis and spectral characterization.

2. Elimination Geometry

Instability arises not from spectral growth but from progressive state elimination under shifting boundary constraints. The operator may be contractive or acyclic while instability emerges from boundary refinement.

3. Infinite-Tail Geometry

Finite truncations remain stable while the limit object exhibits collapse. No finite spectral invariant predicts the instability. This class cannot be reduced to finite spectral criticality.

4. Boundary Variance Collapse

Instability emerges from reduction or coarse-graining rather than intrinsic spectral growth. Cross-scale operator approximations may preserve eigenvalues while failing to preserve boundary behavior.

Structural Equivalence and Reduction

Recursion Geometry investigates when two coherence objects
S₁ = (X₁, T₁, B₁) and S₂ = (X₂, T₂, B₂)
are boundary-preserving equivalent.

A reduction map R : X₁ → X₂ is admissible when:

• R ∘ T₁ ≈ T₂ ∘ R
• Boundary classifications are preserved under R

The research program explores whether instability classes form structural equivalence classes under such reductions and whether spectral phase partitions operator space into geometric strata.

Applied Program

The operator-phase geometry of trained recurrent neural networks provides an empirical instance of this framework. Observed phenomena include:

• Spectral compression during training,
• Stable phase partitions under parameter scaling,
• Detectable cross-scale reduction failures.

These results motivate a broader classification theory of recursive instability.

Ongoing Work

Current directions include:

• Formal development of the Spectral Criticality Principle,
• Non-reducibility results for infinite-tail geometries,
• Operator-phase equivalence classes,
• Cross-domain applications in ecological, economic, and multi-agent systems.

Recursion Geometry aims to establish a structural foundation for the classification of recursive instability across feedback systems.