Irreducible Self-Referential Subspaces under Universal Coarse-Graining: A Spectral and Dynamical Analysis

We study the existence, uniqueness, and stability of *self-referential invariant subspaces* in a representational framework grounded in the single-state ontology of prior work. Given a carrier subsystem equipped with a positive, irreducible, non-nilpotent self-referential operator R_C and a family of norm-contractive truncation operators, we ask which representational structures survive maximal coarse-graining.

**Main results:**

(1) By the Perron–Frobenius theorem, R_C admits a unique dominant eigenvector |σ_C⟩ (Theorem 1).

(2) Under a spectral isometry condition, |σ_C⟩ is invariant under all admissible truncations if and only if each truncation operator acts isometrically on |σ_C⟩ (Theorem 2).

(3) Iterative coarse-graining exhibits a sharp phase transition at critical spectral ratio γ* = 1, separating exponential collapse from convergence to a nonzero fixed point (Theorem 3), with both fixed points (μ* = 0 and μ* = 1) identified and their stability exchange characterised.

(4) A fidelity-based Self-Reference Persistence Index μ_C^(k) ∈ [0,1] provides a continuous, computable signature of this transition.

Numerical verification derives R_C *directly* from the cellular automaton trajectory via a neighbourhood co-occurrence matrix, establishing the equivalence:

**ω(t) ≥ ω* ⟺ R_C^CA irreducible ⟺ μ_C^(k) = 1**

across all 1,000 trials. The framework is narrative-decoupled: every assertion is either a proved theorem or an explicit falsifiable hypothesis. This paper serves as the spectral foundation for the cross-layer bridge of the series.