| _version_ | 1866902324790689792 |
|---|---|
| author | Aisingioro Ollervides, Vinness |
| author_facet | Aisingioro Ollervides, Vinness |
| contents | <div> <div>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.</div> <br> <div>**Main results:**</div> <div>(1) By the Perron–Frobenius theorem, R_C admits a unique dominant eigenvector |σ_C⟩ (Theorem 1).</div> <div>(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).</div> <div>(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.</div> <div>(4) A fidelity-based Self-Reference Persistence Index μ_C^(k) ∈ [0,1] provides a continuous, computable signature of this transition.</div> <br> <div>Numerical verification derives R_C *directly* from the cellular automaton trajectory via a neighbourhood co-occurrence matrix, establishing the equivalence:</div> <br> <div>**ω(t) ≥ ω* ⟺ R_C^CA irreducible ⟺ μ_C^(k) = 1**</div> <br> <div>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.</div> <br> <div> </div> </div> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19223072 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Irreducible Self-Referential Subspaces under Universal Coarse-Graining: A Spectral and Dynamical Analysis Aisingioro Ollervides, Vinness <div> <div>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.</div> <br> <div>**Main results:**</div> <div>(1) By the Perron–Frobenius theorem, R_C admits a unique dominant eigenvector |σ_C⟩ (Theorem 1).</div> <div>(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).</div> <div>(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.</div> <div>(4) A fidelity-based Self-Reference Persistence Index μ_C^(k) ∈ [0,1] provides a continuous, computable signature of this transition.</div> <br> <div>Numerical verification derives R_C *directly* from the cellular automaton trajectory via a neighbourhood co-occurrence matrix, establishing the equivalence:</div> <br> <div>**ω(t) ≥ ω* ⟺ R_C^CA irreducible ⟺ μ_C^(k) = 1**</div> <br> <div>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.</div> <br> <div> </div> </div> |
| title | Irreducible Self-Referential Subspaces under Universal Coarse-Graining: A Spectral and Dynamical Analysis |
| url | https://doi.org/10.5281/zenodo.19223072 |