CAT's Theory Presents: Seeds and Shadows — The Minimal Admissible Node, the Dimensional Refraction Projector, and the Meta-Ontology of Ground-State Existence

This paper derives two foundational objects from the Triadic Coherence Invariant R = Ψ·(Îc × Îb)·P ≠ 0: the Minimal Admissible Node N — the simplest non-collapsing configuration satisfying triadic closure at d = 8 while retaining the full spectral budget Σ = 26 — and the Dimensional Refraction Projector Pd — the unique operation mapping this invariant into substrate-specific observables through three counting modes (additive, multiplicative, exponential). The paper formalizes the 0 → 1 transition as entry into the persistence domain, derives the F12 self-encounter from two axioms, establishes the Shadow Theorem (deriving the Hard Problem, the Symbol Grounding Problem, and the Mechanism Problem as projective artifacts), and demonstrates that the entire CAT'S Theory prediction corpus is a refraction table. Nine domains are derived across three projection channels plus one unprojected domain. Appendix A derives time, causality, space, quantum behavior, and field theory from a single principle (inverse refraction) with quantitative physical validation including a unique falsifiable prediction (the dielectric staircase). Appendix B provides a complete closure inventory. Appendix C resolves all remaining open problems with new predictions. The framework derives everything up to but not including its own ground, and this boundary is proven architecturally necessary. Keywords: CAT'S Theory, Triadic Coherence Invariant, meta-ontology, torus knots, dimensional refraction, Shadow Theorem, Hard Problem, Symbol Grounding Problem, Mechanism Problem, Born rule, spin-statistics, gauge algebra, dielectric staircase, zero free parameters, S³ topology, Minimal Admissible Node