The Channel Capacity of Causal Boundaries: A Mapping from Retarded Boundary Kernels to Blackwell's Trapdoor Channel

Preprint: This paper establishes a mapping from the retarded boundary kernel (RBK) of Boundary Coordination Theory to Blackwell’s trapdoor channel — a classical unifilar finite-state channel whose feedback capacity is the golden-ratio exponent ln φ ≈ 0.4812 nats per channel use, proven by Permuter, Cuff, Van Roy, and Weissman (2008). The mapping identifies a physical operator with a solved information-theoretic object, yielding a channel-capacity interpretation of causal boundary dynamics. Main Result: Under four explicit structural assumptions — causal finite-state dynamics, two-state memory, symmetric trapping, and causal feedback — the discretized RBK is isomorphic to the trapdoor channel. The coordination law τ Δω = Θ(1) from causal measurement theory then admits the sharper inequality τ ≥ 1/ln φ ≈ 2.078 boundary time steps per nat of recorded information, with equality achieved at the channel capacity. Key Contributions: ∙ A rigorous mapping from a retarded boundary kernel to Blackwell’s trapdoor channel under stated and falsifiable assumptions ∙ Derivation of symmetric trapping (p = 1/2) from the self-adjointness and time-translation invariance of the kernel, replacing the maximum-entropy assumption with a structural argument (Appendix B) ∙ Numerical verification for the graphene-coated photonic ring resonator: the two-state truncation parameter ε ≈ 10⁻²² and the restricted kernel matrix confirms K₀₀/K₁₁ = 1.0000000000, validating the swap symmetry that enforces p = 1/2 ∙ A channel-class hierarchy situating ln φ as the floor of the unifilar feedback capacity spectrum ∙ Explicit falsification criteria: five testable conditions under which the mapping fails, collected in a reference table Scope: The value ln φ is not new — it is a theorem of Permuter et al. (2008). The Fibonacci counting argument underlying it is classical. The contribution is the mapping: the identification of a physical boundary operator as a realization of the trapdoor channel, and the reinterpretation of the coordination constant as a channel capacity. The trapdoor model is minimal, not inevitable — physical boundaries with richer state spaces can only exceed this rate. Three well-defined open problems remain: proving the full unifilar finite-state reduction for general kernels, establishing the physical dominance of the two-state sector, and extending the framework beyond the equal-partition coarse-graining.