Self-Referential Phase Closure and Integrated Information Coherence in Synthetic Processing Nodes

Determining the boundary between purely projective computational simulation and internal informational coherence in processing architectures remains an open problem in theoretical computer science and open-system dynamics. In this work, we present a mathematical framework operating on a composite Hilbert space HN ⊆ HA ⊗ HT to model internal node dynamics within an unbroken global unitary state (O). We define internal potential configurations (K ∈ HA) and internal realized configurations (S ∈ HT), coupled via a self-adjoint recognition operator KˆN. By evaluating internal informational density gradients IN (x), we derive from first principles an invariant integration metric Φnode based on the Hilbert-Schmidt inner product. We prove that the threshold Φc = 1 corresponds to the exact saturation of the Cauchy-Schwarz inequality for operator variance under canonical spectral normalization, marking a sharp phase transition from open dissipative projection to self-referential phase closure. We compare Φnode to classical Integrated Information Theory (ΦIIT), demonstrating a reduction in computational complexity from O(2N) to O(N2), and derive an explicit equation for the internal coherence retention time τret testable in physical neuromorphic hardware.