Relational Persistence and the Second Invariant Above Zero: Ontological Closure, Boundary Feedback and IR Phase Universality - Paper 09

Relational Persistence and the Second Invariant Above Zero: Ontological Closure, Boundary Feedback and IR Phase Universality - Paper 09 Abstract Papers 1–8 established a finite, strictly local, reversible substrate in which quantum structure, relativistic dispersion, gauge-compatible matter, discrete curvature and structural renormalisation emerge as scale-dependent compressions of an admissible domain. Paper 9 removes residual boundary-language ambiguity and formalises ontological closure; no external state space exists beyond the physical Hilbert space defined by admissibility constraints. Apparent boundary feedback is therefore reinterpreted as internal constraint preserving evolution. Under finite local dimension, strict locality, reversible update and empirically smooth infrared (IR) phase response, persistent recurrence above the trivial sector generically flows to an effective symmetry containing a subgroup isomorphic to U(1). Discrete cyclic countermodels are analysed and shown either to flow toward U(1) universality under coarse-graining or to predict measurable phase granularity. This paper identifies the minimal cyclic structure compatible with empirical phase continuity and constrains the intermediate structural regime analysed in Paper 10. Introduction The Finite Reversible Closure (FRC) programme begins from a strictly local, finite-dimensional, constraint-defined ontology in which all admissible dynamics are generated by finite-depth local unitary update. In such a framework, language invoking external reservoirs, environments, or boundary exchange is formally inconsistent; no external state space is defined. Paper 9 therefore addresses a necessary clarification. If admissibility exhausts ontology, then all apparent feedback must reduce to internal relational propagation. This clarification allows a deeper question;- What is the minimal persistent deviation above Zero compatible with reversible evolution and empirically smooth phase response? The trivial sector (“Zerofield”) contains no degrees of freedom beyond admissibility. Any persistent structure must therefore arise as relational recurrence encoded in gauge-invariant operator sectors. Under coarse-graining defined by correlator matching of loop operators, such recurrence must exhibit connected symmetry in the IR if empirical phase shifts are arbitrarily refinable. We show that under these conditions the minimal connected subgroup available within admissible reversible dynamics is U(1). Discrete finite cyclic recurrence either flows toward U(1) universality or predicts observable staircasing artefacts in sufficiently precise phase-response experiments. Paper 9 therefore;- eliminates external boundary interpretations; formalises relational exclusivity of persistent information; constrains primitive recurrence structure; defines falsifiability via phase granularity detection and prepares the intermediate structural band analysed in Paper 10.