Structural Renormalisation and the Vacuum Sector in Finite Reversible Closure (Paper 7)

Structural Renormalisation and the Vacuum Sector in Finite Reversible Closure (Paper 7) Abstract This paper extends the 'Finite Reversible Closure' (FRC) framework developed in Papers 1–6 by establishing structural renormalisation within a discrete, finite and strictly local substrate. We show that hierarchical coarse-graining induces a well-defined scale flow that is generically channel-valued, that ultraviolet divergences are structurally excluded by finite local Hilbert dimension and bounded update depth and that running couplings arise from operator mixing under blocking rather than cancellation of infinities. The vacuum is formulated as a sector of invariant states (eigenstate, stationary or finite-cycle) allowing for sector labels and discrete-time periodic structure. A finite uniform residual curvature observable is defined in the gauge sector via plaquette holonomy expectation values. Any mapping of this residual curvature into geometric (gravitational) language is treated as an operational correspondence to be developed separately. Renormalisation is therefore recast as finite hierarchical compression of bounded reversible dynamics. Continuum field descriptions arise as effective infrared limits rather than primitive ontology. Introduction Renormalisation is traditionally presented as a procedure for regulating and subtracting ultraviolet divergences in quantum field theories. In Wilson’s formulation, however, renormalisation is more fundamentally understood as the study of how physical descriptions change under scale transformation and coarse-graining. This paper develops renormalisation from a different starting point: a discrete, finite and strictly local substrate termed finite reversible closure (FRC). In this framework, spacetime is not taken as a primitive continuum. Instead, the primitive structure consists of;- finite-dimensional local Hilbert spaces per site and link; a discrete tick representing primitive sequencing; a finite-depth strictly local unitary update per tick; exact local gauge constraints; curvature defined through closed-loop holonomy. Within this setting, ultraviolet divergences cannot arise in finite regions because the local operator algebra is finite. The question therefore becomes not how to cancel infinities, but how scale-dependent effective descriptions emerge under hierarchical compression. We show that coarse-graining induces a well-defined effective scale flow described by 'Completely Positive Trace Preserving' (CPTP) maps, with unitary effective dynamics appearing only in reversible sub-sectors. Running couplings are identified with operator mixing under blocking and logarithmic behaviour corresponds to marginal directions of the linearised renormalisation map. The vacuum is treated generally as an invariant sector of the discrete update, allowing eigenstate, stationary or finite-cycle structures. Gauge curvature is encoded through plaquette holonomy expectation values; uniform residual curvature in this sector is finite and scale-dependent. Any mapping to geometric curvature language is deferred to a subsequent synthesis. The aim of this paper is not to replace established continuum formalisms, but to demonstrate that renormalisation and scale dependence arise naturally and consistently within a strictly finite, discrete and reversible framework.