Primitive 0: Finite Realisation - A Foundational Admissibility Rule for Physical Structure - (Paper 0) Abstract Modern theoretical physics frequently employs mathematical infinities within formal models. However, no confirmed observation requires that a physically realised configuration itself contain an operational infinity. This paper introduces Primitive 0 - Finite Realisation, an admissibility rule for physical ontology. The primitive states that no realised physical configuration may require an operational infinity for its specification, evolution or measurement. Any realised configuration must therefore admit finite relational description, finite dynamical update and finite observational distinction. Primitive 0 does not prohibit the use of mathematical infinities as analytical tools. Instead, it constrains the ontology of realised states. Within this admissibility framework, singularities and ultraviolet divergences are interpreted as breakdowns of effective descriptions rather than physically realised infinities. Primitive 0 establishes the foundational boundary condition beneath the relational closure framework developed in subsequent papers of the Finite Reversible Closure (FRC) programme. It defines the admissible domain within which closure structure, relational lattice dynamics, gauge symmetry, matter excitations and gravitational geometry can emerge. Introduction Many successful physical theories contain mathematical infinities when equations are extended beyond the domain in which they were originally derived. Examples include singularities in classical gravitational collapse solutions and divergences appearing in quantum field theory calculations. Despite the presence of such infinities within formal mathematics, no direct observation has ever confirmed a realised infinite physical quantity. All empirical measurements yield finite values. This observation motivates a foundational admissibility rule for physical ontology. The rule proposed in this paper is Primitive 0 - Finite Realisation. Primitive 0 states that no physically realised configuration may require an operational infinity for its specification, evolution or measurement. A realised state must therefore admit;- finite relational description; finite dynamical update and finite observational distinction. The primitive does not forbid mathematical infinities as tools of analysis. Rather, it restricts the interpretation of infinities as physically realised states. Primitive 0 therefore defines the admissible domain within which physical structure may exist. The derivation of structural dynamics begins in the following paper with the Primitive Closure Axiom, which introduces finite reversible local update.

Description This paper establishes the admissibility boundary for the Finite Reversible Closure (FRC) programme. Primitive 0 does not derive physical structure by itself. Instead, it defines the conditions under which realised structure may exist. The primitive excludes operational infinities from realised states and therefore restricts candidate physical ontologies to configurations that admit finite relational specification and finite dynamical evolution. The logical flow of the framework is therefore;- Primitive 0Finite RealisationNo operational infinity in any realised state------------------------------------------------Finite relational descriptionFinite dynamical updateFinite observational distinction------------------------------------------------Admissible domain for physical structure↓Paper 1Primitive Closure AxiomFinite reversible local updateFinite Hilbert space per siteFinite depth evolution per tick↓Relational lattice structureGauge symmetryMatter sectorsCurvature and gravity↓Papers 2–20Development of the full FRC framework Within this programme, Primitive 0 performs a strictly foundational role. It establishes that physically realised configurations must remain finite and operationally distinguishable. Divergences appearing in continuum descriptions therefore signal breakdowns of effective models rather than the presence of physically realised infinities. The paper further clarifies that the Zerofield lies outside the domain to which Primitive 0 applies. The Zerofield represents the absence of realised relational structure rather than an infinite measurable magnitude. Primitive 0 therefore governs realised states only, while the Zerofield functions as a non-operational boundary condition from which finite structure departs. By defining this admissibility rule, Paper 0 provides the conceptual and logical boundary beneath the closure framework developed in the remainder of the programme. This paper forms the foundational admissibility layer of the Finite Reversible Closure (FRC) research programme (Papers 0–20).