Finite Closure as the Unified Primitive: Emergence of Quantum Mechanics, Discrete Gravity, and Relativistic Structure (Rev00)

Finite Closure as the Unified Primitive: Emergence of Quantum Mechanics, Discrete Gravity, and Relativistic Structure (Rev00) Abstract This paper completes the structural programme initiated in Reality is What Misspoken Language Cannot Deliver (Rev18) by deriving quantum mechanics, discrete gravity, and relativistic structure from a single finite reversible closure substrate. The previous Rev18 paper established the geometric scaffolding; finite distinguishable states under entropy bounds, closure algebra as primitive, holonomy defect curvature and Regge type discrete gravity converging conditionally to the Einstein / Hilbert action. The present work supplies the representation scaffolding. From finite reversible closure and amplitude additivity, we show that a linear state container is forced. Invariant comparability uniquely selects an inner-product structure. Local tomography selects the complex field, excluding real and quaternionic fundamental scalars under standard composition. The Born rule arises as the unique non-contextual additive measure on projectors consistent with unitary invariance. Tensor composition yields entanglement and operator norm geometry enforces the Tsirelson bound, excluding PR-box correlations within the closure derived Hilbert structure. On the geometric side, closure defects embed into weighted Regge curvature via holonomy-conjugacy classes. Under standard refinement conditions (bounded curvature, non-degenerate simplices) the Regge action converges to the Einstein / Hilbert action. Local Minkowski structure appears as the vanishing-curvature tangent limit. Closure is taken as more primitive than both unitary and “Lorentzian, these are representation languages for different compressions of the same finite reversible substrate. No dual ontology is introduced. Introduction Modern physics employs two highly successful but linguistically separated frameworks; quantum Hilbert structure and Lorentzian spacetime geometry. Their mathematical forms are well established, yet their joint ontological status remains debated. In Reality is What Misspoken Language Cannot Deliver (Rev18). Reality is What Misspoken Language a closure-first ontology was developed in which;- Bounded entropy per bounded region excludes primitive continuum; Finite distinguishable regional states form the primitive substrate; Deterministic, composable, reversible updates form a closure group; Closure defects correspond to holonomy elements; Weighted hinge sums reproduce Regge discrete curvature and Under refinement assumptions, the discrete action converges to Einstein / Hilbert. The prior Rev18 paper therefore provided the geometric scaffolding; discrete gravity from finite closure. The present paper extends that scaffolding upward into the quantum representation layer. It asks: If finite reversible closure is primitive, what state representation is forced when coherent overlap is admitted? The answer is not assumed but derived;- Amplitude additivity forces linear structure; Invariant comparability forces inner product geometry; Local tomography selects the complex field; Non-contextual additive projector measures yield the Born rule; Tensor composition yields entanglement and Operator-norm structure enforces Tsirelson rigidity. Thus the same primitive closure substrate underwrites: Finite reversible closure;- Complex Hilbert representation Born rule Tsirelson bound Weighted Regge curvature Einstein / Hilbert limit Local special relativity Unitary and Lorentzian structures emerge as representation languages of the same finite closure primitive.