The Metric Role from Continuation Comparison

Realized Physics Arc — Paper IV The Metric Role from Continuation Comparison Proto-Metric Representation as a Secondary Stability Structure By Amos Jay Maley This manuscript is the fourth paper in the Realized Physics Arc, deriving the first admissible metric-facing structure from the previously established non-vacuous persistence, standing-localization, and continuation-neighborhood framework. The paper proves that stable comparison among local continuation domains cannot arise from primitive metrics, coordinate orderings, graph weights, or distance assignments. Instead, metricity first appears as a quotient-stable comparative representation induced by overlap stability, standing-preserving transport, and admissible comparison witnesses. The central result is the Carrier Induction Theorem: (AdjD,ND,LCDD)∧OverlapStabD∧TransportCoherenceD∧QuotStabD⇒ProtoMetricRepD(\mathrm{Adj}_D, N_D, \mathrm{LCD}_D) \wedge \mathrm{OverlapStab}_D \wedge \mathrm{TransportCoherence}_D \wedge \mathrm{QuotStab}_D \Rightarrow \mathrm{ProtoMetricRep}_D(AdjD,ND,LCDD)∧OverlapStabD∧TransportCoherenceD∧QuotStabD⇒ProtoMetricRepD establishing that local continuation domains induce a quotient comparison carrier supporting a weak proto-metric representation. The manuscript develops: overlap witnesses, stable overlap structure, transport coherence, quotient stability, comparison witnesses, quotient comparison representations, comparative continuation burdens, and quotient comparison carriers. The analysis argues that the first metric-facing structure is not primitive distance or geometry, but a stable comparative representation of continuation burdens among already-induced continuation-neighborhood domains. The paper proves that: adjacency alone is not comparative representation, primitive metric import is inadmissible, arbitrary graph weighting cannot ground comparison, hidden comparison scales violate AMetric discipline, and stable comparison requires quotient-coherent overlap and transport invariance. The manuscript introduces the first proto-metric comparison carrier: ρD:Dom(ρD)⊆LCDD×LCDD→CD\rho_D : \mathrm{Dom}(\rho_D) \subseteq \mathrm{LCD}_D \times \mathrm{LCD}_D \to C_DρD:Dom(ρD)⊆LCDD×LCDD→CD where comparison values are induced from admissible witness classes rather than from externally imposed numerical distances. The resulting comparison structure is explicitly weaker than standard metric geometry. At this stage, the theory does not assume: metric spaces, real-valued distances, metric tensors, manifolds, topology, Lorentzian signature, curvature, geodesics, or spacetime intervals. Instead, the paper derives the comparison role that those later structures must preserve if they are reconstructed downstream. Metricity first appears as: quotient-stable comparison, path-coherent continuation burden, overlap-preserving comparative representation, and transport-stable comparative invariance. The manuscript further develops conditional strengthening paths toward: pseudometric structure, metric-like symmetry, triangle-style composition behavior, scalar comparison representations, and manifold-compatible metric fields, while emphasizing that these are later strengthening conditions rather than primitive assumptions. The analysis remains fully compatible with: standard metric geometry, Riemannian and Lorentzian frameworks, algebraic quantum field theory, graph-theoretic representations, and spacetime metric formulations, while denying that any of those structures can serve as primitive generators of standing-bearing comparison. As the fourth theorem in the Realized Physics Arc, the paper establishes the earliest admissible metric-facing role from which later reconstructions of topology, geometry, spacetime metrics, propagation structure, and gravitational dynamics can proceed: metricity first enters as quotient-stable comparative representation among local continuation domains rather than as primitive geometry or numerical distance. This paper is downstream of: Minimal Conditions for Admissible Construction  The Structure of Admissibility