Rank-One Representation of Measure Rescalings via Time-Leg Renormalization

Preprint (Version v2). Version note (v2). This version adds a short, readout-only vignette (an exposure–rate example) to clarify how the Lorentzian volume measure enters practical reporting/inference pipelines, and refines the positioning and phrasing throughout. The main representation theorem and its scope remain unchanged. This paper establishes a conditional representation theorem for prescribed positive rescalings of the Lorentzian spacetime volume measure in four-dimensional geometry. Let g be a non-degenerate Lorentzian metric with volume measure μ(g)=sqrt(-g) d^4x. Fix a unit timelike one-form n (a “time leg”), and consider deformations g' that are local, algebraic (derivative-free), symmetric, and constructed from {g,n} only. Imposing (i) preservation of the induced spatial metric on n^⊥ and (ii) the target pointwise rescaling μ(g')=sqrt(1+α) μ(g) with 1+α>0, we prove that the rescaling is realized uniquely by the rank-one update along the time leg:g' = g − α (n ⊗ n).Uniqueness is conditional: we classify how it is lost when assumptions are relaxed, e.g., restoring conformal freedom or allowing additional one-form directions, which reintroduce standard conformal/disformal-type ambiguities. The proof is representation-theoretic and rests on rank-one determinant identities (matrix determinant lemma) in the non-degenerate sector det(g)≠0. Degenerate configurations det(g)=0 lie outside scope, and no claim is made about “repairing” degeneracy. No action principle, modified field equations, new propagating degrees of freedom, or universal matter coupling is assumed. The one-form n is treated as chosen structure (an observer congruence/time leg) without independent dynamics. A tetrad interpretation identifies the rank-one form as a renormalization of the tetrad time leg by sqrt(1+α) while keeping the spatial triad fixed. This provides a clean interface to tetrad/Clifford conventions: the corresponding gamma-matrix representation remains compatible with g' under standard tetrad-based constructions. To make the operational relevance explicit without adding model dependence, we include a minimal vignette showing how a declared readout-layer convention κ=1+α propagates through exposure-based rate inference (e.g., V_obs and T_obs rescalings) as a purely multiplicative accounting rule, while leaving all phenomenology and system-specific modeling to companion studies. A companion manuscript develops an operational readout/estimator protocol that invokes the present theorem as an algebraic interface between “measure rescaling” and “rank-one representation.”