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

Preprint. This paper establishes a conditional representation theorem for scalar rescalings of the spacetime volume measure in four-dimensional Lorentzian geometry. Given a non-degenerate Lorentzian metric g and its associated measure mu(g) = sqrt(-g) d^4x, we study when a prescribed pointwise rescaling mu(g') = sqrt(1+alpha) mu(g) (with 1+alpha > 0) can be realized by a purely algebraic, derivative-free modification of the covariant metric tensor. Under explicit structural assumptions—locality, symmetry, construction from {g, n} only with a fixed unit timelike one-form n, and preservation of the induced spatial metric on the subspace orthogonal to n—we prove that the rescaling is realized by a unique rank-one update along the time leg:g' = g − alpha (n ⊗ n).We also classify how uniqueness is lost when assumptions are relaxed (e.g., by restoring conformal freedom or allowing additional one-form directions, leading to disformal-type ambiguities). A tetrad interpretation identifies the rank-one form as a renormalization of the tetrad time leg while keeping the spatial triad fixed, providing a clean interface to tetrad/Clifford conventions used in relativistic field theory. The results are representation-theoretic: no action principle, equations of motion, or universal matter coupling is assumed, and no claim is made about “degeneracy repair” of det(g) = 0 configurations. A companion manuscript develops an operational readout/estimator protocol that invokes the present theorem as an interface.