Covariant Finite Reclosure and the Emergence of a Local Lorentzian Response Envelope

Abstract

We establish a local structural route from finite quadratic reclosure to a Lorentzian response envelope. First, a faithful and lossless alternating reclosure map Q: Λ²W → V with dim W = dim V = n > 0 selects n = 3 by the dimension identity dim Λ²W = n(n − 1)/2. We then realize this three-dimensional sector by an A₄-covariant four-site carrier on S = 1⊥ ⊂ ℝ⁴ and prove the exact reclosure identity Q_λ = 4λ δ ∘ ⋆_h for a fixed metric and orientation convention. Infinitesimal unitary deformations of the carrier define a reclosure-twist map Θ_λ: H_edge → so(S, h); its A₄ covariance and one explicit nonzero witness imply rank Θ_λ = 3. No A₄ closure of the physical control tangent space is assumed. Instead, one nonzero physical twist is sufficient for the associated connected Lie-group completion of the structural twist algebra to be SO(S, h). A compact calibrated spatial response body then has a symmetry-saturated convex envelope equal to the closed h-ball of radius c_x. After adjoining one temporal response direction, the positive homogeneous extension of this ball determines, by rotational invariance, spatial normalization, and the requirement that the envelope boundary be null, the unique local form g_env = h(P_S ·, P_S ·) − c_x² θ², sig g_env = (−, +, +, +). The result concerns a symmetry-saturated response envelope. It does not identify that envelope with the physical reachable set or an operational causal cone, does not identify c_x with the physical speed of light without additional calibration and propagation input, and does not derive Einstein equations or gravitational dynamics.