Relativistic Momentum Deformation of Oriented Spacetime Topological Bodies: Source Completion of Historical Spacetime Wave Theory

This article gives a conditional source completion for observer-rooted historical spacetime wave theory. At a fixed receiving root \(A\) and on a weak short tube \(\Omega_{A,\rho}\), a massive material core is represented by an oriented relative integral cycle. Its admissible displacement is split into zero modes and an internal deformation driven by an observer-relative relativistic momentum load. A coercive constitutive form and a positive quartic correction yield linear and nonlinear deformation branches, stability estimates, and an explicit sufficient pre-break criterion. A swept-current calculation proves preservation of the supplied relative homology class on every admitted pair flow. The deformed momentum one-form is coupled to an independently supplied encoding line bundle. Closedness, zero-period, integral-period, and Hodge conditions distinguish local, real-valued, and circle-valued phases; no spinorial structure or statistical interpretation is inferred from \(U(1)\) periodicity. A compactly supported smoothing of the oriented current and an explicit interaction action produce $$J_A^{\mathrm{full}}=J_A^{\mathrm{old}}+\Delta J_A^{\mathrm{def}}$$ by first variation. Support, Sobolev regularity, boundary terms, gauge covariance, and the associated Noether exchange identity are recorded. Under a declared closed augmented hyperbolic realization, the completed source has a retarded response with causal support. The common-operator branch admits superposition and a source-completion inheritance theorem; source-dependent propagation instead requires a contraction or resolvent condition. Additional invariant scalar and spectral assumptions give a local Schrödinger envelope with quantified geometric, relativistic, deformation, model, and boundary errors, followed by Klein–Gordon and conditional Dirac representations. A two-channel downstream construction produces the interference cross term without splitting the material core and proves that one intensity record identifies only the combined deformation, propagation, and apparatus phase. Cross-root comparison is constructed componentwise along a declared causal corridor and is followed by target-record reconstruction and target reclosure. Finite chains satisfy an explicit product–sum error bound; holonomy measures path dependence; compact coherent finite protocols have an inverse limit, which is not promoted to a single observer-independent physical state without effective descent data. All conclusions are restricted to the displayed regularity, causal, boundary, coercive, spectral, and reclosure branches. Constitutive coefficients, numerical calibration, nontrivial particle-bearing joint solutions, and empirical adequacy remain independent inputs or open problems. Keywords **Observer relativity; weak short tubes; oriented chains; relativistic momentum deformation; matter waves; source completion; historical spacetime wave theory; retarded Green operators.**