On Trajectory Boundary Cancellations, Dirac Constraint Stability, Wave-Packet Mechanics, and Curved-Ray Dynamics in Static and Stationary Vector-Coupled Fields

We establish an exhaustive, mathematically rigorous analytical architecture governingelectromagnetic radiation propagating through weak gravitational fields coupled to effectivevector-mediated perturbations. In a rectilinear unperturbed null trajectory traversing a static,spherically symmetric potential ψ0(r), we state and prove the generalized Boundary-TermVanishing Lemma: for minimal symmetric gradient coupling Sµν = 12(∇µψν + ∇νψµ), thefirst-order transverse impulse evaluates to an exact total spatial derivative ∫︁ +∞[∂yψ0]+∞−∞ ≡0, vanishing identically at spatial infinity.−∞ ∂x(∂yψ0)dx =We systematically expand this architecture across eight critical physical frontiers: (1) weformulate the explicit mathematical assumptions underlying boundary cancellation, provingthat the vanishing null result serves as a foundational No-Go screening theorem; (2) we conducta complete Hamiltonian Dirac constraint analysis for the massive Proca system, proving thatthe constraints are second-class, algebraically eliminating the scalar mode and guaranteeingunconditional ghost-freedom; (3) we elevate the eikonal ray approximation to full covariantwave mechanics by applying the d’Alembertian operator □geffΨ = 0 on the effective opticalmetric geffµν ≡ gµν −2βSµν, deriving frequency-dependent wave-packet diffraction broadeningδθdiff ∼ λEM/(2πb) and proving zero vacuum birefringence; (4) we overcome the rectilinear rayidealization by developing a dynamic curved trajectory perturbation framework and implementinga 4th-order Runge-Kutta (RK4) numerical integrator, demonstrating analytically that dynamicray bending breaks the incoming/outgoing reflection symmetry and induces an irreduciblesecond-order transverse deflection ∆k(2)y ∼ O(GMβ/b3); (5) we replace idealized uniformfield = 3κ2GM2cores with an astrophysically realistic parabolic polytropic stellar interior density profile ρ(r) =ρc(1−r2/R20), proving smooth C1 boundary matching and regularizing the total field energy massto Mtotal14πR0 ; (6) we generalize the trajectory integration to stationary Kerr geometries,deriving the gravitomagnetic frame-dragging deflection ∆θdrag = ∓4GJ/(c3b2) and showing thatspacetime rotation breaks reflection symmetry, generating an observable kick ∆kKerry ̸=0; (7)we supply a first-principles quantum gravitational effective field theory derivation of the Wilsondeflection coefficient αQG = κQG/(96π) via Seeley-DeWitt heat kernel expansions, elucidatingthe universality of the geometric factor 1/(96π) against microscopic particle contents; and (8)we confirm the constructive curvature summation of isotropic metric Christoffels (+1+1 = +2),reproducing Einstein’s deflection ∆θGR = 4GM/(c2b). Five native vector visualizations and acertified master verification script are appended.