Phase-Space Thermodynamic Gravity (Letter III) Strong-Field Resummation, Black-Hole Shadows, and Binary Pulsars

Building upon our flat-spacetime theory of gravitation grounded in the Planck Force vacuum impedance (µ = c4/G) and the 1PN local Lagrangian established in Letter I [1], together with the 3D Ginzburg–Landau phase-condensation and Yukawa–Meissner screening framework established in Letter II [2], we derive the exact analytical extension of the theory to the strong-field 2PN sector, comparative-mass two-body systems (m1 ∼ m2), and 2.5PN gravitational radiation. First, integrating the differential pair-stress equation from Postulate 2 [1] yields the non-perturbative binding-energy resummation 1 + |Utot|/E0 = eχ(with χ =GM/rc2).In the pre-registered canonical strong-field closure (Branch I), central singularities are eliminated in favor of a core saturated at rc = GM/c2, placing the photon sphere at the Golden Ratio (rph = φ rg ≈ 1.618 rg), the apparent black-hole shadow at bcrit = 4.857 rg (consistent with EHT Sgr A∗ bounds [5]), and the 2PN solar limb light deflection at 10.21 µas [12]. Second, for binary systems (m1 ∼ m2), the flat-spacetime Helmholtz–Darwin propagation delay (∆t = r/c) contributes + 32ν, which exactly cancels the special-relativistic kinetic deficit−32ν, locking the secular periastron advance coefficient at Csec(ν) = 3 for any mass ratio and reproducing the 16.8993◦/yr precession of double pulsar PSR J0737–3039 [6]. Third, under the Pair/Self boundary condition(ηN = 0) [1], monopole and dipole radiation vanish identically [8], while transverse-traceless (TT) shear waves in the µ = c4/G medium yield the 2.5PN quadrupole formula (factor 15µc =G5c5), matching the orbital decay P˙b of PSR B1913+16 [7] and PSR J0737–3039 [6] to within 7 × 10−5. Finally, we examine the October 2026 observational updates regarding the pre-registered sentinels of Letter II [2, 9, 10, 11].