Derivation of the Coherence Stress-Energy Tensor from the PDL Axioms: Explicit Form of C_coh and Resolution of OP2 (D35)
Year: 2026
Type: article
Abstract: The PDL Einstein field equation G[g_eff] = kappa_PDL(N) C_coh, established in D35, contains a coherence stress-energy tensor C_coh whose derivation from first principles was left as OP2 of D35. This document derives the explicit tensorial form of C_coh from axioms C1–C4, corrects four dimensional errors present in versions 1 and 2, and resolves OP-spin as a new result. The coherence volume V_C = (4/3)pi [hbar/(m_p c)]^3 is established as an unconditional theorem via the proper-time counting chain (D10a) and the three-dimensional topology (D23). The coherence density rho_coh(N) = sigma(N) R_surf m_p / (V_C R_tot) follows directly. The mass component C^(mass)_{mu nu} = rho_coh(N) c^2 u_mu u_nu is an unconditional theorem, verified numerically to 0.013%. The spin component C^(spin)_{mu nu} is derived by explicit computation over all 8 coherent configurations of K_4 and the 4 pulsation states T^k|+>. The results = 0 and = (1/4) delta^ij are exact theorems from C1–C4, from which the Weyssenhoff–Raabe formalism gives C^(spin)_{mu nu} = 0 in the homogeneous ground state. This resolves OP-spin. The orbital component C^(orb)_{mu nu} = m_p^2 / rho_coh is derived from the Madelung decomposition of the PDL wave function (D32), with the wave function normalised as a probability amplitude ([psi] = m^{-3/2}). It vanishes in the ground state by the London gauge theorem of D49. The leakage component C^(leak)_{mu nu} = -(Lambda_PDL c^4)/(8 pi G_PDL sigma(N)) g_{mu nu} is given in uniform units J/m^3, so that the full PDL Einstein equation is dimensionally consistent throughout. The factor eta_L^18 entering Lambda_PDL is an unconditional theorem (D17, D30); the constant C is derived in D51–D53. Version 3 corrects four errors from versions 1 and 2: (C1) Chain 2 for V_C was a tautology, not an independent proof; (C2) the v1/v2 formula for C^(spin) was dimensionally incorrect and its claimed magnitude 1e-12 was unverified; (C3) a factor m_p was missing in the formula for C^(orb); (C4) the formula for C^(leak) had units m^{-2}, rendering the Einstein equation dimensionally inconsistent. All corrections are verified numerically to machine precision. This document supersedes D48v1 (10.5281/zenodo.19969831) and D48v2.
Author: Laubscher, Cédric
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7159811079