Equation of State of the PDL Coherence Fluid: Resolution of OP-pressure (Projective Dynamic Logo Framework, Document D54)

The PDL coherence fluid — the source term of the PDL Einstein equation established in D35 — has a well-defined equation of state w = P/(rho_coh c^2) derivable fromaxioms C1–C4 alone. This document resolves OP-pressure by assembling four theorems established in D48v3 and D49 into a single coherent statement. In the homogeneous ground state, the PDL coherence fluid satisfies w_mass = 0 (pressureless dust structure of C^(mass)), w_spin = 0 (perfect isotropy of the K4spin distribution: <S^i S^j> = (1/4) delta^ij over all 8 coherent configurations, OP-spin resolved in D48v3), and w_orb = 0 (C4 forces the London gauge over V_C,hence j^i = 0, proved in D49). The leakage component has w_leak = -1 intrinsically (cosmological constant structure, D48v3 + D51–D53). The complete equation of state is: w(N) = - rho_Lambda / rho_coh(N) an unconditional theorem of C1–C4 with no free parameter. Two physical regimes: (a) At coherence and nuclear densities (N = 1..200, rho_coh ~ 10^17–10^18 kg/m^3): |w| ~ 10⁻⁴⁴. The PDL coherence fluid behaves as pressureless cold matter, indistinguishable in equation of state from cold dark matter. (b) At cosmological scales: the PDL dark energy density rho_Lambda = Lambda_PDL c^2 / (8 pi G_PDL) = 5.835 x 10⁻²⁷ kg/m^3 gives Omega_Lambda^PDL = rho_Lambda / rho_crit = 0.6838, in agreement with the Planck 2020 observed value Omega_Lambda^obs = 0.685 to 0.17%. This 0.17% agreement is derived entirely from the proton quintuplet (24, 28, 930, 10087, 11017) and axioms C1–C4, with no cosmological parameter as input. The PDLcoherence fluid thus provides a unified description of the two dark components of the concordance cosmology — cold dark matter and the cosmological constant — as two density regimes of the same combinatorial structure. All results are verified numerically at 50-decimal precision (Python/mpmath). The complete verification script is provided as supplementary material. Depends on: D48v3 (doi:10.5281/zenodo.20151380), D49 (doi:10.5281/zenodo.20025166), D51 (doi:10.5281/zenodo.20033520), D52 (doi:10.5281/zenodo.20036769), D53 (doi:10.5281/zenodo.20052558).