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/ (rhocoh c²) 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 wₘass = 0 (pressureless dust structure of C^ (mass) ), wₛpin = 0 (perfect isotropy of the K4spin distribution: = (1/4) deltaⁱj over all 8 coherent configurations, OP-spin resolved in D48v3), and wₒrb = 0 (C4 forces the London gauge over VC, hence jⁱ = 0, proved in D49). The leakage component has wₗeak = -1 intrinsically (cosmological constant structure, D48v3 + D51–D53). The complete equation of state is: w (N) = - rhoLambda / rhocoh (N) an unconditional theorem of C1–C4 with no free parameter. Two physical regimes: (a) At coherence and nuclear densities (N = 1. . 200, rhocoh ~ 10¹7–10¹8 kg/m³): |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 rhoLambda = LambdaPDL c² / (8 pi GPDL) = 5. 835 x 10^-27 kg/m³ gives OmegaLambdaPDL = rhoLambda / rhocrit = 0. 6838, in agreement with the Planck 2020 observed value OmegaLambdaᵒbs = 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).