Phase Partition in the First Window of Expansion: From Non-Interacting Planckons to Composite Particles — Sector Ratios, the Acquisition of Mass, and the Proof of Masslessness (ΨD)

Within the ΨD framework (Dimensional Flux Cosmology), this paper builds the first window of expansion on one chain of questions: what fraction of planckons never interacts, what fraction rises to string (d1), surface (d2) and volume (d3), which sectors the relics establish, how mass is acquired in d3 — and how masslessness is possible there. The method uses three tools only — doubling-by-two cubic geometry, the energy–potential ledger, derivative–integral operators; time is not a dimension in any proof, and all dynamics run on the event-time τ = n·tP. The independent actualization of three orthogonal axes forces the biased binomial fd (A) ∝ C (3, d) ·Aᵈ· (1−A) ^ (3−d) ·e^ (j·C (d, 2) ), j = 1/e, with coefficients the vertex count of the 0, 1³ cube (Theorem C2) ; differentiation proves the peak law — f₁ at A = 1/3, f₂ at A = 2/3 (Theorem C3). Below the cold threshold the partition freezes; xf = W (4 ln 2) = 1. 00992, η = 0. 6358, A₀ = 0. 2094 are derived, and the fractions close without calibration: 0. 4640 / 0. 3688 / 0. 1412 / 0. 0260 (Theorem C4). The ledger identifies the sectors — d0 a carrier-less isolated reserve, d1 the dark-energy floor, d2 dark matter; with the registration gate, ΩDE/ΩDM = f₁/f₂ = 2. 6128 against the observed 2. 589 (0. 9%; Theorem C6), and the baryonic lock is e⁻³ ≈ 0. 0498 (Theorem C7). The law of mass is m = Eₖ/c² and its criterion is localization: the flat d3 substrate and the photon are massless despite d3 form (Theorem C9, Corollary C9. 1) ; a massless charged particle is forbidden (Theorem C11). All derivations pass programmatic checks, 38/38 (Appendix A) ; the chain is closed-form exact with zero free parameters, tabulated against observation at 40 digits in Appendix B.