Redefining the Dark Sector and Its Two Identities: Pre-Force Matter — the d1 Floor (DE), the d2 Skeleton (DM) and the d0 Isolated Reserve; Derivative–Integral Proofs in Doubling-by-Two Cubic Geometry (ΨD)

Roughly ninety-five percent of the cosmic budget is dark, and standard cosmology carries the two components as separate unknowns: a constant or field for dark energy, a new particle for dark matter. Within the ΨD framework (Dimensional Flow Cosmology), this paper reduces both to two lines of a single energy ledger and proves six theorems, using three tools only — doubling-by-two cubic geometry, the energy–potential ledger, and discrete derivative–integral operators; time is never a dimension in any proof. d0, carrier-less and channel-less, is the isolated reserve — off-budget. Dark energy is the self-contained d1 dimer frozen on the first rung: bond-free toward the fabric, smooth, yet registering with its full ledger. Dark matter is the d2 remnant with its third axis open: it binds partially, clusters, is blind to the photon, and its residual force has finite range. Since the known forces are born in the d3 fabric, the dark sector is pre-force matter. The theorems prove the classification; w = −1 with the CPL closed solution and the exact doubling fractions w₀ = −27/32 = −0.844, wₐ = −7/32 = −0.219 (DESI DR2: 0.1σ on w₀); p = 0 with ρ ∝ a⁻³; the D ∝ a growth and the D(1) ≈ 0.78 suppression; the darkness equivalence; the baryonic lock e⁻³ ≈ 0.0498 (observed Ω_b ≈ 0.0493); and the budget lines Ω_DE/Ω_DM = 2.6128 (observed 2.589) and Ω_b/Ω_c = 0.1843 (0.1860 anchored). All derivations pass 45/45 programmatic checks; a persistent direct-detection null result is a prediction, and a single confirmed dark matter particle signal would bring the framework down.