The Integrated Theory of the Ladder: The Necessity of Existence, Two-Step Mechanics, and d3 Closure Derivative–Integral Proofs in 2-Fold Cubic Geometry (ΨD)

This paper raises the ΨD (Dimensional-Flow Cosmology) ladder to theorem level in three layers: the necessity of existence, the micro-move, and d3 closure. The method is confined to three tools — 2-fold cubic geometry, the energy–potential ledger (E(d) = (d+1)εP, U(d) = (3−d)εP, E+U = 4εP), and discrete derivative–integral operators; in no proof is time a dimension — every dynamical derivative runs over the event counter n and τ = n·tP. First layer, existence: a discrete derivative needs a positive unit — set it to zero and event, measurement and geometry are erased together (C1); nothingness is not a rival state, the minimal state is a single planckon (C2); a positive reserve cannot stay static (C3); «the first event out of nothingness» is outside the difference operator's domain — existence is unconditioned in event-time (C4). Second layer, step mechanics: phases rise only by equal-order binary fusion — hybrid addition breaks identity (C5); the bond axis is a two-step closed cycle under period-2 (C6); the bond event keeps the pair ledger at 8εP and saturates the channel (C7); the assembly tree closes in three generations, seven fusions, 8 = 2³ (C8). Third layer, closure: staying at d2 is thermal, not structural (D1); d4 is excluded three ways — fuel U(3) = 0 (C9), counting C(3,4) = 0 (C10), geometry 2 ≡ 0 (C11) — and three axes are proved by discrete Gauss and stability (C12–C14). Particles are configurations written onto the completed d3 texture; darkness is the texture/off-texture distinction (D2), the registering rows closing on the companion papers’ pure numbers Ω_DE/Ω_DM = 2.6128 and w₀ = −27/32, wₐ = −7/32. The axiom core is two items: time is not a dimension (B2), 2-fold cubic geometry (B4). A bridge section derives the continuum limit and the effective wave rows — E = pc exact along the axes, the lattice dispersion sin²(ωtP/2) = Σ sin²(kᵢℓP/2) as the falsifiable remainder — reads the built-in U(1) from bond-phase loop invariance, and states the SU(2)×SU(3), spinor and weak-field programme openly (Section 11). All derivations are confirmed by 77/77 algebraic checks and a 9-row numerical lattice bridge (Appendices A–B).