Black-Hole Collapse: Horizon, Compression Ceiling, Event-Time Freezing, and Evaporation — Derivative–Integral Proofs in Folding (Period-2) Cubic Geometry (ΨD)

In the standard picture, black-hole collapse ends in a singularity: density, curvature, and temperature run off to infinity, and the theory falls silent. This paper rebuilds the collapse from start to finish within the ΨD (Dimensional Flow Cosmology) framework and replaces the singular endpoint with a saturated core of finite Planck-scale density. The method is restricted to three tools: folding (period-2) cubic geometry, the energy–potential ledger (E (d) = (d+1) ·εP, U (d) = (3−d) ·εP, E + U = 4εP), and discrete derivative–integral operations. Time is not a dimension in any proof; every derivative is taken over the event counter n. Five theorems are proved. C1 (horizon): the escape condition is ledger algebra — the kinetic capacity of a unit mass is c²/2, the integral of the inverse-square flux law yields Gm/r, and setting them equal solves Rₛ = 2Gm/c²; 2953 m for a solar mass. C2 (compression ceiling): each cell carries at most one indivisible node — ρₘax = εP/ℓP³ = c⁷/ (ħG²), curvature ceiling 1/ℓP², force ceiling c⁴/G; a solar-mass collapse halts in a saturated core of radius rcore ≈ 4. 5×10⁻²³ m: a black hole from outside, a Planck-dense node packet inside. C3 (event-time freezing): relative to the outside counter, the event rate is throttled by the factor (1 − Rₛ/r) ^ (1/2) and drops to zero at the horizon; from inside, the free-fall integral is finite — the core forms in a finite number of event steps; the space-column mate of the counter rule — step densification — derives the light deflection 4Gm/ (c²b) at full coefficient (C3′). C4 (boundary counting): the horizon is a two-state node surface of ℓP² cells — N = 4πRₛ²/ℓP², counting capacity Scount = N·kB·ln2, and the coefficient is read off the ledger: 1/4 = U (d2) /4εP ⇒ S = kB·A/ (4ℓP²) ; T ∝ 1/m is derived. C5 (evaporation): dm/dτ = −α/m² solves in closed form — m³ decreases linearly and the lifetime is m₀³/ (3α) ∝ m₀³; the end is not a singular burst but a drain from the ceiling. The axiom core is two items: time is not a dimension (B2) and folding cubic geometry (B3) ; both are stated in full and motivated in Section 2. A correspondence section (Section 9) fixes the framework’s relation to standard physics: in the continuum reading the flux law becomes the Poisson equation, the counter–step pair (f, 1/f) is the static weak-field pair of the Schwarzschild geometry, and the semiclassical boundary laws are recovered at the structural level — what is not yet derived (the dynamical field equations, the temperature prefactor) is recorded as open. All derivations are verified by 31/31 programmatic checks (Appendix A).