Perturbation Theory for the Bounce: Two Parts Built, One Part Proved Absent, One Part Left Conditional

Paper #131 of my corpus, Inflation, Quantified, priced the largest single debt in its range: this framework's bounce has no perturbation theory. It inventoried four missing parts — (1) a background solution, (2) linearized dynamics, (3) a bounce initial-condition rule, (4) a transfer to observables — and found all four absent. This paper works that debt, part by part, and returns a split verdict. Part 1 is built, conditionally. The frame's own record does print a Friedmann equation in its variables — H² = 8π G/3(ρ - ρ²/ρA + ρC), with the Aldred floor ρA = 9.72×10⁵⁴ g/cm³ — and its dust case integrates in closed form: ρ(t) = ρA/(1+(t/τb)²), a(t) = aA(1+(t/τb)²)1/3, with τb = 2.86×10⁻²⁵ s. The build carries three premises in full: the equation lives in the flat, homogeneous, isotropic (FLRW) dress, and identifying its scale factor with the bubble radius of the bounce picture is an added assumption, not a derivation — the membrane's own stretch law $R(t)$ remains unowned, exactly as the companion debt shared with paper #132 records; the floor's value is the dust spin-fluid pin, with its domain limit printed above; and the bounce stands at its printed numbers only under the coherence condition my corpus derived elsewhere — the bounce exists if and only if the spin-coherence number at the bounce point is N O(1–10²), at its published numbers if and only if N ≈ 1–2, and under any fixed-physical-length coherence there is no bounce. Part 2 is built, with one sub-step fenced. In longitudinal gauge the linearized Einstein equations close with the effective source the background already owns; the tensor sector is complete and regular across the bounce, with computed transfer matrices of determinant exactly 1; the scalar sector is derived and its one degeneracy is mapped, not hidden — at ρ = ρA/2 the effective fluid's enthalpy and density perturbation vanish together, its formal sound speed diverges, and yet the curvature perturbation in bare variables, ζ = Φ - δ/3, is finite there and at the bounce, because the vanishing factor cancels exactly. Part 3 is proved absent. No rule the frame prints determines the modes' state at the bounce: the gate's forced identity transmission is a statement about background quantities, not a mode-by-mode rule, and in any case an identity in one perturbation variable is not an identity in another; a vacuum rule would need a pre-bounce phase, a field to quantize, and a duration, none of which the frame supplies; and the inheritance the frame does print is a statement about composition and spin, not a spectrum — the missing input is a whole function of wavenumber, the same marking-and-profile law my Harrison–Zel'dovich paper already owes. Supplying that function by declaration is the author's door; this paper names it and stops. Part 4 is built as a conditional and fenced as a number. Given any post-bounce curvature spectrum, the transfer to Aₛ, nₛ and r is standard and is written out; but the transfer manufactures no tilt and no amplitude, so the frame still predicts none of the three numbers. The spectrum half of paper #131 therefore stays unproduced — no longer because the machinery is missing, but because its one input is missing, and that is a different, smaller, and proved absence. Grades, part by part: Part 1 — DERIVED-INDEPENDENT (conditional). Part 2 — EXTENSION. Part 3 — DERIVED-INDEPENDENT (an obstruction, at full value). Part 4 — EXTENSION as a map; its numbers NOT PRODUCED.