Paper O: Analytical Structure of Sector-IV Asymptotic Behavior in the Four-Sector Cosmological Model— Derivation of Ω₀ ≈ 1/33, τ* = −τ₀/3, and α = 0.527 from First Principles

                                                      Abstract

We present a complete analytical framework for the asymptotic behavior of
Sector-IV in the Four-Sector Cosmological (FSC) model. The present-day value
ΩIV(0) ≈1/33, the transition log-scale-factor τ∗ = −τ0/3, and the power-law index
α = 0.527 are all derived analytically from the Friedmann equation and the FSC
decay parameters alone, without free parameters. Five theorems constitute the
complete structure: (O-1) closed-form derivation of Ω0 = 1/33 (accuracy 1.16%);
(O-2) geometric cube-root relation a∗3 = a0 (accuracy 0.08%); (O-3) ODE origin
of α = 0.527 (accuracy 0.08%); (O-4) closed-form α =
√3−1
2e (1 + λtot/H0) Ω−1/√3
M
involving radiation sound speed cs = 1/√3 and Euler’s number e (accuracy ≤1.0%);
(O-5) exact invariance of Ω0 under variation of the integration starting point τ∗,
establishing self-consistency of Theorem O-1, and identification of the exact value
Ω0 = 1/33 at Ω∗M = 0.303, within 1σ of the Planck 2018 measurement. The results
demonstrate that the FSC model encodes acoustic and gravitational memory of the
early universe into present-day dark-sector abundances through the radiation sound
speed cs = 1/√3.