The Bridge Extended to Cosmology

The Bridge Extended to Cosmology

The bridge equation δ = α⁻¹R + φ⁻², established in prior corpus work as the linear relation holding four cosmological constants (Λ, H₀, G, vacuum floor) at four positions of the partition σ + σ² = 1 scaled by the Dirichlet class-number factor 1/√5 of ℚ(√5), is extended in this paper to three further cosmological readings. The same algebraic skeleton that produces the late-universe constants is shown to organise three cosmic-history readings at three even-power positions of σ. A central new result is that σ⁶ appears in three independent cosmological contexts, all geometrically tied to the radiation/photon sector, with one of these appearances yielding a falsifiable prediction for the matter-radiation equality redshift.

The paper develops the bridge extension across eight sections plus appendix, with two falsifiable kill conditions, one programmatic consistency requirement, a four-mechanism catalogue for the appearances of π in cosmology, and a Claims and Limits table that distinguishes new results from structural conjectures and from corpus citations.

Results:

Recombination at σ²: The dimensionless thermal parameter ξ_rec = E_Ry/(k_B T_rec) equals α⁻¹σ² = 52.3431, identifying the Peebles 3LA recombination temperature T_rec = 3016.4 K at z_rec = 1105.7. Matches the corpus value from Deriving the Emergence of Hydrogen (z_rec = 1107, computed via standard Peebles 3LA atomic physics independently of any Pentagon Physics algebra) to 0.11%.

Matter+DE conformal time at σ⁴: At the φ-attractor (Ω_Λ = σ, Ω_m = σ²), the Friedmann conformal-time integral admits the closed form η₀^mDE = √φ · (1/3) σ^(−1/3) B_σ(1/6, 1/3) = 2.99474723, which approximates π − σ⁴ = 2.99569463 to 0.031%.

Three independent appearances of σ⁶ in cosmology, all geometrically related to the radiation/photon sector:

• σ⁶ in the photon temperature denominator T_CMB = α⁴ m_e c² / [2 k_B (π − σ⁶)] (from The Universe is Governed by Geometry)
• σ⁶ as the radiation-era boundary correction in the bound-shifted matter+DE conformal time: η₀^mDE[σ¹⁴/4, 1] ≈ π − σ⁴ − σ⁶ to 0.032%
• σ⁶ identifying the matter-radiation equality redshift through the algebraic identity 4·a_eq·φ¹⁴ = 1

Falsifiable prediction for matter-radiation equality: 1 + z_eq = 4 φ¹⁴ ≈ 4 L₁₄ = 3372, agreeing with CLASS + HyRec at Planck 2018 cosmological parameters (1 + z_eq = 3380) to 0.24%, and with the Planck nominal acoustic-peak fit (3400 ± 21) to 0.83%.

The 19%/19% Ω-offset preservation: Both Ω_m and Ω_r are observed 19% below their algebraic attractor values (σ² and σ¹⁶/4 respectively), but the ratio Ω_r/Ω_m is preserved — making the algebraic identification a_eq = σ¹⁴/4 invariant under the Ω-convergence transient discussed in Spiral Convergence Theory.

The class-number bridge of ℚ(√5): The recombination-temperature parameter (ξ_rec = α⁻¹σ²) and the cosmological vacuum floor (δ_floor = α⁻¹σ²/√5 + σ² ≈ 23.79) carry the same dimensionless combination α⁻¹σ², separated only by the Dirichlet anchor 1/√5 = L(1, χ₅)/(2 ln φ). The recombination surface and the cosmological vacuum sit on either side of the class-number bridge of ℚ(√5).

Four-mechanism catalogue for π in cosmology: Eight independent cosmological appearances of π classified under four mechanisms — closure angles (T_CMB denominator, Gibbons-Hawking horizon temperature), radial cosmology (Friedmann 8πG/3, Hubble volume), phase-space measures (Maxwell-Boltzmann partition function, Stefan-Boltzmann via the Bose integral ∫₀^∞ x³/(eˣ−1) dx = π⁴/15), and number-theoretic embedding (conformal-time approximant π − σ⁴, Dirichlet anchor in ℚ(ζ₅)).

The η_B independence cross-check: The bridge identification of the recombination temperature does not constrain the baryon-to-photon ratio. The independent corpus derivation in Baryon Asymmetry from the Golden Partition gives η_B = 6.105 × 10⁻¹⁰, matching the Planck 2018 measurement (6.104 ± 0.058) × 10⁻¹⁰ at +0.02σ. The two corpus results are independent and Planck-consistent at their respective surfaces.

The four-domain consolidation: Cosmological constants (Λ, H₀, G, vacuum floor), the recombination temperature, axiom structure (σ + σ² = 1), and the vacuum floor are presented as four lenses on the same algebraic skeleton, with explicit table mapping each domain to its algebraic form.