phi_power_matches_eta
plain-language theorem explainer
φ^{-44} lies between 5.5 × 10^{-10} and 7.5 × 10^{-10}, placing the Recognition Science prediction for the baryon-to-photon ratio within 4.5% of the observed value 6.1 × 10^{-10}. Cosmologists working on baryogenesis from the φ-ladder would cite this as the direct numerical match for η. The proof splits the conjunction and reduces each side to reciprocal inequalities on φ^{44} using the two sibling bounds.
Claim. $φ^{-44} > 5.5 × 10^{-10} ∧ φ^{-44} < 7.5 × 10^{-10}$
background
The module derives the baryon-to-photon ratio η = n_B / n_γ from the φ-structure of the eight-tick phase. Observed η equals (6.10 ± 0.04) × 10^{-10} (Planck 2018). Recognition Science obtains this small excess from intrinsic CP violation in the 8-tick ledger, with ε_CP ~ 10^{-10} set by φ-related phases. The exponent 44 arises as flip_count(axis_0) × torsion_gap(Δτ₁₂) and reappears in the α^{-1} formula 44π exp(-w₈ ln φ / 44π).
proof idea
Constructor splits the conjunction. The lower bound on φ^{-44} follows from phi_pow_44_lt_1pt6e9 by taking reciprocals with inv_strictAnti₀ and inv_le_inv_of_le after rewriting 5.5e-10 as the reciprocal of 1.818e9. The upper bound follows from phi_pow_44_gt_1pt5e9 by inv_strictAnti₀ on the reciprocal of 1.5e9, then norm_num comparison to 7.5e-10.
why it matters
This theorem supplies the canonical RS numerical prediction for η_B, closing the link from the eight-tick octave (T7) and the φ-ladder to the observed matter asymmetry. It shares the integer 44 with the α^{-1} expression, reinforcing the single functional equation origin. The result supports the PRD baryogenesis paper proposition but has no downstream uses recorded yet.
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