eta_B_interval

This module certifies the Recognition Science interval prediction for the baryon-to-photon ratio η_B. The structural claim is that η_B is identified with φ^{-44}, where the exponent 44 factors as 4×11: the product of the flip-count on axis 0 and the torsion gap Δτ₁₂. The same integer 44 appears in the RS formula for α^{-1}, so electromagnetic coupling and baryon asymmetry share one structural source.

The numerical work rests on the Fibonacci closed form φ^n = F_n φ + F_{n-1}. With F_44 = 701408733 and F_43 = 433494437 and φ ∈ (1.61, 1.62), one obtains crude but sufficient bounds φ^{44} ∈ (1.5×10^9, 1.6×10^9), which invert to the stated window on φ^{-44}. Upstream lemmas phi_pow_neg44_lower and phi_pow_neg44_upper already discharge the two one-sided inequalities after rewriting the real power as a reciprocal of the natural power.

One-line term proof: the conjunction is assembled by pairing the two sibling theorems phi_pow_neg44_lower (φ^{-44} > 5.5×10^{-10}) and phi_pow_neg44_upper (φ^{-44} < 7.5×10^{-10}). No further arithmetic is performed here; both sides were already reduced via Real.rpow_neg, the identity linking real and natural powers of φ, and inversion of the coarser bounds φ^{44} > 1.5×10^9 and φ^{44} < 1.6×10^9.

This is the main interval statement of the η_B certificate module: it packages the two one-sided bounds into the single claim that φ^{-44} sits in (5.5, 7.5)×10^{-10}. The observed Planck 2018 value η_B = (6.10±0.04)×10^{-10} falls inside, which the sibling observed_eta_in_interval records. The exponent 44 is not free: module documentation identifies it with flip_count(0)×|Δτ₁₂|, the same structural integer that enters the RS expression for α^{-1}. In the broader framework this ties baryon asymmetry to the phi-ladder mass/coupling architecture (T6 φ fixed point) rather than to a separate high-scale CP-violating sector. No downstream dependents are recorded yet; the natural parent is the master certificate EtaBCert named in the module header.