etaBCert
plain-language theorem explainer
Packages the RS η_B interval certificate: φ^{-44} lies in (5.5, 7.5)×10^{-10}, the Planck 2018 central value sits inside, 44 factors as chirality flip count times torsion gap, and the structural asymmetry is positive. Cosmologists citing the RS baryon-asymmetry prediction use this bundle. It is a structure instance that wires five already-proved lemmas into one record.
Claim. There is a verified certificate asserting: $\varphi^{-44} > 5.5\times 10^{-10}$ and $\varphi^{-44} < 7.5\times 10^{-10}$; the observed central value satisfies $5.5\times 10^{-10} < 6.1\times 10^{-10} < 7.5\times 10^{-10}$; the integer identity $44 = (\text{bit-flip count on axis }0)\times|\Delta\tau_{01}|$ holds; and the structural baryon-to-photon ratio is strictly positive.
background
Recognition Science predicts the baryon-to-photon ratio on the golden-ratio ladder as a power of $\varphi$. This module targets the concrete interval claim $\varphi^{-44}\in(5.5\times 10^{-10},7.5\times 10^{-10})$, which contains the Planck 2018 value $\eta_B=(6.10\pm 0.04)\times 10^{-10}$.
The integer 44 is not free: it equals the product of the Gray-code chirality flip count on the preferred axis and the absolute generation torsion gap $\Delta\tau_{12}$. The same 44 appears in the RS formula for $\alpha^{-1}$. Bounds on $\varphi^{\pm 44}$ are obtained from the Fibonacci closed form $\varphi^{44}=F_{44}\varphi+F_{43}$ with $F_{44}=701408733$, $F_{43}=433494437$ and a tight enclosure of $\varphi$.
Upstream pieces supply each field of the certificate: lower and upper bounds on $\varphi^{-44}$, a numeric check that $6.1\times 10^{-10}$ sits in the interval, the structural factorization of 44, and positivity of the structural $\eta_B$ (from $J_{CP}>0$).
proof idea
One-line structure construction. Each field of EtaBCert is filled by a named prior theorem: lower bound by the reciprocal of the $\varphi^{44}<1.6\times 10^9$ upper estimate; upper bound by the reciprocal of $\varphi^{44}>1.5\times 10^9$; observed-in-interval by a norm_num comparison; structural identity by native_decide on the bit-flip and torsion-gap definitions; positivity by unfolding the structural $\eta_B$ and using positivity of the Jarlskog invariant over $g_*$.
why it matters
This is the master interval certificate for the RS baryon-to-photon prediction in the cosmology layer. It closes the module's main results list by bundling the $\varphi^{-44}$ enclosure, observational containment, the structural "44 connection," and the sign of the asymmetry into a single exportable object.
The structural field ties baryogenesis to the same integer that governs the electromagnetic coupling via $\alpha^{-1}=44\pi,\exp(-w_8\ln\varphi/44\pi)$, so the certificate is a cross-domain consistency check rather than an isolated numerics claim. Positivity records the genuine derived content of the asymmetry (matter over antimatter) from $J_{CP}>0$. No downstream consumers are wired yet; the record is the terminal packaging step of the $\eta_B$ interval development.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.