eta_B_phi_scale
plain-language theorem explainer
The Recognition Science baryon-to-photon ratio is placed on the golden-ratio ladder at rung −44: η_B := φ^{−44}. Cosmologists and RS auditors cite this as the exact scale that enters the φ^{44}/φ^{45} balance and the baryon-asymmetry certificate. It is a one-line definition, not a derived equality.
Claim. Define the RS baryon-asymmetry scale by $\eta_B := \varphi^{-44}$, where $\varphi$ is the golden ratio (the self-similar fixed point forced by the Recognition Composition Law).
background
In electroweak baryogenesis the baryon-to-photon ratio takes the schematic form $\eta_B = (\varepsilon_{CP}/g_\star)\times$ washout, with every factor RS-derived: Jarlskog $J_{CP}>0$, $g_\star=106.75$ from SM content forced by $Q_3$, and the sphaleron-to-Hubble washout at the electroweak temperature.
The module’s structural claim is that the relevant ladder rung is $44=4\times 11$: generation-0 chirality flip count times the CW torsion gap $\Delta\tau_{12}$. The same product seeds the fine-structure formula $\alpha^{-1}=44\pi,e^{-w_8\ln\varphi/(44\pi)}$. Thus both $\alpha$ and $\eta_B$ sit on a common combinatorial skeleton.
Positive powers of $\varphi$ appear elsewhere as ladder scales (e.g. $\mathrm{scale}(k)=\varphi^k$). Here the negative integer exponent simply places $\eta_B$ on the reciprocal rung.
proof idea
Pure definitional assignment: unfold to $\varphi$ raised to the integer power $-44$. No lemmas, no tactics. Downstream positivity and comparison proofs unfold this name and apply zpow_pos / zpow_neg with $\varphi>1$.
why it matters
This name is the left-hand side of the module’s central identity $\eta_B\times\varphi^{45}=\varphi$ (eta_B_phi45_balance) and of the ratio form $\eta_B=\varphi/\varphi^{45}$. It is packaged into BaryonAsymmetryExactCert (rung product, $\varphi$-power identity, positivity) and into full_derivation_chain, which traces RCL $\to$ J-uniqueness (T5) $\to$ $\varphi$ forced (T6) $\to$ $D=3$ (T8) $\to$ Gray-code flips $\to$ torsion gap 11 $\to$ rung 44 $\to$ $\eta_B\approx\varphi^{-44}$. Higher-order baryon corrections (BaryonCorrectionCert, corrected_decomposition) reuse the same scale. The definition therefore anchors the claim that matter content sits exactly one $\varphi$-rung above the complementary $-44/45$ pair.
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