correction_term_rung
plain-language theorem explainer
The product of the leading baryon-asymmetry scale φ^{-44} and the per-cycle washout rate φ^{-8} equals φ^{-52}. Cosmologists computing the first-order 8-tick correction to η_B cite this identity to place the absolute correction term on the φ-ladder. The proof unfolds the two pure-power definitions and adds the integer exponents.
Claim. The product of the RS baryon-asymmetry scale $\phi^{-44}$ and the washout rate per 8-tick cycle $\phi^{-8}$ equals $\phi^{-52}$.
background
Recognition Science places the baryon-to-photon ratio on a fixed rung of the golden-ratio ladder. The leading-order scale is $\eta_B = \phi^{-44}\approx 6.376\times 10^{-10}$, which exceeds the Planck 2018 CMB central value by roughly 4.5%. Because RS has no free parameters, the gap cannot be tuned; the present module studies the first subleading correction from 8-tick defect propagation during the electroweak phase transition.
The washout rate per 8-tick cycle is defined as $\delta=\phi^{-8}$ (one full octave rung). The product of the leading scale and this rate is the absolute size of the first-order correction term before it is assembled into the multiplicative factor $(1-\delta)$. Both quantities are pure integer powers of $\phi$, so their product is again a pure power on the same ladder.
proof idea
Unfold the two definitions (leading scale $=\phi^{-44}$, washout rate $=\phi^{-8}$). Rewrite the product by the integer power-addition law for nonzero bases ($\phi\neq 0$), then discharge the arithmetic identity $-44+(-8)=-52$ by numeric normalization.
why it matters
The identity is consumed by the baryon-correction certificate, which packages the claim that the first-order prediction is $\phi^{-44}(1-\phi^{-8})$ and thereby halves the 4.5% gap to the CMB value. That certificate formalizes the 8-tick washout hypothesis: sphalerons remain active for roughly $N_{\mathrm{sph}}\approx\phi^8$ recognition cycles, each reducing the baryon excess by one octave rung (T7). The result sits in the cosmology sector of the RS zero-parameter program and sharpens the comparison of $\eta_B$ against precision data. The underlying washout mechanism remains a hypothesis whose explicit falsifier is a measured $\eta_B$ outside $[6.0,6.5]\times 10^{-10}$ at $>3\sigma$.
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