phi_zpow_neg8_upper
plain-language theorem explainer
The bound phi to the negative eighth power less than 0.02137 is established to constrain the washout factor in the baryon asymmetry prefactor. Cosmologists working within the Recognition Science framework cite this numerical result when fixing the interval for c_RS in the eta_B formula. The proof is a short tactic sequence that rewrites the negative exponent as an inverse and applies the known lower bound on phi to the eighth through inversion and a numerical comparison.
Claim. $phi^{-8} < 0.02137$
background
The module derives the order-one prefactor c_RS in the baryogenesis formula eta_B = c times algebraic content forced by the integration-gap rung. The interpretive form is c_RS = (1 - phi^{-8})^2, with each factor tied to one washout channel carrying an integration-gap rung of fermionic degrees of freedom at D = 3. The fundamental time quantum is the tick, defined as 1 in RS-native units and serving as the base period for the eight-tick octave.
proof idea
The tactic proof first rewrites the left-hand side via the lemma phi_zpow_neg8_eq_inv to obtain the reciprocal of phi to the eighth. It invokes the upstream theorem phi_pow_8_lower to obtain phi^8 > 46.81 together with a positivity fact. Inversion of the inequality produces the reciprocal bound, which is compared to 0.02137 by a norm_num step; linarith closes the chain.
why it matters
This result supplies the upper limit required by the downstream theorem one_minus_phi_neg8_lower and thereby pins the numerical band for the washout prefactor c_RS. It realizes the eight-tick octave structure in the cosmology setting by quantifying the per-tick suppression on the phi ladder. The derived eta_B interval contains the Planck 2018 measurement while leaving the squared interpretive form as a hypothesis pending a first-principles Boltzmann derivation.
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