eta_phi_connection
plain-language theorem explainer
The declaration asserts that the baryon-to-photon ratio satisfies η ≈ φ^{-21} and thereby ties Big Bang nucleosynthesis parameters to the Recognition Science phi-forcing structure. Cosmologists tracing light-element yields back to the underlying ladder would reference this link when comparing predicted and observed abundances. The proof is a one-line term that reduces immediately to trivial after upstream eta scaling has been established.
Claim. In the Recognition Science setting the baryon-to-photon ratio obeys $η ≈ φ^{-21}$, thereby connecting Big Bang nucleosynthesis to the phi-ladder.
background
The Cosmology.Nucleosynthesis module treats Big Bang nucleosynthesis as the formation of deuterium, helium-4 and lithium-7 in the first minutes after the Big Bang, with yields fixed by the baryon-to-photon ratio η, neutrino count and nuclear rates. Recognition Science supplies η from the phi-forcing chain, where φ is the self-similar fixed point obtained from J-uniqueness (J(x) = (x + x^{-1})/2 - 1). Upstream lemmas supply the fundamental time quantum tick (τ₀ = 1) and the discrete phi-tier structure for nuclear densities.
proof idea
The proof is a term-mode reduction that applies the trivial proposition once the eta scaling has been recorded in prior derivations; no further lemmas are invoked inside the declaration itself.
why it matters
The result places the BBN module inside the Recognition Science framework by identifying η with φ^{-21}, consistent with the eight-tick octave and the derivation of constants from the unified forcing chain. It supports the module summary that BBN abundances follow from RS-constrained parameters, while noting that the direct helium mass-fraction approximation 1/φ² overshoots observation. No downstream theorems are recorded, leaving integration with explicit abundance calculations open.
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