phi_pow_44_upper
plain-language theorem explainer
The golden ratio raised to the 44th power is strictly less than 1.6×10⁹. Cosmologists certifying the RS baryon-to-photon interval cite this bound when converting φ^44 into a lower bound on φ^(-44). The proof rewrites the power via the Fibonacci closed form and finishes by linear arithmetic from φ < 1.62.
Claim. Let $\varphi = (1+\sqrt{5})/2$. Then $\varphi^{44} < 1.6 \times 10^{9}$.
background
The module certifies the Recognition Science prediction that the baryon-to-photon ratio sits at $\varphi^{-44}$, inside the observational window $(5.5, 7.5)\times 10^{-10}$. The integer 44 is structural: $44 = 4 \times 11$, the product of the axis-0 flip count and the torsion gap $\Delta\tau_{12}$, the same 44 that appears in the $\alpha^{-1}$ formula.
Powers of $\varphi$ admit the exact Fibonacci identity $\varphi^n = F_n\varphi + F_{n-1}$. For $n=44$ this is $\varphi^{44} = 701408733,\varphi + 433494437$, proved as the sibling lemma that expands $\varphi^{44}$ via $F_{44}$ and $F_{43}$. A separate constants lemma supplies the tight decimal bound $\varphi < 1.62$ (from $\sqrt{5} < 2.24$).
Together these reduce a transcendental comparison to a purely rational inequality that nlinarith can discharge.
proof idea
One short tactic proof. First rewrite $\varphi^{44}$ by the Fibonacci identity lemma, obtaining $701408733,\varphi + 433494437$. Introduce the constants fact $\varphi < 1.62$. Then nlinarith closes the strict inequality against $1.6\times 10^9$ from the linear upper envelope $701408733\cdot 1.62 + 433494437 < 1.6\times 10^9$.
why it matters
This is the upper half of the $\varphi^{44}$ sandwich used throughout the $\eta_B$ certificate. Its sole direct consumer is the lower bound on the reciprocal power: $\varphi^{-44} > 5.5\times 10^{-10}$, which inverts this inequality after converting the natural power to a real power. That reciprocal bound, paired with the matching upper reciprocal bound, yields the closed interval certificate $\varphi^{-44}\in(5.5,7.5)\times 10^{-10}$, inside which the Planck 2018 value $(6.10\pm 0.04)\times 10^{-10}$ falls.
In the broader framework the exponent 44 is not free: it is the same chirality-times-torsion integer that organizes $\alpha^{-1}$. Pinning $\varphi^{44}$ by a machine-checked decimal therefore locks both the electromagnetic coupling scale and the baryon asymmetry to one structural rung on the $\varphi$-ladder.
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