phi10_gt_100
plain-language theorem explainer
The inequality φ¹⁰ > 100 follows from rewriting φ¹⁰ as the Fibonacci expression 55φ + 34 and applying the lower bound φ > 1.61. Researchers deriving the weak nuclear force constant G_F from Recognition Science would cite this numerical threshold when certifying the Fermi constant in RS-native units. The proof is a one-line wrapper that rewrites via the Fibonacci identity then invokes linear arithmetic.
Claim. $φ^{10} > 100$
background
In the Weak Nuclear Force from RS module the golden ratio φ satisfies the closed-form relation φ¹⁰ = 55φ + 34, obtained from the Fibonacci recurrence that follows from φ being the self-similar fixed point. The upstream lemma phi_gt_onePointSixOne supplies the tighter bound φ > 1.61, derived from the quadratic minimal polynomial of φ. The local setting states G_F = φ^{-10} / (8 m_W²) together with the five canonical weak decay types, all expressed in RS-native units where c = 1 and ħ = φ^{-5}.
proof idea
The proof is a one-line wrapper. It rewrites the left-hand side with the theorem phi10_fibonacci to obtain 55φ + 34, then applies linarith using the inequality φ > 1.61 supplied by phi_gt_onePointSixOne to conclude the comparison with 100.
why it matters
The bound is referenced inside the definition weakForceCert to certify the weak-force parameters, including the five decay types and the numerical value of φ¹⁰. It supplies the concrete numerical anchor required by the Recognition Science derivation of G_F from the phi-ladder and the eight-tick octave structure. The result closes the numerical verification step in the A1 SM Depth section of the framework.
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