phi5_gt_10
plain-language theorem explainer
Golden ratio to the fifth power exceeds 10. Linguists modeling word half-lives in corpora cite the bound to confirm that the observed eightfold ratio fits inside the phi-ladder interval around 11.1. The proof iterates the quadratic identity to express phi^5 as 5*phi + 3, then applies the lower bound phi > 1.5 via linear arithmetic.
Claim. $phi^5 > 10$ where $phi = (1 + sqrt(5))/2$ is the golden ratio.
background
The Lexical Decay from Phi-Ladder module treats word longevity as scaling with recognition frequency rung on the phi-ladder. Half-life at rung k equals T0 * phi^k, so the ratio across five rungs is exactly phi^5. The module states that the empirical ratio 6000/750 = 8 lies inside the J-cost band around phi^5 approx 11.1. This theorem supplies the strict lower estimate phi^5 > 10 required for that interval claim. It rests on the identity phi^2 = phi + 1 from Constants.phi_sq_eq and the bound phi > 1.5 from Constants.phi_gt_onePointFive.
proof idea
The tactic proof first obtains phi^2 = phi + 1 via phi_sq_eq. It then derives the closed forms phi^3 = 2phi + 1, phi^4 = 3phi + 2, and phi^5 = 5*phi + 3 by three successive nlinarith steps. The final linarith call combines these expressions with the hypothesis phi > 1.5 to conclude the target inequality.
why it matters
The result supplies the phi5_lower field inside lexicalDecayCert, the definition that certifies compatibility between the empirical half-life ratio and the phi-ladder prediction. It closes the lower side of the (10,12) interval that the module aligns with the canonical J-band in Recognition Science. The parent lexicalDecayCert in turn anchors the linguistics application of the phi-ladder scaling.
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