halfLifeRatio
plain-language theorem explainer
Consecutive half-lives on the phi-ladder stand in the golden ratio. Linguists fitting word persistence data from corpora to frequency classes cite this scaling. The proof reduces the ratio directly to phi by unfolding the power definition and applying algebraic identities for exponents.
Claim. For any natural number $k$, the ratio of the half-life at level $k+1$ to the half-life at level $k$ equals $phi$, where the half-life function is given by $phi^k$.
background
The module derives lexical half-lives from the phi-ladder in Recognition Science. Half-life at rung k is defined as phi raised to k. This follows the RS claim that word longevity scales with the recognition frequency rung, yielding T(k) = T0 * phi^k and thus a constant ratio phi between consecutive rungs. The upstream halfLife definition supplies the explicit power expression used here.
proof idea
The proof unfolds the halfLife definition, establishes positivity of the base to justify the division step, rewrites the numerator via the successor power rule, and concludes with ring simplification to obtain phi.
why it matters
This result populates the phi_ratio component of the lexicalDecayCert certificate. It formalizes the per-rung scaling on the phi-ladder that underlies the module's match between observed half-life ratios and phi^5. The certificate in turn supports the broader claim that lexical decay follows the Recognition Science phi-ladder structure.
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