lexicon_ratio_lt_one
plain-language theorem explainer
The theorem establishes that the active fraction of the lexicon is strictly less than one. Researchers modeling token activation via Fibonacci recurrences in cognitive science or linguistics would cite this bound when confirming the steady-state ratio lies in the open unit interval. The proof is a one-line wrapper that reduces directly to the inverse golden ratio inequality from the PhiEmergence module.
Claim. The active fraction $r$ of the lexicon satisfies $r < 1$, where $r := 1/φ$ and $φ$ is the golden ratio.
background
In the Recognition Science model of lexicon evolution, active and passive tokens follow a σ-conserving recurrence derived from the active-edge budget. With $a_n$ the count of active tokens and $p_n$ the count of passive tokens, the rules are $a_{n+1} = a_n + p_n$ and $p_{n+1} = a_n$. The total lexicon size $L_n$ grows asymptotically as $φ^n$, and the steady-state active fraction converges to the unique positive solution of $x + x^2 = 1$, which equals $φ^{-1}$.
proof idea
The proof is a one-line wrapper that applies the theorem phi_inv_lt_one from the PhiEmergence module after the definition lexicon_ratio := phi_inv has been unfolded.
why it matters
This bound is collected into the lexiconRatioCert certificate together with positivity, the numerical band, the Fibonacci fixed-point identity, and the phi-squared relation. It completes the derivation that the active fraction equals the inverse golden ratio forced by the recognition cost and the self-similar fixed point of the recurrence. The result sits downstream of the phi emergence theorems and supports applications of the Recognition Composition Law to linguistic structures.
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