canonicalThreshold
plain-language theorem explainer
The canonical threshold is the fixed real cutoff φ − 3/2 used when certifying that Fibonacci successive ratios approach the golden ratio. Anyone working the RS Fibonacci–φ limit structural theorem cites it as the comparison level against the domain cost. The declaration is a pure definition: the arithmetic expression itself, with no proof obligation.
Claim. Define the canonical threshold to be the real number $\varphi - 3/2$, where $\varphi$ denotes the golden ratio (the self-similar fixed point of the Recognition $J$-cost).
background
The module treats the classical limit $F_{n+1}/F_n \to \varphi$ as the attractor of the $J$-cost recursion in Recognition Science. Here $\varphi$ is the unique positive fixed point of the self-similarity map forced at T6 of the unified forcing chain, equivalently the limit of successive Fibonacci ratios.
The $J$-cost is $J(x)=(x+x^{-1})/2-1$ (also $\cosh(\log x)-1$). Domain cost in this file measures how far a candidate ratio sits from that attractor. The numerical cutoff $\varphi-3/2\approx 0.118$ sits strictly between $0$ and the scale of early Fibonacci defects, so it is a natural positive comparison level for a certificate that the ratios eventually stay inside a $J$-ball about $\varphi$.
Sibling facts in the same module establish non-negativity of domain cost and positivity of this threshold; the present declaration only names the constant.
proof idea
Pure definition: the real constant is introduced by the closed-form expression $\varphi - 3/2$. No tactics, no lemmas, no reduction. Downstream positivity or comparison lemmas simply unfold this abbreviation.
why it matters
Inside the RS Fibonacci–φ limit module (status: structural theorem, zero sorry), this constant is the named numerical gate for the limit certificate. It ties the classical Fibonacci convergence story to the T6 forcing of $\varphi$ as the $J$-cost self-similar fixed point. Parent consumers are the certificate inhabitants and positivity lemmas in the same file (e.g. the inhabited Fib-limit certificate). It does not itself close any open forcing step; it only supplies the cutoff those certificates compare against.
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