IndisputableMonolith.Mathematics.Fibonacci_Phi_Limit_RS
Mathematical support module linking Fibonacci ratio convergence to the golden ratio φ with Recognition Science cost and threshold certificates. Anyone citing the φ-ladder, self-similar fixed point, or rung spacing would land here. It defines a domain cost, a canonical positive threshold, and an inhabited Fib-limit certificate bundle rather than a single deep theorem.
claimPackage for the classical limit $\lim_{n\to\infty} F_{n+1}/F_n = \varphi$ (golden ratio) together with an RS domain cost $C$, a canonical positive threshold $t_*>0$, and an inhabited certificate that the Fibonacci–$\varphi$ limit sits inside the RS cost/threshold framework.
background
Recognition Science forces $\varphi$ as the unique self-similar fixed point (forcing step T6) and builds the mass and coupling ladders on powers of $\varphi$. Classically, $\varphi=(1+\sqrt{5})/2$ is also the limit of consecutive Fibonacci ratios, so a clean bridge from discrete Fibonacci data to the RS $\varphi$-ladder is useful bookkeeping.
The module sits in the Mathematics domain and imports RS Constants (native tick $\tau_0=1$) and Cost (the J-cost calculus whose unique generator is $J(x)=(x+x^{-1})/2-1$). Sibling objects introduce a domain cost functional, prove it is nonnegative and agrees with pointwise evaluation, fix a canonical positive threshold, and wrap these facts in a Fib-limit certificate type that is inhabited.
proof idea
Definition-and-certificate module, not a single deep proof. It introduces the domain cost and canonical threshold as defs, then supplies short lemmas: evaluation agreement, nonnegativity of the cost, and positivity of the threshold. The certificate record bundles those facts; inhabitation is discharged by assembling the preceding lemmas. Heavier Fibonacci-limit analysis is left to Mathlib or upstream classical results; this file only wires them into RS cost language.
why it matters in Recognition Science
Gives a named, reusable certificate that the Fibonacci route to $\varphi$ is compatible with RS cost and threshold conventions. Downstream φ-ladder, rung, and self-similarity arguments can cite the inhabited certificate instead of re-proving nonnegativity or threshold positivity. No direct used-by edges are recorded yet; the module is infrastructure for T6-style φ uniqueness and for any mass or coupling formula that counts Fibonacci-adjacent rungs. It does not itself force φ from the Recognition Composition Law; it only packages the classical limit inside RS units.
scope and limits
- Does not prove φ uniqueness from the Recognition Composition Law or forcing chain T5–T6.
- Does not derive mass, coupling, or eight-tick statements from the Fib limit.
- Does not replace Mathlib’s classical Fibonacci-ratio limit theorems.
- Does not assert numerical bounds on α or dimensional forcing D=3.
- Does not supply a computational closed form beyond the certificate interface.