fib_10_eq
plain-language theorem explainer
The equality asserts that the tenth Fibonacci number equals 55. Inflation modelers in Recognition Science cite it to anchor the hypothesized e-folding count N_e = 55 on the phi-ladder, linking to the spectral index 1 - 2/55. The proof is a one-line reflexivity that evaluates the Fibonacci recursion directly.
Claim. The tenth Fibonacci number satisfies $F_{10} = 55$.
background
The JCostInflaton module identifies the inflaton potential with the J-cost in log coordinates, so that G(t) = cosh(t) - 1. This form follows from the Recognition Composition Law applied to multiplicative recognizers and yields the slow-roll parameters epsilon and eta from the second derivative at the vacuum. The equality fib_10_eq supplies the concrete integer 55 that the module comment conjectures equals N_e = M_pass + eta_B_rung_abs, thereby connecting the phi-ladder Fibonacci sequence to the predicted n_s = 1 - 2/55.
proof idea
The proof is a term-mode reflexivity that unfolds the recursive definition of the Fibonacci sequence at index 10 and reduces directly to the numeral 55.
why it matters
This identity grounds the N_e = 55 hypothesis inside the master certificate InflationFromJCostCert and the n_s_from_jcost result. It sits at the intersection of the phi-forcing chain (T6 fixed point) and the eight-tick octave, supplying the numerical rung that produces the observed spectral index within 0.13 percent of Planck 2018. The remaining open task is to derive the same 55 from the duration of the J-cost slow-roll trajectory rather than postulate it.
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