phiContinuedFraction
The golden ratio admits the infinite continued fraction with all partial quotients equal to 1. This definition records a finite prefix of exactly ten ones for use in summations that relate φ to Euler's number e. Researchers deriving constants from φ-forcing and J-cost exponentials would cite the list when approximating continued-fraction expressions. The definition is a direct literal assignment with no computation or lemmas.
claimThe finite prefix of the continued fraction for the golden ratio is the list of partial quotients $[1,1,1,1,1,1,1,1,1,1]$.
background
The module derives Euler's number from φ-summations and 8-tick probability normalization. J-cost is the non-negative cost of a recognition event, obtained as the derived cost of a multiplicative recognizer's comparator on positive ratios. Probability of a configuration is the squared norm of its amplitude vector. The exponential base e appears in distributions of the form P ∝ exp(-J/J₀) because only the exponential is invariant under differentiation.
proof idea
The definition is a direct literal assignment of the ten-element list of ones. No lemmas or tactics are applied; the value is supplied as a constant for later use in φ-related summations.
why it matters in Recognition Science
The definition supplies the continued-fraction coefficients for φ that the module invokes when exploring links between e and φ through J-cost exponentials and 8-tick normalization. It fills the φ-related continued-fraction slot listed in the module target. No downstream theorems reference it, so its role in larger derivations remains open.
scope and limits
- Does not prove that the infinite continued fraction converges to φ.
- Does not derive any summation formula for e.
- Does not connect the list to probability distributions or J-cost.
- Does not invoke the forcing chain T0-T8 or Recognition Composition Law.
formal statement (Lean)
130def phiContinuedFraction : List ℕ := [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
proof body
Definition body.
131
132/-! ## The J-Cost Connection -/
133
134/-- In RS, e appears in probability distributions:
135
136 Boltzmann: P ∝ exp(-E/kT)
137 J-cost: P ∝ exp(-J/J₀)
138
139 The exponential (base e) is fundamental for probability normalization.
140
141 Why e specifically? Because:
142 d/dx e^x = e^x
143
144 Only exponential maintains shape under differentiation. -/