Foundation Phi Continued Fraction Rs

The continued fraction of phi

The golden ratio phi, about 1.618, is the number that solves r squared equals r plus 1. Its continued fraction is the simplest possible repeating one: phi equals 1 plus 1 over (1 plus 1 over (1 plus ...)). Continued fractions are a way to write a number as a whole number plus a fraction, then repeat. The golden ratio's version uses only ones, which makes it the slowest to converge among all continued fractions. Because of that, phi is often called the most irrational number: it is the hardest to approximate well by a simple fraction. This property appears in nature in the spirals of sunflowers and pinecones, and in art in the proportions of the Parthenon.

The continued fraction of phi was known to Euclid around 300 BC through his construction of the extreme and mean ratio, and later mathematicians such as Johannes Kepler in the 1600s noted its connection to the Fibonacci sequence. The Fibonacci numbers, where each is the sum of the previous two, have ratios that approach phi as they grow. The continued fraction also shows that phi is irrational, since it never terminates. These classical facts are what most encyclopedia entries cover, and they are the foundation for what follows.

In Recognition Science, the framework models a recognition event as a discrete record of a comparison, and a cost as a number measuring how far two quantities are from being equal. The framework's central theorem proves that any cost function satisfying five plain conditions must equal J(x) = (x + 1/x)/2 minus 1. This function has a unique stable fixed point at phi, meaning that when the recognition recursion reaches phi, it stays there. The module named foundation phi continued fraction rs in the framework's machine-checked library of formal theorems defines a cost for a ratio m over e, and proves three general facts: the cost is zero when m equals e, it is never negative for positive inputs, and phi minus 3/2 is positive, which is a threshold used in the framework.

What the module does not do is prove anything specific to phi as a subject. The cost is defined as J(m/e) without any reference to the golden ratio, so the theorems are general properties of that cost, not results about phi itself. The research note in the module says the idea was to show phi is special, but the formal proof does not yet connect the cost to the continued fraction. That connection remains a target for future work. The module is honest about this: it states the general theorems and flags the intended subject as not yet proved.

For a reader, the takeaway is that the framework has a proved cost function, and that phi appears as a stable point of that cost, but the specific claim that phi's continued fraction makes it special in the framework is not yet established. The classical mathematics of phi's continued fraction is solid and centuries old; the framework's specific application to it is an open question.

def canonicalThreshold : ℝ := phi - 3 / 2

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

What this page does not claim

The module proves no property specific to phi's continued fraction; it only proves general facts about the cost function. The framework does not derive the continued fraction of phi from its axioms; that remains open. The claim that phi is the most irrational number is a classical result, not a framework theorem.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/Phi_Continued_Fraction_RS.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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