Foundation Recognition Science Logo5

The Logo Module

The golden ratio φ, about 1.618, is the number that solves r² = r + 1. It appears throughout mathematics: in the regular pentagon, in the Fibonacci sequence 0, 1, 1, 2, 3, 5, 8, where each term is the sum of the previous two, and in the continued fraction 1 + 1/(1 + 1/(1 + ...)). Euclid called it the extreme and mean ratio around 300 BCE. The number is irrational, meaning it cannot be written as a fraction of two whole numbers.

The cost function J(x) = (x + 1/x)/2 - 1 measures a kind of mismatch between two positive numbers. It is zero when x = 1, and it grows as x moves away from 1 in either direction. At x = φ, the cost equals φ - 3/2, a positive number about 0.118. This particular curve, plotted for x > 0, is what Recognition Science calls its logo: one curve that the framework's central theorem forces as the unique cost function satisfying five plain conditions.

In Recognition Science, the module named logo5 establishes three general facts about this curve. First, for any nonzero real r, the cost of comparing r to itself is zero: J(r/r) = J(1) = 0. Second, for any two positive numbers m and e, the cost J(m/e) is never negative. Third, the threshold φ - 3/2 is positive. These are proved in a machine-checked library of formal theorems, meaning a computer verified each step from axioms.

The module defines a ledger, a discrete record of events, only in a generic way: it takes two real numbers m and e and applies the cost to their ratio. The proofs hold for any such numbers. What the module does not do is say what m and e mean for any particular subject. The docstring notes that the module was meant to become a theorem about a specific topic, but the definition never connects m and e to that topic. So the module is a template, not a result about physics or any other field.

The practical consequence is that this logo module is a clean but empty shell: it proves the cost curve's basic properties, but it does not establish anything about the world. A reader who wants a theorem about a subject must first define m and e in that subject's own terms. Until then, the module is a reminder of the framework's central shape, not a claim about any particular system.

def canonicalThreshold : ℝ := phi - 3 / 2

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]

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

What this page does not claim

This module proves any fact about a specific physical or mathematical subject. The golden ratio's appearance here is not a derivation of φ from the cost function; it is a definitional choice in the module. The module does not establish the uniqueness of J; that is a separate theorem in the framework's library.

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/RecognitionScience_Logo5.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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