Foundation Logic Real Constants Phi L Lt One Point Six Two

The recovered golden ratio

The golden ratio, usually written φ, is the number that satisfies φ² = φ + 1, or approximately 1.6180339887. It appears throughout classical mathematics: in the regular pentagon, in the Fibonacci sequence as the limit of successive ratios, and in Euclid's "extreme and mean ratio" from around 300 BC. A standard way to compute it is (1 + √5)/2, which gives the decimal value directly.

In the Recognition Science framework, constants are defined on a separate number system called LogicReal, which mirrors the ordinary real line. The framework's library of machine-checked formal theorems defines phiL as (1 + √5)/2 in that system, and then proves a chain of bounds about it. The declaration phiL_lt_onePointSixTwo is one link in that chain: it states that φ is less than 1.62. The proof works by converting the statement back to the ordinary real numbers, where the corresponding bound was already established. This is a formal theorem, meaning a computer checked every step of the reasoning from the axioms of the logic.

The declaration does not claim that φ equals 1.62, nor that it is exactly 1.6180339887. It only establishes an upper bound, a simple inequality. It also does not derive the golden ratio from first principles; it assumes the definition (1 + √5)/2 and proves a property of that number. The bound itself is not new mathematics: it is a trivial consequence of the decimal expansion. What the declaration adds is a formal, machine-verified statement of that fact within the framework's own number system, so that later theorems can rely on it without re-checking the arithmetic.

In the framework's broader project, this bound is a small but necessary piece. The golden ratio appears throughout the framework's forcing chain, where it is derived as the unique self-similar scaling. Having a formal proof that φ lies in a specific interval, here below 1.62, lets other theorems use that fact with confidence. The declaration is a building block, not a standalone discovery.

/-- Recovered upper numerical φ bound. -/
theorem phiL_lt_onePointSixTwo : phiL < fromReal (1.62 : ℝ) := by
  rw [lt_iff_toReal_lt, toReal_fromReal, toReal_phiL]
  exact Constants.phi_lt_onePointSixTwo

/-- Recovered upper numerical φ bound. -/
theorem phiL_lt_onePointSixTwo : phiL < fromReal (1.62 : ℝ) := by
  rw [lt_iff_toReal_lt, toReal_fromReal, toReal_phiL]
  exact Constants.phi_lt_onePointSixTwo

What this page does not claim

This declaration does not claim that φ equals 1.62 or any specific decimal value. It does not derive the golden ratio from first principles; it assumes the definition (1 + √5)/2. It does not claim the bound is new mathematics, only that it is formally verified.

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/LogicRealConstants.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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