Foundation Logic Real Constants Phi L Gt One Point Five

A narrow bound, precisely

The golden ratio is the number that solves r² = r + 1, roughly 1.618. It appears wherever self-similar growth meets geometry: in the pentagon's diagonals, in the Fibonacci sequence's ratios, and in the spiral that bears its name. Its exact value is (1 + √5)/2, and it is irrational, meaning no fraction of whole numbers equals it.

A machine-checked library of formal theorems records a proof that this constant, under the name phiL, is greater than 1.5 and less than 1.62. The proof does not merely assert the bound; it derives it from the definition of the golden ratio itself, using the tools of formal logic. This is a narrow result about a number's position on the real line, not a statement about why the number matters.

The same library also proves that phiL is positive and greater than 1, and that its reciprocal power appears in a separate constant. These are simple consequences of the definition, each checked step by step. The bound between 1.5 and 1.62 is one such consequence, useful for pinning the value down without computing it exactly.

In Recognition Science, this bound appears as part of a larger structure. The framework models reality as a discrete ledger of recognition events, where each event has a forced cost. From that cost, a chain of theorems derives the golden ratio as a unique scaling constant, and from it, other constants follow. The phiL_gt_onePointFive theorem belongs to this framework's library, but it does not itself prove the framework's claims. It only certifies a numerical fact about the golden ratio within that library's formal system.

What the theorem does not claim is as important as what it does. It does not say the golden ratio is the only number with this property, nor does it say anything about the physical world. It does not assert that the framework's derivation of the golden ratio is correct, only that, given the framework's definitions, the number falls in this range. The bound is a piece of internal consistency, not an external discovery.

/-- Recovered lower numerical φ bound. -/
theorem phiL_gt_onePointFive : fromReal (1.5 : ℝ) < phiL := by
  rw [lt_iff_toReal_lt, toReal_fromReal, toReal_phiL]
  exact Constants.phi_gt_onePointFive

/-- 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 lower numerical φ bound. -/
theorem phiL_gt_onePointFive : fromReal (1.5 : ℝ) < phiL := by
  rw [lt_iff_toReal_lt, toReal_fromReal, toReal_phiL]
  exact Constants.phi_gt_onePointFive

What this page does not claim

The theorem does not prove that the golden ratio is the only number satisfying any particular property. The theorem does not make any claim about the physical world or about measurements. The theorem does not validate the framework's derivation of the golden ratio; it only certifies a numerical bound within the framework's definitions.

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)

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