Constants One Lt Phi Point Six One

The golden ratio's lower bound

The golden ratio φ is the number (1 + √5)/2 ≈ 1.61803, the classical solution to the equation r² = r + 1. It appears in pentagon geometry, Fibonacci sequences, and spiral growth patterns. The declaration one_lt_phiPointSixOne is a formal, machine-checked lemma stating that 1.6 < φ. This is a narrow but exact numerical bound: it proves the golden ratio exceeds 1.6, nothing more and nothing less.

The lemma is part of a larger library of formal theorems in the Recognition Science framework. That framework models reality as a discrete record of recognition events, with a forced cost function J(x) = (x + 1/x)/2 − 1. From this cost function, the framework derives the golden ratio as the unique self-similar scaling, then an eight-tick cycle, then three spatial dimensions. The bound 1.6 < φ is a supporting lemma in this chain: it helps establish that φ is greater than 1, which is needed for later proofs about the framework's constants.

In Recognition Science, the framework defines φ exactly as in classical mathematics, not as an approximation. The lemma one_lt_phiPointSixOne is proved from the classical definition using standard real-number arithmetic. It does not introduce a new value of φ, nor does it claim anything about the golden ratio's role in nature. It is a precise, verified statement that the number lies in a particular range.

What this lemma does not claim is equally important. It does not say that φ is approximately 1.618, though that is true. It does not assert that the golden ratio is the only number satisfying any property, nor does it connect φ to any physical constant. The declaration is a single, isolated inequality: 1.6 < φ. Its value lies in being a verified building block, not in making a grand statement on its own.

/-- Alias matching parallel-work naming convention. -/
lemma one_lt_phiPointSixOne : (1.6 : ℝ) < phi := by linarith [phi_gt_onePointSixOne]

/-- Golden ratio φ as a concrete real. -/
noncomputable def phi : ℝ := (1 + Real.sqrt 5) / 2

/-- Alias matching parallel-work naming convention. -/
lemma one_lt_phiPointSixOne : (1.6 : ℝ) < phi := by linarith [phi_gt_onePointSixOne]

What this page does not claim

The lemma does not state that φ is approximately 1.618, only that it is greater than 1.6. The lemma does not assert any connection between φ and physical constants like hbar or G. The lemma does not prove that φ is the unique solution to any equation on its own.

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

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