Constants Phi Gt One Point Six One

The golden ratio in the framework

The golden ratio, usually written φ (phi), is the number that satisfies φ² = φ + 1. Its exact value is (1 + √5) / 2, approximately 1.6180339887. It is the proportion found in the regular pentagon, in the Fibonacci sequence as the limit of successive ratios, and in the spiral growth patterns of many plants. Euclid called it the extreme and mean ratio around 300 BCE, and it has been studied continuously since.

The number is irrational, meaning it cannot be written as a fraction of two integers. Its continued fraction representation is the simplest possible, an infinite string of ones: [1; 1, 1, 1, ...]. This simplicity is why it is the slowest-converging continued fraction and why it appears in problems about approximation and symmetry. The golden ratio is also the unique positive number whose reciprocal is exactly one less than itself, since 1/φ = φ - 1.

In Recognition Science, the framework models reality as a discrete record of events, called a ledger, and derives a forced cost function for recognition from five plain conditions. The framework's library of machine-checked formal theorems proves that the golden ratio is the unique self-similar scaling for this cost function. One consequence is a proved lower bound: the declaration phi_gt_onePointSixOne establishes that φ is greater than 1.6, a numerical fact that follows from the exact definition of φ and the classical inequality (1 + √5) / 2 > 1.6. This bound is not a new discovery about the number itself; it is a formal checkpoint in the framework's derivation chain, confirming that the golden ratio lies in the expected range before further theorems build on it.

The declaration does not claim that φ is exactly 1.6, nor that the framework invented the golden ratio. It also does not claim that the bound is tight; the framework's library separately proves φ is less than 1.62. The value of the declaration is organizational: it gives later theorems a named, machine-checked fact to cite, so the chain of reasoning that leads from the cost function to physical constants has a verified numerical foundation.

/-- 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]

/-- φ ≈ 1.618 (coarse upper bound used in some modules). -/
lemma phi_approx : phi < (1.62 : ℝ) := phi_lt_onePointSixTwo

What this page does not claim

The declaration does not claim φ equals 1.6 exactly. The declaration does not claim the framework discovered or defined the golden ratio. The declaration does not claim the bound is the tightest possible.

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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