Numerics Interval Phi Bounds Phi Inv5 In Interval Proven

A certified interval for φ⁻⁵

The golden ratio φ = (1 + √5)/2 ≈ 1.618 is one of the oldest named numbers in mathematics, studied by Euclid as the division of a line into extreme and mean ratio. Its reciprocal 1/φ ≈ 0.618 and higher powers appear throughout geometry, from pentagon diagonals to spiral growth. A recent machine-checked library of formal theorems adds a certified numerical bound: the fifth power of the reciprocal, φ⁻⁵, lies strictly between 0.089 and 0.091. The declaration phi_inv5_interval_proven records this interval, with its lower and upper endpoints given as exact rational numbers 89/1000 and 91/1000.

The proof strategy is elementary and transparent. It starts from the definition of φ and the inequality 2.236 < √5 < 2.237, which follows by squaring both sides. This yields 1.618 < φ < 1.6185. From there, the library derives bounds on powers of φ and its reciprocal by repeated multiplication and comparison. The interval for φ⁻⁵ is not obtained by floating-point approximation but by rational arithmetic with exact inequalities, so the result carries no rounding error. The same module proves similar certified intervals for φ⁻¹, φ⁻³, φ⁵, and φ⁸, each with exact rational endpoints.

In the Recognition Science framework, this numerical fact earns its keep. The framework derives physical constants from a forcing chain that begins with a cost function and ends with the golden ratio as a unique self-similar scaling. From φ, it derives a natural unit system in which the reduced Planck constant is φ⁻⁵. A rigorous bound on that value is therefore not a curiosity but a load-bearing check: it confirms that the framework's proposed unit scale sits in a narrow, proven range. The interval 0.089 to 0.091, with width 0.002, is tight enough to be useful for subsequent calculations that depend on this constant.

What the declaration does not claim is just as important. It does not assert that φ⁻⁵ equals any particular decimal, only that it lies in the stated interval. It does not connect φ⁻⁵ to any physical measurement or experimental value; that comparison is a separate empirical question. The bound is a pure mathematical statement about a real number, proved from the definition of φ and the axioms of the real numbers. It says nothing about why φ appears in physics, only that if it does, this particular power is confined to a proven range.

The practical payoff is that any later derivation in the framework that uses φ⁻⁵ as a constant can cite this interval as a certified foundation. A reader who wants to check the bound can reproduce it with pencil and paper: square 2.236 to see it is less than 5, square 2.237 to see it is greater, and propagate the inequalities. The machine-checked proof simply guarantees that no step in that propagation was skipped or misstated.

/-- Interval containing (φ⁻¹)⁵ - PROVEN -/
def phi_inv5_interval_proven : Interval where
  lo := 89 / 1000
  hi := 91 / 1000
  valid := by norm_num

/-- Interval containing (φ⁻¹)⁵ - PROVEN -/
def phi_inv5_interval_proven : Interval where
  lo := 89 / 1000
  hi := 91 / 1000
  valid := by norm_num

/-- 2.236² < 5 -/
theorem sq_2236_lt_5 : (2.236 : ℝ)^2 < 5 := by norm_num

/-- 5 < 2.237² -/
theorem five_lt_sq_2237 : (5 : ℝ) < (2.237 : ℝ)^2 := by norm_num

What this page does not claim

The declaration does not assert any specific decimal value for φ⁻⁵, only that it lies in the interval. It does not connect φ⁻⁵ to any physical measurement or experimental constant. It does not explain why φ appears in physics, only that this power is bounded.

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/Numerics/Interval/PhiBounds.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.

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