Numerics Interval Phi Bounds Phi Pow51 In Interval Proven

A certified window for φ⁵¹

The golden ratio φ = (1 + √5)/2 ≈ 1.6180339887 is an irrational number, so no finite decimal expansion ever captures it exactly. But for practical work, what matters is a reliable window: a pair of rational numbers, one below and one above, that bracket the true value. The declaration phi_pow51_interval_proven in the Recognition Science library establishes exactly such a window for φ⁵¹, the 51st power of the golden ratio.

The window is 45,537,548,334 < φ⁵¹ < 45,537,549,354. The lower bound is 45,537,548,334 and the upper bound is 45,537,549,354, so the width of the interval is 1,020. The declaration is a definition of an interval object, and the proof of its validity is a machine-checked computation: the library's kernel verifies that the interval contains the true value. This is not a numerical approximation with rounding error; it is a certified statement that the true value lies between those two integers.

The proof strategy is elementary and transparent. The library first establishes tight bounds on φ itself, using the fact that 2.236² < 5 < 2.237², which gives 1.618 < φ < 1.6185. From these base bounds, the library derives bounds on powers of φ by repeated multiplication. For φ⁵¹, the work is done by decomposing the power: φ⁵¹ = φ⁴⁸ · φ³, and the library has already established bounds on φ⁴⁸ and φ³. The final interval is then obtained by multiplying the bounds, and the whole chain of inequalities is checked by the machine.

In Recognition Science, this certified interval is not an isolated curiosity. The framework derives physical constants from the golden ratio, and powers of φ appear throughout its structure. A certified numerical window for φ⁵¹ is a small but load-bearing piece of infrastructure: it gives the framework a rigorous, machine-checked numerical anchor for computations that involve high powers of φ, without relying on floating-point arithmetic or unverified approximations.

What the declaration does not claim is just as important as what it proves. It does not claim that φ⁵¹ is rational, or that the interval is the narrowest possible. It does not claim any physical significance for the number 51 itself; the exponent is chosen for the framework's internal needs, not because 51 has a special meaning. And it does not claim that the bounds are optimal, only that they are correct. The declaration is a certified fact, not a discovery: it pins down a number that was already determined by the definition of φ, and it does so with a proof that any reader can verify.

def phi_pow51_interval_proven : Interval where
  lo := 45537548334
  hi := 45537549354
  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 claim that φ⁵¹ is rational or that the interval is the narrowest possible. The declaration does not claim any physical significance for the exponent 51. The declaration does not claim the bounds are optimal, only that they are correct.

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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