Constants Gap Weight F Gap Lower Bound

The gap weight bound

The declaration fGapLowerBound is a rational number: 2993443258792019287026689 / 2500000000000000000000000, approximately 1.197377. In the Recognition Science framework, this number is a certified lower bound for the product of a weight and a logarithm: the gap weight w₈ multiplied by the natural logarithm of the golden ratio φ. The framework defines this product as f_gap, a quantity used in its alpha pipeline.

The weight w₈ itself is not a fitted parameter. It is a parameter-free closed form, approximately 2.49056927545, derived from a Parseval-normalized 64-cell projection of the discrete Fourier transform of the canonical φ-pattern on an 8-tick cycle. The integers in its closed form, (348, 210, 204, 130, 7), are forced by the algebra of the 8-tick structure, not chosen to match any measurement. The framework's machine-checked library proves this equality: w8_projected = w8_from_eight_tick, with zero axioms beyond the standard three.

The lower bound is paired with an upper bound, fGapUpperBound, and together they define a hypothesis: that f_gap lies strictly between these two rationals. This hypothesis is a checkpoint for the numerical computation, not a theorem. The lower bound itself is a definition, a precise statement of where the value sits, and the positivity of w₈ is the only related theorem in the pack. The bound does not, by itself, prove anything about the fine-structure constant α; it is a component in a larger pipeline, and the final comparison against measured α remains an empirical check, not a derivation.

What the declaration does not claim is equally important. It does not claim that the gap weight w₈ is derived from first principles in this file; that derivation is established elsewhere in the framework's library. It does not claim that the bound is tight or optimal, only that f_gap exceeds it. And it does not claim that the alpha pipeline is complete: the bound is a necessary numerical anchor, not a sufficient proof of the alpha value. The framework's own honesty rules require stating this plainly: the bound is a definitional checkpoint, and the alpha prediction itself remains an open comparison against measurement.

def fGapLowerBound : ℚ := 2993443258792019287026689 / 2500000000000000000000000

/-- The canonical gap weight `w₈` (parameter‑free, closed form).

This is the normalized projection weight of the gap onto the fundamental
8-tick basis. Numerically it is approximately `2.49056927545…`. -/
@[simp] noncomputable def w8_from_eight_tick : ℝ :=
  (348 + 210 * Real.sqrt 2 - (204 + 130 * Real.sqrt 2) * phi) / 7

/-- The canonical gap weight `w₈` (parameter‑free, closed form).

This is the normalized projection weight of the gap onto the fundamental
8-tick basis. Numerically it is approximately `2.49056927545…`. -/
@[simp] noncomputable def w8_from_eight_tick : ℝ :=
  (348 + 210 * Real.sqrt 2 - (204 + 130 * Real.sqrt 2) * phi) / 7

/-- Hypothesis for the certified numerical bounds for the gap weight. -/
def f_gap_bounds_hypothesis : Prop :=
  ((fGapLowerBound : ℚ) : ℝ) < f_gap ∧ f_gap < ((fGapUpperBound : ℚ) : ℝ)

What this page does not claim

The lower bound is not a proof of the fine-structure constant α. The bound is not claimed to be tight or optimal. The derivation of w₈ from first principles is not established in this file.

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/GapWeight.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.

Derived articles

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