Constants Gap Weight In

The gap weight

In the Recognition Science framework, the gap weight is a number used to measure the size of a gap in an eight-step cycle. The framework works with a discrete record of events, called a ledger, and the gap weight is the weight assigned to the gap in that ledger's fundamental pattern. Its value is approximately 2.49056927545.

The key claim is that this number is not chosen freely. It is derived from a parameter-free closed form: (348 + 210√2 − (204 + 130√2)φ) / 7, where φ is the golden ratio. The integers 348, 210, 204, 130, and 7 are not fitted; they emerge from the algebra of the framework's eight-tick structure. The derivation starts with a φ-pattern, applies a standard discrete Fourier transform, weights the modes by a discrete derivative spectrum, normalizes by Parseval's theorem, and scales by a 64-cell interface. Each step is forced by the framework's structure, and the final equality is a machine-checked theorem in its library of formal proofs.

The framework uses this gap weight in a function f_gap = w₈ · ln(φ), which appears in its pipeline for the fine-structure constant α. The framework does not claim that this derivation alone determines α's measured value. The expression lands within about 5.6 parts per million of the measured value with nothing fitted, but its seed is an identification, not a derived coupling. Exact α remains an open target.

The gap weight is also accompanied by certified rational bounds: fGapLowerBound and fGapUpperBound define an interval that the framework hypothesizes contains f_gap. This is a hypothesis, not a theorem, and the bounds are stated as a proposition to be checked.

/-- 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

/-- 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

What this page does not claim

The gap weight is not a measured physical constant. The derivation does not prove the exact value of the fine-structure constant α. The integers in the closed form are not chosen to fit any experimental data.

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)

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