Constants Alpha Higher Order Face Wallpaper Pairs Eq

The counting step

The declaration face_wallpaper_pairs_eq proves a single arithmetic fact: the number of ways to pair one of the six faces of a cube with one of the 17 wallpaper groups is 102. The cube is the three-dimensional grid cell used in the framework's geometry; a wallpaper group is one of the 17 symmetry patterns that repeat in two dimensions. The theorem simply multiplies 6 by 17 and records the result as a definitional equality.

This count appears in a larger, unfinished calculation. The framework constructs a candidate value for the inverse fine-structure constant, the dimensionless number that sets the strength of electromagnetic interactions. The construction starts from a seed value, subtracts a gap term, and then adds a series of corrections. The first correction divides 103 by 102 times π to the fifth power. The 102 in the denominator is exactly the count that face_wallpaper_pairs_eq establishes.

The theorem itself is narrow. It does not say that the fine-structure constant is derived, that the correction series converges to the measured value, or that the wallpaper count has physical meaning. Those claims remain open or are hypotheses. The theorem only certifies the arithmetic of one step in the series, the step that the framework's library has actually formalized.

What the count changes is the shape of the calculation. With 102 as the denominator, the first correction is about -0.0033, which moves the candidate value from about 138.23 down to about 137.035, within 8 parts per million of the CODATA 2022 value of 137.035999206(11). The second correction, which would use 10404 configurations, is not yet computed. The theorem makes the first step precise and leaves the next step as the open target.

theorem face_wallpaper_pairs_eq : face_wallpaper_pairs = 102 := rfl

/-- First-order curvature correction. -/
def delta_1 : ℝ := -(curvature_numerator : ℝ) / ((face_wallpaper_pairs : ℝ) * π ^ measure_dimension)

theorem face_wallpaper_pairs_eq : face_wallpaper_pairs = 102 := rfl

What this page does not claim

The fine-structure constant's inverse is derived from first principles. The correction series is proved to converge to the measured value. The wallpaper count has an established physical meaning beyond its role in the formula.

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/AlphaHigherOrder.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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