Constants Alpha Higher Order Curvature Numerator Eq

The curvature numerator

The fine-structure constant α is a fundamental number in physics, roughly 1/137.036, that sets the strength of electromagnetic interactions. The Recognition Science framework constructs a candidate value for its reciprocal, α⁻¹, starting from a geometric seed of 4π × 11 ≈ 138.230. This construction is not a derivation of α; it is a construction whose seed identification remains an open problem. The framework then applies a gap weight and a series of corrections to bring the candidate closer to the measured value.

One of these corrections is the first-order curvature term, δ₁ = -103/(102π⁵) ≈ -0.00330. The declaration curvature_numerator_eq establishes that the numerator 103 is not an arbitrary fitted parameter. It is the sum of two combinatorial counts from a three-dimensional cube: the number of face-wallpaper pairs, which is 6 faces times 17 wallpaper groups, equals 102, plus one active edge, giving 103. This is a proved theorem in the framework's machine-checked library of formal theorems, meaning the counting is verified by the computer.

The declaration does not claim that this correction makes the construction match experiment. The docstring states that the additive formula lands at about -8 ppm from the CODATA value, while an exponential variant lands at about +6 ppm, leaving an open residual. The convergence of the full series of corrections to the CODATA value is a hypothesis, not a theorem. The computation of the second-order correction δ₂ is explicitly open, and is described as the key deliverable for testing the framework's ability to close the gap.

In plain terms, the declaration proves a piece of bookkeeping: the number 103 arises from a specific counting of geometric features on a cube. It does not prove that this counting is physically meaningful, nor that the series converges to the measured value of α⁻¹. Those remain open targets. The framework's library shows the counting is internally consistent; the leap from that consistency to a prediction of a physical constant is not made by this declaration.

theorem curvature_numerator_eq : curvature_numerator = 103 := rfl

theorem face_wallpaper_pairs_eq : face_wallpaper_pairs = 102 := rfl

theorem passive_edges_eq : passive_edges = 11 := rfl

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

What this page does not claim

The declaration does not prove that the curvature correction is physically meaningful or that the series converges to the measured value. The construction of α⁻¹ is not a derivation of the fine-structure constant; its seed identification remains open. The convergence of the correction series to the CODATA value is a hypothesis, not a theorem.

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)

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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: