Constants Alpha Higher Order Exp Minus Add Pos

The exponential correction

The fine-structure constant α is a dimensionless number, about 1/137, that sets the strength of electromagnetic interactions between charged particles. Its measured reciprocal value, 137.035999206(11), comes from CODATA 2022. Recognition Science (RS) does not derive this constant; its exact value is an open boundary datum. Instead, RS constructs an approximation from geometric ingredients, and the declaration exp_minus_add_pos concerns one small step in that construction.

The construction starts with a seed value α_seed = 4π × 11 ≈ 138.230. From this it subtracts a gap weight f_gap = w₈ · ln φ ≈ 1.198, where φ is the golden ratio, and adds a first-order correction δ₁ = −103/(102π⁵) ≈ −0.00330. The additive formula α_seed − f_gap + δ₁ lands near 137.035, about 8 parts per million below the measured value. The exponential formula, α_seed · exp(−f_gap/α_seed), lands near 137.037, about 6 parts per million above. The declaration exp_minus_add_pos establishes, in the machine-checked library, that the exponential expression is positive: a formal guarantee that this particular correction term stays above zero.

The positivity result is a theorem in the framework's library, proved from the definitions of the seed, the gap weight, and the exponential function. It does not assert that the exponential formula equals the measured constant, nor that it is closer than the additive one. The library also proves the cube combinatorics behind δ₁: a cube has 6 faces, 12 edges, 8 vertices, and 17 wallpaper groups, giving 102 face-wallpaper pairs and a numerator of 103. These counts are exact, but they are ingredients, not the final answer.

What remains open is the second-order correction δ₂, which the library explicitly marks as the key deliverable. The series of corrections is proved alternating and convergent, and convergence to the CODATA value is a hypothesis, not a theorem. The positivity of the exponential term is a small but necessary piece: it keeps the correction within the band where the series can be compared with measurement.

/-- The exponential overshoot above CODATA. -/
def exponential_residual (w8_val : ℝ) : ℝ :=
  alpha_seed * Real.exp (-(f_gap w8_val) / alpha_seed) - CODATA_alpha_inv

/-- 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 itself is not derived; its exact value remains open. The exponential formula does not equal the measured constant, only approximates it. Convergence to CODATA is a hypothesis, not a proved 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)

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