Physics Alpha Running Correction Score Card Running Ratio Lt

The running ratio

The fine-structure constant, which measures the strength of electromagnetic interactions, is not truly constant. At higher energies, quantum effects from virtual particles screen the bare charge, so the measured value changes with the energy scale. This is the running of the coupling. The ratio of its inverse at the Z boson mass to its inverse at zero energy is a standard quantity in particle physics, and it is always less than one because the screening weakens the interaction.

In the Recognition Science framework, the inverse fine-structure constant at zero energy is not a free parameter but a predicted number, falling in a certified window between 137.030 and 137.039. The framework's machine-checked library of formal theorems proves this band and then combines it with the measured value at the Z mass to establish a precise band for the running ratio. The theorem running_ratio_lt proves the ratio is below 0.935, and its companion running_ratio_gt proves it is above 0.933. Together they pin the ratio to the interval (0.933, 0.935).

This band is a statement about a construction, not a measurement. The framework's predicted window for the inverse coupling is about 429,000 times wider than the CODATA 2022 measurement of 137.035999177(21), and every value inside that wide window is realized by some choice of the U(1) kinetic normalization. The framework's own theorem band_selects_no_coupling proves that no measurement could have contradicted the band, because it does not select a specific coupling at all. The construction value itself is excluded by the measurement at more than 30,000 sigma.

What the theorem establishes is therefore a consistency check, not a prediction that could fail. The particle content below the Z mass, three charged leptons and five light quarks, is treated as derived within the framework, and the theorem certifies that this content, together with the framework's zero-energy window, produces a running ratio that falls in the expected range. The honest verdict is plain: the framework's band holds the construction and the measurement at once, but it does not derive the fine-structure constant itself.

/-- The running ratio is below 0.935. -/
theorem running_ratio_lt : running_ratio < (0.935 : ℝ) := by
  unfold running_ratio alpha_inv_mz_pdg alpha_inv_0
  rw [div_lt_iff₀ (by linarith [Numerics.alphaInv_gt] : (0 : ℝ) < alphaInv)]
  calc (0.935 : ℝ) * alphaInv > 0.935 * 137.030 := by nlinarith [Numerics.alphaInv_gt]
    _ = 128.12305 := by norm_num
    _ > 127.951 := by norm_num

/-- α⁻¹(0) > 137.030. -/
theorem alpha_inv_0_gt : (137.030 : ℝ) < alpha_inv_0 :=
  Numerics.alphaInv_gt

/-- α⁻¹(0) < 137.039. -/
theorem alpha_inv_0_lt : alpha_inv_0 < (137.039 : ℝ) :=
  Numerics.alphaInv_lt

theorem alphaRunningCorrectionScoreCardCert_holds :
    Nonempty AlphaRunningCorrectionScoreCardCert :=
  ⟨{ alpha_0_band := ⟨alpha_inv_0_gt, alpha_inv_0_lt⟩
     ratio_lt_one := running_ratio_lt_one
     ratio_band := ⟨running_ratio_gt, running_ratio_lt⟩
     leptons := rfl
     quarks := rfl
     zero_params := zero_free_params }⟩

What this page does not claim

This page does not claim the framework derives the fine-structure constant. This page does not claim the running ratio band is a prediction that could have been contradicted by measurement. This page does not claim the framework's zero-energy window is a precise measurement of the inverse coupling.

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/Physics/AlphaRunningCorrectionScoreCard.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: