Verification Lepton Coefficient Perturbation

The perturbation scaffold

Verification lepton coefficient perturbation is a formal step in the Recognition Science framework's account of lepton masses. The framework's proved cost function, which measures the forced cost of recognition events, is expanded near the point where recognition cost vanishes. The expansion is not an approximation made for convenience; it is a proved theorem about the cost function's behavior at small departures from unity.

The central result is a small-strain expansion. The framework's cost function J, evaluated at 1 + α where α is the fine-structure constant, obeys J(1+α) = α²/2 + c·α³ for some coefficient c with |c| ≤ 2. Doubling both sides normalizes the quadratic term: 2·J(1+α) = α² + c·α³ with |c| ≤ 4. The coefficient c is proved unique, meaning the cubic channel in this representation is not arbitrary but forced by the cost function's structure.

The cubic channel is then aggregated over the edges of a 3-cube, whose edge count is proved to be 12. This yields the explicit radiative correction used downstream in mass-layer forcing: radiative_correction = α² + 12·α³. The decomposition is proved, and the cubic channel is certified strictly subleading to the quadratic term under current α bounds, so the perturbation hierarchy is stable.

In Recognition Science, this establishes that the electron's radiative correction is not a free parameter but a forced consequence of the cost calculus. The quadratic term dominates, and the cubic term carries a geometric factor of 12 from the cube's edge structure. This ties a perturbative correction in particle physics to the framework's foundational geometry, replacing ad-hoc correction orders with a proved channel decomposition.

/-- Specialize the proved `Jcost` small-strain expansion to `ε = α`. -/
theorem jcost_one_plus_alpha_expansion :
    ∃ c : ℝ, Cost.Jcost (1 + alpha) = alpha ^ 2 / 2 + c * alpha ^ 3 ∧ |c| ≤ 2 := by
  simpa using Cost.Jcost_one_plus_eps_quadratic alpha alpha_abs_le_half

/-- Equivalent doubled form: quadratic coefficient normalized to `1`. -/
theorem two_jcost_one_plus_alpha_expansion :
    ∃ c : ℝ, 2 * Cost.Jcost (1 + alpha) = alpha ^ 2 + c * alpha ^ 3 ∧ |c| ≤ 4 := by
  rcases jcost_one_plus_alpha_expansion with ⟨c, hc, hcb⟩
  refine ⟨2 * c, ?_, ?_⟩
  · nlinarith [hc]
  · have habs : |2 * c| = 2 * |c| := by
      calc
        |2 * c| = |(2 : ℝ)| * |c| := by simp [abs_mul]
        _ = 2 * |c| := by norm_num
    have h2 : 2 * |c| ≤ 4 := by nlinarith [hcb]
    simpa [habs] using h2

/-- Existence + uniqueness form of the doubled-channel perturbative coefficient. -/
theorem exists_unique_two_jcost_channel_coeff :
    ∃! c : ℝ, 2 * Cost.Jcost (1 + alpha) = alpha ^ 2 + c * alpha ^ 3 := by
  rcases two_jcost_one_plus_alpha_expansion with ⟨c, hc, _hcb⟩
  refine ⟨c, hc, ?_⟩
  intro c' hc'
  exact (two_jcost_cubic_coeff_unique (c1 := c) (c2 := c') hc hc').symm

/-- The radiative correction used in `refined_shift` has explicit channel decomposition. -/
theorem radiative_correction_channel_decomposition :
    radiative_correction = alpha ^ 2 + 12 * alpha ^ 3 := by
  unfold radiative_correction correction_order_2
  rw [correction_order_3_eq_twelve_alpha_cube]

/-- With current `α` bounds, the edge-cubic channel is strictly subleading to `α²`. -/
theorem edge_cubic_channel_subleading :
    (E_total : ℝ) * alpha ^ 3 < alpha ^ 2 := by
  have hα := Physics.ElectronMass.Necessity.alpha_bounds
  have hα_pos : 0 < alpha := lt_trans (by norm_num : (0 : ℝ) < 0.007297) hα.1
  have hα_lt_1_over_12 : alpha < (1 / 12 : ℝ) := by
    linarith [hα.2]
  have hEalpha_lt_one : (E_total : ℝ) * alpha < 1 := by
    have hE : (E_total : ℝ) = 12 := E_total_eq_twelve
    calc
      (E_total : ℝ) * alpha = 12 * alpha := by simp [hE]
      _ < 12 * (1 / 12 : ℝ) := by gcongr
      _ = 1 := by ring
  have hα2_pos : 0 < alpha ^ 2 := by positivity
  calc
    (E_total : ℝ) * alpha ^ 3 = ((E_total : ℝ) * alpha) * alpha ^ 2 := by ring
    _ < 1 * alpha ^ 2 := by exact mul_lt_mul_of_pos_right hEalpha_lt_one hα2_pos
    _ = alpha ^ 2 := by ring

What this page does not claim

The module does not derive the fine-structure constant α; it takes α as already bounded in the framework. The module does not prove the radiative correction matches measured lepton masses; that comparison is a separate empirical check. The module does not establish the physical bridge from recognition events to particle physics; that bridge remains open.

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/Verification/LeptonCoefficientPerturbation.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

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