balanced_count_sum_tendsto_gaussian

Plain-English statement

Actual allN normalized counting pushforward of signed sum divided by sqrtN; N0 Dirac0 and successor weak convergence to mean0 variance1 real Gaussian. Actual coordinate moments and finite CF factorization internally produced. Pinned Mathlib scalar CLT and Levy reused; no entropy, full flip-energy, GaussianLSI/T2 or cost upgrade. Independently verified 4d9e71a6b835b54453a5cfd2d132f9a51f66f69c; paper main and actual cost/composition remain open.

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Lean statement

theorem balanced_count_sum_tendsto_gaussian :
    ∃ laws : ℕ → ProbabilityMeasure ℝ,
      (∀ n : ℕ, (laws n : Measure ℝ) =
        (((Fintype.card (Fin n → Bool) : ℝ≥0∞)⁻¹ •
          (Measure.count : Measure (Fin n → Bool))).map
          (fun ε : Fin n → Bool => (Real.sqrt (n : ℝ))⁻¹ *
            ∑ j : Fin n, if ε j then (1 : ℝ) else -1))) ∧
      (laws 0 : Measure ℝ) = Measure.dirac 0 ∧
      Tendsto (fun n : ℕ => laws (n + 1)) atTop
        (𝓝 (⟨gaussianReal 0 1, inferInstance⟩ : ProbabilityMeasure ℝ)) := by
  refine ⟨actualLaw, ?_, actualLaw_zero, ?_⟩
  · intro n
    rfl
  · exact actualLaw_tendsto.comp (tendsto_add_atTop_nat 1)

end AutoSamplingTheory.TechnicalLemmas.Probability.BalancedRademacherCLT

Open AutoSamplingTheory/TechnicalLemmas/Probability/BalancedRademacherCLT.lean:116published source at ed709a7f320d

Proof architecture

Actual SPHMC omitted Gaussian LSI background; no SALD or second-route admission

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