actual_phase_gibbs_normalization

Plain-English statement

- Positive finite normalization and exact product-law identification for the actual target phase energy. Integrability and all density identities are derived internally; no normalizer or minimizer is an additional hypothesis.

Scope guard. This card records a compiled local declaration. Its mathematical scope is exactly the Lean statement below; the Registry note and source correspondence may describe motivation but do not strengthen it.

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Lean statement

theorem actual_phase_gibbs_normalization
    {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
    [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
    {V : E → ℝ} {α β : ℝ≥0} {η : ℝ}
    (hα : 0 < (α : ℝ)) (hαβ : α ≤ β) (hV : ContDiff ℝ 2 V)
    (hH : ∀ x v : E,
      (α : ℝ) * ‖v‖ ^ 2 ≤ (fderiv ℝ (fderiv ℝ V) x v) v ∧
      (fderiv ℝ (fderiv ℝ V) x v) v ≤ (β : ℝ) * ‖v‖ ^ 2)
    (hη : 0 < η) (hβη : (β : ℝ) * η ≤ 1) : actual_phase_normalization_statement hα hαβ hV hH hη hβη := by

  dsimp only [actual_phase_normalization_statement]
  intro y
  let Vy : E → ℝ := fun x => V x + ‖x - y‖ ^ 2 / (2 * η)
  let U : E × E → ℝ := fun z => V z.1 + ‖z.1 - y‖ ^ 2 / (2 * η) + ‖z.2‖ ^ 2 / 2
  let f : E → ℝ := fun x => Real.exp (-Vy x)
  let g : E → ℝ := fun p => Real.exp (-‖p‖ ^ 2 / 2)
  let C : ℝ := (Real.sqrt (2 * Real.pi)) ^ Module.finrank ℝ E
  let Z : ℝ := ∫ x, f x ∂(volume : Measure E)
  let ell : Measure (E × E) := (volume : Measure E).prod (volume : Measure E)
  let Zp : ℝ≥0∞ := ∫⁻ z, ENNReal.ofReal (Real.exp (-U z)) ∂ell
  change Integrable f volume ∧ 0 < Z ∧ Zp = ENNReal.ofReal (Z * C) ∧
    0 < Zp ∧ Zp < ⊤ ∧
    IsProbabilityMeasure (ell.withDensity (fun z => Zp⁻¹ * ENNReal.ofReal (Real.exp (-U z)))) ∧
    ell.withDensity (fun z => Zp⁻¹ * ENNReal.ofReal (Real.exp (-U z))) =
      ((volume : Measure E).tilted (fun x => -Vy x)).prod (stdGaussian E)
  have hsc := AutoSamplingTheory.TechnicalLemmas.Analysis.HessianStrongConvexity.strongConvexOn_univ_of_fderiv2_lower hV (fun x v => (hH x v).1)
  have hbase := AutoSamplingTheory.TechnicalLemmas.Analysis.StrongConvexGibbsIntegrability.integrable_exp_neg_of_strongConvexOn hα (hV.differentiable (by norm_num)) hsc
  have hVy : Continuous Vy := hV.continuous.add (((continuous_id.sub continuous_const).norm.pow 2).div_const (2 * η))
  have hfc : Continuous f := Real.continuous_exp.comp hVy.neg
  have hgc : Continuous g := Real.continuous_exp.comp ((continuous_norm.pow 2).neg.div_const 2)
  have hf : Integrable f (volume : Measure E) := by
    refine hbase.mono' hfc.aestronglyMeasurable ?_
    filter_upwards with x
    rw [Real.norm_eq_abs, abs_of_pos (Real.exp_pos _)]
    apply Real.exp_le_exp.mpr
    dsimp [Vy]
    have : 0 ≤ ‖x - y‖ ^ 2 / (2 * η) := by positivity
    linarith
  have hZ : 0 < Z := integral_exp_pos hf
  have hC : 0 < C := by dsimp [C]; positivity
  have hCg : (∫⁻ p : E, ENNReal.ofReal (g p) ∂volume) = ENNReal.ofReal C := by
    have h := AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.lintegral_exp_neg_mul_norm_sq_eq (E := E) (a := (1 / 2 : ℝ)) (by norm_num)
    have hn : (Real.pi / (1 / 2 : ℝ)) ^ ((Module.finrank ℝ E : ℝ) / 2) = C := by
      dsimp [C]
      rw [Real.sqrt_eq_rpow, ← Real.rpow_mul_natCast (by positivity)]
      congr 1 <;> ring
    rw [hn] at h
    convert h using 1
    congr 1
    funext p
    congr 2
    dsimp [g]
    ring
  have hw (z : E × E) : Real.exp (-U z) = f z.1 * g z.2 := by
    dsimp [U, f, g, Vy]
    rw [← Real.exp_add]
    congr 1
    ring
  have hphase : Zp = ENNReal.ofReal (Z * C) := by
    calc
      Zp = ∫⁻ z : E × E, ENNReal.ofReal (f z.1) * ENNReal.ofReal (g z.2) ∂ell := by
        apply lintegral_congr
        intro z
        rw [hw, ENNReal.ofReal_mul (Real.exp_nonneg _)]
      _ = (∫⁻ x : E, ENNReal.ofReal (f x) ∂volume) * ∫⁻ p : E, ENNReal.ofReal (g p) ∂volume :=
        lintegral_prod_mul hfc.measurable.ennreal_ofReal.aemeasurable hgc.measurable.ennreal_ofReal.aemeasurable
      _ = ENNReal.ofReal (Z * C) := by
        rw [← ofReal_integral_eq_lintegral_ofReal hf (ae_of_all _ (fun x => Real.exp_nonneg _)), hCg, ← ENNReal.ofReal_mul hZ.le]
  have hpos : 0 < Zp := by rw [hphase]; exact ENNReal.ofReal_pos.mpr (mul_pos hZ hC)
  have hfin : Zp < ⊤ := by rw [hphase]; exact ENNReal.ofReal_lt_top
  have hprob : IsProbabilityMeasure (ell.withDensity (fun z => Zp⁻¹ * ENNReal.ofReal (Real.exp (-U z)))) :=
    AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs.isProbabilityMeasure_withDensity_normalized_gibbs ell U hpos.ne' hfin.ne
  refine ⟨hf, hZ, hphase, hpos, hfin, hprob, ?_⟩
  have hgamma : stdGaussian E = (volume : Measure E).withDensity (fun p => ENNReal.ofReal (C⁻¹ * g p)) := by
    have h := AutoSamplingTheory.TechnicalLemmas.Measure.IsotropicGaussianDensity.map_sqrt_smul_stdGaussian_eq_withDensity (E := E) 1 (by norm_num)
    have hid : (fun z : E => Real.sqrt 1 • z) = id := by funext z; simp
    rw [hid, Measure.map_id] at h
    simpa only [mul_one, inv_pow, C, g] using h
  rw [hgamma, Measure.tilted, prod_withDensity ((hfc.measurable.div_const Z).ennreal_ofReal) ((hgc.measurable.const_mul C⁻¹).ennreal_ofReal)]
  apply withDensity_congr_ae
  filter_upwards with z
  rw [hphase, hw, ENNReal.ofReal_mul hZ.le, ENNReal.ofReal_mul (Real.exp_nonneg _), ENNReal.ofReal_div_of_pos hZ, ENNReal.ofReal_mul (inv_nonneg.mpr hC.le), ENNReal.ofReal_inv_of_pos hC]
  rw [ENNReal.mul_inv (Or.inr ENNReal.ofReal_ne_top) (Or.inl ENNReal.ofReal_ne_top)]
  simp only [div_eq_mul_inv, f]
  ac_rfl

#print axioms actual_phase_gibbs_normalization
end
end AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPhaseNormalization

Open AutoSamplingTheory/ExampleCases/ProximalBPS/ActualPhaseNormalization.lean:61published source at ed709a7f320d

Proof architecture

Actual conditional position-momentum phase law; no SALD claim

Lean proof walkthrough

Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
`intro` introduces quantified hypotheses into the local proof context.
`have` creates a named intermediate mathematical fact.
`calc` records an equality or inequality chain matching a paper calculation.
`rw` rewrites by an established identity.
`simp` normalizes through registered definitional and theorem rewrites.
`apply` reduces the goal to the hypotheses of a reusable theorem.
`refine` instantiates a reusable theorem while leaving explicit subgoals.

Why the statement has this shape

The declaration is kept at the reusable level recorded by its Registry tags and direct consumers. Explicit measures, spaces, wrappers, and regularity hypotheses expose interfaces that paper notation often infers. A theorem card explains those interfaces but never widens the compiled statement.

Hidden assumptions and non-claims

Measurability is represented explicitly or must be supplied by a dependency.
Integrability is an input or proved output; a displayed integral alone does not supply it.
Totalized `fderiv` values must not be read as a differentiability theorem.
Genuine differentiability is localized to the hypotheses shown in the Lean statement.
Almost-everywhere hypotheses depend on the stated measure and representative.
Density statements retain normalization and absolute-continuity prerequisites.
A Gibbs expression is not automatically a probability law or an invariant law.