quadratic_feature_law_gradient_angular_transport_control

Plain-English statement

Actual ambient feature derivatives and visible operator bounds produce genuine mean integrability, potential gradient, tangent uniform bound, finite angular coupling cost and joint law/point control. Concrete feature-model inputs must still be produced. Fresh independent source/compiler and exact candidate audits accepted. Final root/reader gates and postmerge purification separate; no whole-model claim.

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 quadratic_feature_law_gradient_angular_transport_control
    {E H : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
    [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
    [NormedAddCommGroup H] [InnerProductSpace ℝ H] [CompleteSpace H]
    (F : E → H) (D : E → E →L[ℝ] H)
    (hF : ∀ w, HasFDerivAt F (D w) w)
    (M₁ M₂ : ℝ) (hM₁ : 0 ≤ M₁) (hM₂ : 0 ≤ M₂)
    (hBound : ∀ w : E, ‖w‖ ≤ 1 → ‖D w‖ ≤ M₁)
    (hLip : ∀ w u : E, ‖w‖ ≤ 1 → ‖u‖ ≤ 1 → ‖D w-D u‖ ≤ M₂*‖w-u‖)
    (y : H) :
    let S := {w : E // ‖w‖ = 1}
    let K : S → H := fun w => F w.1
    let c : S × S → ℝ := fun z => Real.arccos ⟪z.1.1,z.2.1⟫_ℝ
    let m : Measure S → H := fun μ => ∫ w, K w ∂μ
    let V : Measure S → E → ℝ := fun μ w => 2 * ⟪m μ-y,F w⟫_ℝ
    let G : Measure S → E → E := fun μ w =>
      (2 : ℝ) • ((D w).adjoint (m μ-y) - ⟪(D w).adjoint (m μ-y),w⟫_ℝ • w)
    let Q : Measure S → Measure S → ℝ≥0∞ := fun μ ν =>
      AutoSamplingTheory.TechnicalLemmas.Measure.Transport.transportCost
        (fun z => ENNReal.ofReal ((c z)^2)) μ ν
    ∃ C₁ C₂ : ℝ, 0 < C₁ ∧ 0 < C₂ ∧
      (∀ μ : Measure S, IsProbabilityMeasure μ →
        Integrable K μ ∧
        (∀ w : E, HasGradientAt (V μ) ((2 : ℝ) • (D w).adjoint (m μ-y)) w) ∧
        ∀ w : S, ⟪w.1,G μ w.1⟫_ℝ = 0 ∧ ‖G μ w.1‖ ≤ C₁) ∧
      ∀ μ ν : Measure S, IsProbabilityMeasure μ → IsProbabilityMeasure ν →
        Q μ ν ≠ ∞ ∧ ∀ w u : S,
          ‖G μ w.1-G ν u.1‖ ≤
            C₂ * ((Q μ ν ^ (1 / 2 : ℝ)).toReal + c (w,u)) := by
  let : SecondCountableTopology E := inferInstance
  let S := {w : E // ‖w‖ = 1}
  let K : S → H := fun w => F w.1
  let c : S × S → ℝ := fun z => Real.arccos ⟪z.1.1,z.2.1⟫_ℝ
  let m : Measure S → H := fun μ => ∫ w, K w ∂μ
  let V : Measure S → E → ℝ := fun μ w => 2 * ⟪m μ-y,F w⟫_ℝ
  let G : Measure S → E → E := fun μ w =>
    (2 : ℝ) • ((D w).adjoint (m μ-y) - ⟪(D w).adjoint (m μ-y),w⟫_ℝ • w)
  let Q : Measure S → Measure S → ℝ≥0∞ := fun μ ν =>
    AutoSamplingTheory.TechnicalLemmas.Measure.Transport.transportCost
      (fun z => ENNReal.ofReal ((c z)^2)) μ ν
  change ∃ C₁ C₂ : ℝ, 0 < C₁ ∧ 0 < C₂ ∧
    (∀ μ : Measure S, IsProbabilityMeasure μ →
      Integrable K μ ∧
      (∀ w : E, HasGradientAt (V μ) ((2 : ℝ) • (D w).adjoint (m μ-y)) w) ∧
      ∀ w : S, ⟪w.1,G μ w.1⟫_ℝ = 0 ∧ ‖G μ w.1‖ ≤ C₁) ∧
    ∀ μ ν : Measure S, IsProbabilityMeasure μ → IsProbabilityMeasure ν →
      Q μ ν ≠ ∞ ∧ ∀ w u : S,
        ‖G μ w.1-G ν u.1‖ ≤ C₂ * ((Q μ ν ^ (1 / 2 : ℝ)).toReal + c (w,u))
  have hFc : Continuous F := continuous_iff_continuousAt.mpr fun w => (hF w).continuousAt
  have hKsm : StronglyMeasurable K := (hFc.comp continuous_subtype_val).stronglyMeasurable
  have hChord (w u : E) (hw : ‖w‖ ≤ 1) (hu : ‖u‖ ≤ 1) :
      ‖F w-F u‖ ≤ M₁*‖w-u‖ := by
    apply (convex_closedBall (0 : E) 1).norm_image_sub_le_of_norm_hasFDerivWithin_le
      (fun z _ => (hF z).hasFDerivWithinAt) ?_
      (by simpa [Metric.mem_closedBall,dist_zero_right] using hu)
      (by simpa [Metric.mem_closedBall,dist_zero_right] using hw)
    intro z hz
    exact hBound z (by simpa [Metric.mem_closedBall,dist_zero_right] using hz)
  let B := ‖F (0:E)‖+M₁+1
  have hB : 0 ≤ B := by dsimp [B]; positivity
  have hKb (w : S) : ‖K w‖ ≤ B := by
    have hb := hChord w.1 0 w.property.le (by simp)
    have hn := norm_add_le (F w.1-F 0) (F 0)
    rw [sub_add_cancel] at hn
    simp only [sub_zero,w.property,mul_one] at hb
    change ‖F w.1‖ ≤ B
    dsimp [B]
    linarith
  have hAng := AutoSamplingTheory.TechnicalLemmas.Geometry.UnitSphereAngularCost.unit_sphere_angular_cost_properties (E := E)
  change Continuous c ∧ Measurable c ∧
    (∀ u v : S, 0 ≤ c (u,v) ∧ c (u,v) ≤ Real.pi ∧
      c (u,v) = InnerProductGeometry.angle u.1 v.1 ∧ ‖u.1-v.1‖ ≤ c (u,v) ∧
      (c (u,v) = 0 ↔ u = v) ∧ c (u,v) = c (v,u)) ∧
    ∀ u v w : S, c (u,w) ≤ c (u,v)+c (v,w) at hAng
  have hc0 (z : S × S) : 0 ≤ c z := (hAng.2.2.1 z.1 z.2).1
  have hcB (z : S × S) : c z ≤ Real.pi := (hAng.2.2.1 z.1 z.2).2.1
  have hKlip (w u : S) : ‖K w-K u‖ ≤ M₁*c (w,u) :=
    (hChord w.1 u.1 w.property.le u.property.le).trans
      (mul_le_mul_of_nonneg_left (hAng.2.2.1 w u).2.2.2.1 hM₁)
  have hMean (μ ν : Measure S) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] :=
    AutoSamplingTheory.TechnicalLemmas.Measure.BochnerMeanTransport.bounded_bochner_mean_quadratic_transport_control
      K hKsm B Real.pi M₁ hB Real.pi_pos.le hM₁ hKb c hAng.2.1 hc0 hcB hKlip μ ν
  have hTangent :=
    AutoSamplingTheory.TechnicalLemmas.Geometry.QuadraticFeatureTangentGradient.quadratic_feature_potential_tangent_gradient
      F D hF M₁ M₂ hM₁ hM₂ hBound hLip
  let R₀ := B+‖y‖
  have hR₀ : 0 ≤ R₀ := add_nonneg hB (norm_nonneg y)
  have hResidual (μ : Measure S) [IsProbabilityMeasure μ] : ‖m μ-y‖ ≤ R₀ :=
    ((hMean μ μ).2.2.2.2 y).1
  have hGrad (μ : Measure S) (w : E) :
      HasGradientAt (V μ) ((2 : ℝ) • (D w).adjoint (m μ-y)) w := by
    rw [hasGradientAt_iff_hasFDerivAt]
    have he : InnerProductSpace.toDual ℝ E ((2 : ℝ) • (D w).adjoint (m μ-y)) =
        (2 : ℝ) • innerSL ℝ ((D w).adjoint (m μ-y)) := by
      ext v
      simp [InnerProductSpace.toDual_apply_apply,innerSL_apply_apply]
    rw [he]
    exact hTangent.1 (m μ-y) w
  let A := 2*M₁^2
  let B₂ := 2*(M₂+2*M₁)*R₀
  have hA : 0 ≤ A := by dsimp [A]; positivity
  have hB₂ : 0 ≤ B₂ := by dsimp [B₂]; positivity
  refine ⟨2*M₁*R₀+1,A+B₂+1,by positivity,by positivity,?_,?_⟩
  · intro μ hμ
    let : IsProbabilityMeasure μ := hμ
    refine ⟨(hMean μ μ).1,hGrad μ,?_⟩
    intro w
    have ht := hTangent.2.1 (m μ-y) w.1 w.property
    refine ⟨ht.1,?_⟩
    exact ht.2.2.trans ((mul_le_mul_of_nonneg_left (hResidual μ) (by positivity)).trans (by linarith))
  · intro μ ν hμ hν
    let : IsProbabilityMeasure μ := hμ
    let : IsProbabilityMeasure ν := hν
    refine ⟨(hMean μ ν).2.2.1,?_⟩
    intro w u
    have hm : ‖(m μ-y)-(m ν-y)‖ ≤ M₁*(Q μ ν ^ (1 / 2 : ℝ)).toReal := by
      simpa only [sub_sub_sub_cancel_right] using (hMean μ ν).2.2.2.1
    have hr := hResidual ν
    have ht := hTangent.2.2 (m μ-y) (m ν-y) w.1 u.1 w.property u.property
    have hpoint := (hAng.2.2.1 w u).2.2.2.1
    have ht' : ‖G μ w.1-G ν u.1‖ ≤
        A*(Q μ ν ^ (1 / 2 : ℝ)).toReal+B₂*c (w,u) := by
      calc
        _ ≤ 2*M₁*‖(m μ-y)-(m ν-y)‖+
            2*(M₂+2*M₁)*‖m ν-y‖*‖w.1-u.1‖ := ht
        _ ≤ 2*M₁*(M₁*(Q μ ν ^ (1 / 2 : ℝ)).toReal)+
            2*(M₂+2*M₁)*R₀*c (w,u) := by
          gcongr
        _ = _ := by dsimp [A,B₂]; ring
    apply ht'.trans
    have hs : 0 ≤ (Q μ ν ^ (1 / 2 : ℝ)).toReal := ENNReal.toReal_nonneg
    have hc := hc0 (w,u)
    nlinarith [mul_nonneg hA hc,mul_nonneg hB₂ hs]

end AutoSamplingTheory.TechnicalLemmas.Geometry.QuadraticFeatureLawGradient

Open AutoSamplingTheory/TechnicalLemmas/Geometry/QuadraticFeatureLawGradient.lean:13published source at ed709a7f320d

Proof architecture

Generic geometric optimisation and law-dependent feature prerequisites; actual model compositions remain separate

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.