Plain-English statement
- A single positive constant controls the second iterated Fréchet derivative of every positive-scale radial cutoff by `C / R^2`. The proof first bounds the second derivative of the unit-scale radial cutoff using continuity and compact support. It then writes the radius-`R` cutoff as the unit cutoff composed with scalar dilation and applies Mathlib's exact iterated-derivative composition rule for continuous linear maps.
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Proof architecture
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Lean statement
theorem radialSmoothCutoff_iteratedFDeriv_two_bound
[InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [Nontrivial E] :
∃ C : ℝ, 0 < C ∧ ∀ R : ℝ, 0 < R → ∀ x : E,
‖iteratedFDeriv ℝ 2 (radialSmoothCutoff R : E → ℝ) x‖ ≤ C / R ^ 2 := by
have hunit_smooth : ContDiff ℝ (⊤ : ℕ∞)
(radialSmoothCutoff 1 : E → ℝ) :=
radialSmoothCutoff_contDiff one_pos
have hunit_continuous : Continuous
(iteratedFDeriv ℝ 2 (radialSmoothCutoff 1 : E → ℝ)) :=
hunit_smooth.continuous_iteratedFDeriv
(WithTop.coe_le_coe.mpr (le_top : (2 : ℕ∞) ≤ ⊤))
have hunit_support : HasCompactSupport
(iteratedFDeriv ℝ 2 (radialSmoothCutoff 1 : E → ℝ)) :=
(radialSmoothCutoff_hasCompactSupport one_pos).iteratedFDeriv 2
obtain ⟨C, hC⟩ :=
hunit_support.exists_bound_of_continuous hunit_continuous
let C' : ℝ := max C 1
have hC'_pos : 0 < C' := lt_max_of_lt_right one_pos
refine ⟨C', hC'_pos, ?_⟩
intro R hR x
let L : E →L[ℝ] E := R⁻¹ • ContinuousLinearMap.id ℝ E
have hL_norm : ‖L‖ ≤ 1 / R := by
refine L.opNorm_le_bound (by positivity) ?_
intro y
simp only [L, smul_apply, ContinuousLinearMap.id_apply]
rw [norm_smul, Real.norm_eq_abs, abs_inv, abs_of_pos hR]
simp [one_div]
have hfun : (radialSmoothCutoff R : E → ℝ) =
(radialSmoothCutoff 1 : E → ℝ) ∘ L := by
funext y
simp only [radialSmoothCutoff, Function.comp_apply, L,
smul_apply, ContinuousLinearMap.id_apply, div_one]
rw [norm_smul, Real.norm_eq_abs, abs_inv, abs_of_pos hR]
ring_nf
rw [hfun]
rw [L.iteratedFDeriv_comp_right hunit_smooth x
(i := 2) (WithTop.coe_le_coe.mpr (le_top : (2 : ℕ∞) ≤ ⊤))]
calc
‖(iteratedFDeriv ℝ 2 (radialSmoothCutoff 1 : E → ℝ) (L x)).compContinuousLinearMap
(fun _ => L)‖ ≤
‖iteratedFDeriv ℝ 2 (radialSmoothCutoff 1 : E → ℝ) (L x)‖ *
∏ _ : Fin 2, ‖L‖ :=
ContinuousMultilinearMap.norm_compContinuousLinearMap_le _ _
_ = ‖iteratedFDeriv ℝ 2 (radialSmoothCutoff 1 : E → ℝ) (L x)‖ * ‖L‖ ^ 2 := by
simp
_ ≤ C' * (1 / R) ^ 2 := by
gcongr
exact (hC (L x)).trans (le_max_left C 1)
_ = C' / R ^ 2 := by ring
/-- At each fixed point, the positive-scale radial cutoffs tend to one as the scale diverges. -/Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Cutoff.lean:339published source at 7bcd37294df1
Proof architecture
log-concave sampling Ch.1 second-order exhaustion branch: one constant controls the second iterated Frechet derivative at every positive scale by C/R^2
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.
- `refine` instantiates a reusable theorem while leaving explicit subgoals.
- `exact` closes the current goal with an already typed term.
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
- Totalized `fderiv` values must not be read as a differentiability theorem.
- Pointwise support, topological support, and compact support retain distinct meanings.
- A cutoff lemma does not by itself prove a whole-space integration-by-parts identity.