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Lean gate passed 2026-08-19T06:32:39.895922+00:00 · 7bcd37294df1
Registry leaf card · analysis.calculus.radial-smooth-cutoff-fderiv-zero-of-two-mul-le-norm

radialSmoothCutoff_fderiv_eq_zero_of_two_mul_le_norm

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- The totalized derivative of the radial cutoff vanishes throughout the outer zero region, including its boundary sphere. At the boundary the cutoff is a global minimum rather than locally constant; `IsLocalMin.fderiv_eq_zero` records that distinction.

Plain-English statement

- The totalized derivative of the radial cutoff vanishes throughout the outer zero region, including its boundary sphere. At the boundary the cutoff is a global minimum rather than locally constant; `IsLocalMin.fderiv_eq_zero` records that distinction.

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 radialSmoothCutoff_fderiv_eq_zero_of_two_mul_le_norm [NormedSpace ℝ E]
    {R : ℝ} (hR : 0 < R) {x : E} (hx : 2 * R ≤ ‖x‖) :
    fderiv ℝ (radialSmoothCutoff R : E → ℝ) x = 0 := by
  apply IsLocalMin.fderiv_eq_zero
  change ∀ᶠ y in 𝓝 x, radialSmoothCutoff R x ≤ radialSmoothCutoff R y
  rw [radialSmoothCutoff_eq_zero_of_two_mul_le_norm hR hx]
  exact Filter.Eventually.of_forall fun y => (radialSmoothCutoff_mem_Icc R y).1

/-- The support of the radial cutoff lies in the closed ball of radius `2 * R`. -/

Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Cutoff.lean:296published source at 7bcd37294df1

Proof architecture

log-concave sampling Ch.1 cutoff exhaustion: the totalized first derivative vanishes on the closed outer region with norm at least 2R

Lean proof walkthrough

  • Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
  • `rw` rewrites by an established identity.
  • `apply` reduces the goal to the hypotheses of a reusable theorem.
  • `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.
  • Genuine differentiability is localized to the hypotheses shown in the Lean statement.
  • 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.
Common pitfall. A wrapper or display identity is not a new analytic theorem merely because it has its own Lean name. Check the statement, hypotheses, and downstream consumers before interpreting its mathematical contribution.