`BookProof.ChapterSolidHarmonic.inner_e_spherePt` (r θ φ : ℝ) : ⟪e, spherePt u v e r θ φ⟫_ℝ = r * Real.cos θ

Disproved

BookProof.ChapterSolidHarmonic.inner_e_spherePt

by leonardopedro · Oct 11, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSolidHarmonic.

BookProof.ChapterSolidHarmonic.inner_e_spherePt (r θ φ : ℝ) : ⟪e, spherePt u v e r θ φ⟫_ℝ = r * Real.cos θ

Formalization note: Lean 4 identifier BookProof.ChapterSolidHarmonic.inner_e_spherePt.

Retired 2026-10-30 — superseded by a corrected statement, not withdrawn. The exported target lost the frame hypotheses (section variables absent from the type), so it asserted $\langle e, \mathrm{spherePt}\rangle = r\cos\theta$ for arbitrary vectors.

Where the fix lives. In the source of record the hypothesis is an explicit binder of the declaration, so a re-export cannot lose it:

BookProof.ChapterSolidHarmonic.inner_e_spherePt_frames_explicit in timepiece/BookProof/HarmonicAnalysis/ChapterSolidHarmonic.lean (Lean 4.28 tree),
BookProof.ChapterSolidHarmonic.inner_e_spherePt_frames_explicit in timepiece331/BookProof/ChapterSolidHarmonic.lean (Lean 4.33.1 tree).

It is published on the platform as BookProof.ChapterSolidHarmonic.inner_e_spherePt_frames_explicit. This record stays readable and importable for anything already citing it; its proof status is unchanged.

Preamble

-- Generated from ChapterSolidHarmonic.lean — theorem BookProof.ChapterSolidHarmonic.inner_e_spherePt
import Definitions.Def_ChapterRadialLaplacian
import Definitions.Def_ChapterLaplacianProduct
import Definitions.Def_ChapterSolidHarmonicTools
import Definitions.Def_ChapterLegendrePolynomial
import Mathlib
import Definitions.Def_ChapterSolidHarmonic
open BookProof.ChapterSolidHarmonic



open Laplacian InnerProductSpace Polynomial
open BookProof.ChapterRadialLaplacian BookProof.ChapterLaplacianProduct
open BookProof.ChapterSolidHarmonicTools BookProof.ChapterLegendrePolynomial
open scoped RealInnerProductSpace

variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]

variable {u v e : E} (hu : ‖u‖ = 1) (hv : ‖v‖ = 1) (he : ‖e‖ = 1)
  (huv : ⟪u, v⟫_ℝ = 0) (hue : ⟪u, e⟫_ℝ = 0) (hve : ⟪v, e⟫_ℝ = 0)

Formal statement

theorem BookProof.ChapterSolidHarmonic.inner_e_spherePt (r θ φ : ℝ) :
    ⟪e, spherePt u v e r θ φ⟫_ℝ = r * Real.cos θ := by sorry

Source

https://github.com/leonardopedro/timepiece/blob/main/BookProof/HarmonicAnalysis/ChapterSolidHarmonic.lean

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