unit_sphere_angular_cost_properties

Plain-English statement

Actual unit-subtype arccos cost is continuous/measurable, bounded, chord-dominating, separating, symmetric and triangular. No installed metric or intrinsic path-length identity. 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 unit_sphere_angular_cost_properties
    {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
    [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] :
    let S := {w : E // ‖w‖ = 1}
    let c : S × S → ℝ := fun z => Real.arccos ⟪z.1.1,z.2.1⟫_ℝ
    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) := by
  dsimp only
  letI : SecondCountableTopology E := inferInstance
  have heq (u v : {w : E // ‖w‖ = 1}) :
      Real.arccos ⟪u.1,v.1⟫_ℝ = InnerProductGeometry.angle u.1 v.1 := by
    simp [InnerProductGeometry.angle,u.property,v.property]
  have hc : Continuous (fun z : {w : E // ‖w‖ = 1} × {w : E // ‖w‖ = 1} =>
      Real.arccos ⟪z.1.1,z.2.1⟫_ℝ) := by fun_prop
  refine ⟨hc,hc.measurable,?_,?_⟩
  · intro u v
    have hchord : ‖u.1-v.1‖ ≤ Real.arccos ⟪u.1,v.1⟫_ℝ := by
      rw [heq]
      exact chord_le_angle u.1 v.1 u.property v.property
    refine ⟨Real.arccos_nonneg _,Real.arccos_le_pi _,heq u v,hchord,?_,?_⟩
    · constructor
      · intro hz
        apply Subtype.ext
        apply sub_eq_zero.mp
        apply norm_eq_zero.mp
        exact le_antisymm (by simpa only [hz] using hchord) (norm_nonneg _)
      · rintro rfl
        rw [heq]
        apply InnerProductGeometry.angle_self
        intro hz
        have hh := u.property
        rw [hz,norm_zero] at hh
        norm_num at hh
    · rw [heq,heq,InnerProductGeometry.angle_comm]
  · intro u v w
    rw [heq,heq,heq]
    exact InnerProductGeometry.angle_le_angle_add_angle u.1 v.1 w.1

end AutoSamplingTheory.TechnicalLemmas.Geometry.UnitSphereAngularCost

Open AutoSamplingTheory/TechnicalLemmas/Geometry/UnitSphereAngularCost.lean:27published source at ed709a7f320d

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

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Hidden assumptions and non-claims

Measurability is represented explicitly or must be supplied by a dependency.