A quadratic gradient through the actual sphere differential

1. Symmetry gives the genuine ambient derivative \[Dq_x(v)=\langle v,Ax\rangle+\langle x,Av\rangle=2\langle v,Ax\rangle.\] Hence the real gradient is \(g=2Ax\). The bounded linear map and quadratic operations supply differentiability; the equality is not an inference from a totalized zero derivative.

2. For the actual inclusion, \(\operatorname{range}(D i_x)=K\). The chain rule therefore gives \[D(q|_S)_x(w)=\langle D i_x(w),g\rangle.\] This uses the inclusion-induced pairing, not an assumed isometry from an arbitrary chart metric.

3. Because \(\|x\|=1\), orthogonal projection onto K satisfies \[P_Ku=u-\langle x,u\rangle x.\] Substituting g yields the tangent vector \[G=P_K(2Ax)=2(Ax-\langle x,Ax\rangle x).\]

4. Every \(D i_x(w)\) lies in K, so orthogonal projection preserves its pairing with g. Thus G represents the restricted differential. Conversely, if a tangent vector z has the same pairing for every w, surjectivity onto K makes \(z-G\) orthogonal to every vector in K. Taking its inner product with itself gives \(z=G\), proving the equivalence.

/- SamplingOptimisationGeometryBench public-dev reference.
   Reuses the existing project proof; retain LICENSE and NOTICE. -/
import AutoSamplingTheory.TechnicalLemmas.Geometry.SphereGradient
import Mathlib.Analysis.InnerProductSpace.Rayleigh

noncomputable section
open Metric InnerProductSpace
open scoped Manifold InnerProductSpace
open AutoSamplingTheory.TechnicalLemmas.Geometry
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]

example
    (n : ℕ) [Fact (Module.finrank ℝ E = n + 1)]
    (A : E →L[ℝ] E) (hA : A.IsSymmetric) (x : sphere (0 : E) 1) :
    ∀ z : (ℝ ∙ (x : E))ᗮ,
      ((∀ w : TangentSpace (𝓡 n) x,
        mvfderiv (𝓡 n) (fun y : sphere (0 : E) 1 => ⟪(y : E), A (y : E)⟫_ℝ) x w =
          ⟪mvfderiv (𝓡 n) ((↑) : sphere (0 : E) 1 → E) x w, (z : E)⟫_ℝ) ↔
        (z : E) = (2 : ℝ) • (A (x : E) - ⟪(x : E), A (x : E)⟫_ℝ • (x : E))) := by
  have hx : ‖(x : E)‖ = 1 := mem_sphere_zero_iff_norm.mp x.property
  have hg : HasGradientAt (fun y : E => ⟪y, A y⟫_ℝ) ((2 : ℝ) • A (x : E)) (x : E) := by
    apply hasGradientAt_iff_hasFDerivAt.mpr
    have hraw := (hA.hasStrictFDerivAt_reApplyInnerSelf (x : E)).hasFDerivAt
    have hfun : (fun y : E => ⟪y, A y⟫_ℝ) = A.reApplyInnerSelf := by
      funext y
      simp [ContinuousLinearMap.reApplyInnerSelf_apply, real_inner_comm]
    have hlin : toDual ℝ E ((2 : ℝ) • A (x : E)) =
        (2 : ℕ) • innerSL ℝ (A (x : E)) := by
      ext v
      simp [toDual_apply_apply, two_smul]
    rw [hfun, hlin]
    exact hraw
  let K := (ℝ ∙ (x : E))ᗮ
  have hproj (u : E) : (K.orthogonalProjectionOnto u : E) =
      u - ⟪(x : E), u⟫_ℝ • (x : E) := by
    change (ℝ ∙ (x : E))ᗮ.starProjection u = _
    rw [Submodule.starProjection_orthogonal_val,
      Submodule.starProjection_unit_singleton ℝ hx]
  have hp : (K.orthogonalProjectionOnto ((2 : ℝ) • A (x : E)) : E) =
      (2 : ℝ) • (A (x : E) - ⟪(x : E), A (x : E)⟫_ℝ • (x : E)) := by
    rw [hproj]
    simp [inner_smul_right, smul_sub, smul_smul]
  intro z
  rw [SphereGradient.sphere_gradient_characterization n x hg z]
  constructor
  · intro hz
    exact (congrArg (fun v : K => (v : E)) hz).trans hp
  · intro hz
    apply Subtype.ext
    exact hz.trans hp.symm

1. Uses symmetry and genuine differentiability to obtain ambient gradient 2Ax with the correct factor two. (1 point)
2. Uses the actual sphere inclusion differential and its range K, retaining the induced pairing rather than assuming a chart metric. (1 point)
3. Uses unit norm to derive the projection and the exact Rayleigh coefficient ⟨x,Ax⟩. (1 point)
4. Proves both representation and uniqueness using tangent directions and the range of the inclusion derivative. (1 point)

/- SamplingOptimisationGeometryBench public-dev reference.
   Reuses the existing project proof; retain LICENSE and NOTICE. -/
import AutoSamplingTheory.TechnicalLemmas.Geometry.SphereGradient
import Mathlib.Analysis.InnerProductSpace.Rayleigh

noncomputable section
open Metric InnerProductSpace
open scoped Manifold InnerProductSpace
open AutoSamplingTheory.TechnicalLemmas.Geometry
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]

example
    (n : ℕ) [Fact (Module.finrank ℝ E = n + 1)]
    (A : E →L[ℝ] E) (hA : A.IsSymmetric) (x : sphere (0 : E) 1) :
    ∀ z : (ℝ ∙ (x : E))ᗮ,
      ((∀ w : TangentSpace (𝓡 n) x,
        mvfderiv (𝓡 n) (fun y : sphere (0 : E) 1 => ⟪(y : E), A (y : E)⟫_ℝ) x w =
          ⟪mvfderiv (𝓡 n) ((↑) : sphere (0 : E) 1 → E) x w, (z : E)⟫_ℝ) ↔
        (z : E) = (2 : ℝ) • (A (x : E) - ⟪(x : E), A (x : E)⟫_ℝ • (x : E))) := by
  -- Replace this comment with your proof.

Download task template

The task retains the exact ambient variables and finrank Fact from the source section. The proof first constructs HasGradientAt, using the symmetric quadratic derivative, rather than interpreting a totalized value as differentiability. It then computes the orthogonal projection using unit sphere membership and specializes the imported sphere-gradient characterization. The final subtype/coercion steps identify ambient tangent vectors in both directions. No proof of the target theorem is supplied in the context, and its module is not imported.

/- SamplingOptimisationGeometryBench public-dev reference.
   Reuses the existing project proof; retain LICENSE and NOTICE. -/
import AutoSamplingTheory.TechnicalLemmas.Geometry.SphereGradient
import Mathlib.Analysis.InnerProductSpace.Rayleigh

noncomputable section
open Metric InnerProductSpace
open scoped Manifold InnerProductSpace
open AutoSamplingTheory.TechnicalLemmas.Geometry
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]

example
    (n : ℕ) [Fact (Module.finrank ℝ E = n + 1)]
    (A : E →L[ℝ] E) (hA : A.IsSymmetric) (x : sphere (0 : E) 1) :
    ∀ z : (ℝ ∙ (x : E))ᗮ,
      ((∀ w : TangentSpace (𝓡 n) x,
        mvfderiv (𝓡 n) (fun y : sphere (0 : E) 1 => ⟪(y : E), A (y : E)⟫_ℝ) x w =
          ⟪mvfderiv (𝓡 n) ((↑) : sphere (0 : E) 1 → E) x w, (z : E)⟫_ℝ) ↔
        (z : E) = (2 : ℝ) • (A (x : E) - ⟪(x : E), A (x : E)⟫_ℝ • (x : E))) := by
  have hx : ‖(x : E)‖ = 1 := mem_sphere_zero_iff_norm.mp x.property
  have hg : HasGradientAt (fun y : E => ⟪y, A y⟫_ℝ) ((2 : ℝ) • A (x : E)) (x : E) := by
    apply hasGradientAt_iff_hasFDerivAt.mpr
    have hraw := (hA.hasStrictFDerivAt_reApplyInnerSelf (x : E)).hasFDerivAt
    have hfun : (fun y : E => ⟪y, A y⟫_ℝ) = A.reApplyInnerSelf := by
      funext y
      simp [ContinuousLinearMap.reApplyInnerSelf_apply, real_inner_comm]
    have hlin : toDual ℝ E ((2 : ℝ) • A (x : E)) =
        (2 : ℕ) • innerSL ℝ (A (x : E)) := by
      ext v
      simp [toDual_apply_apply, two_smul]
    rw [hfun, hlin]
    exact hraw
  let K := (ℝ ∙ (x : E))ᗮ
  have hproj (u : E) : (K.orthogonalProjectionOnto u : E) =
      u - ⟪(x : E), u⟫_ℝ • (x : E) := by
    change (ℝ ∙ (x : E))ᗮ.starProjection u = _
    rw [Submodule.starProjection_orthogonal_val,
      Submodule.starProjection_unit_singleton ℝ hx]
  have hp : (K.orthogonalProjectionOnto ((2 : ℝ) • A (x : E)) : E) =
      (2 : ℝ) • (A (x : E) - ⟪(x : E), A (x : E)⟫_ℝ • (x : E)) := by
    rw [hproj]
    simp [inner_smul_right, smul_sub, smul_smul]
  intro z
  rw [SphereGradient.sphere_gradient_characterization n x hg z]
  constructor
  · intro hz
    exact (congrArg (fun v : K => (v : E)) hz).trans hp
  · intro hz
    apply Subtype.ext
    exact hz.trans hp.symm

Download runnable reference