Planar DFP convergence with locally Lipschitz Hessian

The proof of thm:planar-convergence in main-new.tex controls the free coefficient in the planar secant representation. Nonconvergence forces this coefficient to vanish. A corrected logarithmic quantity with summable increments rules out that degeneration under a locally Lipschitz Hessian.

The public raw-orbit theorem derives later positive definiteness from the initial matrix and includes stationary termination. The module also proves the C3 specialization and the failure of local Hessian Lipschitz regularity for planar weak-Wolfe counterexamples.

public sectionopen scoped Matrixopen Filteropen scoped Nat ContDiffopen scoped Topologynamespace DFP.PlanarConvergence

Helper for lem:planar-degeneration: a symmetric planar secant matrix has off-diagonal entry determined by its transverse entry and the secant vector.

theorem secantOffDiagonal {H : Matrix (Fin 2) (Fin 2) ℝ} {κ β : ℝ} (hH : H.IsHermitian) (hκ : κ ≠ 0) (hsec : H *ᵥ ![κ, β] = ![1, 0]) : H 0 1 = -(H 1 1 * β / κ) := by have hsym : H 1 0 = H 0 1 := by simpa using hH.apply 0 1 have hrow := congrFun hsec 1 simp only [Matrix.mulVec, dotProduct, Fin.sum_univ_two, Matrix.cons_val_zero, Matrix.cons_val_one] at hrow rw [hsym] at hrow apply (eq_neg_iff_add_eq_zero).mpr apply (mul_right_cancel₀ hκ) field_simp nlinarith [hrow]

Helper for lem:planar-degeneration: the longitudinal entry follows from the first secant equation.

theorem secantLongitudinal {H : Matrix (Fin 2) (Fin 2) ℝ} {κ β : ℝ} (hH : H.IsHermitian) (hκ : κ ≠ 0) (hsec : H *ᵥ ![κ, β] = ![1, 0]) : H 0 0 = 1 / κ + H 1 1 * β ^ 2 / κ ^ 2 := by have hoff := secantOffDiagonal hH hκ hsec have hrow := congrFun hsec 0 simp only [Matrix.mulVec, dotProduct, Fin.sum_univ_two, Matrix.cons_val_zero, Matrix.cons_val_one] at hrow rw [hoff] at hrow field_simp at hrow ⊢ nlinarith [hrow]

Helper for lem:planar-degeneration: in secant coordinates, the full matrix is determined by the curvature, transverse secant component, and one free entry.

theorem secantMatrixRepresentation {H : Matrix (Fin 2) (Fin 2) ℝ} {κ β : ℝ} (hH : H.IsHermitian) (hκ : κ ≠ 0) (hsec : H *ᵥ ![κ, β] = ![1, 0]) : H = !![1 / κ + H 1 1 * β ^ 2 / κ ^ 2, -(H 1 1 * β / κ); -(H 1 1 * β / κ), H 1 1] := by have hsym : H 1 0 = H 0 1 := by simpa using hH.apply 0 1 ext i j fin_cases i · fin_cases j · exact secantLongitudinal hH hκ hsec · exact secantOffDiagonal hH hκ hsec · fin_cases j · exact hsym.trans (secantOffDiagonal hH hκ hsec) · rfl

Helper for lem:planar-degeneration: the free transverse entry is the secant curvature times the determinant, rather than an eigenvalue.

theorem transverseEntry_eq_curvature_mul_det {H : Matrix (Fin 2) (Fin 2) ℝ} {κ β : ℝ} (hH : H.IsHermitian) (hκ : κ ≠ 0) (hsec : H *ᵥ ![κ, β] = ![1, 0]) : H 1 1 = κ * H.det := by have hsym : H 1 0 = H 0 1 := by simpa using hH.apply 0 1 rw [Matrix.det_fin_two, hsym, secantLongitudinal hH hκ hsec, secantOffDiagonal hH hκ hsec] field_simp ring

Helper for lem:planar-degeneration: positive definiteness makes the free coefficient in the secant representation strictly positive.

theorem curvature_mul_det_pos {H : Matrix (Fin 2) (Fin 2) ℝ} {κ β : ℝ} (hH : H.PosDef) (hκ : κ ≠ 0) (hsec : H *ᵥ ![κ, β] = ![1, 0]) : 0 < κ * H.det := by rw [← transverseEntry_eq_curvature_mul_det hH.isHermitian hκ hsec] exact hH.diag_pos

Helper for lem:planar-degeneration: absorption of the small current-step term yields the product bound on the free secant coefficient.

theorem freeCoefficientProductBound {a C u v lam : ℝ} (ha : 0 < a) (hlam : 0 ≤ lam) (hsmall : C * v ≤ a / 2) (hbound : a * lam ≤ C * v * (u + lam)) : lam ≤ (2 * C / a) * u * v := by have habsorb : C * v * lam ≤ (a / 2) * lam := mul_le_mul_of_nonneg_right hsmall hlam rw [div_mul_eq_mul_div, div_mul_eq_mul_div] apply (le_div_iff₀ ha).mpr nlinarith

Helper for thm:planar-convergence: summable increments of the corrected logarithm, a vanishing correction, and an upper bound on the gradient norms give a positive eventual lower bound for the free secant coefficient.

theorem positiveLowerBoundOfCorrectedLog {lam eta N : ℕ → ℝ} {Nmax : ℝ} (hlam : ∀ k, 0 < lam k) (hN : ∀ k, 0 < N k) (hNupper : ∀ k, N k ≤ Nmax) (heta : Tendsto eta atTop (𝓝 0)) (hincrements : Summable (fun k ↦ |(Real.log (lam (k + 1)) - eta (k + 1) + 2 * Real.log (N (k + 1))) - (Real.log (lam k) - eta k + 2 * Real.log (N k))|)) : ∃ δ > 0, ∀ᶠ k in atTop, δ ≤ lam k := by let invariant : ℕ → ℝ := fun k ↦ Real.log (lam k) - eta k + 2 * Real.log (N k) have hdist : Summable (fun k ↦ dist (invariant k) (invariant (k + 1))) := by simpa only [invariant, Real.dist_eq, abs_sub_comm] using hincrements obtain ⟨L, hL⟩ := cauchySeq_tendsto_of_complete (cauchySeq_of_summable_dist hdist) have hInvariantLower : ∀ᶠ k in atTop, L - 1 < invariant k := hL.eventually (eventually_gt_nhds (sub_lt_self L zero_lt_one)) have hEtaLower : ∀ᶠ k in atTop, (-1 : ℝ) < eta k := heta.eventually (eventually_gt_nhds (neg_lt_zero.mpr zero_lt_one)) refine ⟨Real.exp (L - 2 - 2 * Real.log Nmax), Real.exp_pos _, ?_⟩ filter_upwards [hInvariantLower, hEtaLower] with k hk hetaLower have hlogN : Real.log (N k) ≤ Real.log Nmax := Real.log_le_log (hN k) (hNupper k) have hlogLower : L - 2 - 2 * Real.log Nmax ≤ Real.log (lam k) := by dsimp only [invariant] at hk linarith have hexp := Real.exp_le_exp.mpr hlogLower rwa [Real.exp_log (hlam k)] at hexp

Helper for thm:planar-convergence: a positive eventual lower bound is incompatible with the degeneration of the free secant coefficient to zero.

theorem not_tendsto_zero_of_eventually_positive_lower_bound {lam : ℕ → ℝ} {δ : ℝ} (hδ : 0 < δ) (hlower : ∀ᶠ k in atTop, δ ≤ lam k) : ¬ Tendsto lam atTop (𝓝 0) := by intro hzero have hle : δ ≤ 0 := ge_of_tendsto hzero hlower exact (not_le_of_gt hδ) hle

Helper for thm:planar-convergence: the logarithm has a uniform quadratic remainder on the half-unit interval.

theorem logOneAddQuadraticRemainder {r : ℝ} (hr : |r| ≤ 1 / 2) : |Real.log (1 + r) - r| ≤ 2 * r ^ 2 := by have hrlt : |-r| < 1 := by rw [abs_neg] linarith have hseries := Real.abs_log_sub_add_sum_range_le hrlt 1 have htaylor : |Real.log (1 + r) - r| ≤ r ^ 2 / (1 - |r|) := by simpa [Finset.sum_range_succ, sub_eq_add_neg, add_comm] using hseries have hden : 0 < 1 - |r| := by linarith have hquotient : r ^ 2 / (1 - |r|) ≤ 2 * r ^ 2 := by apply (div_le_iff₀ hden).mpr nlinarith [mul_le_mul_of_nonneg_left hr (sq_nonneg r)] exact htaylor.trans hquotient

Helper for thm:planar-convergence: after the first-order terms are removed, the logarithmic determinant increment has a quadratic remainder.

theorem logDeterminantQuadraticRemainder {P Q : ℝ} (hP : |P| ≤ 1 / 2) (hQ : |Q| ≤ 1 / 2) : |2 * Real.log (1 + P) - Real.log (1 + Q) - (2 * P - Q)| ≤ 4 * P ^ 2 + 2 * Q ^ 2 := by have hsplit : 2 * Real.log (1 + P) - Real.log (1 + Q) - (2 * P - Q) = 2 * (Real.log (1 + P) - P) - (Real.log (1 + Q) - Q) := by ring rw [hsplit] calc |2 * (Real.log (1 + P) - P) - (Real.log (1 + Q) - Q)| ≤ |2 * (Real.log (1 + P) - P)| + |Real.log (1 + Q) - Q| := abs_sub _ _ _ = 2 * |Real.log (1 + P) - P| + |Real.log (1 + Q) - Q| := by norm_num [abs_mul] _ ≤ 4 * P ^ 2 + 2 * Q ^ 2 := by nlinarith [logOneAddQuadraticRemainder hP, logOneAddQuadraticRemainder hQ]

Helper for thm:planar-convergence: the old secant equation eliminates the matrix from the two linear terms in a perturbed quadratic form.

theorem secantQuadraticPerturbation {ι : Type*} [Fintype ι] {H : Matrix ι ι ℝ} {u v : ι → ℝ} (hH : H.IsHermitian) (hsec : H *ᵥ v = u) (dv : ι → ℝ) : (v + dv) ⬝ᵥ (H *ᵥ (v + dv)) = u ⬝ᵥ v + 2 * (u ⬝ᵥ dv) + dv ⬝ᵥ (H *ᵥ dv) := by have hrow : v ᵥ* H = H *ᵥ v := by have hstar := Matrix.star_mulVec H v rw [hH.eq] at hstar simpa using hstar.symm rw [Matrix.mulVec_add, dotProduct_add, add_dotProduct, add_dotProduct, hsec, Matrix.dotProduct_mulVec v H dv, hrow, hsec] rw [dotProduct_comm v u, dotProduct_comm dv u] ring

Helper for thm:planar-convergence: perturbing both oriented secant vectors gives their exact curvature increment.

theorem secantCurvaturePerturbation {ι : Type*} [Fintype ι] (u v du dv : ι → ℝ) : (u + du) ⬝ᵥ (v + dv) = u ⬝ᵥ v + v ⬝ᵥ du + u ⬝ᵥ dv + du ⬝ᵥ dv := by rw [dotProduct_add, add_dotProduct, add_dotProduct, dotProduct_comm du v] ring

Helper for lem:planar-degeneration: the secant-energy estimate and vanishing step lengths force the nonnegative free coefficient to vanish.

theorem freeCoefficientTendstoZero {lam ell : ℕ → ℝ} {a C : ℝ} (ha : 0 < a) (hell : Tendsto ell atTop (𝓝 0)) (hlam : ∀ᶠ k in atTop, 0 ≤ lam k) (henergy : ∀ᶠ k in atTop, a * lam k ≤ C * ell (k + 1) * (ell k + lam k)) : Tendsto lam atTop (𝓝 0) := by have hnext : Tendsto (fun k ↦ ell (k + 1)) atTop (𝓝 0) := hell.comp (tendsto_add_atTop_nat 1) have hscaled : Tendsto (fun k ↦ C * ell (k + 1)) atTop (𝓝 0) := by simpa using hnext.const_mul C have hsmall : ∀ᶠ k in atTop, C * ell (k + 1) < a / 2 := hscaled.eventually (eventually_lt_nhds (half_pos ha)) have hbound : ∀ᶠ k in atTop, lam k ≤ (2 * C / a) * ell k * ell (k + 1) := by filter_upwards [hlam, hsmall, henergy] with k hk hs he exact freeCoefficientProductBound ha hk hs.le he have hproduct : Tendsto (fun k ↦ (2 * C / a) * ell k * ell (k + 1)) atTop (𝓝 0) := by simpa using (hell.const_mul (2 * C / a)).mul hnext exact squeeze_zero' hlam hbound hproduct

Helper for thm:planar-convergence: the first-order secant-vector variation cancels from twice the curvature increment minus the quadratic form increment.

theorem normalizedSecantFirstOrderCancellation {ι : Type*} [Fintype ι] (H : Matrix ι ι ℝ) (u v du dv : ι → ℝ) (κ : ℝ) : 2 * ((v ⬝ᵥ du + u ⬝ᵥ dv + du ⬝ᵥ dv) / κ) - ((2 * (u ⬝ᵥ dv) + dv ⬝ᵥ (H *ᵥ dv)) / κ) = 2 * (v ⬝ᵥ du) / κ + (2 * (du ⬝ᵥ dv) - dv ⬝ᵥ (H *ᵥ dv)) / κ := by ring

Helper for thm:planar-convergence: the normalized determinant recurrence has precisely the logarithmic form used in the corrected invariant.

theorem normalizedDeterminantLogIncrement {lam lamNext P Q : ℝ} (hlam : 0 < lam) (hP : 0 < 1 + P) (hQ : 0 < 1 + Q) (hrec : lamNext = lam * (1 + P) ^ 2 / (1 + Q)) : Real.log lamNext - Real.log lam = 2 * Real.log (1 + P) - Real.log (1 + Q) := by rw [hrec, Real.log_div (mul_ne_zero (ne_of_gt hlam) (pow_ne_zero 2 (ne_of_gt hP))) (ne_of_gt hQ), Real.log_mul (ne_of_gt hlam) (pow_ne_zero 2 (ne_of_gt hP)), Real.log_pow] ring

Helper for thm:planar-convergence: under the nonconvergent-orbit energy estimate, no vanishing correction can make the corrected log increments absolutely summable while the positive gradient norms remain bounded above.

theorem not_summable_correctedLogIncrements {lam eta N ell : ℕ → ℝ} {a C Nmax : ℝ} (ha : 0 < a) (hell : Tendsto ell atTop (𝓝 0)) (hlam : ∀ k, 0 < lam k) (hN : ∀ k, 0 < N k) (hNupper : ∀ k, N k ≤ Nmax) (heta : Tendsto eta atTop (𝓝 0)) (henergy : ∀ᶠ k in atTop, a * lam k ≤ C * ell (k + 1) * (ell k + lam k)) : ¬ Summable (fun k ↦ |(Real.log (lam (k + 1)) - eta (k + 1) + 2 * Real.log (N (k + 1))) - (Real.log (lam k) - eta k + 2 * Real.log (N k))|) := by intro hincrements obtain ⟨δ, hδ, hlower⟩ := positiveLowerBoundOfCorrectedLog hlam hN hNupper heta hincrements have hnonnegative : ∀ᶠ k in atTop, 0 ≤ lam k := Eventually.of_forall (fun k ↦ (hlam k).le) have hzero := freeCoefficientTendstoZero ha hell hnonnegative henergy exact not_tendsto_zero_of_eventually_positive_lower_bound hδ hlower hzero

Helper for lem:planar-degeneration: a global positive Hessian lower bound supplies an explicit global lower bound for the objective.

theorem objectiveLowerBoundOfHessian {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] (f : E → ℝ) {μ : ℝ} (hf : ContDiff ℝ 2 f) (hμ : 0 < μ) (hHessian : ∀ x v : E, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient f) x v) v) (x : E) : f 0 - ‖gradient f 0‖ ^ 2 / (2 * μ) ≤ f x := by have hfirst := hf.firstOrderOfHessianLowerBound f μ hμ hHessian 0 x simp only [sub_zero] at hfirst have hinner : -(‖gradient f 0‖ * ‖x‖) ≤ inner ℝ (gradient f 0) x := neg_le_of_abs_le (abs_real_inner_le_norm _ _) have hden : 0 < 2 * μ := by positivity have hquadratic : -(‖gradient f 0‖ ^ 2) / (2 * μ) ≤ inner ℝ (gradient f 0) x + μ / 2 * ‖x‖ ^ 2 := by apply (div_le_iff₀ hden).mpr nlinarith [mul_le_mul_of_nonneg_left hinner hden.le, sq_nonneg (μ * ‖x‖ - ‖gradient f 0‖)] rw [neg_div] at hquadratic linarith

Helper for lem:planar-degeneration: a lower-bounded objective that decreases by a fixed positive multiple of nonnegative costs has summable costs.

theorem summableCostsOfDescent {F q : ℕ → ℝ} {c B : ℝ} (hc : 0 < c) (hq : ∀ k, 0 ≤ q k) (hbelow : ∀ k, B ≤ F k) (hdescent : ∀ k, F (k + 1) ≤ F k - c * q k) : Summable q := by have hpartial : ∀ n, c * ∑ k ∈ Finset.range n, q k ≤ F 0 - F n := by intro n induction n with | zero => simp | succ n ih => rw [Finset.sum_range_succ, mul_add] linarith [hdescent n] apply summable_of_sum_range_le (c := (F 0 - B) / c) hq intro n apply (le_div_iff₀ hc).mpr have hbound := hpartial n have hfloor := hbelow n nlinarith

Helper for lem:planar-degeneration: the actual predicted decreases of a positive-step weak-Wolfe DFP orbit are summable under a global positive Hessian lower bound.

theorem summablePredictedDecrease {n : ℕ} (iteration : DFP.InverseIteration (Fin n)) {μ c₁ c₂ : ℝ} (hf : ContDiff ℝ 2 iteration.objective) (hμ : 0 < μ) (hHessian : ∀ x v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient iteration.objective) x v) v) (hStep : ∀ k, 0 < iteration.stepLength k) (hWolfe : ∀ k, LineSearch.IsWeakWolfe c₁ c₂ iteration.objective (iteration.point k) (iteration.point (k + 1) - iteration.point k)) : Summable (fun k ↦ -inner ℝ (gradient iteration.objective (iteration.point k)) (iteration.point (k + 1) - iteration.point k)) := by have hnonnegative : ∀ k, 0 ≤ -inner ℝ (gradient iteration.objective (iteration.point k)) (iteration.point (k + 1) - iteration.point k) := by intro k have hneg := iteration.gradientInnerDisplacementNeg k (hStep k) simpa only [DFP.gradients_apply] using (neg_pos.mpr hneg).le have hbelow := objectiveLowerBoundOfHessian iteration.objective hf hμ hHessian apply summableCostsOfDescent (F := fun k ↦ iteration.objective (iteration.point k)) (hWolfe 0).c₁_pos hnonnegative · intro k exact hbelow (iteration.point k) · intro k have harmijo := (hWolfe k).armijo simpa only [add_sub_cancel, mul_neg, sub_neg_eq_add] using harmijo

Helper for lem:planar-degeneration: strong convexity and Armijo bound the squared physical displacement by its predicted decrease.

theorem stepSquareBound {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] {f : E → ℝ} {x y : E} {μ c₁ c₂ : ℝ} (hf : ContDiff ℝ 2 f) (hμ : 0 < μ) (hHessian : ∀ z v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient f) z v) v) (hWolfe : LineSearch.IsWeakWolfe c₁ c₂ f x (y - x)) : ‖y - x‖ ^ 2 ≤ (2 * (1 - c₁) / μ) * (-inner ℝ (gradient f x) (y - x)) := by have hfirst := hf.firstOrderOfHessianLowerBound f μ hμ hHessian x y have harmijo := hWolfe.armijo rw [add_sub_cancel] at harmijo rw [div_mul_eq_mul_div] apply (le_div_iff₀ hμ).mpr nlinarith

Helper for lem:planar-degeneration: the squared physical step lengths of the classical DFP orbit are summable.

theorem summableStepSquares {n : ℕ} (iteration : DFP.InverseIteration (Fin n)) {μ c₁ c₂ : ℝ} (hf : ContDiff ℝ 2 iteration.objective) (hμ : 0 < μ) (hHessian : ∀ x v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient iteration.objective) x v) v) (hStep : ∀ k, 0 < iteration.stepLength k) (hWolfe : ∀ k, LineSearch.IsWeakWolfe c₁ c₂ iteration.objective (iteration.point k) (iteration.point (k + 1) - iteration.point k)) : Summable (fun k ↦ ‖iteration.point (k + 1) - iteration.point k‖ ^ 2) := by have hsum := (summablePredictedDecrease iteration hf hμ hHessian hStep hWolfe).mul_left (2 * (1 - c₁) / μ) apply Summable.of_nonneg_of_le (fun k ↦ sq_nonneg ‖iteration.point (k + 1) - iteration.point k‖) (fun k ↦ stepSquareBound hf hμ hHessian (hWolfe k)) hsum

Helper for lem:planar-degeneration: the physical DFP step lengths tend to zero, by square summability derived from strong convexity and Armijo.

theorem stepNormTendstoZero {n : ℕ} (iteration : DFP.InverseIteration (Fin n)) {μ c₁ c₂ : ℝ} (hf : ContDiff ℝ 2 iteration.objective) (hμ : 0 < μ) (hHessian : ∀ x v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient iteration.objective) x v) v) (hStep : ∀ k, 0 < iteration.stepLength k) (hWolfe : ∀ k, LineSearch.IsWeakWolfe c₁ c₂ iteration.objective (iteration.point k) (iteration.point (k + 1) - iteration.point k)) : Tendsto (fun k ↦ ‖iteration.point (k + 1) - iteration.point k‖) atTop (𝓝 0) := by have hsum := summableStepSquares iteration hf hμ hHessian hStep hWolfe have hsqrt := hsum.tendsto_atTop_zero.sqrt simpa only [Real.sqrt_sq (norm_nonneg _), Real.sqrt_zero] using hsqrt

Helper for lem:planar-degeneration: a strong-convexity sublevel set lies in an explicit ball about its initial point.

theorem sublevelRadiusBound {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] {f : E → ℝ} {μ : ℝ} (hf : ContDiff ℝ 2 f) (hμ : 0 < μ) (hHessian : ∀ z v : E, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient f) z v) v) {x₀ x : E} (hx : f x ≤ f x₀) : ‖x - x₀‖ ≤ 2 * ‖gradient f x₀‖ / μ := by have hfirst := hf.firstOrderOfHessianLowerBound f μ hμ hHessian x₀ x have hinner : -(‖gradient f x₀‖ * ‖x - x₀‖) ≤ inner ℝ (gradient f x₀) (x - x₀) := neg_le_of_abs_le (abs_real_inner_le_norm _ _) apply (le_div_iff₀ hμ).mpr by_contra hlarge have hstrict : 2 * ‖gradient f x₀‖ < ‖x - x₀‖ * μ := lt_of_not_ge hlarge have hnorm : 0 < ‖x - x₀‖ := by by_contra hnot have hz : ‖x - x₀‖ = 0 := le_antisymm (le_of_not_gt hnot) (norm_nonneg _) rw [hz, zero_mul] at hstrict nlinarith [norm_nonneg (gradient f x₀)] nlinarith [mul_lt_mul_of_pos_right hstrict hnorm]

Helper for lem:planar-degeneration: in a proper Hilbert space, a positive global Hessian lower bound makes each initial sublevel compact.

theorem compactInitialSublevel {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] [ProperSpace E] {f : E → ℝ} {μ : ℝ} (hf : ContDiff ℝ 2 f) (hμ : 0 < μ) (hHessian : ∀ z v : E, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient f) z v) v) (x₀ : E) : IsCompact {x | f x ≤ f x₀} := by have hclosed : IsClosed {x | f x ≤ f x₀} := isClosed_le hf.continuous continuous_const apply (isCompact_closedBall x₀ (2 * ‖gradient f x₀‖ / μ)).of_isClosed_subset hclosed intro x hx rw [Metric.mem_closedBall, dist_eq_norm] exact sublevelRadiusBound hf hμ hHessian hx

Helper for thm:planar-convergence: the strongly convex objective attains a global minimum, obtained by minimizing on an initial sublevel.

theorem existsGlobalMinimizerOfHessian {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] [ProperSpace E] {f : E → ℝ} {μ : ℝ} (hf : ContDiff ℝ 2 f) (hμ : 0 < μ) (hHessian : ∀ z v : E, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient f) z v) v) : ∃ xstar, ∀ x, f xstar ≤ f x := by have hcompact := compactInitialSublevel hf hμ hHessian (0 : E) have hzero : (0 : E) ∈ {x | f x ≤ f 0} := by simp have hnonempty : ({x | f x ≤ f 0} : Set E).Nonempty := ⟨0, hzero⟩ obtain ⟨xstar, hxstar, hmin⟩ := hcompact.exists_isMinOn hnonempty hf.continuous.continuousOn refine ⟨xstar, ?_⟩ intro x by_cases hx : f x ≤ f 0 · exact hmin hx · have hlevel : f xstar ≤ f 0 := hxstar exact hlevel.trans (le_of_lt (lt_of_not_ge hx))

Helper for thm:planar-convergence: a global minimizer has zero gradient, by Fermat's theorem and the canonical gradient-dual identity.

theorem gradientZeroOfGlobalMinimizer {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] {f : E → ℝ} {xstar : E} (hmin : ∀ x, f xstar ≤ f x) : gradient f xstar = 0 := by have hlocal : IsLocalMin f xstar := Eventually.of_forall hmin apply (InnerProductSpace.toDual ℝ E).injective rw [toDual_gradient, hlocal.fderiv_eq_zero, map_zero]

Helper for thm:planar-convergence: strong convexity turns the objective gap above a stationary point into a bound on the squared point distance.

theorem distanceSquareBoundOfStationaryPoint {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] {f : E → ℝ} {μ : ℝ} (hf : ContDiff ℝ 2 f) (hμ : 0 < μ) (hHessian : ∀ z v : E, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient f) z v) v) {xstar : E} (hstar : gradient f xstar = 0) (x : E) : ‖x - xstar‖ ^ 2 ≤ (2 / μ) * (f x - f xstar) := by have hfirst := hf.firstOrderOfHessianLowerBound f μ hμ hHessian xstar x rw [hstar, inner_zero_left, add_zero] at hfirst rw [div_mul_eq_mul_div] apply (le_div_iff₀ hμ).mpr nlinarith

Helper for thm:planar-convergence: the minimizer of an objective with a positive global Hessian lower bound is unique.

theorem existsUniqueGlobalMinimizerOfHessian {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] [ProperSpace E] {f : E → ℝ} {μ : ℝ} (hf : ContDiff ℝ 2 f) (hμ : 0 < μ) (hHessian : ∀ z v : E, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient f) z v) v) : ∃! xstar, ∀ x, f xstar ≤ f x := by obtain ⟨xstar, hmin⟩ := existsGlobalMinimizerOfHessian hf hμ hHessian refine ⟨xstar, hmin, ?_⟩ intro y hy have hvalues : f y = f xstar := le_antisymm (hy xstar) (hmin y) have hdistance := distanceSquareBoundOfStationaryPoint hf hμ hHessian (gradientZeroOfGlobalMinimizer hmin) y rw [hvalues, sub_self, mul_zero] at hdistance have hnorm : ‖y - xstar‖ = 0 := by nlinarith [norm_nonneg (y - xstar)] exact sub_eq_zero.mp (norm_eq_zero.mp hnorm)

Helper for thm:planar-convergence: convergence of objective values to the minimum implies convergence of points, using the strong-convexity gap.

theorem pointTendstoOfObjectiveTendsto {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] {f : E → ℝ} {μ : ℝ} (hf : ContDiff ℝ 2 f) (hμ : 0 < μ) (hHessian : ∀ z v : E, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient f) z v) v) {xstar : E} (hstar : gradient f xstar = 0) {x : ℕ → E} (hvalues : Tendsto (fun k ↦ f (x k)) atTop (𝓝 (f xstar))) : Tendsto x atTop (𝓝 xstar) := by have hgap : Tendsto (fun k ↦ (2 / μ) * (f (x k) - f xstar)) atTop (𝓝 0) := by simpa using (hvalues.sub_const (f xstar)).const_mul (2 / μ) have hsq : Tendsto (fun k ↦ ‖x k - xstar‖ ^ 2) atTop (𝓝 0) := squeeze_zero (fun k ↦ sq_nonneg ‖x k - xstar‖) (fun k ↦ distanceSquareBoundOfStationaryPoint hf hμ hHessian hstar (x k)) hgap have hnorm : Tendsto (fun k ↦ ‖x k - xstar‖) atTop (𝓝 0) := by simpa only [Real.sqrt_sq (norm_nonneg _), Real.sqrt_zero] using hsq.sqrt exact tendsto_iff_norm_sub_tendsto_zero.mpr hnorm

Helper for lem:planar-degeneration: on the compact initial sublevel, both the gradient and the Hessian have a common finite positive norm bound.

theorem uniformDifferentialBoundsOnSublevel {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] [ProperSpace E] {f : E → ℝ} {μ : ℝ} (hf : ContDiff ℝ 2 f) (hμ : 0 < μ) (hHessian : ∀ z v : E, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient f) z v) v) (x₀ : E) : ∃ M > 0, ∀ x, f x ≤ f x₀ → ‖gradient f x‖ ≤ M ∧ ‖fderiv ℝ (gradient f) x‖ ≤ M := by have hcompact := compactInitialSublevel hf hμ hHessian x₀ have hfOrder : ContDiff ℝ ((1 : ℕ∞) + 1) f := by convert hf using 1 norm_num have hgradient : ContDiff ℝ 1 (gradient f) := hfOrder.gradient_succ (n := 1) have hHessianContinuous : Continuous (fderiv ℝ (gradient f)) := hgradient.continuous_fderiv one_ne_zero obtain ⟨G, hG⟩ := hcompact.exists_bound_of_continuousOn hgradient.continuous.continuousOn obtain ⟨B, hB⟩ := hcompact.exists_bound_of_continuousOn hHessianContinuous.continuousOn refine ⟨max 1 (max G B), ?_, ?_⟩ · exact lt_of_lt_of_le zero_lt_one (le_max_left _ _) · intro x hx constructor · exact (hG x hx).trans ((le_max_left G B).trans (le_max_right _ _)) · exact (hB x hx).trans ((le_max_right G B).trans (le_max_right _ _))

Helper for lem:planar-degeneration: the global Hessian lower bound controls every objective gap by the squared gradient norm.

theorem objectiveGapBoundByGradient {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] {f : E → ℝ} {μ : ℝ} (hf : ContDiff ℝ 2 f) (hμ : 0 < μ) (hHessian : ∀ z v : E, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient f) z v) v) (x y : E) : 2 * μ * (f x - f y) ≤ ‖gradient f x‖ ^ 2 := by have hfirst := hf.firstOrderOfHessianLowerBound f μ hμ hHessian x y have hinner : -(‖gradient f x‖ * ‖y - x‖) ≤ inner ℝ (gradient f x) (y - x) := neg_le_of_abs_le (abs_real_inner_le_norm _ _) have hfactor : 0 ≤ 2 * μ := by positivity have hfirstScaled := mul_le_mul_of_nonneg_left hfirst hfactor have hinnerScaled := mul_le_mul_of_nonneg_left hinner hfactor nlinarith [sq_nonneg (μ * ‖y - x‖ - ‖gradient f x‖)]

Helper for lem:planar-degeneration: objective values along an admissible DFP sequence decrease to a finite limit bounded above by every iterate value.

theorem objectiveValuesHaveLimit {n : ℕ} (iteration : DFP.InverseIteration (Fin n)) {c₁ c₂ : ℝ} (hStep : ∀ k, 0 < iteration.stepLength k) (hWolfe : ∀ k, LineSearch.IsWeakWolfe c₁ c₂ iteration.objective (iteration.point k) (iteration.point (k + 1) - iteration.point k)) {xstar : EuclideanSpace ℝ (Fin n)} (hmin : ∀ x, iteration.objective xstar ≤ iteration.objective x) : ∃ F, Tendsto (fun k ↦ iteration.objective (iteration.point k)) atTop (𝓝 F) ∧ iteration.objective xstar ≤ F ∧ ∀ k, F ≤ iteration.objective (iteration.point k) := by have hanti := iteration.objectiveValuesAntitoneOfWeakWolfe hStep hWolfe have hbounded : BddBelow (Set.range (fun k ↦ iteration.objective (iteration.point k))) := by refine ⟨iteration.objective xstar, ?_⟩ rintro y ⟨k, rfl⟩ exact hmin (iteration.point k) refine ⟨⨅ k, iteration.objective (iteration.point k), tendsto_atTop_ciInf hanti hbounded, ?_, ?_⟩ · exact le_ciInf (fun k ↦ hmin (iteration.point k)) · intro k exact ciInf_le hbounded k

Helper for lem:planar-degeneration: a nonconvergent strongly convex DFP orbit has gradient norms bounded uniformly away from zero.

theorem gradientLowerBoundOfNonconvergence {n : ℕ} (iteration : DFP.InverseIteration (Fin n)) {μ c₁ c₂ : ℝ} (hf : ContDiff ℝ 2 iteration.objective) (hμ : 0 < μ) (hHessian : ∀ x v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient iteration.objective) x v) v) (hStep : ∀ k, 0 < iteration.stepLength k) (hWolfe : ∀ k, LineSearch.IsWeakWolfe c₁ c₂ iteration.objective (iteration.point k) (iteration.point (k + 1) - iteration.point k)) {xstar : EuclideanSpace ℝ (Fin n)} (hmin : ∀ x, iteration.objective xstar ≤ iteration.objective x) (hnot : ¬ Tendsto iteration.point atTop (𝓝 xstar)) : ∃ Nmin > 0, ∀ k, Nmin ≤ ‖gradient iteration.objective (iteration.point k)‖ := by obtain ⟨F, hF, hminF, hFk⟩ := objectiveValuesHaveLimit iteration hStep hWolfe hmin have hstrict : iteration.objective xstar < F := by apply lt_of_le_of_ne hminF intro heq apply hnot apply pointTendstoOfObjectiveTendsto hf hμ hHessian (gradientZeroOfGlobalMinimizer hmin) rwa [← heq] at hF have hgap : 0 < 2 * μ * (F - iteration.objective xstar) := by positivity refine ⟨Real.sqrt (2 * μ * (F - iteration.objective xstar)), Real.sqrt_pos.mpr hgap, ?_⟩ intro k have hbound := objectiveGapBoundByGradient hf hμ hHessian (iteration.point k) xstar have hfactor : 0 ≤ 2 * μ := by positivity have hvalue := mul_le_mul_of_nonneg_left (hFk k) hfactor have hsqrt := Real.sq_sqrt hgap.le nlinarith [norm_nonneg (gradient iteration.objective (iteration.point k))]

Helper for lem:planar-degeneration: a global positive Hessian lower bound makes the initial sublevel convex, so all accepted step segments stay inside it.

theorem convexInitialSublevel {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] {f : E → ℝ} {μ : ℝ} (hf : ContDiff ℝ 2 f) (hμ : 0 < μ) (hHessian : ∀ z v : E, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient f) z v) v) (x₀ : E) : Convex ℝ {x | f x ≤ f x₀} := by have hstrong := hf.strongConvexOnOfHessianLowerBound f μ hμ hHessian have hconvex := (hstrong.strictConvexOn hμ).convexOn simpa only [Set.mem_univ, true_and] using hconvex.convex_le (f x₀)

Helper for lem:planar-degeneration: on the actual DFP orbit, secant norms and predicted decreases have common quadratic-scale bounds.

theorem secantBoundsAlongOrbit {n : ℕ} (iteration : DFP.InverseIteration (Fin n)) {μ c₁ c₂ : ℝ} (hf : ContDiff ℝ 2 iteration.objective) (hμ : 0 < μ) (hHessian : ∀ x v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient iteration.objective) x v) v) (hStep : ∀ k, 0 < iteration.stepLength k) (hWolfe : ∀ k, LineSearch.IsWeakWolfe c₁ c₂ iteration.objective (iteration.point k) (iteration.point (k + 1) - iteration.point k)) : ∃ M > 0, ∀ k, ‖gradient iteration.objective (iteration.point (k + 1)) - gradient iteration.objective (iteration.point k)‖ ≤ M * ‖iteration.point (k + 1) - iteration.point k‖ ∧ -inner ℝ (gradient iteration.objective (iteration.point k)) (iteration.point (k + 1) - iteration.point k) ≤ (M / (1 - c₂)) * ‖iteration.point (k + 1) - iteration.point k‖ ^ 2 := by obtain ⟨M, hM, hbound⟩ := uniformDifferentialBoundsOnSublevel hf hμ hHessian (iteration.point 0) have hconvex := convexInitialSublevel hf hμ hHessian (iteration.point 0) have hmem := iteration.pointMemObjectiveSublevelOfWeakWolfe hStep hWolfe have hfOrder : ContDiff ℝ ((1 : ℕ∞) + 1) iteration.objective := by convert hf using 1 norm_num have hgradient : ContDiff ℝ 1 (gradient iteration.objective) := hfOrder.gradient_succ (n := 1) refine ⟨M, hM, ?_⟩ intro k have hsec : ‖gradient iteration.objective (iteration.point (k + 1)) - gradient iteration.objective (iteration.point k)‖ ≤ M * ‖iteration.point (k + 1) - iteration.point k‖ := hconvex.norm_image_sub_le_of_norm_fderiv_le (fun x _ ↦ hgradient.differentiable_one.differentiableAt) (fun x hx ↦ (hbound x hx).2) ((DFP.mem_objectiveSublevel_iff iteration (iteration.point k)).mp (hmem k)) ((DFP.mem_objectiveSublevel_iff iteration (iteration.point (k + 1))).mp (hmem (k + 1))) refine ⟨hsec, ?_⟩ have hcurvature := (hWolfe k).weakCurvature rw [add_sub_cancel] at hcurvature have hpair := real_inner_le_norm (gradient iteration.objective (iteration.point (k + 1)) - gradient iteration.objective (iteration.point k)) (iteration.point (k + 1) - iteration.point k) rw [inner_sub_left] at hpair have hsecScaled := mul_le_mul_of_nonneg_right hsec (norm_nonneg (iteration.point (k + 1) - iteration.point k)) have hden : 0 < 1 - c₂ := sub_pos.mpr (hWolfe k).c₂_lt_one rw [div_mul_eq_mul_div] apply (le_div_iff₀ hden).mpr nlinarith

Helper for lem:planar-degeneration: orthogonal coordinate changes preserve positive definiteness of the inverse Hessian.

theorem orthogonalCoordinatesPosDef {H R : Matrix (Fin 2) (Fin 2) ℝ} (hH : H.PosDef) (hR : R ∈ Matrix.orthogonalGroup (Fin 2) ℝ) : (R.transpose * H * R).PosDef := by have hcancel : R.transpose * R = 1 := (Matrix.mem_orthogonalGroup_iff' (Fin 2) ℝ).mp hR have hleft : Function.LeftInverse R.transpose.mulVec R.mulVec := by intro v rw [Matrix.mulVec_mulVec, hcancel, Matrix.one_mulVec] simpa only [Matrix.conjTranspose_eq_transpose_of_trivial] using hH.conjTranspose_mul_mul_same hleft.injective

Helper for lem:planar-degeneration: orthogonal coordinate changes preserve the determinant used to define the free secant coefficient.

theorem orthogonalCoordinatesDet (H R : Matrix (Fin 2) (Fin 2) ℝ) (hR : R ∈ Matrix.orthogonalGroup (Fin 2) ℝ) : (R.transpose * H * R).det = H.det := by have hcancel : R.transpose * R = 1 := (Matrix.mem_orthogonalGroup_iff' (Fin 2) ℝ).mp hR have hdet : R.det * R.det = 1 := by simpa only [Matrix.det_mul, Matrix.det_transpose, Matrix.det_one] using congrArg Matrix.det hcancel rw [Matrix.det_mul, Matrix.det_mul, Matrix.det_transpose] calc R.det * H.det * R.det = H.det * (R.det * R.det) := by ring _ = H.det := by rw [hdet, mul_one]

Helper for lem:planar-degeneration: the physical secant equation transports to the corresponding equation in an orthonormal frame.

theorem orthogonalCoordinatesSecant {H R : Matrix (Fin 2) (Fin 2) ℝ} (hR : R ∈ Matrix.orthogonalGroup (Fin 2) ℝ) {u v : Fin 2 → ℝ} (hsec : H *ᵥ (R *ᵥ v) = R *ᵥ u) : (R.transpose * H * R) *ᵥ v = u := by have hcancel : R.transpose * R = 1 := (Matrix.mem_orthogonalGroup_iff' (Fin 2) ℝ).mp hR calc (R.transpose * H * R) *ᵥ v = R.transpose *ᵥ (H *ᵥ (R *ᵥ v)) := by rw [Matrix.mulVec_mulVec, Matrix.mulVec_mulVec, Matrix.mul_assoc] _ = R.transpose *ᵥ (R *ᵥ u) := by rw [hsec] _ = u := by rw [Matrix.mulVec_mulVec, hcancel, Matrix.one_mulVec]

Helper for lem:planar-degeneration: the transverse matrix entry in any secant-aligned orthonormal frame equals curvature times the physical determinant.

theorem orthogonalSecantFreeCoefficient {H R : Matrix (Fin 2) (Fin 2) ℝ} {κ β : ℝ} (hH : H.PosDef) (hR : R ∈ Matrix.orthogonalGroup (Fin 2) ℝ) (hκ : κ ≠ 0) (hsec : H *ᵥ (R *ᵥ ![κ, β]) = R *ᵥ ![1, 0]) : (R.transpose * H * R) 1 1 = κ * H.det := by have hK := orthogonalCoordinatesPosDef hH hR have hsecK := orthogonalCoordinatesSecant hR hsec rw [transverseEntry_eq_curvature_mul_det hK.isHermitian hκ hsecK, orthogonalCoordinatesDet H R hR]

Helper for lem:planar-degeneration: the secant-coordinate quadratic form splits into a nonnegative longitudinal square and a free-coefficient square.

theorem secantQuadraticDecomposition {H : Matrix (Fin 2) (Fin 2) ℝ} {κ β : ℝ} (hH : H.IsHermitian) (hκ : κ ≠ 0) (hsec : H *ᵥ ![κ, β] = ![1, 0]) (g : Fin 2 → ℝ) : g ⬝ᵥ (H *ᵥ g) = (g 0) ^ 2 / κ + H 1 1 * (g 1 - β / κ * g 0) ^ 2 := by have hsym : H 1 0 = H 0 1 := by simpa using hH.apply 0 1 simp only [Matrix.mulVec, dotProduct, Fin.sum_univ_two] rw [hsym, secantLongitudinal hH hκ hsec, secantOffDiagonal hH hκ hsec] field_simp ring

Helper for lem:planar-degeneration: the transverse square gives a lower bound on the secant-coordinate gradient energy.

theorem secantEnergyLowerBound {H : Matrix (Fin 2) (Fin 2) ℝ} {κ β : ℝ} (hH : H.IsHermitian) (hκ : 0 < κ) (hsec : H *ᵥ ![κ, β] = ![1, 0]) (g : Fin 2 → ℝ) : H 1 1 * (g 1 - β / κ * g 0) ^ 2 ≤ g ⬝ᵥ (H *ᵥ g) := by rw [secantQuadraticDecomposition hH hκ.ne' hsec g] exact le_add_of_nonneg_left (div_nonneg (sq_nonneg _) hκ.le)

Helper for lem:planar-degeneration: the secant-coordinate search direction is the sum of its longitudinal and free-coefficient components.

theorem secantActionDecomposition {H : Matrix (Fin 2) (Fin 2) ℝ} {κ β : ℝ} (hH : H.IsHermitian) (hκ : κ ≠ 0) (hsec : H *ᵥ ![κ, β] = ![1, 0]) (g : Fin 2 → ℝ) : (WithLp.toLp 2 (H *ᵥ g) : EuclideanSpace ℝ (Fin 2)) = (g 0 / κ) • WithLp.toLp 2 ![1, 0] + (H 1 1 * (g 1 - β / κ * g 0)) • WithLp.toLp 2 ![-β / κ, 1] := by have hsym : H 1 0 = H 0 1 := by simpa using hH.apply 0 1 ext i fin_cases i · norm_num [Matrix.mulVec, dotProduct, Fin.sum_univ_two, PiLp.toLp_apply, PiLp.add_apply, PiLp.smul_apply, secantLongitudinal hH hκ hsec, secantOffDiagonal hH hκ hsec] field_simp ring · norm_num [Matrix.mulVec, dotProduct, Fin.sum_univ_two, PiLp.toLp_apply, PiLp.add_apply, PiLp.smul_apply, hsym, secantOffDiagonal hH hκ hsec] field_simp ring

Helper for lem:planar-degeneration: the exact secant decomposition bounds the search-direction norm by the longitudinal component and the free coefficient.

theorem secantActionNormBound {H : Matrix (Fin 2) (Fin 2) ℝ} {κ β : ℝ} (hH : H.PosDef) (hκ : 0 < κ) (hsec : H *ᵥ ![κ, β] = ![1, 0]) (g : Fin 2 → ℝ) : ‖(WithLp.toLp 2 (H *ᵥ g) : EuclideanSpace ℝ (Fin 2))‖ ≤ |g 0| / κ + H 1 1 * |g 1 - β / κ * g 0| * ‖(WithLp.toLp 2 ![-β / κ, 1] : EuclideanSpace ℝ (Fin 2))‖ := by have hunit : ‖(WithLp.toLp 2 ![1, 0] : EuclideanSpace ℝ (Fin 2))‖ = 1 := by simp [EuclideanSpace.norm_eq, Fin.sum_univ_two] rw [secantActionDecomposition hH.isHermitian hκ.ne' hsec g] have htriangle := norm_add_le ((g 0 / κ) • (WithLp.toLp 2 ![1, 0] : EuclideanSpace ℝ (Fin 2))) ((H 1 1 * (g 1 - β / κ * g 0)) • (WithLp.toLp 2 ![-β / κ, 1] : EuclideanSpace ℝ (Fin 2))) simpa only [norm_smul, Real.norm_eq_abs, abs_div, abs_mul, abs_of_pos hκ, abs_of_pos hH.diag_pos, hunit, mul_one] using htriangle

Helper for lem:planar-degeneration: a quadratic bound on the predicted decrease of a positive multiple of -v gives the corresponding energy-to-direction bound.

theorem energyDirectionBound {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {g v s : E} {α C : ℝ} (hα : 0 < α) (hstep : s = α • (-v)) (hdescent : -inner ℝ g s ≤ C * ‖s‖ ^ 2) : inner ℝ g v ≤ C * ‖s‖ * ‖v‖ := by have hnorm : ‖s‖ = α * ‖v‖ := by rw [hstep, norm_smul, norm_neg, Real.norm_eq_abs, abs_of_pos hα] have hpair : -inner ℝ g s = α * inner ℝ g v := by rw [hstep, inner_smul_right, inner_neg_right] ring rw [hpair, hnorm] at hdescent rw [hnorm] apply (mul_le_mul_iff_right₀ hα).mp nlinarith [hdescent]

Helper for lem:planar-degeneration: the Euclidean norm of a planar vector is bounded by the sum of its coordinate absolute values.

theorem planarNormLeCoordinateAbsSum (g : Fin 2 → ℝ) : ‖(WithLp.toLp 2 g : EuclideanSpace ℝ (Fin 2))‖ ≤ |g 0| + |g 1| := by have hsq := EuclideanSpace.real_norm_sq_eq (WithLp.toLp 2 g) simp only [Fin.sum_univ_two] at hsq nlinarith [sq_abs (g 0), sq_abs (g 1), abs_nonneg (g 0), abs_nonneg (g 1), norm_nonneg (WithLp.toLp 2 g), mul_nonneg (abs_nonneg (g 0)) (abs_nonneg (g 1))]

Helper for lem:planar-degeneration: a bounded secant tilt and a small longitudinal gradient component force a lower bound on the transverse pairing.

theorem transversePairingLowerBound (g : Fin 2 → ℝ) {b B N : ℝ} (htilt : |b| ≤ B) (hgradient : N ≤ ‖(WithLp.toLp 2 g : EuclideanSpace ℝ (Fin 2))‖) (hlongitudinal : (1 + B) * |g 0| ≤ N / 2) : N / 2 ≤ |g 1 - b * g 0| := by have hnorm := planarNormLeCoordinateAbsSum g have htriangle := abs_add_le (g 1 - b * g 0) (b * g 0) rw [sub_add_cancel, abs_mul] at htriangle have htiltProduct := mul_le_mul_of_nonneg_right htilt (abs_nonneg (g 0)) linarith

Helper for lem:planar-degeneration: a nonvanishing gradient and a small longitudinal component give a uniform multiple of the free coefficient as an energy lower bound.

theorem secantEnergyLowerBoundOfGradientNorm {H : Matrix (Fin 2) (Fin 2) ℝ} {κ β B N : ℝ} (hH : H.PosDef) (hκ : 0 < κ) (hsec : H *ᵥ ![κ, β] = ![1, 0]) (g : Fin 2 → ℝ) (hN : 0 ≤ N) (htilt : |β / κ| ≤ B) (hgradient : N ≤ ‖(WithLp.toLp 2 g : EuclideanSpace ℝ (Fin 2))‖) (hlongitudinal : (1 + B) * |g 0| ≤ N / 2) : (N ^ 2 / 4) * H 1 1 ≤ g ⬝ᵥ (H *ᵥ g) := by have hpairing := transversePairingLowerBound g htilt hgradient hlongitudinal have hsquare : N ^ 2 / 4 ≤ (g 1 - β / κ * g 0) ^ 2 := by nlinarith [sq_abs (g 1 - β / κ * g 0)] have hscaled := mul_le_mul_of_nonneg_left hsquare (hH.diag_pos (i := 1)).le have henergy := secantEnergyLowerBound hH.isHermitian hκ hsec g nlinarith

Helper for lem:planar-degeneration: the transpose of an orthogonal frame is orthogonal as well.

theorem orthogonalTranspose {R : Matrix (Fin 2) (Fin 2) ℝ} (hR : R ∈ Matrix.orthogonalGroup (Fin 2) ℝ) : R.transpose ∈ Matrix.orthogonalGroup (Fin 2) ℝ := by apply (Matrix.mem_orthogonalGroup_iff (Fin 2) ℝ).mpr rw [Matrix.transpose_transpose] exact (Matrix.mem_orthogonalGroup_iff' (Fin 2) ℝ).mp hR

Helper for lem:planar-degeneration: the inverse-Hessian action commutes with passage to orthonormal secant coordinates.

theorem orthogonalCoordinatesAction (H R : Matrix (Fin 2) (Fin 2) ℝ) (hR : R ∈ Matrix.orthogonalGroup (Fin 2) ℝ) (g : Fin 2 → ℝ) : (R.transpose * H * R) *ᵥ (R.transpose *ᵥ g) = R.transpose *ᵥ (H *ᵥ g) := by simpa only [Matrix.transpose_transpose] using Matrix.conjugate_mulVec_of_mem_orthogonalGroup R.transpose H (orthogonalTranspose hR) g

Helper for lem:planar-degeneration: the physical gradient energy agrees with its expression in orthonormal secant coordinates.

theorem orthogonalCoordinatesEnergy (H R : Matrix (Fin 2) (Fin 2) ℝ) (hR : R ∈ Matrix.orthogonalGroup (Fin 2) ℝ) (g : Fin 2 → ℝ) : (R.transpose *ᵥ g) ⬝ᵥ ((R.transpose * H * R) *ᵥ (R.transpose *ᵥ g)) = g ⬝ᵥ (H *ᵥ g) := by rw [orthogonalCoordinatesAction H R hR g] exact Matrix.dotProduct_mulVec_eq_of_mem_orthogonalGroup R.transpose (orthogonalTranspose hR) g (H *ᵥ g)

Helper for lem:planar-degeneration: the physical search-direction norm agrees with its expression in orthonormal secant coordinates.

theorem orthogonalCoordinatesActionNorm (H R : Matrix (Fin 2) (Fin 2) ℝ) (hR : R ∈ Matrix.orthogonalGroup (Fin 2) ℝ) (g : Fin 2 → ℝ) : ‖(WithLp.toLp 2 ((R.transpose * H * R) *ᵥ (R.transpose *ᵥ g)) : EuclideanSpace ℝ (Fin 2))‖ = ‖WithLp.toLp 2 (H *ᵥ g)‖ := by rw [orthogonalCoordinatesAction H R hR g] exact Matrix.norm_toLp_mulVec_eq_of_mem_orthogonalGroup R.transpose (orthogonalTranspose hR) (H *ᵥ g)

Helper for lem:planar-degeneration: the weak-Wolfe upper descent bound controls the actual inverse-Hessian energy relative to its direction norm.

theorem orbitEnergyDirectionBound {n : ℕ} (iteration : DFP.InverseIteration (Fin n)) {μ c₁ c₂ : ℝ} (hf : ContDiff ℝ 2 iteration.objective) (hμ : 0 < μ) (hHessian : ∀ x v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient iteration.objective) x v) v) (hStep : ∀ k, 0 < iteration.stepLength k) (hWolfe : ∀ k, LineSearch.IsWeakWolfe c₁ c₂ iteration.objective (iteration.point k) (iteration.point (k + 1) - iteration.point k)) : ∃ C > 0, ∀ k, let g := gradient iteration.objective (iteration.point k) let v : EuclideanSpace ℝ (Fin n) := WithLp.toLp 2 (iteration.inverseHessian k *ᵥ WithLp.ofLp g) inner ℝ g v ≤ C * ‖iteration.point (k + 1) - iteration.point k‖ * ‖v‖ := by obtain ⟨M, hM, hbounds⟩ := secantBoundsAlongOrbit iteration hf hμ hHessian hStep hWolfe have hden : 0 < 1 - c₂ := sub_pos.mpr (hWolfe 0).c₂_lt_one refine ⟨M / (1 - c₂), div_pos hM hden, ?_⟩ intro k have hstep : iteration.point (k + 1) - iteration.point k = iteration.stepLength k • (-(WithLp.toLp 2 (iteration.inverseHessian k *ᵥ WithLp.ofLp (gradient iteration.objective (iteration.point k))) : EuclideanSpace ℝ (Fin n))) := by rw [iteration.pointSucc k, DFP.steps_apply, DFP.directions_apply, DFP.gradients_apply] abel exact energyDirectionBound (hStep k) hstep (hbounds k).2

Helper for lem:planar-degeneration: every step of an infinite positive-step DFP orbit is nonzero.

theorem orbitStepNormPos {n : ℕ} (iteration : DFP.InverseIteration (Fin n)) (k : ℕ) (hStep : 0 < iteration.stepLength k) : 0 < ‖iteration.point (k + 1) - iteration.point k‖ := by apply norm_pos_iff.mpr intro hzero have hdescent := iteration.gradientInnerDisplacementNeg k hStep rw [hzero, inner_zero_right] at hdescent exact lt_irrefl 0 hdescent

Helper for lem:planar-degeneration: the original DFP recurrence implies the exact secant equation between actual point and gradient differences.

theorem orbitSecantEquation {n : ℕ} (iteration : DFP.InverseIteration (Fin n)) (k : ℕ) : iteration.inverseHessian (k + 1) *ᵥ WithLp.ofLp (gradient iteration.objective (iteration.point (k + 1)) - gradient iteration.objective (iteration.point k)) = WithLp.ofLp (iteration.point (k + 1) - iteration.point k) := by have hden := iteration.denominators_spec k have hsec := Matrix.inverseDFPUpdate_mulVec_secant hden.1 hden.2 rw [← iteration.inverseHessianSucc k] at hsec have hstep : iteration.point (k + 1) - iteration.point k = DFP.steps iteration.stepLength (DFP.directions iteration.inverseHessian (DFP.gradients iteration.objective iteration.point)) k := by rw [iteration.pointSucc k] abel simpa only [DFP.gradientChanges_apply, DFP.gradients_apply, hstep] using hsec

Helper for lem:planar-degeneration: normalizing the displacement and gradient change by the same step length preserves the exact secant equation.

theorem normalizedOrbitSecantEquation {n : ℕ} (iteration : DFP.InverseIteration (Fin n)) (k : ℕ) : iteration.inverseHessian (k + 1) *ᵥ WithLp.ofLp (‖iteration.point (k + 1) - iteration.point k‖⁻¹ • (gradient iteration.objective (iteration.point (k + 1)) - gradient iteration.objective (iteration.point k))) = WithLp.ofLp (‖iteration.point (k + 1) - iteration.point k‖⁻¹ • (iteration.point (k + 1) - iteration.point k)) := by simp only [WithLp.ofLp_smul, Matrix.mulVec_smul, orbitSecantEquation]

Helper for lem:planar-degeneration: the new gradient's component along a normalized old step is small whenever descent and secants have quadratic-scale bounds.

theorem longitudinalGradientBound {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] (g gNext s : E) {C M : ℝ} (hdescent : inner ℝ g s ≤ 0) (hdecrease : -inner ℝ g s ≤ C * ‖s‖ ^ 2) (hsecant : ‖gNext - g‖ ≤ M * ‖s‖) : |inner ℝ (‖s‖⁻¹ • s) gNext| ≤ (C + M) * ‖s‖ := by by_cases hs : s = 0 · simp [hs] have hnorm : 0 < ‖s‖ := norm_pos_iff.mpr hs have hsplit : inner ℝ s gNext = inner ℝ g s + inner ℝ (gNext - g) s := by rw [inner_sub_left, real_inner_comm s gNext] ring have htriangle := abs_add_le (inner ℝ g s) (inner ℝ (gNext - g) s) rw [abs_of_nonpos hdescent] at htriangle have hsecantPair := abs_real_inner_le_norm (gNext - g) s have hsecantScaled := mul_le_mul_of_nonneg_right hsecant (norm_nonneg s) have htotal : |inner ℝ s gNext| ≤ (C + M) * ‖s‖ ^ 2 := by rw [hsplit] nlinarith rw [real_inner_smul_left, abs_mul, abs_inv, abs_norm, ← div_eq_inv_mul] apply (div_le_iff₀ hnorm).mpr nlinarith

Helper for lem:planar-degeneration: on the actual DFP orbit the new gradient has a uniformly small component along the previous normalized step.

theorem orbitLongitudinalGradientBound {n : ℕ} (iteration : DFP.InverseIteration (Fin n)) {μ c₁ c₂ : ℝ} (hf : ContDiff ℝ 2 iteration.objective) (hμ : 0 < μ) (hHessian : ∀ x v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient iteration.objective) x v) v) (hStep : ∀ k, 0 < iteration.stepLength k) (hWolfe : ∀ k, LineSearch.IsWeakWolfe c₁ c₂ iteration.objective (iteration.point k) (iteration.point (k + 1) - iteration.point k)) : ∃ C > 0, ∀ k, |inner ℝ (‖iteration.point (k + 1) - iteration.point k‖⁻¹ • (iteration.point (k + 1) - iteration.point k)) (gradient iteration.objective (iteration.point (k + 1)))| ≤ C * ‖iteration.point (k + 1) - iteration.point k‖ := by obtain ⟨M, hM, hbounds⟩ := secantBoundsAlongOrbit iteration hf hμ hHessian hStep hWolfe have hden : 0 < 1 - c₂ := sub_pos.mpr (hWolfe 0).c₂_lt_one have hC : 0 < M / (1 - c₂) + M := add_pos (div_pos hM hden) hM refine ⟨M / (1 - c₂) + M, hC, ?_⟩ intro k have hdescent : inner ℝ (gradient iteration.objective (iteration.point k)) (iteration.point (k + 1) - iteration.point k) ≤ 0 := by simpa only [DFP.gradients_apply] using (iteration.gradientInnerDisplacementNeg k (hStep k)).le exact longitudinalGradientBound _ _ _ hdescent (hbounds k).2 (hbounds k).1

Helper for lem:planar-degeneration: simultaneous secant normalization expresses curvature as the secant pairing divided by squared step length.

theorem normalizedSecantCurvature {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] (s y : E) : inner ℝ (‖s‖⁻¹ • s) (‖s‖⁻¹ • y) = inner ℝ y s / ‖s‖ ^ 2 := by rw [real_inner_smul_left, real_inner_smul_right, real_inner_comm s y] ring

Helper for lem:planar-degeneration: the normalized secant curvature of every physical DFP step is bounded below by the strong-convexity constant.

theorem normalizedOrbitCurvatureLowerBound {n : ℕ} (iteration : DFP.InverseIteration (Fin n)) {μ : ℝ} (hf : ContDiff ℝ 2 iteration.objective) (hμ : 0 < μ) (hHessian : ∀ x v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient iteration.objective) x v) v) (k : ℕ) (hStep : 0 < iteration.stepLength k) : let s := iteration.point (k + 1) - iteration.point k let y := gradient iteration.objective (iteration.point (k + 1)) - gradient iteration.objective (iteration.point k) μ ≤ inner ℝ (‖s‖⁻¹ • s) (‖s‖⁻¹ • y) := by dsimp only rw [normalizedSecantCurvature] apply (le_div_iff₀ (sq_pos_of_pos (orbitStepNormPos iteration k hStep))).mpr exact inner_gradient_sub_ge_of_hessian_lower_bound iteration.objective μ hf hμ hHessian (iteration.point k) (iteration.point (k + 1))

Helper for lem:planar-degeneration: the first column of the canonical oriented planar frame is its defining vector.

theorem planarFrameFirstColumn (u : EuclideanSpace ℝ (Fin 2)) (i : Fin 2) : EuclideanPlane.frame u i 0 = u i := by have hcolumn := congrFun (EuclideanPlane.frame_mulVec u 1 0) i simpa [Matrix.mulVec, dotProduct, Fin.sum_univ_two] using hcolumn

Helper for lem:planar-degeneration: the second column of the canonical oriented planar frame is the positively rotated defining vector.

theorem planarFrameSecondColumn (u : EuclideanSpace ℝ (Fin 2)) (i : Fin 2) : EuclideanPlane.frame u i 1 = EuclideanPlane.perp u i := by have hcolumn := congrFun (EuclideanPlane.frame_mulVec u 0 1) i simpa [Matrix.mulVec, dotProduct, Fin.sum_univ_two] using hcolumn

Helper for lem:planar-degeneration: coordinates in the canonical planar frame are the two inner products with its columns.

theorem planarFrameCoordinates (u v : EuclideanSpace ℝ (Fin 2)) : (EuclideanPlane.frame u).transpose *ᵥ WithLp.ofLp v = ![inner ℝ u v, inner ℝ (EuclideanPlane.perp u) v] := by ext i fin_cases i · simp [Matrix.mulVec, dotProduct, Fin.sum_univ_two, Matrix.transpose_apply, planarFrameFirstColumn, EuclideanSpace.inner_eq_star_dotProduct] ring · simp [Matrix.mulVec, dotProduct, Fin.sum_univ_two, Matrix.transpose_apply, planarFrameSecondColumn, EuclideanSpace.inner_eq_star_dotProduct] ring

Helper for lem:planar-degeneration: normalizing a nonzero step gives an orthogonal frame with that step as its first direction.

theorem normalizedStepFrameOrthogonal (s : EuclideanSpace ℝ (Fin 2)) (hs : s ≠ 0) : EuclideanPlane.frame (‖s‖⁻¹ • s) ∈ Matrix.orthogonalGroup (Fin 2) ℝ := by have hnorm : ‖‖s‖⁻¹ • s‖ = 1 := by rw [norm_smul, Real.norm_eq_abs, abs_inv, abs_norm, inv_mul_cancel₀ (norm_ne_zero_iff.mpr hs)] have hspecial := (EuclideanPlane.frame_mem_specialOrthogonalGroup_iff _).mpr hnorm exact (Matrix.mem_specialOrthogonalGroup_iff.mp hspecial).1

Helper for lem:planar-degeneration: a unit planar step and its secant vector give the standard secant equation in the step's oriented frame.

theorem planarSecantFrameSpecification (H : Matrix (Fin 2) (Fin 2) ℝ) (u v : EuclideanSpace ℝ (Fin 2)) (hu : ‖u‖ = 1) (hsec : H *ᵥ WithLp.ofLp v = WithLp.ofLp u) : ((EuclideanPlane.frame u).transpose * H * EuclideanPlane.frame u) *ᵥ ![inner ℝ u v, inner ℝ (EuclideanPlane.perp u) v] = ![1, 0] := by have hspecial := (EuclideanPlane.frame_mem_specialOrthogonalGroup_iff u).mpr hu have hR := (Matrix.mem_specialOrthogonalGroup_iff.mp hspecial).1 have hcancel : EuclideanPlane.frame u * (EuclideanPlane.frame u).transpose = 1 := (Matrix.mem_orthogonalGroup_iff (Fin 2) ℝ).mp hR have hvector : EuclideanPlane.frame u *ᵥ ![inner ℝ u v, inner ℝ (EuclideanPlane.perp u) v] = WithLp.ofLp v := by rw [← planarFrameCoordinates, Matrix.mulVec_mulVec, hcancel, Matrix.one_mulVec] have hfirst : EuclideanPlane.frame u *ᵥ ![1, 0] = WithLp.ofLp u := by ext i simpa [Matrix.mulVec, dotProduct, Fin.sum_univ_two] using planarFrameFirstColumn u i have hsource : H *ᵥ (EuclideanPlane.frame u *ᵥ ![inner ℝ u v, inner ℝ (EuclideanPlane.perp u) v]) = EuclideanPlane.frame u *ᵥ ![1, 0] := by rw [hvector, hfirst] exact hsec exact orthogonalCoordinatesSecant hR hsource

Helper for lem:planar-degeneration: in the previous-step frame, the actual DFP matrix satisfies the standard planar secant equation.

theorem orbitPlanarSecantFrameSpecification (iteration : DFP.InverseIteration (Fin 2)) (k : ℕ) (hStep : 0 < iteration.stepLength k) : let s := iteration.point (k + 1) - iteration.point k let u := ‖s‖⁻¹ • s let v := ‖s‖⁻¹ • (gradient iteration.objective (iteration.point (k + 1)) - gradient iteration.objective (iteration.point k)) ((EuclideanPlane.frame u).transpose * iteration.inverseHessian (k + 1) * EuclideanPlane.frame u) *ᵥ ![inner ℝ u v, inner ℝ (EuclideanPlane.perp u) v] = ![1, 0] := by dsimp only have hnorm : ‖‖iteration.point (k + 1) - iteration.point k‖⁻¹ • (iteration.point (k + 1) - iteration.point k)‖ = 1 := by rw [norm_smul, Real.norm_eq_abs, abs_inv, abs_norm, inv_mul_cancel₀ (orbitStepNormPos iteration k hStep).ne'] exact planarSecantFrameSpecification _ _ _ hnorm (normalizedOrbitSecantEquation iteration k)

Helper for lem:planar-degeneration: a positive curvature lower bound and a bound on the transverse secant control its normalized tilt.

theorem secantTiltBound {μ κ β M : ℝ} (hμ : 0 < μ) (hκ : μ ≤ κ) (hβ : |β| ≤ M) : |β / κ| ≤ M / μ := by have hκpos := hμ.trans_le hκ have hM : 0 ≤ M := (abs_nonneg β).trans hβ rw [abs_div, abs_of_pos hκpos] apply (div_le_iff₀ hκpos).mpr calc |β| ≤ M := hβ _ = (M / μ) * μ := (div_mul_cancel₀ M hμ.ne').symm _ ≤ (M / μ) * κ := mul_le_mul_of_nonneg_left hκ (div_nonneg hM hμ.le)

Helper for lem:planar-degeneration: uniformly bounded gradients and secant tilts bound the matrix action by the old step scale plus the free coefficient.

theorem secantActionUniformBound {H : Matrix (Fin 2) (Fin 2) ℝ} {κ β μ A B G ell : ℝ} (hH : H.PosDef) (hμ : 0 < μ) (hκ : μ ≤ κ) (hsec : H *ᵥ ![κ, β] = ![1, 0]) (g : Fin 2 → ℝ) (hA : 0 ≤ A) (hB : 0 ≤ B) (hG : 0 ≤ G) (hell : 0 ≤ ell) (htilt : |β / κ| ≤ B) (hgradient : ‖(WithLp.toLp 2 g : EuclideanSpace ℝ (Fin 2))‖ ≤ G) (hlongitudinal : |g 0| ≤ A * ell) : ‖(WithLp.toLp 2 (H *ᵥ g) : EuclideanSpace ℝ (Fin 2))‖ ≤ (A / μ + G * (1 + B) ^ 2) * (ell + H 1 1) := by have hκpos := hμ.trans_le hκ have hfree : 0 ≤ H 1 1 := (hH.diag_pos (i := 1)).le have hcoord (i : Fin 2) : |g i| ≤ G := by have hbound := (PiLp.norm_apply_le (WithLp.toLp 2 g) i).trans hgradient simpa only [Real.norm_eq_abs] using hbound have hpair : |g 1 - β / κ * g 0| ≤ (1 + B) * G := by have htriangle := abs_sub (g 1) (β / κ * g 0) rw [abs_mul] at htriangle have hproduct := mul_le_mul htilt (hcoord 0) (abs_nonneg (g 0)) hB nlinarith [hcoord 1] have hvector : ‖(WithLp.toLp 2 ![-β / κ, 1] : EuclideanSpace ℝ (Fin 2))‖ ≤ 1 + B := by have hnorm := planarNormLeCoordinateAbsSum ![-β / κ, 1] simp only [Matrix.cons_val_zero, Matrix.cons_val_one, abs_one, neg_div, abs_neg] at hnorm have hsum : |β / κ| + 1 ≤ 1 + B := by linarith simpa only [neg_div] using hnorm.trans hsum have hlong : |g 0| / κ ≤ (A / μ) * ell := by apply (div_le_iff₀ hκpos).mpr calc |g 0| ≤ A * ell := hlongitudinal _ = ((A / μ) * ell) * μ := by field_simp _ ≤ ((A / μ) * ell) * κ := mul_le_mul_of_nonneg_left hκ (mul_nonneg (div_nonneg hA hμ.le) hell) have htransverse : H 1 1 * |g 1 - β / κ * g 0| * ‖(WithLp.toLp 2 ![-β / κ, 1] : EuclideanSpace ℝ (Fin 2))‖ ≤ H 1 1 * (G * (1 + B) ^ 2) := by have hpositive : 0 ≤ H 1 1 * ((1 + B) * G) := by positivity have hbound := mul_le_mul (mul_le_mul_of_nonneg_left hpair hfree) hvector (norm_nonneg _) hpositive nlinarith [hbound] have haction := secantActionNormBound hH hκpos hsec g have hcross₁ := mul_nonneg (div_nonneg hA hμ.le) hfree have hcross₂ := mul_nonneg (mul_nonneg hG (sq_nonneg (1 + B))) hell nlinarith

Helper for lem:planar-degeneration: uniform secant-coordinate estimates yield an eventual product bound and vanishing of the free matrix entry.

theorem secantSequenceDegeneration (H : ℕ → Matrix (Fin 2) (Fin 2) ℝ) (g : ℕ → Fin 2 → ℝ) (κ β ell : ℕ → ℝ) {μ A B G N C : ℝ} (hμ : 0 < μ) (hA : 0 < A) (hB : 0 ≤ B) (hG : 0 < G) (hN : 0 < N) (hC : 0 < C) (hH : ∀ k, (H k).PosDef) (hκ : ∀ k, μ ≤ κ k) (hsec : ∀ k, H k *ᵥ ![κ k, β k] = ![1, 0]) (hell : ∀ k, 0 ≤ ell k) (hellLimit : Tendsto ell atTop (𝓝 0)) (htilt : ∀ k, |β k / κ k| ≤ B) (hgradientLower : ∀ k, N ≤ ‖(WithLp.toLp 2 (g k) : EuclideanSpace ℝ (Fin 2))‖) (hgradientUpper : ∀ k, ‖(WithLp.toLp 2 (g k) : EuclideanSpace ℝ (Fin 2))‖ ≤ G) (hlongitudinal : ∀ k, |g k 0| ≤ A * ell k) (henergy : ∀ k, g k ⬝ᵥ (H k *ᵥ g k) ≤ C * ell (k + 1) * ‖(WithLp.toLp 2 (H k *ᵥ g k) : EuclideanSpace ℝ (Fin 2))‖) : (∃ K > 0, ∀ᶠ k in atTop, 0 < H k 1 1 ∧ H k 1 1 ≤ K * ell k * ell (k + 1)) ∧ Tendsto (fun k ↦ H k 1 1) atTop (𝓝 0) := by let D := A / μ + G * (1 + B) ^ 2 have hD : 0 < D := by dsimp only [D] positivity have ha : 0 < N ^ 2 / 4 := by positivity have hlongSmall : ∀ᶠ k in atTop, (1 + B) * (A * ell k) < N / 2 := by have hlim : Tendsto (fun k ↦ (1 + B) * (A * ell k)) atTop (𝓝 0) := by simpa using (hellLimit.const_mul A).const_mul (1 + B) exact hlim.eventually (eventually_lt_nhds (half_pos hN)) have henergyBound : ∀ᶠ k in atTop, (N ^ 2 / 4) * H k 1 1 ≤ (C * D) * ell (k + 1) * (ell k + H k 1 1) := by filter_upwards [hlongSmall] with k hk have hlong : (1 + B) * |g k 0| ≤ N / 2 := by have hfactor : 0 ≤ 1 + B := by positivity exact (mul_le_mul_of_nonneg_left (hlongitudinal k) hfactor).trans hk.le have hlower := secantEnergyLowerBoundOfGradientNorm (hH k) (hμ.trans_le (hκ k)) (hsec k) (g k) hN.le (htilt k) (hgradientLower k) hlong have hupper := secantActionUniformBound (hH k) hμ (hκ k) (hsec k) (g k) hA.le hB hG.le (hell k) (htilt k) (hgradientUpper k) (hlongitudinal k) have hfactor : 0 ≤ C * ell (k + 1) := mul_nonneg hC.le (hell (k + 1)) have hscaled := mul_le_mul_of_nonneg_left hupper hfactor dsimp only [D] nlinarith [henergy k] have hnonnegative : ∀ᶠ k in atTop, 0 ≤ H k 1 1 := Eventually.of_forall (fun k ↦ (hH k).diag_pos.le) have hzero := freeCoefficientTendstoZero ha hellLimit hnonnegative henergyBound refine ⟨?_, hzero⟩ have hconstant : 0 < 2 * (C * D) / (N ^ 2 / 4) := by positivity refine ⟨2 * (C * D) / (N ^ 2 / 4), hconstant, ?_⟩ have hsmall : ∀ᶠ k in atTop, (C * D) * ell (k + 1) < (N ^ 2 / 4) / 2 := by have hlim : Tendsto (fun k ↦ (C * D) * ell (k + 1)) atTop (𝓝 0) := by simpa using (hellLimit.comp (tendsto_add_atTop_nat 1)).const_mul (C * D) exact hlim.eventually (eventually_lt_nhds (half_pos ha)) filter_upwards [henergyBound, hsmall] with k hk hs exact ⟨(hH k).diag_pos, freeCoefficientProductBound ha (hH k).diag_pos.le hs.le hk⟩

Helper for lem:planar-degeneration: failure of convergence of the actual planar DFP orbit forces its free secant coefficient to vanish, with an eventual bound by the product of adjacent physical step lengths. The index k here represents the paper's coefficient at iteration k + 1.

theorem orbitFreeCoefficientDegeneration (iteration : DFP.InverseIteration (Fin 2)) {μ c₁ c₂ : ℝ} (hf : ContDiff ℝ 2 iteration.objective) (hμ : 0 < μ) (hHessian : ∀ x v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient iteration.objective) x v) v) (hStep : ∀ k, 0 < iteration.stepLength k) (hWolfe : ∀ k, LineSearch.IsWeakWolfe c₁ c₂ iteration.objective (iteration.point k) (iteration.point (k + 1) - iteration.point k)) {xstar : EuclideanSpace ℝ (Fin 2)} (hmin : ∀ x, iteration.objective xstar ≤ iteration.objective x) (hnot : ¬ Tendsto iteration.point atTop (𝓝 xstar)) : let s := fun k ↦ iteration.point (k + 1) - iteration.point k let y := fun k ↦ gradient iteration.objective (iteration.point (k + 1)) - gradient iteration.objective (iteration.point k) let lam := fun k ↦ (inner ℝ (y k) (s k) / ‖s k‖ ^ 2) * (iteration.inverseHessian (k + 1)).det (∃ K > 0, ∀ᶠ k in atTop, 0 < lam k ∧ lam k ≤ K * ‖s k‖ * ‖s (k + 1)‖) ∧ Tendsto lam atTop (𝓝 0) := by let s := fun k ↦ iteration.point (k + 1) - iteration.point k let y := fun k ↦ gradient iteration.objective (iteration.point (k + 1)) - gradient iteration.objective (iteration.point k) let ell := fun k ↦ ‖s k‖ let u := fun k ↦ (ell k)⁻¹ • s k let v := fun k ↦ (ell k)⁻¹ • y k let R := fun k ↦ EuclideanPlane.frame (u k) let K := fun k ↦ (R k).transpose * iteration.inverseHessian (k + 1) * R k let g := fun k ↦ (R k).transpose *ᵥ WithLp.ofLp (gradient iteration.objective (iteration.point (k + 1))) let κ := fun k ↦ inner ℝ (u k) (v k) let β := fun k ↦ inner ℝ (EuclideanPlane.perp (u k)) (v k) let lam := fun k ↦ (inner ℝ (y k) (s k) / ‖s k‖ ^ 2) * (iteration.inverseHessian (k + 1)).det obtain ⟨M, hM, hsecBounds⟩ := secantBoundsAlongOrbit iteration hf hμ hHessian hStep hWolfe obtain ⟨G, hG, hdiffBounds⟩ := uniformDifferentialBoundsOnSublevel hf hμ hHessian (iteration.point 0) obtain ⟨N, hN, hgradLower⟩ := gradientLowerBoundOfNonconvergence iteration hf hμ hHessian hStep hWolfe hmin hnot obtain ⟨A, hA, hlongBound⟩ := orbitLongitudinalGradientBound iteration hf hμ hHessian hStep hWolfe obtain ⟨C, hC, henergyBound⟩ := orbitEnergyDirectionBound iteration hf hμ hHessian hStep hWolfe have hmem := iteration.pointMemObjectiveSublevelOfWeakWolfe hStep hWolfe have hellPos (k : ℕ) : 0 < ell k := orbitStepNormPos iteration k (hStep k) have hellLimit : Tendsto ell atTop (𝓝 0) := stepNormTendstoZero iteration hf hμ hHessian hStep hWolfe have hu (k : ℕ) : ‖u k‖ = 1 := by dsimp only [u] rw [norm_smul, Real.norm_eq_abs, abs_inv, abs_norm] exact inv_mul_cancel₀ (hellPos k).ne' have hR (k : ℕ) : R k ∈ Matrix.orthogonalGroup (Fin 2) ℝ := by have hspecial := (EuclideanPlane.frame_mem_specialOrthogonalGroup_iff (u k)).mpr (hu k) exact (Matrix.mem_specialOrthogonalGroup_iff.mp hspecial).1 have hK (k : ℕ) : (K k).PosDef := orthogonalCoordinatesPosDef (iteration.inverseHessianPosDef (k + 1)) (hR k) have hsec (k : ℕ) : K k *ᵥ ![κ k, β k] = ![1, 0] := orbitPlanarSecantFrameSpecification iteration k (hStep k) have hκ (k : ℕ) : μ ≤ κ k := normalizedOrbitCurvatureLowerBound iteration hf hμ hHessian k (hStep k) have hv (k : ℕ) : ‖v k‖ ≤ M := by dsimp only [v] rw [norm_smul, Real.norm_eq_abs, abs_inv, abs_of_pos (hellPos k), ← div_eq_inv_mul] apply (div_le_iff₀ (hellPos k)).mpr exact (hsecBounds k).1 have hβ (k : ℕ) : |β k| ≤ M := by have hpair := abs_real_inner_le_norm (EuclideanPlane.perp (u k)) (v k) rw [EuclideanPlane.perp.norm_map, hu k, one_mul] at hpair exact hpair.trans (hv k) have htilt (k : ℕ) : |β k / κ k| ≤ M / μ := secantTiltBound hμ (hκ k) (hβ k) have hgNorm (k : ℕ) : ‖(WithLp.toLp 2 (g k) : EuclideanSpace ℝ (Fin 2))‖ = ‖gradient iteration.objective (iteration.point (k + 1))‖ := by exact Matrix.norm_toLp_mulVec_eq_of_mem_orthogonalGroup (R k).transpose (orthogonalTranspose (hR k)) _ have hgLower (k : ℕ) : N ≤ ‖(WithLp.toLp 2 (g k) : EuclideanSpace ℝ (Fin 2))‖ := by rw [hgNorm] exact hgradLower (k + 1) have hgUpper (k : ℕ) : ‖(WithLp.toLp 2 (g k) : EuclideanSpace ℝ (Fin 2))‖ ≤ G := by rw [hgNorm] have hlevel := (DFP.mem_objectiveSublevel_iff iteration _).mp (hmem (k + 1)) exact (hdiffBounds _ hlevel).1 have hgFirst (k : ℕ) : g k 0 = inner ℝ (u k) (gradient iteration.objective (iteration.point (k + 1))) := by have hcoordinates := congrFun (planarFrameCoordinates (u k) (gradient iteration.objective (iteration.point (k + 1)))) 0 simpa only [Matrix.cons_val_zero] using hcoordinates have hlong (k : ℕ) : |g k 0| ≤ A * ell k := by rw [hgFirst] exact hlongBound k have henergy (k : ℕ) : g k ⬝ᵥ (K k *ᵥ g k) ≤ C * ell (k + 1) * ‖(WithLp.toLp 2 (K k *ᵥ g k) : EuclideanSpace ℝ (Fin 2))‖ := by dsimp only [g, K] rw [orthogonalCoordinatesEnergy _ _ (hR k), orthogonalCoordinatesActionNorm _ _ (hR k)] have hphysical := henergyBound (k + 1) dsimp only at hphysical rw [EuclideanSpace.inner_eq_star_dotProduct] at hphysical simp only [star_trivial] at hphysical rw [dotProduct_comm] at hphysical exact hphysical have hfree (k : ℕ) : K k 1 1 = lam k := by calc K k 1 1 = κ k * (K k).det := transverseEntry_eq_curvature_mul_det (hK k).isHermitian (hμ.trans_le (hκ k)).ne' (hsec k) _ = κ k * (iteration.inverseHessian (k + 1)).det := by rw [orthogonalCoordinatesDet _ _ (hR k)] _ = lam k := by dsimp only [κ, u, v, ell, lam] rw [normalizedSecantCurvature] have hresult := secantSequenceDegeneration K g κ β ell hμ hA (div_nonneg hM.le hμ.le) hG hN hC hK hκ hsec (fun k ↦ (hellPos k).le) hellLimit htilt hgLower hgUpper hlong henergy simpa only [hfree, lam, ell, s, y] using hresult

Helper for lem:planar-degeneration: the free secant coefficient controls the Euclidean operator norm of the whole secant-coordinate matrix.

theorem secantOperatorNormBound {H : Matrix (Fin 2) (Fin 2) ℝ} {κ β μ B : ℝ} (hH : H.PosDef) (hμ : 0 < μ) (hκ : μ ≤ κ) (hsec : H *ᵥ ![κ, β] = ![1, 0]) (hB : 0 ≤ B) (htilt : |β / κ| ≤ B) : ‖Matrix.toEuclideanCLM (𝕜 := ℝ) H‖ ≤ (1 / μ + (1 + B) ^ 2) * (1 + H 1 1) := by have hfree : 0 < H 1 1 := hH.diag_pos have hconstant : 0 ≤ (1 / μ + (1 + B) ^ 2) * (1 + H 1 1) := by positivity apply ContinuousLinearMap.opNorm_le_of_unit_norm hconstant intro x hx have hcoord : |x 0| ≤ 1 := by simpa only [Real.norm_eq_abs, hx] using PiLp.norm_apply_le x 0 have hnorm : ‖(WithLp.toLp 2 (WithLp.ofLp x) : EuclideanSpace ℝ (Fin 2))‖ ≤ 1 := hx.le have hlong : |WithLp.ofLp x 0| ≤ (1 : ℝ) * 1 := by simpa using hcoord have hbound := secantActionUniformBound hH hμ hκ hsec (WithLp.ofLp x) zero_le_one hB zero_le_one zero_le_one htilt hnorm hlong have hrepr : Matrix.toEuclideanCLM (𝕜 := ℝ) H x = WithLp.toLp 2 (H *ᵥ WithLp.ofLp x) := by exact Matrix.toEuclideanCLM_toLp H (WithLp.ofLp x) simpa only [one_mul, ← hrepr] using hbound

Helper for lem:planar-degeneration: an operator norm bound in an orthonormal coordinate system is also a bound in physical coordinates.

theorem physicalOperatorNormBoundOfCoordinates (H R : Matrix (Fin 2) (Fin 2) ℝ) (hR : R ∈ Matrix.orthogonalGroup (Fin 2) ℝ) {B : ℝ} (hB : 0 ≤ B) (hbound : ‖Matrix.toEuclideanCLM (𝕜 := ℝ) (R.transpose * H * R)‖ ≤ B) : ‖Matrix.toEuclideanCLM (𝕜 := ℝ) H‖ ≤ B := by apply ContinuousLinearMap.opNorm_le_of_unit_norm hB intro x hx have hnorm : ‖(WithLp.toLp 2 (R.transpose *ᵥ WithLp.ofLp x) : EuclideanSpace ℝ (Fin 2))‖ = 1 := by rw [Matrix.norm_toLp_mulVec_eq_of_mem_orthogonalGroup R.transpose (orthogonalTranspose hR)] exact hx have haction := (Matrix.toEuclideanCLM (𝕜 := ℝ) (R.transpose * H * R)).le_of_opNorm_le hbound (WithLp.toLp 2 (R.transpose *ᵥ WithLp.ofLp x)) rw [Matrix.toEuclideanCLM_toLp, orthogonalCoordinatesActionNorm H R hR, hnorm, mul_one] at haction exact haction

Helper for lem:planar-degeneration: the physical inverse Hessian is bounded in terms of its normalized secant curvature and determinant.

theorem planarPhysicalSecantOperatorBound (H : Matrix (Fin 2) (Fin 2) ℝ) (u v : EuclideanSpace ℝ (Fin 2)) {μ M : ℝ} (hH : H.PosDef) (hu : ‖u‖ = 1) (hsec : H *ᵥ WithLp.ofLp v = WithLp.ofLp u) (hμ : 0 < μ) (hκ : μ ≤ inner ℝ u v) (hv : ‖v‖ ≤ M) : ‖Matrix.toEuclideanCLM (𝕜 := ℝ) H‖ ≤ (1 / μ + (1 + M / μ) ^ 2) * (1 + inner ℝ u v * H.det) := by have hM : 0 ≤ M := (norm_nonneg v).trans hv have hspecial := (EuclideanPlane.frame_mem_specialOrthogonalGroup_iff u).mpr hu have hR := (Matrix.mem_specialOrthogonalGroup_iff.mp hspecial).1 have hK := orthogonalCoordinatesPosDef hH hR have hsecK := planarSecantFrameSpecification H u v hu hsec have hβ : |inner ℝ (EuclideanPlane.perp u) v| ≤ M := by have hpair := abs_real_inner_le_norm (EuclideanPlane.perp u) v rw [EuclideanPlane.perp.norm_map, hu, one_mul] at hpair exact hpair.trans hv have htilt := secantTiltBound hμ hκ hβ have hbound := secantOperatorNormBound hK hμ hκ hsecK (div_nonneg hM hμ.le) htilt have hfree : ((EuclideanPlane.frame u).transpose * H * EuclideanPlane.frame u) 1 1 = inner ℝ u v * H.det := by rw [transverseEntry_eq_curvature_mul_det hK.isHermitian (hμ.trans_le hκ).ne' hsecK, orthogonalCoordinatesDet H (EuclideanPlane.frame u) hR] rw [hfree] at hbound have hlam : 0 < inner ℝ u v * H.det := by rw [← hfree] exact hK.diag_pos have hconstant : 0 ≤ (1 / μ + (1 + M / μ) ^ 2) * (1 + inner ℝ u v * H.det) := by positivity exact physicalOperatorNormBoundOfCoordinates H (EuclideanPlane.frame u) hR hconstant hbound

Helper for lem:planar-degeneration: if a strongly convex planar DFP orbit fails to converge, its inverse-Hessian operators remain uniformly bounded.

theorem orbitInverseHessianBoundedOfNonconvergence (iteration : DFP.InverseIteration (Fin 2)) {μ c₁ c₂ : ℝ} (hf : ContDiff ℝ 2 iteration.objective) (hμ : 0 < μ) (hHessian : ∀ x v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient iteration.objective) x v) v) (hStep : ∀ k, 0 < iteration.stepLength k) (hWolfe : ∀ k, LineSearch.IsWeakWolfe c₁ c₂ iteration.objective (iteration.point k) (iteration.point (k + 1) - iteration.point k)) {xstar : EuclideanSpace ℝ (Fin 2)} (hmin : ∀ x, iteration.objective xstar ≤ iteration.objective x) (hnot : ¬ Tendsto iteration.point atTop (𝓝 xstar)) : ∃ B > 0, ∀ k, ‖Matrix.toEuclideanCLM (𝕜 := ℝ) (iteration.inverseHessian k)‖ ≤ B := by let s := fun k ↦ iteration.point (k + 1) - iteration.point k let y := fun k ↦ gradient iteration.objective (iteration.point (k + 1)) - gradient iteration.objective (iteration.point k) let lam := fun k ↦ (inner ℝ (y k) (s k) / ‖s k‖ ^ 2) * (iteration.inverseHessian (k + 1)).det have hlim : Tendsto lam atTop (𝓝 0) := (orbitFreeCoefficientDegeneration iteration hf hμ hHessian hStep hWolfe hmin hnot).2 obtain ⟨L, hL⟩ := hlim.bddAbove_range obtain ⟨M, hM, hsecBounds⟩ := secantBoundsAlongOrbit iteration hf hμ hHessian hStep hWolfe let a := 1 / μ + (1 + M / μ) ^ 2 have ha : 0 < a := by dsimp only [a] positivity have hsuccessor (k : ℕ) : ‖Matrix.toEuclideanCLM (𝕜 := ℝ) (iteration.inverseHessian (k + 1))‖ ≤ a * (1 + max 0 L) := by have hnorm : 0 < ‖s k‖ := orbitStepNormPos iteration k (hStep k) have hu : ‖‖s k‖⁻¹ • s k‖ = 1 := by rw [norm_smul, Real.norm_eq_abs, abs_inv, abs_norm, inv_mul_cancel₀ hnorm.ne'] have hv : ‖‖s k‖⁻¹ • y k‖ ≤ M := by rw [norm_smul, Real.norm_eq_abs, abs_inv, abs_norm, ← div_eq_inv_mul] apply (div_le_iff₀ hnorm).mpr exact (hsecBounds k).1 have hκ : μ ≤ inner ℝ (‖s k‖⁻¹ • s k) (‖s k‖⁻¹ • y k) := normalizedOrbitCurvatureLowerBound iteration hf hμ hHessian k (hStep k) have hbound := planarPhysicalSecantOperatorBound (iteration.inverseHessian (k + 1)) (‖s k‖⁻¹ • s k) (‖s k‖⁻¹ • y k) (iteration.inverseHessianPosDef (k + 1)) hu (normalizedOrbitSecantEquation iteration k) hμ hκ hv have hbound' : ‖Matrix.toEuclideanCLM (𝕜 := ℝ) (iteration.inverseHessian (k + 1))‖ ≤ a * (1 + lam k) := by simpa only [normalizedSecantCurvature] using hbound have hlam : lam k ≤ max 0 L := (hL (Set.mem_range_self k)).trans (le_max_right _ _) have hscaled := mul_le_mul_of_nonneg_left hlam ha.le nlinarith let B := max ‖Matrix.toEuclideanCLM (𝕜 := ℝ) (iteration.inverseHessian 0)‖ (a * (1 + max 0 L)) + 1 have hB : 0 < B := by dsimp only [B] positivity refine ⟨B, hB, ?_⟩ intro k cases k with | zero => have hmax := le_max_left ‖Matrix.toEuclideanCLM (𝕜 := ℝ) (iteration.inverseHessian 0)‖ (a * (1 + max 0 L)) dsimp only [B] linarith | succ k => have hmax := le_max_right ‖Matrix.toEuclideanCLM (𝕜 := ℝ) (iteration.inverseHessian 0)‖ (a * (1 + max 0 L)) have hbound := hsuccessor k dsimp only [B] exact hbound.trans (hmax.trans (le_add_of_nonneg_right zero_le_one))

Helper for thm:planar-convergence: a Lipschitz derivative on a segment gives a uniform quadratic first-order remainder.

theorem quadraticRemainderOfLipschitzDerivative {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {f : E → F} {x y : E} {L : ℝ} (hL : 0 ≤ L) (hf : ∀ z ∈ segment ℝ x y, DifferentiableAt ℝ f z) (hderiv : ∀ z ∈ segment ℝ x y, ‖fderiv ℝ f z - fderiv ℝ f x‖ ≤ L * ‖z - x‖) : ‖f y - f x - fderiv ℝ f x (y - x)‖ ≤ L * ‖y - x‖ ^ 2 := by have hbound : ∀ z ∈ segment ℝ x y, ‖fderiv ℝ f z - fderiv ℝ f x‖ ≤ L * ‖y - x‖ := by intro z hz exact (hderiv z hz).trans (mul_le_mul_of_nonneg_left (norm_sub_le_of_mem_segment hz) hL) have hmean := (convex_segment x y).norm_image_sub_le_of_norm_fderiv_le' hf hbound (left_mem_segment ℝ x y) (right_mem_segment ℝ x y) simpa only [pow_two, mul_assoc] using hmean

Helper for thm:planar-convergence: local Lipschitz Hessian regularity on a compact initial sublevel yields one finite Lipschitz constant there.

theorem hessianLipschitzOnInitialSublevel {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] [ProperSpace E] {f : E → ℝ} {μ : ℝ} (hf : ContDiff ℝ 2 f) (hμ : 0 < μ) (hHessian : ∀ z v : E, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient f) z v) v) (x₀ : E) (hLip : LocallyLipschitzOn {x | f x ≤ f x₀} (fderiv ℝ (gradient f))) : ∃ L ≥ 0, ∀ x, f x ≤ f x₀ → ∀ y, f y ≤ f x₀ → ‖fderiv ℝ (gradient f) y - fderiv ℝ (gradient f) x‖ ≤ L * ‖y - x‖ := by obtain ⟨L, hL⟩ := hLip.exists_lipschitzOnWith_of_compact (compactInitialSublevel hf hμ hHessian x₀) refine ⟨(L : ℝ), L.coe_nonneg, ?_⟩ intro x hx y hy simpa only [dist_eq_norm] using hL.dist_le_mul y hy x hx

Helper for thm:planar-convergence: the actual gradient secants along the DFP orbit have a uniform quadratic Hessian-linearization remainder.

theorem orbitGradientTaylorRemainder {n : ℕ} (iteration : DFP.InverseIteration (Fin n)) {μ c₁ c₂ : ℝ} (hf : ContDiff ℝ 2 iteration.objective) (hμ : 0 < μ) (hHessian : ∀ x v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient iteration.objective) x v) v) (hStep : ∀ k, 0 < iteration.stepLength k) (hWolfe : ∀ k, LineSearch.IsWeakWolfe c₁ c₂ iteration.objective (iteration.point k) (iteration.point (k + 1) - iteration.point k)) (hLip : LocallyLipschitzOn (DFP.objectiveSublevel iteration) (fderiv ℝ (gradient iteration.objective))) : ∃ L ≥ 0, ∀ k, ‖gradient iteration.objective (iteration.point (k + 1)) - gradient iteration.objective (iteration.point k) - fderiv ℝ (gradient iteration.objective) (iteration.point k) (iteration.point (k + 1) - iteration.point k)‖ ≤ L * ‖iteration.point (k + 1) - iteration.point k‖ ^ 2 := by have hlevelEq : DFP.objectiveSublevel iteration = {x | iteration.objective x ≤ iteration.objective (iteration.point 0)} := by ext x exact DFP.mem_objectiveSublevel_iff iteration x rw [hlevelEq] at hLip obtain ⟨L, hL, hbound⟩ := hessianLipschitzOnInitialSublevel hf hμ hHessian (iteration.point 0) hLip have hconvex := convexInitialSublevel hf hμ hHessian (iteration.point 0) have hmem := iteration.pointMemObjectiveSublevelOfWeakWolfe hStep hWolfe rw [hlevelEq] at hmem have hfOrder : ContDiff ℝ ((1 : ℕ∞) + 1) iteration.objective := by convert hf using 1 norm_num have hgradient : ContDiff ℝ 1 (gradient iteration.objective) := hfOrder.gradient_succ (n := 1) refine ⟨L, hL, ?_⟩ intro k apply quadraticRemainderOfLipschitzDerivative hL · intro z _ exact hgradient.differentiable_one.differentiableAt · intro z hz have hzLevel := hconvex.segment_subset (hmem k) (hmem (k + 1)) hz exact hbound _ (hmem k) z hzLevel

Helper for thm:planar-convergence: the reciprocal correction cancels the linear term of the logarithm, leaving a uniform quadratic remainder.

theorem logReciprocalQuadraticRemainder {r : ℝ} (hr : |r| ≤ 1 / 2) : |Real.log (1 + r) + (1 + r)⁻¹ - 1| ≤ 4 * r ^ 2 := by have hden : 0 < 1 + r := by have hlow := (abs_le.mp hr).1 linarith have hsplit : Real.log (1 + r) + (1 + r)⁻¹ - 1 = (Real.log (1 + r) - r) + r ^ 2 / (1 + r) := by field_simp ring have hrational : |r ^ 2 / (1 + r)| ≤ 2 * r ^ 2 := by rw [abs_div, abs_of_nonneg (sq_nonneg r), abs_of_pos hden] apply (div_le_iff₀ hden).mpr have hlow := (abs_le.mp hr).1 nlinarith [mul_le_mul_of_nonneg_left hlow (sq_nonneg r)] rw [hsplit] have htriangle := abs_add_le (Real.log (1 + r) - r) (r ^ 2 / (1 + r)) linarith [logOneAddQuadraticRemainder hr]

Helper for thm:planar-convergence: normalized vector increments have an exact finite-difference identity that isolates the tangent cancellation.

theorem normalizedIncrementPairingIdentity {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] (a t h : E) {N Nnext : ℝ} (hpair : inner ℝ a t = 2) : inner ℝ a ((N / Nnext) • t + Nnext⁻¹ • h - t) = inner ℝ a h / Nnext + 2 * (N / Nnext - 1) := by rw [inner_sub_right, inner_add_right, real_inner_smul_right, real_inner_smul_right, hpair] ring

Helper for thm:planar-convergence: the exact normalized increment and logarithmic amplitude increment cancel to quadratic order.

theorem normalizedIncrementLogBound {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] (a t tNext h : E) {N Nnext : ℝ} (hN : 0 < N) (hNnext : 0 < Nnext) (hpair : inner ℝ a t = 2) (hnext : tNext = (N / Nnext) • t + Nnext⁻¹ • h) (hr : |(Nnext - N) / N| ≤ 1 / 2) : |inner ℝ a (tNext - t) + 2 * (Real.log Nnext - Real.log N)| ≤ |inner ℝ a h| / Nnext + 8 * ((Nnext - N) / N) ^ 2 := by let r := (Nnext - N) / N have hratio : 1 + r = Nnext / N := by dsimp only [r] field_simp ring have hinverse : N / Nnext = (1 + r)⁻¹ := by rw [hratio, inv_div] have hlog : Real.log Nnext - Real.log N = Real.log (1 + r) := by rw [hratio, Real.log_div hNnext.ne' hN.ne'] have hidentity : inner ℝ a (tNext - t) + 2 * (Real.log Nnext - Real.log N) = inner ℝ a h / Nnext + 2 * (Real.log (1 + r) + (1 + r)⁻¹ - 1) := by rw [hnext, normalizedIncrementPairingIdentity a t h hpair, hinverse, hlog] ring have hremainder := logReciprocalQuadraticRemainder hr rw [hidentity] have htriangle := abs_add_le (inner ℝ a h / Nnext) (2 * (Real.log (1 + r) + (1 + r)⁻¹ - 1)) rw [abs_div, abs_of_pos hNnext, abs_mul] at htriangle norm_num only at htriangle dsimp only [r] at htriangle ⊢ linarith

Helper for thm:planar-convergence: right-angle rotation is orthogonal to its argument.

theorem innerPerpSelf (x : EuclideanSpace ℝ (Fin 2)) : inner ℝ x (EuclideanPlane.perp x) = 0 := by rw [EuclideanPlane.perp_apply] simp [EuclideanSpace.inner_eq_star_dotProduct, dotProduct, Fin.sum_univ_two] ring

Helper for thm:planar-convergence: the coefficient field that corrects the logarithmic determinant along the gradient's unit tangent.

noncomputable def tangentCoefficient (A : EuclideanSpace ℝ (Fin 2) →L[ℝ] EuclideanSpace ℝ (Fin 2)) (t : EuclideanSpace ℝ (Fin 2)) : EuclideanSpace ℝ (Fin 2) := (2 / inner ℝ t (A t)) • A t

Helper for thm:planar-convergence: the coefficient field pairs with the unit tangent to give exactly two.

theorem tangentCoefficientPairing (A : EuclideanSpace ℝ (Fin 2) →L[ℝ] EuclideanSpace ℝ (Fin 2)) (t : EuclideanSpace ℝ (Fin 2)) (hcurv : inner ℝ t (A t) ≠ 0) : inner ℝ (tangentCoefficient A t) t = 2 := by rw [tangentCoefficient, real_inner_smul_left] have hpair : inner ℝ (A t) t = inner ℝ t (A t) := real_inner_comm _ _ calc (2 / inner ℝ t (A t)) * inner ℝ (A t) t = (2 / inner ℝ t (A t)) * inner ℝ t (A t) := congrArg (fun z : ℝ ↦ (2 / inner ℝ t (A t)) * z) hpair _ = 2 := div_mul_cancel₀ 2 hcurv

Helper for thm:planar-convergence: the coefficient field annihilates the rotated tangential Hessian image exactly.

theorem tangentCoefficientCancellation (A : EuclideanSpace ℝ (Fin 2) →L[ℝ] EuclideanSpace ℝ (Fin 2)) (t : EuclideanSpace ℝ (Fin 2)) : inner ℝ (tangentCoefficient A t) (EuclideanPlane.perp (A t)) = 0 := by rw [tangentCoefficient, real_inner_smul_left, innerPerpSelf, mul_zero]

Helper for thm:planar-convergence: the tangential secant term cancels and only the Taylor remainder and departure from a tangent step remain.

theorem tangentCoefficientSecantBound (A : EuclideanSpace ℝ (Fin 2) →L[ℝ] EuclideanSpace ℝ (Fin 2)) (t s y : EuclideanSpace ℝ (Fin 2)) (d : ℝ) : |inner ℝ (tangentCoefficient A t) (EuclideanPlane.perp y)| ≤ ‖tangentCoefficient A t‖ * (‖y - A s‖ + ‖A‖ * ‖s - d • t‖) := by have hsplit : EuclideanPlane.perp y = EuclideanPlane.perp (y - A s) + EuclideanPlane.perp (A (s - d • t)) + d • EuclideanPlane.perp (A t) := by simp only [map_sub, map_smul] module have hpair : inner ℝ (tangentCoefficient A t) (EuclideanPlane.perp y) = inner ℝ (tangentCoefficient A t) (EuclideanPlane.perp (y - A s) + EuclideanPlane.perp (A (s - d • t))) := by rw [hsplit, inner_add_right, real_inner_smul_right, tangentCoefficientCancellation, mul_zero, add_zero] rw [hpair] have hinner := abs_real_inner_le_norm (tangentCoefficient A t) (EuclideanPlane.perp (y - A s) + EuclideanPlane.perp (A (s - d • t))) have hsum := norm_add_le (EuclideanPlane.perp (y - A s)) (EuclideanPlane.perp (A (s - d • t))) rw [EuclideanPlane.perp.norm_map, EuclideanPlane.perp.norm_map] at hsum have haction := A.le_opNorm (s - d • t) have hnorm : ‖EuclideanPlane.perp (y - A s) + EuclideanPlane.perp (A (s - d • t))‖ ≤ ‖y - A s‖ + ‖A‖ * ‖s - d • t‖ := by linarith exact hinner.trans (mul_le_mul_of_nonneg_left hnorm (norm_nonneg _))

Helper for thm:planar-convergence: any planar vector decomposes exactly into its normal and positively oriented tangent components.

theorem planarNormalTangentDecomposition (ν s : EuclideanSpace ℝ (Fin 2)) (hν : ‖ν‖ = 1) : s = inner ℝ ν s • ν + inner ℝ (EuclideanPlane.perp ν) s • EuclideanPlane.perp ν := by have hunit := EuclideanSpace.real_norm_sq_eq ν rw [hν, one_pow, Fin.sum_univ_two] at hunit have hfirst := congrArg (fun r : ℝ ↦ r * s 0) hunit have hsecond := congrArg (fun r : ℝ ↦ r * s 1) hunit ext i fin_cases i · simp [PiLp.add_apply, PiLp.smul_apply, EuclideanSpace.inner_eq_star_dotProduct, EuclideanPlane.perp_apply, dotProduct, Fin.sum_univ_two] nlinarith · simp [PiLp.add_apply, PiLp.smul_apply, EuclideanSpace.inner_eq_star_dotProduct, EuclideanPlane.perp_apply, dotProduct, Fin.sum_univ_two] nlinarith

Helper for thm:planar-convergence: the distance from a planar step to its tangential projection is its normal component's absolute value.

theorem normNormalResidual (ν s : EuclideanSpace ℝ (Fin 2)) (hν : ‖ν‖ = 1) : ‖s - inner ℝ (EuclideanPlane.perp ν) s • EuclideanPlane.perp ν‖ = |inner ℝ ν s| := by have hdecomp := planarNormalTangentDecomposition ν s hν have hresidual : s - inner ℝ (EuclideanPlane.perp ν) s • EuclideanPlane.perp ν = inner ℝ ν s • ν := by exact sub_eq_iff_eq_add.mpr hdecomp rw [hresidual, norm_smul, Real.norm_eq_abs, hν, mul_one]

Helper for thm:planar-convergence: rotating a normalized gradient increment gives the exact normalization identity used by tangent cancellation.

theorem normalizedPerpIncrement (g gNext : EuclideanSpace ℝ (Fin 2)) (hg : g ≠ 0) : ‖gNext‖⁻¹ • EuclideanPlane.perp gNext = (‖g‖ / ‖gNext‖) • (‖g‖⁻¹ • EuclideanPlane.perp g) + ‖gNext‖⁻¹ • EuclideanPlane.perp (gNext - g) := by have hnorm : ‖g‖ ≠ 0 := norm_ne_zero_iff.mpr hg have hcancel : (‖g‖ / ‖gNext‖) * ‖g‖⁻¹ = ‖gNext‖⁻¹ := by field_simp rw [smul_smul, hcancel, map_sub, smul_sub] module

Helper for thm:planar-convergence: for two nonzero gradients, the rotated normalized-gradient increment and twice the log-norm increment cancel up to the rotated secant pairing and a quadratic relative norm change.

theorem gradientTangentLogBound (g gNext : EuclideanSpace ℝ (Fin 2)) (A : EuclideanSpace ℝ (Fin 2) →L[ℝ] EuclideanSpace ℝ (Fin 2)) (hg : g ≠ 0) (hgNext : gNext ≠ 0) (hcurv : inner ℝ (‖g‖⁻¹ • EuclideanPlane.perp g) (A (‖g‖⁻¹ • EuclideanPlane.perp g)) ≠ 0) (hrelative : |(‖gNext‖ - ‖g‖) / ‖g‖| ≤ 1 / 2) : let t := ‖g‖⁻¹ • EuclideanPlane.perp g let tNext := ‖gNext‖⁻¹ • EuclideanPlane.perp gNext |inner ℝ (tangentCoefficient A t) (tNext - t) + 2 * (Real.log ‖gNext‖ - Real.log ‖g‖)| ≤ |inner ℝ (tangentCoefficient A t) (EuclideanPlane.perp (gNext - g))| / ‖gNext‖ + 8 * ((‖gNext‖ - ‖g‖) / ‖g‖) ^ 2 := by dsimp only exact normalizedIncrementLogBound _ _ _ _ (norm_pos_iff.mpr hg) (norm_pos_iff.mpr hgNext) (tangentCoefficientPairing A _ hcurv) (normalizedPerpIncrement g gNext hg) hrelative

Helper for thm:planar-convergence: positive tangential curvature and a bounded Hessian give a uniform norm bound on the correcting coefficient.

theorem tangentCoefficientNormBound (A : EuclideanSpace ℝ (Fin 2) →L[ℝ] EuclideanSpace ℝ (Fin 2)) (t : EuclideanSpace ℝ (Fin 2)) {μ M : ℝ} (ht : ‖t‖ = 1) (hμ : 0 < μ) (hcurv : μ ≤ inner ℝ t (A t)) (hA : ‖A‖ ≤ M) : ‖tangentCoefficient A t‖ ≤ 2 * M / μ := by have hratio : |2 / inner ℝ t (A t)| ≤ 2 / μ := by have htwo : |(2 : ℝ)| ≤ 2 := by norm_num exact secantTiltBound hμ hcurv htwo have himage : ‖A t‖ ≤ M := by simpa only [ht, mul_one] using A.le_of_opNorm_le hA t rw [tangentCoefficient, norm_smul, Real.norm_eq_abs] have hpositive : 0 ≤ 2 / μ := by positivity have hproduct := mul_le_mul hratio himage (norm_nonneg (A t)) hpositive simpa only [div_mul_eq_mul_div] using hproduct

Helper for thm:planar-convergence: the descent bound makes the normal part of the step quadratic in its length when the gradient stays nonzero.

theorem stepNormalResidualBound (g s : EuclideanSpace ℝ (Fin 2)) {N C : ℝ} (hN : 0 < N) (hg : N ≤ ‖g‖) (hC : 0 ≤ C) (hdescent : inner ℝ g s ≤ 0) (hdecrease : -inner ℝ g s ≤ C * ‖s‖ ^ 2) : let t := ‖g‖⁻¹ • EuclideanPlane.perp g ‖s - inner ℝ t s • t‖ ≤ (C / N) * ‖s‖ ^ 2 := by dsimp only have hgpos := hN.trans_le hg have hunit : ‖‖g‖⁻¹ • g‖ = 1 := by rw [norm_smul, Real.norm_eq_abs, abs_inv, abs_norm, inv_mul_cancel₀ hgpos.ne'] have hresidual := normNormalResidual (‖g‖⁻¹ • g) s hunit rw [map_smul] at hresidual rw [hresidual, real_inner_smul_left, abs_mul, abs_inv, abs_norm, abs_of_nonpos hdescent, ← div_eq_inv_mul] apply (div_le_iff₀ hgpos).mpr have hfactor : 0 ≤ (C / N) * ‖s‖ ^ 2 := by positivity calc -inner ℝ g s ≤ C * ‖s‖ ^ 2 := hdecrease _ = ((C / N) * ‖s‖ ^ 2) * N := by field_simp _ ≤ ((C / N) * ‖s‖ ^ 2) * ‖g‖ := mul_le_mul_of_nonneg_left hg hfactor

Helper for thm:planar-convergence: a gradient secant bound and a positive gradient lower bound control relative changes of gradient norm.

theorem relativeGradientNormBound {E : Type*} [NormedAddCommGroup E] (g gNext : E) {N M ell : ℝ} (hN : 0 < N) (hg : N ≤ ‖g‖) (hsec : ‖gNext - g‖ ≤ M * ell) : |(‖gNext‖ - ‖g‖) / ‖g‖| ≤ (M / N) * ell := by have hnorm := (abs_norm_sub_norm_le gNext g).trans hsec have hquotient := secantTiltBound hN hg hnorm simpa only [div_mul_eq_mul_div] using hquotient

Helper for thm:planar-convergence: a quadratic gradient Taylor remainder and quadratic normal step component give quadratic cancellation between tangent transport and logarithmic gradient-norm change.

theorem gradientTangentLogQuadraticBound (g gNext s : EuclideanSpace ℝ (Fin 2)) (A : EuclideanSpace ℝ (Fin 2) →L[ℝ] EuclideanSpace ℝ (Fin 2)) {μ N M C L : ℝ} (hμ : 0 < μ) (hN : 0 < N) (hg : N ≤ ‖g‖) (hgNext : N ≤ ‖gNext‖) (hC : 0 ≤ C) (hA : ‖A‖ ≤ M) (hcurv : μ ≤ inner ℝ (‖g‖⁻¹ • EuclideanPlane.perp g) (A (‖g‖⁻¹ • EuclideanPlane.perp g))) (hsec : ‖gNext - g‖ ≤ M * ‖s‖) (hTaylor : ‖gNext - g - A s‖ ≤ L * ‖s‖ ^ 2) (hdescent : inner ℝ g s ≤ 0) (hdecrease : -inner ℝ g s ≤ C * ‖s‖ ^ 2) (hsmall : (M / N) * ‖s‖ ≤ 1 / 2) : let t := ‖g‖⁻¹ • EuclideanPlane.perp g let tNext := ‖gNext‖⁻¹ • EuclideanPlane.perp gNext |inner ℝ (tangentCoefficient A t) (tNext - t) + 2 * (Real.log ‖gNext‖ - Real.log ‖g‖)| ≤ ((2 * M / μ) * (L + M * (C / N)) / N + 8 * (M / N) ^ 2) * ‖s‖ ^ 2 := by let t := ‖g‖⁻¹ • EuclideanPlane.perp g have hgpos := hN.trans_le hg have hgNextPos := hN.trans_le hgNext have hM : 0 ≤ M := (norm_nonneg A).trans hA have ht : ‖t‖ = 1 := by dsimp only [t] rw [norm_smul, Real.norm_eq_abs, abs_inv, abs_norm, EuclideanPlane.perp.norm_map, inv_mul_cancel₀ hgpos.ne'] have hcoef := tangentCoefficientNormBound A t ht hμ hcurv hA have hnormal := stepNormalResidualBound g s hN hg hC hdescent hdecrease have hraw := tangentCoefficientSecantBound A t s (gNext - g) (inner ℝ t s) have hcoefPos : 0 ≤ 2 * M / μ := by positivity have hnormalScaled := mul_le_mul hA hnormal (norm_nonneg _) hM have hins : ‖gNext - g - A s‖ + ‖A‖ * ‖s - inner ℝ t s • t‖ ≤ (L + M * (C / N)) * ‖s‖ ^ 2 := by nlinarith have hinsPos : 0 ≤ ‖gNext - g - A s‖ + ‖A‖ * ‖s - inner ℝ t s • t‖ := by positivity have hnum : |inner ℝ (tangentCoefficient A t) (EuclideanPlane.perp (gNext - g))| ≤ ((2 * M / μ) * (L + M * (C / N))) * ‖s‖ ^ 2 := by have hproduct := mul_le_mul hcoef hins hinsPos hcoefPos nlinarith have hnumDiv := secantTiltBound hN hgNext hnum rw [abs_div, abs_of_pos hgNextPos] at hnumDiv have hrelative := relativeGradientNormBound g gNext hN hg hsec have hrelativeSmall := hrelative.trans hsmall have hcurvNe : inner ℝ t (A t) ≠ 0 := (hμ.trans_le hcurv).ne' have hbound := gradientTangentLogBound g gNext A (norm_pos_iff.mp hgpos) (norm_pos_iff.mp hgNextPos) hcurvNe hrelativeSmall have hratioPos : 0 ≤ (M / N) * ‖s‖ := by positivity have hsquare : ((‖gNext‖ - ‖g‖) / ‖g‖) ^ 2 ≤ (M / N) ^ 2 * ‖s‖ ^ 2 := by have hproduct := mul_le_mul hrelative hrelative (abs_nonneg ((‖gNext‖ - ‖g‖) / ‖g‖)) hratioPos simpa only [← pow_two, sq_abs, mul_pow] using hproduct dsimp only at hbound ⊢ dsimp only [t] at hnumDiv rw [← div_mul_eq_mul_div] at hnumDiv nlinarith

Helper for thm:planar-convergence: the positive unit tangent to the level curve is the right-angle rotation of the normalized gradient.

noncomputable def gradientUnitTangent (f : EuclideanSpace ℝ (Fin 2) → ℝ) (x : EuclideanSpace ℝ (Fin 2)) : EuclideanSpace ℝ (Fin 2) := ‖gradient f x‖⁻¹ • EuclideanPlane.perp (gradient f x)

Helper for thm:planar-convergence: the gradient transport coefficient is evaluated at the physical Hessian and the unit level-set tangent.

noncomputable def gradientTransportCoefficient (f : EuclideanSpace ℝ (Fin 2) → ℝ) (x : EuclideanSpace ℝ (Fin 2)) : EuclideanSpace ℝ (Fin 2) := tangentCoefficient (fderiv ℝ (gradient f) x) (gradientUnitTangent f x)

Helper for thm:planar-convergence: at a nonstationary point, the gradient tangent has unit norm.

theorem gradientUnitTangentNorm (f : EuclideanSpace ℝ (Fin 2) → ℝ) (x : EuclideanSpace ℝ (Fin 2)) (hg : gradient f x ≠ 0) : ‖gradientUnitTangent f x‖ = 1 := by rw [gradientUnitTangent, norm_smul, Real.norm_eq_abs, abs_inv, abs_norm, EuclideanPlane.perp.norm_map, inv_mul_cancel₀ (norm_ne_zero_iff.mpr hg)]

Helper for thm:planar-convergence: local Lipschitz Hessian regularity gives the quadratic tangent-log cancellation estimate along every sufficiently late step of the actual nonconvergent DFP orbit.

theorem orbitTangentLogQuadraticBound (iteration : DFP.InverseIteration (Fin 2)) {μ c₁ c₂ : ℝ} (hf : ContDiff ℝ 2 iteration.objective) (hμ : 0 < μ) (hHessian : ∀ x v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient iteration.objective) x v) v) (hStep : ∀ k, 0 < iteration.stepLength k) (hWolfe : ∀ k, LineSearch.IsWeakWolfe c₁ c₂ iteration.objective (iteration.point k) (iteration.point (k + 1) - iteration.point k)) (hLip : LocallyLipschitzOn (DFP.objectiveSublevel iteration) (fderiv ℝ (gradient iteration.objective))) {xstar : EuclideanSpace ℝ (Fin 2)} (hmin : ∀ x, iteration.objective xstar ≤ iteration.objective x) (hnot : ¬ Tendsto iteration.point atTop (𝓝 xstar)) : ∃ Q ≥ 0, ∀ᶠ k in atTop, |inner ℝ (gradientTransportCoefficient iteration.objective (iteration.point k)) (gradientUnitTangent iteration.objective (iteration.point (k + 1)) - gradientUnitTangent iteration.objective (iteration.point k)) + 2 * (Real.log ‖gradient iteration.objective (iteration.point (k + 1))‖ - Real.log ‖gradient iteration.objective (iteration.point k)‖)| ≤ Q * ‖iteration.point (k + 1) - iteration.point k‖ ^ 2 := by obtain ⟨N, hN, hgradient⟩ := gradientLowerBoundOfNonconvergence iteration hf hμ hHessian hStep hWolfe hmin hnot obtain ⟨M₁, hM₁, hsecBounds⟩ := secantBoundsAlongOrbit iteration hf hμ hHessian hStep hWolfe obtain ⟨M₂, hM₂, hdiffBounds⟩ := uniformDifferentialBoundsOnSublevel hf hμ hHessian (iteration.point 0) obtain ⟨L, hL, hTaylor⟩ := orbitGradientTaylorRemainder iteration hf hμ hHessian hStep hWolfe hLip let M := M₁ + M₂ let C := M₁ / (1 - c₂) have hM : 0 < M := add_pos hM₁ hM₂ have hC : 0 < C := div_pos hM₁ (sub_pos.mpr (hWolfe 0).c₂_lt_one) have hM₁le : M₁ ≤ M := le_add_of_nonneg_right hM₂.le have hM₂le : M₂ ≤ M := le_add_of_nonneg_left hM₁.le let Q := (2 * M / μ) * (L + M * (C / N)) / N + 8 * (M / N) ^ 2 have hQ : 0 ≤ Q := by dsimp only [Q] positivity refine ⟨Q, hQ, ?_⟩ have hsmall : ∀ᶠ k in atTop, (M / N) * ‖iteration.point (k + 1) - iteration.point k‖ ≤ 1 / 2 := by have hlimit := (stepNormTendstoZero iteration hf hμ hHessian hStep hWolfe).const_mul (M / N) have hlimitZero : Tendsto (fun k ↦ (M / N) * ‖iteration.point (k + 1) - iteration.point k‖) atTop (𝓝 0) := by simpa using hlimit have hhalf : (0 : ℝ) < 1 / 2 := by norm_num exact (hlimitZero.eventually (eventually_lt_nhds hhalf)).mono (fun _ hk ↦ hk.le) have hmem := iteration.pointMemObjectiveSublevelOfWeakWolfe hStep hWolfe filter_upwards [hsmall] with k hk have hg : gradient iteration.objective (iteration.point k) ≠ 0 := norm_pos_iff.mp (hN.trans_le (hgradient k)) have ht := gradientUnitTangentNorm iteration.objective (iteration.point k) hg have hcurv := hHessian (iteration.point k) (gradientUnitTangent iteration.objective (iteration.point k)) rw [ht, one_pow, mul_one, real_inner_comm] at hcurv have hlevel := (DFP.mem_objectiveSublevel_iff iteration _).mp (hmem k) have hA := ((hdiffBounds _ hlevel).2).trans hM₂le have hsec := ((hsecBounds k).1).trans (mul_le_mul_of_nonneg_right hM₁le (norm_nonneg (iteration.point (k + 1) - iteration.point k))) have hdescent : inner ℝ (gradient iteration.objective (iteration.point k)) (iteration.point (k + 1) - iteration.point k) ≤ 0 := by simpa only [DFP.gradients_apply] using (iteration.gradientInnerDisplacementNeg k (hStep k)).le exact gradientTangentLogQuadraticBound _ _ _ _ hμ hN (hgradient k) (hgradient (k + 1)) hC.le hA hcurv hsec (hTaylor k) hdescent (hsecBounds k).2 hk

Helper for thm:planar-convergence: orient a unit step toward the chosen tangent, using the positive sign when the tangential pairing is zero.

noncomputable def tangentOrientation (t u : EuclideanSpace ℝ (Fin 2)) : ℝ := if inner ℝ t u < 0 then -1 else 1

Helper for thm:planar-convergence: the tangent orientation is a sign.

theorem tangentOrientationSign (t u : EuclideanSpace ℝ (Fin 2)) : tangentOrientation t u = 1 ∨ tangentOrientation t u = -1 := by by_cases h : inner ℝ t u < 0 · exact Or.inr (if_pos h) · exact Or.inl (if_neg h)

Helper for thm:planar-convergence: tangent orientation preserves norms.

theorem tangentOrientationAbs (t u : EuclideanSpace ℝ (Fin 2)) : |tangentOrientation t u| = 1 := by rcases tangentOrientationSign t u with h | h · rw [h, abs_one] · rw [h, abs_neg, abs_one]

Helper for thm:planar-convergence: orientation replaces the tangential component by its absolute value.

theorem tangentOrientationPairing (t u : EuclideanSpace ℝ (Fin 2)) : inner ℝ (tangentOrientation t u • u) t = |inner ℝ t u| := by have hscalar : tangentOrientation t u * inner ℝ t u = |inner ℝ t u| := by by_cases h : inner ℝ t u < 0 · rw [tangentOrientation, if_pos h, neg_one_mul, abs_of_neg h] · rw [tangentOrientation, if_neg h, one_mul, abs_of_nonneg (le_of_not_gt h)] rw [real_inner_smul_left] have hcomm : inner ℝ u t = inner ℝ t u := real_inner_comm _ _ exact (congrArg (fun r : ℝ ↦ tangentOrientation t u * r) hcomm).trans hscalar

Helper for thm:planar-convergence: squared normal and tangent components of a planar unit vector add to one.

theorem planarUnitComponentSquares (ν u : EuclideanSpace ℝ (Fin 2)) (hν : ‖ν‖ = 1) (hu : ‖u‖ = 1) : (inner ℝ ν u) ^ 2 + (inner ℝ (EuclideanPlane.perp ν) u) ^ 2 = 1 := by have hspecial := (EuclideanPlane.frame_mem_specialOrthogonalGroup_iff ν).mpr hν have hR := (Matrix.mem_specialOrthogonalGroup_iff.mp hspecial).1 have hcoordinates : ‖(WithLp.toLp 2 ![inner ℝ ν u, inner ℝ (EuclideanPlane.perp ν) u] : EuclideanSpace ℝ (Fin 2))‖ = 1 := by rw [← planarFrameCoordinates] have hnorm := Matrix.norm_toLp_mulVec_eq_of_mem_orthogonalGroup (EuclideanPlane.frame ν).transpose (orthogonalTranspose hR) (WithLp.ofLp u) exact hnorm.trans hu have hsquare := EuclideanSpace.real_norm_sq_eq (WithLp.toLp 2 ![inner ℝ ν u, inner ℝ (EuclideanPlane.perp ν) u]) rw [hcoordinates, one_pow, Fin.sum_univ_two] at hsquare exact hsquare.symm

Helper for thm:planar-convergence: after orienting a planar unit vector toward the tangent, its distance to that tangent is controlled by twice its normal component, including either original step direction.

theorem orientedUnitStepTangentBound (ν u : EuclideanSpace ℝ (Fin 2)) (hν : ‖ν‖ = 1) (hu : ‖u‖ = 1) : ‖tangentOrientation (EuclideanPlane.perp ν) u • u - EuclideanPlane.perp ν‖ ≤ 2 * |inner ℝ ν u| := by have hnorm : ‖tangentOrientation (EuclideanPlane.perp ν) u • u‖ = 1 := by rw [norm_smul, Real.norm_eq_abs, tangentOrientationAbs, hu, mul_one] have ht : ‖EuclideanPlane.perp ν‖ = 1 := (EuclideanPlane.perp.norm_map ν).trans hν have hcomponents := planarUnitComponentSquares ν u hν hu have hdistance := norm_sub_sq_real (tangentOrientation (EuclideanPlane.perp ν) u • u) (EuclideanPlane.perp ν) rw [hnorm, ht, tangentOrientationPairing, one_pow] at hdistance have hnormalSq := sq_abs (inner ℝ ν u) have htangentSq := sq_abs (inner ℝ (EuclideanPlane.perp ν) u) have htangent : |inner ℝ (EuclideanPlane.perp ν) u| ≤ 1 := by nlinarith [sq_nonneg (inner ℝ ν u), abs_nonneg (inner ℝ (EuclideanPlane.perp ν) u)] have hproduct := mul_nonneg (abs_nonneg (inner ℝ (EuclideanPlane.perp ν) u)) (sub_nonneg.mpr htangent) nlinarith [norm_nonneg (tangentOrientation (EuclideanPlane.perp ν) u • u - EuclideanPlane.perp ν), abs_nonneg (inner ℝ ν u)]

Helper for thm:planar-convergence: the normalized physical step has a small normal component under the quadratic descent bound.

theorem normalizedStepNormalBound (g s : EuclideanSpace ℝ (Fin 2)) {N C : ℝ} (hN : 0 < N) (hg : N ≤ ‖g‖) (hC : 0 ≤ C) (hs : s ≠ 0) (hdescent : inner ℝ g s ≤ 0) (hdecrease : -inner ℝ g s ≤ C * ‖s‖ ^ 2) : |inner ℝ (‖g‖⁻¹ • g) (‖s‖⁻¹ • s)| ≤ (C / N) * ‖s‖ := by have hgpos := hN.trans_le hg have hspos := norm_pos_iff.mpr hs have hunit : ‖‖g‖⁻¹ • g‖ = 1 := by rw [norm_smul, Real.norm_eq_abs, abs_inv, abs_norm, inv_mul_cancel₀ hgpos.ne'] have hnormal := stepNormalResidualBound g s hN hg hC hdescent hdecrease dsimp only at hnormal have hresidual := normNormalResidual (‖g‖⁻¹ • g) s hunit rw [map_smul] at hresidual rw [hresidual] at hnormal rw [real_inner_smul_right, abs_mul, abs_inv, abs_norm, ← div_eq_inv_mul] apply (div_le_iff₀ hspos).mpr nlinarith

Helper for thm:planar-convergence: orienting the physical step toward the level-set tangent gives an error linear in its step length.

theorem orientedStepTangentBound (g s : EuclideanSpace ℝ (Fin 2)) {N C : ℝ} (hN : 0 < N) (hg : N ≤ ‖g‖) (hC : 0 ≤ C) (hs : s ≠ 0) (hdescent : inner ℝ g s ≤ 0) (hdecrease : -inner ℝ g s ≤ C * ‖s‖ ^ 2) : let t := ‖g‖⁻¹ • EuclideanPlane.perp g let u := ‖s‖⁻¹ • s ‖tangentOrientation t u • u - t‖ ≤ (2 * C / N) * ‖s‖ := by have hgpos := hN.trans_le hg have hunitG : ‖‖g‖⁻¹ • g‖ = 1 := by rw [norm_smul, Real.norm_eq_abs, abs_inv, abs_norm, inv_mul_cancel₀ hgpos.ne'] have hunitS : ‖‖s‖⁻¹ • s‖ = 1 := by rw [norm_smul, Real.norm_eq_abs, abs_inv, abs_norm, inv_mul_cancel₀ (norm_ne_zero_iff.mpr hs)] have hbound := orientedUnitStepTangentBound (‖g‖⁻¹ • g) (‖s‖⁻¹ • s) hunitG hunitS rw [map_smul] at hbound have hnormal := normalizedStepNormalBound g s hN hg hC hs hdescent hdecrease dsimp only calc _ ≤ 2 * ((C / N) * ‖s‖) := hbound.trans (mul_le_mul_of_nonneg_left hnormal zero_le_two) _ = (2 * C / N) * ‖s‖ := by ring

Helper for thm:planar-convergence: a common orientation sign preserves the normalized secant equation and does not enlarge the Taylor remainder.

theorem orientedSecantTaylorBound (A : EuclideanSpace ℝ (Fin 2) →L[ℝ] EuclideanSpace ℝ (Fin 2)) (s y t : EuclideanSpace ℝ (Fin 2)) {σ M L E : ℝ} (hs : s ≠ 0) (hσ : |σ| = 1) (hA : ‖A‖ ≤ M) (hTaylor : ‖y - A s‖ ≤ L * ‖s‖ ^ 2) (herror : ‖σ • (‖s‖⁻¹ • s) - t‖ ≤ E * ‖s‖) : ‖σ • (‖s‖⁻¹ • y) - A t‖ ≤ (L + M * E) * ‖s‖ := by have hspos := norm_pos_iff.mpr hs have hM : 0 ≤ M := (norm_nonneg A).trans hA have hsplit : σ • (‖s‖⁻¹ • y) - A t = σ • (‖s‖⁻¹ • (y - A s)) + A (σ • (‖s‖⁻¹ • s) - t) := by simp only [map_sub, map_smul, smul_sub] module have hfirst : ‖σ • (‖s‖⁻¹ • (y - A s))‖ ≤ L * ‖s‖ := by rw [norm_smul, norm_smul, Real.norm_eq_abs, Real.norm_eq_abs, hσ, one_mul, abs_inv, abs_norm, ← div_eq_inv_mul] apply (div_le_iff₀ hspos).mpr nlinarith have hsecond : ‖A (σ • (‖s‖⁻¹ • s) - t)‖ ≤ M * (E * ‖s‖) := by exact (A.le_opNorm _).trans (mul_le_mul hA herror (norm_nonneg _) hM) rw [hsplit] have htriangle := norm_add_le (σ • (‖s‖⁻¹ • (y - A s))) (A (σ • (‖s‖⁻¹ • s) - t)) nlinarith

Helper for thm:planar-convergence: the canonical signed normalized step direction is oriented toward the current gradient tangent.

noncomputable def orientedOrbitStep (iteration : DFP.InverseIteration (Fin 2)) (k : ℕ) : EuclideanSpace ℝ (Fin 2) := let s := iteration.point (k + 1) - iteration.point k let u := ‖s‖⁻¹ • s tangentOrientation (gradientUnitTangent iteration.objective (iteration.point k)) u • u

Helper for thm:planar-convergence: orient the normalized gradient secant by the same sign used for the physical step direction.

noncomputable def orientedOrbitSecant (iteration : DFP.InverseIteration (Fin 2)) (k : ℕ) : EuclideanSpace ℝ (Fin 2) := let s := iteration.point (k + 1) - iteration.point k let u := ‖s‖⁻¹ • s let y := gradient iteration.objective (iteration.point (k + 1)) - gradient iteration.objective (iteration.point k) tangentOrientation (gradientUnitTangent iteration.objective (iteration.point k)) u • (‖s‖⁻¹ • y)

Helper for thm:planar-convergence: the canonically oriented physical step direction has unit norm.

theorem orientedOrbitStepNorm (iteration : DFP.InverseIteration (Fin 2)) (k : ℕ) (hStep : 0 < iteration.stepLength k) : ‖orientedOrbitStep iteration k‖ = 1 := by rw [orientedOrbitStep] rw [norm_smul, Real.norm_eq_abs, tangentOrientationAbs, one_mul, norm_smul, Real.norm_eq_abs, abs_inv, abs_norm, inv_mul_cancel₀ (orbitStepNormPos iteration k hStep).ne']

Helper for thm:planar-convergence: orientation preserves the exact secant equation for the actual inverse-Hessian sequence.

theorem orientedOrbitSecantEquation (iteration : DFP.InverseIteration (Fin 2)) (k : ℕ) : iteration.inverseHessian (k + 1) *ᵥ WithLp.ofLp (orientedOrbitSecant iteration k) = WithLp.ofLp (orientedOrbitStep iteration k) := by simp only [orientedOrbitStep, orientedOrbitSecant, WithLp.ofLp_smul, Matrix.mulVec_smul] have hsec := normalizedOrbitSecantEquation iteration k simp only [WithLp.ofLp_smul, Matrix.mulVec_smul] at hsec rw [hsec]

Helper for thm:planar-convergence: a common sign preserves real inner products, so orientation leaves normalized secant curvature unchanged.

theorem commonSignInner {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] (u v : E) {σ : ℝ} (hσ : |σ| = 1) : inner ℝ (σ • u) (σ • v) = inner ℝ u v := by have hsquare : σ ^ 2 = 1 := by have h := sq_abs σ rw [hσ, one_pow] at h exact h.symm rw [real_inner_smul_left, real_inner_smul_right] calc σ * (σ * inner ℝ u v) = σ ^ 2 * inner ℝ u v := by ring _ = inner ℝ u v := by rw [hsquare, one_mul]

Helper for thm:planar-convergence: oriented orbit curvature is exactly the original normalized gradient secant curvature.

theorem orientedOrbitCurvature (iteration : DFP.InverseIteration (Fin 2)) (k : ℕ) : inner ℝ (orientedOrbitStep iteration k) (orientedOrbitSecant iteration k) = inner ℝ (gradient iteration.objective (iteration.point (k + 1)) - gradient iteration.objective (iteration.point k)) (iteration.point (k + 1) - iteration.point k) / ‖iteration.point (k + 1) - iteration.point k‖ ^ 2 := by rw [orientedOrbitStep, orientedOrbitSecant] rw [commonSignInner _ _ (tangentOrientationAbs _ _), normalizedSecantCurvature]

Helper for thm:planar-convergence: on the actual DFP orbit, the canonically oriented step approximates the gradient tangent and the oriented secant approximates its Hessian image, uniformly to first order in step length.

theorem orbitOrientedSecantApproximation (iteration : DFP.InverseIteration (Fin 2)) {μ c₁ c₂ : ℝ} (hf : ContDiff ℝ 2 iteration.objective) (hμ : 0 < μ) (hHessian : ∀ x v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient iteration.objective) x v) v) (hStep : ∀ k, 0 < iteration.stepLength k) (hWolfe : ∀ k, LineSearch.IsWeakWolfe c₁ c₂ iteration.objective (iteration.point k) (iteration.point (k + 1) - iteration.point k)) (hLip : LocallyLipschitzOn (DFP.objectiveSublevel iteration) (fderiv ℝ (gradient iteration.objective))) {xstar : EuclideanSpace ℝ (Fin 2)} (hmin : ∀ x, iteration.objective xstar ≤ iteration.objective x) (hnot : ¬ Tendsto iteration.point atTop (𝓝 xstar)) : ∃ E > 0, ∃ V ≥ 0, ∀ k, ‖orientedOrbitStep iteration k - gradientUnitTangent iteration.objective (iteration.point k)‖ ≤ E * ‖iteration.point (k + 1) - iteration.point k‖ ∧ ‖orientedOrbitSecant iteration k - fderiv ℝ (gradient iteration.objective) (iteration.point k) (gradientUnitTangent iteration.objective (iteration.point k))‖ ≤ V * ‖iteration.point (k + 1) - iteration.point k‖ := by obtain ⟨N, hN, hgradient⟩ := gradientLowerBoundOfNonconvergence iteration hf hμ hHessian hStep hWolfe hmin hnot obtain ⟨M₁, hM₁, hsecBounds⟩ := secantBoundsAlongOrbit iteration hf hμ hHessian hStep hWolfe obtain ⟨M₂, hM₂, hdiffBounds⟩ := uniformDifferentialBoundsOnSublevel hf hμ hHessian (iteration.point 0) obtain ⟨L, hL, hTaylor⟩ := orbitGradientTaylorRemainder iteration hf hμ hHessian hStep hWolfe hLip let C := M₁ / (1 - c₂) let E := 2 * C / N have hC : 0 < C := div_pos hM₁ (sub_pos.mpr (hWolfe 0).c₂_lt_one) have hE : 0 < E := by dsimp only [E] positivity have hV : 0 ≤ L + M₂ * E := by positivity refine ⟨E, hE, L + M₂ * E, hV, ?_⟩ have hmem := iteration.pointMemObjectiveSublevelOfWeakWolfe hStep hWolfe intro k have hs : iteration.point (k + 1) - iteration.point k ≠ 0 := norm_pos_iff.mp (orbitStepNormPos iteration k (hStep k)) have hdescent : inner ℝ (gradient iteration.objective (iteration.point k)) (iteration.point (k + 1) - iteration.point k) ≤ 0 := by simpa only [DFP.gradients_apply] using (iteration.gradientInnerDisplacementNeg k (hStep k)).le have herror : ‖orientedOrbitStep iteration k - gradientUnitTangent iteration.objective (iteration.point k)‖ ≤ E * ‖iteration.point (k + 1) - iteration.point k‖ := orientedStepTangentBound _ _ hN (hgradient k) hC.le hs hdescent (hsecBounds k).2 refine ⟨herror, ?_⟩ have hlevel := (DFP.mem_objectiveSublevel_iff iteration _).mp (hmem k) have hA := (hdiffBounds _ hlevel).2 exact orientedSecantTaylorBound _ _ _ _ hs (tangentOrientationAbs _ _) hA (hTaylor k) herror

Helper for thm:planar-convergence: normalization is Lipschitz between vectors whose norms have a common positive lower bound.

theorem normalizationDifferenceBound {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] (g gNext : E) {N : ℝ} (hN : 0 < N) (hg : N ≤ ‖g‖) (hgNext : N ≤ ‖gNext‖) : ‖‖gNext‖⁻¹ • gNext - ‖g‖⁻¹ • g‖ ≤ (2 / N) * ‖gNext - g‖ := by have hgpos := hN.trans_le hg have hnpos := hN.trans_le hgNext have hcoeff : ‖gNext‖ * ‖g‖⁻¹ = 1 + (‖gNext‖ - ‖g‖) / ‖g‖ := by field_simp ring have hsplit : ‖gNext‖ • (‖gNext‖⁻¹ • gNext - ‖g‖⁻¹ • g) = (gNext - g) - ((‖gNext‖ - ‖g‖) / ‖g‖) • g := by rw [smul_sub, smul_smul, smul_smul, mul_inv_cancel₀ hnpos.ne', one_smul, hcoeff, add_smul, one_smul] module have hcorrection : ‖((‖gNext‖ - ‖g‖) / ‖g‖) • g‖ = |‖gNext‖ - ‖g‖| := by rw [norm_smul, Real.norm_eq_abs, abs_div, abs_norm, div_mul_cancel₀ _ hgpos.ne'] have htriangle := norm_sub_le (gNext - g) (((‖gNext‖ - ‖g‖) / ‖g‖) • g) rw [hcorrection, ← hsplit, norm_smul, Real.norm_eq_abs, abs_norm] at htriangle have hscale := mul_le_mul_of_nonneg_right hgNext (norm_nonneg (‖gNext‖⁻¹ • gNext - ‖g‖⁻¹ • g)) rw [div_mul_eq_mul_div] apply (le_div_iff₀ hN).mpr nlinarith [abs_norm_sub_norm_le gNext g]

Helper for thm:planar-convergence: right-angle rotation preserves the Lipschitz estimate for the normalized gradient tangent.

theorem tangentDifferenceBound (g gNext : EuclideanSpace ℝ (Fin 2)) {N : ℝ} (hN : 0 < N) (hg : N ≤ ‖g‖) (hgNext : N ≤ ‖gNext‖) : ‖‖gNext‖⁻¹ • EuclideanPlane.perp gNext - ‖g‖⁻¹ • EuclideanPlane.perp g‖ ≤ (2 / N) * ‖gNext - g‖ := by have hbound := normalizationDifferenceBound g gNext hN hg hgNext have hnorm := EuclideanPlane.perp.norm_map (‖gNext‖⁻¹ • gNext - ‖g‖⁻¹ • g) rw [map_sub, map_smul, map_smul] at hnorm exact hnorm.le.trans hbound

Helper for thm:planar-convergence: the actual unit tangent changes by at most a fixed multiple of the physical step length on a nonconvergent orbit.

theorem orbitTangentVariationBound (iteration : DFP.InverseIteration (Fin 2)) {μ c₁ c₂ : ℝ} (hf : ContDiff ℝ 2 iteration.objective) (hμ : 0 < μ) (hHessian : ∀ x v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient iteration.objective) x v) v) (hStep : ∀ k, 0 < iteration.stepLength k) (hWolfe : ∀ k, LineSearch.IsWeakWolfe c₁ c₂ iteration.objective (iteration.point k) (iteration.point (k + 1) - iteration.point k)) {xstar : EuclideanSpace ℝ (Fin 2)} (hmin : ∀ x, iteration.objective xstar ≤ iteration.objective x) (hnot : ¬ Tendsto iteration.point atTop (𝓝 xstar)) : ∃ T > 0, ∀ k, ‖gradientUnitTangent iteration.objective (iteration.point (k + 1)) - gradientUnitTangent iteration.objective (iteration.point k)‖ ≤ T * ‖iteration.point (k + 1) - iteration.point k‖ := by obtain ⟨N, hN, hgradient⟩ := gradientLowerBoundOfNonconvergence iteration hf hμ hHessian hStep hWolfe hmin hnot obtain ⟨M, hM, hsec⟩ := secantBoundsAlongOrbit iteration hf hμ hHessian hStep hWolfe have hT : 0 < (2 / N) * M := by positivity refine ⟨(2 / N) * M, hT, ?_⟩ intro k have hbound := tangentDifferenceBound _ _ hN (hgradient k) (hgradient (k + 1)) have hfactor : 0 ≤ 2 / N := by positivity have hscaled := mul_le_mul_of_nonneg_left (hsec k).1 hfactor exact hbound.trans (hscaled.trans_eq (mul_assoc _ _ _).symm)

Helper for thm:planar-convergence: variations of a unit tangent and a bounded linear operator control the variation of their product.

theorem operatorUnitVectorVariation {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] (A B : E →L[ℝ] E) (t u : E) (hu : ‖u‖ = 1) : ‖B u - A t‖ ≤ ‖B - A‖ + ‖A‖ * ‖u - t‖ := by have hsplit : B u - A t = (B - A) u + A (u - t) := by rw [sub_apply, map_sub] module rw [hsplit] have htriangle := norm_add_le ((B - A) u) (A (u - t)) have hfirst := (B - A).le_opNorm u rw [hu, mul_one] at hfirst have hsecond := A.le_opNorm (u - t) linarith

Helper for thm:planar-convergence: pointwise first-order approximation and controlled target variation give controlled approximant variation.

theorem variationBoundOfApproximation {E : Type*} [NormedAddCommGroup E] (p q : ℕ → E) (ell : ℕ → ℝ) {A T : ℝ} (hT : 0 ≤ T) (hell : ∀ k, 0 ≤ ell k) (herror : ∀ k, ‖p k - q k‖ ≤ A * ell k) (hvariation : ∀ k, ‖q (k + 1) - q k‖ ≤ T * ell k) : ∀ k, ‖p (k + 1) - p k‖ ≤ (A + T) * (ell k + ell (k + 1)) := by intro k have htriangle := dist_triangle (p (k + 1)) (q (k + 1)) (p k) have htriangle' := dist_triangle (q (k + 1)) (q k) (p k) simp only [dist_eq_norm] at htriangle htriangle' have hsym : ‖q k - p k‖ = ‖p k - q k‖ := norm_sub_rev _ _ have hcross := mul_nonneg hT (hell (k + 1)) nlinarith [herror k, herror (k + 1), hvariation k]

Helper for thm:planar-convergence: local Lipschitz Hessian regularity controls changes of its action on the actual unit tangent.

theorem orbitHessianTangentVariationBound (iteration : DFP.InverseIteration (Fin 2)) {μ c₁ c₂ : ℝ} (hf : ContDiff ℝ 2 iteration.objective) (hμ : 0 < μ) (hHessian : ∀ x v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient iteration.objective) x v) v) (hStep : ∀ k, 0 < iteration.stepLength k) (hWolfe : ∀ k, LineSearch.IsWeakWolfe c₁ c₂ iteration.objective (iteration.point k) (iteration.point (k + 1) - iteration.point k)) (hLip : LocallyLipschitzOn (DFP.objectiveSublevel iteration) (fderiv ℝ (gradient iteration.objective))) {xstar : EuclideanSpace ℝ (Fin 2)} (hmin : ∀ x, iteration.objective xstar ≤ iteration.objective x) (hnot : ¬ Tendsto iteration.point atTop (𝓝 xstar)) : ∃ S ≥ 0, ∀ k, ‖fderiv ℝ (gradient iteration.objective) (iteration.point (k + 1)) (gradientUnitTangent iteration.objective (iteration.point (k + 1))) - fderiv ℝ (gradient iteration.objective) (iteration.point k) (gradientUnitTangent iteration.objective (iteration.point k))‖ ≤ S * ‖iteration.point (k + 1) - iteration.point k‖ := by obtain ⟨T, hT, htangent⟩ := orbitTangentVariationBound iteration hf hμ hHessian hStep hWolfe hmin hnot obtain ⟨M, hM, hdiffBounds⟩ := uniformDifferentialBoundsOnSublevel hf hμ hHessian (iteration.point 0) have hlevelEq : DFP.objectiveSublevel iteration = {x | iteration.objective x ≤ iteration.objective (iteration.point 0)} := by ext x exact DFP.mem_objectiveSublevel_iff iteration x rw [hlevelEq] at hLip obtain ⟨L, hL, hhessianLip⟩ := hessianLipschitzOnInitialSublevel hf hμ hHessian (iteration.point 0) hLip have hS : 0 ≤ L + M * T := by positivity refine ⟨L + M * T, hS, ?_⟩ intro k have hmem := iteration.pointMemObjectiveSublevelOfWeakWolfe hStep hWolfe rw [hlevelEq] at hmem have hg : gradient iteration.objective (iteration.point (k + 1)) ≠ 0 := by simpa only [DFP.gradients_apply] using iteration.gradientNeZeroOfSecantDenominator (k + 1) have ht := gradientUnitTangentNorm iteration.objective (iteration.point (k + 1)) hg have hbound := operatorUnitVectorVariation (fderiv ℝ (gradient iteration.objective) (iteration.point k)) (fderiv ℝ (gradient iteration.objective) (iteration.point (k + 1))) (gradientUnitTangent iteration.objective (iteration.point k)) (gradientUnitTangent iteration.objective (iteration.point (k + 1))) ht have hHdiff := hhessianLip _ (hmem k) _ (hmem (k + 1)) have hHnorm := (hdiffBounds _ (hmem k)).2 have hproduct := mul_le_mul hHnorm (htangent k) (norm_nonneg _) hM.le nlinarith

Helper for thm:planar-convergence: both oriented normalized secant vectors vary at the sum of adjacent step scales along the actual orbit.

theorem orbitOrientedSecantVariationBound (iteration : DFP.InverseIteration (Fin 2)) {μ c₁ c₂ : ℝ} (hf : ContDiff ℝ 2 iteration.objective) (hμ : 0 < μ) (hHessian : ∀ x v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient iteration.objective) x v) v) (hStep : ∀ k, 0 < iteration.stepLength k) (hWolfe : ∀ k, LineSearch.IsWeakWolfe c₁ c₂ iteration.objective (iteration.point k) (iteration.point (k + 1) - iteration.point k)) (hLip : LocallyLipschitzOn (DFP.objectiveSublevel iteration) (fderiv ℝ (gradient iteration.objective))) {xstar : EuclideanSpace ℝ (Fin 2)} (hmin : ∀ x, iteration.objective xstar ≤ iteration.objective x) (hnot : ¬ Tendsto iteration.point atTop (𝓝 xstar)) : ∃ D > 0, ∀ k, ‖orientedOrbitStep iteration (k + 1) - orientedOrbitStep iteration k‖ + ‖orientedOrbitSecant iteration (k + 1) - orientedOrbitSecant iteration k‖ ≤ D * (‖iteration.point (k + 1) - iteration.point k‖ + ‖iteration.point (k + 2) - iteration.point (k + 1)‖) := by obtain ⟨E, hE, V, hV, happrox⟩ := orbitOrientedSecantApproximation iteration hf hμ hHessian hStep hWolfe hLip hmin hnot obtain ⟨T, hT, htangent⟩ := orbitTangentVariationBound iteration hf hμ hHessian hStep hWolfe hmin hnot obtain ⟨S, hS, hHessianTangent⟩ := orbitHessianTangentVariationBound iteration hf hμ hHessian hStep hWolfe hLip hmin hnot have hstep := variationBoundOfApproximation (orientedOrbitStep iteration) (fun k ↦ gradientUnitTangent iteration.objective (iteration.point k)) (fun k ↦ ‖iteration.point (k + 1) - iteration.point k‖) hT.le (fun k ↦ norm_nonneg _) (fun k ↦ (happrox k).1) htangent have hsecant := variationBoundOfApproximation (orientedOrbitSecant iteration) (fun k ↦ fderiv ℝ (gradient iteration.objective) (iteration.point k) (gradientUnitTangent iteration.objective (iteration.point k))) (fun k ↦ ‖iteration.point (k + 1) - iteration.point k‖) hS (fun k ↦ norm_nonneg _) (fun k ↦ (happrox k).2) hHessianTangent have hD : 0 < E + T + V + S := by positivity refine ⟨E + T + V + S, hD, ?_⟩ intro k have hu := hstep k have hv := hsecant k nlinarith

Helper for thm:planar-convergence: global strong convexity makes the physical secant pairing strictly positive at every positive DFP step.

theorem orbitSecantPairingPos {n : ℕ} (iteration : DFP.InverseIteration (Fin n)) {μ : ℝ} (hf : ContDiff ℝ 2 iteration.objective) (hμ : 0 < μ) (hHessian : ∀ x v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient iteration.objective) x v) v) (k : ℕ) (hStep : 0 < iteration.stepLength k) : 0 < inner ℝ (gradient iteration.objective (iteration.point (k + 1)) - gradient iteration.objective (iteration.point k)) (iteration.point (k + 1) - iteration.point k) := by have hlower := inner_gradient_sub_ge_of_hessian_lower_bound iteration.objective μ hf hμ hHessian (iteration.point k) (iteration.point (k + 1)) have hpositive := mul_pos hμ (sq_pos_of_pos (orbitStepNormPos iteration k hStep)) exact hpositive.trans_le hlower

Helper for thm:planar-convergence: the original DFP iteration has the exact determinant ratio expressed in physical gradient and point differences.

theorem orbitDeterminantRecurrence {n : ℕ} (iteration : DFP.InverseIteration (Fin n)) {μ : ℝ} (hf : ContDiff ℝ 2 iteration.objective) (hμ : 0 < μ) (hHessian : ∀ x v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient iteration.objective) x v) v) (k : ℕ) (hStep : 0 < iteration.stepLength k) : let s := iteration.point (k + 1) - iteration.point k let y := gradient iteration.objective (iteration.point (k + 1)) - gradient iteration.objective (iteration.point k) (iteration.inverseHessian (k + 1)).det = (iteration.inverseHessian k).det * (inner ℝ y s / inner ℝ y (Matrix.toEuclideanCLM (𝕜 := ℝ) (iteration.inverseHessian k) y)) := by have hstep : iteration.point (k + 1) - iteration.point k = DFP.steps iteration.stepLength (DFP.directions iteration.inverseHessian (DFP.gradients iteration.objective iteration.point)) k := by rw [iteration.pointSucc k] abel have hpair := orbitSecantPairingPos iteration hf hμ hHessian k hStep have hsy : 0 < WithLp.ofLp (DFP.steps iteration.stepLength (DFP.directions iteration.inverseHessian (DFP.gradients iteration.objective iteration.point)) k) ⬝ᵥ WithLp.ofLp (DFP.gradientChanges (DFP.gradients iteration.objective iteration.point) k) := by rw [← hstep, DFP.gradientChanges_apply, DFP.gradients_apply, DFP.gradients_apply] simpa only [EuclideanSpace.inner_eq_star_dotProduct, star_trivial] using hpair dsimp only rw [iteration.inverseHessianSucc k, Matrix.det_inverseDFPUpdate (iteration.inverseHessianPosDef k) hsy] rw [Matrix.inner_toEuclideanCLM, EuclideanSpace.inner_eq_star_dotProduct] simp only [star_trivial, DFP.gradientChanges_apply, DFP.gradients_apply, hstep]

Helper for thm:planar-convergence: positive curvature lower bounds control the difference of reciprocal curvatures.

theorem reciprocalCurvatureDifference {a b μ : ℝ} (hμ : 0 < μ) (ha : μ ≤ a) (hb : μ ≤ b) : |a⁻¹ - b⁻¹| ≤ |a - b| / μ ^ 2 := by have haPos := hμ.trans_le ha have hbPos := hμ.trans_le hb have hidentity : a⁻¹ - b⁻¹ = (b - a) / (a * b) := by field_simp have hproduct : μ ^ 2 ≤ a * b := by simpa only [pow_two] using mul_le_mul ha hb hμ.le haPos.le have hnumerator : |b - a| ≤ |a - b| := le_of_eq (abs_sub_comm _ _) rw [hidentity] exact secantTiltBound (sq_pos_of_pos hμ) hproduct hnumerator

Helper for thm:planar-convergence: dividing nearby bounded vectors by uniformly positive curvatures preserves a quantitative variation bound.

theorem scaledSecantQuotientDifference {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] (v w : E) {a b μ M : ℝ} (hμ : 0 < μ) (ha : μ ≤ a) (hb : μ ≤ b) (hw : ‖w‖ ≤ M) : ‖(2 / a) • v - (2 / b) • w‖ ≤ (2 / μ) * ‖v - w‖ + (2 * M / μ ^ 2) * |a - b| := by have hM : 0 ≤ M := (norm_nonneg w).trans hw have hsplit : (2 / a) • v - (2 / b) • w = (2 / a) • (v - w) + (2 / a - 2 / b) • w := by rw [smul_sub, sub_smul] module have htwo : |(2 : ℝ)| ≤ 2 := by norm_num have hratio := secantTiltBound hμ ha htwo have hreciprocal := reciprocalCurvatureDifference hμ ha hb have hcoeff : |2 / a - 2 / b| ≤ (2 / μ ^ 2) * |a - b| := by have hidentity : 2 / a - 2 / b = 2 * (a⁻¹ - b⁻¹) := by ring have habs : |2 / a - 2 / b| = 2 * |a⁻¹ - b⁻¹| := by rw [hidentity, abs_mul] norm_num rw [habs] calc 2 * |a⁻¹ - b⁻¹| ≤ 2 * (|a - b| / μ ^ 2) := mul_le_mul_of_nonneg_left hreciprocal zero_le_two _ = (2 / μ ^ 2) * |a - b| := by ring have htriangle := norm_add_le ((2 / a) • (v - w)) ((2 / a - 2 / b) • w) simp only [norm_smul, Real.norm_eq_abs] at htriangle have hfirst := mul_le_mul_of_nonneg_right hratio (norm_nonneg (v - w)) have hcoeffPos : 0 ≤ (2 / μ ^ 2) * |a - b| := by positivity have hsecond := mul_le_mul hcoeff hw (norm_nonneg w) hcoeffPos have hnormalize : (2 / μ ^ 2) * |a - b| * M = (2 * M / μ ^ 2) * |a - b| := by ring rw [hnormalize] at hsecond rw [hsplit] nlinarith

Helper for thm:planar-convergence: curvature differences are controlled by changes of the unit direction and its bounded secant image.

theorem secantCurvatureVariation {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] (u v t w : E) {M : ℝ} (hu : ‖u‖ = 1) (hw : ‖w‖ ≤ M) : |inner ℝ u v - inner ℝ t w| ≤ ‖v - w‖ + M * ‖u - t‖ := by have hidentity : inner ℝ u v - inner ℝ t w = inner ℝ u (v - w) + inner ℝ (u - t) w := by rw [inner_sub_right, inner_sub_left] ring have htriangle := abs_add_le (inner ℝ u (v - w)) (inner ℝ (u - t) w) have hfirst := abs_real_inner_le_norm u (v - w) rw [hu, one_mul] at hfirst have hsecond := abs_real_inner_le_norm (u - t) w have hproduct := mul_le_mul_of_nonneg_left hw (norm_nonneg (u - t)) rw [hidentity] nlinarith

Helper for thm:planar-convergence: first-order direction and secant-image errors give a first-order error in the correcting coefficient.

theorem secantCoefficientApproximationBound {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] (u v t w : E) {μ M U V ell : ℝ} (hμ : 0 < μ) (hu : ‖u‖ = 1) (hw : ‖w‖ ≤ M) (hcurv : μ ≤ inner ℝ u v) (href : μ ≤ inner ℝ t w) (huError : ‖u - t‖ ≤ U * ell) (hvError : ‖v - w‖ ≤ V * ell) : ‖(2 / inner ℝ u v) • v - (2 / inner ℝ t w) • w‖ ≤ ((2 / μ) * V + (2 * M / μ ^ 2) * (V + M * U)) * ell := by have hM : 0 ≤ M := (norm_nonneg w).trans hw have hcurvature := secantCurvatureVariation u v t w hu hw have herrorProduct := mul_le_mul_of_nonneg_left huError hM have hcurvError : |inner ℝ u v - inner ℝ t w| ≤ (V + M * U) * ell := by nlinarith have hbound := scaledSecantQuotientDifference v w hμ hcurv href hw have hfactor₁ : 0 ≤ 2 / μ := by positivity have hfactor₂ : 0 ≤ 2 * M / μ ^ 2 := by positivity have hfirst := mul_le_mul_of_nonneg_left hvError hfactor₁ have hsecond := mul_le_mul_of_nonneg_left hcurvError hfactor₂ nlinarith

Helper for thm:planar-convergence: the Hessian lower bound specializes to positive curvature in the unit tangent direction.

theorem gradientTangentCurvatureLower (f : EuclideanSpace ℝ (Fin 2) → ℝ) (μ : ℝ) (hHessian : ∀ x v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient f) x v) v) (x : EuclideanSpace ℝ (Fin 2)) (hg : gradient f x ≠ 0) : μ ≤ inner ℝ (gradientUnitTangent f x) (fderiv ℝ (gradient f) x (gradientUnitTangent f x)) := by have hbound := hHessian x (gradientUnitTangent f x) rw [gradientUnitTangentNorm f x hg, one_pow, mul_one, real_inner_comm] at hbound exact hbound

Helper for thm:planar-convergence: orienting the physical secant preserves its positive curvature lower bound.

theorem orientedOrbitCurvatureLower (iteration : DFP.InverseIteration (Fin 2)) {μ : ℝ} (hf : ContDiff ℝ 2 iteration.objective) (hμ : 0 < μ) (hHessian : ∀ x v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient iteration.objective) x v) v) (k : ℕ) (hStep : 0 < iteration.stepLength k) : μ ≤ inner ℝ (orientedOrbitStep iteration k) (orientedOrbitSecant iteration k) := by rw [orientedOrbitCurvature] have hbound := normalizedOrbitCurvatureLowerBound iteration hf hμ hHessian k hStep simpa only [normalizedSecantCurvature] using hbound

Helper for thm:planar-convergence: the oriented secant coefficient approximates the Hessian tangent coefficient to first order along the actual orbit.

theorem orbitSecantCoefficientApproximation (iteration : DFP.InverseIteration (Fin 2)) {μ c₁ c₂ : ℝ} (hf : ContDiff ℝ 2 iteration.objective) (hμ : 0 < μ) (hHessian : ∀ x v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient iteration.objective) x v) v) (hStep : ∀ k, 0 < iteration.stepLength k) (hWolfe : ∀ k, LineSearch.IsWeakWolfe c₁ c₂ iteration.objective (iteration.point k) (iteration.point (k + 1) - iteration.point k)) (hLip : LocallyLipschitzOn (DFP.objectiveSublevel iteration) (fderiv ℝ (gradient iteration.objective))) {xstar : EuclideanSpace ℝ (Fin 2)} (hmin : ∀ x, iteration.objective xstar ≤ iteration.objective x) (hnot : ¬ Tendsto iteration.point atTop (𝓝 xstar)) : ∃ Q ≥ 0, ∀ k, ‖(2 / inner ℝ (orientedOrbitStep iteration k) (orientedOrbitSecant iteration k)) • orientedOrbitSecant iteration k - gradientTransportCoefficient iteration.objective (iteration.point k)‖ ≤ Q * ‖iteration.point (k + 1) - iteration.point k‖ := by obtain ⟨E, hE, V, hV, happrox⟩ := orbitOrientedSecantApproximation iteration hf hμ hHessian hStep hWolfe hLip hmin hnot obtain ⟨M, hM, hdiffBounds⟩ := uniformDifferentialBoundsOnSublevel hf hμ hHessian (iteration.point 0) have hQ : 0 ≤ (2 / μ) * V + (2 * M / μ ^ 2) * (V + M * E) := by positivity refine ⟨(2 / μ) * V + (2 * M / μ ^ 2) * (V + M * E), hQ, ?_⟩ intro k have hg : gradient iteration.objective (iteration.point k) ≠ 0 := by simpa only [DFP.gradients_apply] using iteration.gradientNeZeroOfSecantDenominator k have ht := gradientUnitTangentNorm iteration.objective (iteration.point k) hg have hmem := iteration.pointMemObjectiveSublevelOfWeakWolfe hStep hWolfe k have hlevel := (DFP.mem_objectiveSublevel_iff iteration _).mp hmem have hA := (hdiffBounds _ hlevel).2 have himage : ‖fderiv ℝ (gradient iteration.objective) (iteration.point k) (gradientUnitTangent iteration.objective (iteration.point k))‖ ≤ M := by simpa only [ht, mul_one] using (fderiv ℝ (gradient iteration.objective) (iteration.point k)).le_of_opNorm_le hA (gradientUnitTangent iteration.objective (iteration.point k)) have hcurv := orientedOrbitCurvatureLower iteration hf hμ hHessian k (hStep k) have href := gradientTangentCurvatureLower iteration.objective μ hHessian (iteration.point k) hg exact secantCoefficientApproximationBound _ _ _ _ hμ (orientedOrbitStepNorm iteration k (hStep k)) himage hcurv href (happrox k).1 (happrox k).2

Helper for thm:planar-convergence: the actual Hessian tangent coefficient varies by at most a fixed multiple of physical step length.

theorem orbitTransportCoefficientVariation (iteration : DFP.InverseIteration (Fin 2)) {μ c₁ c₂ : ℝ} (hf : ContDiff ℝ 2 iteration.objective) (hμ : 0 < μ) (hHessian : ∀ x v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient iteration.objective) x v) v) (hStep : ∀ k, 0 < iteration.stepLength k) (hWolfe : ∀ k, LineSearch.IsWeakWolfe c₁ c₂ iteration.objective (iteration.point k) (iteration.point (k + 1) - iteration.point k)) (hLip : LocallyLipschitzOn (DFP.objectiveSublevel iteration) (fderiv ℝ (gradient iteration.objective))) {xstar : EuclideanSpace ℝ (Fin 2)} (hmin : ∀ x, iteration.objective xstar ≤ iteration.objective x) (hnot : ¬ Tendsto iteration.point atTop (𝓝 xstar)) : ∃ Q ≥ 0, ∀ k, ‖gradientTransportCoefficient iteration.objective (iteration.point (k + 1)) - gradientTransportCoefficient iteration.objective (iteration.point k)‖ ≤ Q * ‖iteration.point (k + 1) - iteration.point k‖ := by obtain ⟨T, hT, htangent⟩ := orbitTangentVariationBound iteration hf hμ hHessian hStep hWolfe hmin hnot obtain ⟨S, hS, hHessianTangent⟩ := orbitHessianTangentVariationBound iteration hf hμ hHessian hStep hWolfe hLip hmin hnot obtain ⟨M, hM, hdiffBounds⟩ := uniformDifferentialBoundsOnSublevel hf hμ hHessian (iteration.point 0) have hQ : 0 ≤ (2 / μ) * S + (2 * M / μ ^ 2) * (S + M * T) := by positivity refine ⟨(2 / μ) * S + (2 * M / μ ^ 2) * (S + M * T), hQ, ?_⟩ intro k have hg (j : ℕ) : gradient iteration.objective (iteration.point j) ≠ 0 := by simpa only [DFP.gradients_apply] using iteration.gradientNeZeroOfSecantDenominator j have ht := gradientUnitTangentNorm iteration.objective (iteration.point k) (hg k) have htNext := gradientUnitTangentNorm iteration.objective (iteration.point (k + 1)) (hg (k + 1)) have hmem := iteration.pointMemObjectiveSublevelOfWeakWolfe hStep hWolfe k have hlevel := (DFP.mem_objectiveSublevel_iff iteration _).mp hmem have hA := (hdiffBounds _ hlevel).2 have himage : ‖fderiv ℝ (gradient iteration.objective) (iteration.point k) (gradientUnitTangent iteration.objective (iteration.point k))‖ ≤ M := by simpa only [ht, mul_one] using (fderiv ℝ (gradient iteration.objective) (iteration.point k)).le_of_opNorm_le hA (gradientUnitTangent iteration.objective (iteration.point k)) have hcurv := gradientTangentCurvatureLower iteration.objective μ hHessian (iteration.point (k + 1)) (hg (k + 1)) have href := gradientTangentCurvatureLower iteration.objective μ hHessian (iteration.point k) (hg k) exact secantCoefficientApproximationBound _ _ _ _ hμ htNext himage hcurv href (htangent k) (hHessianTangent k)

Helper for thm:planar-convergence: the correcting coefficient field stays uniformly bounded along the actual DFP orbit.

theorem orbitTransportCoefficientBound (iteration : DFP.InverseIteration (Fin 2)) {μ c₁ c₂ : ℝ} (hf : ContDiff ℝ 2 iteration.objective) (hμ : 0 < μ) (hHessian : ∀ x v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient iteration.objective) x v) v) (hStep : ∀ k, 0 < iteration.stepLength k) (hWolfe : ∀ k, LineSearch.IsWeakWolfe c₁ c₂ iteration.objective (iteration.point k) (iteration.point (k + 1) - iteration.point k)) : ∃ B > 0, ∀ k, ‖gradientTransportCoefficient iteration.objective (iteration.point k)‖ ≤ B := by obtain ⟨M, hM, hdiffBounds⟩ := uniformDifferentialBoundsOnSublevel hf hμ hHessian (iteration.point 0) have hB : 0 < 2 * M / μ := by positivity refine ⟨2 * M / μ, hB, ?_⟩ intro k have hg : gradient iteration.objective (iteration.point k) ≠ 0 := by simpa only [DFP.gradients_apply] using iteration.gradientNeZeroOfSecantDenominator k have ht := gradientUnitTangentNorm iteration.objective (iteration.point k) hg have hcurv := gradientTangentCurvatureLower iteration.objective μ hHessian (iteration.point k) hg have hmem := iteration.pointMemObjectiveSublevelOfWeakWolfe hStep hWolfe k have hlevel := (DFP.mem_objectiveSublevel_iff iteration _).mp hmem exact tangentCoefficientNormBound _ _ ht hμ hcurv (hdiffBounds _ hlevel).2

Helper for thm:planar-convergence: the relative curvature increment of two successive oriented secant pairs.

noncomputable def relativeCurvatureIncrement {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] (u v du dv : E) : ℝ := (inner ℝ v du + inner ℝ u dv + inner ℝ du dv) / inner ℝ u v

Helper for thm:planar-convergence: the relative quadratic-form increment measured by the preceding inverse Hessian.

noncomputable def relativeQuadraticIncrement {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] (H : E →L[ℝ] E) (u v dv : E) : ℝ := (2 * inner ℝ u dv + inner ℝ dv (H dv)) / inner ℝ u v

Helper for thm:planar-convergence: the relative curvature increment recovers the next curvature exactly.

theorem relativeCurvatureIncrementSpec {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] (u v du dv : E) (hκ : inner ℝ u v ≠ 0) : inner ℝ (u + du) (v + dv) = inner ℝ u v * (1 + relativeCurvatureIncrement u v du dv) := by have hcomm : inner ℝ du v = inner ℝ v du := real_inner_comm _ _ rw [inner_add_left, inner_add_right, inner_add_right, hcomm, relativeCurvatureIncrement] field_simp ring

Helper for thm:planar-convergence: symmetry and the old secant equation identify the new quadratic form with its relative increment.

theorem relativeQuadraticIncrementSpec {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] (H : E →L[ℝ] E) (u v dv : E) (hκ : inner ℝ u v ≠ 0) (hSymm : ∀ x y, inner ℝ x (H y) = inner ℝ (H x) y) (hsec : H v = u) : inner ℝ (v + dv) (H (v + dv)) = inner ℝ u v * (1 + relativeQuadraticIncrement H u v dv) := by have hcomm : inner ℝ v u = inner ℝ u v := real_inner_comm _ _ have hcomm' : inner ℝ dv u = inner ℝ u dv := real_inner_comm _ _ rw [map_add, inner_add_left, inner_add_right, inner_add_right, hsec, hcomm, hcomm', hSymm v dv, hsec, relativeQuadraticIncrement] field_simp ring

Helper for thm:planar-convergence: the first-order change in the secant vector cancels exactly from the normalized determinant increment.

theorem relativeIncrementCancellation {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] (H : E →L[ℝ] E) (u v du dv : E) : 2 * relativeCurvatureIncrement u v du dv - relativeQuadraticIncrement H u v dv = 2 * inner ℝ v du / inner ℝ u v + (2 * inner ℝ du dv - inner ℝ dv (H dv)) / inner ℝ u v := by rw [relativeCurvatureIncrement, relativeQuadraticIncrement] ring

Helper for thm:planar-convergence: the quadratic-form remainder is bounded without any positive lower bound on the inverse Hessian.

theorem secantQuadraticRemainderBound {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] (H : E →L[ℝ] E) (du dv : E) {B : ℝ} (hH : ‖H‖ ≤ B) : |2 * inner ℝ du dv - inner ℝ dv (H dv)| ≤ 2 * ‖du‖ * ‖dv‖ + B * ‖dv‖ ^ 2 := by have htriangle := abs_sub (2 * inner ℝ du dv) (inner ℝ dv (H dv)) have hnormTwo : |2 * inner ℝ du dv| = 2 * |inner ℝ du dv| := by norm_num [abs_mul] rw [hnormTwo] at htriangle have hcross := abs_real_inner_le_norm du dv have hquadratic := abs_real_inner_le_norm dv (H dv) have hoperator := H.le_of_opNorm_le hH dv have hscaled := mul_le_mul_of_nonneg_left hoperator (norm_nonneg dv) nlinarith only [htriangle, hcross, hquadratic, hscaled]

Helper for thm:planar-convergence: the logarithmic determinant increment differs from its tangent-linear part by quadratic secant errors.

theorem secantLogIncrementRemainderBound {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] (H : E →L[ℝ] E) (u v du dv : E) {μ B lam lamNext : ℝ} (hμ : 0 < μ) (hκ : μ ≤ inner ℝ u v) (hH : ‖H‖ ≤ B) (hlam : 0 < lam) (hP : |relativeCurvatureIncrement u v du dv| ≤ 1 / 2) (hQ : |relativeQuadraticIncrement H u v dv| ≤ 1 / 2) (hrec : lamNext = lam * (1 + relativeCurvatureIncrement u v du dv) ^ 2 / (1 + relativeQuadraticIncrement H u v dv)) : |Real.log lamNext - Real.log lam - 2 * inner ℝ v du / inner ℝ u v| ≤ 4 * relativeCurvatureIncrement u v du dv ^ 2 + 2 * relativeQuadraticIncrement H u v dv ^ 2 + (2 * ‖du‖ * ‖dv‖ + B * ‖dv‖ ^ 2) / μ := by have hPpos : 0 < 1 + relativeCurvatureIncrement u v du dv := by have hlow := (abs_le.mp hP).1 linarith have hQpos : 0 < 1 + relativeQuadraticIncrement H u v dv := by have hlow := (abs_le.mp hQ).1 linarith have hlog := normalizedDeterminantLogIncrement hlam hPpos hQpos hrec have hlogError := logDeterminantQuadraticRemainder hP hQ have hquadratic := secantQuadraticRemainderBound H du dv hH have hquotient := secantTiltBound hμ hκ hquadratic have hsplit : Real.log lamNext - Real.log lam - 2 * inner ℝ v du / inner ℝ u v = (2 * Real.log (1 + relativeCurvatureIncrement u v du dv) - Real.log (1 + relativeQuadraticIncrement H u v dv) - (2 * relativeCurvatureIncrement u v du dv - relativeQuadraticIncrement H u v dv)) + (2 * inner ℝ du dv - inner ℝ dv (H dv)) / inner ℝ u v := by rw [hlog, relativeIncrementCancellation] ring rw [hsplit] exact (abs_add_le _ _).trans (add_le_add hlogError hquotient)

Helper for thm:planar-convergence: a common nonzero normalization and orientation scale cancels from the exact DFP determinant ratio.

theorem commonScaledSecantRatio {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] (H : E →L[ℝ] E) (s y : E) {r : ℝ} (hr : r ≠ 0) : inner ℝ (r • s) (r • y) / inner ℝ (r • y) (H (r • y)) = inner ℝ y s / inner ℝ y (H y) := by simp only [map_smul, real_inner_smul_left, real_inner_smul_right] have hcomm : inner ℝ s y = inner ℝ y s := real_inner_comm _ _ rw [hcomm] calc (r * (r * inner ℝ y s)) / (r * (r * inner ℝ y (H y))) = ((r * r) * inner ℝ y s) / ((r * r) * inner ℝ y (H y)) := by ring _ = inner ℝ y s / inner ℝ y (H y) := mul_div_mul_left _ _ (mul_ne_zero hr hr)

Helper for thm:planar-convergence: the oriented normalized secant ratio equals the physical DFP determinant ratio.

theorem orientedOrbitSecantRatio (iteration : DFP.InverseIteration (Fin 2)) (k : ℕ) (hStep : 0 < iteration.stepLength k) : inner ℝ (orientedOrbitStep iteration k) (orientedOrbitSecant iteration k) / inner ℝ (orientedOrbitSecant iteration k) (Matrix.toEuclideanCLM (𝕜 := ℝ) (iteration.inverseHessian k) (orientedOrbitSecant iteration k)) = let s := iteration.point (k + 1) - iteration.point k let y := gradient iteration.objective (iteration.point (k + 1)) - gradient iteration.objective (iteration.point k) inner ℝ y s / inner ℝ y (Matrix.toEuclideanCLM (𝕜 := ℝ) (iteration.inverseHessian k) y) := by simp only [orientedOrbitStep, orientedOrbitSecant, smul_smul] apply commonScaledSecantRatio apply mul_ne_zero · intro hzero have habs := tangentOrientationAbs (gradientUnitTangent iteration.objective (iteration.point k)) (‖iteration.point (k + 1) - iteration.point k‖⁻¹ • (iteration.point (k + 1) - iteration.point k)) rw [hzero, abs_zero] at habs exact zero_ne_one habs · exact inv_ne_zero (orbitStepNormPos iteration k hStep).ne'

Helper for thm:planar-convergence: the actual DFP determinant recurrence is unchanged by canonical secant orientation and normalization.

theorem orientedOrbitDeterminantRecurrence (iteration : DFP.InverseIteration (Fin 2)) {μ : ℝ} (hf : ContDiff ℝ 2 iteration.objective) (hμ : 0 < μ) (hHessian : ∀ x v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient iteration.objective) x v) v) (k : ℕ) (hStep : 0 < iteration.stepLength k) : (iteration.inverseHessian (k + 1)).det = (iteration.inverseHessian k).det * (inner ℝ (orientedOrbitStep iteration k) (orientedOrbitSecant iteration k) / inner ℝ (orientedOrbitSecant iteration k) (Matrix.toEuclideanCLM (𝕜 := ℝ) (iteration.inverseHessian k) (orientedOrbitSecant iteration k))) := by rw [orientedOrbitSecantRatio iteration k hStep] exact orbitDeterminantRecurrence iteration hf hμ hHessian k hStep

Helper for thm:planar-convergence: the free coefficient of the matrix after step k, equal to the paper's coefficient with index k + 1.

noncomputable def orbitFreeCoefficient (iteration : DFP.InverseIteration (Fin 2)) (k : ℕ) : ℝ := inner ℝ (orientedOrbitStep iteration k) (orientedOrbitSecant iteration k) * (iteration.inverseHessian (k + 1)).det

Helper for thm:planar-convergence: the oriented free coefficient has the same physical secant formula used by the degeneration theorem.

theorem orbitFreeCoefficientPhysicalFormula (iteration : DFP.InverseIteration (Fin 2)) (k : ℕ) : orbitFreeCoefficient iteration k = (inner ℝ (gradient iteration.objective (iteration.point (k + 1)) - gradient iteration.objective (iteration.point k)) (iteration.point (k + 1) - iteration.point k) / ‖iteration.point (k + 1) - iteration.point k‖ ^ 2) * (iteration.inverseHessian (k + 1)).det := by rw [orbitFreeCoefficient, orientedOrbitCurvature]

Helper for thm:planar-convergence: every free coefficient of the positive-step strongly convex DFP orbit is strictly positive.

theorem orbitFreeCoefficientPos (iteration : DFP.InverseIteration (Fin 2)) {μ : ℝ} (hf : ContDiff ℝ 2 iteration.objective) (hμ : 0 < μ) (hHessian : ∀ x v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient iteration.objective) x v) v) (k : ℕ) (hStep : 0 < iteration.stepLength k) : 0 < orbitFreeCoefficient iteration k := by rw [orbitFreeCoefficient] exact mul_pos (hμ.trans_le (orientedOrbitCurvatureLower iteration hf hμ hHessian k hStep)) (iteration.inverseHessianPosDef (k + 1)).det_pos

Helper for thm:planar-convergence: the free coefficient in oriented coordinates tends to zero under nonconvergence.

theorem orbitFreeCoefficientTendstoZero (iteration : DFP.InverseIteration (Fin 2)) {μ c₁ c₂ : ℝ} (hf : ContDiff ℝ 2 iteration.objective) (hμ : 0 < μ) (hHessian : ∀ x v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient iteration.objective) x v) v) (hStep : ∀ k, 0 < iteration.stepLength k) (hWolfe : ∀ k, LineSearch.IsWeakWolfe c₁ c₂ iteration.objective (iteration.point k) (iteration.point (k + 1) - iteration.point k)) {xstar : EuclideanSpace ℝ (Fin 2)} (hmin : ∀ x, iteration.objective xstar ≤ iteration.objective x) (hnot : ¬ Tendsto iteration.point atTop (𝓝 xstar)) : Tendsto (orbitFreeCoefficient iteration) atTop (𝓝 0) := by have hlimit := (orbitFreeCoefficientDegeneration iteration hf hμ hHessian hStep hWolfe hmin hnot).2 exact (tendsto_congr (fun k ↦ orbitFreeCoefficientPhysicalFormula iteration k)).mpr hlimit

Helper for thm:planar-convergence: the oriented matrix secant equation also holds in the canonical Euclidean operator representation.

theorem orientedOrbitOperatorSecantEquation (iteration : DFP.InverseIteration (Fin 2)) (k : ℕ) : Matrix.toEuclideanCLM (𝕜 := ℝ) (iteration.inverseHessian (k + 1)) (orientedOrbitSecant iteration k) = orientedOrbitStep iteration k := by apply WithLp.ofLp_injective 2 rw [Matrix.ofLp_toEuclideanCLM] exact orientedOrbitSecantEquation iteration k

Helper for thm:planar-convergence: positive-definite matrices act as symmetric operators on Euclidean space.

theorem positiveMatrixOperatorSymmetry {H : Matrix (Fin 2) (Fin 2) ℝ} (hH : H.PosDef) (x y : EuclideanSpace ℝ (Fin 2)) : inner ℝ x (Matrix.toEuclideanCLM (𝕜 := ℝ) H y) = inner ℝ (Matrix.toEuclideanCLM (𝕜 := ℝ) H x) y := by have hself : IsSelfAdjoint (Matrix.toEuclideanCLM (𝕜 := ℝ) H) := hH.isHermitian.isSelfAdjoint.map Matrix.toEuclideanCLM exact (hself.isSymmetric x y).symm

Helper for thm:planar-convergence: the relative curvature increment is controlled by the total secant variation when the secant change is small.

theorem relativeCurvatureIncrementBound {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] (u v du dv : E) {μ M : ℝ} (hμ : 0 < μ) (hκ : μ ≤ inner ℝ u v) (hu : ‖u‖ = 1) (hv : ‖v‖ ≤ M) (hdv : ‖dv‖ ≤ 1) : |relativeCurvatureIncrement u v du dv| ≤ ((M + 2) / μ) * (‖du‖ + ‖dv‖) := by have hM : 0 ≤ M := (norm_nonneg v).trans hv have htriangle := abs_add_le (inner ℝ v du + inner ℝ u dv) (inner ℝ du dv) have htriangle' := abs_add_le (inner ℝ v du) (inner ℝ u dv) have hfirst := abs_real_inner_le_norm v du have hfirst' := mul_le_mul_of_nonneg_right hv (norm_nonneg du) have hsecond := abs_real_inner_le_norm u dv rw [hu, one_mul] at hsecond have hcross := abs_real_inner_le_norm du dv have hsmall := mul_le_mul_of_nonneg_left hdv (norm_nonneg du) have hMdv := mul_nonneg hM (norm_nonneg dv) have hnum : |inner ℝ v du + inner ℝ u dv + inner ℝ du dv| ≤ (M + 2) * (‖du‖ + ‖dv‖) := by nlinarith only [htriangle, htriangle', hfirst, hfirst', hsecond, hcross, hsmall, hMdv, norm_nonneg du, norm_nonneg dv] have hquotient := secantTiltBound hμ hκ hnum simpa only [relativeCurvatureIncrement, div_mul_eq_mul_div] using hquotient

Helper for thm:planar-convergence: the relative quadratic-form increment is uniformly controlled by the secant change and the operator norm.

theorem relativeQuadraticIncrementBound {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] (H : E →L[ℝ] E) (u v dv : E) {μ B : ℝ} (hμ : 0 < μ) (hκ : μ ≤ inner ℝ u v) (hH : ‖H‖ ≤ B) (hu : ‖u‖ = 1) (hdv : ‖dv‖ ≤ 1) : |relativeQuadraticIncrement H u v dv| ≤ ((2 + B) / μ) * ‖dv‖ := by have hB : 0 ≤ B := (norm_nonneg H).trans hH have htriangle := abs_add_le (2 * inner ℝ u dv) (inner ℝ dv (H dv)) have htwo : |2 * inner ℝ u dv| = 2 * |inner ℝ u dv| := by norm_num [abs_mul] rw [htwo] at htriangle have hfirst := abs_real_inner_le_norm u dv rw [hu, one_mul] at hfirst have hsecond := abs_real_inner_le_norm dv (H dv) have hoperator := H.le_of_opNorm_le hH dv have hscaled := mul_le_mul_of_nonneg_left hoperator (norm_nonneg dv) have hdvSquare : ‖dv‖ ^ 2 ≤ ‖dv‖ := by nlinarith [norm_nonneg dv] have hsmall := mul_le_mul_of_nonneg_left hdvSquare hB have hnum : |2 * inner ℝ u dv + inner ℝ dv (H dv)| ≤ (2 + B) * ‖dv‖ := by nlinarith only [htriangle, hfirst, hsecond, hscaled, hsmall] have hquotient := secantTiltBound hμ hκ hnum simpa only [relativeQuadraticIncrement, div_mul_eq_mul_div] using hquotient

Helper for thm:planar-convergence: the bounded gradient secants give a uniform bound for the actual oriented normalized secant vectors.

theorem orientedOrbitSecantBound (iteration : DFP.InverseIteration (Fin 2)) {μ c₁ c₂ : ℝ} (hf : ContDiff ℝ 2 iteration.objective) (hμ : 0 < μ) (hHessian : ∀ x v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient iteration.objective) x v) v) (hStep : ∀ k, 0 < iteration.stepLength k) (hWolfe : ∀ k, LineSearch.IsWeakWolfe c₁ c₂ iteration.objective (iteration.point k) (iteration.point (k + 1) - iteration.point k)) : ∃ M > 0, ∀ k, ‖orientedOrbitSecant iteration k‖ ≤ M := by obtain ⟨M, hM, hbounds⟩ := secantBoundsAlongOrbit iteration hf hμ hHessian hStep hWolfe refine ⟨M, hM, ?_⟩ intro k rw [orientedOrbitSecant, norm_smul, Real.norm_eq_abs, tangentOrientationAbs, one_mul, norm_smul, Real.norm_eq_abs, abs_inv, abs_norm, ← div_eq_inv_mul] apply (div_le_iff₀ (orbitStepNormPos iteration k (hStep k))).mpr exact (hbounds k).1

Helper for thm:planar-convergence: the relative curvature increment between successive canonically oriented orbit secants.

noncomputable def orbitRelativeCurvature (iteration : DFP.InverseIteration (Fin 2)) (k : ℕ) : ℝ := relativeCurvatureIncrement (orientedOrbitStep iteration k) (orientedOrbitSecant iteration k) (orientedOrbitStep iteration (k + 1) - orientedOrbitStep iteration k) (orientedOrbitSecant iteration (k + 1) - orientedOrbitSecant iteration k)

Helper for thm:planar-convergence: the relative quadratic-form increment measured by the actual inverse Hessian following the old secant.

noncomputable def orbitRelativeQuadratic (iteration : DFP.InverseIteration (Fin 2)) (k : ℕ) : ℝ := relativeQuadraticIncrement (Matrix.toEuclideanCLM (𝕜 := ℝ) (iteration.inverseHessian (k + 1))) (orientedOrbitStep iteration k) (orientedOrbitSecant iteration k) (orientedOrbitSecant iteration (k + 1) - orientedOrbitSecant iteration k)

Helper for thm:planar-convergence: the actual free coefficient satisfies the normalized determinant recurrence on the logarithm's local domain.

theorem orbitFreeCoefficientNormalizedRecurrence (iteration : DFP.InverseIteration (Fin 2)) {μ : ℝ} (hf : ContDiff ℝ 2 iteration.objective) (hμ : 0 < μ) (hHessian : ∀ x v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient iteration.objective) x v) v) (hStep : ∀ k, 0 < iteration.stepLength k) (k : ℕ) (hQ : |orbitRelativeQuadratic iteration k| ≤ 1 / 2) : orbitFreeCoefficient iteration (k + 1) = orbitFreeCoefficient iteration k * (1 + orbitRelativeCurvature iteration k) ^ 2 / (1 + orbitRelativeQuadratic iteration k) := by have hκ : inner ℝ (orientedOrbitStep iteration k) (orientedOrbitSecant iteration k) ≠ 0 := (hμ.trans_le (orientedOrbitCurvatureLower iteration hf hμ hHessian k (hStep k))).ne' have hcurvature := relativeCurvatureIncrementSpec (orientedOrbitStep iteration k) (orientedOrbitSecant iteration k) (orientedOrbitStep iteration (k + 1) - orientedOrbitStep iteration k) (orientedOrbitSecant iteration (k + 1) - orientedOrbitSecant iteration k) hκ simp only [add_sub_cancel] at hcurvature have hquadratic := relativeQuadraticIncrementSpec (Matrix.toEuclideanCLM (𝕜 := ℝ) (iteration.inverseHessian (k + 1))) (orientedOrbitStep iteration k) (orientedOrbitSecant iteration k) (orientedOrbitSecant iteration (k + 1) - orientedOrbitSecant iteration k) hκ (positiveMatrixOperatorSymmetry (iteration.inverseHessianPosDef (k + 1))) (orientedOrbitOperatorSecantEquation iteration k) simp only [add_sub_cancel] at hquadratic have hQpos : 0 < 1 + orbitRelativeQuadratic iteration k := by have hlow := (abs_le.mp hQ).1 linarith rw [orbitFreeCoefficient, orbitFreeCoefficient, orientedOrbitDeterminantRecurrence iteration hf hμ hHessian (k + 1) (hStep (k + 1)), hcurvature, hquadratic] dsimp only [orbitRelativeCurvature, orbitRelativeQuadratic] at hQpos ⊢ field_simp

Helper for thm:planar-convergence: uniform secant and operator bounds make the logarithmic determinant error quadratic in total secant variation.

theorem secantLogIncrementUniformBound {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] (H : E →L[ℝ] E) (u v du dv : E) {μ M B lam lamNext : ℝ} (hμ : 0 < μ) (hκ : μ ≤ inner ℝ u v) (hH : ‖H‖ ≤ B) (hu : ‖u‖ = 1) (hv : ‖v‖ ≤ M) (hdv : ‖dv‖ ≤ 1) (hlam : 0 < lam) (hP : |relativeCurvatureIncrement u v du dv| ≤ 1 / 2) (hQ : |relativeQuadraticIncrement H u v dv| ≤ 1 / 2) (hrec : lamNext = lam * (1 + relativeCurvatureIncrement u v du dv) ^ 2 / (1 + relativeQuadraticIncrement H u v dv)) : |Real.log lamNext - Real.log lam - 2 * inner ℝ v du / inner ℝ u v| ≤ (4 * ((M + 2) / μ) ^ 2 + 2 * ((2 + B) / μ) ^ 2 + (2 + B) / μ) * (‖du‖ + ‖dv‖) ^ 2 := by have hM : 0 ≤ M := (norm_nonneg v).trans hv have hB : 0 ≤ B := (norm_nonneg H).trans hH have hAP : 0 ≤ (M + 2) / μ := by positivity have hAQ : 0 ≤ (2 + B) / μ := by positivity have hδ : 0 ≤ ‖du‖ + ‖dv‖ := add_nonneg (norm_nonneg du) (norm_nonneg dv) have hPbound := relativeCurvatureIncrementBound u v du dv hμ hκ hu hv hdv have hQbound : |relativeQuadraticIncrement H u v dv| ≤ ((2 + B) / μ) * (‖du‖ + ‖dv‖) := by have hbound := relativeQuadraticIncrementBound H u v dv hμ hκ hH hu hdv have hcross := mul_nonneg hAQ (norm_nonneg du) nlinarith only [hbound, hcross] have hPsquare : relativeCurvatureIncrement u v du dv ^ 2 ≤ ((M + 2) / μ) ^ 2 * (‖du‖ + ‖dv‖) ^ 2 := by have hproduct := mul_le_mul hPbound hPbound (abs_nonneg _) (mul_nonneg hAP hδ) simpa only [← pow_two, sq_abs, mul_pow] using hproduct have hQsquare : relativeQuadraticIncrement H u v dv ^ 2 ≤ ((2 + B) / μ) ^ 2 * (‖du‖ + ‖dv‖) ^ 2 := by have hproduct := mul_le_mul hQbound hQbound (abs_nonneg _) (mul_nonneg hAQ hδ) simpa only [← pow_two, sq_abs, mul_pow] using hproduct have hcross : ‖du‖ * ‖dv‖ ≤ (‖du‖ + ‖dv‖) ^ 2 := by nlinarith [sq_nonneg ‖du‖, sq_nonneg ‖dv‖, mul_nonneg (norm_nonneg du) (norm_nonneg dv)] have hvsquare : ‖dv‖ ^ 2 ≤ (‖du‖ + ‖dv‖) ^ 2 := by nlinarith [sq_nonneg ‖du‖, mul_nonneg (norm_nonneg du) (norm_nonneg dv)] have hBscaled := mul_le_mul_of_nonneg_left hvsquare hB have hnumerator : 2 * ‖du‖ * ‖dv‖ + B * ‖dv‖ ^ 2 ≤ (2 + B) * (‖du‖ + ‖dv‖) ^ 2 := by nlinarith only [hcross, hBscaled] have hquadratic : (2 * ‖du‖ * ‖dv‖ + B * ‖dv‖ ^ 2) / μ ≤ ((2 + B) / μ) * (‖du‖ + ‖dv‖) ^ 2 := by rw [div_mul_eq_mul_div] exact div_le_div_of_nonneg_right hnumerator hμ.le have hbound := secantLogIncrementRemainderBound H u v du dv hμ hκ hH hlam hP hQ hrec nlinarith only [hbound, hPsquare, hQsquare, hquadratic]

Helper for thm:planar-convergence: the actual logarithmic free coefficient increment differs from its oriented secant-linear term by a uniform multiple of the two adjacent squared step lengths.

theorem orbitLogIncrementRemainderBound (iteration : DFP.InverseIteration (Fin 2)) {μ c₁ c₂ : ℝ} (hf : ContDiff ℝ 2 iteration.objective) (hμ : 0 < μ) (hHessian : ∀ x v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient iteration.objective) x v) v) (hStep : ∀ k, 0 < iteration.stepLength k) (hWolfe : ∀ k, LineSearch.IsWeakWolfe c₁ c₂ iteration.objective (iteration.point k) (iteration.point (k + 1) - iteration.point k)) (hLip : LocallyLipschitzOn (DFP.objectiveSublevel iteration) (fderiv ℝ (gradient iteration.objective))) {xstar : EuclideanSpace ℝ (Fin 2)} (hmin : ∀ x, iteration.objective xstar ≤ iteration.objective x) (hnot : ¬ Tendsto iteration.point atTop (𝓝 xstar)) : ∃ R ≥ 0, ∀ᶠ k in atTop, |Real.log (orbitFreeCoefficient iteration (k + 1)) - Real.log (orbitFreeCoefficient iteration k) - inner ℝ ((2 / inner ℝ (orientedOrbitStep iteration k) (orientedOrbitSecant iteration k)) • orientedOrbitSecant iteration k) (orientedOrbitStep iteration (k + 1) - orientedOrbitStep iteration k)| ≤ R * (‖iteration.point (k + 1) - iteration.point k‖ ^ 2 + ‖iteration.point (k + 2) - iteration.point (k + 1)‖ ^ 2) := by obtain ⟨M, hM, hsecant⟩ := orientedOrbitSecantBound iteration hf hμ hHessian hStep hWolfe obtain ⟨B, hB, hoperator⟩ := orbitInverseHessianBoundedOfNonconvergence iteration hf hμ hHessian hStep hWolfe hmin hnot obtain ⟨D, hD, hvariation⟩ := orbitOrientedSecantVariationBound iteration hf hμ hHessian hStep hWolfe hLip hmin hnot let ell := fun k ↦ ‖iteration.point (k + 1) - iteration.point k‖ let du := fun k ↦ orientedOrbitStep iteration (k + 1) - orientedOrbitStep iteration k let dv := fun k ↦ orientedOrbitSecant iteration (k + 1) - orientedOrbitSecant iteration k let δ := fun k ↦ ‖du k‖ + ‖dv k‖ let AP := (M + 2) / μ let AQ := (2 + B) / μ let F := 4 * AP ^ 2 + 2 * AQ ^ 2 + AQ have hAP : 0 ≤ AP := by dsimp only [AP] positivity have hAQ : 0 ≤ AQ := by dsimp only [AQ] positivity have hF : 0 ≤ F := by dsimp only [F] positivity have hδ (k : ℕ) : 0 ≤ δ k := add_nonneg (norm_nonneg _) (norm_nonneg _) have hδbound (k : ℕ) : δ k ≤ D * (ell k + ell (k + 1)) := hvariation k have hellLimit : Tendsto ell atTop (𝓝 0) := stepNormTendstoZero iteration hf hμ hHessian hStep hWolfe have hδlimit : Tendsto δ atTop (𝓝 0) := by have hupper : Tendsto (fun k ↦ D * (ell k + ell (k + 1))) atTop (𝓝 0) := by simpa using (hellLimit.add (hellLimit.comp (tendsto_add_atTop_nat 1))).const_mul D exact squeeze_zero hδ hδbound hupper have hsmall (a : ℝ) : ∀ᶠ k in atTop, a * δ k < 1 / 2 := by have hlimit : Tendsto (fun k ↦ a * δ k) atTop (𝓝 0) := by simpa using hδlimit.const_mul a have hhalf : (0 : ℝ) < 1 / 2 := by norm_num exact hlimit.eventually (eventually_lt_nhds hhalf) have hR : 0 ≤ 2 * F * D ^ 2 := by positivity refine ⟨2 * F * D ^ 2, hR, ?_⟩ filter_upwards [hsmall 1, hsmall AP, hsmall AQ] with k hk hkP hkQ have hdv : ‖dv k‖ ≤ 1 := by dsimp only [δ] at hk nlinarith [norm_nonneg (du k)] have hκ := orientedOrbitCurvatureLower iteration hf hμ hHessian k (hStep k) have hu := orientedOrbitStepNorm iteration k (hStep k) have hPbound : |orbitRelativeCurvature iteration k| ≤ AP * δ k := relativeCurvatureIncrementBound _ _ _ _ hμ hκ hu (hsecant k) hdv have hQbound : |orbitRelativeQuadratic iteration k| ≤ AQ * ‖dv k‖ := relativeQuadraticIncrementBound _ _ _ _ hμ hκ (hoperator (k + 1)) hu hdv have hP : |orbitRelativeCurvature iteration k| ≤ 1 / 2 := hPbound.trans hkP.le have hQ : |orbitRelativeQuadratic iteration k| ≤ 1 / 2 := by dsimp only [δ] at hkQ have hcross := mul_nonneg hAQ (norm_nonneg (du k)) nlinarith only [hQbound, hkQ, hcross] have hrec := orbitFreeCoefficientNormalizedRecurrence iteration hf hμ hHessian hStep k hQ have hbound : |Real.log (orbitFreeCoefficient iteration (k + 1)) - Real.log (orbitFreeCoefficient iteration k) - 2 * inner ℝ (orientedOrbitSecant iteration k) (du k) / inner ℝ (orientedOrbitStep iteration k) (orientedOrbitSecant iteration k)| ≤ F * δ k ^ 2 := secantLogIncrementUniformBound _ _ _ _ _ hμ hκ (hoperator (k + 1)) hu (hsecant k) hdv (orbitFreeCoefficientPos iteration hf hμ hHessian k (hStep k)) hP hQ hrec have hell (j : ℕ) : 0 ≤ ell j := norm_nonneg _ have hscalePos : 0 ≤ D * (ell k + ell (k + 1)) := by positivity have hδsquare : δ k ^ 2 ≤ D ^ 2 * (ell k + ell (k + 1)) ^ 2 := by have hproduct := mul_le_mul (hδbound k) (hδbound k) (hδ k) hscalePos simpa only [← pow_two, mul_pow] using hproduct have hsum : (ell k + ell (k + 1)) ^ 2 ≤ 2 * (ell k ^ 2 + ell (k + 1) ^ 2) := by nlinarith [sq_nonneg (ell k - ell (k + 1))] have hscaledSum := mul_le_mul_of_nonneg_left hsum (sq_nonneg D) have hδfinal := hδsquare.trans hscaledSum have hscaled := mul_le_mul_of_nonneg_left hδfinal hF rw [real_inner_smul_left, div_mul_eq_mul_div] nlinarith only [hbound, hscaled]

Helper for thm:planar-convergence: the error between the oriented step and the physical gradient tangent.

noncomputable def orbitTangentError (iteration : DFP.InverseIteration (Fin 2)) (k : ℕ) : EuclideanSpace ℝ (Fin 2) := orientedOrbitStep iteration k - gradientUnitTangent iteration.objective (iteration.point k)

Helper for thm:planar-convergence: the small scalar correction that compensates for orientation errors in logarithmic determinant transport.

noncomputable def orbitLogCorrection (iteration : DFP.InverseIteration (Fin 2)) (k : ℕ) : ℝ := inner ℝ (gradientTransportCoefficient iteration.objective (iteration.point k)) (orbitTangentError iteration k)

Helper for thm:planar-convergence: the corrected logarithmic quantity whose increments are summable on a hypothetical nonconvergent orbit.

noncomputable def orbitCorrectedLog (iteration : DFP.InverseIteration (Fin 2)) (k : ℕ) : ℝ := Real.log (orbitFreeCoefficient iteration k) - orbitLogCorrection iteration k + 2 * Real.log ‖gradient iteration.objective (iteration.point k)‖

Helper for thm:planar-convergence: the logarithmic correction tends to zero because the coefficient stays bounded and the orientation error vanishes.

theorem orbitLogCorrectionTendstoZero (iteration : DFP.InverseIteration (Fin 2)) {μ c₁ c₂ : ℝ} (hf : ContDiff ℝ 2 iteration.objective) (hμ : 0 < μ) (hHessian : ∀ x v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient iteration.objective) x v) v) (hStep : ∀ k, 0 < iteration.stepLength k) (hWolfe : ∀ k, LineSearch.IsWeakWolfe c₁ c₂ iteration.objective (iteration.point k) (iteration.point (k + 1) - iteration.point k)) (hLip : LocallyLipschitzOn (DFP.objectiveSublevel iteration) (fderiv ℝ (gradient iteration.objective))) {xstar : EuclideanSpace ℝ (Fin 2)} (hmin : ∀ x, iteration.objective xstar ≤ iteration.objective x) (hnot : ¬ Tendsto iteration.point atTop (𝓝 xstar)) : Tendsto (orbitLogCorrection iteration) atTop (𝓝 0) := by obtain ⟨E, hE, V, hV, happrox⟩ := orbitOrientedSecantApproximation iteration hf hμ hHessian hStep hWolfe hLip hmin hnot obtain ⟨B, hB, hcoef⟩ := orbitTransportCoefficientBound iteration hf hμ hHessian hStep hWolfe have hnorm (k : ℕ) : ‖orbitLogCorrection iteration k‖ ≤ (B * E) * ‖iteration.point (k + 1) - iteration.point k‖ := by rw [orbitLogCorrection, Real.norm_eq_abs] have hinner := abs_real_inner_le_norm (gradientTransportCoefficient iteration.objective (iteration.point k)) (orbitTangentError iteration k) have hproduct := mul_le_mul (hcoef k) (happrox k).1 (norm_nonneg _) hB.le exact hinner.trans (hproduct.trans_eq (mul_assoc _ _ _).symm) have hupper : Tendsto (fun k ↦ (B * E) * ‖iteration.point (k + 1) - iteration.point k‖) atTop (𝓝 0) := by simpa using (stepNormTendstoZero iteration hf hμ hHessian hStep hWolfe).const_mul (B * E) have hnormLimit := squeeze_zero (fun k ↦ norm_nonneg (orbitLogCorrection iteration k)) hnorm hupper exact tendsto_zero_iff_norm_tendsto_zero.mpr hnormLimit

Helper for thm:planar-convergence: the corrected log increment splits exactly into determinant error, coefficient approximation error, tangent cancellation, and the coefficient-variation correction.

theorem correctedLogIncrementIdentity {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] (l n : ℕ → ℝ) (u t a c : ℕ → E) (k : ℕ) : (l (k + 1) - inner ℝ (a (k + 1)) (u (k + 1) - t (k + 1)) + 2 * n (k + 1)) - (l k - inner ℝ (a k) (u k - t k) + 2 * n k) = (l (k + 1) - l k - inner ℝ (c k) (u (k + 1) - u k)) + inner ℝ (c k - a k) (u (k + 1) - u k) + (inner ℝ (a k) (t (k + 1) - t k) + 2 * (n (k + 1) - n k)) - inner ℝ (a (k + 1) - a k) (u (k + 1) - t (k + 1)) := by simp only [inner_sub_left, inner_sub_right] ring

Helper for thm:planar-convergence: the four error bounds in the corrected logarithmic identity yield one quadratic increment bound.

theorem correctedLogIncrementBound {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] (l n : ℕ → ℝ) (u t a c : ℕ → E) (ell : ℕ → ℝ) (k : ℕ) {R C D T A V : ℝ} (hC : 0 ≤ C) (hD : 0 ≤ D) (hT : 0 ≤ T) (hA : 0 ≤ A) (hV : 0 ≤ V) (hell : ∀ j, 0 ≤ ell j) (hlog : |l (k + 1) - l k - inner ℝ (c k) (u (k + 1) - u k)| ≤ R * (ell k ^ 2 + ell (k + 1) ^ 2)) (hcoef : ‖c k - a k‖ ≤ C * ell k) (hstep : ‖u (k + 1) - u k‖ ≤ D * (ell k + ell (k + 1))) (htangent : |inner ℝ (a k) (t (k + 1) - t k) + 2 * (n (k + 1) - n k)| ≤ T * ell k ^ 2) (hvariation : ‖a (k + 1) - a k‖ ≤ A * ell k) (herror : ‖u (k + 1) - t (k + 1)‖ ≤ V * ell (k + 1)) : |(l (k + 1) - inner ℝ (a (k + 1)) (u (k + 1) - t (k + 1)) + 2 * n (k + 1)) - (l k - inner ℝ (a k) (u k - t k) + 2 * n k)| ≤ (R + 2 * C * D + T + A * V) * (ell k ^ 2 + ell (k + 1) ^ 2) := by have hcoefInner := abs_real_inner_le_norm (c k - a k) (u (k + 1) - u k) have hcoefProduct := mul_le_mul hcoef hstep (norm_nonneg _) (mul_nonneg hC (hell k)) have hcross : ell k * (ell k + ell (k + 1)) ≤ 2 * (ell k ^ 2 + ell (k + 1) ^ 2) := by nlinarith [sq_nonneg (ell k - ell (k + 1)), sq_nonneg (ell (k + 1))] have hcrossScaled := mul_le_mul_of_nonneg_left hcross (mul_nonneg hC hD) have hcoefBound : |inner ℝ (c k - a k) (u (k + 1) - u k)| ≤ (2 * C * D) * (ell k ^ 2 + ell (k + 1) ^ 2) := by nlinarith only [hcoefInner, hcoefProduct, hcrossScaled] have hcorrectionInner := abs_real_inner_le_norm (a (k + 1) - a k) (u (k + 1) - t (k + 1)) have hcorrectionProduct := mul_le_mul hvariation herror (norm_nonneg _) (mul_nonneg hA (hell k)) have hadjacent : ell k * ell (k + 1) ≤ ell k ^ 2 + ell (k + 1) ^ 2 := by nlinarith [sq_nonneg (ell k - ell (k + 1)), sq_nonneg (ell k), sq_nonneg (ell (k + 1))] have hadjacentScaled := mul_le_mul_of_nonneg_left hadjacent (mul_nonneg hA hV) have hcorrectionBound : |inner ℝ (a (k + 1) - a k) (u (k + 1) - t (k + 1))| ≤ (A * V) * (ell k ^ 2 + ell (k + 1) ^ 2) := by nlinarith only [hcorrectionInner, hcorrectionProduct, hadjacentScaled] have htangentExtra := mul_nonneg hT (sq_nonneg (ell (k + 1))) rw [correctedLogIncrementIdentity l n u t a c k] have htriangle := abs_sub ((l (k + 1) - l k - inner ℝ (c k) (u (k + 1) - u k)) + inner ℝ (c k - a k) (u (k + 1) - u k) + (inner ℝ (a k) (t (k + 1) - t k) + 2 * (n (k + 1) - n k))) (inner ℝ (a (k + 1) - a k) (u (k + 1) - t (k + 1))) have htriangle' := abs_add_le ((l (k + 1) - l k - inner ℝ (c k) (u (k + 1) - u k)) + inner ℝ (c k - a k) (u (k + 1) - u k)) (inner ℝ (a k) (t (k + 1) - t k) + 2 * (n (k + 1) - n k)) have htriangle'' := abs_add_le (l (k + 1) - l k - inner ℝ (c k) (u (k + 1) - u k)) (inner ℝ (c k - a k) (u (k + 1) - u k)) nlinarith only [htriangle, htriangle', htriangle'', hlog, hcoefBound, htangent, htangentExtra, hcorrectionBound]

Helper for thm:planar-convergence: the corrected logarithmic quantity on the actual DFP orbit has uniformly quadratic eventual increments.

theorem orbitCorrectedLogIncrementBound (iteration : DFP.InverseIteration (Fin 2)) {μ c₁ c₂ : ℝ} (hf : ContDiff ℝ 2 iteration.objective) (hμ : 0 < μ) (hHessian : ∀ x v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient iteration.objective) x v) v) (hStep : ∀ k, 0 < iteration.stepLength k) (hWolfe : ∀ k, LineSearch.IsWeakWolfe c₁ c₂ iteration.objective (iteration.point k) (iteration.point (k + 1) - iteration.point k)) (hLip : LocallyLipschitzOn (DFP.objectiveSublevel iteration) (fderiv ℝ (gradient iteration.objective))) {xstar : EuclideanSpace ℝ (Fin 2)} (hmin : ∀ x, iteration.objective xstar ≤ iteration.objective x) (hnot : ¬ Tendsto iteration.point atTop (𝓝 xstar)) : ∃ K ≥ 0, ∀ᶠ k in atTop, |orbitCorrectedLog iteration (k + 1) - orbitCorrectedLog iteration k| ≤ K * (‖iteration.point (k + 1) - iteration.point k‖ ^ 2 + ‖iteration.point (k + 2) - iteration.point (k + 1)‖ ^ 2) := by obtain ⟨R, hR, hlog⟩ := orbitLogIncrementRemainderBound iteration hf hμ hHessian hStep hWolfe hLip hmin hnot obtain ⟨C, hC, hcoef⟩ := orbitSecantCoefficientApproximation iteration hf hμ hHessian hStep hWolfe hLip hmin hnot obtain ⟨D, hD, hstep⟩ := orbitOrientedSecantVariationBound iteration hf hμ hHessian hStep hWolfe hLip hmin hnot obtain ⟨T, hT, htangent⟩ := orbitTangentLogQuadraticBound iteration hf hμ hHessian hStep hWolfe hLip hmin hnot obtain ⟨A, hA, hvariation⟩ := orbitTransportCoefficientVariation iteration hf hμ hHessian hStep hWolfe hLip hmin hnot obtain ⟨V, hV, _, _, herror⟩ := orbitOrientedSecantApproximation iteration hf hμ hHessian hStep hWolfe hLip hmin hnot have hK : 0 ≤ R + 2 * C * D + T + A * V := by positivity refine ⟨R + 2 * C * D + T + A * V, hK, ?_⟩ filter_upwards [hlog, htangent] with k hkLog hkTangent have hstepOnly : ‖orientedOrbitStep iteration (k + 1) - orientedOrbitStep iteration k‖ ≤ D * (‖iteration.point (k + 1) - iteration.point k‖ + ‖iteration.point (k + 2) - iteration.point (k + 1)‖) := by have htotal := hstep k nlinarith only [htotal, norm_nonneg (orientedOrbitSecant iteration (k + 1) - orientedOrbitSecant iteration k)] exact correctedLogIncrementBound (fun j ↦ Real.log (orbitFreeCoefficient iteration j)) (fun j ↦ Real.log ‖gradient iteration.objective (iteration.point j)‖) (orientedOrbitStep iteration) (fun j ↦ gradientUnitTangent iteration.objective (iteration.point j)) (fun j ↦ gradientTransportCoefficient iteration.objective (iteration.point j)) (fun j ↦ (2 / inner ℝ (orientedOrbitStep iteration j) (orientedOrbitSecant iteration j)) • orientedOrbitSecant iteration j) (fun j ↦ ‖iteration.point (j + 1) - iteration.point j‖) k hC hD.le hT hA hV.le (fun j ↦ norm_nonneg _) hkLog (hcoef k) hstepOnly hkTangent (hvariation k) (herror (k + 1)).1

Helper for thm:planar-convergence: the absolute increments of the actual corrected logarithmic quantity form a summable series.

theorem orbitCorrectedLogIncrementsSummable (iteration : DFP.InverseIteration (Fin 2)) {μ c₁ c₂ : ℝ} (hf : ContDiff ℝ 2 iteration.objective) (hμ : 0 < μ) (hHessian : ∀ x v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient iteration.objective) x v) v) (hStep : ∀ k, 0 < iteration.stepLength k) (hWolfe : ∀ k, LineSearch.IsWeakWolfe c₁ c₂ iteration.objective (iteration.point k) (iteration.point (k + 1) - iteration.point k)) (hLip : LocallyLipschitzOn (DFP.objectiveSublevel iteration) (fderiv ℝ (gradient iteration.objective))) {xstar : EuclideanSpace ℝ (Fin 2)} (hmin : ∀ x, iteration.objective xstar ≤ iteration.objective x) (hnot : ¬ Tendsto iteration.point atTop (𝓝 xstar)) : Summable (fun k ↦ |orbitCorrectedLog iteration (k + 1) - orbitCorrectedLog iteration k|) := by obtain ⟨K, hK, hbound⟩ := orbitCorrectedLogIncrementBound iteration hf hμ hHessian hStep hWolfe hLip hmin hnot have hsquares := summableStepSquares iteration hf hμ hHessian hStep hWolfe have hnext : Summable (fun k ↦ ‖iteration.point (k + 2) - iteration.point (k + 1)‖ ^ 2) := hsquares.comp_injective Nat.succ_injective have hmajorant : Summable (fun k ↦ K * (‖iteration.point (k + 1) - iteration.point k‖ ^ 2 + ‖iteration.point (k + 2) - iteration.point (k + 1)‖ ^ 2)) := (hsquares.add hnext).mul_left K apply hmajorant.of_norm_bounded_eventually_nat filter_upwards [hbound] with k hk simpa only [Real.norm_eq_abs, abs_abs] using hk

Helper for thm:planar-convergence: the corrected logarithmic quantity rules out nonconvergence of a planar DFP orbit with locally Lipschitz Hessian.

theorem tendstoMinimizerOfLocallyLipschitzHessian (iteration : DFP.InverseIteration (Fin 2)) {μ c₁ c₂ : ℝ} (hf : ContDiff ℝ 2 iteration.objective) (hμ : 0 < μ) (hHessian : ∀ x v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient iteration.objective) x v) v) (hStep : ∀ k, 0 < iteration.stepLength k) (hWolfe : ∀ k, LineSearch.IsWeakWolfe c₁ c₂ iteration.objective (iteration.point k) (iteration.point (k + 1) - iteration.point k)) (hLip : LocallyLipschitzOn (DFP.objectiveSublevel iteration) (fderiv ℝ (gradient iteration.objective))) {xstar : EuclideanSpace ℝ (Fin 2)} (hmin : ∀ x, iteration.objective xstar ≤ iteration.objective x) : Tendsto iteration.point atTop (𝓝 xstar) ∧ Tendsto (fun k ↦ ‖gradient iteration.objective (iteration.point k)‖) atTop (𝓝 0) := by have hpoint : Tendsto iteration.point atTop (𝓝 xstar) := by by_contra hnot have hpositive (k : ℕ) : 0 < orbitFreeCoefficient iteration k := orbitFreeCoefficientPos iteration hf hμ hHessian k (hStep k) have hgradient (k : ℕ) : 0 < ‖gradient iteration.objective (iteration.point k)‖ := by apply norm_pos_iff.mpr simpa only [DFP.gradients_apply] using iteration.gradientNeZeroOfSecantDenominator k obtain ⟨M, _, hdiffBounds⟩ := uniformDifferentialBoundsOnSublevel hf hμ hHessian (iteration.point 0) have hupper (k : ℕ) : ‖gradient iteration.objective (iteration.point k)‖ ≤ M := by have hmem := iteration.pointMemObjectiveSublevelOfWeakWolfe hStep hWolfe k have hlevel := (DFP.mem_objectiveSublevel_iff iteration _).mp hmem exact (hdiffBounds _ hlevel).1 have hcorrection := orbitLogCorrectionTendstoZero iteration hf hμ hHessian hStep hWolfe hLip hmin hnot have hincrements := orbitCorrectedLogIncrementsSummable iteration hf hμ hHessian hStep hWolfe hLip hmin hnot obtain ⟨δ, hδ, hlower⟩ := positiveLowerBoundOfCorrectedLog hpositive hgradient hupper hcorrection hincrements have hzero := orbitFreeCoefficientTendstoZero iteration hf hμ hHessian hStep hWolfe hmin hnot exact not_tendsto_zero_of_eventually_positive_lower_bound hδ hlower hzero have hfOrder : ContDiff ℝ ((1 : ℕ∞) + 1) iteration.objective := by convert hf using 1 norm_num have hgradientCont : ContDiff ℝ 1 (gradient iteration.objective) := hfOrder.gradient_succ (n := 1) have hgradientLimit := (hgradientCont.continuous.tendsto xstar).comp hpoint have hstationary := gradientZeroOfGlobalMinimizer hmin refine ⟨hpoint, ?_⟩ simpa only [hstationary, norm_zero, Function.comp_def] using hgradientLimit.norm

Helper for thm:planar-convergence (infinite-orbit branch): every positive-step classical planar DFP orbit on a globally strongly convex C2 objective with locally Lipschitz Hessian on its initial sublevel converges to the unique minimizer, and its gradient norms converge to zero. The weak Wolfe coefficients are arbitrary fixed admissible coefficients.

theorem existsUniqueLimitOfLocallyLipschitzHessian (iteration : DFP.InverseIteration (Fin 2)) {μ c₁ c₂ : ℝ} (hf : ContDiff ℝ 2 iteration.objective) (hμ : 0 < μ) (hHessian : ∀ x v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient iteration.objective) x v) v) (hStep : ∀ k, 0 < iteration.stepLength k) (hWolfe : ∀ k, LineSearch.IsWeakWolfe c₁ c₂ iteration.objective (iteration.point k) (iteration.point (k + 1) - iteration.point k)) (hLip : LocallyLipschitzOn (DFP.objectiveSublevel iteration) (fderiv ℝ (gradient iteration.objective))) : ∃! xstar, (∀ x, iteration.objective xstar ≤ iteration.objective x) ∧ Tendsto iteration.point atTop (𝓝 xstar) ∧ Tendsto (fun k ↦ ‖gradient iteration.objective (iteration.point k)‖) atTop (𝓝 0) := by obtain ⟨xstar, hmin, hunique⟩ := existsUniqueGlobalMinimizerOfHessian hf hμ hHessian have hconvergence := tendstoMinimizerOfLocallyLipschitzHessian iteration hf hμ hHessian hStep hWolfe hLip hmin refine ⟨xstar, ⟨hmin, hconvergence⟩, ?_⟩ intro y hy exact hunique y hy.1

Helper for thm:planar-convergence: every stationary point of the globally strongly convex objective is a global minimizer. This verifies the finite-termination branch independently of any infinite-orbit encoding.

theorem stationaryPointMinimizes {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] {f : E → ℝ} {μ : ℝ} (hf : ContDiff ℝ 2 f) (hμ : 0 < μ) (hHessian : ∀ x v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient f) x v) v) {xstar : E} (hstationary : gradient f xstar = 0) : ∀ x, f xstar ≤ f x := by intro x have hfirst := hf.firstOrderOfHessianLowerBound f μ hμ hHessian xstar x rw [hstationary, inner_zero_left, add_zero] at hfirst have hnonnegative : 0 ≤ μ / 2 * ‖x - xstar‖ ^ 2 := by positivity linarith

Helper for thm:planar-convergence: a strongly convex C3 objective satisfies the Hessian regularity needed by the planar convergence theorem.

theorem existsUniqueLimitOfC3 (iteration : DFP.InverseIteration (Fin 2)) {μ c₁ c₂ : ℝ} (hf : ContDiff ℝ 3 iteration.objective) (hμ : 0 < μ) (hHessian : ∀ x v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient iteration.objective) x v) v) (hStep : ∀ k, 0 < iteration.stepLength k) (hWolfe : ∀ k, LineSearch.IsWeakWolfe c₁ c₂ iteration.objective (iteration.point k) (iteration.point (k + 1) - iteration.point k)) : ∃! xstar, (∀ x, iteration.objective xstar ≤ iteration.objective x) ∧ Tendsto iteration.point atTop (𝓝 xstar) ∧ Tendsto (fun k ↦ ‖gradient iteration.objective (iteration.point k)‖) atTop (𝓝 0) := by have htwoThree : (2 : WithTop ℕ∞) ≤ 3 := by norm_num have hfOrder : ContDiff ℝ ((2 : ℕ∞) + 1) iteration.objective := by convert hf using 1 norm_num have hgradient : ContDiff ℝ 2 (gradient iteration.objective) := hfOrder.gradient_succ (n := 2) have hderivativeOrder : (1 : WithTop ℕ∞) + 1 ≤ 2 := by norm_num have hHessianSmooth : ContDiff ℝ 1 (fderiv ℝ (gradient iteration.objective)) := hgradient.fderiv_right hderivativeOrder have hLip : LocallyLipschitzOn (DFP.objectiveSublevel iteration) (fderiv ℝ (gradient iteration.objective)) := hHessianSmooth.locallyLipschitz.locallyLipschitzOn exact existsUniqueLimitOfLocallyLipschitzHessian iteration (hf.of_le htwoThree) hμ hHessian hStep hWolfe hLip

Helper for eq:counterexample-nonlipschitz: a planar uniformly convex weak-Wolfe counterexample cannot have a locally Lipschitz Hessian on its initial objective sublevel.

theorem counterexampleHessianNotLocallyLipschitz {m M c₁ c₂ : ℝ} (counterexample : DFP.WolfeCounterexample (Fin 2) m M c₁ c₂) (hm : 0 < m) : ¬ LocallyLipschitzOn (DFP.objectiveSublevel counterexample.iteration) (fderiv ℝ (gradient counterexample.iteration.objective)) := by intro hLip have hHessian (x v : EuclideanSpace ℝ (Fin 2)) : m * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient counterexample.iteration.objective) x v) v := by have hbound := ((counterexample.hessianBounds.at x).quadraticForm v).1 simpa only [hessian_def] using hbound obtain ⟨xstar, hresult, _⟩ := existsUniqueLimitOfLocallyLipschitzHessian counterexample.iteration counterexample.objectiveContDiff hm hHessian counterexample.stepLengthPos counterexample.weakWolfe hLip have hzero : Tendsto (fun k ↦ ‖DFP.gradients counterexample.iteration.objective counterexample.iteration.point k‖) atTop (𝓝 0) := by simpa only [DFP.gradients_apply] using hresult.2.2 exact DFP.not_tendsto_zero_of_pos_limit counterexample.gradientLimitPos counterexample.gradientNormTendsto hzero

Helper for thm:planar-convergence: a nonstationary raw DFP step from a positive-definite matrix has strictly positive secant curvature under weak Wolfe.

theorem rawOrbitSecantCurvaturePos {f : EuclideanSpace ℝ (Fin 2) → ℝ} {α : ℕ → ℝ} {x g : ℕ → EuclideanSpace ℝ (Fin 2)} {H : ℕ → Matrix (Fin 2) (Fin 2) ℝ} (orbit : DFP.IsOrbit f α x g H) {c₁ c₂ : ℝ} (k : ℕ) (hH : (H k).PosDef) (hg : g k ≠ 0) (hWolfe : LineSearch.IsWeakWolfe c₁ c₂ f (x k) (x (k + 1) - x k)) : 0 < WithLp.ofLp (DFP.steps α (DFP.directions H g) k) ⬝ᵥ WithLp.ofLp (DFP.gradientChanges g k) := by have hcoordinates : WithLp.ofLp (g k) ≠ 0 := by intro hzero apply hg apply WithLp.ofLp_injective 2 simpa only [WithLp.ofLp_zero] using hzero have henergy : 0 < WithLp.ofLp (g k) ⬝ᵥ (H k *ᵥ WithLp.ofLp (g k)) := by simpa only [star_trivial] using hH.dotProduct_mulVec_pos hcoordinates have hstep : x (k + 1) - x k = DFP.steps α (DFP.directions H g) k := by rw [orbit.pointSucc k] abel have hdescent : inner ℝ (g k) (x (k + 1) - x k) < 0 := by rw [hstep, DFP.steps_apply, DFP.directions_apply, real_inner_smul_right, inner_neg_right, EuclideanSpace.inner_eq_star_dotProduct] simp only [star_trivial] have hcomm : (H k *ᵥ WithLp.ofLp (g k)) ⬝ᵥ WithLp.ofLp (g k) = WithLp.ofLp (g k) ⬝ᵥ (H k *ᵥ WithLp.ofLp (g k)) := dotProduct_comm _ _ rw [hcomm] nlinarith [orbit.stepLengthPos k] have hcurvature := hWolfe.weakCurvature rw [add_sub_cancel, (orbit.gradientAt k).gradient, (orbit.gradientAt (k + 1)).gradient] at hcurvature have hpair : 0 < inner ℝ (g (k + 1) - g k) (x (k + 1) - x k) := by rw [inner_sub_left] have hstrict := mul_neg_of_pos_of_neg (sub_pos.mpr hWolfe.c₂_lt_one) hdescent nlinarith rw [hstep, EuclideanSpace.inner_eq_star_dotProduct] at hpair simpa only [star_trivial, DFP.gradientChanges_apply] using hpair

Helper for thm:planar-convergence: initial positive definiteness propagates along every nonstationary raw DFP orbit satisfying weak Wolfe.

theorem rawOrbitPositiveDefinite {f : EuclideanSpace ℝ (Fin 2) → ℝ} {α : ℕ → ℝ} {x g : ℕ → EuclideanSpace ℝ (Fin 2)} {H : ℕ → Matrix (Fin 2) (Fin 2) ℝ} (orbit : DFP.IsOrbit f α x g H) {c₁ c₂ : ℝ} (hH₀ : (H 0).PosDef) (hg : ∀ k, g k ≠ 0) (hWolfe : ∀ k, LineSearch.IsWeakWolfe c₁ c₂ f (x k) (x (k + 1) - x k)) : ∀ k, (H k).PosDef := by intro k induction k with | zero => exact hH₀ | succ k ih => rw [orbit.inverseHessianSucc k] exact ih.inverseDFPUpdate (rawOrbitSecantCurvaturePos orbit k ih (hg k) (hWolfe k))

Helper for thm:planar-convergence: the nonstationary raw orbit converges using only initial positive definiteness, with all later well-definedness conditions derived from the DFP and Wolfe equations.

theorem rawOrbitConvergenceOfNonstationary {f : EuclideanSpace ℝ (Fin 2) → ℝ} {α : ℕ → ℝ} {x g : ℕ → EuclideanSpace ℝ (Fin 2)} {H : ℕ → Matrix (Fin 2) (Fin 2) ℝ} (orbit : DFP.IsOrbit f α x g H) {μ c₁ c₂ : ℝ} (hf : ContDiff ℝ 2 f) (hμ : 0 < μ) (hHessian : ∀ z v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient f) z v) v) (hH₀ : (H 0).PosDef) (hg : ∀ k, g k ≠ 0) (hWolfe : ∀ k, LineSearch.IsWeakWolfe c₁ c₂ f (x k) (x (k + 1) - x k)) (hLip : LocallyLipschitzOn {z | f z ≤ f (x 0)} (fderiv ℝ (gradient f))) : ∃! xstar, (∀ z, f xstar ≤ f z) ∧ Tendsto x atTop (𝓝 xstar) ∧ Tendsto (fun k ↦ ‖gradient f (x k)‖) atTop (𝓝 0) := by have hPosDef := rawOrbitPositiveDefinite orbit hH₀ hg hWolfe have hDenominator (k : ℕ) : WithLp.ofLp (DFP.steps α (DFP.directions H g) k) ⬝ᵥ WithLp.ofLp (DFP.gradientChanges g k) ≠ 0 := (rawOrbitSecantCurvaturePos orbit k (hPosDef k) (hg k) (hWolfe k)).ne' let iteration := orbit.toInverseIteration hPosDef hDenominator have hfEq : iteration.objective = f := orbit.toInverseIteration_objective hPosDef hDenominator have hxEq : iteration.point = x := orbit.toInverseIteration_point_eq hPosDef hDenominator have hαEq : iteration.stepLength = α := orbit.toInverseIteration_stepLength hPosDef hDenominator have hSmooth : ContDiff ℝ 2 iteration.objective := by rw [hfEq] exact hf have hLower (z v : EuclideanSpace ℝ (Fin 2)) : μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient iteration.objective) z v) v := by rw [hfEq] exact hHessian z v have hStep (k : ℕ) : 0 < iteration.stepLength k := by rw [hαEq] exact orbit.stepLengthPos k have hWeak (k : ℕ) : LineSearch.IsWeakWolfe c₁ c₂ iteration.objective (iteration.point k) (iteration.point (k + 1) - iteration.point k) := by rw [hfEq, hxEq] exact hWolfe k have hlevel : DFP.objectiveSublevel iteration = {z | f z ≤ f (x 0)} := by ext z rw [DFP.mem_objectiveSublevel_iff, hfEq, hxEq, Set.mem_setOf_eq] have hLocal : LocallyLipschitzOn (DFP.objectiveSublevel iteration) (fderiv ℝ (gradient iteration.objective)) := by rw [hlevel, hfEq] exact hLip have hresult := existsUniqueLimitOfLocallyLipschitzHessian iteration hSmooth hμ hLower hStep hWeak hLocal simpa only [hfEq, hxEq] using hresult

Helper for thm:planar-convergence: after a raw orbit reaches a stationary point, its algebraically continued point sequence remains there.

theorem rawOrbitStationaryTail {f : EuclideanSpace ℝ (Fin 2) → ℝ} {α : ℕ → ℝ} {x g : ℕ → EuclideanSpace ℝ (Fin 2)} {H : ℕ → Matrix (Fin 2) (Fin 2) ℝ} (orbit : DFP.IsOrbit f α x g H) {k : ℕ} (hg : g k = 0) : ∀ j, x (k + j) = x k := by have hstationary : gradient f (x k) = 0 := (orbit.gradientAt k).gradient.trans hg intro j induction j with | zero => simp | succ j ih => have hzero : g (k + j) = 0 := by rw [← (orbit.gradientAt (k + j)).gradient, ih] exact hstationary rw [Nat.add_succ, Nat.succ_eq_add_one, orbit.pointSucc, DFP.steps_apply, DFP.directions_apply, hzero] simp only [WithLp.ofLp_zero, Matrix.mulVec_zero, WithLp.toLp_zero, neg_zero, smul_zero, add_zero] exact ih

thm:planar-convergence: from an arbitrary positive-definite initial inverse Hessian, a classical planar DFP orbit satisfying fixed weak Wolfe conditions converges to the unique minimizer of a globally strongly convex C2 objective with locally Lipschitz Hessian on its initial sublevel. The statement includes a stationary hit through the constant algebraic continuation of the point sequence; later matrix positivity is not assumed.

theorem rawOrbitConvergence {f : EuclideanSpace ℝ (Fin 2) → ℝ} {α : ℕ → ℝ} {x g : ℕ → EuclideanSpace ℝ (Fin 2)} {H : ℕ → Matrix (Fin 2) (Fin 2) ℝ} (orbit : DFP.IsOrbit f α x g H) {μ c₁ c₂ : ℝ} (hf : ContDiff ℝ 2 f) (hμ : 0 < μ) (hHessian : ∀ z v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient f) z v) v) (hH₀ : (H 0).PosDef) (hWolfe : ∀ k, LineSearch.IsWeakWolfe c₁ c₂ f (x k) (x (k + 1) - x k)) (hLip : LocallyLipschitzOn {z | f z ≤ f (x 0)} (fderiv ℝ (gradient f))) : ∃! xstar, (∀ z, f xstar ≤ f z) ∧ Tendsto x atTop (𝓝 xstar) ∧ Tendsto (fun k ↦ ‖gradient f (x k)‖) atTop (𝓝 0) := by classical by_cases hnonstationary : ∀ k, g k ≠ 0 · exact rawOrbitConvergenceOfNonstationary orbit hf hμ hHessian hH₀ hnonstationary hWolfe hLip · push Not at hnonstationary obtain ⟨k, hg⟩ := hnonstationary have hstationary : gradient f (x k) = 0 := (orbit.gradientAt k).gradient.trans hg have hminHit := stationaryPointMinimizes hf hμ hHessian hstationary obtain ⟨xstar, hmin, hunique⟩ := existsUniqueGlobalMinimizerOfHessian hf hμ hHessian have hhit : x k = xstar := hunique (x k) hminHit have htail := rawOrbitStationaryTail orbit hg have heventual : ∀ᶠ j in atTop, x j = xstar := by filter_upwards [eventually_ge_atTop k] with j hj obtain ⟨d, rfl⟩ := Nat.exists_eq_add_of_le hj exact (htail d).trans hhit have hpoint : Tendsto x atTop (𝓝 xstar) := (tendsto_congr' heventual).mpr tendsto_const_nhds have hfOrder : ContDiff ℝ ((1 : ℕ∞) + 1) f := by convert hf using 1 norm_num have hgradient : ContDiff ℝ 1 (gradient f) := hfOrder.gradient_succ (n := 1) have hgradientLimit := (hgradient.continuous.tendsto xstar).comp hpoint have hzero := gradientZeroOfGlobalMinimizer hmin have hnorm : Tendsto (fun j ↦ ‖gradient f (x j)‖) atTop (𝓝 0) := by simpa only [Function.comp_def, hzero, norm_zero] using hgradientLimit.norm refine ⟨xstar, ⟨hmin, hpoint, hnorm⟩, ?_⟩ intro y hy exact hunique y hy.1

Helper for thm:planar-convergence: the manuscript's hypothesis of a locally Lipschitz Hessian on an open neighborhood of the initial sublevel implies the sublevel regularity used by the convergence proof.

theorem rawOrbitConvergenceOfNeighborhood {f : EuclideanSpace ℝ (Fin 2) → ℝ} {α : ℕ → ℝ} {x g : ℕ → EuclideanSpace ℝ (Fin 2)} {H : ℕ → Matrix (Fin 2) (Fin 2) ℝ} (orbit : DFP.IsOrbit f α x g H) {μ c₁ c₂ : ℝ} (hf : ContDiff ℝ 2 f) (hμ : 0 < μ) (hHessian : ∀ z v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient f) z v) v) (hH₀ : (H 0).PosDef) (hWolfe : ∀ k, LineSearch.IsWeakWolfe c₁ c₂ f (x k) (x (k + 1) - x k)) (hNeighborhood : ∃ U : Set (EuclideanSpace ℝ (Fin 2)), IsOpen U ∧ {z | f z ≤ f (x 0)} ⊆ U ∧ LocallyLipschitzOn U (fderiv ℝ (gradient f))) : ∃! xstar, (∀ z, f xstar ≤ f z) ∧ Tendsto x atTop (𝓝 xstar) ∧ Tendsto (fun k ↦ ‖gradient f (x k)‖) atTop (𝓝 0) := by obtain ⟨U, _, hsubset, hLip⟩ := hNeighborhood exact rawOrbitConvergence orbit hf hμ hHessian hH₀ hWolfe (hLip.mono hsubset)

Helper for thm:planar-convergence: the C3 specialization applies to raw DFP trajectories with only initial positive definiteness.

theorem rawOrbitConvergenceOfC3 {f : EuclideanSpace ℝ (Fin 2) → ℝ} {α : ℕ → ℝ} {x g : ℕ → EuclideanSpace ℝ (Fin 2)} {H : ℕ → Matrix (Fin 2) (Fin 2) ℝ} (orbit : DFP.IsOrbit f α x g H) {μ c₁ c₂ : ℝ} (hf : ContDiff ℝ 3 f) (hμ : 0 < μ) (hHessian : ∀ z v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient f) z v) v) (hH₀ : (H 0).PosDef) (hWolfe : ∀ k, LineSearch.IsWeakWolfe c₁ c₂ f (x k) (x (k + 1) - x k)) : ∃! xstar, (∀ z, f xstar ≤ f z) ∧ Tendsto x atTop (𝓝 xstar) ∧ Tendsto (fun k ↦ ‖gradient f (x k)‖) atTop (𝓝 0) := by have htwoThree : (2 : WithTop ℕ∞) ≤ 3 := by norm_num have hfOrder : ContDiff ℝ ((2 : ℕ∞) + 1) f := by convert hf using 1 norm_num have hgradient : ContDiff ℝ 2 (gradient f) := hfOrder.gradient_succ (n := 2) have hderivativeOrder : (1 : WithTop ℕ∞) + 1 ≤ 2 := by norm_num have hHessianSmooth : ContDiff ℝ 1 (fderiv ℝ (gradient f)) := hgradient.fderiv_right hderivativeOrder exact rawOrbitConvergence orbit (hf.of_le htwoThree) hμ hHessian hH₀ hWolfe hHessianSmooth.locallyLipschitz.locallyLipschitzOn

Helper for thm:planar-convergence: Armijo decrease on a strongly convex objective forces the accepted displacement to have nonpositive initial directional derivative.

theorem armijoDescentOfHessianLowerBound {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] {f : E → ℝ} {μ c₁ : ℝ} (hf : ContDiff ℝ 2 f) (hμ : 0 < μ) (hHessian : ∀ z v, μ * ‖v‖ ^ 2 ≤ inner ℝ (fderiv ℝ (gradient f) z v) v) {x s : E} (hc₁ : c₁ < 1) (hArmijo : f (x + s) ≤ f x + c₁ * inner ℝ (gradient f x) s) : inner ℝ (gradient f x) s ≤ 0 := by have hfirst := hf.firstOrderOfHessianLowerBound f μ hμ hHessian x (x + s) rw [add_sub_cancel_left] at hfirst have hnonnegative : 0 ≤ μ / 2 * ‖s‖ ^ 2 := by positivity by_contra hpositive have hproduct := mul_pos (sub_pos.mpr hc₁) (lt_of_not_ge hpositive) nlinarith

The next interface records the exact two-dimensional residual geometry used in the manuscript's Hölder sharpness argument.

Helper for eq:counterexample-nonlipschitz: in two dimensions, the residual of the symmetric secant matrix 1 + εS from the identity has Euclidean norm |ε| times the norm of the secant step, where S exchanges the coordinates.

theorem planarSecantResidualNorm (ε : ℝ) (v : EuclideanSpace ℝ (Fin 2)) : ‖WithLp.toLp 2 (((1 + ε • (!![0, 1; 1, 0] : Matrix (Fin 2) (Fin 2) ℝ)) - 1) *ᵥ WithLp.ofLp v)‖ = |ε| * ‖v‖ := by let S : Matrix (Fin 2) (Fin 2) ℝ := !![0, 1; 1, 0] have hS : S ∈ Matrix.orthogonalGroup (Fin 2) ℝ := by apply (Matrix.mem_orthogonalGroup_iff (Fin 2) ℝ).mpr ext i j fin_cases i <;> fin_cases j <;> simp [S, Matrix.mul_apply, Fin.sum_univ_two] have hsub : (1 + ε • S - 1 : Matrix (Fin 2) (Fin 2) ℝ) = ε • S := by ext i j fin_cases i <;> fin_cases j <;> simp [S] rw [hsub, Matrix.smul_mulVec, WithLp.toLp_smul, norm_smul, Real.norm_eq_abs, Matrix.norm_toLp_mulVec_eq_of_mem_orthogonalGroup S hS]

Helper for eq:counterexample-nonlipschitz: an abstract DFP phase whose secant matrix is 1 + εS has gradient-change residual exactly |ε| times its step norm.

theorem abstractSecantResidualNorm {z : DFP.AbstractSecantStep (Fin 2)} (ε : ℝ) (hA : z.secantMatrix = !![1, ε; ε, 1]) : ‖WithLp.toLp 2 (z.gradientChange - z.displacement)‖ = |ε| * ‖WithLp.toLp 2 z.displacement‖ := by rw [z.gradientChange_def, hA] have hvec : (!![1, ε; ε, 1] : Matrix (Fin 2) (Fin 2) ℝ) *ᵥ z.displacement - z.displacement = ((1 + ε • (!![0, 1; 1, 0] : Matrix (Fin 2) (Fin 2) ℝ)) - 1) *ᵥ z.displacement := by ext i fin_cases i <;> simp [Matrix.mulVec, dotProduct, Fin.sum_univ_two] rw [hvec] simpa only [WithLp.ofLp_toLp] using planarSecantResidualNorm ε (WithLp.toLp 2 z.displacement)

Helper for eq:counterexample-nonlipschitz: a planar secant residual of size εₖ‖sₖ‖, together with ‖sₖ‖/εₖ² → C > 0, cannot satisfy a Hölder remainder bound of order 1 + β when β > 1/2.

theorem holderExponent_le_half_of_planarSecantResidual {ε sn : ℕ → ℝ} {β L C : ℝ} (hε : Tendsto ε atTop (𝓝 0)) (hεpos : ∀ᶠ k in atTop, 0 < ε k) (hsn : Tendsto (fun k ↦ sn k / (ε k) ^ 2) atTop (𝓝 C)) (hC : 0 < C) (hβ : 1 / 2 < β) (hL : 0 < L) (hresidual : ∀ᶠ k in atTop, ε k * sn k ≤ L * (sn k) ^ (1 + β)) : False := by have hδ : 0 < 2 * β - 1 := by linarith have hApos : ∀ᶠ k in atTop, 0 < sn k / (ε k) ^ 2 := hsn.eventually (eventually_gt_nhds hC) have hratio : Tendsto (fun k ↦ (sn k) ^ β / ε k) atTop (𝓝 0) := by have hApow : Tendsto (fun k ↦ (sn k / (ε k) ^ 2) ^ β) atTop (𝓝 (C ^ β)) := by exact hsn.rpow tendsto_const_nhds (Or.inl (ne_of_gt hC)) have hEpow : Tendsto (fun k ↦ (ε k) ^ (2 * β - 1)) atTop (𝓝 0) := by have h := hε.rpow tendsto_const_nhds (Or.inr hδ) simpa only [Real.zero_rpow (ne_of_gt hδ)] using h have hprod : Tendsto (fun k ↦ (sn k / (ε k) ^ 2) ^ β * (ε k) ^ (2 * β - 1)) atTop (𝓝 0) := by simpa only [mul_zero] using hApow.mul hEpow have heq : (fun k ↦ (sn k) ^ β / ε k) =ᶠ[atTop] (fun k ↦ (sn k / (ε k) ^ 2) ^ β * (ε k) ^ (2 * β - 1)) := by filter_upwards [hApos, hεpos] with k hAk hεk have hden : 0 < (ε k) ^ 2 := sq_pos_of_pos hεk have hsnpos : 0 < sn k := by have hmul := mul_pos hAk hden rwa [div_mul_cancel₀ _ (ne_of_gt hden)] at hmul have hsnFactor : sn k = (sn k / (ε k) ^ 2) * (ε k) ^ 2 := by exact (div_mul_cancel₀ _ (ne_of_gt hden)).symm have hsnPow : (sn k) ^ β = (sn k / (ε k) ^ 2) ^ β * ((ε k) ^ 2) ^ β := by conv_lhs => rw [hsnFactor] rw [Real.mul_rpow (div_nonneg (le_of_lt hsnpos) hden.le) hden.le] have hratioEq : (sn k) ^ β / ε k = (sn k / (ε k) ^ 2) ^ β * (ε k) ^ (2 * β - 1) := by rw [hsnPow] have hpowtwo : (ε k) ^ 2 = ε k ^ (2 : ℝ) := by exact (Real.rpow_natCast (ε k) 2).symm rw [hpowtwo, ← Real.rpow_mul hεk.le] rw [Real.rpow_sub hεk (2 * β) 1, Real.rpow_one] ring exact hratioEq exact hprod.congr' heq.symm have hone : ∀ᶠ k in atTop, (1 : ℝ) ≤ L * ((sn k) ^ β / ε k) := by filter_upwards [hresidual, hApos, hεpos] with k hk hAk hεk have hden : 0 < (ε k) ^ 2 := sq_pos_of_pos hεk have hsnpos : 0 < sn k := by have hmul := mul_pos hAk hden rwa [div_mul_cancel₀ _ (ne_of_gt hden)] at hmul have hresidual' : ε k * sn k ≤ (L * (sn k) ^ β) * sn k := by simpa [Real.rpow_add hsnpos, mul_assoc, mul_left_comm, mul_comm] using hk have hdivSn : ε k ≤ L * (sn k) ^ β := by exact le_of_mul_le_mul_right hresidual' hsnpos have hdesired : (1 : ℝ) ≤ (L * (sn k) ^ β) / ε k := (le_div_iff₀ hεk).2 (by simpa using hdivSn) simpa [mul_div_assoc] using hdesired have hsmall : ∀ᶠ k in atTop, (sn k) ^ β / ε k < 1 / (L + 1) := by exact hratio.eventually (eventually_lt_nhds (by positivity)) obtain ⟨k, hk⟩ := (hone.and hsmall).exists have hden : 0 < L + 1 := by linarith have hupper : L * ((sn k) ^ β / ε k) < 1 := by calc L * ((sn k) ^ β / ε k) < L * (1 / (L + 1)) := mul_lt_mul_of_pos_left hk.2 hL _ < 1 := by field_simp linarith exact (not_lt_of_ge hk.1) hupper

Helper for eq:holder-hessian-bound: the third derivative of a scaled linear bump is controlled by the third and second derivatives of its cutoff, with the expected quadratic inverse-scale factor.

theorem norm_iteratedFDeriv_three_scaledLinearBump_le {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] [Nontrivial E] (χ : E → ℝ) (hχ : ContDiff ℝ (3 : WithTop ℕ∞) χ) {M0 M2 M3 : ℝ} (hM0 : ∀ y, ‖χ y‖ ≤ M0) (hM2 : ∀ y, ‖iteratedFDeriv ℝ 2 χ y‖ ≤ M2) (hM3 : ∀ y, ‖iteratedFDeriv ℝ 3 χ y‖ ≤ M3) (hχ_support : tsupport χ ⊆ Metric.ball 0 1) (x : E) (ρ : ℝ) (a z : E) (hρ : 0 < ρ) (hz : z ∈ tsupport (AffineBump.scaledLinearBump χ x ρ a)) : ‖iteratedFDeriv ℝ 3 (AffineBump.scaledLinearBump χ x ρ a) z‖ ≤ (M3 + 3 * M2) * ‖a‖ / ρ ^ 2 := by let A : E →L[ℝ] E := ρ⁻¹ • ContinuousLinearMap.id ℝ E let b : E := -(ρ⁻¹ • x) have hA : ∀ w : E, A w + b = ρ⁻¹ • (w - x) := by intro w dsimp only [A, b] simp only [smul_apply, ContinuousLinearMap.id_apply] rw [smul_sub] simp only [sub_eq_add_neg] have hAnorm : ‖A‖ ≤ ρ⁻¹ := by dsimp only [A] rw [norm_smul, ContinuousLinearMap.norm_id] rw [Real.norm_eq_abs, abs_inv, abs_of_pos hρ] simp have hχ2 : ContDiff ℝ (2 : WithTop ℕ∞) χ := by apply hχ.of_le norm_num have hχ3 : ContDiff ℝ (3 : WithTop ℕ∞) χ := hχ have hf : ContDiff ℝ (3 : WithTop ℕ∞) (fun w : E => χ (A w + b)) := by exact hχ3.comp ((A.contDiff).add contDiff_const) let glin : E → ℝ := fun w => inner ℝ a (w - x) have hg : ContDiff ℝ (3 : WithTop ℕ∞) glin := by dsimp only [glin] exact (contDiff_const : ContDiff ℝ (3 : WithTop ℕ∞) (fun _ : E ↦ a)).inner ℝ (contDiff_id.sub (contDiff_const : ContDiff ℝ (3 : WithTop ℕ∞) (fun _ : E ↦ x))) have hfg : (fun w => χ (A w + b) * glin w) = AffineBump.scaledLinearBump χ x ρ a := by funext w rw [AffineBump.scaledLinearBump_apply, hA] have hM2nonneg : 0 ≤ M2 := (norm_nonneg _).trans (hM2 0) have hM3nonneg : 0 ≤ M3 := (norm_nonneg _).trans (hM3 0) have hcomp2 : ∀ w, ‖iteratedFDeriv ℝ 2 (fun t : E => χ (A t + b)) w‖ ≤ M2 * (ρ⁻¹) ^ 2 := by intro w have hcomp := norm_iteratedFDeriv_comp_affine_le 2 χ A b hχ2 w have hcomp' : ‖iteratedFDeriv ℝ 2 (fun t : E => χ (A t + b)) w‖ ≤ ‖iteratedFDeriv ℝ 2 χ (ρ⁻¹ • (w - x))‖ * ‖A‖ ^ 2 := by simpa only [hA w] using hcomp calc ‖iteratedFDeriv ℝ 2 (fun t : E => χ (A t + b)) w‖ ≤ ‖iteratedFDeriv ℝ 2 χ (ρ⁻¹ • (w - x))‖ * ‖A‖ ^ 2 := hcomp' _ ≤ M2 * (ρ⁻¹) ^ 2 := by exact mul_le_mul (hM2 _) (pow_le_pow_left₀ (norm_nonneg _) hAnorm 2) (pow_nonneg (norm_nonneg _) 2) hM2nonneg have hcomp3 : ∀ w, ‖iteratedFDeriv ℝ 3 (fun t : E => χ (A t + b)) w‖ ≤ M3 * (ρ⁻¹) ^ 3 := by intro w have hcomp := norm_iteratedFDeriv_comp_affine_le 3 χ A b hχ3 w have hcomp' : ‖iteratedFDeriv ℝ 3 (fun t : E => χ (A t + b)) w‖ ≤ ‖iteratedFDeriv ℝ 3 χ (ρ⁻¹ • (w - x))‖ * ‖A‖ ^ 3 := by simpa only [hA w] using hcomp calc ‖iteratedFDeriv ℝ 3 (fun t : E => χ (A t + b)) w‖ ≤ ‖iteratedFDeriv ℝ 3 χ (ρ⁻¹ • (w - x))‖ * ‖A‖ ^ 3 := hcomp' _ ≤ M3 * (ρ⁻¹) ^ 3 := by exact mul_le_mul (hM3 _) (pow_le_pow_left₀ (norm_nonneg _) hAnorm 3) (pow_nonneg (norm_nonneg _) 3) hM3nonneg have hcomp0 : ∀ w, ‖iteratedFDeriv ℝ 0 (fun t : E => χ (A t + b)) w‖ ≤ M0 := by intro w rw [norm_iteratedFDeriv_zero] exact hM0 _ have hlinDeriv : fderiv ℝ glin = fun _ => innerSL ℝ a := by funext w have hsub : HasFDerivAt (fun y : E ↦ y - x) (ContinuousLinearMap.id ℝ E) w := by convert (hasFDerivAt_id (𝕜 := ℝ) w).sub (hasFDerivAt_const x w) using 1 · ext y rfl · simp have hinner := fderiv_inner_apply (f := fun _ : E ↦ a) (g := fun y : E ↦ y - x) ℝ (differentiableAt_const (c := a)) hsub.differentiableAt rw [hsub.fderiv] at hinner ext y simpa [Function.id_def, innerSL_apply_apply] using hinner y have hlin1 : ∀ w, ‖iteratedFDeriv ℝ 1 glin w‖ ≤ ‖a‖ := by intro w rw [norm_iteratedFDeriv_one, hlinDeriv] exact (innerSL_apply_norm ℝ a).le have hlin0 : ‖iteratedFDeriv ℝ 0 glin z‖ ≤ ‖a‖ * ρ := by rw [norm_iteratedFDeriv_zero] have hzclosed := AffineBump.tsupport_scaledLinearBump_subset_closedBall χ hχ_support x ρ a hρ hz have hdist : ‖z - x‖ ≤ ρ := by simpa [Metric.mem_closedBall, dist_eq_norm] using hzclosed exact (norm_inner_le_norm _ _).trans (mul_le_mul_of_nonneg_left hdist (norm_nonneg a)) have hlin2 : ∀ w, ‖iteratedFDeriv ℝ 2 glin w‖ = 0 := by intro w rw [← norm_iteratedFDeriv_fderiv (𝕜 := ℝ) (f := glin) (n := 1), hlinDeriv] simp [iteratedFDeriv_const_of_ne (by norm_num : 1 ≠ 0)] have hlin3 : ∀ w, ‖iteratedFDeriv ℝ 3 glin w‖ = 0 := by intro w rw [← norm_iteratedFDeriv_fderiv (𝕜 := ℝ) (f := glin) (n := 2), hlinDeriv] simp [iteratedFDeriv_const_of_ne (by norm_num : 2 ≠ 0)] have hprod := norm_iteratedFDeriv_mul_le hf hg z (n := 3) (by rfl : (3 : WithTop ℕ∞) ≤ (3 : WithTop ℕ∞)) have hprod' : ‖iteratedFDeriv ℝ 3 (AffineBump.scaledLinearBump χ x ρ a) z‖ ≤ ∑ i ∈ Finset.range (3 + 1), (Nat.choose 3 i : ℝ) * ‖iteratedFDeriv ℝ i (fun t : E => χ (A t + b)) z‖ * ‖iteratedFDeriv ℝ (3 - i) glin z‖ := by rw [← hfg] exact hprod calc ‖iteratedFDeriv ℝ 3 (AffineBump.scaledLinearBump χ x ρ a) z‖ ≤ |χ (A z + b)| * ‖iteratedFDeriv ℝ (3 - 0) glin z‖ + 3 * ‖fderiv ℝ (fun t : E => χ (A t + b)) z‖ * ‖iteratedFDeriv ℝ (3 - 1) glin z‖ + 3 * ‖iteratedFDeriv ℝ 2 (fun t : E => χ (A t + b)) z‖ * ‖iteratedFDeriv ℝ (3 - 2) glin z‖ + ‖iteratedFDeriv ℝ 3 (fun t : E => χ (A t + b)) z‖ * ‖iteratedFDeriv ℝ (3 - 3) glin z‖ := by norm_num [Finset.sum_range_succ] at hprod' exact hprod' _ ≤ 3 * (M2 * (ρ⁻¹) ^ 2) * ‖a‖ + (M3 * (ρ⁻¹) ^ 3) * (‖a‖ * ρ) := by rw [hlin3 z, hlin2 z] simp only [mul_zero, add_zero, zero_add] gcongr · exact hcomp2 z · exact hlin1 z · exact hcomp3 z _ = (M3 + 3 * M2) * ‖a‖ / ρ ^ 2 := by field_simp [hρ.ne'] ring

Helper for eq:holder-hessian-bound: a uniform third-derivative bound yields a Lipschitz estimate for the second Fréchet derivative.

theorem norm_hessian_sub_le_of_thirdFDeriv {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] {f : E → ℝ} {B : ℝ} (hf : ContDiff ℝ (3 : WithTop ℕ∞) f) (hbound : ∀ z, ‖iteratedFDeriv ℝ 3 f z‖ ≤ B) (x y : E) : ‖fderiv ℝ (fderiv ℝ f) y - fderiv ℝ (fderiv ℝ f) x‖ ≤ B * ‖y - x‖ := by have horder₁ : (2 : WithTop ℕ∞) + 1 ≤ (3 : WithTop ℕ∞) := by norm_num have hf2 : ContDiff ℝ (2 : WithTop ℕ∞) (fderiv ℝ f) := hf.fderiv_right horder₁ have horder₂ : (1 : WithTop ℕ∞) + 1 ≤ (2 : WithTop ℕ∞) := by norm_num have hf1 : ContDiff ℝ (1 : WithTop ℕ∞) (fderiv ℝ (fderiv ℝ f)) := hf2.fderiv_right horder₂ have hbound' : ∀ z, failed to synthesize instance of type class Norm (E →L[ℝ] E →L[ℝ] E →L[ℝ] ℝ) Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.‖fderiv ℝ (fderiv ℝ (fderiv ℝ f)) z‖ ≤ B := by intro z calc failed to synthesize instance of type class Norm (E →L[ℝ] E →L[ℝ] E →L[ℝ] ℝ) Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.‖fderiv ℝ (fderiv ℝ (fderiv ℝ f)) z‖ = ‖iteratedFDeriv ℝ 1 (fderiv ℝ (fderiv ℝ f)) z‖ := (norm_iteratedFDeriv_one (𝕜 := ℝ) (fderiv ℝ (fderiv ℝ f))).symm _ = ‖iteratedFDeriv ℝ 2 (fderiv ℝ f) z‖ := by rw [norm_iteratedFDeriv_fderiv] _ = ‖iteratedFDeriv ℝ 3 f z‖ := by rw [norm_iteratedFDeriv_fderiv] _ ≤ B := hbound z exact Convex.norm_image_sub_le_of_norm_fderiv_le (s := Set.univ) (f := fderiv ℝ (fderiv ℝ f)) (C := B) (x := x) (y := y) (fun z _ ↦ hf1.differentiable_one z) (fun z _ ↦ hbound' z) convex_univ (Set.mem_univ x) (Set.mem_univ y)

Helper for eq:holder-hessian-bound: the Hessian of each endpoint bump is Lipschitz with the inverse scale predicted by the shrinking construction.

theorem endpointBumpHessianLipschitz (curve : DFP.TwoLeg.SlowCurve) : ∃ εbar ∈ Set.Ioo (0 : ℝ) (1 / 4), ∃ K > 0, ∀ ε₀ ∈ Set.Ioc 0 εbar, let orbit := DFP.TwoPhaseOrbit.ofSlowCurve curve.shape curve.high ε₀ ∀ Clim : EuclideanSpace ℝ (Fin 2), Tendsto (fun j : ℕ ↦ (orbit.state j).center) atTop (𝓝 Clim) → ∀ Glim > 0, Tendsto (fun j : ℕ ↦ (orbit.state j).amplitude) atTop (𝓝 Glim) → ∀ k z w, ‖fderiv ℝ (fderiv ℝ (orbit.endpointBump Clim Glim k)) z - fderiv ℝ (fderiv ℝ (orbit.endpointBump Clim Glim k)) w‖ ≤ K / (orbit.state (k / 2)).ε * ‖z - w‖ := by obtain ⟨ηRadius, hηRadius, cLower, hcLower, cUpper, hcUpper, hRadius⟩ := curve.interpolationRadiusUniformBounds obtain ⟨ηCorrection, hηCorrection, Kcorrection, hKcorrection, hCorrection⟩ := DFP.TwoPhaseOrbit.slowCurveEndpointCorrectionUniformBound curve.shape curve.high curve.isInvariant curve.shapeRemainder curve.highRemainder obtain ⟨ηPos, hηPos, hPos⟩ := DFP.TwoLeg.slowCurveForwardOrbitPos curve.shape curve.high curve.isInvariant curve.shapeRemainder curve.highRemainder let εbar := min ηRadius (min ηCorrection ηPos) have hεbarPos : 0 < εbar := by dsimp only [εbar] exact lt_min hηRadius.1 (lt_min hηCorrection.1 hηPos) have hεbarLt : εbar < (1 / 4 : ℝ) := (min_le_left _ _).trans_lt hηRadius.2 let M0 := EuclideanPlane.smoothCutoffDerivBound 0 let M2 := EuclideanPlane.smoothCutoffDerivBound 2 let M3 := EuclideanPlane.smoothCutoffDerivBound 3 have hM2 : 0 ≤ M2 := by dsimp only [M2] exact EuclideanPlane.smoothCutoffDerivBound_nonneg 2 have hM3 : 0 ≤ M3 := by dsimp only [M3] exact EuclideanPlane.smoothCutoffDerivBound_nonneg 3 let Kthird := (M3 + 3 * M2) * Kcorrection / cLower ^ 2 let K := Kthird + 1 have hKthird : 0 ≤ Kthird := by dsimp only [Kthird] positivity have hK : 0 < K := by dsimp only [K] linarith refine ⟨εbar, ⟨hεbarPos, hεbarLt⟩, K, hK, ?_⟩ intro ε₀ hε₀ dsimp only let orbit := DFP.TwoPhaseOrbit.ofSlowCurve curve.shape curve.high ε₀ have hεRadius : ε₀ ∈ Set.Ioc 0 ηRadius := ⟨hε₀.1, hε₀.2.trans (min_le_left _ _ )⟩ have hεCorrection : ε₀ ∈ Set.Ioc 0 ηCorrection := ⟨hε₀.1, hε₀.2.trans ((min_le_right _ _).trans (min_le_left _ _))⟩ have hεPos : ε₀ ∈ Set.Ioc 0 ηPos := ⟨hε₀.1, hε₀.2.trans ((min_le_right _ _).trans (min_le_right _ _))⟩ intro Clim hClim Glim hGlim hGlimTendsto k z w have hRadiusData := hRadius ε₀ hεRadius Clim hClim Glim hGlim hGlimTendsto k have hCorrectionData := hCorrection ε₀ hεCorrection Clim hClim k have hcoord (j : ℕ) : (orbit.state j).ε = (DFP.TwoLeg.stateMap^[j] (ε₀, curve.shape ε₀, curve.high ε₀)).1 := by have hc := DFP.TwoPhaseOrbit.ofSlowCurve_coordinates curve.shape curve.high ε₀ j have hc' : (orbit.state j).coordinates = DFP.TwoLeg.stateMap^[j] (ε₀, curve.shape ε₀, curve.high ε₀) := by simpa only [orbit] using hc simpa only [DFP.TwoPhaseOrbit.State.coordinates_def] using congrArg Prod.fst hc' have hscalePos (j : ℕ) : 0 < (orbit.state j).ε := by rw [hcoord j] exact (hPos ε₀ hεPos j).1 let e := (orbit.state (k / 2)).ε let R := orbit.endpointRadius k let ρ := orbit.interpolationRadius Clim Glim k let a := orbit.endpointCorrection Clim k have he : 0 < e := by simpa only [e] using hscalePos (k / 2) have hR : R = e ^ 2 := by simpa only [R, e] using DFP.TwoPhaseOrbit.endpointRadius_def orbit k have hRpos : 0 < R := by rw [hR] positivity have hρLower : cLower * R ≤ ρ := by simpa only [ρ, R] using hRadiusData.1 have hρ : 0 < ρ := by exact (mul_pos hcLower hRpos).trans_le hρLower have ha : ‖a‖ ≤ Kcorrection * e ^ 3 := by simpa only [a, e] using hCorrectionData have hthree : (3 : WithTop ℕ∞) ≤ ∞ := by have hnat : (3 : ℕ∞) ≤ ⊤ := le_top exact WithTop.coe_le_coe.mpr hnat have hcutoff : ContDiff ℝ (3 : WithTop ℕ∞) EuclideanPlane.smoothCutoff := by exact EuclideanPlane.contDiff_smoothCutoff.of_le hthree have hcoef : 0 ≤ M3 + 3 * M2 := by positivity have hρLower' : cLower * e ^ 2 ≤ ρ := by simpa only [hR] using hρLower have hρsq : (cLower * e ^ 2) ^ 2 ≤ ρ ^ 2 := by exact pow_le_pow_left₀ (mul_nonneg hcLower.le (sq_nonneg e)) hρLower' 2 have hρsqPos : 0 < ρ ^ 2 := pow_pos hρ 2 have hthird : ∀ u, ‖iteratedFDeriv ℝ 3 (orbit.endpointBump Clim Glim k) u‖ ≤ Kthird / e := by intro u by_cases hu : u ∈ tsupport (orbit.endpointBump Clim Glim k) · have hbase := norm_iteratedFDeriv_three_scaledLinearBump_le EuclideanPlane.smoothCutoff hcutoff (fun y ↦ by simpa only [M0] using EuclideanPlane.norm_smoothCutoff_le y) (fun y ↦ by simpa only [M2] using EuclideanPlane.norm_iteratedFDeriv_smoothCutoff_le 2 y) (fun y ↦ by simpa only [M3] using EuclideanPlane.norm_iteratedFDeriv_smoothCutoff_le 3 y) EuclideanPlane.tsupport_smoothCutoff_subset (orbit.endpoint k) ρ a u hρ (by simpa only [DFP.TwoPhaseOrbit.endpointBump_eq_scaledLinearBump] using hu) rw [DFP.TwoPhaseOrbit.endpointBump_eq_scaledLinearBump] change ‖iteratedFDeriv ℝ 3 (AffineBump.scaledLinearBump EuclideanPlane.smoothCutoff (orbit.endpoint k) ρ a) u‖ ≤ Kthird / e calc ‖iteratedFDeriv ℝ 3 (AffineBump.scaledLinearBump EuclideanPlane.smoothCutoff (orbit.endpoint k) ρ a) u‖ ≤ (M3 + 3 * M2) * ‖a‖ / ρ ^ 2 := by simpa only [M3, M2] using hbase _ ≤ Kthird / e := by apply (div_le_iff₀ hρsqPos).2 calc (M3 + 3 * M2) * ‖a‖ ≤ (M3 + 3 * M2) * (Kcorrection * e ^ 3) := mul_le_mul_of_nonneg_left ha hcoef _ = (Kthird / e) * (cLower * e ^ 2) ^ 2 := by dsimp only [Kthird] field_simp [hcLower.ne', he.ne'] _ ≤ (Kthird / e) * ρ ^ 2 := by exact mul_le_mul_of_nonneg_left hρsq (div_nonneg hKthird (le_of_lt he)) · rw [iteratedFDeriv_eq_zero_of_notMem_tsupport (orbit.endpointBump Clim Glim k) 3 hu, norm_zero] exact div_nonneg hKthird (le_of_lt he) have hf3 : ContDiff ℝ (3 : WithTop ℕ∞) (orbit.endpointBump Clim Glim k) := by exact (DFP.TwoPhaseOrbit.contDiff_endpointBump orbit Clim Glim k).of_le hthree have hL := norm_hessian_sub_le_of_thirdFDeriv hf3 hthird w z calc ‖fderiv ℝ (fderiv ℝ (orbit.endpointBump Clim Glim k)) z - fderiv ℝ (fderiv ℝ (orbit.endpointBump Clim Glim k)) w‖ ≤ Kthird / e * ‖z - w‖ := hL _ ≤ K / e * ‖z - w‖ := by gcongr dsimp only [K] linarith

Helper for eq:holder-hessian-bound: the minimum of a uniform size bound and an inverse-scale Lipschitz bound has the square-root interpolation estimate used in the bump sum.

theorem min_scaled_le_sqrt {A B e d : ℝ} (hA : 0 ≤ A) (hB : 0 ≤ B) (he : 0 < e) (hd : 0 ≤ d) : min (2 * A * e) (B / e * d) ≤ Real.sqrt (2 * A * B) * Real.sqrt d := by by_cases hd0 : d = 0 · subst d simp only [mul_zero, Real.sqrt_zero, mul_zero] simp have hdpos : 0 < d := lt_of_le_of_ne hd (Ne.symm hd0) have hsqrtAB : 0 ≤ Real.sqrt (2 * A * B) := Real.sqrt_nonneg _ have hsqrtd : 0 ≤ Real.sqrt d := Real.sqrt_nonneg _ have hsqAB : (Real.sqrt (2 * A * B)) ^ 2 = 2 * A * B := by rw [Real.sq_sqrt] positivity have hsqd : (Real.sqrt d) ^ 2 = d := by rw [Real.sq_sqrt hd] have hsqrtProd : (Real.sqrt (2 * A * B) * Real.sqrt d) ^ 2 = 2 * A * B * d := by calc (Real.sqrt (2 * A * B) * Real.sqrt d) ^ 2 = (Real.sqrt (2 * A * B)) ^ 2 * (Real.sqrt d) ^ 2 := by rw [mul_pow] _ = 2 * A * B * d := by rw [hsqAB, hsqd] by_cases hab : 2 * A * e ≤ B / e * d · apply le_trans (le_of_eq (min_eq_left hab)) have hmul : (2 * A * e) * (2 * A * e) ≤ (B / e * d) * (2 * A * e) := mul_le_mul_of_nonneg_right hab (by positivity) have hsq : (2 * A * e) ^ 2 ≤ (Real.sqrt (2 * A * B) * Real.sqrt d) ^ 2 := by rw [hsqrtProd] calc (2 * A * e) ^ 2 ≤ (B / e * d) * (2 * A * e) := by simpa only [pow_two] using hmul _ = 2 * A * B * d := by field_simp [he.ne'] exact le_of_sq_le_sq hsq (by positivity) · have hba : B / e * d ≤ 2 * A * e := le_of_not_ge hab apply le_trans (le_of_eq (min_eq_right hba)) have hmul : (B / e * d) * (B / e * d) ≤ (2 * A * e) * (B / e * d) := mul_le_mul_of_nonneg_right hba (by positivity) have hsq : (B / e * d) ^ 2 ≤ (Real.sqrt (2 * A * B) * Real.sqrt d) ^ 2 := by rw [hsqrtProd] calc (B / e * d) ^ 2 ≤ (2 * A * e) * (B / e * d) := by simpa only [pow_two] using hmul _ = 2 * A * B * d := by field_simp [he.ne'] exact le_of_sq_le_sq hsq (by positivity)

Helper for eq:holder-hessian-bound: equality of second Fréchet derivatives follows from equality of the corresponding second iterated derivatives.

theorem fderiv_fderiv_eq_of_iteratedFDeriv_two_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {f g : E → ℝ} {x : E} (h : iteratedFDeriv ℝ 2 f x = iteratedFDeriv ℝ 2 g x) : fderiv ℝ (fderiv ℝ f) x = fderiv ℝ (fderiv ℝ g) x := by ext u v have h' := congrArg (fun T => T ![u, v]) h simpa [iteratedFDeriv_two_apply] using h'

Helper for eq:holder-hessian-bound: disjoint shrinking bumps with a uniform local square-root Hessian modulus have the same modulus after zero-extension.

theorem holder_of_disjoint_finsum_hessian {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] (Γ : Set E) (x : ℕ → E) (ρ : ℕ → ℝ) (ψ : ℕ → E → ℝ) (F : E → ℝ) (L : ℝ) (hΓ : IsClosed Γ) (hcluster : ∀ y, MapClusterPt y atTop x → y ∈ Γ) (hρ0 : Tendsto ρ atTop (𝓝 0)) (hballs : Set.univ.PairwiseDisjoint (fun k ↦ Metric.closedBall (x k) (ρ k))) (hsupport : ∀ k, tsupport (ψ k) ⊆ Metric.closedBall (x k) (ρ k)) (hdisjointΓ : ∀ k, Disjoint (tsupport (ψ k)) Γ) (hHessianEq : fderiv ℝ (fderiv ℝ F) = Γᶜ.indicator (fderiv ℝ (fderiv ℝ (fun z ↦ ∑ᶠ k, ψ k z)))) (hlocal : ∀ k u v, ‖fderiv ℝ (fderiv ℝ (ψ k)) u - fderiv ℝ (fderiv ℝ (ψ k)) v‖ ≤ L * ‖u - v‖ ^ (1 / 2 : ℝ)) (hL : 0 ≤ L) (u v : E) : ‖fderiv ℝ (fderiv ℝ F) u - fderiv ℝ (fderiv ℝ F) v‖ ≤ 2 * L * ‖u - v‖ ^ (1 / 2 : ℝ) := by have hsumHessianAt : ∀ z, z ∈ Γᶜ → (∃ k, z ∈ tsupport (ψ k)) → ∀ k, z ∈ tsupport (ψ k) → fderiv ℝ (fderiv ℝ (fun y ↦ ∑ᶠ n, ψ n y)) z = fderiv ℝ (fderiv ℝ (ψ k)) z := by intro z hzΓ hactive k hzk have hz0 := iteratedFDeriv_finsum_eq_of_mem_tsupport Γ x ρ ψ hcluster hρ0 hballs hsupport hzΓ hzk (j := 2) exact fderiv_fderiv_eq_of_iteratedFDeriv_two_eq hz0 have hsumHessianZero : ∀ z, z ∈ Γᶜ → (¬ ∃ k, z ∈ tsupport (ψ k)) → fderiv ℝ (fderiv ℝ (fun y ↦ ∑ᶠ n, ψ n y)) z = 0 := by intro z hzΓ hna have hz0 := iteratedFDeriv_finsum_eq_zero Γ x ρ ψ hcluster hρ0 hsupport hzΓ (fun k ↦ fun hzk ↦ hna ⟨k, hzk⟩) (j := 2) have hz0' : iteratedFDeriv ℝ 2 (fun y ↦ ∑ᶠ n, ψ n y) z = iteratedFDeriv ℝ 2 (fun _ : E ↦ (0 : ℝ)) z := by simpa using hz0 exact (fderiv_fderiv_eq_of_iteratedFDeriv_two_eq hz0').trans (by simp) have hzero (k : ℕ) (z : E) (hz : z ∉ tsupport (ψ k)) : fderiv ℝ (fderiv ℝ (ψ k)) z = 0 := by have hz0 : iteratedFDeriv ℝ 2 (ψ k) z = iteratedFDeriv ℝ 2 (fun _ : E ↦ (0 : ℝ)) z := by rw [iteratedFDeriv_eq_zero_of_notMem_tsupport (ψ k) 2 hz] simp exact (fderiv_fderiv_eq_of_iteratedFDeriv_two_eq hz0).trans (by simp) have hglobalCircle (z : E) (hz : z ∈ Γ) : fderiv ℝ (fderiv ℝ F) z = 0 := by rw [hHessianEq] simp [hz] have hglobalActive (z : E) (hzΓ : z ∈ Γᶜ) (k : ℕ) (hzk : z ∈ tsupport (ψ k)) : fderiv ℝ (fderiv ℝ F) z = fderiv ℝ (fderiv ℝ (ψ k)) z := by rw [hHessianEq] simp only [Set.indicator_of_mem hzΓ] exact hsumHessianAt z hzΓ ⟨k, hzk⟩ k hzk have hglobalZero (z : E) (hzΓ : z ∈ Γᶜ) (hna : ¬ ∃ k, z ∈ tsupport (ψ k)) : fderiv ℝ (fderiv ℝ F) z = 0 := by rw [hHessianEq] simp only [Set.indicator_of_mem hzΓ] exact hsumHessianZero z hzΓ hna have hglobalNoActive (z : E) (hna : ¬ ∃ k, z ∈ tsupport (ψ k)) : fderiv ℝ (fderiv ℝ F) z = 0 := by by_cases hzΓ : z ∈ Γ · exact hglobalCircle z hzΓ · exact hglobalZero z hzΓ hna by_cases hu : ∃ k, u ∈ tsupport (ψ k) · obtain ⟨ku, hku⟩ := hu have huΓ : u ∉ Γ := by intro huΓ exact Set.disjoint_left.mp (hdisjointΓ ku) hku huΓ by_cases hv : ∃ k, v ∈ tsupport (ψ k) · obtain ⟨kv, hkv⟩ := hv have hvΓ : v ∉ Γ := by intro hvΓ exact Set.disjoint_left.mp (hdisjointΓ kv) hkv hvΓ have huΓ' : u ∈ Γᶜ := huΓ have hvΓ' : v ∈ Γᶜ := hvΓ by_cases hEq : ku = kv · subst kv rw [hglobalActive u huΓ' ku hku, hglobalActive v hvΓ' ku hkv] calc ‖fderiv ℝ (fderiv ℝ (ψ ku)) u - fderiv ℝ (fderiv ℝ (ψ ku)) v‖ ≤ L * ‖u - v‖ ^ (1 / 2 : ℝ) := hlocal ku u v _ ≤ 2 * L * ‖u - v‖ ^ (1 / 2 : ℝ) := by exact mul_le_mul_of_nonneg_right (by linarith) (Real.rpow_nonneg (norm_nonneg _) _) · have hkv_u : u ∉ tsupport (ψ kv) := by intro hkuv have heq := hballs.elim_set (Set.mem_univ ku) (Set.mem_univ kv) u (hsupport ku hku) (hsupport kv hkuv) exact hEq heq have hku_v : v ∉ tsupport (ψ ku) := by intro hkuv have heq := hballs.elim_set (Set.mem_univ ku) (Set.mem_univ kv) v (hsupport ku hkuv) (hsupport kv hkv) exact hEq heq rw [hglobalActive u huΓ' ku hku, hglobalActive v hvΓ' kv hkv] let Hk := fderiv ℝ (fderiv ℝ (ψ ku)) let Hl := fderiv ℝ (fderiv ℝ (ψ kv)) have hkuZero := hzero ku v hku_v have hkvZero := hzero kv u hkv_u change ‖Hk u - Hl v‖ ≤ _ calc ‖Hk u - Hl v‖ ≤ ‖Hk u - 0‖ + ‖0 - Hl v‖ := by convert norm_add_le (Hk u - 0) (0 - Hl v) using 1 <;> Try this: [apply] abel_nfabel _ = ‖Hk u - Hk v‖ + ‖Hl u - Hl v‖ := by dsimp [Hk, Hl] rw [hkuZero, hkvZero] _ ≤ L * ‖u - v‖ ^ (1 / 2 : ℝ) + L * ‖u - v‖ ^ (1 / 2 : ℝ) := add_le_add (hlocal ku u v) (hlocal kv u v) _ = 2 * L * ‖u - v‖ ^ (1 / 2 : ℝ) := by ring · rw [hglobalActive u huΓ ku hku, hglobalNoActive v hv] let Hk := fderiv ℝ (fderiv ℝ (ψ ku)) have hkuZero := hzero ku v (fun hvk ↦ hv ⟨ku, hvk⟩) change ‖Hk u - 0‖ ≤ _ calc ‖Hk u - 0‖ = ‖Hk u - Hk v‖ := by dsimp [Hk] rw [hkuZero] _ ≤ L * ‖u - v‖ ^ (1 / 2 : ℝ) := hlocal ku u v _ ≤ 2 * L * ‖u - v‖ ^ (1 / 2 : ℝ) := by exact mul_le_mul_of_nonneg_right (by linarith) (Real.rpow_nonneg (norm_nonneg _) _) · by_cases hv : ∃ k, v ∈ tsupport (ψ k) · obtain ⟨kv, hkv⟩ := hv have hvΓ : v ∉ Γ := by intro hvΓ exact Set.disjoint_left.mp (hdisjointΓ kv) hkv hvΓ rw [hglobalNoActive u hu, hglobalActive v hvΓ kv hkv] let Hk := fderiv ℝ (fderiv ℝ (ψ kv)) have hkvZero := hzero kv u (fun huk ↦ hu ⟨kv, huk⟩) change ‖0 - Hk v‖ ≤ _ calc ‖0 - Hk v‖ = ‖Hk u - Hk v‖ := by dsimp [Hk] rw [hkvZero] _ ≤ L * ‖u - v‖ ^ (1 / 2 : ℝ) := hlocal kv u v _ ≤ 2 * L * ‖u - v‖ ^ (1 / 2 : ℝ) := by exact mul_le_mul_of_nonneg_right (by linarith) (Real.rpow_nonneg (norm_nonneg _) _) · rw [hglobalNoActive u hu, hglobalNoActive v hv] simpa only [sub_self, norm_zero] using (mul_nonneg (mul_nonneg (unsolved goals E:Type u_1inst✝²:NormedAddCommGroup Einst✝¹:InnerProductSpace ℝ Einst✝:CompleteSpace EΓ:Set Ex:ℕ → Eρ:ℕ → ℝψ:ℕ → E → ℝF:E → ℝL:ℝhΓ:IsClosed Γhcluster:∀ (y : E), MapClusterPt y atTop x → y ∈ Γhρ0:Tendsto ρ atTop (𝓝 0)hballs:Set.univ.PairwiseDisjoint fun k => Metric.closedBall (x k) (ρ k)hsupport:∀ (k : ℕ), tsupport (ψ k) ⊆ Metric.closedBall (x k) (ρ k)hdisjointΓ:∀ (k : ℕ), Disjoint (tsupport (ψ k)) ΓhHessianEq:fderiv ℝ (fderiv ℝ F) = Γᶜ.indicator (fderiv ℝ (fderiv ℝ fun z => ∑ᶠ (k : ℕ), ψ k z))hlocal:∀ (k : ℕ) (u v : E), ‖fderiv ℝ (fderiv ℝ (ψ k)) u - fderiv ℝ (fderiv ℝ (ψ k)) v‖ ≤ L * ‖u - v‖ ^ (1 / 2)hL:0 ≤ Lu v:EhsumHessianAt:∀ z ∈ Γᶜ, (∃ k, z ∈ tsupport (ψ k)) → ∀ (k : ℕ), z ∈ tsupport (ψ k) → fderiv ℝ (fderiv ℝ fun y => ∑ᶠ (n : ℕ), ψ n y) z = fderiv ℝ (fderiv ℝ (ψ k)) zhsumHessianZero:∀ z ∈ Γᶜ, (¬∃ k, z ∈ tsupport (ψ k)) → fderiv ℝ (fderiv ℝ fun y => ∑ᶠ (n : ℕ), ψ n y) z = 0hzero:∀ (k : ℕ), ∀ z ∉ tsupport (ψ k), fderiv ℝ (fderiv ℝ (ψ k)) z = 0hglobalCircle:∀ z ∈ Γ, fderiv ℝ (fderiv ℝ F) z = 0hglobalActive:∀ z ∈ Γᶜ, ∀ (k : ℕ), z ∈ tsupport (ψ k) → fderiv ℝ (fderiv ℝ F) z = fderiv ℝ (fderiv ℝ (ψ k)) zhglobalZero:∀ z ∈ Γᶜ, (¬∃ k, z ∈ tsupport (ψ k)) → fderiv ℝ (fderiv ℝ F) z = 0hglobalNoActive:∀ (z : E), (¬∃ k, z ∈ tsupport (ψ k)) → fderiv ℝ (fderiv ℝ F) z = 0hu:¬∃ k, u ∈ tsupport (ψ k)hv:¬∃ k, v ∈ tsupport (ψ k)⊢ 0 ≤ ?m.1610by norm_num) hL) (Real.rpow_nonneg (norm_nonneg _) _))

Helper for eq:holder-hessian-bound: the endpoint-bump family has a uniform local square-root Hessian modulus after combining its scale and inverse-scale estimates.

theorem endpointBumpHessianHolder (curve : DFP.TwoLeg.SlowCurve) : ∃ εbar ∈ Set.Ioo (0 : ℝ) (1 / 4), ∃ L > 0, ∀ ε₀ ∈ Set.Ioc 0 εbar, let orbit := DFP.TwoPhaseOrbit.ofSlowCurve curve.shape curve.high ε₀ ∀ Clim : EuclideanSpace ℝ (Fin 2), Tendsto (fun j : ℕ ↦ (orbit.state j).center) atTop (𝓝 Clim) → ∀ Glim > 0, Tendsto (fun j : ℕ ↦ (orbit.state j).amplitude) atTop (𝓝 Glim) → ∀ k u v, ‖fderiv ℝ (fderiv ℝ (orbit.endpointBump Clim Glim k)) u - fderiv ℝ (fderiv ℝ (orbit.endpointBump Clim Glim k)) v‖ ≤ L * ‖u - v‖ ^ (1 / 2 : ℝ) := by obtain ⟨ηLip, hηLip, KLip, hKLip, hLip⟩ := endpointBumpHessianLipschitz curve obtain ⟨ηBounds, hηBounds, Kvalue, hKvalue, Kgradient, hKgradient, Khessian, hKhessian, hBounds⟩ := curve.endpointBumpUniformBounds obtain ⟨ηPos, hηPos, hPos⟩ := DFP.TwoLeg.slowCurveForwardOrbitPos curve.shape curve.high curve.isInvariant curve.shapeRemainder curve.highRemainder let εbar := min ηLip (min ηBounds ηPos) have hεbarPos : 0 < εbar := by dsimp only [εbar] exact lt_min hηLip.1 (lt_min hηBounds.1 hηPos) have hεbarLt : εbar < (1 / 4 : ℝ) := (min_le_left _ _).trans_lt hηLip.2 let L := Real.sqrt (2 * Khessian * KLip) have hL : 0 < L := by dsimp only [L] have hprod : 0 < 2 * Khessian * KLip := by positivity exact Real.sqrt_pos.2 hprod refine ⟨εbar, ⟨hεbarPos, hεbarLt⟩, L, hL, ?_⟩ intro ε₀ hε₀ dsimp only let orbit := DFP.TwoPhaseOrbit.ofSlowCurve curve.shape curve.high ε₀ have hεLip : ε₀ ∈ Set.Ioc 0 ηLip := ⟨hε₀.1, hε₀.2.trans (min_le_left _ _ )⟩ have hεBounds : ε₀ ∈ Set.Ioc 0 ηBounds := ⟨hε₀.1, hε₀.2.trans ((min_le_right _ _).trans (min_le_left _ _))⟩ have hεPos : ε₀ ∈ Set.Ioc 0 ηPos := ⟨hε₀.1, hε₀.2.trans ((min_le_right _ _).trans (min_le_right _ _))⟩ intro Clim hClim Glim hGlim hGlimTendsto k u v have hLipData := hLip ε₀ hεLip Clim hClim Glim hGlim hGlimTendsto k u v have hBoundsData := hBounds ε₀ hεBounds Clim hClim Glim hGlim hGlimTendsto have hcoord (j : ℕ) : (orbit.state j).ε = (DFP.TwoLeg.stateMap^[j] (ε₀, curve.shape ε₀, curve.high ε₀)).1 := by have hc := DFP.TwoPhaseOrbit.ofSlowCurve_coordinates curve.shape curve.high ε₀ j have hc' : (orbit.state j).coordinates = DFP.TwoLeg.stateMap^[j] (ε₀, curve.shape ε₀, curve.high ε₀) := by simpa only [orbit] using hc simpa only [DFP.TwoPhaseOrbit.State.coordinates_def] using congrArg Prod.fst hc' have hscalePos (j : ℕ) : 0 < (orbit.state j).ε := by rw [hcoord j] exact (hPos ε₀ hεPos j).1 let e := (orbit.state (k / 2)).ε have he : 0 < e := by simpa only [e] using hscalePos (k / 2) have hHessBound (z : EuclideanSpace ℝ (Fin 2)) : ‖fderiv ℝ (fderiv ℝ (orbit.endpointBump Clim Glim k)) z‖ ≤ Khessian * e := by by_cases hz : z ∈ tsupport (orbit.endpointBump Clim Glim k) · simpa only [e] using (hBoundsData k z hz).2.2 · have hz0 : iteratedFDeriv ℝ 2 (orbit.endpointBump Clim Glim k) z = iteratedFDeriv ℝ 2 (fun _ : EuclideanSpace ℝ (Fin 2) ↦ (0 : ℝ)) z := by rw [iteratedFDeriv_eq_zero_of_notMem_tsupport (orbit.endpointBump Clim Glim k) 2 hz] simp have hzero := fderiv_fderiv_eq_of_iteratedFDeriv_two_eq hz0 rw [hzero] Type mismatch: After simplification, term mul_nonneg (LT.lt.le hKhessian) (le_of_lt he) has type 0 ≤ Khessian * e but is expected to have type ‖0‖ ≤ Khessian * esimpa only [fderiv_fun_const, fderiv_zero, Pi.zero_apply, norm_zero, ge_iff_le] using (mul_nonneg hKhessian.le (le_of_lt he)) have hmin : ‖fderiv ℝ (fderiv ℝ (orbit.endpointBump Clim Glim k)) u - fderiv ℝ (fderiv ℝ (orbit.endpointBump Clim Glim k)) v‖ ≤ min (2 * Khessian * e) (KLip / e * ‖u - v‖) := by refine le_min ?_ ?_ · calc ‖fderiv ℝ (fderiv ℝ (orbit.endpointBump Clim Glim k)) u - fderiv ℝ (fderiv ℝ (orbit.endpointBump Clim Glim k)) v‖ ≤ ‖fderiv ℝ (fderiv ℝ (orbit.endpointBump Clim Glim k)) u‖ + ‖fderiv ℝ (fderiv ℝ (orbit.endpointBump Clim Glim k)) v‖ := Type mismatch norm_sub_le ?m.867 ?m.868 has type ‖@HSub.hSub ?m.865 ?m.865 ?m.865 (@instHSub ?m.865 SeminormedAddGroup.toAddGroup.toSub) ?m.867 ?m.868‖ ≤ ‖?m.867‖ + ‖?m.868‖ but is expected to have type ‖@HSub.hSub (EuclideanSpace ℝ (Fin 2) →L[ℝ] EuclideanSpace ℝ (Fin 2) →L[ℝ] ℝ) (EuclideanSpace ℝ (Fin 2) →L[ℝ] EuclideanSpace ℝ (Fin 2) →L[ℝ] ℝ) (EuclideanSpace ℝ (Fin 2) →L[ℝ] EuclideanSpace ℝ (Fin 2) →L[ℝ] ℝ) (@instHSub (EuclideanSpace ℝ (Fin 2) →L[ℝ] EuclideanSpace ℝ (Fin 2) →L[ℝ] ℝ) ContinuousLinearMap.sub) (fderiv ℝ (fderiv ℝ (orbit.endpointBump Clim Glim k)) u) (fderiv ℝ (fderiv ℝ (orbit.endpointBump Clim Glim k)) v)‖ ≤ ‖fderiv ℝ (fderiv ℝ (orbit.endpointBump Clim Glim k)) u‖ + ‖fderiv ℝ (fderiv ℝ (orbit.endpointBump Clim Glim k)) v‖norm_sub_le _ _ _ ≤ Khessian * e + Khessian * e := add_le_add (hHessBound u) (hHessBound v) _ = 2 * Khessian * e := by ring · exact hLipData calc ‖fderiv ℝ (fderiv ℝ (orbit.endpointBump Clim Glim k)) u - fderiv ℝ (fderiv ℝ (orbit.endpointBump Clim Glim k)) v‖ ≤ min (2 * Khessian * e) (KLip / e * ‖u - v‖) := hmin _ ≤ Real.sqrt (2 * Khessian * KLip) * Real.sqrt ‖u - v‖ := by exact min_scaled_le_sqrt (A := Khessian) (B := KLip) (e := e) (d := ‖u - v‖) hKhessian.le hKLip.le he (norm_nonneg _) _ = L * ‖u - v‖ ^ (1 / 2 : ℝ) := by dsimp only [L] simp only [Real.sqrt_eq_rpow]

Helper for eq:holder-hessian-bound: the Euclidean-plane Hessian norm of a difference agrees with the norm of the corresponding difference of second Fréchet derivatives.

theorem norm_euclideanHessian_sub_eq_norm_secondFDeriv_sub (f : EuclideanSpace ℝ (Fin 2) → ℝ) (x y : EuclideanSpace ℝ (Fin 2)) : ‖EuclideanPlane.hessian f x - EuclideanPlane.hessian f y‖ = ‖fderiv ℝ (fderiv ℝ f) x - fderiv ℝ (fderiv ℝ f) y‖ := by let E := EuclideanSpace ℝ (Fin 2) let R : StrongDual ℝ E →L[ℝ] E := (InnerProductSpace.toDual ℝ E).symm.toContinuousLinearMap have hRiesz (z : E) : fderiv ℝ (gradient f) z = R ∘L fderiv ℝ (fderiv ℝ f) z := by unfold gradient have hfun : (fun w ↦ (InnerProductSpace.toDual ℝ E).symm (fderiv ℝ f w)) = (InnerProductSpace.toDual ℝ E).symm ∘ fderiv ℝ f := rfl rw [hfun] exact LinearIsometryEquiv.comp_fderiv (InnerProductSpace.toDual ℝ E).symm rw [EuclideanPlane.hessian_def f x, EuclideanPlane.hessian_def f y, hRiesz x, hRiesz y] have hcomp : R ∘L fderiv ℝ (fderiv ℝ f) x - R ∘L fderiv ℝ (fderiv ℝ f) y = R ∘L (fderiv ℝ (fderiv ℝ f) x - fderiv ℝ (fderiv ℝ f) y) := by ext u v simp [ContinuousLinearMap.comp_apply] rw [hcomp] exact LinearIsometry.norm_toContinuousLinearMap_comp (InnerProductSpace.toDual ℝ E).symm.toLinearIsometry

Helper for eq:holder-hessian-bound: the full endpoint-bump correction has a global one-half Hölder Hessian modulus after zero-extension across the limit circle.

theorem bumpCorrectionHessianHolder (curve : DFP.TwoLeg.SlowCurve) : ∃ εbar ∈ Set.Ioo (0 : ℝ) (1 / 4), ∃ L > 0, ∀ ε₀ ∈ Set.Ioc 0 εbar, let orbit := DFP.TwoPhaseOrbit.ofSlowCurve curve.shape curve.high ε₀ ∀ Clim : EuclideanSpace ℝ (Fin 2), Tendsto (fun j : ℕ ↦ (orbit.state j).center) atTop (𝓝 Clim) → ∀ Glim > 0, Tendsto (fun j : ℕ ↦ (orbit.state j).amplitude) atTop (𝓝 Glim) → ∀ u v, ‖EuclideanPlane.hessian (orbit.bumpCorrection Clim Glim) u - EuclideanPlane.hessian (orbit.bumpCorrection Clim Glim) v‖ ≤ L * ‖u - v‖ ^ (1 / 2 : ℝ) := by obtain ⟨ηRadius, hηRadius, hRadius⟩ := curve.interpolationRadius_pos obtain ⟨ηShrink, hηShrink, hShrink⟩ := curve.interpolationRadiusTendstoZero obtain ⟨ηCluster, hηCluster, hCluster⟩ := curve.endpointClusterPt_mem_limitCircle obtain ⟨ηBalls, hηBalls, hBalls⟩ := curve.pairwiseDisjointInterpolationClosedBalls obtain ⟨ηCircle, hηCircle, hCircle⟩ := curve.interpolationClosedBallDisjointLimitCircle obtain ⟨ηJets, hηJets, hJets⟩ := curve.endpointBumpSecondOrderJetsVanish obtain ⟨ηHolder, hηHolder, Llocal, hLlocal, hHolder⟩ := endpointBumpHessianHolder curve let εbar := min ηRadius (min ηShrink (min ηCluster (min ηBalls (min ηCircle (min ηJets ηHolder))))) have hεbarPos : 0 < εbar := by dsimp only [εbar] exact lt_min hηRadius.1 (lt_min hηShrink.1 (lt_min hηCluster.1 (lt_min hηBalls.1 (lt_min hηCircle.1 (lt_min hηJets.1 hηHolder.1))))) have hεbarLt : εbar < (1 / 4 : ℝ) := (min_le_left _ _).trans_lt hηRadius.2 have hεbarLeRadius : εbar ≤ ηRadius := by dsimp only [εbar] exact min_le_left _ _ have hεbarLeShrink : εbar ≤ ηShrink := by dsimp only [εbar] exact (min_le_right _ _).trans (min_le_left _ _) have hεbarLeCluster : εbar ≤ ηCluster := by dsimp only [εbar] exact (min_le_right _ _).trans ((min_le_right _ _).trans (min_le_left _ _)) have hεbarLeBalls : εbar ≤ ηBalls := by dsimp only [εbar] exact (min_le_right _ _).trans ((min_le_right _ _).trans ((min_le_right _ _).trans (min_le_left _ _))) have hεbarLeCircle : εbar ≤ ηCircle := by dsimp only [εbar] exact (min_le_right _ _).trans ((min_le_right _ _).trans ((min_le_right _ _).trans ((min_le_right _ _).trans (min_le_left _ _)))) have hεbarLeJets : εbar ≤ ηJets := by dsimp only [εbar] exact (((((min_le_right _ _).trans (min_le_right _ _)).trans (min_le_right _ _)).trans (min_le_right _ _)).trans (min_le_right _ _)).trans (min_le_left _ _) have hεbarLeHolder : εbar ≤ ηHolder := by dsimp only [εbar] exact (((((min_le_right _ _).trans (min_le_right _ _)).trans (min_le_right _ _)).trans (min_le_right _ _)).trans (min_le_right _ _)).trans (min_le_right _ _) let L := 2 * Llocal have hL : 0 < L := by dsimp only [L] linarith refine ⟨εbar, ⟨hεbarPos, hεbarLt⟩, L, hL, ?_⟩ intro ε₀ hε₀ dsimp only let orbit := DFP.TwoPhaseOrbit.ofSlowCurve curve.shape curve.high ε₀ have hεRadius : ε₀ ∈ Set.Ioc 0 ηRadius := ⟨hε₀.1, hε₀.2.trans hεbarLeRadius⟩ have hεShrink : ε₀ ∈ Set.Ioc 0 ηShrink := ⟨hε₀.1, hε₀.2.trans hεbarLeShrink⟩ have hεCluster : ε₀ ∈ Set.Ioc 0 ηCluster := ⟨hε₀.1, hε₀.2.trans hεbarLeCluster⟩ have hεBalls : ε₀ ∈ Set.Ioc 0 ηBalls := ⟨hε₀.1, hε₀.2.trans hεbarLeBalls⟩ have hεCircle : ε₀ ∈ Set.Ioc 0 ηCircle := ⟨hε₀.1, hε₀.2.trans hεbarLeCircle⟩ have hεJets : ε₀ ∈ Set.Ioc 0 ηJets := ⟨hε₀.1, hε₀.2.trans hεbarLeJets⟩ have hεHolder : ε₀ ∈ Set.Ioc 0 ηHolder := ⟨hε₀.1, hε₀.2.trans hεbarLeHolder⟩ intro Clim hClim Glim hGlim hGlimTendsto u v have hRadiusPos : ∀ k : ℕ, 0 < orbit.interpolationRadius Clim Glim k := by simpa only [orbit] using hRadius ε₀ hεRadius Clim hClim Glim hGlim hGlimTendsto have hRadiusZero : Tendsto (fun k : ℕ ↦ orbit.interpolationRadius Clim Glim k) atTop (𝓝 0) := by simpa only [orbit] using hShrink ε₀ hεShrink Clim hClim Glim hGlim hGlimTendsto have hBallsData : Set.univ.PairwiseDisjoint (fun k : ℕ ↦ Metric.closedBall (orbit.endpoint k) (orbit.interpolationRadius Clim Glim k)) := by simpa only [orbit] using hBalls ε₀ hεBalls Clim hClim Glim hGlim hGlimTendsto have hCircleData : ∀ k : ℕ, Disjoint (Metric.closedBall (orbit.endpoint k) (orbit.interpolationRadius Clim Glim k)) (DFP.TwoPhaseOrbit.limitCircle Clim Glim) := by simpa only [orbit] using hCircle ε₀ hεCircle Clim hClim Glim hGlim hGlimTendsto let Γ : Set (EuclideanSpace ℝ (Fin 2)) := DFP.TwoPhaseOrbit.limitCircle Clim Glim have hClosed : IsClosed Γ := by simpa only [Γ] using DFP.TwoPhaseOrbit.isClosed_limitCircle Clim Glim hGlim have hClusterData : ∀ y, MapClusterPt y atTop orbit.endpoint → y ∈ Γ := by intro y hy simpa only [Γ, orbit] using (hCluster ε₀ hεCluster Clim hClim Glim hGlim hGlimTendsto y hy) have hSupport (k : ℕ) : tsupport (orbit.endpointBump Clim Glim k) ⊆ Metric.closedBall (orbit.endpoint k) (orbit.interpolationRadius Clim Glim k) := DFP.TwoPhaseOrbit.endpointBump_tsupport_subset_interpolationClosedBall orbit Clim Glim hRadiusPos k have hTwoLe : (2 : WithTop ℕ∞) ≤ ∞ := by have hTwoNat : (2 : ℕ∞) ≤ ⊤ := le_top exact WithTop.coe_le_coe.mpr hTwoNat have hSmooth (k : ℕ) : ContDiff ℝ 2 (orbit.endpointBump Clim Glim k) := (DFP.TwoPhaseOrbit.contDiff_endpointBump orbit Clim Glim k).of_le hTwoLe have hJetsData := hJets ε₀ hεJets Clim hClim Glim hGlim hGlimTendsto have hValueDecay : ∀ η > 0, ∃ δ > 0, ∀ k z, z ∈ Γᶜ → z ∈ tsupport (orbit.endpointBump Clim Glim k) → Metric.infDist z Γ < δ → ‖orbit.endpointBump Clim Glim k z‖ / Metric.infDist z Γ ^ 2 < η := by intro η hη obtain ⟨δ, hδ, hbound⟩ := hJetsData η hη refine ⟨δ, hδ, ?_⟩ intro k z _ hz hzδ simpa only [Γ, orbit] using (hbound k z hz hzδ).1 have hGradientDecay : ∀ η > 0, ∃ δ > 0, ∀ k z, z ∈ Γᶜ → z ∈ tsupport (orbit.endpointBump Clim Glim k) → Metric.infDist z Γ < δ → ‖fderiv ℝ (orbit.endpointBump Clim Glim k) z‖ / Metric.infDist z Γ < η := by intro η hη obtain ⟨δ, hδ, hbound⟩ := hJetsData η hη refine ⟨δ, hδ, ?_⟩ intro k z _ hz hzδ simpa only [Γ, orbit] using (hbound k z hz hzδ).2.1 have hValueDecayIterated : ∀ η > 0, ∃ δ > 0, ∀ k z, z ∈ Γᶜ → z ∈ tsupport (orbit.endpointBump Clim Glim k) → Metric.infDist z Γ < δ → ‖iteratedFDeriv ℝ 0 (orbit.endpointBump Clim Glim k) z‖ / Metric.infDist z Γ ^ 2 < η := by intro η hη obtain ⟨δ, hδ, hbound⟩ := hValueDecay η hη refine ⟨δ, hδ, ?_⟩ intro k z hzΓ hzk hzδ simpa only [norm_iteratedFDeriv_zero] using hbound k z hzΓ hzk hzδ have hGradientDecayIterated : ∀ η > 0, ∃ δ > 0, ∀ k z, z ∈ Γᶜ → z ∈ tsupport (orbit.endpointBump Clim Glim k) → Metric.infDist z Γ < δ → ‖iteratedFDeriv ℝ 1 (orbit.endpointBump Clim Glim k) z‖ / Metric.infDist z Γ ^ 1 < η := by intro η hη obtain ⟨δ, hδ, hbound⟩ := hGradientDecay η hη refine ⟨δ, hδ, ?_⟩ intro k z hzΓ hzk hzδ simpa only [norm_iteratedFDeriv_one, pow_one] using hbound k z hzΓ hzk hzδ have hvalueIterated := tendsto_norm_iteratedFDeriv_finsum_div_infDist_pow 2 0 2 Γ orbit.endpoint (fun k ↦ orbit.interpolationRadius Clim Glim k) (fun k ↦ orbit.endpointBump Clim Glim k) hClosed hClusterData (fun k ↦ (hRadiusPos k).le) hRadiusZero hBallsData hSupport hSmooth (by omega) hValueDecayIterated have hderivIterated := tendsto_norm_iteratedFDeriv_finsum_div_infDist_pow 2 1 1 Γ orbit.endpoint (fun k ↦ orbit.interpolationRadius Clim Glim k) (fun k ↦ orbit.endpointBump Clim Glim k) hClosed hClusterData (fun k ↦ (hRadiusPos k).le) hRadiusZero hBallsData hSupport hSmooth (by omega) hGradientDecayIterated have hvalue : Tendsto (fun z ↦ ‖∑ᶠ k, orbit.endpointBump Clim Glim k z‖ / Metric.infDist z Γ ^ 2) (Filter.comap (fun z ↦ Metric.infDist z Γ) (𝓝 0) ⊓ Filter.principal Γᶜ) (𝓝 0) := by simpa only [norm_iteratedFDeriv_zero] using hvalueIterated have hderiv : Tendsto (fun z ↦ ‖fderiv ℝ (fun w ↦ ∑ᶠ k, orbit.endpointBump Clim Glim k w) z‖ / Metric.infDist z Γ) (Filter.comap (fun z ↦ Metric.infDist z Γ) (𝓝 0) ⊓ Filter.principal Γᶜ) (𝓝 0) := by simpa only [norm_iteratedFDeriv_one, pow_one] using hderivIterated have hsmoothOutside : ContDiffOn ℝ 2 (fun z ↦ ∑ᶠ k, orbit.endpointBump Clim Glim k z) Γᶜ := contDiffOn_finsum_outside 2 Γ orbit.endpoint (fun k ↦ orbit.interpolationRadius Clim Glim k) (fun k ↦ orbit.endpointBump Clim Glim k) hClosed hClusterData hRadiusZero hSupport hSmooth have hFderivEq := IsClosed.fderiv_fderiv_indicator_compl Γ (fun z ↦ ∑ᶠ k, orbit.endpointBump Clim Glim k z) hClosed hsmoothOutside hvalue hderiv have hFunctionEq : Γᶜ.indicator (fun z ↦ ∑ᶠ k, orbit.endpointBump Clim Glim k z) = orbit.bumpCorrection Clim Glim := by funext z by_cases hz : z ∈ Γ · have hzComplement : z ∉ Γᶜ := by simpa only [Set.mem_compl_iff, not_not] using hz rw [Set.indicator_of_notMem hzComplement] have hZero (k : ℕ) : orbit.endpointBump Clim Glim k z = 0 := by by_contra hk have hzSupport : z ∈ tsupport (orbit.endpointBump Clim Glim k) := subset_tsupport (orbit.endpointBump Clim Glim k) hk have hzBall := hSupport k hzSupport exact Set.disjoint_left.mp (hCircleData k) hzBall hz rw [DFP.TwoPhaseOrbit.bumpCorrection_apply] exact (finsum_eq_zero_of_forall_eq_zero (fun k ↦ hZero k)).symm · have hzComplement : z ∈ Γᶜ := by simpa only [Set.mem_compl_iff] using hz rw [Set.indicator_of_mem hzComplement] exact (DFP.TwoPhaseOrbit.bumpCorrection_apply orbit Clim Glim z).symm have hHessianEq : fderiv ℝ (fderiv ℝ (orbit.bumpCorrection Clim Glim)) = Γᶜ.indicator (fderiv ℝ (fderiv ℝ (fun z ↦ ∑ᶠ k, orbit.endpointBump Clim Glim k z))) := by calc fderiv ℝ (fderiv ℝ (orbit.bumpCorrection Clim Glim)) = fderiv ℝ (fderiv ℝ (Γᶜ.indicator (fun z ↦ ∑ᶠ k, orbit.endpointBump Clim Glim k z))) := by rw [hFunctionEq] _ = _ := hFderivEq have hdisjointΓ : ∀ k, Disjoint (tsupport (orbit.endpointBump Clim Glim k)) Γ := by intro k exact Set.disjoint_of_subset_left (hSupport k) (hCircleData k) have hlocal : ∀ k a b, ‖fderiv ℝ (fderiv ℝ (orbit.endpointBump Clim Glim k)) a - fderiv ℝ (fderiv ℝ (orbit.endpointBump Clim Glim k)) b‖ ≤ Llocal * ‖a - b‖ ^ (1 / 2 : ℝ) := by intro k a b simpa only [orbit] using hHolder ε₀ hεHolder Clim hClim Glim hGlim hGlimTendsto k a b have hgeneric := holder_of_disjoint_finsum_hessian Γ orbit.endpoint (fun k ↦ orbit.interpolationRadius Clim Glim k) (fun k ↦ orbit.endpointBump Clim Glim k) (orbit.bumpCorrection Clim Glim) Llocal hClosed hClusterData hRadiusZero hBallsData hSupport hdisjointΓ hHessianEq hlocal hLlocal.le u v calc ‖EuclideanPlane.hessian (orbit.bumpCorrection Clim Glim) u - EuclideanPlane.hessian (orbit.bumpCorrection Clim Glim) v‖ = ‖fderiv ℝ (fderiv ℝ (orbit.bumpCorrection Clim Glim)) u - fderiv ℝ (fderiv ℝ (orbit.bumpCorrection Clim Glim)) v‖ := norm_euclideanHessian_sub_eq_norm_secondFDeriv_sub (orbit.bumpCorrection Clim Glim) u v _ ≤ 2 * Llocal * ‖u - v‖ ^ (1 / 2 : ℝ) := hgeneric _ = L * ‖u - v‖ ^ (1 / 2 : ℝ) := by rfl

Helper for eq:holder-hessian-bound: adding the translated quadratic preserves the one-half Hölder modulus of the concrete endpoint-bump Hessian.

theorem realizedObjectiveHessianHolder (curve : DFP.TwoLeg.SlowCurve) : ∃ εbar ∈ Set.Ioo (0 : ℝ) (1 / 4), ∃ L > 0, ∀ ε₀ ∈ Set.Ioc 0 εbar, let orbit := DFP.TwoPhaseOrbit.ofSlowCurve curve.shape curve.high ε₀ ∀ Clim : EuclideanSpace ℝ (Fin 2), Tendsto (fun j : ℕ ↦ (orbit.state j).center) atTop (𝓝 Clim) → ∀ Glim > 0, Tendsto (fun j : ℕ ↦ (orbit.state j).amplitude) atTop (𝓝 Glim) → ∀ u v, ‖EuclideanPlane.hessian (orbit.realizedObjective Clim Glim) u - EuclideanPlane.hessian (orbit.realizedObjective Clim Glim) v‖ ≤ L * ‖u - v‖ ^ (1 / 2 : ℝ) := by obtain ⟨ηHolder, hηHolder, L, hL, hHolder⟩ := bumpCorrectionHessianHolder curve obtain ⟨ηRegular, hηRegular, hRegular⟩ := curve.bumpCorrectionContDiffTwoAndZeroJets let εbar := min ηHolder ηRegular have hεbarPos : 0 < εbar := by dsimp only [εbar] exact lt_min hηHolder.1 hηRegular.1 have hεbarLt : εbar < (1 / 4 : ℝ) := (min_le_left _ _).trans_lt hηHolder.2 have hεbarLeHolder : εbar ≤ ηHolder := by dsimp only [εbar] exact min_le_left _ _ have hεbarLeRegular : εbar ≤ ηRegular := by dsimp only [εbar] exact min_le_right _ _ refine ⟨εbar, ⟨hεbarPos, hεbarLt⟩, L, hL, ?_⟩ intro ε₀ hε₀ dsimp only let orbit := DFP.TwoPhaseOrbit.ofSlowCurve curve.shape curve.high ε₀ have hεHolder : ε₀ ∈ Set.Ioc 0 ηHolder := ⟨hε₀.1, hε₀.2.trans hεbarLeHolder⟩ have hεRegular : ε₀ ∈ Set.Ioc 0 ηRegular := ⟨hε₀.1, hε₀.2.trans hεbarLeRegular⟩ intro Clim hClim Glim hGlim hGlimTendsto u v have hPsi : ContDiff ℝ 2 (orbit.bumpCorrection Clim Glim) := by simpa only [orbit] using (hRegular ε₀ hεRegular Clim hClim Glim hGlim hGlimTendsto).1 have hobj : orbit.realizedObjective Clim Glim = (fun z ↦ (1 / 2 : ℝ) * ‖z - Clim‖ ^ 2 + orbit.bumpCorrection Clim Glim z) := by funext z exact DFP.TwoPhaseOrbit.realizedObjective_apply orbit Clim Glim z have hHessianObj (z : EuclideanSpace ℝ (Fin 2)) : EuclideanPlane.hessian (orbit.realizedObjective Clim Glim) z = (1 : EuclideanSpace ℝ (Fin 2) →L[ℝ] EuclideanSpace ℝ (Fin 2)) + EuclideanPlane.hessian (orbit.bumpCorrection Clim Glim) z := by calc EuclideanPlane.hessian (orbit.realizedObjective Clim Glim) z = fderiv ℝ (gradient (orbit.realizedObjective Clim Glim)) z := by rw [EuclideanPlane.hessian_def] _ = fderiv ℝ (gradient (fun y ↦ (1 / 2 : ℝ) * ‖y - Clim‖ ^ 2 + orbit.bumpCorrection Clim Glim y)) z := by rw [hobj] _ = 1 + fderiv ℝ (gradient (orbit.bumpCorrection Clim Glim)) z := HessianPerturbation.fderiv_gradient_halfNormSq_sub_add Clim (orbit.bumpCorrection Clim Glim) z hPsi _ = (1 : EuclideanSpace ℝ (Fin 2) →L[ℝ] EuclideanSpace ℝ (Fin 2)) + EuclideanPlane.hessian (orbit.bumpCorrection Clim Glim) z := by rw [EuclideanPlane.hessian_def] rw [hHessianObj u, hHessianObj v] simpa only [add_sub_add_left_eq_sub] using (hHolder ε₀ hεHolder Clim hClim Glim hGlim hGlimTendsto u v)

Helper for eq:holder-hessian-bound: the canonical invariant slow curve supplied by the two-phase construction admits the concrete global one-half Hölder Hessian bound for its realized objective.

theorem existsRealizedObjectiveHessianHolder : ∃ curve : DFP.TwoLeg.SlowCurve, ∃ εbar ∈ Set.Ioo (0 : ℝ) (1 / 4), ∃ L > 0, ∀ ε₀ ∈ Set.Ioc 0 εbar, let orbit := DFP.TwoPhaseOrbit.ofSlowCurve curve.shape curve.high ε₀ ∀ Clim : EuclideanSpace ℝ (Fin 2), Tendsto (fun j : ℕ ↦ (orbit.state j).center) atTop (𝓝 Clim) → ∀ Glim > 0, Tendsto (fun j : ℕ ↦ (orbit.state j).amplitude) atTop (𝓝 Glim) → ∀ u v, ‖EuclideanPlane.hessian (orbit.realizedObjective Clim Glim) u - EuclideanPlane.hessian (orbit.realizedObjective Clim Glim) v‖ ≤ L * ‖u - v‖ ^ (1 / 2 : ℝ) := by obtain ⟨p, h, _, _, _, hInvariant, hpJet, hhJet, _⟩ := DFP.TwoLeg.exists_localForwardInvariantSlowCurve let curve := DFP.TwoLeg.SlowCurve.ofAsymptotics p h hpJet hhJet hInvariant obtain ⟨εbar, hεbar, L, hL, hHolder⟩ := realizedObjectiveHessianHolder curve exact ⟨curve, εbar, hεbar, L, hL, hHolder⟩
end DFP.PlanarConvergence