theorem compact_segment_diameter_achieved (φ : ℝ → ℝ × ℝ) (h_cont : Continuous φ) (A B : ℝ) (h_le : A ≤ B) : ∃ (x : ℝ) (y : ℝ), A ≤ x ∧ x ≤ B ∧ A ≤ y ∧ y ≤ B ∧ Metric.ediam (φ '' Set.Icc A B) = edist (φ x) (φ y)

The goal of this file is to prove that for any continuous parameterization φ, its 1D Hausdorff measure is bounded strictly by its sequential trajectory variation. Instead of relying on unavailable arc-length reparameterization calculus, we use Topological Sequence Covers using compactness and ediam.

For any continuous curve segment φ([A, B]), the geometric diameter of that segment is physically achieved by two explicit sequence points x, y ∈ [A, B].

theorem diameter_bounds_monotone_sequence (u_n u_next x y : ℝ) (hx : u_n ≤ x ∧ x ≤ u_next) (hy : u_n ≤ y ∧ y ≤ u_next) : u_n ≤ min x y ∧ min x y ≤ max x y ∧ max x y ≤ u_next

Mapping extracted geometric extrema limit points x and y back into ordered sorting dynamically forms a monotonic sequence subset perfectly bounded inside eVariationOn boundaries!

theorem delta_cover_sequence (δ : ℝ) (hδ : 0 < δ) : ∃ (u : ℕ → ℝ), u 0 = 0 ∧ (∀ (i : ℕ), u i ≤ u (i + 1)) ∧ (∀ (i : ℕ), u (i + 1) - u i ≤ δ) ∧ (∀ (i : ℕ), 0 ≤ u i) ∧ (∀ (i : ℕ), u i ≤ 1) ∧ ∃ (N : ℕ), ∀ i ≥ N, u i = 1

Explicit sequence generating function slicing the [0, 1] continuum into consecutive sub-intervals where no interval expands geometrically wider than generic metric step bounds δ.

theorem ediam_le_eVariationOn (φ : ℝ → ℝ × ℝ) (h_cont : Continuous φ) (A B : ℝ) (h_le : A ≤ B) : Metric.ediam (φ '' Set.Icc A B) ≤ eVariationOn φ (Set.Icc A B)

Verifies that the geometric maximum diameter spanning any continuous parameter space mathematically can never exceed the sequence supremum distance tracked sequentially by its total variation sum limits.

theorem eVariationOn_sequence_sum_bounded (φ : ℝ → ℝ × ℝ) (u : ℕ → ℝ) (hu_mono : ∀ (i : ℕ), u i ≤ u (i + 1)) (N : ℕ) : ∑ i ∈ Finset.range N, eVariationOn φ (Set.Icc (u i) (u (i + 1))) ≤ eVariationOn φ (Set.Icc (u 0) (u N))

Resolves continuous variation summations iteratively across parameter space topologies, proving that bounded sequence closures strictly collapse perfectly inside sequential monotonic metric loops.

theorem cover_sequence_subset (φ : ℝ → ℝ × ℝ) (u : ℕ → ℝ) (h0 : u 0 = 0) (hM : ∀ (i : ℕ), u i ≤ u (i + 1)) (N : ℕ) (hN : ∀ i ≥ N, u i = 1) : φ '' Set.Icc 0 1 ⊆ ⋃ (n : ℕ), φ '' Set.Icc (u n) (u (n + 1))

Mathematically guarantees that the sequential extrema mappings defined by the delta steps provide a complete, unbroken metric cover of the subset sequence space [0, 1].

theorem uniform_segment_diameter_bounded (φ : ℝ → ℝ × ℝ) (h_cont : Continuous φ) (u : ℕ → ℝ) (n : ℕ) (δ r_eff : ℝ) (hδ_pos : 0 < δ) (hu_mono : ∀ (i : ℕ), u i ≤ u (i + 1)) (hu_bound : ∀ (i : ℕ), u (i + 1) - u i ≤ δ / 2) (hu_mem : ∀ (i : ℕ), u i ∈ Set.Icc 0 1) (h_uc : ∀ x ∈ Set.Icc 0 1, ∀ y ∈ Set.Icc 0 1, dist x y < δ → dist (φ x) (φ y) < r_eff) : Metric.ediam (φ '' Set.Icc (u n) (u (n + 1))) ≤ ENNReal.ofReal r_eff

Verifies that sequence boundaries built at exactly half-metric constraints force inner segment topological bounds to satisfy open strict measure closures unconditionally.

theorem continuous_forces_uniform_geometric_bounds (φ : ℝ → ℝ × ℝ) (h_cont : Continuous φ) (ε : ℝ) : ε > 0 → ∃ δ > 0, ∀ x ∈ Set.Icc 0 1, ∀ y ∈ Set.Icc 0 1, dist x y < δ → dist (φ x) (φ y) < ε

By the topological consequence of continuous mappings on compact sequences, any distance limitation ε corresponds necessarily to a rigorous sequential chunk constraint δ.

theorem continuous_variation_bounds_hausdorff (φ : ℝ → ℝ × ℝ) (h_cont : Continuous φ) : (MeasureTheory.Measure.hausdorffMeasure 1) (φ '' Set.Icc 0 1) ≤ eVariationOn φ (Set.Icc 0 1)