stable_existence_requires_discrete

formal source

 492    Note: RSExists_stable x config_space means x has defect 0 and is isolated
 493    from config_space. This doesn't require x ∈ config_space, but in practice
 494    we're interested in cases where x = 1 (the unique defect minimum). -/
 495theorem stable_existence_requires_discrete :
 496    (∃ x config_space, RSExists_stable x config_space) →
 497    ∃ config_space : Set ℝ, ∃ x, RSExists_stable x config_space := by
 498  intro ⟨x, config_space, hstable⟩
 499  exact ⟨config_space, x, hstable⟩
 500
 501/-! ## Summary -/
 502
 503/-- **THE DISCRETENESS FORCING PRINCIPLE**
 504
 505    The cost functional J(x) = ½(x + x⁻¹) - 1 forces discrete ontology:
 506
 507    1. J has a unique minimum at x = 1 with J(1) = 0
 508    2. J''(1) = 1 sets the minimum "step cost" for discrete configurations
 509    3. In continuous spaces, configurations drift (infinitesimal cost for infinitesimal perturbation)
 510    4. In discrete spaces, configurations are trapped (finite cost for any step)
 511
 512    Therefore: **Stable existence (RSExists) requires discrete configuration space**
 513
 514    This is Level 2 of the forcing chain:
 515    Composition law → J unique → Discreteness forced → Ledger → φ → D=3 → physics
 516-/
 517theorem discreteness_forcing_principle :
 518    (∀ x : ℝ, 0 < x → defect x ≥ 0) ∧                    -- J ≥ 0
 519    (∀ x : ℝ, 0 < x → (defect x = 0 ↔ x = 1)) ∧         -- Unique minimum
 520    (deriv (deriv J_log) 0 = 1) ∧                        -- Curvature = 1
 521    (∀ x : ℝ, 0 < x → defect x = 0 →                     -- Continuous → no isolation
 522      ∀ ε > 0, ∃ y : ℝ, y ≠ x ∧ |y - x| < ε) :=
 523  ⟨fun x hx => defect_nonneg hx,
 524   fun x hx => defect_zero_iff_one hx,
 525   J_log_second_deriv_at_zero,