IndisputableMonolith.Foundation.HierarchyRealizationObstruction

   1import Mathlib
   2import IndisputableMonolith.Foundation.ClosedObservableFramework
   3
   4namespace IndisputableMonolith
   5namespace Foundation
   6namespace HierarchyRealizationObstruction
   7
   8open ClosedFramework
   9
  10/-!
  11# Obstruction: `ClosedObservableFramework` Alone Does Not Force Hierarchy Fields
  12
  13This module formalizes the key honesty check for the T5→T6 bridge:
  14the current earlier primitive `ClosedObservableFramework` is too weak
  15to derive either
  16
  17* `ratio_self_similar`, or
  18* `additive_posting`
  19
  20for the orbit-defined levels `k ↦ r (T^[k] baseState)`.
  21
  22We exhibit an explicit finite counterexample. This means any honest
  23derivation must use stronger earlier structure than `ClosedObservableFramework`
  24alone.
  25-/
  26
  27/-- Any map `ℝ → Bool` fails to be injective. -/
  28theorem no_injective_real_to_bool (embed : ℝ → Bool) :
  29    ¬ Function.Injective embed := by
  30  intro h_inj
  31  by_cases h01 : embed 0 = embed 1
  32  · exact zero_ne_one (h_inj h01)
  33  · have hrep : embed 0 = embed 2 ∨ embed 1 = embed 2 := by
  34      cases h0 : embed 0 <;> cases h1 : embed 1 <;> cases h2 : embed 2 <;> simp_all
  35    rcases hrep with h02 | h12
  36    · have : (0 : ℝ) = 2 := h_inj h02
  37      norm_num at this
  38    · have : (1 : ℝ) = 2 := h_inj h12
  39      norm_num at this
  40
  41/-- A finite closed-observable framework whose orbit alternates between
  42observable values `1` and `2`. -/
  43def boolFramework : ClosedObservableFramework where
  44  S := Bool
  45  T := not
  46  r := fun b => if b then 2 else 1
  47  r_pos := by
  48    intro b
  49    cases b <;> norm_num
  50  nontrivial := by
  51    refine ⟨false, true, ?_⟩
  52    norm_num
  53  S_countable := by
  54    refine ⟨fun n => if n % 2 = 0 then false else true, ?_⟩
  55    intro b
  56    cases b
  57    · refine ⟨0, ?_⟩
  58      simp
  59    · refine ⟨1, ?_⟩
  60      simp
  61  no_continuous_moduli := no_injective_real_to_bool
  62  charge := fun _ => 0
  63  charge_conserved := by
  64    intro s
  65    rfl
  66
  67/-- The base state used for the obstruction. -/
  68def baseState : boolFramework.S := false
  69
  70/-- The orbit-defined levels of the counterexample framework. -/
  71def orbitLevels (k : ℕ) : ℝ := boolFramework.r (boolFramework.T^[k] baseState)
  72
  73@[simp] theorem orbitLevels_zero : orbitLevels 0 = 1 := by
  74  simp [orbitLevels, boolFramework, baseState]
  75
  76@[simp] theorem orbitLevels_one : orbitLevels 1 = 2 := by
  77  simp [orbitLevels, boolFramework, baseState]
  78
  79@[simp] theorem orbitLevels_two : orbitLevels 2 = 1 := by
  80  simp [orbitLevels, boolFramework, baseState]
  81
  82/-- The counterexample orbit does not satisfy ratio self-similarity. -/
  83theorem orbit_not_ratio_self_similar :
  84    ¬ (∀ k,
  85      orbitLevels (k + 2) / orbitLevels (k + 1) =
  86        orbitLevels (k + 1) / orbitLevels k) := by
  87  intro h
  88  have h0 := h 0
  89  simp [orbitLevels, boolFramework, baseState] at h0
  90  norm_num at h0
  91
  92/-- The counterexample orbit does not satisfy additive posting. -/
  93theorem orbit_not_additive_posting :
  94    ¬ (orbitLevels 2 = orbitLevels 1 + orbitLevels 0) := by
  95  simp [orbitLevels, boolFramework, baseState]
  96
  97/-- Therefore `ClosedObservableFramework` alone cannot force
  98`ratio_self_similar`. -/
  99theorem closedFramework_does_not_force_ratio_self_similar :
 100    ∃ (F : ClosedObservableFramework) (base : F.S),
 101      ¬ (∀ k,
 102        F.r (F.T^[k + 2] base) / F.r (F.T^[k + 1] base) =
 103          F.r (F.T^[k + 1] base) / F.r (F.T^[k] base)) := by
 104  exact ⟨boolFramework, baseState, orbit_not_ratio_self_similar⟩
 105
 106/-- Therefore `ClosedObservableFramework` alone cannot force
 107`additive_posting`. -/
 108theorem closedFramework_does_not_force_additive_posting :
 109    ∃ (F : ClosedObservableFramework) (base : F.S),
 110      ¬ (F.r (F.T^[2] base) = F.r (F.T^[1] base) + F.r base) := by
 111  exact ⟨boolFramework, baseState, orbit_not_additive_posting⟩
 112
 113/-- Combined obstruction theorem: the earlier primitive layer admits
 114models where both target fields fail. -/
 115theorem closedFramework_does_not_force_realizedHierarchy_fields :
 116    ∃ (F : ClosedObservableFramework) (base : F.S),
 117      (¬ (∀ k,
 118        F.r (F.T^[k + 2] base) / F.r (F.T^[k + 1] base) =
 119          F.r (F.T^[k + 1] base) / F.r (F.T^[k] base))) ∧
 120      (¬ (F.r (F.T^[2] base) = F.r (F.T^[1] base) + F.r base)) := by
 121  exact ⟨boolFramework, baseState, orbit_not_ratio_self_similar, orbit_not_additive_posting⟩
 122
 123end HierarchyRealizationObstruction
 124end Foundation
 125end IndisputableMonolith
 126