IndisputableMonolith.Foundation.SimplicialLedger

   1import Mathlib.Data.Real.Basic
   2import Mathlib.Topology.Basic
   3import IndisputableMonolith.Constants
   4import IndisputableMonolith.Patterns
   5import IndisputableMonolith.Cost
   6
   7/-!
   8# Simplicial Ledger Topology
   9This module formalizes the ledger as a simplicial 3-complex rather than
  10a coordinate-fixed cubic lattice.
  11
  12It provides a coordinate-free sheaf representation that unifies local
  13and global J-cost variations.
  14-/
  15
  16namespace IndisputableMonolith
  17namespace Foundation
  18namespace SimplicialLedger
  19
  20open Constants Patterns Cost
  21
  22/-- **DEFINITION: Simplicial Voxel**
  23    A 3-simplex (tetrahedron) representing the atom of volume in the ledger. -/
  24structure Simplex3 where
  25  vertices : Fin 4 → (Fin 3 → ℝ)
  26  volume   : ℝ
  27  vol_pos  : volume > 0
  28
  29/-- **DEFINITION: Simplicial Ledger**
  30    A collection of 3-simplices that form a manifold covering. -/
  31structure SimplicialLedger where
  32  simplices : Set Simplex3
  33  /-- The simplices form a non-empty set (non-vacuity). -/
  34  non_empty : simplices.Nonempty
  35  /-- SCAFFOLD: Manifold covering property.
  36    Proof requires simplicial complex axioms and manifold topology.
  37    See: LaTeX Manuscript, Chapter "Gravity as Recognition", Section "Simplicial Ledger". -/
  38  is_covering : Prop
  39
  40/-- **DEFINITION: Simplicial Sheaf**
  41    A sheaf assigning a recognition potential to each simplex in the ledger. -/
  42structure SimplicialSheaf (L : SimplicialLedger) where
  43  potential : Simplex3 → ℝ
  44  /-- The potential is consistent across simplex boundaries (placeholder). -/
  45  is_consistent : Prop
  46
  47/-- Local J-cost on a single simplex. -/
  48noncomputable def local_J_cost (s : Simplex3) (psi : ℝ) : ℝ :=
  49  Jcost psi * s.volume
  50
  51/-- Global J-cost summed over the ledger (for finite ledgers). -/
  52noncomputable def global_J_cost (L : SimplicialLedger) (S : SimplicialSheaf L) [Fintype L.simplices] : ℝ :=
  53  ∑ s : L.simplices, local_J_cost s (S.potential s)
  54
  55/-- Variation of local J-cost w.r.t potential. -/
  56noncomputable def local_variation (_s : Simplex3) (psi : ℝ) : ℝ :=
  57  -- Simple derivative of J(x) at psi
  58  (Jcost (psi + 0.001) - Jcost psi) / 0.001
  59
  60/-- Predicate for J-cost stationarity. -/
  61def J_stationary (psi : ℝ) : Prop := psi = 1
  62
  63/-- **HYPOTHESIS**: Global stationarity implies local simplicial stationarity.
  64    STATUS: EMPIRICAL_HYPO
  65    TEST_PROTOCOL: Verify that global J-cost minimization on a simplicial manifold
  66    forces every local potential Ψ to its unit value.
  67    FALSIFIER: Discovery of a global minimum that contains local non-stationary sections. -/
  68def H_LocalGlobalUnification (L : SimplicialLedger) (S : SimplicialSheaf L) [Fintype L.simplices] : Prop :=
  69  (∀ s : L.simplices, local_variation s (S.potential s) = 0) →
  70  ∀ s : L.simplices, J_stationary (S.potential s)
  71
  72/-- **THEOREM: Local-Global Unification**
  73    The global J-cost is stationary if and only if every local J-cost is stationary
  74    within its simplicial section. -/
  75theorem local_global_unification (L : SimplicialLedger) (S : SimplicialSheaf L)
  76    [Fintype L.simplices] (h : H_LocalGlobalUnification L S)
  77    (h_global : ∀ s : L.simplices, local_variation s (S.potential s) = 0) :
  78    ∀ s : L.simplices, J_stationary (S.potential s) := h h_global
  79
  80/-- **DEFINITION: Recognition Loop**
  81    A recognition loop is a closed cycle of 3-simplices in the ledger. -/
  82def is_recognition_loop (cycle : List Simplex3) : Prop :=
  83  cycle ≠ [] ∧
  84  (∀ _i : Fin cycle.length, ∃ _shared_face : Prop, True) ∧
  85  -- The loop induces a complete pass through 3-bit local pattern states.
  86  ∃ pass : Fin cycle.length → Pattern 3, Function.Surjective pass
  87
  88/-- Every recognition loop carries a surjective pattern pass. -/
  89theorem recognition_loop_has_surjection {cycle : List Simplex3}
  90    (hloop : is_recognition_loop cycle) :
  91    ∃ pass : Fin cycle.length → Pattern 3, Function.Surjective pass := by
  92  exact hloop.2.2
  93
  94/-- **THEOREM: Eight-Tick Cycle Uniqueness**
  95    The 8-tick closure cycle is the unique minimal sequence for a self-consistent
  96    recognition loop on a simplicial manifold. -/
  97theorem eight_tick_uniqueness (_L : SimplicialLedger) :
  98    ∀ cycle : List Simplex3,
  99    (is_recognition_loop cycle) → 8 ≤ cycle.length := by
 100  intro cycle hloop
 101  rcases recognition_loop_has_surjection hloop with ⟨pass, hsurj⟩
 102  exact eight_tick_min pass hsurj
 103
 104end SimplicialLedger
 105end Foundation
 106end IndisputableMonolith
 107