IndisputableMonolith.Foundation.SubstrateAxioms

   1import Mathlib
   2
   3/-!
   4# T7.5 Substrate Axioms
   5
   6This module records the substrate-side structural inputs used by the
   7T7/T8 dimension route.
   8
   9The statements are deliberately predicate-level, matching the current
  10`Foundation.AlexanderDuality` style: the Lean framework names the
  11load-bearing commitments without pretending that Mathlib currently supplies
  12the full smooth-topology proof chain (cellular completions, Thom isomorphism,
  13and Alexander/Lefschetz duality for the required class of substrates).
  14-/
  15
  16namespace IndisputableMonolith
  17namespace Foundation
  18namespace SubstrateAxioms
  19
  20/-- Spatial dimension parameter. -/
  21abbrev Dimension := ℕ
  22
  23/-- T7.5a: a cellular completion of the cube graph in dimension `D`.
  24
  25The three fields are proof placeholders for the three structural clauses used
  26in the dimension paper: a closed orientable smooth `D`-manifold substrate, a
  27tame cube-graph embedding, and a retraction back to the cube graph. -/
  28structure CellularCompletion (D : Dimension) : Prop where
  29  closed_orientable_smooth : True
  30  cube_graph_embeds : True
  31  retraction_back_to_cube_graph : True
  32
  33/-- The current framework admits a predicate-level completion witness in every
  34dimension. This is the Lean counterpart of the paper's `S^D` witness at the
  35structural-interface level. -/
  36theorem cellular_completion_trivial (D : Dimension) :
  37    CellularCompletion D where
  38  closed_orientable_smooth := trivial
  39  cube_graph_embeds := trivial
  40  retraction_back_to_cube_graph := trivial
  41
  42/-- T7.5c: integral `1`-acyclicity of the substrate. -/
  43structure OneAcyclicSubstrate (D : Dimension) : Prop where
  44  H1_vanishes : True
  45
  46/-- Predicate-level `1`-acyclic witness. -/
  47theorem one_acyclic_trivial (D : Dimension) :
  48    OneAcyclicSubstrate D where
  49  H1_vanishes := trivial
  50
  51/-- Dimension-uniform loop-entanglement: there is some recognized sphere
  52dimension `p ≥ 1` whose complement carries the required nontrivial
  53homological separator. -/
  54structure LoopEntanglement (D : Dimension) : Prop where
  55  exists_p : ∃ p : ℕ, 1 ≤ p ∧ True
  56
  57/-- The circle case (`p = 1`) supplies the predicate-level witness. -/
  58theorem loop_entanglement_circle_witness (D : Dimension) :
  59    LoopEntanglement D where
  60  exists_p := ⟨1, by decide, trivial⟩
  61
  62/-- Compatibility with the realized recognition cycle: the topological witness
  63is the closed walk produced by the T7 recognition cycle. -/
  64structure CompatibilityWithRealizedCycle (D : Dimension) : Prop where
  65  witness_is_closed_walk : True
  66
  67/-- Predicate-level compatibility witness. -/
  68theorem compatibility_trivial (D : Dimension) :
  69    CompatibilityWithRealizedCycle D where
  70  witness_is_closed_walk := trivial
  71
  72/-- Bundled T7.5/loop substrate package. -/
  73structure T75SubstratePackage (D : Dimension) : Prop where
  74  cellular_completion : CellularCompletion D
  75  one_acyclic : OneAcyclicSubstrate D
  76  loop_entanglement : LoopEntanglement D
  77  compatibility : CompatibilityWithRealizedCycle D
  78
  79/-- Predicate-level package witness. -/
  80theorem substrate_package_trivial (D : Dimension) :
  81    T75SubstratePackage D where
  82  cellular_completion := cellular_completion_trivial D
  83  one_acyclic := one_acyclic_trivial D
  84  loop_entanglement := loop_entanglement_circle_witness D
  85  compatibility := compatibility_trivial D
  86
  87end SubstrateAxioms
  88end Foundation
  89end IndisputableMonolith
  90