IndisputableMonolith.Verification.T6T8SpineAudit

   1import Mathlib
   2import IndisputableMonolith.Foundation.HierarchyRealizationObstruction
   3import IndisputableMonolith.Foundation.AlexanderDuality
   4import IndisputableMonolith.Foundation.DimensionForcing
   5import IndisputableMonolith.Foundation.T7CycleRealization
   6import IndisputableMonolith.Foundation.MathlibCohomologyBridge
   7import IndisputableMonolith.Foundation.CircleWindingChain
   8import IndisputableMonolith.Verification.DimensionLinking
   9import IndisputableMonolith.Foundation.UnifiedForcingChain
  10
  11/-!
  12# T6–T8 spine honesty audit (July 2026 internal report)
  13
  14Machine-checked summary of the internal audit *T6 through T8 — what I read,
  15what I asked, and whether each tier is forced* (2026-07-06).
  16
  17**Tier tags (honest):**
  18- **THEOREM**: proved content with no hidden premise in the statement.
  19- **FORCED-CONDITIONAL**: correct consequence once named hypotheses are supplied.
  20- **MODEL / ENCODING**: definitional packaging of a classical fact or modeling choice.
  21- **OPEN**: missing bridge the program still owes.
  22
  23This module does not upgrade any tier; it records what the repository already proves
  24about its own gaps.
  25-/
  26
  27namespace IndisputableMonolith
  28namespace Verification
  29namespace T6T8SpineAudit
  30
  31open Foundation
  32open Foundation.HierarchyRealizationObstruction
  33open Foundation.AlexanderDuality
  34open Foundation.DimensionForcing
  35open Foundation.T7CycleRealization
  36open Foundation.MathlibCohomologyBridge
  37open DimensionLinking
  38
  39/-! ## T6: closure is supplied, not derived from T5 alone -/
  40
  41/-- **AUDIT (THEOREM).** `ClosedObservableFramework` alone does not force the
  42hierarchy fields consumed by the internal T5→T6 bridge. -/
  43theorem t6_obstruction_closed_framework :
  44    ∃ (F : ClosedFramework.ClosedObservableFramework) (base : F.S),
  45      (¬ (∀ k,
  46        F.r (F.T^[k + 2] base) / F.r (F.T^[k + 1] base) =
  47          F.r (F.T^[k + 1] base) / F.r (F.T^[k] base))) ∧
  48      (¬ (F.r (F.T^[2] base) = F.r (F.T^[1] base) + F.r base)) :=
  49  closedFramework_does_not_force_realizedHierarchy_fields
  50
  51/-- **AUDIT (THEOREM).** Quadratic uniqueness `r² = r + 1 ∧ r > 0 ⇒ r = φ` is
  52available without importing T5 (standalone `t6_holds`). -/
  53theorem t6_quadratic_algebra_standalone : UnifiedForcingChain.T6_Phi_Forced :=
  54  UnifiedForcingChain.t6_holds
  55
  56/-! ## T7: combinatorics real; realization layer predicate-level -/
  57
  58/-- **AUDIT (MODEL).** `EdgeDistinct` is definitionally `True` (placeholder predicate). -/
  59theorem t7_edge_distinct_is_placeholder (D : ℕ) (W : ClosedWalkOnCube D) :
  60    EdgeDistinct W ↔ True :=
  61  Iff.rfl
  62
  63/-- **AUDIT (MODEL).** `RealizedDefect` is definitionally `Circle` for every walk. -/
  64theorem t7_realized_defect_by_definition (D : ℕ)
  65    (cell : SubstrateAxioms.CellularCompletion D) (W : ClosedWalkOnCube D) :
  66    RealizedDefect cell W = T7CycleRealization.Circle :=
  67  rfl
  68
  69/-! ## T8: linking predicate is an encoding; H₁(S¹) is proved separately -/
  70
  71/-- **AUDIT (ENCODING).** After unfolding, circle linking is the arithmetic
  72condition `D - 2 = 1`, not a Mathlib Alexander-duality computation. -/
  73theorem t8_linking_predicate_unfolds_to_arithmetic (D : ℕ) :
  74    SphereAdmitsCircleLinking D ↔ (D : ℤ) - 2 = 1 := by
  75  unfold SphereAdmitsCircleLinking
  76  rw [circle_reduced_cohomology_iff]
  77
  78/-- **AUDIT (THEOREM).** `dimension_unique` discharges from the `linking` field
  79alone; `eight_tick`, `gap_sync`, and substrate fields are not used in the proof.
  80The genuine `H₁(S¹;ℤ) ≅ ℤ` certificate is proved separately and is **OPEN** as a
  81premise of this discharge (not machine-checked here). -/
  82theorem t8_dimension_unique_from_linking (D : Dimension)
  83    (hlink : SupportsNontrivialLinking D) : D = 3 :=
  84  linking_requires_D3 D hlink
  85
  86/-- **AUDIT (THEOREM).** Same-sector linking arithmetic permits every odd
  87`D ≥ 3`; the loop-loop specialization `p = 1` is an additional choice. -/
  88theorem t8_same_sector_allows_odd_dimensions (D : ℕ) :
  89    (D ≥ 3 ∧ ¬ 2 ∣ D) ↔
  90      ∃ p : ℕ, p ≥ 1 ∧ D = 2 * p + 1 :=
  91  (allowed_set_A_characterization D).symm
  92
  93/-- **AUDIT (THEOREM).** `H₁(S¹; ℤ) ≅ ℤ` is proved against Mathlib singular
  94homology (`CircleWindingChain.circleH1ZIsoInt_holds`). This certificate is
  95not yet a premise of `linking_requires_D3`. -/
  96theorem t8_circle_h1_iso_proved : circleH1ZIsoInt :=
  97  CircleWindingChain.circleH1ZIsoInt_holds
  98
  99/-- **AUDIT (ENCODING).** The Mathlib backend builder still sets
 100`supportsLinking := fun D => D = 3` when given only circle-H1 nonvanishing. -/
 101theorem t8_backend_still_encodes_D3 (hH1 : circleH1ZNonzero) (D : ℕ) :
 102    (mathlibCircleLinkingBackend_from_circleH1ZNonzero hH1).supportsLinking D ↔ D = 3 := by
 103  dsimp [mathlibCircleLinkingBackend_from_circleH1ZNonzero]
 104  constructor <;> intro h <;> simpa using h
 105
 106structure T6T8SpineAuditCert : Prop where
 107  t6_obstruction : ∃ (F : ClosedFramework.ClosedObservableFramework) (base : F.S),
 108    (¬ (∀ k,
 109      F.r (F.T^[k + 2] base) / F.r (F.T^[k + 1] base) =
 110        F.r (F.T^[k + 1] base) / F.r (F.T^[k] base))) ∧
 111    (¬ (F.r (F.T^[2] base) = F.r (F.T^[1] base) + F.r base))
 112  t6_algebra_standalone : UnifiedForcingChain.T6_Phi_Forced
 113  t7_edge_distinct_placeholder : ∀ (D : ℕ) (W : ClosedWalkOnCube D), EdgeDistinct W ↔ True
 114  t7_placeholder_realization : ∀ (D : ℕ)
 115    (cell : SubstrateAxioms.CellularCompletion D) (W : ClosedWalkOnCube D),
 116    RealizedDefect cell W = T7CycleRealization.Circle
 117  t8_linking_encoding : ∀ D : ℕ, SphereAdmitsCircleLinking D ↔ (D : ℤ) - 2 = 1
 118  t8_odd_dimensions_allowed : ∀ D : ℕ,
 119    (D ≥ 3 ∧ ¬ 2 ∣ D) ↔ ∃ p : ℕ, p ≥ 1 ∧ D = 2 * p + 1
 120  t8_h1_proved : circleH1ZIsoInt
 121  t8_dimension_unique_from_linking :
 122    ∀ D : Dimension, SupportsNontrivialLinking D → D = 3
 123  t8_backend_encodes_D3 : ∀ (hH1 : circleH1ZNonzero) (D : ℕ),
 124    (mathlibCircleLinkingBackend_from_circleH1ZNonzero hH1).supportsLinking D ↔ D = 3
 125
 126/-- Checked audit certificate bundling the July 2026 T6–T8 honesty report. -/
 127theorem t6t8_spine_audit_cert : T6T8SpineAuditCert where
 128  t6_obstruction := t6_obstruction_closed_framework
 129  t6_algebra_standalone := t6_quadratic_algebra_standalone
 130  t7_edge_distinct_placeholder := fun D W => t7_edge_distinct_is_placeholder D W
 131  t7_placeholder_realization := fun D cell W =>
 132    t7_realized_defect_by_definition D cell W
 133  t8_linking_encoding := t8_linking_predicate_unfolds_to_arithmetic
 134  t8_odd_dimensions_allowed := t8_same_sector_allows_odd_dimensions
 135  t8_h1_proved := t8_circle_h1_iso_proved
 136  t8_dimension_unique_from_linking := linking_requires_D3
 137  t8_backend_encodes_D3 := fun hH1 D => by
 138    dsimp [mathlibCircleLinkingBackend_from_circleH1ZNonzero]
 139    constructor <;> intro h <;> simpa using h
 140
 141end T6T8SpineAudit
 142end Verification
 143end IndisputableMonolith
 144