IndisputableMonolith.Foundation.TMinus1ToT8Bridge

   1import Mathlib
   2import IndisputableMonolith.Foundation.NothingToDistinction
   3import IndisputableMonolith.Foundation.TMinus1ToT1Bridge
   4import IndisputableMonolith.Foundation.LogicRealization
   5import IndisputableMonolith.Foundation.UniversalForcing
   6import IndisputableMonolith.Foundation.UniversalInstantiationFromDistinction
   7import IndisputableMonolith.Foundation.LogicAsFunctionalEquation
   8import IndisputableMonolith.Foundation.DiscretenessForcing
   9import IndisputableMonolith.Foundation.LedgerForcing
  10import IndisputableMonolith.Foundation.PhiForcing
  11import IndisputableMonolith.Foundation.PhiForcingDerived
  12import IndisputableMonolith.Foundation.HierarchyDynamics
  13import IndisputableMonolith.Foundation.DimensionForcing
  14import IndisputableMonolith.Foundation.CircleWindingChain
  15import IndisputableMonolith.Foundation.RecognitionForcing
  16import IndisputableMonolith.Recognition
  17import IndisputableMonolith.Cost
  18import IndisputableMonolith.CostUniqueness
  19
  20/-!
  21# Public T-1 through T8 Forcing Spine
  22
  23This module exposes the public, theory-only T-1 through T8 forcing spine:
  24
  25* T-1: absolute distinguishability floor.
  26* T0: Boolean recognition-work split.
  27* T1: cost-form Meta-Principle.
  28* T2: two-state discreteness of the floor.
  29* T3: additive ledger bookkeeping.
  30* T4: recognition witness on the discrete floor.
  31* T5: uniqueness of the canonical reciprocal cost.
  32* T6: φ forced by realized self-similar hierarchy.
  33* T7: eight-tick cadence from dimension.
  34* T8: D = 3 from linking / eight-tick / gap-sync compatibility.
  35
  36It deliberately stops before the private operator / measurement layers
  37that live in the `/reality` repository.
  38-/
  39
  40namespace IndisputableMonolith
  41namespace Foundation
  42namespace TMinus1ToT8Bridge
  43
  44open Real
  45open CostFromDistinction
  46
  47namespace T01
  48
  49/-- Compatibility alias for the Boolean recognition-work cost used by the
  50public T-1 through T8 spine. -/
  51abbrev boolRecognitionCost : CostFromDistinction.CostFunction Bool :=
  52  TMinus1ToT1Bridge.boolRecognitionCost
  53
  54end T01
  55
  56/-! ## T-1, T0, T1 aliases from the public first bridge -/
  57
  58abbrev TMinus1_AbsoluteFloor := TMinus1ToT1Bridge.TMinus1_AbsoluteFloor.{0, 0}
  59abbrev T0_Logic_Forced := TMinus1ToT1Bridge.T0_Logic_Forced
  60abbrev T1_MP_Forced := TMinus1ToT1Bridge.T1_MetaPrinciple_Forced
  61abbrev TMinus1_To_T0_Bridge := TMinus1ToT1Bridge.TMinus1_To_T0_Bridge
  62abbrev T0_To_T1_Bridge := TMinus1ToT1Bridge.T0_To_T1_Bridge
  63
  64def tminus1_holds : TMinus1_AbsoluteFloor := @TMinus1ToT1Bridge.tminus1_holds.{0, 0}
  65def tminus1_to_t0_bridge : TMinus1_AbsoluteFloor → TMinus1_To_T0_Bridge :=
  66  TMinus1ToT1Bridge.tminus1_to_t0_bridge
  67def t0_to_t1_bridge_holds : (h0 : T0_Logic_Forced) → T0_To_T1_Bridge h0 :=
  68  TMinus1ToT1Bridge.t0_to_t1_bridge_holds
  69
  70/-! ## Normalized two-point floor audit -/
  71
  72/-- A normalized two-point recognition floor.  This is the abstract version
  73of the Boolean floor: one empty/consistent point, one marked inconsistent
  74point, a unit-normalized recognition-work cost, and an equivalence to `Bool`
  75showing that `Bool` is only the canonical representative. -/
  76structure NormalizedTwoPointRecognitionFloor
  77    (Config : Type) [CostFromDistinction.ConfigSpace Config]
  78    (mark : Config) (cost : CostFromDistinction.CostFunction Config)
  79    (toBoolEquiv : Config ≃ Bool) : Prop where
  80  mark_ne_emp : mark ≠ CostFromDistinction.ConfigSpace.emp
  81  exhaustive :
  82    ∀ Γ : Config, Γ = CostFromDistinction.ConfigSpace.emp ∨ Γ = mark
  83  consistent_iff_emp :
  84    ∀ Γ : Config,
  85      CostFromDistinction.ConfigSpace.IsConsistent Γ ↔
  86        Γ = CostFromDistinction.ConfigSpace.emp
  87  cost_emp_zero : cost.C CostFromDistinction.ConfigSpace.emp = 0
  88  cost_mark_one : cost.C mark = 1
  89  toBool_emp : toBoolEquiv CostFromDistinction.ConfigSpace.emp = false
  90  toBool_mark : toBoolEquiv mark = true
  91
  92/-- The concrete Boolean floor is the canonical normalized two-point
  93recognition floor. -/
  94theorem bool_normalized_two_point_floor :
  95    NormalizedTwoPointRecognitionFloor Bool true
  96      TMinus1ToT1Bridge.boolRecognitionCost (Equiv.refl Bool) where
  97  mark_ne_emp := by
  98    intro h
  99    change true = false at h
 100    exact Bool.noConfusion h
 101  exhaustive := by
 102    intro Γ
 103    cases Γ
 104    · exact Or.inl rfl
 105    · exact Or.inr rfl
 106  consistent_iff_emp := by
 107    intro Γ
 108    rfl
 109  cost_emp_zero := rfl
 110  cost_mark_one := rfl
 111  toBool_emp := rfl
 112  toBool_mark := rfl
 113
 114instance NormalizedTwoPointRecognitionFloor.instSubsingleton
 115    {Config : Type} [CostFromDistinction.ConfigSpace Config]
 116    {mark : Config} {cost : CostFromDistinction.CostFunction Config}
 117    {toBoolEquiv : Config ≃ Bool} :
 118    Subsingleton (NormalizedTwoPointRecognitionFloor Config mark cost toBoolEquiv) where
 119  allEq _ _ := by rfl
 120
 121theorem normalized_two_point_floor_unique
 122    {Config : Type} [CostFromDistinction.ConfigSpace Config]
 123    {mark : Config} {cost : CostFromDistinction.CostFunction Config}
 124    {toBoolEquiv : Config ≃ Bool}
 125    (h1 h2 : NormalizedTwoPointRecognitionFloor Config mark cost toBoolEquiv) :
 126    h1 = h2 :=
 127  Subsingleton.elim _ _
 128
 129/-- On a normalized two-point floor, the cost is forced to be the `0/1`
 130indicator pulled back along the equivalence to `Bool`.  This is the
 131theorem-level form of "unit recognition work" rather than a hidden definition
 132of the Boolean representative. -/
 133theorem normalized_two_point_cost_eq_indicator
 134    {Config : Type} [CostFromDistinction.ConfigSpace Config]
 135    {mark : Config} {cost : CostFromDistinction.CostFunction Config}
 136    {toBoolEquiv : Config ≃ Bool}
 137    (h : NormalizedTwoPointRecognitionFloor Config mark cost toBoolEquiv)
 138    (Γ : Config) :
 139    cost.C Γ = if toBoolEquiv Γ = false then 0 else 1 := by
 140  rcases h.exhaustive Γ with hΓ | hΓ
 141  · rw [hΓ, h.toBool_emp, h.cost_emp_zero]
 142    simp
 143  · rw [hΓ, h.toBool_mark, h.cost_mark_one]
 144    simp
 145
 146/-- For fixed empty/marked states, the equivalence-to-`Bool` of a normalized
 147two-point floor is unique. -/
 148theorem normalized_two_point_equiv_unique
 149    {Config : Type} [CostFromDistinction.ConfigSpace Config]
 150    {mark : Config}
 151    {cost₁ cost₂ : CostFromDistinction.CostFunction Config}
 152    {toBoolEquiv₁ toBoolEquiv₂ : Config ≃ Bool}
 153    (h₁ : NormalizedTwoPointRecognitionFloor Config mark cost₁ toBoolEquiv₁)
 154    (h₂ : NormalizedTwoPointRecognitionFloor Config mark cost₂ toBoolEquiv₂) :
 155    toBoolEquiv₁ = toBoolEquiv₂ := by
 156  ext Γ
 157  rcases h₁.exhaustive Γ with hΓ | hΓ
 158  · rw [hΓ, h₁.toBool_emp, h₂.toBool_emp]
 159  · rw [hΓ, h₁.toBool_mark, h₂.toBool_mark]
 160
 161/-- Any two normalized two-point recognition costs over the same two-point
 162shape agree pointwise. -/
 163theorem normalized_two_point_cost_unique_up_to_equiv
 164    {Config : Type} [CostFromDistinction.ConfigSpace Config]
 165    {mark : Config}
 166    {cost₁ cost₂ : CostFromDistinction.CostFunction Config}
 167    {toBoolEquiv₁ toBoolEquiv₂ : Config ≃ Bool}
 168    (h₁ : NormalizedTwoPointRecognitionFloor Config mark cost₁ toBoolEquiv₁)
 169    (h₂ : NormalizedTwoPointRecognitionFloor Config mark cost₂ toBoolEquiv₂) :
 170    toBoolEquiv₁ = toBoolEquiv₂ ∧ ∀ Γ : Config, cost₁.C Γ = cost₂.C Γ := by
 171  have heq : toBoolEquiv₁ = toBoolEquiv₂ :=
 172    normalized_two_point_equiv_unique h₁ h₂
 173  constructor
 174  · exact heq
 175  · intro Γ
 176    rw [normalized_two_point_cost_eq_indicator h₁ Γ]
 177    rw [normalized_two_point_cost_eq_indicator h₂ Γ]
 178    rw [heq]
 179
 180/-- Any normalized Boolean two-point floor with marked state `true` is the
 181canonical Boolean floor: the equivalence is `Equiv.refl Bool` and the cost
 182agrees pointwise with `boolRecognitionCost`. -/
 183theorem bool_normalized_two_point_floor_unique
 184    {cost : CostFromDistinction.CostFunction Bool}
 185    {toBoolEquiv : Bool ≃ Bool}
 186    (h : NormalizedTwoPointRecognitionFloor Bool true cost toBoolEquiv) :
 187    toBoolEquiv = Equiv.refl Bool ∧
 188      ∀ Γ : Bool, cost.C Γ = TMinus1ToT1Bridge.boolRecognitionCost.C Γ := by
 189  exact normalized_two_point_cost_unique_up_to_equiv
 190    h bool_normalized_two_point_floor
 191
 192theorem absolute_bool_floor_unique_normalized_01
 193    (_floor : AbsoluteFloorClosure.AbsoluteFloorWitness Bool)
 194    {cost : CostFromDistinction.CostFunction Bool}
 195    {toBoolEquiv : Bool ≃ Bool}
 196    (h : NormalizedTwoPointRecognitionFloor Bool true cost toBoolEquiv) :
 197    toBoolEquiv = Equiv.refl Bool ∧
 198      ∀ Γ : Bool, cost.C Γ = TMinus1ToT1Bridge.boolRecognitionCost.C Γ :=
 199  bool_normalized_two_point_floor_unique h
 200
 201/-! ## T2: discreteness from the floor split -/
 202
 203structure T2_Discreteness_Forced : Prop where
 204  state_dichotomy : ∀ Γ : Bool, Γ = false ∨ Γ = true
 205  states_distinct : (false : Bool) ≠ true
 206  zero_cost_selects_consistency :
 207    ∀ Γ : Bool, T01.boolRecognitionCost.C Γ = 0 → Γ = false
 208  positive_cost_selects_marked :
 209    ∀ Γ : Bool, 0 < T01.boolRecognitionCost.C Γ → Γ = true
 210
 211structure T1_To_T2_Bridge (b01 : TMinus1_To_T0_Bridge) (h1 : T1_MP_Forced) :
 212    Prop where
 213  floor_used : AbsoluteFloorClosure.AbsoluteFloorWitness Bool
 214  floor_dichotomy : ∀ Γ : Bool, Γ = false ∨ Γ = true
 215  floor_states_distinct : (false : Bool) ≠ true
 216  consistency_is_false :
 217    ∀ Γ : Bool, CostFromDistinction.ConfigSpace.IsConsistent Γ → Γ = false
 218  positive_cost_selects_marked :
 219    ∀ Γ : Bool, 0 < T01.boolRecognitionCost.C Γ → Γ = true
 220  t2 : T2_Discreteness_Forced
 221
 222theorem t1_to_t2_bridge_holds
 223    (b01 : TMinus1_To_T0_Bridge) (h1 : T1_MP_Forced) :
 224    T1_To_T2_Bridge b01 h1 where
 225  floor_used := b01.bool_floor
 226  floor_dichotomy := b01.floor_config.floor_dichotomy
 227  floor_states_distinct := b01.floor_config.false_true_distinct
 228  consistency_is_false := fun Γ hΓ =>
 229    (b01.floor_config.consistency_iff_false Γ).mp hΓ
 230  positive_cost_selects_marked := by
 231    intro Γ hpos
 232    have hinc : ¬CostFromDistinction.ConfigSpace.IsConsistent Γ :=
 233      (b01.positive_iff_inconsistent Γ).mp hpos
 234    rcases b01.floor_config.floor_dichotomy Γ with hΓ | hΓ
 235    · exfalso
 236      exact hinc ((b01.floor_config.consistency_iff_false Γ).mpr hΓ)
 237    · exact hΓ
 238  t2 := {
 239    state_dichotomy := b01.floor_config.floor_dichotomy
 240    states_distinct := b01.floor_config.false_true_distinct
 241    zero_cost_selects_consistency := fun Γ hzero =>
 242      (b01.floor_config.consistency_iff_false Γ).mp
 243        (h1.zero_cost_consistent Γ hzero)
 244    positive_cost_selects_marked := by
 245      intro Γ hpos
 246      have hinc : ¬CostFromDistinction.ConfigSpace.IsConsistent Γ :=
 247        (b01.positive_iff_inconsistent Γ).mp hpos
 248      rcases b01.floor_config.floor_dichotomy Γ with hΓ | hΓ
 249      · exfalso
 250        exact hinc ((b01.floor_config.consistency_iff_false Γ).mpr hΓ)
 251      · exact hΓ
 252  }
 253
 254/-! ## T3: ledger from additive recognition work -/
 255
 256structure T3_Ledger_Forced : Prop where
 257  empty_balanced : T01.boolRecognitionCost.C false = 0
 258  empty_join_left :
 259    ∀ Γ : Bool, CostFromDistinction.ConfigSpace.join false Γ = Γ
 260  empty_join_cost_neutral :
 261    ∀ Γ : Bool,
 262      T01.boolRecognitionCost.C (CostFromDistinction.ConfigSpace.join false Γ) =
 263        T01.boolRecognitionCost.C Γ
 264  independent_join_additive :
 265    ∀ Γ₁ Γ₂ : Bool,
 266      CostFromDistinction.ConfigSpace.Independent Γ₁ Γ₂ →
 267        T01.boolRecognitionCost.C
 268          (CostFromDistinction.ConfigSpace.join Γ₁ Γ₂) =
 269        T01.boolRecognitionCost.C Γ₁ + T01.boolRecognitionCost.C Γ₂
 270
 271structure T0_T2_To_T3_Bridge
 272    (b01 : TMinus1_To_T0_Bridge) (h0 : T0_Logic_Forced) (h2 : T2_Discreteness_Forced) :
 273    Prop where
 274  floor_empty_join :
 275    ∀ Γ : Bool, CostFromDistinction.ConfigSpace.join false Γ = Γ
 276  t0_additivity :
 277    ∀ Γ₁ Γ₂ : Bool,
 278      CostFromDistinction.ConfigSpace.Independent Γ₁ Γ₂ →
 279        T01.boolRecognitionCost.C
 280          (CostFromDistinction.ConfigSpace.join Γ₁ Γ₂) =
 281        T01.boolRecognitionCost.C Γ₁ + T01.boolRecognitionCost.C Γ₂
 282  t2_floor_split : ∀ Γ : Bool, Γ = false ∨ Γ = true
 283  t3 : T3_Ledger_Forced
 284
 285theorem t0_t2_to_t3_bridge_holds
 286    (b01 : TMinus1_To_T0_Bridge) (h0 : T0_Logic_Forced) (h2 : T2_Discreteness_Forced) :
 287    T0_T2_To_T3_Bridge b01 h0 h2 where
 288  floor_empty_join := b01.floor_config.empty_join_left
 289  t0_additivity := h0.additive_indep
 290  t2_floor_split := h2.state_dichotomy
 291  t3 := {
 292    empty_balanced := h0.consistency_zero
 293    empty_join_left := by
 294      intro Γ
 295      rcases h2.state_dichotomy Γ with hΓ | hΓ
 296      · simpa [hΓ] using b01.floor_config.empty_join_left Γ
 297      · simpa [hΓ] using b01.floor_config.empty_join_left Γ
 298    empty_join_cost_neutral := by
 299      intro Γ
 300      rw [b01.floor_config.empty_join_left Γ]
 301    independent_join_additive := h0.additive_indep
 302  }
 303
 304/-! ## T4: recognition from the discrete balanced floor -/
 305
 306structure T4_Recognition_Forced : Prop where
 307  floor_distinction : ∃ a b : Bool, a ≠ b
 308  floor_recognition : Nonempty (Recognition.Recognize Bool Bool)
 309  floor_recognition_structure :
 310    ∃ R : Recognition.RecognitionStructure, R.U = Bool
 311  zero_cost_recognition :
 312    T01.boolRecognitionCost.C false = 0 →
 313      Nonempty (Recognition.Recognize Bool Bool)
 314
 315structure BalancedFloorRecognition
 316    (hbalanced : T01.boolRecognitionCost.C false = 0) : Prop where
 317  source_balance : T01.boolRecognitionCost.C false = 0
 318  recognition : Nonempty (Recognition.Recognize Bool Bool)
 319
 320theorem balanced_floor_recognition
 321    (hbalanced : T01.boolRecognitionCost.C false = 0) :
 322    BalancedFloorRecognition hbalanced where
 323  source_balance := hbalanced
 324  recognition := ⟨⟨false, false⟩⟩
 325
 326theorem recognition_from_balanced_floor_ledger :
 327    T01.boolRecognitionCost.C false = 0 →
 328      Nonempty (Recognition.Recognize Bool Bool) :=
 329  fun hbalanced => (balanced_floor_recognition hbalanced).recognition
 330
 331structure T2_T3_To_T4_Bridge (h2 : T2_Discreteness_Forced) (h3 : T3_Ledger_Forced) :
 332    Prop where
 333  distinction_from_t2 : ∃ a b : Bool, a ≠ b
 334  balanced_ledger_from_t3 : T01.boolRecognitionCost.C false = 0
 335  balanced_floor_recognition_cert :
 336    BalancedFloorRecognition balanced_ledger_from_t3
 337  recognition_from_balanced_ledger :
 338    T01.boolRecognitionCost.C false = 0 →
 339      Nonempty (Recognition.Recognize Bool Bool)
 340  t4 : T4_Recognition_Forced
 341
 342theorem t2_t3_to_t4_bridge_holds
 343    (h2 : T2_Discreteness_Forced) (h3 : T3_Ledger_Forced) :
 344    T2_T3_To_T4_Bridge h2 h3 where
 345  distinction_from_t2 := ⟨false, true, h2.states_distinct⟩
 346  balanced_ledger_from_t3 := h3.empty_balanced
 347  balanced_floor_recognition_cert := balanced_floor_recognition h3.empty_balanced
 348  recognition_from_balanced_ledger := recognition_from_balanced_floor_ledger
 349  t4 := {
 350    floor_distinction := ⟨false, true, h2.states_distinct⟩
 351    floor_recognition := recognition_from_balanced_floor_ledger h3.empty_balanced
 352    floor_recognition_structure := ⟨{ U := Bool, R := fun a b => a = b }, rfl⟩
 353    zero_cost_recognition := fun hzero => recognition_from_balanced_floor_ledger hzero
 354  }
 355
 356/-! ## T5: unique reciprocal cost from continuous positive-ratio realization -/
 357
 358namespace T4ToT5
 359
 360open LogicAsFunctionalEquation
 361
 362noncomputable def floorRealization : LogicRealization.{0, 0} :=
 363  UniversalInstantiationFromDistinction.logicRealizationOfDistinction
 364    Bool false true (by decide)
 365
 366noncomputable def positiveRatioRealization
 367    (C : ComparisonOperator) (h : SatisfiesLawsOfLogic C) :
 368    LogicRealization.{0, 0} :=
 369  LogicRealization.ofPositiveRatioComparison C h
 370
 371noncomputable def floor_to_positive_ratio_arithmetic
 372    (C : ComparisonOperator) (h : SatisfiesLawsOfLogic C) :
 373    (UniversalForcing.arithmeticOf floorRealization).peano.carrier ≃
 374      (UniversalForcing.arithmeticOf (positiveRatioRealization C h)).peano.carrier :=
 375  by
 376    change floorRealization.Orbit ≃ (positiveRatioRealization C h).Orbit
 377    exact floorRealization.orbitEquivLogicNat.trans
 378      (positiveRatioRealization C h).orbitEquivLogicNat.symm
 379
 380end T4ToT5
 381
 382structure T4_To_T5_Realization_Bridge (h4 : T4_Recognition_Forced) : Prop where
 383  t4_floor_recognition : Nonempty (Recognition.Recognize Bool Bool)
 384  t4_floor_distinction : ∃ a b : Bool, a ≠ b
 385  floor_realization : Nonempty LogicRealization.{0, 0}
 386  positive_ratio_realization :
 387    ∀ (C : LogicAsFunctionalEquation.ComparisonOperator)
 388      (h : LogicAsFunctionalEquation.SatisfiesLawsOfLogic C),
 389      Nonempty LogicRealization.{0, 0}
 390  arithmetic_invariant :
 391    ∀ (C : LogicAsFunctionalEquation.ComparisonOperator)
 392      (h : LogicAsFunctionalEquation.SatisfiesLawsOfLogic C),
 393      Nonempty
 394        ((UniversalForcing.arithmeticOf T4ToT5.floorRealization).peano.carrier ≃
 395          (UniversalForcing.arithmeticOf
 396            (T4ToT5.positiveRatioRealization C h)).peano.carrier)
 397  rcl_surface :
 398    ∀ (C : LogicAsFunctionalEquation.ComparisonOperator)
 399      (h : LogicAsFunctionalEquation.SatisfiesLawsOfLogic C),
 400      ∃ (P : ℝ → ℝ → ℝ) (c : ℝ),
 401        DAlembert.Inevitability.HasMultiplicativeConsistency
 402          (LogicAsFunctionalEquation.derivedCost C) P ∧
 403        (∀ u v, P u v = 2*u + 2*v + c*u*v)
 404
 405noncomputable def t4_to_t5_bridge_holds (h4 : T4_Recognition_Forced) :
 406    T4_To_T5_Realization_Bridge h4 where
 407  t4_floor_recognition := h4.floor_recognition
 408  t4_floor_distinction := h4.floor_distinction
 409  floor_realization := ⟨T4ToT5.floorRealization⟩
 410  positive_ratio_realization := fun C h => ⟨T4ToT5.positiveRatioRealization C h⟩
 411  arithmetic_invariant := fun C h =>
 412    ⟨by
 413      change T4ToT5.floorRealization.Orbit ≃
 414        (T4ToT5.positiveRatioRealization C h).Orbit
 415      exact T4ToT5.floorRealization.orbitEquivLogicNat.trans
 416        (T4ToT5.positiveRatioRealization C h).orbitEquivLogicNat.symm⟩
 417  rcl_surface := fun C h =>
 418    LogicAsFunctionalEquation.RCL_is_unique_functional_form_of_logic C h
 419
 420structure T5_J_Unique : Prop where
 421  J_reciprocal : Cost.FunctionalEquation.IsReciprocalCost Cost.Jcost
 422  J_normalized : Cost.FunctionalEquation.IsNormalized Cost.Jcost
 423  J_composition : Cost.FunctionalEquation.SatisfiesCompositionLaw Cost.Jcost
 424  J_calibrated : Cost.FunctionalEquation.IsCalibrated Cost.Jcost
 425  J_continuous : ContinuousOn Cost.Jcost (Set.Ioi 0)
 426  uniqueness :
 427    ∀ (F : ℝ → ℝ),
 428      Cost.FunctionalEquation.AczelSmoothnessPackage →
 429      Cost.FunctionalEquation.IsReciprocalCost F →
 430      Cost.FunctionalEquation.IsNormalized F →
 431      Cost.FunctionalEquation.SatisfiesCompositionLaw F →
 432      Cost.FunctionalEquation.IsCalibrated F →
 433      ContinuousOn F (Set.Ioi 0) →
 434      ∀ {x : ℝ}, 0 < x → F x = Cost.Jcost x
 435
 436structure T4_To_T5_Cost_Bridge
 437    {h4 : T4_Recognition_Forced} (bridge : T4_To_T5_Realization_Bridge h4) :
 438    Prop where
 439  rcl_surface_available :
 440    ∀ (C : LogicAsFunctionalEquation.ComparisonOperator)
 441      (h : LogicAsFunctionalEquation.SatisfiesLawsOfLogic C),
 442      ∃ (P : ℝ → ℝ → ℝ) (c : ℝ),
 443        DAlembert.Inevitability.HasMultiplicativeConsistency
 444          (LogicAsFunctionalEquation.derivedCost C) P ∧
 445        (∀ u v, P u v = 2*u + 2*v + c*u*v)
 446  rcl_surface_is_bridge_surface :
 447    rcl_surface_available = bridge.rcl_surface
 448  t5 : T5_J_Unique
 449
 450/-- **HONESTY NOTE (2026 audit, T4→T5 arrow).** The T5 record below is
 451proved entirely from `CostUniqueness` lemmas about `Jcost` and from
 452`law_of_logic_forces_jcost`; it consumes NOTHING from `bridge` beyond
 453re-exporting `bridge.rcl_surface` as a field. An earlier revision bound
 454`bridge.rcl_surface` inside the uniqueness proof as an unused variable
 455(`_rcl_surface`), which cosmetically suggested the T−1..T4 floor feeds the
 456T5 proof. It does not: deleting T−1..T4 would break no T5 proof. The
 457substantive gap — that the floor's own cost provably CANNOT satisfy the
 458composition law T5 needs (`PrimitiveDistinction.lean`), so the continuous
 459positive-ratio comparison surface is an imported hypothesis (SI2/C6), not
 460a consequence of the floor — is recorded in the paper as open problems.
 461This structure packages the conditional chain; it is not a forcing proof
 462of T5 from T4.
 463
 464Repair pointer: `Foundation.RecognitionLedgerFloor` builds the upgraded
 465carrier (free additive defect ledger `I →₀ ℕ` with kernel-derived
 466observable equivalence and unconditional additivity) that answers the
 467audit's kernel/cokernel gaps at the floor level. That module is NOT yet
 468wired into this chain: no field of this bridge consumes it. Integrating
 469it — i.e. replacing the imported comparison-surface hypothesis with a
 470theorem from the ledger floor, if that is possible at all — is an open
 471task, not a completed step. -/
 472theorem t4_to_t5_cost_bridge_holds
 473    {h4 : T4_Recognition_Forced} (bridge : T4_To_T5_Realization_Bridge h4) :
 474    T4_To_T5_Cost_Bridge bridge where
 475  rcl_surface_available := bridge.rcl_surface
 476  rcl_surface_is_bridge_surface := rfl
 477  t5 := {
 478    J_reciprocal := CostUniqueness.Jcost_is_reciprocal
 479    J_normalized := CostUniqueness.Jcost_is_normalized
 480    J_composition := CostUniqueness.Jcost_satisfies_composition_law
 481    J_calibrated := CostUniqueness.Jcost_is_calibrated
 482    J_continuous := CostUniqueness.Jcost_continuous_pos
 483    uniqueness := fun F hAczel hRecip hNorm hComp hCalib hCont => by
 484      let _ : Cost.FunctionalEquation.AczelSmoothnessPackage := hAczel
 485      exact Cost.FunctionalEquation.law_of_logic_forces_jcost F
 486        hRecip hNorm hComp hCalib hCont
 487  }
 488
 489/-! ## T6: φ from realized self-similar hierarchy -/
 490
 491structure T5_To_T6_SelfSimilarity_Bridge (h5 : T5_J_Unique) : Prop where
 492  t5_uniqueness_available :
 493    ∀ (F : ℝ → ℝ),
 494      Cost.FunctionalEquation.AczelSmoothnessPackage →
 495      Cost.FunctionalEquation.IsReciprocalCost F →
 496      Cost.FunctionalEquation.IsNormalized F →
 497      Cost.FunctionalEquation.SatisfiesCompositionLaw F →
 498      Cost.FunctionalEquation.IsCalibrated F →
 499      ContinuousOn F (Set.Ioi 0) →
 500      ∀ {x : ℝ}, 0 < x → F x = Cost.Jcost x
 501  internal_hierarchy_forces_phi :
 502    ∀ (F : ClosedFramework.ClosedObservableFramework)
 503      (H : HierarchyRealization.RealizedHierarchy F),
 504      (HierarchyRealization.realized_to_ladder F H).ratio = PhiForcing.φ
 505  self_similar_forces_golden :
 506    ∀ S : PhiForcing.SelfSimilar,
 507      PhiForcing.satisfies_golden_constraint S.ratio
 508  golden_constraint_unique :
 509    ∀ r : ℝ, 0 < r → PhiForcing.satisfies_golden_constraint r → r = PhiForcing.φ
 510  discrete_ledger_ratio_phi :
 511    ∀ (L : PhiForcing.DiscreteLedger) (r : ℝ),
 512      PhiForcing.is_self_similar L r → r = PhiForcing.φ
 513
 514theorem t5_to_t6_bridge_holds (h5 : T5_J_Unique) :
 515    T5_To_T6_SelfSimilarity_Bridge h5 where
 516  t5_uniqueness_available := h5.uniqueness
 517  internal_hierarchy_forces_phi := HierarchyDynamics.bridge_T5_T6_internal
 518  self_similar_forces_golden := PhiForcing.self_similar_forces_golden_constraint
 519  golden_constraint_unique := fun r hr hgold =>
 520    PhiForcing.phi_unique_self_similar hr hgold
 521  discrete_ledger_ratio_phi := PhiForcing.phi_forced
 522
 523structure T6_Phi_Forced : Prop where
 524  phi_equation : PhiForcing.φ^2 = PhiForcing.φ + 1
 525  phi_positive : PhiForcing.φ > 0
 526  phi_unique : ∀ r : ℝ, 0 < r → r^2 = r + 1 → r = PhiForcing.φ
 527
 528theorem t6_phi_unique_from_derived :
 529    ∀ r : ℝ, 0 < r → r^2 = r + 1 → r = PhiForcing.φ := by
 530  intro r hr hgolden
 531  have hr_ne_one : r ≠ 1 := by
 532    intro hr1
 533    rw [hr1] at hgolden
 534    norm_num at hgolden
 535  have hclosure : 1 + r = r^2 := by linarith [hgolden]
 536  have hphi : r = Constants.phi :=
 537    PhiForcingDerived.phi_forcing_complete r hr hr_ne_one hclosure
 538  simpa [PhiForcing.φ, Constants.phi] using hphi
 539
 540theorem t6_holds : T6_Phi_Forced := {
 541  phi_equation := PhiForcing.phi_equation
 542  phi_positive := PhiForcing.phi_pos
 543  phi_unique := t6_phi_unique_from_derived
 544}
 545
 546structure T5_To_T6_Forced_Bridge (h5 : T5_J_Unique) : Prop where
 547  self_similarity : T5_To_T6_SelfSimilarity_Bridge h5
 548  t6 : T6_Phi_Forced
 549
 550theorem t5_to_t6_forced_bridge_holds (h5 : T5_J_Unique) :
 551    T5_To_T6_Forced_Bridge h5 where
 552  self_similarity := t5_to_t6_bridge_holds h5
 553  t6 := t6_holds
 554
 555/-! ## T7/T8: eight-tick and dimension -/
 556
 557structure T7_EightTick_Forced : Prop where
 558  eight_is_2_cubed : DimensionForcing.eight_tick = 2^3
 559  from_dimension : DimensionForcing.EightTickFromDimension 3 = DimensionForcing.eight_tick
 560
 561structure T8_Dimension_Forced : Prop where
 562  linking_forces_D3 : ∀ D, DimensionForcing.SupportsNontrivialLinking D → D = 3
 563  eight_tick_forces_D3 :
 564    ∀ D, DimensionForcing.EightTickFromDimension D = DimensionForcing.eight_tick → D = 3
 565  unique_dimension : ∃! D, DimensionForcing.RSCompatibleDimension D
 566
 567theorem t8_holds : T8_Dimension_Forced := {
 568  linking_forces_D3 := DimensionForcing.linking_requires_D3
 569  eight_tick_forces_D3 := DimensionForcing.eight_tick_forces_D3
 570  unique_dimension := DimensionForcing.dimension_forced
 571}
 572
 573structure T8_To_T7_EightTick_Bridge (h8 : T8_Dimension_Forced) : Prop where
 574  compatible_dimension_three :
 575    ∀ D : DimensionForcing.Dimension,
 576      DimensionForcing.RSCompatibleDimension D → D = 3
 577  compatible_dimension_eight_tick :
 578    ∀ D : DimensionForcing.Dimension,
 579      DimensionForcing.RSCompatibleDimension D →
 580        DimensionForcing.EightTickFromDimension D = DimensionForcing.eight_tick
 581  dimension_three_eight_tick :
 582    DimensionForcing.EightTickFromDimension 3 = DimensionForcing.eight_tick
 583
 584theorem t8_to_t7_bridge_holds (h8 : T8_Dimension_Forced) :
 585    T8_To_T7_EightTick_Bridge h8 where
 586  compatible_dimension_three := by
 587    intro D hD
 588    exact h8.linking_forces_D3 D hD.linking
 589  compatible_dimension_eight_tick := by
 590    intro D hD
 591    exact hD.eight_tick
 592  dimension_three_eight_tick := rfl
 593
 594theorem t7_from_t8 (h8 : T8_Dimension_Forced) : T7_EightTick_Forced := {
 595  eight_is_2_cubed := DimensionForcing.eight_tick_is_2_cubed
 596  from_dimension := (t8_to_t7_bridge_holds h8).dimension_three_eight_tick
 597}
 598
 599/-! ## Complete public T-1 through T8 certificate -/
 600
 601structure CompleteForcingChainT8 where
 602  tminus1 : TMinus1_AbsoluteFloor
 603  tminus1_to_t0 : TMinus1_To_T0_Bridge
 604  t0 : T0_Logic_Forced
 605  t0_to_t1 : T0_To_T1_Bridge t0
 606  t1 : T1_MP_Forced
 607  t1_to_t2 : T1_To_T2_Bridge tminus1_to_t0 t1
 608  t2 : T2_Discreteness_Forced
 609  t0_t2_to_t3 : T0_T2_To_T3_Bridge tminus1_to_t0 t0 t2
 610  t3 : T3_Ledger_Forced
 611  t2_t3_to_t4 : T2_T3_To_T4_Bridge t2 t3
 612  t4 : T4_Recognition_Forced
 613  t4_to_t5 : T4_To_T5_Realization_Bridge t4
 614  t4_to_t5_cost : T4_To_T5_Cost_Bridge t4_to_t5
 615  t5 : T5_J_Unique
 616  t5_to_t6 : T5_To_T6_Forced_Bridge t5
 617  t6 : T6_Phi_Forced
 618  t8 : T8_Dimension_Forced
 619  t8_to_t7 : T8_To_T7_EightTick_Bridge t8
 620  t7 : T7_EightTick_Forced
 621
 622noncomputable def complete_forcing_chain_t8 : CompleteForcingChainT8 :=
 623  let hm1 := tminus1_holds
 624  let b01 := tminus1_to_t0_bridge hm1
 625  let h0 := b01.t0
 626  let b12 := t0_to_t1_bridge_holds h0
 627  let h1 := b12.t1
 628  let b23 := t1_to_t2_bridge_holds b01 h1
 629  let h2 := b23.t2
 630  let b03 := t0_t2_to_t3_bridge_holds b01 h0 h2
 631  let h3 := b03.t3
 632  let b34 := t2_t3_to_t4_bridge_holds h2 h3
 633  let h4 := b34.t4
 634  let b45 := t4_to_t5_bridge_holds h4
 635  let b45c := t4_to_t5_cost_bridge_holds b45
 636  let h5 := b45c.t5
 637  let b56 := t5_to_t6_forced_bridge_holds h5
 638  let h6 := b56.t6
 639  let h8 := t8_holds
 640  let b87 := t8_to_t7_bridge_holds h8
 641  let h7 := t7_from_t8 h8
 642  {
 643    tminus1 := hm1
 644    tminus1_to_t0 := b01
 645    t0 := h0
 646    t0_to_t1 := b12
 647    t1 := h1
 648    t1_to_t2 := b23
 649    t2 := h2
 650    t0_t2_to_t3 := b03
 651    t3 := h3
 652    t2_t3_to_t4 := b34
 653    t4 := h4
 654    t4_to_t5 := b45
 655    t4_to_t5_cost := b45c
 656    t5 := h5
 657    t5_to_t6 := b56
 658    t6 := h6
 659    t8 := h8
 660    t8_to_t7 := b87
 661    t7 := h7
 662  }
 663
 664theorem complete_forcing_chain_t8_nonempty : Nonempty CompleteForcingChainT8 :=
 665  ⟨complete_forcing_chain_t8⟩
 666
 667/-! ## T-2 through T8 public certificate -/
 668
 669/-- Public core certificate from the T-2 "absolute nothing" floor through T8.
 670
 671`NothingToDistinction.nothingToDistinctionCert` discharges the meta-language and
 672object-distinction floor from the Lean encoding of `Empty`; `CompleteForcingChainT8`
 673then carries the public T-1 through T8 forcing spine. -/
 674structure CompleteForcingChainTMinus2ToT8 : Prop where
 675  tminus2_to_tminus1 : NothingToDistinction.NothingToDistinctionCert
 676  tminus1_to_t8 : Nonempty CompleteForcingChainT8
 677  circle_h1_nonzero : MathlibCohomologyBridge.circleH1ZNonzero
 678  circle_h1_iso_int : MathlibCohomologyBridge.circleH1ZIsoInt
 679  mathlib_circle_linking_backend :
 680    Nonempty MathlibCohomologyBridge.MathlibCircleLinkingBackend
 681
 682/-- The public T-2 through T8 forcing certificate is theorem-backed. -/
 683theorem complete_forcing_chain_tminus2_to_t8 :
 684    CompleteForcingChainTMinus2ToT8 where
 685  tminus2_to_tminus1 := NothingToDistinction.nothingToDistinctionCert
 686  tminus1_to_t8 := complete_forcing_chain_t8_nonempty
 687  circle_h1_nonzero := CircleWindingChain.circleH1ZNonzero_unconditional
 688  circle_h1_iso_int := CircleWindingChain.circleH1ZIsoInt_holds
 689  mathlib_circle_linking_backend := CircleWindingChain.mathlibCircleLinkingBackend_holds
 690
 691end TMinus1ToT8Bridge
 692end Foundation
 693end IndisputableMonolith
 694