IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.Kernel

   1/-
   2  PrimitiveRecognitionCalculus/Kernel.lean
   3
   4  Round-trip source:
   5    PRC_Kernel_Spec_20260526.html
   6
   7  Aggregate import for the first PRC kernel pass:
   8
   9    δ → trace → SameT/DiffT → substitution → quotient → ℕ → ℤ → ℚ
  10
  11  The ℤ and ℚ stages are PRC-owned quotient surfaces with internal
  12  equivalence relations (balanced length for `PRCInt`; cross-multiplication
  13  for `PRCRat`). The maps into Lean's `ℤ` and `ℚ` are conservative verifier
  14  displays, proved well-defined from the internal characterizations.
  15-/
  16
  17import IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.Strength
  18import IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.Basic
  19import IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.SameDiff
  20import IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.TraceLogic
  21import IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.FormalSystem
  22import IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.Inevitability
  23import IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.Quotient
  24import IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.Orbit
  25import IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.OrbitArithmetic
  26import IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.IntegerRational
  27import IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.IntegerOrder
  28import IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.OrbitDivisibility
  29import IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.OrbitEuclidean
  30import IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.PRCJCost
  31import IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.RationalField
  32import IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.RecognizerBridge
  33import IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.TraceClosure
  34import IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.RealCauchy
  35import IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.RealNullSetoid
  36import IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.PRCJCostDistanceTriangle
  37import IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.PRCJCostDistanceVerifierTriangle
  38import IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.PRCJCostDistanceIncrementTriangle
  39import IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.RealCompleteOrderedField
  40import IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.RealMulBoundedContinuity
  41import IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.RealBoundednessModulus
  42import IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.RealProductContinuity
  43import IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.RealOrderCongruence
  44import IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.RealCompleteness
  45import IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.RealCompleteOrderedFieldPromoted
  46import IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.RealCompletion
  47
  48namespace IndisputableMonolith
  49namespace Foundation
  50namespace PrimitiveRecognitionCalculus
  51
  52/-- K7/A2. First-pass PRC kernel certificate:
  53the analytic specification has concrete Lean objects for each stage in the
  54first theorem chain. This is a bundling certificate, not yet the final
  55inevitability theorem. -/
  56structure KernelFirstPassCertificate : Prop where
  57  strength_tags_exist : Nonempty StrengthTag
  58  trace_syntax_exists : Nonempty Trace
  59  judgment_surface_exists : Nonempty TraceJudgment
  60  /-- Stable finite-trace predicates carry the first PRC logic surface. -/
  61  trace_logic : TraceLogicCertificate
  62  /-- Expressive formal systems admit a PRC trace embedding. -/
  63  formal_system : FormalSystemCertificate
  64  /-- Every admissible foundation carries a PRC trace core. -/
  65  inevitability : PRCInevitabilityCertificate
  66  quotient_surface_exists :
  67    ∀ (J : TraceJudgment) (T : Trace), Nonempty (EndpointClass J T)
  68  orbit_nat_equivalence : Nonempty (DistinctionNat ≃ Nat)
  69  /-- Orbit addition is verifier-faithful. -/
  70  orbit_add_faithful :
  71    ∀ a b : DistinctionNat, (a + b).toNat = a.toNat + b.toNat
  72  /-- Orbit multiplication is verifier-faithful. -/
  73  orbit_mul_faithful :
  74    ∀ a b : DistinctionNat, (a * b).toNat = a.toNat * b.toNat
  75  prc_integer_surface_exists : Nonempty PRCInt
  76  prc_rational_surface_exists : Nonempty PRCRat
  77  /-- The internal balanced-length relation characterizes signed-orbit
  78  equivalence in PRC integers (K4.9). -/
  79  prc_int_balanced_iff_display :
  80    ∀ a b : SignedOrbit,
  81      SignedOrbit.balanced a b ↔ a.toInt = b.toInt
  82  /-- The internal cross-multiplication relation characterizes ratio-orbit
  83  equivalence in PRC rationals (K4.10). -/
  84  prc_rat_cross_iff_display :
  85    ∀ a b : RatioOrbit,
  86      RatioOrbit.crossEq a b ↔ a.toRat = b.toRat
  87  /-- The PRC integer display is injective on the quotient. -/
  88  prc_int_display_injective : Function.Injective PRCInt.toInt
  89  /-- The PRC rational display is injective on the quotient. -/
  90  prc_rat_display_injective : Function.Injective PRCRat.toRat
  91  /-- The PRC integer surface is isomorphic to verifier `ℤ`. -/
  92  prc_int_equiv_int : Nonempty (PRCInt ≃ ℤ)
  93  /-- Internal signed-orbit order and absolute value are closed. -/
  94  integer_order : IntegerOrderCertificate
  95  /-- Native orbit divisibility, units, factorization, and prime-orbit predicates are closed. -/
  96  orbit_divisibility : DistinctionNat.OrbitDivisibilityCertificate
  97  /-- Native Euclidean quotient/remainder, GCD, and coprime predicates are closed. -/
  98  orbit_euclidean : DistinctionNat.OrbitEuclideanCertificate
  99  /-- PRC rational J-cost, canonical RCL, and bridge to existing real uniqueness are closed. -/
 100  prc_jcost : PRCJCost.PRCJCostCertificate
 101  /-- PRC rational field-style laws, division, positivity, and quotient-level J-cost are packaged. -/
 102  rational_field : RationalFieldCertificate
 103  /-- Positive PRC ratios carry recognition cost and bridge to the Law-of-Logic J-cost chain. -/
 104  recognizer_bridge : PRCRecognizerBridgeCertificate
 105  /-- Addition on PRC integers is commutative. -/
 106  prc_int_add_comm : ∀ a b : PRCInt, a + b = b + a
 107  /-- Addition on PRC integers is associative. -/
 108  prc_int_add_assoc : ∀ a b c : PRCInt, a + b + c = a + (b + c)
 109  /-- Multiplication on PRC integers is commutative. -/
 110  prc_int_mul_comm : ∀ a b : PRCInt, a * b = b * a
 111  /-- Multiplication on PRC integers is associative. -/
 112  prc_int_mul_assoc : ∀ a b c : PRCInt, a * b * c = a * (b * c)
 113  /-- Multiplication distributes over addition. -/
 114  prc_int_left_distrib :
 115    ∀ a b c : PRCInt, a * (b + c) = a * b + a * c
 116  /-- Negation is the additive inverse on PRC integers. -/
 117  prc_int_add_negate : ∀ a : PRCInt, a + (-a) = 0
 118  /-- The PRC rational display preserves addition. -/
 119  prc_rat_display_add :
 120    ∀ a b : PRCRat, (a + b).toRat = a.toRat + b.toRat
 121  /-- The PRC rational display preserves multiplication. -/
 122  prc_rat_display_mul :
 123    ∀ a b : PRCRat, (a * b).toRat = a.toRat * b.toRat
 124  /-- The PRC rational display preserves reciprocal. -/
 125  prc_rat_display_inv :
 126    ∀ a : PRCRat, (a⁻¹).toRat = (a.toRat)⁻¹
 127  /-- Ratio reciprocal uses internal signed-orbit absolute value for the denominator. -/
 128  ratio_recip_internal_den :
 129    ∀ (a : RatioOrbit)
 130      (h : ¬ SignedOrbit.balanced a.num SignedOrbit.zero),
 131      (RatioOrbit.recipNonzero a h).den = a.num.abs
 132  /-- Addition on PRC rationals is commutative. -/
 133  prc_rat_add_comm : ∀ a b : PRCRat, a + b = b + a
 134  /-- Multiplication on PRC rationals is commutative. -/
 135  prc_rat_mul_comm : ∀ a b : PRCRat, a * b = b * a
 136  /-- Multiplication distributes over addition on PRC rationals. -/
 137  prc_rat_left_distrib :
 138    ∀ a b c : PRCRat, a * (b + c) = a * b + a * c
 139  /-- Nonzero PRC rationals multiply by their reciprocal to one. -/
 140  prc_rat_mul_recip_cancel :
 141    ∀ a : PRCRat, a.toRat ≠ 0 → a * a⁻¹ = 1
 142  /-- The completed trace boundary is inhabited and trace-closure tagged. -/
 143  trace_closure_boundary : TraceClosureCertificate
 144  /-- Internal Cauchy ledgers and the first PRC real quotient carrier are trace-closure tagged. -/
 145  real_cauchy : PRCRealCauchyCertificate
 146  /-- The final null-distance setoid is reduced to the exact J-cost triangle-modulus target. -/
 147  real_null_setoid : PRCRealNullSetoidConditionalCertificate
 148  /-- The J-cost distance triangle target is reduced to an exact rational display inequality. -/
 149  jcost_distance_triangle : PRCJCostDistanceTriangleConditionalCertificate
 150  /-- The verifier-rational triangle target is reduced to an increment-only modulus. -/
 151  jcost_distance_verifier_triangle : PRCJCostDistanceVerifierTriangleConditionalCertificate
 152  /-- The increment-only J-cost modulus closes the null-distance setoid chain. -/
 153  jcost_distance_increment_triangle : PRCJCostDistanceIncrementTriangleCertificate
 154  /-- The first complete ordered field pass closes add/neg and names mul/order/completeness targets. -/
 155  real_complete_ordered_field : PRCRealCompleteOrderedFieldConditionalCertificate
 156  /-- Multiplication on the null quotient is reduced to boundedness and bounded product continuity. -/
 157  real_mul_bounded_continuity : PRCRealMulBoundedContinuityConditionalCertificate
 158  /-- Every J-cost Cauchy ledger is eventually PRC-bounded. -/
 159  real_boundedness_modulus : PRCRealBoundednessModulusCertificate
 160  /-- Bounded product-continuity closes multiplication on the null quotient. -/
 161  real_product_continuity : PRCRealProductContinuityCertificate
 162  /-- Eventual non-strict order descends to the null-distance quotient. -/
 163  real_order_congruence : PRCRealOrderCongruenceCertificate
 164  /-- Internal completeness is sharpened to the exact Cauchy-of-Cauchy diagonal target. -/
 165  real_completeness : PRCRealCompletenessSharpenedCertificate
 166  /-- The complete ordered-field certificate surface now carries proved mul, order, and completeness targets. -/
 167  real_complete_ordered_field_promoted :
 168    PRCRealCompleteOrderedFieldPromotedCertificate
 169  /-- The real-completion boundary is inhabited and classical-extension tagged. -/
 170  real_completion_boundary : RealCompletionBoundaryCertificate
 171  signed_orbit_display : Nonempty (SignedOrbit → ℤ)
 172  ratio_orbit_display : Nonempty (RatioOrbit → ℚ)
 173
 174/-- K7/A2. The first-pass kernel certificate is inhabited. -/
 175theorem kernel_first_pass_certificate :
 176    KernelFirstPassCertificate where
 177  strength_tags_exist := ⟨StrengthTag.deltaOnly⟩
 178  trace_syntax_exists := ⟨Trace.empty⟩
 179  judgment_surface_exists := ⟨verifierEqualityJudgment⟩
 180  trace_logic := trace_logic_certificate
 181  formal_system := formal_system_certificate
 182  inevitability := prc_inevitability_certificate
 183  quotient_surface_exists := by
 184    intro J T
 185    exact ⟨endpointClassOf J T Endpoint.left⟩
 186  orbit_nat_equivalence := ⟨DistinctionNat.equivNat⟩
 187  orbit_add_faithful := DistinctionNat.toNat_add
 188  orbit_mul_faithful := DistinctionNat.toNat_mul
 189  prc_integer_surface_exists := ⟨PRCInt.zero⟩
 190  prc_rational_surface_exists := by
 191    let one : DistinctionNat := DistinctionNat.succ DistinctionNat.zero
 192    have hone : one ≠ DistinctionNat.zero := by
 193      intro h
 194      exact DistinctionNat.zero_ne_succ DistinctionNat.zero h.symm
 195    exact ⟨PRCRat.mk ⟨SignedOrbit.zero, one, hone⟩⟩
 196  prc_int_balanced_iff_display := SignedOrbit.balanced_iff_toInt_eq
 197  prc_rat_cross_iff_display := RatioOrbit.crossEq_iff_toRat_eq
 198  prc_int_display_injective := PRCInt.toInt_injective
 199  prc_rat_display_injective := PRCRat.toRat_injective
 200  prc_int_equiv_int := ⟨PRCInt.equivInt⟩
 201  integer_order := integer_order_certificate
 202  orbit_divisibility := DistinctionNat.orbit_divisibility_certificate
 203  orbit_euclidean := DistinctionNat.orbit_euclidean_certificate
 204  prc_jcost := PRCJCost.prc_jcost_certificate
 205  rational_field := rational_field_certificate
 206  recognizer_bridge := prc_recognizer_bridge_certificate
 207  prc_int_add_comm := PRCInt.add_comm
 208  prc_int_add_assoc := PRCInt.add_assoc
 209  prc_int_mul_comm := PRCInt.mul_comm
 210  prc_int_mul_assoc := PRCInt.mul_assoc
 211  prc_int_left_distrib := PRCInt.left_distrib
 212  prc_int_add_negate := PRCInt.add_negate
 213  prc_rat_display_add := PRCRat.toRat_add'
 214  prc_rat_display_mul := PRCRat.toRat_mul'
 215  prc_rat_display_inv := PRCRat.toRat_inv'
 216  ratio_recip_internal_den := by
 217    intro a h
 218    rfl
 219  prc_rat_add_comm := PRCRat.add_comm
 220  prc_rat_mul_comm := PRCRat.mul_comm
 221  prc_rat_left_distrib := PRCRat.left_distrib
 222  prc_rat_mul_recip_cancel := by
 223    intro a h
 224    simpa using PRCRat.mul_recip_cancel (a := a) h
 225  trace_closure_boundary := trace_closure_certificate
 226  real_cauchy := real_cauchy_certificate
 227  real_null_setoid := real_null_setoid_conditional_certificate
 228  jcost_distance_triangle := prc_jcost_distance_triangle_conditional_certificate
 229  jcost_distance_verifier_triangle :=
 230    prc_jcost_distance_verifier_triangle_conditional_certificate
 231  jcost_distance_increment_triangle :=
 232    prc_jcost_distance_increment_triangle_certificate
 233  real_complete_ordered_field :=
 234    prc_real_complete_ordered_field_conditional_certificate
 235  real_mul_bounded_continuity :=
 236    prc_real_mul_bounded_continuity_conditional_certificate
 237  real_boundedness_modulus :=
 238    prc_real_boundedness_modulus_certificate
 239  real_product_continuity :=
 240    prc_real_product_continuity_certificate
 241  real_order_congruence :=
 242    prc_real_order_congruence_certificate
 243  real_completeness :=
 244    prc_real_completeness_sharpened_certificate
 245  real_complete_ordered_field_promoted :=
 246    prc_real_complete_ordered_field_promoted_certificate
 247  real_completion_boundary := real_completion_boundary_certificate
 248  signed_orbit_display := ⟨SignedOrbit.toInt⟩
 249  ratio_orbit_display := ⟨RatioOrbit.toRat⟩
 250
 251end PrimitiveRecognitionCalculus
 252end Foundation
 253end IndisputableMonolith
 254