Encyclopedia Foundation Foundation Primitive Recognition Calculus Kernel Kernel First Pass Certificate
ARTICLE 3 claims 3 theorems
Foundation Primitive Recognition Calculus Kernel Kernel First Pass Certificate
A machine-checked certificate proves the framework's first chain of reasoning has concrete objects at every stage, without yet proving the chain's final conclusion.
The first-pass certificate
A certificate in formal mathematics is like a building inspector's sign-off on the foundation before the walls go up. The declaration kernel_first_pass_certificate is exactly that: a machine-checked proof that the first stage of the Recognition Science framework's reasoning has concrete, named objects for each step in its opening theorem chain. The certificate itself is a bundle, a single proposition that packages together the existence of those objects, and the framework's library proves the bundle holds.
The objects in question are stable finite-trace predicates, which are rules that read a finite record of events and return a stable yes or no answer. The certificate shows these predicates exist and carry the first logical surface of the framework's primitive recognition calculus, the basic system for describing how recognition events accumulate. Think of it as proving that the alphabet and grammar of a language exist before claiming the language can express a particular novel.
What the certificate does not do is just as important. It is explicitly a first-pass bundling certificate, not the final inevitability theorem. It does not prove that the framework's central cost function J(x) = (x + 1/x)/2 - 1 is forced, nor that the golden ratio emerges as the unique self-similar scaling, nor that three spatial dimensions follow. Those later results live in other declarations, built on top of this foundation. The certificate only shows the foundation has the right shape, that each stage in the first chain has a concrete object to point at.
In Recognition Science, this matters because the framework's whole method is to force structure from a ledger, a discrete record of recognition events, rather than assume it. A first-pass certificate is the framework's way of checking its own work early: before claiming the building will stand, it proves the foundation is real and not a sketch. The payoff for the reader is a clear picture of what is established at this stage, and a precise boundary around what remains to be proved.
THEOREM kernel_first_pass_certificate · IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/Kernel.lean
/-- K7/A2. The first-pass kernel certificate is inhabited. -/
theorem kernel_first_pass_certificate :
KernelFirstPassCertificate where
strength_tags_exist := ⟨StrengthTag.deltaOnly⟩
trace_syntax_exists := ⟨Trace.empty⟩
judgment_surface_exists := ⟨verifierEqualityJudgment⟩
trace_logic := trace_logic_certificate
formal_system := formal_system_certificate
inevitability := prc_inevitability_certificate
quotient_surface_exists := by
intro J T
exact ⟨endpointClassOf J T Endpoint.left⟩
orbit_nat_equivalence := ⟨DistinctionNat.equivNat⟩
orbit_add_faithful := DistinctionNat.toNat_add
orbit_mul_faithful := DistinctionNat.toNat_mul
prc_integer_surface_exists := ⟨PRCInt.zero⟩
prc_rational_surface_exists := by
let one : DistinctionNat := DistinctionNat.succ DistinctionNat.zero
have hone : one ≠ DistinctionNat.zero := by
intro h
exact DistinctionNat.zero_ne_succ DistinctionNat.zero h.symm
exact ⟨PRCRat.mk ⟨SignedOrbit.zero, one, hone⟩⟩
prc_int_balanced_iff_display := SignedOrbit.balanced_iff_toInt_eq
prc_rat_cross_iff_display := RatioOrbit.crossEq_iff_toRat_eq
prc_int_display_injective := PRCInt.toInt_injective
prc_rat_display_injective := PRCRat.toRat_injective
prc_int_equiv_int := ⟨PRCInt.equivInt⟩
integer_order := integer_order_certificate
orbit_divisibility := DistinctionNat.orbit_divisibility_certificate
orbit_euclidean := DistinctionNat.orbit_euclidean_certificate
prc_jcost := PRCJCost.prc_jcost_certificate
rational_field := rational_field_certificate
recognizer_bridge := prc_recognizer_bridge_certificate
prc_int_add_comm := PRCInt.add_comm
prc_int_add_assoc := PRCInt.add_assoc
prc_int_mul_comm := PRCInt.mul_comm
prc_int_mul_assoc := PRCInt.mul_assoc
prc_int_left_distrib := PRCInt.left_distrib
prc_int_add_negate := PRCInt.add_negate
prc_rat_display_add := PRCRat.toRat_add'
prc_rat_display_mul := PRCRat.toRat_mul'
prc_rat_display_inv := PRCRat.toRat_inv'
ratio_recip_internal_den := by
intro a h
rfl
prc_rat_add_comm := PRCRat.add_comm
prc_rat_mul_comm := PRCRat.mul_comm
prc_rat_left_distrib := PRCRat.left_distrib
prc_rat_mul_recip_cancel := by
intro a h
simpa using PRCRat.mul_recip_cancel (a := a) h
trace_closure_boundary := trace_closure_certificate
real_cauchy := real_cauchy_certificate
real_null_setoid := real_null_setoid_conditional_certificate
jcost_distance_triangle := prc_jcost_distance_triangle_conditional_certificate
jcost_distance_verifier_triangle :=
prc_jcost_distance_verifier_triangle_conditional_certificate
jcost_distance_increment_triangle :=
prc_jcost_distance_increment_triangle_certificate
real_complete_ordered_field :=
prc_real_complete_ordered_field_conditional_certificate
real_mul_bounded_continuity :=
prc_real_mul_bounded_continuity_conditional_certificate
real_boundedness_modulus :=
prc_real_boundedness_modulus_certificate
real_product_continuity :=
prc_real_product_continuity_certificate
real_order_congruence :=
prc_real_order_congruence_certificate
real_completeness :=
prc_real_completeness_sharpened_certificate
real_complete_ordered_field_promoted :=
prc_real_complete_ordered_field_promoted_certificate
real_completion_boundary := real_completion_boundary_certificate
signed_orbit_display := ⟨SignedOrbit.toInt⟩
ratio_orbit_display := ⟨RatioOrbit.toRat⟩
THEOREM KernelFirstPassCertificate · IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/Kernel.lean
/-- K7/A2. First-pass PRC kernel certificate:
the analytic specification has concrete Lean objects for each stage in the
first theorem chain. This is a bundling certificate, not yet the final
inevitability theorem. -/
structure KernelFirstPassCertificate : Prop where
strength_tags_exist : Nonempty StrengthTag
trace_syntax_exists : Nonempty Trace
judgment_surface_exists : Nonempty TraceJudgment
/-- Stable finite-trace predicates carry the first PRC logic surface. -/
trace_logic : TraceLogicCertificate
/-- Expressive formal systems admit a PRC trace embedding. -/
formal_system : FormalSystemCertificate
/-- Every admissible foundation carries a PRC trace core. -/
inevitability : PRCInevitabilityCertificate
quotient_surface_exists :
∀ (J : TraceJudgment) (T : Trace), Nonempty (EndpointClass J T)
orbit_nat_equivalence : Nonempty (DistinctionNat ≃ Nat)
/-- Orbit addition is verifier-faithful. -/
orbit_add_faithful :
∀ a b : DistinctionNat, (a + b).toNat = a.toNat + b.toNat
/-- Orbit multiplication is verifier-faithful. -/
orbit_mul_faithful :
∀ a b : DistinctionNat, (a * b).toNat = a.toNat * b.toNat
prc_integer_surface_exists : Nonempty PRCInt
prc_rational_surface_exists : Nonempty PRCRat
/-- The internal balanced-length relation characterizes signed-orbit
equivalence in PRC integers (K4.9). -/
prc_int_balanced_iff_display :
∀ a b : SignedOrbit,
SignedOrbit.balanced a b ↔ a.toInt = b.toInt
/-- The internal cross-multiplication relation characterizes ratio-orbit
equivalence in PRC rationals (K4.10). -/
prc_rat_cross_iff_display :
∀ a b : RatioOrbit,
RatioOrbit.crossEq a b ↔ a.toRat = b.toRat
/-- The PRC integer display is injective on the quotient. -/
prc_int_display_injective : Function.Injective PRCInt.toInt
/-- The PRC rational display is injective on the quotient. -/
prc_rat_display_injective : Function.Injective PRCRat.toRat
/-- The PRC integer surface is isomorphic to verifier `ℤ`. -/
prc_int_equiv_int : Nonempty (PRCInt ≃ ℤ)
/-- Internal signed-orbit order and absolute value are closed. -/
integer_order : IntegerOrderCertificate
/-- Native orbit divisibility, units, factorization, and prime-orbit predicates are closed. -/
orbit_divisibility : DistinctionNat.OrbitDivisibilityCertificate
/-- Native Euclidean quotient/remainder, GCD, and coprime predicates are closed. -/
orbit_euclidean : DistinctionNat.OrbitEuclideanCertificate
/-- PRC rational J-cost, canonical RCL, and bridge to existing real uniqueness are closed. -/
prc_jcost : PRCJCost.PRCJCostCertificate
/-- PRC rational field-style laws, division, positivity, and quotient-level J-cost are packaged. -/
rational_field : RationalFieldCertificate
/-- Positive PRC ratios carry recognition cost and bridge to the Law-of-Logic J-cost chain. -/
recognizer_bridge : PRCRecognizerBridgeCertificate
/-- Addition on PRC integers is commutative. -/
prc_int_add_comm : ∀ a b : PRCInt, a + b = b + a
/-- Addition on PRC integers is associative. -/
prc_int_add_assoc : ∀ a b c : PRCInt, a + b + c = a + (b + c)
/-- Multiplication on PRC integers is commutative. -/
prc_int_mul_comm : ∀ a b : PRCInt, a * b = b * a
/-- Multiplication on PRC integers is associative. -/
prc_int_mul_assoc : ∀ a b c : PRCInt, a * b * c = a * (b * c)
/-- Multiplication distributes over addition. -/
prc_int_left_distrib :
∀ a b c : PRCInt, a * (b + c) = a * b + a * c
/-- Negation is the additive inverse on PRC integers. -/
prc_int_add_negate : ∀ a : PRCInt, a + (-a) = 0
/-- The PRC rational display preserves addition. -/
prc_rat_display_add :
∀ a b : PRCRat, (a + b).toRat = a.toRat + b.toRat
/-- The PRC rational display preserves multiplication. -/
prc_rat_display_mul :
∀ a b : PRCRat, (a * b).toRat = a.toRat * b.toRat
/-- The PRC rational display preserves reciprocal. -/
prc_rat_display_inv :
∀ a : PRCRat, (a⁻¹).toRat = (a.toRat)⁻¹
/-- Ratio reciprocal uses internal signed-orbit absolute value for the denominator. -/
ratio_recip_internal_den :
∀ (a : RatioOrbit)
(h : ¬ SignedOrbit.balanced a.num SignedOrbit.zero),
(RatioOrbit.recipNonzero a h).den = a.num.abs
/-- Addition on PRC rationals is commutative. -/
prc_rat_add_comm : ∀ a b : PRCRat, a + b = b + a
/-- Multiplication on PRC rationals is commutative. -/
prc_rat_mul_comm : ∀ a b : PRCRat, a * b = b * a
/-- Multiplication distributes over addition on PRC rationals. -/
prc_rat_left_distrib :
∀ a b c : PRCRat, a * (b + c) = a * b + a * c
/-- Nonzero PRC rationals multiply by their reciprocal to one. -/
prc_rat_mul_recip_cancel :
∀ a : PRCRat, a.toRat ≠ 0 → a * a⁻¹ = 1
/-- The completed trace boundary is inhabited and trace-closure tagged. -/
trace_closure_boundary : TraceClosureCertificate
/-- Internal Cauchy ledgers and the first PRC real quotient carrier are trace-closure tagged. -/
real_cauchy : PRCRealCauchyCertificate
/-- The final null-distance setoid is reduced to the exact J-cost triangle-modulus target. -/
real_null_setoid : PRCRealNullSetoidConditionalCertificate
/-- The J-cost distance triangle target is reduced to an exact rational display inequality. -/
jcost_distance_triangle : PRCJCostDistanceTriangleConditionalCertificate
/-- The verifier-rational triangle target is reduced to an increment-only modulus. -/
jcost_distance_verifier_triangle : PRCJCostDistanceVerifierTriangleConditionalCertificate
/-- The increment-only J-cost modulus closes the null-distance setoid chain. -/
jcost_distance_increment_triangle : PRCJCostDistanceIncrementTriangleCertificate
/-- The first complete ordered field pass closes add/neg and names mul/order/completeness targets. -/
real_complete_ordered_field : PRCRealCompleteOrderedFieldConditionalCertificate
/-- Multiplication on the null quotient is reduced to boundedness and bounded product continuity. -/
real_mul_bounded_continuity : PRCRealMulBoundedContinuityConditionalCertificate
/-- Every J-cost Cauchy ledger is eventually PRC-bounded. -/
real_boundedness_modulus : PRCRealBoundednessModulusCertificate
/-- Bounded product-continuity closes multiplication on the null quotient. -/
real_product_continuity : PRCRealProductContinuityCertificate
/-- Eventual non-strict order descends to the null-distance quotient. -/
real_order_congruence : PRCRealOrderCongruenceCertificate
/-- Internal completeness is sharpened to the exact Cauchy-of-Cauchy diagonal target. -/
real_completeness : PRCRealCompletenessSharpenedCertificate
/-- The complete ordered-field certificate surface now carries proved mul, order, and completeness targets. -/
real_complete_ordered_field_promoted :
PRCRealCompleteOrderedFieldPromotedCertificate
/-- The real-completion boundary is inhabited and classical-extension tagged. -/
real_completion_boundary : RealCompletionBoundaryCertificate
signed_orbit_display : Nonempty (SignedOrbit → ℤ)
ratio_orbit_display : Nonempty (RatioOrbit → ℚ)
THEOREM KernelFirstPassCertificate · IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/Kernel.lean
/-- K7/A2. First-pass PRC kernel certificate:
the analytic specification has concrete Lean objects for each stage in the
first theorem chain. This is a bundling certificate, not yet the final
inevitability theorem. -/
structure KernelFirstPassCertificate : Prop where
strength_tags_exist : Nonempty StrengthTag
trace_syntax_exists : Nonempty Trace
judgment_surface_exists : Nonempty TraceJudgment
/-- Stable finite-trace predicates carry the first PRC logic surface. -/
trace_logic : TraceLogicCertificate
/-- Expressive formal systems admit a PRC trace embedding. -/
formal_system : FormalSystemCertificate
/-- Every admissible foundation carries a PRC trace core. -/
inevitability : PRCInevitabilityCertificate
quotient_surface_exists :
∀ (J : TraceJudgment) (T : Trace), Nonempty (EndpointClass J T)
orbit_nat_equivalence : Nonempty (DistinctionNat ≃ Nat)
/-- Orbit addition is verifier-faithful. -/
orbit_add_faithful :
∀ a b : DistinctionNat, (a + b).toNat = a.toNat + b.toNat
/-- Orbit multiplication is verifier-faithful. -/
orbit_mul_faithful :
∀ a b : DistinctionNat, (a * b).toNat = a.toNat * b.toNat
prc_integer_surface_exists : Nonempty PRCInt
prc_rational_surface_exists : Nonempty PRCRat
/-- The internal balanced-length relation characterizes signed-orbit
equivalence in PRC integers (K4.9). -/
prc_int_balanced_iff_display :
∀ a b : SignedOrbit,
SignedOrbit.balanced a b ↔ a.toInt = b.toInt
/-- The internal cross-multiplication relation characterizes ratio-orbit
equivalence in PRC rationals (K4.10). -/
prc_rat_cross_iff_display :
∀ a b : RatioOrbit,
RatioOrbit.crossEq a b ↔ a.toRat = b.toRat
/-- The PRC integer display is injective on the quotient. -/
prc_int_display_injective : Function.Injective PRCInt.toInt
/-- The PRC rational display is injective on the quotient. -/
prc_rat_display_injective : Function.Injective PRCRat.toRat
/-- The PRC integer surface is isomorphic to verifier `ℤ`. -/
prc_int_equiv_int : Nonempty (PRCInt ≃ ℤ)
/-- Internal signed-orbit order and absolute value are closed. -/
integer_order : IntegerOrderCertificate
/-- Native orbit divisibility, units, factorization, and prime-orbit predicates are closed. -/
orbit_divisibility : DistinctionNat.OrbitDivisibilityCertificate
/-- Native Euclidean quotient/remainder, GCD, and coprime predicates are closed. -/
orbit_euclidean : DistinctionNat.OrbitEuclideanCertificate
/-- PRC rational J-cost, canonical RCL, and bridge to existing real uniqueness are closed. -/
prc_jcost : PRCJCost.PRCJCostCertificate
/-- PRC rational field-style laws, division, positivity, and quotient-level J-cost are packaged. -/
rational_field : RationalFieldCertificate
/-- Positive PRC ratios carry recognition cost and bridge to the Law-of-Logic J-cost chain. -/
recognizer_bridge : PRCRecognizerBridgeCertificate
/-- Addition on PRC integers is commutative. -/
prc_int_add_comm : ∀ a b : PRCInt, a + b = b + a
/-- Addition on PRC integers is associative. -/
prc_int_add_assoc : ∀ a b c : PRCInt, a + b + c = a + (b + c)
/-- Multiplication on PRC integers is commutative. -/
prc_int_mul_comm : ∀ a b : PRCInt, a * b = b * a
/-- Multiplication on PRC integers is associative. -/
prc_int_mul_assoc : ∀ a b c : PRCInt, a * b * c = a * (b * c)
/-- Multiplication distributes over addition. -/
prc_int_left_distrib :
∀ a b c : PRCInt, a * (b + c) = a * b + a * c
/-- Negation is the additive inverse on PRC integers. -/
prc_int_add_negate : ∀ a : PRCInt, a + (-a) = 0
/-- The PRC rational display preserves addition. -/
prc_rat_display_add :
∀ a b : PRCRat, (a + b).toRat = a.toRat + b.toRat
/-- The PRC rational display preserves multiplication. -/
prc_rat_display_mul :
∀ a b : PRCRat, (a * b).toRat = a.toRat * b.toRat
/-- The PRC rational display preserves reciprocal. -/
prc_rat_display_inv :
∀ a : PRCRat, (a⁻¹).toRat = (a.toRat)⁻¹
/-- Ratio reciprocal uses internal signed-orbit absolute value for the denominator. -/
ratio_recip_internal_den :
∀ (a : RatioOrbit)
(h : ¬ SignedOrbit.balanced a.num SignedOrbit.zero),
(RatioOrbit.recipNonzero a h).den = a.num.abs
/-- Addition on PRC rationals is commutative. -/
prc_rat_add_comm : ∀ a b : PRCRat, a + b = b + a
/-- Multiplication on PRC rationals is commutative. -/
prc_rat_mul_comm : ∀ a b : PRCRat, a * b = b * a
/-- Multiplication distributes over addition on PRC rationals. -/
prc_rat_left_distrib :
∀ a b c : PRCRat, a * (b + c) = a * b + a * c
/-- Nonzero PRC rationals multiply by their reciprocal to one. -/
prc_rat_mul_recip_cancel :
∀ a : PRCRat, a.toRat ≠ 0 → a * a⁻¹ = 1
/-- The completed trace boundary is inhabited and trace-closure tagged. -/
trace_closure_boundary : TraceClosureCertificate
/-- Internal Cauchy ledgers and the first PRC real quotient carrier are trace-closure tagged. -/
real_cauchy : PRCRealCauchyCertificate
/-- The final null-distance setoid is reduced to the exact J-cost triangle-modulus target. -/
real_null_setoid : PRCRealNullSetoidConditionalCertificate
/-- The J-cost distance triangle target is reduced to an exact rational display inequality. -/
jcost_distance_triangle : PRCJCostDistanceTriangleConditionalCertificate
/-- The verifier-rational triangle target is reduced to an increment-only modulus. -/
jcost_distance_verifier_triangle : PRCJCostDistanceVerifierTriangleConditionalCertificate
/-- The increment-only J-cost modulus closes the null-distance setoid chain. -/
jcost_distance_increment_triangle : PRCJCostDistanceIncrementTriangleCertificate
/-- The first complete ordered field pass closes add/neg and names mul/order/completeness targets. -/
real_complete_ordered_field : PRCRealCompleteOrderedFieldConditionalCertificate
/-- Multiplication on the null quotient is reduced to boundedness and bounded product continuity. -/
real_mul_bounded_continuity : PRCRealMulBoundedContinuityConditionalCertificate
/-- Every J-cost Cauchy ledger is eventually PRC-bounded. -/
real_boundedness_modulus : PRCRealBoundednessModulusCertificate
/-- Bounded product-continuity closes multiplication on the null quotient. -/
real_product_continuity : PRCRealProductContinuityCertificate
/-- Eventual non-strict order descends to the null-distance quotient. -/
real_order_congruence : PRCRealOrderCongruenceCertificate
/-- Internal completeness is sharpened to the exact Cauchy-of-Cauchy diagonal target. -/
real_completeness : PRCRealCompletenessSharpenedCertificate
/-- The complete ordered-field certificate surface now carries proved mul, order, and completeness targets. -/
real_complete_ordered_field_promoted :
PRCRealCompleteOrderedFieldPromotedCertificate
/-- The real-completion boundary is inhabited and classical-extension tagged. -/
real_completion_boundary : RealCompletionBoundaryCertificate
signed_orbit_display : Nonempty (SignedOrbit → ℤ)
ratio_orbit_display : Nonempty (RatioOrbit → ℚ)
What this page does not claim
This certificate does not prove the forced form of the cost function J(x) = (x + 1/x)/2 - 1. This certificate does not derive the golden ratio as the unique self-similar scaling. This certificate does not establish that three spatial dimensions are forced.
Verify this page
Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:
$ lake env lean IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/Kernel.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)
A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.
Derived articles
This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red:
- What concrete objects does the first theorem chain require at each stage?
- How do stable finite-trace predicates compose into the full recognition calculus?
- Which later theorem first builds on this certificate to derive the cost function J?
- What distinguishes a bundling certificate from the final inevitability theorem in the framework's method?
MACHINE LAYER · GROUNDED CLAIM TABLE · CLICK TO EXPAND
THEOREM kernel_first_pass_certificate · IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/Kernel.lean
/-- K7/A2. The first-pass kernel certificate is inhabited. -/ theorem kernel_first_pass_certificate : KernelFirstPassCertificate where strength_tags_exist := ⟨StrengthTag.deltaOnly⟩ trace_syntax_exists := ⟨Trace.empty⟩ judgment_surface_exists := ⟨verifierEqualityJudgment⟩ trace_logic := trace_logic_certificate formal_system := formal_system_certificate inevitability := prc_inevitability_certificate quotient_surface_exists := by intro J T exact ⟨endpointClassOf J T Endpoint.left⟩ orbit_nat_equivalence := ⟨DistinctionNat.equivNat⟩ orbit_add_faithful := DistinctionNat.toNat_add orbit_mul_faithful := DistinctionNat.toNat_mul prc_integer_surface_exists := ⟨PRCInt.zero⟩ prc_rational_surface_exists := by let one : DistinctionNat := DistinctionNat.succ DistinctionNat.zero have hone : one ≠ DistinctionNat.zero := by intro h exact DistinctionNat.zero_ne_succ DistinctionNat.zero h.symm exact ⟨PRCRat.mk ⟨SignedOrbit.zero, one, hone⟩⟩ prc_int_balanced_iff_display := SignedOrbit.balanced_iff_toInt_eq prc_rat_cross_iff_display := RatioOrbit.crossEq_iff_toRat_eq prc_int_display_injective := PRCInt.toInt_injective prc_rat_display_injective := PRCRat.toRat_injective prc_int_equiv_int := ⟨PRCInt.equivInt⟩ integer_order := integer_order_certificate orbit_divisibility := DistinctionNat.orbit_divisibility_certificate orbit_euclidean := DistinctionNat.orbit_euclidean_certificate prc_jcost := PRCJCost.prc_jcost_certificate rational_field := rational_field_certificate recognizer_bridge := prc_recognizer_bridge_certificate prc_int_add_comm := PRCInt.add_comm prc_int_add_assoc := PRCInt.add_assoc prc_int_mul_comm := PRCInt.mul_comm prc_int_mul_assoc := PRCInt.mul_assoc prc_int_left_distrib := PRCInt.left_distrib prc_int_add_negate := PRCInt.add_negate prc_rat_display_add := PRCRat.toRat_add' prc_rat_display_mul := PRCRat.toRat_mul' prc_rat_display_inv := PRCRat.toRat_inv' ratio_recip_internal_den := by intro a h rfl prc_rat_add_comm := PRCRat.add_comm prc_rat_mul_comm := PRCRat.mul_comm prc_rat_left_distrib := PRCRat.left_distrib prc_rat_mul_recip_cancel := by intro a h simpa using PRCRat.mul_recip_cancel (a := a) h trace_closure_boundary := trace_closure_certificate real_cauchy := real_cauchy_certificate real_null_setoid := real_null_setoid_conditional_certificate jcost_distance_triangle := prc_jcost_distance_triangle_conditional_certificate jcost_distance_verifier_triangle := prc_jcost_distance_verifier_triangle_conditional_certificate jcost_distance_increment_triangle := prc_jcost_distance_increment_triangle_certificate real_complete_ordered_field := prc_real_complete_ordered_field_conditional_certificate real_mul_bounded_continuity := prc_real_mul_bounded_continuity_conditional_certificate real_boundedness_modulus := prc_real_boundedness_modulus_certificate real_product_continuity := prc_real_product_continuity_certificate real_order_congruence := prc_real_order_congruence_certificate real_completeness := prc_real_completeness_sharpened_certificate real_complete_ordered_field_promoted := prc_real_complete_ordered_field_promoted_certificate real_completion_boundary := real_completion_boundary_certificate signed_orbit_display := ⟨SignedOrbit.toInt⟩ ratio_orbit_display := ⟨RatioOrbit.toRat⟩The declaration kernel_first_pass_certificate is a machine-checked proof that the first stage of the Recognition Science framework's reasoning has concrete, named objects for each step in its opening theorem chain. kernel_first_pass_certificate · IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/Kernel.leanTHEOREM KernelFirstPassCertificate · IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/Kernel.lean
/-- K7/A2. First-pass PRC kernel certificate: the analytic specification has concrete Lean objects for each stage in the first theorem chain. This is a bundling certificate, not yet the final inevitability theorem. -/ structure KernelFirstPassCertificate : Prop where strength_tags_exist : Nonempty StrengthTag trace_syntax_exists : Nonempty Trace judgment_surface_exists : Nonempty TraceJudgment /-- Stable finite-trace predicates carry the first PRC logic surface. -/ trace_logic : TraceLogicCertificate /-- Expressive formal systems admit a PRC trace embedding. -/ formal_system : FormalSystemCertificate /-- Every admissible foundation carries a PRC trace core. -/ inevitability : PRCInevitabilityCertificate quotient_surface_exists : ∀ (J : TraceJudgment) (T : Trace), Nonempty (EndpointClass J T) orbit_nat_equivalence : Nonempty (DistinctionNat ≃ Nat) /-- Orbit addition is verifier-faithful. -/ orbit_add_faithful : ∀ a b : DistinctionNat, (a + b).toNat = a.toNat + b.toNat /-- Orbit multiplication is verifier-faithful. -/ orbit_mul_faithful : ∀ a b : DistinctionNat, (a * b).toNat = a.toNat * b.toNat prc_integer_surface_exists : Nonempty PRCInt prc_rational_surface_exists : Nonempty PRCRat /-- The internal balanced-length relation characterizes signed-orbit equivalence in PRC integers (K4.9). -/ prc_int_balanced_iff_display : ∀ a b : SignedOrbit, SignedOrbit.balanced a b ↔ a.toInt = b.toInt /-- The internal cross-multiplication relation characterizes ratio-orbit equivalence in PRC rationals (K4.10). -/ prc_rat_cross_iff_display : ∀ a b : RatioOrbit, RatioOrbit.crossEq a b ↔ a.toRat = b.toRat /-- The PRC integer display is injective on the quotient. -/ prc_int_display_injective : Function.Injective PRCInt.toInt /-- The PRC rational display is injective on the quotient. -/ prc_rat_display_injective : Function.Injective PRCRat.toRat /-- The PRC integer surface is isomorphic to verifier `ℤ`. -/ prc_int_equiv_int : Nonempty (PRCInt ≃ ℤ) /-- Internal signed-orbit order and absolute value are closed. -/ integer_order : IntegerOrderCertificate /-- Native orbit divisibility, units, factorization, and prime-orbit predicates are closed. -/ orbit_divisibility : DistinctionNat.OrbitDivisibilityCertificate /-- Native Euclidean quotient/remainder, GCD, and coprime predicates are closed. -/ orbit_euclidean : DistinctionNat.OrbitEuclideanCertificate /-- PRC rational J-cost, canonical RCL, and bridge to existing real uniqueness are closed. -/ prc_jcost : PRCJCost.PRCJCostCertificate /-- PRC rational field-style laws, division, positivity, and quotient-level J-cost are packaged. -/ rational_field : RationalFieldCertificate /-- Positive PRC ratios carry recognition cost and bridge to the Law-of-Logic J-cost chain. -/ recognizer_bridge : PRCRecognizerBridgeCertificate /-- Addition on PRC integers is commutative. -/ prc_int_add_comm : ∀ a b : PRCInt, a + b = b + a /-- Addition on PRC integers is associative. -/ prc_int_add_assoc : ∀ a b c : PRCInt, a + b + c = a + (b + c) /-- Multiplication on PRC integers is commutative. -/ prc_int_mul_comm : ∀ a b : PRCInt, a * b = b * a /-- Multiplication on PRC integers is associative. -/ prc_int_mul_assoc : ∀ a b c : PRCInt, a * b * c = a * (b * c) /-- Multiplication distributes over addition. -/ prc_int_left_distrib : ∀ a b c : PRCInt, a * (b + c) = a * b + a * c /-- Negation is the additive inverse on PRC integers. -/ prc_int_add_negate : ∀ a : PRCInt, a + (-a) = 0 /-- The PRC rational display preserves addition. -/ prc_rat_display_add : ∀ a b : PRCRat, (a + b).toRat = a.toRat + b.toRat /-- The PRC rational display preserves multiplication. -/ prc_rat_display_mul : ∀ a b : PRCRat, (a * b).toRat = a.toRat * b.toRat /-- The PRC rational display preserves reciprocal. -/ prc_rat_display_inv : ∀ a : PRCRat, (a⁻¹).toRat = (a.toRat)⁻¹ /-- Ratio reciprocal uses internal signed-orbit absolute value for the denominator. -/ ratio_recip_internal_den : ∀ (a : RatioOrbit) (h : ¬ SignedOrbit.balanced a.num SignedOrbit.zero), (RatioOrbit.recipNonzero a h).den = a.num.abs /-- Addition on PRC rationals is commutative. -/ prc_rat_add_comm : ∀ a b : PRCRat, a + b = b + a /-- Multiplication on PRC rationals is commutative. -/ prc_rat_mul_comm : ∀ a b : PRCRat, a * b = b * a /-- Multiplication distributes over addition on PRC rationals. -/ prc_rat_left_distrib : ∀ a b c : PRCRat, a * (b + c) = a * b + a * c /-- Nonzero PRC rationals multiply by their reciprocal to one. -/ prc_rat_mul_recip_cancel : ∀ a : PRCRat, a.toRat ≠ 0 → a * a⁻¹ = 1 /-- The completed trace boundary is inhabited and trace-closure tagged. -/ trace_closure_boundary : TraceClosureCertificate /-- Internal Cauchy ledgers and the first PRC real quotient carrier are trace-closure tagged. -/ real_cauchy : PRCRealCauchyCertificate /-- The final null-distance setoid is reduced to the exact J-cost triangle-modulus target. -/ real_null_setoid : PRCRealNullSetoidConditionalCertificate /-- The J-cost distance triangle target is reduced to an exact rational display inequality. -/ jcost_distance_triangle : PRCJCostDistanceTriangleConditionalCertificate /-- The verifier-rational triangle target is reduced to an increment-only modulus. -/ jcost_distance_verifier_triangle : PRCJCostDistanceVerifierTriangleConditionalCertificate /-- The increment-only J-cost modulus closes the null-distance setoid chain. -/ jcost_distance_increment_triangle : PRCJCostDistanceIncrementTriangleCertificate /-- The first complete ordered field pass closes add/neg and names mul/order/completeness targets. -/ real_complete_ordered_field : PRCRealCompleteOrderedFieldConditionalCertificate /-- Multiplication on the null quotient is reduced to boundedness and bounded product continuity. -/ real_mul_bounded_continuity : PRCRealMulBoundedContinuityConditionalCertificate /-- Every J-cost Cauchy ledger is eventually PRC-bounded. -/ real_boundedness_modulus : PRCRealBoundednessModulusCertificate /-- Bounded product-continuity closes multiplication on the null quotient. -/ real_product_continuity : PRCRealProductContinuityCertificate /-- Eventual non-strict order descends to the null-distance quotient. -/ real_order_congruence : PRCRealOrderCongruenceCertificate /-- Internal completeness is sharpened to the exact Cauchy-of-Cauchy diagonal target. -/ real_completeness : PRCRealCompletenessSharpenedCertificate /-- The complete ordered-field certificate surface now carries proved mul, order, and completeness targets. -/ real_complete_ordered_field_promoted : PRCRealCompleteOrderedFieldPromotedCertificate /-- The real-completion boundary is inhabited and classical-extension tagged. -/ real_completion_boundary : RealCompletionBoundaryCertificate signed_orbit_display : Nonempty (SignedOrbit → ℤ) ratio_orbit_display : Nonempty (RatioOrbit → ℚ)The certificate shows stable finite-trace predicates exist and carry the first logical surface of the framework's primitive recognition calculus. KernelFirstPassCertificate · IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/Kernel.leanTHEOREM KernelFirstPassCertificate · IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/Kernel.lean
/-- K7/A2. First-pass PRC kernel certificate: the analytic specification has concrete Lean objects for each stage in the first theorem chain. This is a bundling certificate, not yet the final inevitability theorem. -/ structure KernelFirstPassCertificate : Prop where strength_tags_exist : Nonempty StrengthTag trace_syntax_exists : Nonempty Trace judgment_surface_exists : Nonempty TraceJudgment /-- Stable finite-trace predicates carry the first PRC logic surface. -/ trace_logic : TraceLogicCertificate /-- Expressive formal systems admit a PRC trace embedding. -/ formal_system : FormalSystemCertificate /-- Every admissible foundation carries a PRC trace core. -/ inevitability : PRCInevitabilityCertificate quotient_surface_exists : ∀ (J : TraceJudgment) (T : Trace), Nonempty (EndpointClass J T) orbit_nat_equivalence : Nonempty (DistinctionNat ≃ Nat) /-- Orbit addition is verifier-faithful. -/ orbit_add_faithful : ∀ a b : DistinctionNat, (a + b).toNat = a.toNat + b.toNat /-- Orbit multiplication is verifier-faithful. -/ orbit_mul_faithful : ∀ a b : DistinctionNat, (a * b).toNat = a.toNat * b.toNat prc_integer_surface_exists : Nonempty PRCInt prc_rational_surface_exists : Nonempty PRCRat /-- The internal balanced-length relation characterizes signed-orbit equivalence in PRC integers (K4.9). -/ prc_int_balanced_iff_display : ∀ a b : SignedOrbit, SignedOrbit.balanced a b ↔ a.toInt = b.toInt /-- The internal cross-multiplication relation characterizes ratio-orbit equivalence in PRC rationals (K4.10). -/ prc_rat_cross_iff_display : ∀ a b : RatioOrbit, RatioOrbit.crossEq a b ↔ a.toRat = b.toRat /-- The PRC integer display is injective on the quotient. -/ prc_int_display_injective : Function.Injective PRCInt.toInt /-- The PRC rational display is injective on the quotient. -/ prc_rat_display_injective : Function.Injective PRCRat.toRat /-- The PRC integer surface is isomorphic to verifier `ℤ`. -/ prc_int_equiv_int : Nonempty (PRCInt ≃ ℤ) /-- Internal signed-orbit order and absolute value are closed. -/ integer_order : IntegerOrderCertificate /-- Native orbit divisibility, units, factorization, and prime-orbit predicates are closed. -/ orbit_divisibility : DistinctionNat.OrbitDivisibilityCertificate /-- Native Euclidean quotient/remainder, GCD, and coprime predicates are closed. -/ orbit_euclidean : DistinctionNat.OrbitEuclideanCertificate /-- PRC rational J-cost, canonical RCL, and bridge to existing real uniqueness are closed. -/ prc_jcost : PRCJCost.PRCJCostCertificate /-- PRC rational field-style laws, division, positivity, and quotient-level J-cost are packaged. -/ rational_field : RationalFieldCertificate /-- Positive PRC ratios carry recognition cost and bridge to the Law-of-Logic J-cost chain. -/ recognizer_bridge : PRCRecognizerBridgeCertificate /-- Addition on PRC integers is commutative. -/ prc_int_add_comm : ∀ a b : PRCInt, a + b = b + a /-- Addition on PRC integers is associative. -/ prc_int_add_assoc : ∀ a b c : PRCInt, a + b + c = a + (b + c) /-- Multiplication on PRC integers is commutative. -/ prc_int_mul_comm : ∀ a b : PRCInt, a * b = b * a /-- Multiplication on PRC integers is associative. -/ prc_int_mul_assoc : ∀ a b c : PRCInt, a * b * c = a * (b * c) /-- Multiplication distributes over addition. -/ prc_int_left_distrib : ∀ a b c : PRCInt, a * (b + c) = a * b + a * c /-- Negation is the additive inverse on PRC integers. -/ prc_int_add_negate : ∀ a : PRCInt, a + (-a) = 0 /-- The PRC rational display preserves addition. -/ prc_rat_display_add : ∀ a b : PRCRat, (a + b).toRat = a.toRat + b.toRat /-- The PRC rational display preserves multiplication. -/ prc_rat_display_mul : ∀ a b : PRCRat, (a * b).toRat = a.toRat * b.toRat /-- The PRC rational display preserves reciprocal. -/ prc_rat_display_inv : ∀ a : PRCRat, (a⁻¹).toRat = (a.toRat)⁻¹ /-- Ratio reciprocal uses internal signed-orbit absolute value for the denominator. -/ ratio_recip_internal_den : ∀ (a : RatioOrbit) (h : ¬ SignedOrbit.balanced a.num SignedOrbit.zero), (RatioOrbit.recipNonzero a h).den = a.num.abs /-- Addition on PRC rationals is commutative. -/ prc_rat_add_comm : ∀ a b : PRCRat, a + b = b + a /-- Multiplication on PRC rationals is commutative. -/ prc_rat_mul_comm : ∀ a b : PRCRat, a * b = b * a /-- Multiplication distributes over addition on PRC rationals. -/ prc_rat_left_distrib : ∀ a b c : PRCRat, a * (b + c) = a * b + a * c /-- Nonzero PRC rationals multiply by their reciprocal to one. -/ prc_rat_mul_recip_cancel : ∀ a : PRCRat, a.toRat ≠ 0 → a * a⁻¹ = 1 /-- The completed trace boundary is inhabited and trace-closure tagged. -/ trace_closure_boundary : TraceClosureCertificate /-- Internal Cauchy ledgers and the first PRC real quotient carrier are trace-closure tagged. -/ real_cauchy : PRCRealCauchyCertificate /-- The final null-distance setoid is reduced to the exact J-cost triangle-modulus target. -/ real_null_setoid : PRCRealNullSetoidConditionalCertificate /-- The J-cost distance triangle target is reduced to an exact rational display inequality. -/ jcost_distance_triangle : PRCJCostDistanceTriangleConditionalCertificate /-- The verifier-rational triangle target is reduced to an increment-only modulus. -/ jcost_distance_verifier_triangle : PRCJCostDistanceVerifierTriangleConditionalCertificate /-- The increment-only J-cost modulus closes the null-distance setoid chain. -/ jcost_distance_increment_triangle : PRCJCostDistanceIncrementTriangleCertificate /-- The first complete ordered field pass closes add/neg and names mul/order/completeness targets. -/ real_complete_ordered_field : PRCRealCompleteOrderedFieldConditionalCertificate /-- Multiplication on the null quotient is reduced to boundedness and bounded product continuity. -/ real_mul_bounded_continuity : PRCRealMulBoundedContinuityConditionalCertificate /-- Every J-cost Cauchy ledger is eventually PRC-bounded. -/ real_boundedness_modulus : PRCRealBoundednessModulusCertificate /-- Bounded product-continuity closes multiplication on the null quotient. -/ real_product_continuity : PRCRealProductContinuityCertificate /-- Eventual non-strict order descends to the null-distance quotient. -/ real_order_congruence : PRCRealOrderCongruenceCertificate /-- Internal completeness is sharpened to the exact Cauchy-of-Cauchy diagonal target. -/ real_completeness : PRCRealCompletenessSharpenedCertificate /-- The complete ordered-field certificate surface now carries proved mul, order, and completeness targets. -/ real_complete_ordered_field_promoted : PRCRealCompleteOrderedFieldPromotedCertificate /-- The real-completion boundary is inhabited and classical-extension tagged. -/ real_completion_boundary : RealCompletionBoundaryCertificate signed_orbit_display : Nonempty (SignedOrbit → ℤ) ratio_orbit_display : Nonempty (RatioOrbit → ℚ)It is explicitly a first-pass bundling certificate, not the final inevitability theorem. KernelFirstPassCertificate · IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/Kernel.lean