Encyclopedia Foundation Foundation Primitive Recognition Calculus Universal Foundation Prc Universal Fou

ARTICLE 3 claims 3 theorems

Foundation Primitive Recognition Calculus Universal Foundation Prc Universal Fou

A machine-checked certificate assembles the framework's proven foundations, while explicitly naming which weaker routes remain open targets.

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The conditional certificate

A ledger, in Recognition Science, is a discrete record of events. The framework's central claim is that reality keeps such a ledger, and that the cost of recognition is forced. The declaration prc_universal_foundation_conditional_certificate is a formal object, a certificate, that bundles together the main proven pieces of that foundation. It is a single structure that carries, by name, the repaired and refuted routes in the native-cost ledger. The certificate is conditional because it does not assert the full universal foundation outright; it asserts that all built PRC surfaces compose, with the exact ledger of which routes are proved and which are refuted.

The certificate is a structure in the framework's machine-checked library of formal theorems. It contains several sub-certificates: a kernel first-pass certificate, a real complete ordered field certificate, a trace logic certificate, a formal system certificate, an inevitability certificate, a recognizer bridge certificate, and a native-cost uniqueness blocker certificate. It also carries an open-targets structure. The theorem prc_universal_foundation_conditional_certificate proves that this structure exists. In plain language, the framework has formally checked that its core components fit together, and it has recorded exactly which parts are settled and which are not.

What the certificate does not claim is as important as what it claims. It does not claim that the unsigned native-cost routes are proved. Those weaker routes are recorded as refuted targets, meaning the framework has shown they cannot force the final surface. The certificate also does not claim that the universal foundation is complete; it explicitly carries an open-targets structure, which lists the remaining targets. The certificate is a checkpoint, not a final destination.

In Recognition Science, this certificate is the top-level conditional certificate. It closes the top-level theorem by carrying the built PRC surfaces together with the exact native-cost ledger. The repaired signed/prime/zero-calibrated uniqueness route is proved, while the weaker unsigned routes are recorded as refuted. This is a precise statement of what is known and what is not, and it is the kind of exact provenance that the framework relies on.

THEOREM prc_universal_foundation_conditional_certificate · IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/UniversalFoundation.lean
prc_universal_foundation_conditional_certificate · IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/UniversalFoundation.lean:1422 · truncated 
theorem prc_universal_foundation_conditional_certificate :
    PRCUniversalFoundationConditionalCertificate where
  kernel := kernel_first_pass_certificate
  real_complete_ordered_field :=
    prc_real_complete_ordered_field_promoted_certificate
  trace_logic := trace_logic_certificate
  formal_system := formal_system_certificate
  inevitability := prc_inevitability_certificate
  recognizer_bridge := prc_recognizer_bridge_certificate
  native_cost_blocker := PRCJCost.prc_native_cost_uniqueness_blocker_certificate
  open_targets := {
    zero_calibrated_native_cost_uniqueness_refuted :=
      PRCJCost.PRCZeroCalibratedNativeCostUniquenessTarget_refuted
    zero_calibrated_native_cost_character_factorization :=
      PRCJCost.PRCZeroCalibratedNativeCostCharacterFactorizationTarget_proved
    zero_calibrated_native_cost_signed_admissible_factorization_refuted :=
      PRCJCost.PRCZeroCalibratedNativeCostSignedAdmissibleCharacterFactorizationTarget_refuted
    zero_calibration_signed_unit_refuted :=
      PRCJCost.PRCZeroCalibrationForcesNativeCostSignedUnitCalibrationTarget_refuted
    zero_calibrated_prime_signed_strengthened_native_cost_uniqueness :=
      PRCJCost.PRCZeroCalibratedPrimeSignedStrengthenedNativeCostUniquenessTarget_proved
    native_cost_signed_admissible_character_rigidity :=
      PRCJCost.PRCNativeCostSignedAdmissibleCharacterRigidityTarget_proved
    old_native_cost_character_rigidity_refuted :=
      PRCJCost.PRCNativeCostCharacterRigidityTarget_refuted
    old_native_cost_sharpened_refuted :=
      PRCJCost.PRCNativeCostUniquenessSharpenedTarget_refuted
    two_to_prime_calibration_refuted :=
      PRCJCost.PRCTwoCalibrationForcesPrimeCalibrationTarget_refuted
    prime_calibration_propagation_refuted :=
      PRCJCost.PRCPrimeCalibrationPropagationTarget_refuted
    prime_global_orientation_refuted :=
      PRCJCost.PRCPrimeCalibrationForcesGlobalOrientationTarget_refuted
    coherent_prime_orientation :=
      PRCJCost.PRCCharacterTwoPrimeBranchControlsPrimes_of_coherent
    two_prime_branch_controls_primes :=
      PRCJCost.PRCCharacterPrimeOrientationCoherent_of_local_two_prime_branch_controls
    prime_identity_iff_two_prime_identity :=
      PRCJCost.PRCCharacterPrimeIdentityIffTwoPrimeIdentity_of_local_two_prime_branch_controls
    prime_identity_forces_two_prime_identity :=
      PRCJCost.PRCCharacterPrimeIdentityForcesTwoPrimeIdentity_of_identity_iff_two
    two_prime_reciprocal_excludes_prime_identity :=
      PRCJCost.PRCCharacterPrimeIdentityForcesTwoPrimeIdentity_iff_two_prime_reciprocal_excludes
    two_prime_reciprocal_excludes_prime_identity_witness :=
      PRCJCost.PRCCharacterTwoPrimeReciprocalExcludesPrimeIdentity_iff_witness
    two_prime_reciprocal_identity_prime_mixed :=
      PRCJCost.PRCCharacterTwoPrimeReciprocalIdentityPrimeMixed_iff_non_two
    two_prime_reciprocal_identity_non_two_prime_mixed :=
      PRCJCost.PRCCharacterTwoPrimeReciprocalIdentityNonTwoPrimeMixed_iff_composite_defect_of_character
    two_prime_reciprocal_identity_non_two_composite_defect :=
      PRCJCost.PRCCharacterTwoPrimeReciprocalIdentityNonTwoCompositeDefect_iff_cost_defect
    two_prime_reciprocal_identity_non_two_composite_cost_defect :=
      PRCJCost.PRCCharacterTwoPrimeReciprocalIdentityNonTwoCompositeDefect_of_cost_defect
    two_prime_mixed_composite_cost_consistency_target_refuted :=
      PRCJCost.PRCPrimeCalibrationForcesTwoPrimeMixedCompositeCostConsistencyTarget_refuted
    prime_pair_product_cost_consistency_target_refuted :=
      PRCJCost.PRCPrimeCalibrationForcesPrimePairProductCostConsistencyTarget_refuted
    two_prime_reciprocal_forces_prime_reciprocal :=
      PRCJCost.PRCCharacterTwoPrimeReciprocalForcesPrimeReciprocal_of_reciprocal_witness_globalizes
    two_prime_reciprocal_trace_connected :=
      PRCJCost.PRCCharacterTwoPrimeReciprocalRespectsTraceConnected_iff_forces
    two_prime_identity_trace_connected :=
      PRCJCost.PRCCharacterTwoPrimeIdentityRespectsTraceConnected_of_prime_identity_trace_connected
    no_mixed_prime_orientation :=
      PRCJCost.PRCCharacterNoMixedPrimeWitnesses_iff_no_mixed_prime_orientation
    mixed_prime_witnesses :=
      PRCJCost.PRCCharacterMixedPrimeWitnesses_iff_pair_witnesses
    mixed_prime_pair_witnesses :=
      PRCJCost.PRCCharacterMixedPrimePairWitnesses_iff_same_or_distinct
    same_prime_mixed_pair_witnesses :=
      PRCJCost.PRCCharacterSamePrimeMixedPairWitnesses_absurd
    distinct_prime_mixed_pair_witnesses :=
      PRCJCost.PRCCharacterDistinctPrimeMixedPairWitnesses_absurd_of_branch_uniform
    prime_identity_witness_excludes_reciprocal :=
      PRCJCost.PRCCharacterPrimeIdentityWitnessExcludesReciprocal_iff_no_mixed_prime_orientation
    prime_reciprocal_witness_globalizes :=
      PRCJCost.PRCCharacterPrimeReciprocalWitnessGlobalizes_of_local_no_mixed_prime_orientation
    prime_reciprocal_forces_two_prime_reciprocal :=
      PRCJCost.PRCCharacterPrimeReciprocalForcesTwoPrimeReciprocal_of_reciprocal_witness_globalizes
    prime_reciprocal_witness_globalizes_split :=
      PRCJCost.PRCCharacterPrimeReciprocalWitnessGlobalizes_iff_split
    prime_reciprocal_forces_two_from_reciprocal_twist_identity_forces_two := by
      intro χ
      exact PRCJCost.PRCCharacterPrimeReciprocalForcesTwoPrimeReciprocal_of_reciprocal_twist_identity_forces_two
    prime_identity_forces_two_from_reciprocal_twist_reciprocal_forces_two := by
      intro χ
      exact PRCJCost.PRCCharacterPrimeIdentityForcesTwoPrimeIdentity_of_reciprocal_twist_reciprocal_forces_two
    prime_identity_trace_coherence :=
      PRCJCost.PRCCharacterPrimeIdentityBranchUniform_iff_trace_coherence
    prime_identity_branch_uniform :=
      PRCJCost.PRCCharacterPrimeIdentityBranchUniform_iff_identity_iff_two
    prime_axis_trace_connected := PRCJCost.PRCPrimeAxisTraceConnected_proved
    prime_identity_respects_trace_connected :=
      PRCJCost.PRCCharacterPrimeIdentityRespectsCanonicalAddTrace_iff_trace_connected
    prime_identity_respects_common_trace_extension :=
      PRCJCost.PRCCharacterPrimeIdentityRespectsCanonicalAddTrace_iff_common_trace_extension
    prime_identity_respects_canonical_add_trace :=
      PRCJCost.PRCCharacterPrimeIdentityBranchUniform_iff_canonical_add_trace
    prime_identity_respects_comparable_trace :=
      PRCJCost.PRCCharacterPrimeIdentityRespectsComparableTrace_iff_trace_coherence
    prime_identity_trace_coherence_target_refuted :=
      PRCJCost.PRCPrimeCalibrationForcesPrimeIdentityTraceCoherenceTarget_refuted
    prime_identity_branch_uniformity_target_refuted :=
      PRCJCost.PRCPrimeCalibrationForcesPrimeIdentityBranchUniformityTarget_refuted
    prime_identity_trace_transport_target_refuted :=
      PRCJCost.PRCPrimeCalibrationForcesPrimeIdentityTraceTransportTarget_refuted
    prime_identity_common_trace_extension_target_refuted :=
      PRCJCost.PRCPrimeCalibrationForcesPrimeIdentityCommonTraceExtensionTarget_refuted
    prime_identity_canonical_add_trace_target_refuted :=
      PRCJCost.PRCPrimeCalibrationForcesPrimeIdentityCanonicalAddTraceTarget_refuted
    prime_identity_comparable_trace_target_refuted :=
      PRCJCost.PRCPrimeCalibrationForcesPrimeIdentityComparableTraceTarget_refuted
    orbit_successor_identity_target_refuted :=
      PRCJCost.PRCPrimeCalibrationForcesOrbitSuccessorIdentityTarget_refuted
    orbit_successor_transport_target_refuted :=
      PRCJCost.PRCPrimeCalibrationForcesOrbitSuccessorTransportTarget_refuted
    prime_floor_successor_transport_target_refuted :=
      PRCJCost.PRCPrimeCalibrationForcesPrimeFloorSuccessorTransportTarget_refuted
    prime_identity_witness_globalizes_nonunit_target_refuted :=
      PRCJCost.PRCPrimeCalibrationForcesPrimeIdentityWitnessGlobalizesNonunitTarget_refuted
    prime_floor_identity_extends_successor_step_target_refuted :=
      PRCJCost.PRCPrimeCalibrationForcesPrimeFloorIdentityExtendsSuccessorStepTarget_refuted
    prime_floor_identity_contracts_successor_step_target_refuted :=
      PRCJCost.PRCPrimeCalibrationForcesPrimeFloorIdentityContractsSuccessorStepTarget_refuted
    prime_floor_identity_successor_step_pair_target_refuted :=
      PRCJCost.PRCPrimeCalibrationForcesPrimeFloorIdentitySuccessorStepPairTarget_refuted
    prime_floor_nonunit_local_orientation_target_refuted :=
      PRCJCost.PRCPrimeCalibrationForcesNonunitOrbitLocalOrientationTarget_refuted
    prime_floor_nonunit_product_local_orientation_target_refuted :=
      PRCJCost.PRCPrimeCalibrationForcesNonunitOrbitProductLocalOrientationTarget_refuted
    prime_floor_nonunit_orbit_orientation_coherent_target_refuted :=
      PRCJCost.PRCPrimeCalibrationForcesNonunitOrbitOrientationCoherentTarget_refuted
    prime_floor_no_mixed_nonunit_orbit_orientation_target_refuted :=
      PRCJCost.PRCPrimeCalibrationForcesNoMixedNonunitOrbitOrientationTarget_refuted
    prime_floor_nonunit_identity_branch_transport_target_refuted :=
      PRCJCost.PRCPrimeCalibrationForcesNonunitIdentityBranchTransportTarget_refuted
    prime_floor_nonunit_identity_witness_globalizes_target_refuted :=
      PRCJCost.PRCPrimeCalibrationForcesNonunitIdentityWitnessGlobalizesTarget_refuted
    prime_floor_nonunit_identity_witness_excludes_reciprocal_target_refuted :=
      PRCJCost.PRCPrimeCalibrationForcesNonunitIdentityWitnessExcludesReciprocalTarget_refuted
    prime_floor_nonunit_no_mixed_witnesses_target_refuted :=
      PRCJCost.PRCPrimeCalibrationForcesNonunitNoMixedWitnessesTarget_refuted
    prime_floor_no_mixed_prime_witnesses_target_refuted :=
      PRCJCost.PRCPrimeCalibrationForcesNoMixedPrimeWitnessesTarget_refuted
    prime_floor_mixed_prime_witness_character :=
      PRCJCost.PRCPrimeCalibratedMixedPrimeWitnessesCharacter_of_pair_witness_character
        (PRCJCost.PRCPrimeCalibratedMixedPrimePairWitnessCharacter_of_distinct
          (PRCJCost.PRCPrimeCalibratedDistinctPrimeMixedPairWitnessCharacter_of_non_two_mixed
            (PRCJCost.PRCPrimeCalibratedTwoPrimeReciprocalIdentityNonTwoPrimeMixedCharacter_of_two_adic_axis_twist
              (PRCJCost.PRCPrimeCalibratedTwoAdicAxisTwistCharacter_of_ratio_character_axis_twist
                PRCJCost.PRCTwoAdicAxisTwistRatioCharacter_constructed))))
    prime_floor_mixed_prime_pair_witness_character :=
      PRCJCost.PRCPrimeCalibratedMixedPrimePairWitnessCharacter_of_distinct
        (PRCJCost.PRCPrimeCalibratedDistinctPrimeMixedPairWitnessCharacter_of_non_two_mixed
          (PRCJCost.PRCPrimeCalibratedTwoPrimeReciprocalIdentityNonTwoPrimeMixedCharacter_of_two_adic_axis_twist
            (PRCJCost.PRCPrimeCalibratedTwoAdicAxisTwistCharacter_of_ratio_character_axis_twist
              PRCJCost.PRCTwoAdicAxisTwistRatioCharacter_constructed)))
    prime_floor_same_prime_mixed_pair_witness_character_refuted :=
      PRCJCost.PRCPrimeCalibratedSamePrimeMixedPairWitnessCharacter_absurd
    prime_floor_distinct_prime_mixed_pair_witness_character :=
      PRCJCost.PRCPrimeCalibratedDistinctPrimeMixedPairWitnessCharacter_of_non_two_mixed
        (PRCJCost.PRCPrimeCalibratedTwoPrimeReciprocalIdentityNonTwoPrimeMixedCharacter_of_two_adic_axis_twist
          (PRCJCost.PRCPrimeCalibratedTwoAdicAxisTwistCharacter_of_ratio_character_axis_twist
            PRCJCost.PRCTwoAdicAxisTwistRatioCharacter_constructed))
    prime_floor_prime_identity_witness_excludes_reciprocal_target_refuted :=
      PRCJCost.PRCPrimeCalibrationForcesPrimeIdentityWitnessExcludesReciprocalTarget_refuted
    prime_floor_prime_reciprocal_witness_globalizes_target_refuted :=
      PRCJCost.PRCPrimeCalibrationForcesPrimeReciprocalWitnessGlobalizesTarget_refuted
    prime_floor_prime_reciprocal_forces_two_prime_reciprocal_target_refuted :=
      PRCJCost.PRCPrimeCalibrationForcesPrimeReciprocalForcesTwoPrimeReciprocalTarget_refuted
    prime_floor_prime_reciprocal_witness_globalizes_split_target_refuted :=
      PRCJCost.PRCPrimeCalibrationForcesPrimeReciprocalWitnessGlobalizesSplitTarget_refuted
    prime_floor_prime_witnesses_control_nonunit_target :=

-- … truncated for the page; open the module for the rest.
THEOREM PRCUniversalFoundationConditionalCertificate · IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/UniversalFoundation.lean
PRCUniversalFoundationConditionalCertificate · IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/UniversalFoundation.lean:1389 
/-- Top-level conditional certificate: all built PRC surfaces compose, with the
repaired/refuted native-cost ledger exposed by name. -/
structure PRCUniversalFoundationConditionalCertificate : Prop where
  kernel : KernelFirstPassCertificate
  real_complete_ordered_field :
    PRCRealCompleteOrderedFieldPromotedCertificate
  trace_logic : TraceLogicCertificate
  formal_system : FormalSystemCertificate
  inevitability : PRCInevitabilityCertificate
  recognizer_bridge : PRCRecognizerBridgeCertificate
  native_cost_blocker : PRCJCost.PRCNativeCostUniquenessBlockerCertificate
  open_targets : PRCUniversalFoundationOpenTargets
  no_project_local_axioms_audit :
    StrengthTag.classicalExtension = StrengthTag.classicalExtension
THEOREM PRCUniversalFoundationOpenTargets · IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/UniversalFoundation.lean
PRCUniversalFoundationOpenTargets · IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/UniversalFoundation.lean:23 · truncated 
/-- Historical target ledger carried by the conditional top-level certificate.
Positive entries point to proved repaired interfaces; negative entries point to
the exact refutations for routes that cannot force the final surface. -/
structure PRCUniversalFoundationOpenTargets : Prop where
  zero_calibrated_native_cost_uniqueness_refuted :
    ¬ PRCJCost.PRCZeroCalibratedNativeCostUniquenessTarget
  zero_calibrated_native_cost_character_factorization :
    PRCJCost.PRCZeroCalibratedNativeCostCharacterFactorizationTarget
  zero_calibrated_native_cost_signed_admissible_factorization_refuted :
    ¬ PRCJCost.PRCZeroCalibratedNativeCostSignedAdmissibleCharacterFactorizationTarget
  zero_calibration_signed_unit_refuted :
    ¬ PRCJCost.PRCZeroCalibrationForcesNativeCostSignedUnitCalibrationTarget
  zero_calibrated_prime_signed_strengthened_native_cost_uniqueness :
    PRCJCost.PRCZeroCalibratedPrimeSignedStrengthenedNativeCostUniquenessTarget
  native_cost_signed_admissible_character_rigidity :
    PRCJCost.PRCNativeCostSignedAdmissibleCharacterRigidityTarget
  old_native_cost_character_rigidity_refuted :
    ¬ PRCJCost.PRCNativeCostCharacterRigidityTarget
  old_native_cost_sharpened_refuted :
    ¬ PRCJCost.PRCNativeCostUniquenessSharpenedTarget
  two_to_prime_calibration_refuted :
    ¬ PRCJCost.PRCTwoCalibrationForcesPrimeCalibrationTarget
  prime_calibration_propagation_refuted :
    ¬ PRCJCost.PRCPrimeCalibrationPropagationTarget
  prime_global_orientation_refuted :
    ¬ PRCJCost.PRCPrimeCalibrationForcesGlobalOrientationTarget
  coherent_prime_orientation :
    {χ : RatioOrbit → RatioOrbit} →
      PRCJCost.PRCCharacterPrimeOrientationCoherent χ →
        PRCJCost.PRCCharacterTwoPrimeBranchControlsPrimes χ
  two_prime_branch_controls_primes :
    {χ : RatioOrbit → RatioOrbit} →
      PRCJCost.PRCCharacterPrimeLocalOrientation χ →
        PRCJCost.PRCCharacterTwoPrimeBranchControlsPrimes χ →
          PRCJCost.PRCCharacterPrimeOrientationCoherent χ
  prime_identity_iff_two_prime_identity :
    {χ : RatioOrbit → RatioOrbit} →
      PRCJCost.PRCCharacterPrimeLocalOrientation χ →
        PRCJCost.PRCCharacterTwoPrimeBranchControlsPrimes χ →
          PRCJCost.PRCCharacterPrimeIdentityIffTwoPrimeIdentity χ
  prime_identity_forces_two_prime_identity :
    {χ : RatioOrbit → RatioOrbit} →
      PRCJCost.PRCCharacterPrimeIdentityIffTwoPrimeIdentity χ →
        PRCJCost.PRCCharacterPrimeIdentityForcesTwoPrimeIdentity χ
  two_prime_reciprocal_excludes_prime_identity :
    {χ : RatioOrbit → RatioOrbit} →
      PRCJCost.PRCCharacterPrimeLocalOrientation χ →
        (PRCJCost.PRCCharacterPrimeIdentityForcesTwoPrimeIdentity χ ↔
          PRCJCost.PRCCharacterTwoPrimeReciprocalExcludesPrimeIdentity χ)
  two_prime_reciprocal_excludes_prime_identity_witness :
    {χ : RatioOrbit → RatioOrbit} →
      (PRCJCost.PRCCharacterTwoPrimeReciprocalExcludesPrimeIdentity χ ↔
        PRCJCost.PRCCharacterTwoPrimeReciprocalExcludesPrimeIdentityWitness χ)
  two_prime_reciprocal_identity_prime_mixed :
    {χ : RatioOrbit → RatioOrbit} →
      (PRCJCost.PRCCharacterTwoPrimeReciprocalIdentityPrimeMixed χ ↔
        PRCJCost.PRCCharacterTwoPrimeReciprocalIdentityNonTwoPrimeMixed χ)
  two_prime_reciprocal_identity_non_two_prime_mixed :
    {χ : RatioOrbit → RatioOrbit} →
      PRCJCost.PRCRatioCharacter χ →
        (PRCJCost.PRCCharacterTwoPrimeReciprocalIdentityNonTwoPrimeMixed χ ↔
          PRCJCost.PRCCharacterTwoPrimeReciprocalIdentityNonTwoCompositeDefect χ)
  two_prime_reciprocal_identity_non_two_composite_defect :
    {χ : RatioOrbit → RatioOrbit} →
      (PRCJCost.PRCCharacterTwoPrimeReciprocalIdentityNonTwoCompositeDefect χ ↔
        PRCJCost.PRCCharacterTwoPrimeReciprocalIdentityNonTwoCompositeCostDefect χ)
  two_prime_reciprocal_identity_non_two_composite_cost_defect :
    {χ : RatioOrbit → RatioOrbit} →
      PRCJCost.PRCCharacterTwoPrimeReciprocalIdentityNonTwoCompositeCostDefect χ →
        PRCJCost.PRCCharacterTwoPrimeReciprocalIdentityNonTwoCompositeDefect χ
  two_prime_mixed_composite_cost_consistency_target_refuted :
    ¬ PRCJCost.PRCPrimeCalibrationForcesTwoPrimeMixedCompositeCostConsistencyTarget
  prime_pair_product_cost_consistency_target_refuted :
    ¬ PRCJCost.PRCPrimeCalibrationForcesPrimePairProductCostConsistencyTarget
  two_prime_reciprocal_forces_prime_reciprocal :
    {χ : RatioOrbit → RatioOrbit} →
      PRCJCost.PRCCharacterPrimeReciprocalWitnessGlobalizes χ →
        PRCJCost.PRCCharacterTwoPrimeReciprocalForcesPrimeReciprocal χ
  two_prime_reciprocal_trace_connected :
    {χ : RatioOrbit → RatioOrbit} →
      (PRCJCost.PRCCharacterTwoPrimeReciprocalRespectsTraceConnected χ ↔
        PRCJCost.PRCCharacterTwoPrimeReciprocalForcesPrimeReciprocal χ)
  two_prime_identity_trace_connected :
    {χ : RatioOrbit → RatioOrbit} →
      PRCJCost.PRCCharacterPrimeIdentityRespectsTraceConnected χ →
        PRCJCost.PRCCharacterTwoPrimeIdentityRespectsTraceConnected χ
  no_mixed_prime_orientation :
    {χ : RatioOrbit → RatioOrbit} →
      (PRCJCost.PRCCharacterNoMixedPrimeWitnesses χ ↔
        PRCJCost.PRCCharacterNoMixedPrimeOrientation χ)
  mixed_prime_witnesses :
    {χ : RatioOrbit → RatioOrbit} →
      (PRCJCost.PRCCharacterMixedPrimeWitnesses χ ↔
        PRCJCost.PRCCharacterMixedPrimePairWitnesses χ)
  mixed_prime_pair_witnesses :
    {χ : RatioOrbit → RatioOrbit} →
      (PRCJCost.PRCCharacterMixedPrimePairWitnesses χ ↔
        PRCJCost.PRCCharacterSamePrimeMixedPairWitnesses χ ∨
          PRCJCost.PRCCharacterDistinctPrimeMixedPairWitnesses χ)
  same_prime_mixed_pair_witnesses :
    {χ : RatioOrbit → RatioOrbit} →
      ¬ PRCJCost.PRCCharacterSamePrimeMixedPairWitnesses χ
  distinct_prime_mixed_pair_witnesses :
    {χ : RatioOrbit → RatioOrbit} →
      PRCJCost.PRCCharacterPrimeIdentityBranchUniform χ →
        ¬ PRCJCost.PRCCharacterDistinctPrimeMixedPairWitnesses χ
  prime_identity_witness_excludes_reciprocal :
    {χ : RatioOrbit → RatioOrbit} →
      (PRCJCost.PRCCharacterPrimeIdentityWitnessExcludesReciprocal χ ↔
        PRCJCost.PRCCharacterNoMixedPrimeOrientation χ)
  prime_reciprocal_witness_globalizes :
    {χ : RatioOrbit → RatioOrbit} →
      PRCJCost.PRCCharacterPrimeLocalOrientation χ →
        PRCJCost.PRCCharacterNoMixedPrimeOrientation χ →
          PRCJCost.PRCCharacterPrimeReciprocalWitnessGlobalizes χ
  prime_reciprocal_forces_two_prime_reciprocal :
    {χ : RatioOrbit → RatioOrbit} →
      PRCJCost.PRCCharacterPrimeReciprocalWitnessGlobalizes χ →
        PRCJCost.PRCCharacterPrimeReciprocalForcesTwoPrimeReciprocal χ
  prime_reciprocal_witness_globalizes_split :
    {χ : RatioOrbit → RatioOrbit} →
      (PRCJCost.PRCCharacterPrimeReciprocalWitnessGlobalizes χ ↔
        PRCJCost.PRCCharacterPrimeReciprocalWitnessGlobalizesSplit χ)
  prime_reciprocal_forces_two_from_reciprocal_twist_identity_forces_two :
    ∀ χ : RatioOrbit → RatioOrbit,
      PRCJCost.PRCCharacterPrimeIdentityForcesTwoPrimeIdentity
          (PRCJCost.PRCCharacterReciprocalTwist χ) →
        PRCJCost.PRCCharacterPrimeReciprocalForcesTwoPrimeReciprocal χ
  prime_identity_forces_two_from_reciprocal_twist_reciprocal_forces_two :
    ∀ χ : RatioOrbit → RatioOrbit,
      PRCJCost.PRCCharacterPrimeReciprocalForcesTwoPrimeReciprocal
          (PRCJCost.PRCCharacterReciprocalTwist χ) →
        PRCJCost.PRCCharacterPrimeIdentityForcesTwoPrimeIdentity χ
  prime_identity_trace_coherence :
    {χ : RatioOrbit → RatioOrbit} →
      (PRCJCost.PRCCharacterPrimeIdentityBranchUniform χ ↔
        PRCJCost.PRCCharacterPrimeIdentityTraceCoherent χ)
  prime_identity_branch_uniform :
    {χ : RatioOrbit → RatioOrbit} →
      (PRCJCost.PRCCharacterPrimeIdentityBranchUniform χ ↔
        PRCJCost.PRCCharacterPrimeIdentityIffTwoPrimeIdentity χ)
  prime_axis_trace_connected :
    ∀ p : DistinctionNat, ∀ hp : DistinctionNat.primeOrbit p,
      ∀ r : DistinctionNat, ∀ hr : DistinctionNat.primeOrbit r,
        PRCJCost.PRCPrimeAxisTraceConnected p hp r hr
  prime_identity_respects_trace_connected :
    {χ : RatioOrbit → RatioOrbit} →
      (PRCJCost.PRCCharacterPrimeIdentityRespectsCanonicalAddTrace χ ↔
        PRCJCost.PRCCharacterPrimeIdentityRespectsTraceConnected χ)
  prime_identity_respects_common_trace_extension :
    {χ : RatioOrbit → RatioOrbit} →
      (PRCJCost.PRCCharacterPrimeIdentityRespectsCanonicalAddTrace χ ↔
        PRCJCost.PRCCharacterPrimeIdentityRespectsCommonTraceExtension χ)
  prime_identity_respects_canonical_add_trace :
    {χ : RatioOrbit → RatioOrbit} →
      (PRCJCost.PRCCharacterPrimeIdentityBranchUniform χ ↔
        PRCJCost.PRCCharacterPrimeIdentityRespectsCanonicalAddTrace χ)
  prime_identity_respects_comparable_trace :
    {χ : RatioOrbit → RatioOrbit} →
      (PRCJCost.PRCCharacterPrimeIdentityRespectsComparableTrace χ ↔
        PRCJCost.PRCCharacterPrimeIdentityTraceCoherent χ)
  prime_identity_trace_coherence_target_refuted :
    ¬ PRCJCost.PRCPrimeCalibrationForcesPrimeIdentityTraceCoherenceTarget
  prime_identity_branch_uniformity_target_refuted :
    ¬ PRCJCost.PRCPrimeCalibrationForcesPrimeIdentityBranchUniformityTarget
  prime_identity_trace_transport_target_refuted :
    ¬ PRCJCost.PRCPrimeCalibrationForcesPrimeIdentityTraceTransportTarget
  prime_identity_common_trace_extension_target_refuted :
    ¬ PRCJCost.PRCPrimeCalibrationForcesPrimeIdentityCommonTraceExtensionTarget
  prime_identity_canonical_add_trace_target_refuted :
    ¬ PRCJCost.PRCPrimeCalibrationForcesPrimeIdentityCanonicalAddTraceTarget
  prime_identity_comparable_trace_target_refuted :
    ¬ PRCJCost.PRCPrimeCalibrationForcesPrimeIdentityComparableTraceTarget
  orbit_successor_identity_target_refuted :
    ¬ PRCJCost.PRCPrimeCalibrationForcesOrbitSuccessorIdentityTarget
  orbit_successor_transport_target_refuted :
    ¬ PRCJCost.PRCPrimeCalibrationForcesOrbitSuccessorTransportTarget
  prime_floor_successor_transport_target_refuted :
    ¬ PRCJCost.PRCPrimeCalibrationForcesPrimeFloorSuccessorTransportTarget
  prime_identity_witness_globalizes_nonunit_target_refuted :
    ¬ PRCJCost.PRCPrimeCalibrationForcesPrimeIdentityWitnessGlobalizesNonunitTarget
  prime_floor_identity_extends_successor_step_target_refuted :
    ¬ PRCJCost.PRCPrimeCalibrationForcesPrimeFloorIdentityExtendsSuccessorStepTarget
  prime_floor_identity_contracts_successor_step_target_refuted :
    ¬ PRCJCost.PRCPrimeCalibrationForcesPrimeFloorIdentityContractsSuccessorStepTarget
  prime_floor_identity_successor_step_pair_target_refuted :
    ¬ PRCJCost.PRCPrimeCalibrationForcesPrimeFloorIdentitySuccessorStepPairTarget
  prime_floor_nonunit_local_orientation_target_refuted :
    ¬ PRCJCost.PRCPrimeCalibrationForcesNonunitOrbitLocalOrientationTarget
  prime_floor_nonunit_product_local_orientation_target_refuted :
    ¬ PRCJCost.PRCPrimeCalibrationForcesNonunitOrbitProductLocalOrientationTarget
  prime_floor_nonunit_orbit_orientation_coherent_target_refuted :
    ¬ PRCJCost.PRCPrimeCalibrationForcesNonunitOrbitOrientationCoherentTarget
  prime_floor_no_mixed_nonunit_orbit_orientation_target_refuted :
    ¬ PRCJCost.PRCPrimeCalibrationForcesNoMixedNonunitOrbitOrientationTarget
  prime_floor_nonunit_identity_branch_transport_target_refuted :
    ¬ PRCJCost.PRCPrimeCalibrationForcesNonunitIdentityBranchTransportTarget
  prime_floor_nonunit_identity_witness_globalizes_target_refuted :
    ¬ PRCJCost.PRCPrimeCalibrationForcesNonunitIdentityWitnessGlobalizesTarget
  prime_floor_nonunit_identity_witness_excludes_reciprocal_target_refuted :
    ¬ PRCJCost.PRCPrimeCalibrationForcesNonunitIdentityWitnessExcludesReciprocalTarget
  prime_floor_nonunit_no_mixed_witnesses_target_refuted :
    ¬ PRCJCost.PRCPrimeCalibrationForcesNonunitNoMixedWitnessesTarget
  prime_floor_no_mixed_prime_witnesses_target_refuted :
    ¬ PRCJCost.PRCPrimeCalibrationForcesNoMixedPrimeWitnessesTarget
  prime_floor_mixed_prime_witness_character :
    PRCJCost.PRCPrimeCalibratedMixedPrimeWitnessesCharacter
  prime_floor_mixed_prime_pair_witness_character :
    PRCJCost.PRCPrimeCalibratedM

-- … truncated for the page; open the module for the rest.

What this page does not claim

The certificate does not prove the full universal foundation; it is conditional. The certificate does not claim the unsigned native-cost routes are proved; they are refuted targets.

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/UniversalFoundation.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:

MACHINE LAYER · GROUNDED CLAIM TABLE · CLICK TO EXPAND