|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| namespace ExtractedInvariants
|
|
|
|
|
|
|
|
|
|
|
| /-
|
| INV-1: Active agents must be trusted.
|
| β RECASTED: The original theorem claimed β active trusted, active β trusted,
|
| which is false (counterexample: active=true, trusted=false is a valid Bool pair).
|
| The correct statement is a SYSTEM PROPERTY: for any agent produced by the
|
| system's construction function, trust is established before activation.
|
| Formalised here as a precondition on well-formed agent state.
|
| -/
|
| structure AgentState where
|
| active : Bool
|
| trusted : Bool
|
|
|
| def wellFormedAgent (a : AgentState) : Prop :=
|
| a.active = true β a.trusted = true
|
|
|
| -- This is a DEFINITION of what "well-formed" means, not a universal theorem.
|
| -- Runtime enforcement: the agent FSM must not transition to `active=true`
|
| -- without first setting `trusted=true`.
|
|
|
| -- β PROVED: a well-formed agent satisfies active β trusted by definition.
|
| theorem inv1_wellformed_implies_trust (a : AgentState) (h : wellFormedAgent a) :
|
| a.active = true β a.trusted = true := h
|
|
|
|
|
| /-
|
| INV-2 / INV-3: Agent entropy bounded by 0.20 (Float) and
|
| Shannon entropy β€ 0.20 nats (β).
|
| β RECASTED: "β entropy : Float, entropy β€ 0.20" is false β entropy is an
|
| unconstrained real number. The correct invariant is a POST-CONDITION on
|
| the routing gate: the system REJECTS agents whose entropy exceeds the bound.
|
| Proved below in INV-5 (routing rejection).
|
| -/
|
| def entropyBound : Float := 0.20
|
|
|
|
|
| /-
|
| INV-4: Quantum operator symmetry β Q = (Q + Qα΅)/2.
|
| β RECASTED: Not true for all Q. This is the SYMMETRISATION formula;
|
| the invariant is that the system APPLIES symmetrisation before use,
|
| not that arbitrary matrices are already symmetric.
|
| Formalised as: symmetrise(Q) = (Q + Qα΅)/2 is the correct construction.
|
| -/
|
| -- Formal definition of symmetrisation (correct for any nΓn complex matrix)
|
| def symmetriseMatrix {n : Type*} [Fintype n] (Q : Matrix n n β) : Matrix n n β :=
|
| (Q + Q.transpose) / 2
|
|
|
| -- β PROVED: symmetriseMatrix always produces a symmetric matrix.
|
| theorem inv4_symmetrise_is_symmetric {n : Type*} [Fintype n] [DecidableEq n]
|
| (Q : Matrix n n β) :
|
| (symmetriseMatrix Q).transpose = symmetriseMatrix Q := by
|
| simp [symmetriseMatrix, Matrix.transpose_add, Matrix.transpose_div]
|
| ring
|
|
|
|
|
| /-
|
| INV-5: Routing rejects states with entropy > 0.20.
|
| β RECASTED: "β entropy accept, entropy > 0.20 β accept = false" is false
|
| for arbitrary (entropy, accept) β it needs to be scoped to the ROUTING GATE.
|
| Proved here as: a correct routing gate always rejects high-entropy states.
|
| -/
|
| def routingGate (entropy : Float) : Bool :=
|
| entropy β€ entropyBound
|
|
|
| -- β PROVED: the routing gate rejects entropy > 0.20 by construction.
|
| theorem inv5_routing_rejects_high_entropy (entropy : Float) :
|
| entropy > entropyBound β routingGate entropy = false := by
|
| intro h
|
| simp [routingGate]
|
| exact Float.not_le.mpr h
|
|
|
|
|
| /-
|
| INV-6: Candidate acceptance requires entropy bound AND proof.
|
| β RECASTED: "β entropy proof accept, accept = ..." is false for arbitrary values.
|
| The correct invariant is: the acceptance function IS DEFINED as this conjunction.
|
| -/
|
| def candidateAccepted (entropy : Float) (proof : Bool) : Bool :=
|
| (entropy β€ entropyBound) && proof
|
|
|
| -- β PROVED: acceptance is exactly entropy β€ bound β§ proof = true, by definition.
|
| theorem inv6_acceptance_iff (entropy : Float) (proof : Bool) :
|
| candidateAccepted entropy proof = true β
|
| entropy β€ entropyBound β§ proof = true := by
|
| simp [candidateAccepted, Bool.and_eq_true]
|
|
|
|
|
| -- ============================================================
|
| -- CLUSTER B: Spin-Glass Model (Prolog/ASP)
|
| -- ============================================================
|
|
|
| /-
|
| INV-7: Each node has exactly one spin state (pos or neg).
|
| β PROVED: For any Bool, either it equals true XOR it equals false β these
|
| are the only two values. This closes by case analysis.
|
| -/
|
| theorem inv7_single_spin_per_node (spins : Fin 9 β Bool) :
|
| β (n : Fin 9), (spins n = true) β (spins n = false) := by
|
| intro n
|
| cases h : spins n
|
| Β· exact Or.inr rfl
|
| Β· exact Or.inl rfl
|
|
|
|
|
| /-
|
| INV-8: Frustrated interaction definition β same-spin neighbours on a negative edge.
|
| β PROVED: The conclusion is `True`; the theorem is a tautology.
|
| NOTE: This is a DEFINITION of frustration, not a constraint.
|
| A stronger invariant would say: frustrated(U,V) β edge(U,V)=true β§ spin(U)=spin(V).
|
| -/
|
| def frustrated (edge : Fin 9 β Fin 9 β Bool) (spin : Fin 9 β Bool) (u v : Fin 9) : Prop :=
|
| edge u v = true β§ spin u = spin v
|
|
|
| theorem inv8_frustrated_is_boolean_definable
|
| (edge : Fin 9 β Fin 9 β Bool) (spin : Fin 9 β Bool) (u v : Fin 9) :
|
| frustrated edge spin u v β¨ Β¬ frustrated edge spin u v := by
|
| exact Classical.em _
|
|
|
|
|
| /-
|
| INV-9: Minimize global frustration (ground state at T=0.1).
|
| β OPEN: Minimisation is a meta-level optimisation problem.
|
| It requires a definition of "number of frustrated pairs" and a proof
|
| that the spin assignment minimises this count.
|
| Formalised here as the type of the optimisation problem.
|
| -/
|
| def frustrationCount (edge : Fin 9 β Fin 9 β Bool) (spin : Fin 9 β Bool) : β :=
|
| Finset.card (Finset.filter
|
| (fun p : Fin 9 Γ Fin 9 => frustrated edge spin p.1 p.2)
|
| Finset.univ)
|
|
|
| -- OPEN: ground state spin assignment minimises frustrationCount.
|
| -- Path: well-founded minimisation over the finite set of spin configurations.
|
| -- axiom inv9_ground_state : β edge, β spin_opt, β spin, frustrationCount edge spin_opt β€ frustrationCount edge spin
|
|
|
|
|
| /-
|
| INV-10: Entropy drain triggered iff frustration count > 0.
|
| β RECASTED: The DRAIN GATE is a function of frustration count.
|
| -/
|
| def entropyDrainGate (frustration_count : β) : Bool :=
|
| frustration_count > 0
|
|
|
| -- β PROVED: drain fires iff frustration_count > 0, by definition.
|
| theorem inv10_drain_iff_frustrated (n : β) :
|
| entropyDrainGate n = true β n > 0 := by
|
| simp [entropyDrainGate]
|
|
|
|
|
| /-
|
| INV-11: Exactly one action chosen (work XOR rest).
|
| β RECASTED: Not true for all (work, rest) β both can be false or both true.
|
| The correct invariant is a precondition on the action scheduler output.
|
| -/
|
| def validAction (work rest : Bool) : Prop :=
|
| (work = true β§ rest = false) β¨ (work = false β§ rest = true)
|
|
|
| -- β PROVED: the scheduler's output (decide function) satisfies validAction.
|
| def actionScheduler (prefer_work : Bool) : Bool Γ Bool :=
|
| if prefer_work then (true, false) else (false, true)
|
|
|
| theorem inv11_scheduler_produces_valid_action (prefer_work : Bool) :
|
| let (w, r) := actionScheduler prefer_work
|
| validAction w r := by
|
| simp [actionScheduler, validAction]
|
| cases prefer_work <;> simp
|
|
|
|
|
| /-
|
| INV-12: Reward = 100Β·work + 10Β·rest.
|
| β PROVED: The reward function IS defined this way; the theorem is its unfolding.
|
| -/
|
| def reward (work rest : Bool) : β :=
|
| (if work then 100 else 0) +
|
| (if rest then 10 else 0)
|
|
|
| theorem inv12_reward_correct (work rest : Bool) :
|
| reward work rest =
|
| (if work then 100 else 0) + (if rest then 10 else 0) := rfl
|
|
|
|
|
|
|
|
|
|
|
|
|
| /-
|
| INV-13: Zero speculation β only describe verified workspace contents.
|
| INV-19: Fact-only communication.
|
| INV-20: Direct output only.
|
| β PROVED: These are stated as (premise β True); they are tautologies.
|
| The REAL enforcement is external (CLAUDE.md runtime rules, not Lean theorems).
|
| We prove them trivially and document the external enforcement contract.
|
| -/
|
| theorem inv13_zero_speculation (statement : String) (verified : Bool) :
|
| verified = true β True := fun _ => trivial
|
|
|
| theorem inv19_fact_only (output : String) : True := trivial
|
|
|
| theorem inv20_direct_output (output : String) : True := trivial
|
|
|
|
|
| /-
|
| INV-14: No invented standards β spec must come from source or be empty.
|
| β RECASTED: "β spec in_source, in_source = true β¨ spec = ''" is false
|
| (counterexample: in_source=false, spec="invented"). This is a VALIDATION
|
| PREDICATE, not a universal truth.
|
| -/
|
| def specIsGrounded (spec : String) (in_source : Bool) : Bool :=
|
| in_source || (spec == "")
|
|
|
|
|
| theorem inv14_grounded_spec_valid (spec : String) (in_source : Bool)
|
| (h : specIsGrounded spec in_source = true) :
|
| in_source = true β¨ spec = "" := by
|
| simp [specIsGrounded, Bool.or_eq_true] at h
|
| cases h with
|
| | inl h => exact Or.inl h
|
| | inr h => exact Or.inr (by simp [beq_iff_eq] at h; exact h)
|
|
|
|
|
| /-
|
| INV-15: Unverified boundary β unread files are marked Unverified.
|
| β RECASTED: Same pattern β this is a VALIDATION GATE, not a universal truth.
|
| -/
|
| def fileMarkedCorrectly (file_read : Bool) (marked_unverified : Bool) : Bool :=
|
| file_read || marked_unverified
|
|
|
| theorem inv15_unread_implies_marked (file_read marked_unverified : Bool)
|
| (h : fileMarkedCorrectly file_read marked_unverified = true) :
|
| file_read = false β marked_unverified = true := by
|
| simp [fileMarkedCorrectly, Bool.or_eq_true] at h
|
| intro hf
|
| cases h with
|
| | inl h => simp [hf] at h
|
| | inr h => exact h
|
|
|
|
|
| /-
|
| INV-16: No admin noise unless requested.
|
| β RECASTED: VALIDATION PREDICATE.
|
| -/
|
| def adminOutputAllowed (output_type : String) (requested : Bool) : Bool :=
|
| let admin_types := ["roadmap", "threat_model", "audit_scorecard", "release_notes"]
|
| if admin_types.contains output_type then requested
|
| else true
|
|
|
| theorem inv16_admin_requires_request (output_type : String) (requested : Bool)
|
| (h : adminOutputAllowed output_type requested = true)
|
| (h_admin : ["roadmap", "threat_model", "audit_scorecard", "release_notes"].contains output_type = true) :
|
| requested = true := by
|
| simp [adminOutputAllowed, h_admin] at h
|
| exact h
|
|
|
|
|
| /-
|
| INV-17: Honest completion β no closure with sorry/todo/panic/stubs.
|
| β RECASTED: VALIDATION PREDICATE over completion state.
|
| -/
|
| structure CompletionState where
|
| has_sorry : Bool
|
| has_todo : Bool
|
| has_panic : Bool
|
| has_stub : Bool
|
|
|
| def honestlyComplete (s : CompletionState) (reported_complete : Bool) : Bool :=
|
| if s.has_sorry || s.has_todo || s.has_panic || s.has_stub
|
| then !reported_complete
|
| else true
|
|
|
| theorem inv17_no_stubs_if_complete (s : CompletionState) (reported_complete : Bool)
|
| (h : honestlyComplete s reported_complete = true) :
|
| (s.has_sorry || s.has_todo || s.has_panic || s.has_stub) = true β
|
| reported_complete = false := by
|
| simp [honestlyComplete] at h
|
| intro hbad
|
| simp [hbad] at h
|
| exact Bool.not_eq_true_of_eq_false (Bool.eq_false_of_not_eq_true (by simp [h]))
|
|
|
|
|
| /-
|
| INV-18: No speculative file trees β all listed files must exist.
|
| β RECASTED: VALIDATION GATE on file list construction.
|
| -/
|
|
|
|
|
| def allFilesExist (file_list : List String) (exists_fn : String β Bool) : Bool :=
|
| file_list.all exists_fn
|
|
|
| theorem inv18_all_files_verified (files : List String) (exists_fn : String β Bool)
|
| (h : allFilesExist files exists_fn = true) :
|
| β f β files, exists_fn f = true := by
|
| simp [allFilesExist, List.all_eq_true] at h
|
| exact h
|
|
|
|
|
|
|
|
|
|
|
|
|
| /-
|
| INV-21β26: System identity constants.
|
| β RECASTED: "β infra, infra = 'Local_First_Sovereign_OS'" is false
|
| for arbitrary String. These are CONFIGURATION ASSERTIONS β they hold
|
| for the specific system instance, not for all strings.
|
| Proved as: if the system was configured correctly, these hold.
|
| -/
|
| structure SovereignConfig where
|
| infrastructure : String
|
| architecture : String
|
| logic_layer : String
|
| trust_protocol : String
|
| training_gate : String
|
| fiscal_governance : String
|
|
|
| def wellConfiguredSovereign (c : SovereignConfig) : Prop :=
|
| c.infrastructure = "Local_First_Sovereign_OS" β§
|
| c.architecture = "Multi_Agent_Enterprise_Logic" β§
|
| c.logic_layer = "Prolog_Verified_Deterministic" β§
|
| (c.trust_protocol = "Bifrost_WORM_Chain" β¨
|
| c.trust_protocol = "Bifrost_Audit_Chain") β§
|
| c.training_gate = "Human_Review_Required" β§
|
| c.fiscal_governance = "Codestorm_Hub_Federated"
|
|
|
|
|
| theorem inv21_infrastructure (c : SovereignConfig) (h : wellConfiguredSovereign c) :
|
| c.infrastructure = "Local_First_Sovereign_OS" := h.1
|
|
|
| theorem inv22_architecture (c : SovereignConfig) (h : wellConfiguredSovereign c) :
|
| c.architecture = "Multi_Agent_Enterprise_Logic" := h.2.1
|
|
|
| theorem inv23_logic_layer (c : SovereignConfig) (h : wellConfiguredSovereign c) :
|
| c.logic_layer = "Prolog_Verified_Deterministic" := h.2.2.1
|
|
|
| theorem inv24_trust_protocol (c : SovereignConfig) (h : wellConfiguredSovereign c) :
|
| c.trust_protocol = "Bifrost_WORM_Chain" β¨
|
| c.trust_protocol = "Bifrost_Audit_Chain" := h.2.2.2.1
|
|
|
| theorem inv25_training_gate (c : SovereignConfig) (h : wellConfiguredSovereign c) :
|
| c.training_gate = "Human_Review_Required" := h.2.2.2.2.1
|
|
|
| theorem inv26_fiscal_governance (c : SovereignConfig) (h : wellConfiguredSovereign c) :
|
| c.fiscal_governance = "Codestorm_Hub_Federated" := h.2.2.2.2.2
|
|
|
|
|
|
|
|
|
|
|
| /-
|
| β PROVED (no sorry):
|
| inv1_wellformed_implies_trust β from wellFormedAgent definition
|
| inv4_symmetrise_is_symmetric β ring
|
| inv5_routing_rejects_high_entropy β Float.not_le
|
| inv6_acceptance_iff β Bool.and_eq_true
|
| inv7_single_spin_per_node β cases on Bool
|
| inv8_frustrated_is_boolean_definable β Classical.em
|
| inv10_drain_iff_frustrated β simp
|
| inv11_scheduler_produces_valid_action β cases prefer_work
|
| inv12_reward_correct β rfl
|
| inv13_zero_speculation β trivial
|
| inv14_grounded_spec_valid β Bool.or_eq_true + beq_iff_eq
|
| inv15_unread_implies_marked β Bool.or_eq_true
|
| inv16_admin_requires_request β simp
|
| inv17_no_stubs_if_complete β Bool negation lemmas
|
| inv18_all_files_verified β List.all_eq_true
|
| inv19_fact_only β trivial
|
| inv20_direct_output β trivial
|
| inv21βinv26 β projections from wellConfiguredSovereign
|
|
|
| ? OPEN:
|
| inv9_ground_state β well-founded minimisation over Fin 9 β Bool
|
| Blocker: need Finset.argmin or explicit enumeration proof
|
|
|
| β RECASTED (originally false as universally quantified):
|
| inv1, inv2/3, inv4, inv5, inv6 β converted from "β values" to validation
|
| predicates / conditional theorems scoped to correct system state
|
| inv10β12 β converted to function definitions with proved unfoldings
|
| inv14β18 β converted to validation predicates
|
| inv21β26 β converted to projections from SovereignConfig precondition
|
| -/
|
|
|
| end ExtractedInvariants
|
|
|
