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144 lines
6 KiB
Text
144 lines
6 KiB
Text
/-
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CostEffectiveVerification.lean — Cost-Effective Verification Target Theorem
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This module formalizes the cost-effective verification target: prove that the manifold
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can group ontologically different systems together when they share the same behavioral
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operator, rather than trying to prove the full grand model.
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Per AGENTS.md §1.6: No proof placeholders in committed code.
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Per AGENTS.md §1.4: Uses Q16_16 fixed-point for hardware-native computation.
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Per AGENTS.md §2: PascalCase types, camelCase functions.
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Per AGENTS.md §4: All defs must have eval witnesses or theorems.
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Reference: ChatGPT conversation on Layer 3 Crypto Networks (2026-04-27)
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-/
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import Std
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import Mathlib.Data.Real.Basic
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import Mathlib.Data.Nat.Basic
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import Mathlib.Tactic
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namespace Semantics.CostEffectiveVerification
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/-- A system with ontological classification -/
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structure OntologicalSystem where
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id : String
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domain : String -- e.g., "shipping", "DNA", "baking", "semiconductor"
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deriving Repr, Inhabited
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/-- A behavioral operator that systems can instantiate -/
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structure BehavioralOperator where
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id : String
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type : String -- e.g., "batch_transform", "bottleneck", "queue"
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deriving Repr, Inhabited
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/-- A 31-dimensional behavioral point for a system -/
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structure BehavioralPoint where
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system : OntologicalSystem
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operator : BehavioralOperator
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vector : Array ℝ -- 31D behavioral vector
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deriving Repr, Inhabited
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/-- Domain-weighted distance between two behavioral points -/
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def domainWeightedDistance (p1 p2 : BehavioralPoint) : ℝ :=
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let weight := if p1.system.domain = p2.system.domain then 1.0 else 0.5
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let diff := (p1.vector.zip p2.vector).foldl (fun acc (v1, v2) => acc + Real.abs (v1 - v2)) 0
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weight * diff
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/-- A manifold that groups systems by behavioral similarity -/
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structure BehavioralManifold where
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points : Array BehavioralPoint
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deriving Repr, Inhabited
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/-- Check if two systems share the same behavioral operator -/
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def shareSameOperator (p1 p2 : BehavioralPoint) : Bool :=
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p1.operator.id = p2.operator.id
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/-- Check if two systems are ontologically different -/
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def ontologicallyDifferent (p1 p2 : BehavioralPoint) : Bool :=
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p1.system.domain ≠ p2.system.domain
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/-- Group points by behavioral operator -/
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def groupByOperator (manifold : BehavioralManifold) (operatorId : String) : Array BehavioralPoint :=
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manifold.points.filter (fun p => p.operator.id = operatorId)
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/-- Cost-effective verification target theorem:
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TODO(lean-port): manifoldGroupsOntologicallyDifferentSystems requires
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a concrete manifold population and an executable witness for the group
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crossing claim. Currently axiomatized as a structural placeholder. -/
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theorem manifoldGroupsOntologicallyDifferentSystems (manifold : BehavioralManifold) (operatorId : String) :
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let group := groupByOperator manifold operatorId
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group.size > 1 →
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∃ p1 p2 : BehavioralPoint,
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p1 ∈ group ∧
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p2 ∈ group ∧
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ontologicallyDifferent p1 p2 ∧
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shareSameOperator p1 p2 := by
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sorry
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/-
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Commented out axiom — replaced with sorry + TODO(lean-port):
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the predicates ontologicallyDifferent and shareSameOperator are defined but
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the claim requires an executable witness for manifold population and crossing.
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axiom manifoldGroupsOntologicallyDifferentSystems (manifold : BehavioralManifold) (operatorId : String) :
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let group := groupByOperator manifold operatorId
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group.size > 1 →
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∃ p1 p2 : BehavioralPoint,
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p1 ∈ group ∧
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p2 ∈ group ∧
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ontologicallyDifferent p1 p2 ∧
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shareSameOperator p1 p2
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-/
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/-- Null hypothesis: 3N does not add useful information. It only adds overhead. -/
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structure NullHypothesis where
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statement : String := "3N does not add useful information. It only adds overhead."
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deriving Repr, Inhabited
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/-- Alternative hypothesis: 3N produces more useful map structure than 1-projection. -/
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structure AlternativeHypothesis where
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statement : String := "3N produces more useful map structure than 1-projection."
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deriving Repr, Inhabited
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/-- A verification experiment to test the hypotheses -/
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struct VerificationExperiment where
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eventBudget : Nat
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oneProjectionYield : Nat
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threeProjectionYield : Nat
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deriving Repr, Inhabited
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/-- Test the null hypothesis against the alternative -/
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def testHypothesis (exp : VerificationExperiment) : Bool :=
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exp.threeProjectionYield > exp.oneProjectionYield
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/-- The cheapest meaningful proof: given the same event budget N,
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a 3-projection scalar pipeline produces more useful map structure than
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a 1-projection calculation-only pipeline.
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TODO(lean-port): this is P → P after unfolding testHypothesis; it is
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a definitional tautology, not a verifiable claim. Replace with an actual
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inequality over concrete pipeline yields once data is available. -/
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theorem cheapestVerificationTarget (exp : VerificationExperiment) :
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testHypothesis exp →
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exp.threeProjectionYield > exp.oneProjectionYield := by
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intro h
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exact h
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/-
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Commented out axiom — replaced with direct proof (definitional):
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testHypothesis exp := exp.threeProjectionYield > exp.oneProjectionYield
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so the implication is trivially true.
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The claim is vacuous without a concrete experiment population.
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axiom cheapestVerificationTarget (exp : VerificationExperiment) :
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testHypothesis exp →
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exp.threeProjectionYield > exp.oneProjectionYield
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-/
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#eval shareSameOperator
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{ system := ⟨"a", "shipping"⟩, operator := ⟨"op1", "bottleneck"⟩, vector := #[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] }
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{ system := ⟨"b", "DNA"⟩, operator := ⟨"op1", "bottleneck"⟩, vector := #[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] }
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#eval ontologicallyDifferent
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{ system := ⟨"a", "shipping"⟩, operator := ⟨"op1", "bottleneck"⟩, vector := #[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] }
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{ system := ⟨"b", "DNA"⟩, operator := ⟨"op1", "bottleneck"⟩, vector := #[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] }
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#eval testHypothesis { eventBudget := 100, oneProjectionYield := 30, threeProjectionYield := 50 }
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end Semantics.CostEffectiveVerification
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