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108 lines
3.9 KiB
Text
108 lines
3.9 KiB
Text
/-
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Curvature.lean - Ollivier-Ricci Curvature on Graphs
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Implements the Intelligence Ladder metric (ORC).
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ORC(x, y) = 1 - W(m_x, m_y) / d(x, y)
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where W is the Wasserstein-1 distance (optimal transport).
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-/
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import Semantics.FixedPoint
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import Semantics.Graph
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import Semantics.Bind
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namespace Semantics.Curvature
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open Semantics.Q16_16
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open Semantics.ENE
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/--
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Representation of a probability measure on a graph neighborhood.
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Stored as a list of (node_id, weight) pairs where sum(weights) = 1.0.
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-/
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structure GraphMeasure where
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support : List (Nat × Semantics.Q16_16)
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deriving Repr
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/-- Wasserstein-1 distance (Earth Mover's Distance) shim.
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In a full implementation, this would involve a linear programming solver.
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For the verification core, we use the upper bound: Σ |m_x(i) - m_y(i)| * dist(i, target).
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-/
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def wasserstein1Shim (_g : Graph) (m1 m2 : GraphMeasure) : Semantics.Q16_16 :=
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-- Simplified shim for formal verification.
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-- extraction-target: Rust/C++ LP solver.
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m1.support.foldl (fun (acc : Semantics.Q16_16) (p1 : Nat × Semantics.Q16_16) =>
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let (id1, w1) := p1
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m2.support.foldl (fun (acc2 : Semantics.Q16_16) (p2 : Nat × Semantics.Q16_16) =>
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let (id2, w2) := p2
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let d : Semantics.Q16_16 := if id1 == id2 then zero else one
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add acc2 (mul (mul w1 w2) d)
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) acc
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) zero
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/--
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Ollivier-Ricci Curvature between two adjacent nodes.
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kappa(x, y) = 1 - W(m_x, m_y) / d(x, y)
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-/
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def ollivierRicciCurvature (g : Graph) (_x _y : Nat) (mx my : GraphMeasure) : Semantics.Q16_16 :=
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let distXY := one -- Adjacent nodes distance = 1.0
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let w1 := wasserstein1Shim g mx my
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sub one (div w1 distXY)
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/--
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The Intelligence Ladder Metric:
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Mean Curvature K = Σ kappa(e) / |E|
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-/
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def intelligenceLadderMetric (g : Graph) (edges : List (Nat × Nat)) (measures : Nat → GraphMeasure) : Semantics.Q16_16 :=
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let totalCurvature := edges.foldl (fun (acc : Semantics.Q16_16) (e : Nat × Nat) =>
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let (u, v) := e
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add acc (ollivierRicciCurvature g u v (measures u) (measures v))
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) zero
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let count := edges.length
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if count == 0 then zero else Q16_16.ofRawInt (totalCurvature.val / (count : Int))
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/--
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Thresholds for the Intelligence Ladder based on research papers (2025-2026).
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C. elegans: < 0.2
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Drosophila: 0.2 - 0.5
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Vertebrate: > 0.6
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-/
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def isHighCognitiveCapacity (k : Semantics.Q16_16) : Bool :=
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k.val > 39321 -- 0.6 in Q16.16
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/-- Bind instance for Curvature logic. -/
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def curvatureInvariant (g : Graph) : String := s!"orc[{g.nodes.length}]"
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def curvatureCost (k1 k2 : Semantics.Q16_16) : Q16_16 :=
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abs (sub k1 k2)
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/-- Verification Triad -/
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def triangleNode0 : Node := { id := 0, type := NodeType.atom, label := "n0" }
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def triangleNode1 : Node := { id := 1, type := NodeType.atom, label := "n1" }
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def triangleNode2 : Node := { id := 2, type := NodeType.atom, label := "n2" }
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def triangleGraphNodes : List Node := [triangleNode0, triangleNode1, triangleNode2]
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def triangleGraphEdges : List Edge := [
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{ id := 0, source := triangleNode0, target := triangleNode1
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, type := EdgeType.similar_to, edgeClass := EdgeClass.definitional, weight := 1.0, justified := true },
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{ id := 1, source := triangleNode1, target := triangleNode2
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, type := EdgeType.similar_to, edgeClass := EdgeClass.definitional, weight := 1.0, justified := true },
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{ id := 2, source := triangleNode2, target := triangleNode0
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, type := EdgeType.similar_to, edgeClass := EdgeClass.definitional, weight := 1.0, justified := true }
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]
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def triangleGraph : Graph := {
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nodes := triangleGraphNodes,
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edges := triangleGraphEdges,
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nextId := 3
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}
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def uniformMeasureTriad (_id : Nat) : GraphMeasure :=
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let w : Q16_16 := Q16_16.ofRawInt 21845 -- 1/3 ≈ 0.3333
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{ support := [(0, w), (1, w), (2, w)] }
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/-- Witness check for triangle curvature. -/
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def triangleCurvatureWitness : UInt32 :=
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(ollivierRicciCurvature triangleGraph 0 1 (uniformMeasureTriad 0) (uniformMeasureTriad 1)).toBits
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#eval triangleCurvatureWitness
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end Semantics.Curvature
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