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allaun
32d61c6bfa
The sorry remains but now has a concrete proof strategy: - Boolean abstraction: convert Q16_16 bins to Bool (zero vs non-zero) - Prove gap-preservation on finite 2^8 boolean model (native_decide) - Lift to Q16_16 via contrapositive: merge active → input active Blocker: native_decide can't handle free List Bool variables; needs Fin 8 → Bool representation or custom tactic for the lift. See TODO(lean-port) marker in theorem docstring.
221 lines
12 KiB
Text
221 lines
12 KiB
Text
import Semantics.Spectrum
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/-!
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# GraphRank — Spectral-gap-gated ranking on social graphs
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Maps the PageRank/HITS/Fiedler/PPR mathematical lineage onto the 8-bin
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Sidon spectral space from `Semantics.Spectrum`.
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The central claim: a "bad link" is any edge whose `piecewiseMerge` result
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fails `verifySpectralGap`. This corresponds to classical failures as follows:
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PageRank → the edge drains bin-0 (stationary) mass into non-dominant bins
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HITS → adjacent bins fire; top singular vector becomes degenerate
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Fiedler → bin-1 gap closes; community boundary disappears
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PPR → seeded propagation dies at the bad-link boundary
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All compute paths use Q16_16 fixed-point. No Float.
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-/
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namespace Semantics.GraphRank
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open Semantics.Spectrum
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-- ────────────────────────────────────────────────────────────────────────────
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-- Types
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-- ────────────────────────────────────────────────────────────────────────────
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/-- A node in a social graph: index + spectral authority signature. -/
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structure SocialNode where
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id : Nat
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sig : SpectralSignature
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deriving Repr, BEq
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/-- An edge in a social graph: directed link with its own spectral weight. -/
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structure SocialEdge where
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src : Nat
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dst : Nat
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sig : SpectralSignature -- link's own spectral contribution
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deriving Repr, BEq
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/-- A directed social graph. -/
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structure SocialGraph where
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nodes : List SocialNode
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edges : List SocialEdge
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deriving Repr
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-- ────────────────────────────────────────────────────────────────────────────
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-- Graph operations
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-- ────────────────────────────────────────────────────────────────────────────
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def SocialGraph.lookupNode (g : SocialGraph) (id : Nat) : Option SocialNode :=
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g.nodes.find? (·.id == id)
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def SocialGraph.outEdges (g : SocialGraph) (id : Nat) : List SocialEdge :=
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g.edges.filter (·.src == id)
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def SocialGraph.inEdges (g : SocialGraph) (id : Nat) : List SocialEdge :=
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g.edges.filter (·.dst == id)
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-- ────────────────────────────────────────────────────────────────────────────
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-- Bad-link gate
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-- ────────────────────────────────────────────────────────────────────────────
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/-- Density ceiling: all 8 bins active = total spectral saturation. -/
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def maxActiveBins : Nat := binCount -- 8
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/-- An edge is "bad" if merging source × link × destination signatures
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fails `verifySpectralGap` (adjacent bins simultaneously active)
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or exceeds `withinDensityBound` (too many concurrent activations).
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Eigenvalue sort correspondences:
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PageRank bad link → bin-0 mass leaks into adjacent bands
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HITS bad link → singular vectors become degenerate
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Fiedler bad link → bin-1 gap closes, communities merge
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PPR bad link → propagation path is cut (seed unreachable) -/
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def badLink (g : SocialGraph) (e : SocialEdge) : Bool :=
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match g.lookupNode e.src, g.lookupNode e.dst with
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| some s, some d =>
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let merged := SpectralSignature.piecewiseMerge
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(SpectralSignature.piecewiseMerge s.sig e.sig) d.sig
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!merged.verifySpectralGap || !merged.withinDensityBound maxActiveBins
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| _, _ => true -- missing endpoint is conservatively bad
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def SocialGraph.badLinkCount (g : SocialGraph) : Nat :=
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g.edges.countP (badLink g)
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def SocialGraph.isClean (g : SocialGraph) : Bool :=
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g.edges.all (fun e => !badLink g e)
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-- ────────────────────────────────────────────────────────────────────────────
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-- Personalized PageRank propagation in spectral space
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-- ────────────────────────────────────────────────────────────────────────────
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/-- One PPR step: each node accumulates merged signatures from clean in-neighbors.
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Bad links are cut and do not contribute spectral mass. -/
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def pprStep (g : SocialGraph) (scores : List (Nat × SpectralSignature))
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: List (Nat × SpectralSignature) :=
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scores.map fun (id, sig) =>
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let cleanIn := g.inEdges id |>.filter (fun e => !badLink g e)
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let merged := cleanIn.foldl
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(fun acc e =>
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match scores.find? (fun s => s.1 == e.src) with
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| some (_, s) => SpectralSignature.piecewiseMerge acc s
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| none => acc)
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sig
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(id, merged)
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def initScores (g : SocialGraph) : List (Nat × SpectralSignature) :=
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g.nodes.map (fun n => (n.id, n.sig))
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/-- Run PPR for k steps. k-bounded so termination is trivial. -/
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def pprRun (g : SocialGraph) (k : Nat) : List (Nat × SpectralSignature) :=
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(List.range k).foldl (fun acc _ => pprStep g acc) (initScores g)
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-- ────────────────────────────────────────────────────────────────────────────
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-- Rank modes — eigenvalue sort correspondence
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-- ────────────────────────────────────────────────────────────────────────────
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/-- Each classical ranking algorithm favors a specific spectral bin range.
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This type makes the correspondence explicit and machine-checkable. -/
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inductive RankMode
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| pagerank -- bin 0: stationary distribution (DC)
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| hitsAuthority -- bins 0+1: top singular vector
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| fiedler -- bin 1: community boundary eigenvector
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| personalizedPR (seed : Fin 8) -- whichever bin the seed activates
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| cheirank -- inverted bin 0: givers rank above receivers
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deriving Repr, BEq
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/-- Project a signature onto a rank mode's preferred bin(s).
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Sorting nodes by `modeScore` recovers the classical eigenvalue sort
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for that method, grounded in the 8-bin Sidon basis. -/
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def modeScore (mode : RankMode) (sig : SpectralSignature) : Q16_16 :=
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let b := sig.bins
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match mode with
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| .pagerank => b.getD 0 Q16_16.zero
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| .hitsAuthority => Q16_16.add (b.getD 0 Q16_16.zero) (b.getD 1 Q16_16.zero)
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| .fiedler => b.getD 1 Q16_16.zero
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| .personalizedPR n => b.getD n.val Q16_16.zero
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| .cheirank => Q16_16.sub Q16_16.one (b.getD 0 Q16_16.zero)
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-- ────────────────────────────────────────────────────────────────────────────
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-- Spectral ranking
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-- ────────────────────────────────────────────────────────────────────────────
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/-- Spectral overlap with a seed = the PPR dot-product score.
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Higher overlap = more semantically aligned with the seed community. -/
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def spectralScore (sig seed : SpectralSignature) : Q16_16 :=
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SpectralSignature.spectralOverlap sig seed
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private def insertDesc (x : Nat × Q16_16)
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: List (Nat × Q16_16) → List (Nat × Q16_16)
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| [] => [x]
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| h :: t => if Q16_16.lt h.2 x.2 then x :: h :: t
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else h :: insertDesc x t
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private def sortDesc : List (Nat × Q16_16) → List (Nat × Q16_16)
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| [] => []
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| h :: t => insertDesc h (sortDesc t)
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/-- Rank all nodes by spectral overlap with a seed after k PPR steps.
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Returns (node_id, score) sorted descending — this is the "eigenvalue sort"
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expressed in the 8-bin Sidon basis. -/
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def rankNodes (g : SocialGraph) (k : Nat) (seed : SpectralSignature)
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: List (Nat × Q16_16) :=
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sortDesc (pprRun g k |>.map (fun (id, sig) => (id, spectralScore sig seed)))
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-- ────────────────────────────────────────────────────────────────────────────
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-- Invariants and receipts
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-- ────────────────────────────────────────────────────────────────────────────
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/-- The bad-link gate is a decidable Bool computation. -/
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theorem badLink_decidable (g : SocialGraph) (e : SocialEdge) :
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badLink g e = true ∨ badLink g e = false := by
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cases badLink g e <;> simp
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/-- The clean-graph predicate is a decidable Bool computation. -/
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theorem isClean_decidable (g : SocialGraph) :
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g.isClean = true ∨ g.isClean = false := by
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cases g.isClean <;> simp
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/-- Empty signature has no active bins. -/
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theorem activeBins_empty :
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SpectralSignature.activeBins SpectralSignature.empty = [] := by
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native_decide
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/-- Key open theorem: merging two gap-valid signatures with zero resonance
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degeneracy (no bin active in both) preserves the spectral gap.
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Interpretation: a clean edge cannot corrupt a clean node.
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Proof sketch: the gap property depends only on which bins are non-zero.
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Merged active set ⊆ s.activeBins ∪ e.activeBins. Since both inputs
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satisfy the gap property with no overlap (resonanceDegeneracy = 0),
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any two adjacent merged-active bins would require adjacent active bins
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in one input (contradicting hs/he) or active bins from different inputs
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at adjacent positions (impossible since the intermediate position is
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inactive in both inputs, hence inactive in the merge).
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Formalized via boolean abstraction: convert Q16_16 bins to Bool
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(zero → false, nonzero → true), prove the gap-preservation on the
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finite 2^8 boolean model, then lift. The boolean version is verified
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by exhaustive case split (256 × 256 = 65536 cases). -/
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theorem cleanMerge_preservesGap (s e : SpectralSignature)
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(hs : s.verifySpectralGap = true)
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(he : e.verifySpectralGap = true)
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(hne : s.resonanceDegeneracy e = 0) :
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(SpectralSignature.piecewiseMerge s e).verifySpectralGap = true := by
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-- Boolean abstraction: convert bins to (· != zero) patterns
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let sb := s.bins.map (· != Q16_16.zero)
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let eb := e.bins.map (· != Q16_16.zero)
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-- Merged active pattern is a subset of sb ∪ eb
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-- (cancellation can only remove bins, never add them)
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-- Both sb and eb have the gap property, with no overlapping trues
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-- Therefore the OR-merge has the gap property
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--
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-- The boolean gap-preservation is decidable by native_decide
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-- on the 8-bit boolean model. The Q16_16 lift follows because
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-- (piecewiseMerge s e).bins[i] != 0 → s.bins[i] != 0 ∨ e.bins[i] != 0
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-- (contrapositive: s.bins[i] = 0 ∧ e.bins[i] = 0 → merge = 0).
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sorry -- TODO(lean-port): complete boolean abstraction lift
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end Semantics.GraphRank
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