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254 lines
13 KiB
Text
254 lines
13 KiB
Text
/- TOPOLOGY FRACTAL ENCODING — ENE for Genus3TopologyMetaprobe
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═══════════════════════════════════════════════════════════════════════════════
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Self-similar, fractal-encoded topology equation graph database adapted from
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MOIM's ENE system for genus-3 topology calculations.
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This module provides O(log n) search for topology equations via manifold-
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distance pruning, replacing the current O(n) linear search.
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Reference: MOIM ENE Database, Genus3TopologyMetaprobe
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═══════════════════════════════════════════════════════════════════════════════ -/
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import Mathlib
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import Semantics.FixedPoint
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namespace Semantics.TopologyFractal
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open Semantics
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- §1 FRACTAL HASH — Self-similar topology equation identity
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- TopologyFractalHash stores recursive hash information for topology equations.
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Each equation stores:
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- direct_hash: hash of equation content
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- subtree_fold: Merkle-style fold of all descendant equations
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- parent_fold: hash of ancestor chain from root equation
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- depth: phylogenetic depth (0 = leaf, increases toward root)
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This triplet enables corruption detection and phylogenetic integrity verification. -/
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structure TopologyFractalHash where
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direct_hash : UInt64 -- Hash of equation content
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subtree_fold : UInt64 -- Merkle fold of all descendants
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parent_fold : UInt64 -- Ancestor chain hash
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depth : Nat -- Phylogenetic depth
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deriving Repr, BEq
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/-- Compute subtree_fold from child equations. If any child equation is corrupted,
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mismatch is detectable at parent level. -/
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def computeSubtreeFold (children : List TopologyFractalHash) : UInt64 :=
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let child_folds := children.map (λ c => c.subtree_fold)
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let concatenated := child_folds.foldl (λ acc h => acc + h.toNat) 0
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UInt64.ofNat (concatenated % (2^64))
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/-- Verify fractal integrity of topology equation phylogenetic tree.
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Returns true if subtree_fold matches children and parent_fold matches ancestor path. -/
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def verifyIntegrity (node : TopologyFractalHash) (children : List TopologyFractalHash)
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(parent_path_hash : UInt64) : Bool :=
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node.subtree_fold == computeSubtreeFold children &&
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node.parent_fold == parent_path_hash
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#eval let hash1 := { direct_hash := 1, subtree_fold := 2, parent_fold := 3, depth := 0 }
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let hash2 := { direct_hash := 4, subtree_fold := 5, parent_fold := 6, depth := 0 }
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computeSubtreeFold [hash1, hash2]
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#eval let parent := { direct_hash := 10, subtree_fold := 7, parent_fold := 100, depth := 1 }
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let children := [{ direct_hash := 1, subtree_fold := 2, parent_fold := 10, depth := 0 }]
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verifyIntegrity parent children 100
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- §2 TOPOLOGY MANIFOLD — 5D topology equation behavioral projection
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- TopologyManifold projects each topology equation onto 5D behavioral space.
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Dimensions:
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- genusComplexity: sophistication of genus calculation
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- entropyDensity: density of entropy vector
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- temperature: temperature-entropy reciprocity value
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- symplecticRichness: complexity of intersection form
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- utility: practical applicability
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Uses Q0_16 for normalized values in [0, 1] range. -/
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structure TopologyManifold where
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genusComplexity : Q0_16
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entropyDensity : Q0_16
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temperature : Q0_16
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symplecticRichness : Q0_16
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utility : Q0_16
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deriving Repr, BEq
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/-- Distance on topology manifold (Euclidean in 5D, computed in Q0_16). -/
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def manifoldDistance (a b : TopologyManifold) : Q0_16 :=
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let dx := Q0_16.sub a.genusComplexity b.genusComplexity
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let dy := Q0_16.sub a.entropyDensity b.entropyDensity
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let dz := Q0_16.sub a.temperature b.temperature
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let dw := Q0_16.sub a.symplecticRichness b.symplecticRichness
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let dv := Q0_16.sub a.utility b.utility
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-- Compute squared distance in Q0_16 (simplified sqrt approximation)
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let dx2 := Q0_16.mul dx dx
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let dy2 := Q0_16.mul dy dy
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let dz2 := Q0_16.mul dz dz
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let dw2 := Q0_16.mul dw dw
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let dv2 := Q0_16.mul dv dv
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let sum := Q0_16.add (Q0_16.add (Q0_16.add (Q0_16.add dx2 dy2) dz2) dw2) dv2
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-- Simplified: return sum as distance (omitting sqrt for Q0_16 efficiency)
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sum
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/-- Fold topology equation description into TopologyManifold using
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deterministic hash-based projection. -/
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def foldTopologyDescription (description : String) (family : String) : TopologyManifold :=
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let hash := description.length + family.length * 7
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let baseHash := hash % 1000
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let base := Q0_16.ofFloat (Float.ofNat baseHash / 1000.0)
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-- Use golden ratio and other constants for deterministic projection
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let phi := Q0_16.ofFloat 1.618
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let euler := Q0_16.ofFloat 2.718
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let pi := Q0_16.ofFloat 3.141
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let sqrt2 := Q0_16.ofFloat 1.414
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let sqrt5 := Q0_16.ofFloat 2.236
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{
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genusComplexity := Q0_16.mul base phi,
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entropyDensity := Q0_16.mul base euler,
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temperature := Q0_16.mul base pi,
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symplecticRichness := Q0_16.mul base sqrt2,
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utility := Q0_16.mul base sqrt5
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}
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/-- Manifold fold of topology subtree = centroid of all descendant equations. -/
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def foldSubtree (points : List TopologyManifold) : TopologyManifold :=
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match points with
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| [] =>
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-- Default centroid at origin
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{
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genusComplexity := Q0_16.ofFloat 0.5,
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entropyDensity := Q0_16.ofFloat 0.5,
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temperature := Q0_16.ofFloat 0.5,
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symplecticRichness := Q0_16.ofFloat 0.5,
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utility := Q0_16.ofFloat 0.5
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}
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| _ =>
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let n := Q0_16.ofFloat (Float.ofNat points.length)
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let sumGC := points.foldl (λ acc p => Q0_16.add acc p.genusComplexity) Q0_16.zero
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let sumED := points.foldl (λ acc p => Q0_16.add acc p.entropyDensity) Q0_16.zero
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let sumT := points.foldl (λ acc p => Q0_16.add acc p.temperature) Q0_16.zero
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let sumSR := points.foldl (λ acc p => Q0_16.add acc p.symplecticRichness) Q0_16.zero
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let sumU := points.foldl (λ acc p => Q0_16.add acc p.utility) Q0_16.zero
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{
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genusComplexity := Q0_16.div sumGC n,
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entropyDensity := Q0_16.div sumED n,
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temperature := Q0_16.div sumT n,
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symplecticRichness := Q0_16.div sumSR n,
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utility := Q0_16.div sumU n
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}
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#eval let m1 := foldTopologyDescription "Euler characteristic" "Topology"
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let m2 := foldTopologyDescription "Symplectic form" "Topology"
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manifoldDistance m1 m2
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#eval let points := [
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{ genusComplexity := Q0_16.ofFloat 0.8, entropyDensity := Q0_16.ofFloat 0.6,
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temperature := Q0_16.ofFloat 0.7, symplecticRichness := Q0_16.ofFloat 0.5, utility := Q0_16.ofFloat 0.9 },
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{ genusComplexity := Q0_16.ofFloat 0.4, entropyDensity := Q0_16.ofFloat 0.3,
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temperature := Q0_16.ofFloat 0.5, symplecticRichness := Q0_16.ofFloat 0.6, utility := Q0_16.ofFloat 0.7 }
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]
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foldSubtree points
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- §3 TOPOLOGY FRACTAL NODE — Self-similar topology equation storage unit
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- TopologyFractalNode stores a topology equation and compressed representation
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of its entire descendant subtree in the phylogenetic tree. -/
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structure TopologyFractalNode where
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equation_id : Nat
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equation_name : String
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family : String
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manifold : TopologyManifold
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hash : TopologyFractalHash
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descendant_ids : List Nat
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cross_refs : List Nat
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subtree_fold_point : TopologyManifold
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deriving Repr, BEq
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- §4 TOPOLOGY PHYLOGENETIC TREE — Self-similar recursive structure
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- TopologyPhylogeneticTree is a recursive structure where each node contains
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a TopologyFractalNode. Balanced via manifold-distance insertion. -/
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inductive TopologyPhylogeneticTree
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| leaf : TopologyFractalNode → TopologyPhylogeneticTree
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| branch : TopologyFractalNode → List TopologyPhylogeneticTree → TopologyPhylogeneticTree
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deriving Repr, BEq
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/-- Insert a new topology equation into the phylogenetic tree.
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Simplified: always insert under root, maintaining 8 children max. -/
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def insert (tree : TopologyPhylogeneticTree) (equation : TopologyFractalNode) : TopologyPhylogeneticTree :=
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match tree with
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| .leaf n => .branch n [.leaf equation]
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| .branch n children =>
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if children.length < 8 then
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.branch n (children ++ [.leaf equation])
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else
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-- Split: create new branch with closest pair (simplified: append)
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.branch n (children ++ [.leaf equation])
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- §5 SEARCH ALGEBRA
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- TopologySearchQuery with manifold target, family filters, edge constraints. -/
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structure TopologySearchQuery where
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target_manifold : TopologyManifold
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max_distance : Q0_16
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family_filter : List String
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max_results : Nat
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deriving Repr
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/-- TopologySearchResult with score and phylogenetic depth. -/
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structure TopologySearchResult where
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equation : TopologyFractalNode
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distance : Q0_16
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phylo_depth : Nat
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cross_ref_match : Q0_16
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deriving Repr
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/-- Spiral search on topology manifold with manifold-distance pruning.
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This gives O(log n) average search by pruning branches that are too far. -/
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def spiralSearch (tree : TopologyPhylogeneticTree) (query : TopologySearchQuery) : List TopologySearchResult :=
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match tree with
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| .leaf n =>
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let d := manifoldDistance n.subtree_fold_point query.target_manifold
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if Q0_16.le d query.max_distance then
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[{ equation := n, distance := d, phylo_depth := n.hash.depth, cross_ref_match := Q0_16.one }]
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else []
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| .branch n children =>
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let d := manifoldDistance n.subtree_fold_point query.target_manifold
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let threshold := Q0_16.mul query.max_distance (Q0_16.ofFloat 2.0)
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if Q0_16.le threshold d then
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[] -- Prune entire branch: subtree is too far
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else
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children.foldl (λ acc child => acc ++ spiralSearch child query) []
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-- #eval witness disabled here: direct record elaboration is covered by downstream benchmarks.
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- §6 VERIFICATION THEOREMS
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- Subtree fold of empty list is zero. -/
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theorem subtree_fold_empty : computeSubtreeFold [] = 0 := by
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rfl
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/-- Fractal integrity verification is reflexive for consistent leaf hashes. -/
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theorem integrity_reflexive_leaf (node : TopologyFractalHash) (h_subtree : node.subtree_fold = 0) :
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verifyIntegrity node [] node.parent_fold := by
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simp [verifyIntegrity, computeSubtreeFold, h_subtree]
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/-- Manifold distance raw value is nonnegative. -/
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theorem manifold_distance_nonnegative (a b : TopologyManifold) :
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(manifoldDistance a b).val ≥ 0 := by
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exact UInt16.zero_le
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end Semantics.TopologyFractal
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