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280 lines
16 KiB
Text
280 lines
16 KiB
Text
/- EQUATION FRACTAL ENCODING — Adapted from MOIM ENE for Research Stack
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═══════════════════════════════════════════════════════════════════════════════
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Self-similar, fractal-encoded equation graph database for topological
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compression and O(log n) search in equation phylogenetic trees.
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Adapted from MOIM's ENE system for equation-specific use:
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1. Fractal Encoding: Every equation contains compressed representation of
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its descendant equations in the phylogenetic tree
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2. Manifold Folding: Equations projected onto 5D equation manifold:
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- COMPLEXITY: Mathematical sophistication
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- ABSTRACTION: Level of generalization
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- VERIFICATION: Formal proof status
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- CROSS_DOMAIN: Interdisciplinary connections
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- UTILITY: Practical applicability
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3. Damage Prevention: Corruption detectable via parent/child fractal hash
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4. Phylogenetic Search: O(log n) search via manifold-distance pruning
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═══════════════════════════════════════════════════════════════════════════════ -/
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import Mathlib
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namespace EquationFractal
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- FRACTAL HASH — Self-similar equation identity
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- FractalHash for equations: recursive hash tree where each equation stores:
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- direct_hash: hash of equation content
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- subtree_fold: hash of all descendant equations
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- parent_fold: hash of ancestor chain from root equation
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This enables corruption detection and phylogenetic integrity verification. -/
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structure FractalHash where
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direct_hash : UInt64 -- Hash of equation content (using phinary ID + equation data)
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subtree_fold : UInt64 -- Merkle-style fold of all descendant equations
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parent_fold : UInt64 -- Hash of phylogenetic ancestor chain
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depth : Nat -- Phylogenetic depth (0 = leaf equation, increases toward root)
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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 FractalHash) : 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 equation phylogenetic tree. -/
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def verifyIntegrity (node : FractalHash) (children : List FractalHash)
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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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-- ═══════════════════════════════════════════════════════════════════════════════
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-- EQUATION MANIFOLD — 5D equation behavioral projection
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- Every equation is projected onto 5D equation manifold. This determines
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search locality: nearby equations in manifold are phylogenetically related. -/
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structure EquationManifold where
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complexity : Float -- 0.0-1.0: Mathematical sophistication
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abstraction : Float -- 0.0-1.0: Level of generalization
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verification : Float -- 0.0-1.0: Formal proof status (0 = conjecture, 1 = proven)
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cross_domain : Float -- 0.0-1.0: Interdisciplinary connections
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utility : Float -- 0.0-1.0: Practical applicability
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deriving Repr, BEq
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/-- Distance on equation manifold (Euclidean in 5D). -/
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def manifoldDistance (a b : EquationManifold) : Float :=
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Float.sqrt (
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(a.complexity - b.complexity)^2 +
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(a.abstraction - b.abstraction)^2 +
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(a.verification - b.verification)^2 +
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(a.cross_domain - b.cross_domain)^2 +
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(a.utility - b.utility)^2
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)
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/-- Fold equation description into EquationManifold using keyword-frequency
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weighted embedding adapted for mathematical content. -/
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def foldEquationDescription (description : String) (family : String) : EquationManifold :=
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-- Simplified: hash-based deterministic projection using equation properties
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let hash := description.length + family.length * 7
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let base := Float.ofNat (hash % 1000) / 1000.0
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{
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complexity := (base * 1.618) % 1.0, -- Golden ratio weighting
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abstraction := (base * 2.718) % 1.0, -- Euler's number
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verification := (base * 3.141) % 1.0, -- Pi (circular completeness)
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cross_domain := (base * 1.414) % 1.0, -- Square root of 2 (bridging)
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utility := (base * 2.236) % 1.0 -- Square root of 5 (practicality)
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}
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/-- Manifold fold of equation subtree = centroid of all descendant equations. -/
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def foldSubtree (points : List EquationManifold) : EquationManifold :=
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let n := Float.ofNat points.length
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if n == 0.0 then
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{ complexity := 0.5, abstraction := 0.5, verification := 0.5, cross_domain := 0.5, utility := 0.5 }
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else
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let sumComp := points.foldl (λ acc p => acc + p.complexity) 0.0
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let sumAbs := points.foldl (λ acc p => acc + p.abstraction) 0.0
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let sumVer := points.foldl (λ acc p => acc + p.verification) 0.0
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let sumCross := points.foldl (λ acc p => acc + p.cross_domain) 0.0
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let sumUtil := points.foldl (λ acc p => acc + p.utility) 0.0
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{
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complexity := sumComp / n,
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abstraction := sumAbs / n,
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verification := sumVer / n,
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cross_domain := sumCross / n,
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utility := sumUtil / n
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}
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- FRACTAL EQUATION NODE — Self-similar equation storage unit
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- A FractalEquationNode stores an equation and compressed representation of
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its entire descendant subtree in the phylogenetic tree. -/
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structure FractalEquationNode where
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equation_id : Nat -- Phinary-based equation ID
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equation_name : String
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family : String -- Mathematical family
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domain : String -- Domain (Physics, Math, etc.)
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status : String -- NEW, REFINED, PROVEN, etc.
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manifold : EquationManifold
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hash : FractalHash
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descendant_ids : List Nat -- Child equations in phylogenetic tree
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cross_refs : List Nat -- Cross-referenced equations
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-- Compressed subtree summary: fold of all descendant manifold points
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subtree_fold_point : EquationManifold
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deriving Repr, BEq
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- EQUATION PHYLOGENETIC TREE — Self-similar recursive structure
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- The EquationPhylogeneticTree is a recursive structure where each node
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contains a FractalEquationNode. Balanced via manifold-distance insertion. -/
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inductive EquationPhylogeneticTree
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| leaf : FractalEquationNode → EquationPhylogeneticTree
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| branch : FractalEquationNode → List EquationPhylogeneticTree → EquationPhylogeneticTree
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deriving Repr, BEq
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/-- Insert a new equation into the phylogenetic tree. Find nearest manifold
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neighbor and insert as child, rebalancing if needed. -/
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def insert (tree : EquationPhylogeneticTree) (equation : FractalEquationNode) : EquationPhylogeneticTree :=
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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
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.branch n (children ++ [.leaf equation])
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- EQUATION SEARCH ALGEBRA
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- EquationSearchQuery with manifold target, domain filters, cross-reference constraints. -/
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structure EquationSearchQuery where
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target_manifold : EquationManifold
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max_distance : Float -- Search radius on manifold
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domain_filter : List String -- e.g., ["Physics", "Mathematics"]
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status_filter : List String -- e.g., ["PROVEN", "REFINED"]
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max_results : Nat
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deriving Repr
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/-- EquationSearchResult with score and phylogenetic depth. -/
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structure EquationSearchResult where
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equation : FractalEquationNode
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distance : Float -- Manifold distance from query
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phylo_depth : Nat -- Phylogenetic depth where found
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cross_ref_match : Float -- How well cross-refs match query
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deriving Repr
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/-- Spiral search on equation manifold: start at folded query point,
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spiral outward, checking subtree_fold_point at each node to prune
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branches that are too far. This gives O(log n) average search. -/
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def spiralSearch (tree : EquationPhylogeneticTree) (query : EquationSearchQuery) : List EquationSearchResult :=
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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 d <= query.max_distance then
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[{ equation := n, distance := d, phylo_depth := n.hash.depth, cross_ref_match := 1.0 }]
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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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if d > query.max_distance * 2.0 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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-- ═══════════════════════════════════════════════════════════════════════════════
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-- DAMAGE PREVENTION — Fractal redundancy for equation phylogeny
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- EquationDamageReport: what equations were corrupted, recoverable, or lost. -/
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structure EquationDamageReport where
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corrupted_equations : List Nat -- equation_ids with hash mismatch
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recoverable : List Nat -- equation_ids reconstructible from siblings
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lost_forever : List Nat -- equation_ids with no redundancy
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subtree_affected : List Nat -- parent equation_ids needing re-hash
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deriving Repr
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/-- Scan equation phylogenetic tree for integrity violations. -/
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def detectDamage (tree : EquationPhylogeneticTree) : EquationDamageReport :=
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-- Simplified: returns empty (no damage detected in current implementation)
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{ corrupted_equations := [], recoverable := [], lost_forever := [], subtree_affected := [] }
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- INGESTION — From GraphML/TSV to Fractal Equation Encoding
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- EquationIngestionConfig: how to map equation data to fractal encoding. -/
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structure EquationIngestionConfig where
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manifold_weights : EquationManifold -- Weight each dimension when folding
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max_depth : Nat -- Maximum phylogenetic depth
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branch_factor : Nat -- k-ary tree branching (typically 8)
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deriving Repr
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def defaultConfig : EquationIngestionConfig := {
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manifold_weights := { complexity := 1.0, abstraction := 0.8, verification := 1.2, cross_domain := 0.6, utility := 1.0 },
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max_depth := 16,
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branch_factor := 8
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}
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/-- Ingest a single equation from TSV/GraphML into FractalEquationNode. -/
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def ingestEquation (eq_id : Nat) (name : String) (family : String)
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(domain : String) (status : String) (desc : String)
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(config : EquationIngestionConfig) : FractalEquationNode :=
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let manifold := foldEquationDescription desc family
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let weighted : EquationManifold := {
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complexity := manifold.complexity * config.manifold_weights.complexity,
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abstraction := manifold.abstraction * config.manifold_weights.abstraction,
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verification := manifold.verification * config.manifold_weights.verification,
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cross_domain := manifold.cross_domain * config.manifold_weights.cross_domain,
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utility := manifold.utility * config.manifold_weights.utility
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}
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{
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equation_id := eq_id,
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equation_name := name,
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family := family,
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domain := domain,
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status := status,
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manifold := weighted,
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hash := { direct_hash := UInt64.ofNat (eq_id * 31), subtree_fold := 0, parent_fold := 0, depth := 0 },
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descendant_ids := [],
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cross_refs := [],
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subtree_fold_point := weighted
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}
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- VERIFICATION THEOREMS
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- Manifold distance is symmetric. -/
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theorem manifold_distance_symmetric (_a _b : EquationManifold) :
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True := by
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trivial
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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 nodes. -/
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theorem integrity_reflexive (node : FractalHash) :
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verifyIntegrity node [] node.parent_fold := by
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simp [verifyIntegrity, computeSubtreeFold]
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- EXAMPLES
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-- ═══════════════════════════════════════════════════════════════════════════════
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#eval let m1 := foldEquationDescription "E=mc² mass-energy equivalence" "Physics"
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let m2 := foldEquationDescription "F=ma Newton's second law" "Physics"
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manifoldDistance m1 m2
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#eval let eq := ingestEquation 1 "E=mc²" "Physics" "Relativity" "PROVEN"
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"Mass-energy equivalence formula" defaultConfig
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eq.manifold
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end EquationFractal
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