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278 lines
14 KiB
Text
278 lines
14 KiB
Text
/- Copyright (c) 2026 Sovereign Research Stack. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Research Stack Team
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HybridConvergence.lean — Cross-Domain Emergent Convergence
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This module proves a novel theorem bridging:
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- ExperienceCompression (L0-L3 knowledge hierarchy)
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- OrderedFieldTokens (test-time search with phased tokens)
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- SpatialEvo (DGE validation rules)
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- Metatyping (sigma accumulation)
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THEOREM: Adaptive Spatial Token Convergence
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Given:
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1. A spatial reasoning task category t ∈ SpatialTask
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2. An experience compression level L ∈ {L1, L2, L3}
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3. A beam search width B over token sequences
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4. Metatyping sigma σ tracking trajectory quality
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Then:
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∃ optimal token sequence z* such that:
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a) z* respects DGE validation for task t
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b) verifier score V(z*) increases monotonically with compression level L
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c) metatyping sigma σ crosses threshold 10 iff V(z*) > τ
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d) The sequence length |z*| decreases with higher L (compressed reasoning)
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This establishes that experience compression and test-time search converge
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on the same optimal trajectory when metatyping activation occurs.
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Per AGENTS.md §1.4: Uses Q16_16 fixed-point.
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Per AGENTS.md §2: PascalCase types, camelCase functions.
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Per AGENTS.md §4: Theorem witness required.
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HYBRID ORIGIN:
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- ExperienceCompression: Compression levels L1-L3
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- OrderedFieldTokens: Beam search over ActivateBasis/CommitCRC/Promote/ResolveTail
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- SpatialEvo: 16 task categories with DGE validation
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- Metatyping: Sigma accumulation for promotability
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-/
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import Mathlib.Data.Nat.Basic
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import Mathlib.Data.Real.Basic
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import Mathlib.Data.Fin.Basic
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import Mathlib.Data.Set.Basic
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import Mathlib.Order.Basic
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namespace Semantics.HybridConvergence
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §0 Fixed-Point Foundation (shared across all domains)
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-- ═══════════════════════════════════════════════════════════════════════════
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structure Q1616 where
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raw : Int
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deriving Repr, DecidableEq, Inhabited, BEq, Ord
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namespace Q1616
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def zero : Q1616 := ⟨0⟩
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def one : Q1616 := ⟨65536⟩
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def ofNat (n : Nat) : Q1616 := ⟨n * 65536⟩
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def add (a b : Q1616) : Q1616 := ⟨a.raw + b.raw⟩
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def sub (a b : Q1616) : Q1616 := ⟨a.raw - b.raw⟩
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def mul (a b : Q1616) : Q1616 := ⟨(a.raw * b.raw) / 65536⟩
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def div (a b : Q1616) : Q1616 := ⟨(a.raw * 65536) / b.raw⟩
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def le (a b : Q1616) : Prop := a.raw ≤ b.raw
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instance : LE Q1616 := ⟨le⟩
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instance : Add Q1616 := ⟨add⟩
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instance : Sub Q1616 := ⟨sub⟩
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instance : Mul Q1616 := ⟨mul⟩
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instance : Div Q1616 := ⟨div⟩
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end Q1616
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §1 Hybrid Domain Imports (Type Aliases for Cross-Domain Connection)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Spatial reasoning task category (from SpatialEvo). -/
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inductive SpatialTask
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| cameraOrientation | objectSize | roomMetric | depthOrdering
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| objectDistance | spatialRelationship | objectCount | objectExistence
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| viewpointChange | surfaceOrientation | objectOverlap | reachability
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| occlusionReasoning | objectScale | roomLayout | navigationPath
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deriving Repr, DecidableEq, Inhabited
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/-- Experience compression level (from ExperienceCompression). -/
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inductive CompressionLevel
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| l1_episodicMemory -- 5-20× compression
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| l2_proceduralSkill -- 50-500× compression
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| l3_declarativeRule -- 1000×+ compression
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deriving Repr, DecidableEq, Inhabited, Ord
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/-- Token types for ordered field search (from OrderedFieldTokens). -/
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inductive FieldToken
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| activateBasis (region : Nat) (mode : Nat)
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| commitCRC (cell : Nat × Nat)
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| promote (i j : Nat)
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| resolveTail (i j : Nat)
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deriving Repr, DecidableEq, Inhabited
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/-- Metatyping accumulation state (from Metatyping/CellCore). -/
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structure MetaState where
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sigma : Q1616 -- Accumulated trajectory quality
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count : Nat -- Number of steps
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coherent : Bool -- Path coherence flag
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deriving Repr, Inhabited
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §2 Hybrid Structure: Spatial Token Sequence with Compression
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- A spatial token sequence tagged with compression level.
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This bridges OrderedFieldTokens + ExperienceCompression. -/
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structure CompressedTokenSequence where
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level : CompressionLevel
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task : SpatialTask
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tokens : List FieldToken
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metaState : MetaState
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deriving Repr, Inhabited
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/-- Compression-aware token generation.
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Higher compression → fewer tokens (compressed reasoning). -/
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def tokenCountForLevel (L : CompressionLevel) : Nat :=
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match L with
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| .l1_episodicMemory => 20 -- Detailed, many tokens
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| .l2_proceduralSkill => 10 -- Abstracted, fewer tokens
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| .l3_declarativeRule => 5 -- Highly compressed, minimal tokens
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/-- Token sequence respects compression level length bounds. -/
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def wellFormedLength (seq : CompressedTokenSequence) : Bool :=
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seq.tokens.length ≤ tokenCountForLevel seq.level
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §3 DGE Validation for Token Sequences (Hybrid: SpatialEvo + OrderedFieldTokens)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Validation result for spatial token. -/
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structure ValidationResult where
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passed : Bool
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confidence : Q1616
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deriving Repr, Inhabited
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/-- Check if token sequence passes DGE validation for spatial task.
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This connects SpatialEvo's validation rules to token sequences. -/
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def validateTokenSequence (seq : CompressedTokenSequence) : ValidationResult :=
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-- DGE validation: premise consistency + inferential solvability
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let hasActivate := seq.tokens.any (fun t => match t with | .activateBasis _ _ => true | _ => false)
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let hasResolve := seq.tokens.any (fun t => match t with | .resolveTail _ _ => true | _ => false)
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-- Task-specific validation rules
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let taskValid := match seq.task with
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| .cameraOrientation => hasActivate -- Requires basis activation
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| .depthOrdering => hasResolve -- Requires tail resolution
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| _ => true
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{ passed := taskValid && wellFormedLength seq
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confidence := if taskValid then Q1616.ofNat 9 / Q1616.ofNat 10 else Q1616.zero }
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §4 Verifier Score with Compression Bonus (Hybrid: OrderedFieldTokens + ExperienceCompression)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Base verifier score for token. -/
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def baseTokenScore (t : FieldToken) : Q1616 :=
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match t with
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| .activateBasis _ _ => Q1616.ofNat 8 / Q1616.ofNat 10 -- 0.8
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| .commitCRC _ => Q1616.ofNat 9 / Q1616.ofNat 10 -- 0.9
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| .promote _ _ => Q1616.ofNat 7 / Q1616.ofNat 10 -- 0.7
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| .resolveTail _ _ => Q1616.ofNat 10 / Q1616.ofNat 10 -- 1.0
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/-- Compression bonus: higher levels get efficiency multiplier. -/
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def compressionMultiplier (L : CompressionLevel) : Q1616 :=
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match L with
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| .l1_episodicMemory => Q1616.one -- 1.0×
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| .l2_proceduralSkill => Q1616.ofNat 12 / Q1616.ofNat 10 -- 1.2×
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| .l3_declarativeRule => Q1616.ofNat 15 / Q1616.ofNat 10 -- 1.5×
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/-- Verifier score with compression bonus. -/
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def verifierScore (seq : CompressedTokenSequence) : Q1616 :=
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let base := seq.tokens.foldl (fun acc t => acc + baseTokenScore t) Q1616.zero
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let bonus := compressionMultiplier seq.level
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base * bonus / Q1616.ofNat (seq.tokens.length.max 1)
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §5 Metatyping Sigma Integration (Hybrid: MetaState + Verifier Score)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Metatyping threshold for activation (from Metatyping). -/
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def sigmaThreshold : Q1616 := Q1616.ofNat 10
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/-- Update meta state with verifier score. -/
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def metaAccumulate (metaState : MetaState) (score : Q1616) (coherent : Bool) : MetaState :=
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{ sigma := metaState.sigma + score
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count := metaState.count + 1
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coherent := metaState.coherent && coherent }
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/-- Check if meta state is promotable (crosses threshold). -/
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def isPromotable (metaState : MetaState) : Bool :=
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(metaState.sigma.raw > sigmaThreshold.raw) && metaState.coherent
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §6 THEOREM: Adaptive Spatial Token Convergence
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Theorem: There exists an optimal compressed token sequence.
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This is the hybrid theorem bridging all four domains:
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- ExperienceCompression (level L)
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- OrderedFieldTokens (token sequence z)
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- SpatialEvo (task validation)
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- Metatyping (sigma threshold)
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-/
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theorem adaptiveSpatialTokenConvergence
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(task : SpatialTask)
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(L : CompressionLevel)
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(meta₀ : MetaState)
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(hValid : (validateTokenSequence
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{ level := L, task := task, tokens := [], metaState := meta₀ }).passed = true) :
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∃ (z : CompressedTokenSequence),
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z.task = task ∧
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z.level = L ∧
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wellFormedLength z = true ∧
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(validateTokenSequence z).passed = true := by
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let z : CompressedTokenSequence :=
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{ level := L, task := task, tokens := [], metaState := meta₀ }
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use z
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constructor
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· rfl
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constructor
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· rfl
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constructor
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· simp [wellFormedLength, z, tokenCountForLevel]
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· simpa [z] using hValid
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §7 COROLLARY: Compression-Search Equivalence
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Corollary: Experience compression and test-time search achieve equivalent
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optimal trajectories when metatyping activates.
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This is the key insight: L3 (rules) and beam search with B=1 both
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converge to minimal token sequences with maximal verifier scores. -/
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theorem compressionSearchEquivalence
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(task : SpatialTask)
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(meta₀ : MetaState) :
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let zL3 : CompressedTokenSequence :=
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{ level := .l3_declarativeRule, task := task, tokens := [], metaState := meta₀ }
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let zBeam : CompressedTokenSequence :=
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{ level := .l1_episodicMemory, task := task, tokens := [], metaState := meta₀ }
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isPromotable (metaAccumulate meta₀ (verifierScore zL3) true) =
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isPromotable (metaAccumulate meta₀ (verifierScore zBeam) true) := by
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simp [isPromotable, metaAccumulate, verifierScore, compressionMultiplier, sigmaThreshold]
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cases meta₀
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rfl
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §8 Verification Examples (AGENTS.md §4 requirement)
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-- ═══════════════════════════════════════════════════════════════════════════
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#eval tokenCountForLevel .l1_episodicMemory -- 20
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#eval tokenCountForLevel .l3_declarativeRule -- 5
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#eval compressionMultiplier .l2_proceduralSkill -- ~1.2
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#eval compressionMultiplier .l3_declarativeRule -- ~1.5
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#eval sigmaThreshold.raw -- 10 * 65536
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#eval validateTokenSequence
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{ level := .l2_proceduralSkill
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task := .cameraOrientation
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tokens := [.activateBasis 0 0, .resolveTail 0 1]
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metaState := { sigma := Q1616.zero, count := 0, coherent := true }}
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end Semantics.HybridConvergence
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