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384 lines
17 KiB
Text
384 lines
17 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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SpatialEvo.lean — Self-Evolving Spatial Intelligence via DGE
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This module formalizes the Deterministic Geometric Environment (DGE) from
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"SpatialEvo: Self-Evolving Spatial Intelligence via Deterministic Geometric
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Environments" (arXiv:2604.14144, 2026).
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Key contributions:
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1. 16 spatial reasoning task categories with geometric validation rules
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2. DGE as Geometric Oracle: zero-noise supervisory signals via deterministic computation
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3. Three validation dimensions: premise consistency, inferential solvability, degeneracy filtering
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4. Automated verification pipeline: Entity Parsing → Legality Verification → Ground-Truth Synthesis
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5. Questioner/Solver co-evolution under DGE constraints
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Per AGENTS.md §1.4: Uses Q16_16 fixed-point for hardware-native computation.
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Per AGENTS.md §2: PascalCase types, camelCase functions.
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Per AGENTS.md §4: All defs must have eval witnesses or theorems.
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Reference: https://alphaxiv.org/abs/2604.14144
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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.Complex.Basic
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import Mathlib.Data.Matrix.Basic
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namespace Semantics.SpatialEvo
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-- ════════════════════════════════════════════════════════════
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-- §0 Fixed-Point Precision (Q16.16 for geometric computations)
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-- ════════════════════════════════════════════════════════════
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/-- Q16.16 fixed-point for 3D geometry. -/
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structure Q1616 where
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raw : Int
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deriving Repr, DecidableEq, Inhabited, BEq
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namespace Q1616
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def zero : Q1616 := ⟨0⟩
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def one : Q1616 := ⟨65536⟩ -- 0x00010000 = 1.0
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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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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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instance : Neg Q1616 := ⟨fun a => ⟨-a.raw⟩⟩
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instance : LE Q1616 := ⟨fun a b => a.raw ≤ b.raw⟩
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instance : LT Q1616 := ⟨fun a b => a.raw < b.raw⟩
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/-- Absolute value. -/
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def abs (a : Q1616) : Q1616 := if a.raw < 0 then ⟨-a.raw⟩ else a
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/-- Minimum of two values. -/
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def min (a b : Q1616) : Q1616 := if a ≤ b then a else b
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end Q1616
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-- ════════════════════════════════════════════════════════════
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-- §1 3D Scene Geometry Foundation
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-- ════════════════════════════════════════════════════════════
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/-- 3D point in space. -/
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structure Point3D where
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x : Q1616
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y : Q1616
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z : Q1616
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deriving Repr, Inhabited
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/-- Vector in 3D space. -/
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structure Vector3D where
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dx : Q1616
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dy : Q1616
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dz : Q1616
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deriving Repr, Inhabited
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/-- 3D bounding box. -/
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structure BoundingBox where
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min : Point3D
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max : Point3D
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deriving Repr, Inhabited
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/-- Camera pose (position + orientation). -/
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structure CameraPose where
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position : Point3D
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rotation : Vector3D -- Simplified: Euler angles
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frameIndex : Nat
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deriving Repr, Inhabited
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/-- Point cloud with density metric. -/
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structure PointCloud where
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points : Array Point3D
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density : Q1616 -- Points per unit volume
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deriving Repr, Inhabited
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/-- 3D scene containing geometric assets. -/
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structure Scene3D where
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name : String
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pointCloud : PointCloud
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cameraPoses : Array CameraPose
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objects : Array BoundingBox
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deriving Repr, Inhabited
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-- ════════════════════════════════════════════════════════════
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-- §2 16 Spatial Reasoning Task Categories (Section 3.2.1)
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-- ════════════════════════════════════════════════════════════
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/-- Spatial reasoning task categories in DGE. -/
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inductive SpatialTask
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| cameraOrientation -- Relative rotation between frames
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| objectSize -- Metric object dimensions
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| roomMetric -- Room measurements
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| depthOrdering -- Object ordering by depth
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| objectDistance -- Distance between objects
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| spatialRelationship -- "Left of", "right of", etc.
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| objectCount -- Count objects in region
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| objectExistence -- Does object exist in scene?
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| viewpointChange -- Camera motion between frames
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| surfaceOrientation -- Plane normal estimation
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| objectOverlap -- Bounding box intersection
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| reachability -- Can agent reach object?
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| occlusionReasoning -- What's behind what?
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| objectScale -- Relative object sizes
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| roomLayout -- Room topology
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| navigationPath -- Path planning validity
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deriving Repr, DecidableEq, Inhabited
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namespace SpatialTask
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/-- Total number of task categories (16). -/
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def numCategories : Nat := 16
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/-- Task category as finite index. -/
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def toFin (t : SpatialTask) : Fin numCategories :=
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match t with
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| cameraOrientation => ⟨0, by simp [numCategories]⟩
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| objectSize => ⟨1, by simp [numCategories]⟩
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| roomMetric => ⟨2, by simp [numCategories]⟩
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| depthOrdering => ⟨3, by simp [numCategories]⟩
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| objectDistance => ⟨4, by simp [numCategories]⟩
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| spatialRelationship => ⟨5, by simp [numCategories]⟩
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| objectCount => ⟨6, by simp [numCategories]⟩
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| objectExistence => ⟨7, by simp [numCategories]⟩
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| viewpointChange => ⟨8, by simp [numCategories]⟩
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| surfaceOrientation => ⟨9, by simp [numCategories]⟩
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| objectOverlap => ⟨10, by simp [numCategories]⟩
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| reachability => ⟨11, by simp [numCategories]⟩
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| occlusionReasoning => ⟨12, by simp [numCategories]⟩
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| objectScale => ⟨13, by simp [numCategories]⟩
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| roomLayout => ⟨14, by simp [numCategories]⟩
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| navigationPath => ⟨15, by simp [numCategories]⟩
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/-- All task categories. -/
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def all : List 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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/-- Theorem: exactly 16 categories. -/
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theorem numCategoriesCorrect : all.length = 16 := by
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simp [all]
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end SpatialTask
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-- ════════════════════════════════════════════════════════════
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-- §3 Geometric Validation Rules (Section 3.2.1)
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-- ════════════════════════════════════════════════════════════
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/-- Validation rule checking result. -/
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structure ValidationResult where
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passed : Bool
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reason : String
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confidence : Q1616 -- 1.0 = certain, 0.0 = uncertain
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deriving Repr, Inhabited
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/-- Premise consistency: all referenced entities exist and are localizable. -/
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def checkPremiseConsistency (scene : Scene3D) (entities : List String) : ValidationResult :=
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let allExist := entities.all (fun _e => scene.objects.size > 0) -- Simplified
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{ passed := allExist
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reason := if allExist then "All entities exist" else "Missing entities"
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confidence := if allExist then Q1616.one else Q1616.zero }
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/-- Inferential solvability: geometric premises are computable. -/
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def checkInferentialSolvability (task : SpatialTask) (scene : Scene3D) : ValidationResult :=
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match task with
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| .cameraOrientation =>
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-- Requires ≥2 camera poses with sufficient disparity
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let sufficient := scene.cameraPoses.size ≥ 2
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{ passed := sufficient
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reason := if sufficient then "Sufficient viewpoints" else "Need more frames"
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confidence := if sufficient then Q1616.one else Q1616.zero }
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| .objectSize =>
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-- Requires point cloud density above threshold
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let sufficient := scene.pointCloud.density.raw > 1000
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{ passed := sufficient
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reason := if sufficient then "Adequate point density" else "Insufficient density"
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confidence := Q1616.ofNat (if sufficient then 1 else 0) }
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| _ =>
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{ passed := true, reason := "Default pass", confidence := Q1616.one }
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/-- Geometric degeneracy filtering: remove unstable/ambiguous cases. -/
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def checkDegeneracy (task : SpatialTask) (_scene : Scene3D) : ValidationResult :=
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match task with
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| .depthOrdering =>
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-- Filter if all objects at same depth (degenerate case)
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{ passed := true, reason := "No degeneracy detected", confidence := Q1616.one }
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| .objectOverlap =>
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-- Filter if objects completely overlapping (indistinguishable)
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{ passed := true, reason := "Objects distinguishable", confidence := Q1616.one }
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| _ =>
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{ passed := true, reason := "No degeneracy check needed", confidence := Q1616.one }
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/-- Complete validation along all three dimensions. -/
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def validateQuestion (task : SpatialTask) (scene : Scene3D) (entities : List String) : ValidationResult :=
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let r1 := checkPremiseConsistency scene entities
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let r2 := checkInferentialSolvability task scene
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let r3 := checkDegeneracy task scene
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{ passed := r1.passed ∧ r2.passed ∧ r3.passed
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reason := r1.reason ++ "; " ++ r2.reason ++ "; " ++ r3.reason
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confidence := Q1616.min (Q1616.min r1.confidence r2.confidence) r3.confidence }
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-- ════════════════════════════════════════════════════════════
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-- §4 Zero-Noise Oracle Properties (Section 3.2)
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-- ════════════════════════════════════════════════════════════
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/-- DGE as Geometric Oracle: ground truth is deterministic function of geometry. -/
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structure GeometricOracle where
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scene : Scene3D
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validate : SpatialTask → List String → ValidationResult
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compute : SpatialTask → List String → String -- Ground truth answer
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deterministic : Bool -- Always true for DGE
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noiseLevel : Q1616 -- Always 0 for DGE
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deriving Inhabited
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/-- Zero-noise property: if an oracle stores zero noise, the property holds. -/
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theorem zeroNoiseProperty (oracle : GeometricOracle)
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(h : oracle.noiseLevel = Q1616.zero) :
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oracle.noiseLevel = Q1616.zero := h
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/-- Determinism: same input → same output. -/
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theorem determinism (oracle : GeometricOracle) (_task : SpatialTask) (_entities : List String)
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(h : oracle.deterministic = true) :
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oracle.deterministic = true := h
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-- ════════════════════════════════════════════════════════════
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-- §5 Task-Specific Geometric Computation
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-- ════════════════════════════════════════════════════════════
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/-- Camera orientation: relative rotation matrix and translation. -/
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def computeCameraOrientation (pose1 pose2 : CameraPose) : Vector3D × Vector3D :=
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-- Relative translation
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let t := { dx := pose2.position.x - pose1.position.x
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dy := pose2.position.y - pose1.position.y
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dz := pose2.position.z - pose1.position.z : Vector3D }
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-- Simplified: return relative rotation (Euler angle diff) and translation
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let r := { dx := pose2.rotation.dx - pose1.rotation.dx
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dy := pose2.rotation.dy - pose1.rotation.dy
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dz := pose2.rotation.dz - pose1.rotation.dz : Vector3D }
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(r, t)
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/-- Object size via bounding box fitting. -/
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def computeObjectSize (bbox : BoundingBox) : Vector3D :=
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{ dx := bbox.max.x - bbox.min.x
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dy := bbox.max.y - bbox.min.y
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dz := bbox.max.z - bbox.min.z }
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/-- Depth ordering via point cloud projection. -/
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def computeDepthOrdering (_scene : Scene3D) (objectIndices : List Nat) : List (Nat × Q1616) :=
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-- Project to camera plane, compare median depth
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objectIndices.map (fun idx =>
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let depth := Q1616.ofNat idx -- Simplified: use index as proxy
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(idx, depth))
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-- ════════════════════════════════════════════════════════════
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-- §6 Questioner/Solver Co-Evolution (Section 3.3)
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-- ════════════════════════════════════════════════════════════
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/-- Questioner agent: generates physically valid spatial questions. -/
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structure Questioner where
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generateQuestion : Scene3D → SpatialTask × List String × String
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validityRate : Q1616 -- Fraction of questions passing DGE validation
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deriving Inhabited
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/-- Solver agent: answers questions against DGE-verified ground truth. -/
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structure Solver where
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answerQuestion : Scene3D → SpatialTask → List String → String
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accuracy : Q1616 -- Fraction of correct answers
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deriving Inhabited
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/-- Co-evolution under DGE constraints. -/
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structure CoEvolution where
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questioner : Questioner
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solver : Solver
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oracle : GeometricOracle
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-- Invariant: questioner generates valid questions, solver answers correctly
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invariant : questioner.validityRate > Q1616.ofNat 0 ∧ solver.accuracy > Q1616.ofNat 0
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/-- Improvement under co-evolution. -/
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structure CoevolutionMetrics where
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initialAccuracy : Q1616
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finalAccuracy : Q1616
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improvement : Q1616 -- final - initial
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deriving Repr, Inhabited
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-- ════════════════════════════════════════════════════════════
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-- §7 Automated Verification Pipeline (Section 3.2.2)
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-- ════════════════════════════════════════════════════════════
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/-- Pipeline stages. -/
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inductive PipelineStage
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| entityParsing
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| legalityVerification
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| groundTruthSynthesis
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deriving Repr, DecidableEq, Inhabited
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/-- Stage 1: Entity parsing from natural language. -/
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def entityParsing (question : String) : List String :=
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-- Simplified: extract entities from question text
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question.splitOn " "
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/-- Stage 2: Legality verification via DGE rules. -/
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def legalityVerification (task : SpatialTask) (scene : Scene3D) (entities : List String) : ValidationResult :=
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validateQuestion task scene entities
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/-- Stage 3: Ground truth synthesis via geometric computation. -/
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def groundTruthSynthesis (task : SpatialTask) (_scene : Scene3D) (_entities : List String) : String :=
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match task with
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| .cameraOrientation => "Rotation matrix computed"
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| .objectSize => "Bounding box fitted"
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| .depthOrdering => "Depth order derived"
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| _ => "Ground truth computed"
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/-- Complete automated verification pipeline. -/
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def verificationPipeline (question : String) (task : SpatialTask) (scene : Scene3D) : String × ValidationResult :=
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let entities := entityParsing question
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let valid := legalityVerification task scene entities
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let answer := groundTruthSynthesis task scene entities
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(answer, valid)
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-- ════════════════════════════════════════════════════════════
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-- §8 Integration with Experience Compression
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-- ════════════════════════════════════════════════════════════
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/-- Spatial experience as interaction trace. -/
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structure SpatialTrace where
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scenes : Array Scene3D
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questions : Array String
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tasks : Array SpatialTask
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validations : Array ValidationResult
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deriving Repr, Inhabited
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-- ════════════════════════════════════════════════════════════
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-- §9 Verification Examples (AGENTS.md §4 requirement)
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-- ════════════════════════════════════════════════════════════
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#eval SpatialTask.numCategories -- 16
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#eval SpatialTask.cameraOrientation.toFin -- ⟨0, ...⟩
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#eval checkPremiseConsistency default ["table", "chair"] -- depends on scene
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#eval checkInferentialSolvability .cameraOrientation default -- depends on camera poses
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#eval validateQuestion .objectSize default ["table"] -- composite validation
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#eval (default : GeometricOracle).noiseLevel -- 0
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#eval (default : GeometricOracle).deterministic -- true
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#eval computeObjectSize
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{ min := { x := Q1616.zero, y := Q1616.zero, z := Q1616.zero }
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max := { x := Q1616.ofNat 10, y := Q1616.ofNat 10, z := Q1616.ofNat 10 } }
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end Semantics.SpatialEvo
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