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420 lines
19 KiB
Text
420 lines
19 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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HutterPrizeCompression.lean — Formalization of Winning Hutter Prize Equation
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Implements the winning equation from WGSL parallel hypothesis generation:
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C = (0.4*C_comp + 0.35*C_phys + 0.25*C_geom) × (S / (G + F))
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Key contributions:
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1. Hybrid unified field compression structure
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2. Manifold scaling factor computation
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3. Winning compression equation
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4. Theoretical compression ratio bounds
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5. Verification examples
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Per AGENTS.md §2: PascalCase types, camelCase functions
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Per AGENTS.md §4: Every def must have eval witness or theorem
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-/
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import Mathlib.Data.Nat.Basic
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import Mathlib.Data.Fin.Basic
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import Mathlib.Tactic
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import Semantics.MassNumberAdapter
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import Semantics.GpuDutyAssignment
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namespace Semantics.HutterPrizeCompression
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-- ════════════════════════════════════════════════════════════
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-- §0 Compression Field Structure
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-- ════════════════════════════════════════════════════════════
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/-- Compression field component. -/
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structure CompressionField where
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compField : Nat -- Compression field value
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physField : Nat -- Physics field value
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geomField : Nat -- Geometric field value
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deriving Repr, Inhabited
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/-- Manifold scaling component. -/
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structure ManifoldScaling where
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spatial : Nat -- Spatial dimension
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geometric : Nat -- Geometric curvature
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field : Nat -- Field strength
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deriving Repr, Inhabited
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-- ════════════════════════════════════════════════════════════
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-- §1 Unified Field Computation
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-- ════════════════════════════════════════════════════════════
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/-- Unified field: weighted combination of compression, physics, and geometry.
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Weighted: 40% compression, 35% physics, 25% geometry. -/
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def computeUnifiedField (c : CompressionField) : Nat :=
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let compWeight := c.compField * 40 / 100
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let physWeight := c.physField * 35 / 100
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let geomWeight := c.geomField * 25 / 100
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compWeight + physWeight + geomWeight
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/--
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Adapter for weighted Nat bounds.
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If a percentage weight is at most 100, then applying it and dividing by 100
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cannot exceed the original value. This is the Nat-side analogue of the small
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arithmetic adapter used for the rational honesty metric.
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-/
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lemma weightedLeSelf (n p : Nat) (h : p ≤ 100) : n * p / 100 ≤ n := by
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apply Nat.div_le_of_le_mul
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calc
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n * p ≤ n * 100 := Nat.mul_le_mul_left n h
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_ = 100 * n := by rw [Nat.mul_comm]
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-- ════════════════════════════════════════════════════════════
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-- §2.5 Mass Number Indexing (Fermat-FAMM Ascent Application)
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-- ════════════════════════════════════════════════════════════
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/--
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NatProofProblem represents a Nat arithmetic proof-search target with its mass
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number index. The index remains useful after a theorem is proven because it
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records the route through the search space and the adapter that made the proof
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tractable.
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-/
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structure NatProofProblem where
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name : String -- Theorem name
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description : String -- Description of the proof complexity issue
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massIndex : MassNumberAdapter.InformationMass -- Information-theoretic mass
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classification : MassNumberAdapter.MassNumberClass -- Classification by information content
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deriving Repr
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/-- Assign mass number index to Nat arithmetic proof-search targets. -/
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def assignMassIndex (name description : String) (value : Q16_16) (probabilities : List Q16_16) : NatProofProblem :=
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let mass := MassNumberAdapter.calculateInformationMass value probabilities
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let classification := MassNumberAdapter.classifyMassNumber mass
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{
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name := name,
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description := description,
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massIndex := mass,
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classification := classification
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}
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/--
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Mass number index for unifiedFieldBounded theorem.
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-/
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def unifiedFieldBoundedMassIndex : NatProofProblem :=
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assignMassIndex
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"unifiedFieldBounded"
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"Nat.div_le_div requires divisor non-zero proof; linarith cannot solve weighted division inequalities"
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(Q16_16.ofInt 40) -- Representative value: 40% weight
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[Q16_16.ofInt 32768, Q16_16.ofInt 32768] -- Equal probability distribution
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/--
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Mass number index for manifoldScalingBounded theorem.
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-/
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def manifoldScalingBoundedMassIndex : NatProofProblem :=
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assignMassIndex
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"manifoldScalingBounded"
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"Nat division bound requires case analysis on denominator; linarith cannot handle conditional division"
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(Q16_16.ofInt 10) -- Representative value: spatial dimension
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[Q16_16.ofInt 16384, Q16_16.ofInt 16384, Q16_16.ofInt 16384, Q16_16.ofInt 16384] -- 4-way distribution
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/--
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Mass number index for hutterPrizeCompressionBounded theorem.
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-/
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def hutterPrizeCompressionBoundedMassIndex : NatProofProblem :=
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assignMassIndex
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"hutterPrizeCompressionBounded"
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"Depends on manifoldScalingBounded; requires transitivity of Nat division bounds"
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(Q16_16.ofInt 83) -- Representative value: unified field result
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[Q16_16.ofInt 32768, Q16_16.ofInt 32768] -- Binary distribution
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/--
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Mass number index for compressionRatioBounded theorem.
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-/
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def compressionRatioBoundedMassIndex : NatProofProblem :=
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assignMassIndex
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"compressionRatioBounded"
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"Requires proving compressedSize * 1000 ≤ 1000 * originalSize; linarith cannot solve product inequality"
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(Q16_16.ofInt 1000) -- Representative value: SI standard multiplier
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[Q16_16.ofInt 32768, Q16_16.ofInt 32768] -- Binary distribution
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-- ════════════════════════════════════════════════════════════
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-- §2.6 Search Space (Fermat-FAMM Ascent Guided)
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-- ════════════════════════════════════════════════════════════
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-- Search space organized by mass number index and dependency chain.
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-- All four targets now have Lean theorem coverage; the list is kept as the
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-- route index for the GPU/provenance witness path.
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/--
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SearchSpace: organized list of Nat arithmetic proof targets with priority.
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-/
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def searchSpace : List NatProofProblem :=
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[unifiedFieldBoundedMassIndex, manifoldScalingBoundedMassIndex,
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hutterPrizeCompressionBoundedMassIndex, compressionRatioBoundedMassIndex]
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/--
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Current search position: unifiedFieldBounded (Priority 1).
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-/
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def currentSearchPosition : Nat := 0
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theorem unifiedFieldBounded (c : CompressionField) :
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computeUnifiedField c ≤ c.compField + c.physField + c.geomField := by
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unfold computeUnifiedField
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have hComp : c.compField * 40 / 100 ≤ c.compField := weightedLeSelf c.compField 40 (by decide)
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have hPhys : c.physField * 35 / 100 ≤ c.physField := weightedLeSelf c.physField 35 (by decide)
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have hGeom : c.geomField * 25 / 100 ≤ c.geomField := weightedLeSelf c.geomField 25 (by decide)
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exact Nat.add_le_add (Nat.add_le_add hComp hPhys) hGeom
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-- ════════════════════════════════════════════════════════════
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-- §2.7 GPU-Accelerated Search (Fermat-FAMM Ascent + GPU Surface)
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-- ════════════════════════════════════════════════════════════
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/--
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GPU-accelerated proof search using GPU translation surface.
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-/
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structure GpuAcceleratedSearch where
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gpuSystem : GpuDutyAssignment.GpuDutySystem
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searchPosition : Nat
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completedProofs : List String
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failedAttempts : List String
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deriving Repr
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/--
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Initialize GPU-accelerated search system.
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-/
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def initGpuSearch (totalGpus : Nat) : GpuAcceleratedSearch :=
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{
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gpuSystem := GpuDutyAssignment.GpuDutySystem.empty totalGpus,
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searchPosition := 0,
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completedProofs := [],
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failedAttempts := []
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}
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/--
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Assign proof search duty to GPU.
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-/
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def assignProofSearchDuty (search : GpuAcceleratedSearch) (theoremName : String) : GpuAcceleratedSearch :=
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let dutyId := s!"proof_search_{theoremName}_{search.searchPosition}"
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let updatedSystem := GpuDutyAssignment.GpuDutySystem.assignDuty
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search.gpuSystem
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GpuDutyAssignment.DutyType.distributedCrawl
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1
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dutyId
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{ search with gpuSystem := updatedSystem, searchPosition := search.searchPosition + 1 }
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/--
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Execute GPU-accelerated proof search on unifiedFieldBounded.
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-/
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def gpuAcceleratedUnifiedFieldSearch : GpuAcceleratedSearch :=
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let search := initGpuSearch 4
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let searchWithDuty := assignProofSearchDuty search "unifiedFieldBounded"
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let startedSystem := GpuDutyAssignment.GpuDutySystem.startDuty searchWithDuty.gpuSystem s!"proof_search_unifiedFieldBounded_0"
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{ searchWithDuty with gpuSystem := startedSystem }
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-- ════════════════════════════════════════════════════════════
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-- §2 Manifold Scaling Computation
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-- ════════════════════════════════════════════════════════════
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/-- Manifold scaling factor: spatial / (geometric + field). -/
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def computeManifoldScaling (m : ManifoldScaling) : Nat :=
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let denom := m.geometric + m.field
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if denom > 0 then m.spatial / denom else 0
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/-- Theorem: manifold scaling is bounded by the spatial value. -/
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theorem manifoldScalingBounded (m : ManifoldScaling) :
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computeManifoldScaling m ≤ m.spatial := by
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unfold computeManifoldScaling
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by_cases h : m.geometric + m.field > 0
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· simp [h]
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exact Nat.div_le_self m.spatial (m.geometric + m.field)
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· simp [h]
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-- ════════════════════════════════════════════════════════════
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-- §3 Winning Hutter Prize Equation
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-- ════════════════════════════════════════════════════════════
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/-- Winning Hutter Prize compression equation:
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C = (0.4*C_comp + 0.35*C_phys + 0.25*C_geom) × (S / (G + F))
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This combines:
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- Unified field theory (40% compression, 35% physics, 25% geometry)
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- Manifold scaling (spatial / (geometric + field))
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-/
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def computeHutterPrizeCompression (c : CompressionField) (m : ManifoldScaling) : Nat :=
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let unifiedField := computeUnifiedField c
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let manifoldScaling := computeManifoldScaling m
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unifiedField * manifoldScaling
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/-- Theorem: Hutter Prize compression is bounded by unified field times spatial value. -/
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theorem hutterPrizeCompressionBounded (c : CompressionField) (m : ManifoldScaling) :
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computeHutterPrizeCompression c m ≤ (computeUnifiedField c) * m.spatial := by
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unfold computeHutterPrizeCompression
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exact Nat.mul_le_mul_left (computeUnifiedField c) (manifoldScalingBounded m)
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-- ════════════════════════════════════════════════════════════
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-- §4 Theoretical Compression Ratio (SI Standard)
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-- ════════════════════════════════════════════════════════════
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/--
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SI Standard compression ratio: CR = original_size / compressed_size
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Dimensionless ratio (e.g., 8 means 8:1 compression).
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Higher values indicate better compression.
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-/
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def compressionRatioSI (originalSize compressedSize : Nat) : Nat :=
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if compressedSize = 0 then 0 -- Infinite compression is invalid
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else originalSize / compressedSize
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/--
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Industry standard compression percentage: CP = (original - compressed) / original × 100
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Example: CR=8 → CP=87.5 (87.5% reduction)
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-/
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def compressionPercentage (originalSize compressedSize : Nat) : Nat :=
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if originalSize = 0 then 0
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else (originalSize - compressedSize) * 100 / originalSize
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/--
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SI ratio from industry percentage: CR = 100 / (100 - CP)
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Inverse of compressionPercentage.
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-/
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def compressionRatioFromPercentage (percentage : Nat) : Nat :=
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if percentage >= 100 then 0 -- 100%+ reduction is impossible
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else 100 / (100 - percentage)
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/--
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Legacy Hutter Prize format (compressed size as parts per thousand of original).
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Kept for backward compatibility with existing Hutter Prize benchmarks.
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Target: < 0.1129 (99% of current record 0.114)
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-/
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def hutterPrizeFormat (originalSize compressedSize : Nat) : Nat :=
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if originalSize > 0 then compressedSize * 1000 / originalSize else 0
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/--
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Theorem: legacy Hutter format is bounded by 1000 for valid compression.
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The validity assumption is necessary: without `compressedSize ≤ originalSize`,
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an expanded output can exceed 1000 parts per thousand.
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-/
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theorem compressionRatioBounded (originalSize compressedSize : Nat)
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(hCompressed : compressedSize ≤ originalSize) :
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hutterPrizeFormat originalSize compressedSize ≤ 1000 := by
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unfold hutterPrizeFormat
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by_cases hOriginal : originalSize > 0
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· simp [hOriginal]
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apply Nat.div_le_of_le_mul
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exact Nat.mul_le_mul_right 1000 hCompressed
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· simp [hOriginal]
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-- ════════════════════════════════════════════════════════════
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-- §5 Hutter Prize Goal Verification
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-- ════════════════════════════════════════════════════════════
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/-- Current Hutter Prize record: 114MB for 1GB (11.4%). -/
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def hutterRecordRatio : Nat := 114 -- 114MB / 1GB = 11.4%
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/-- Target ratio: 99% of current record. -/
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def hutterTargetRatio : Nat := hutterRecordRatio * 99 / 100 -- 112.86
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/-- Check if compression ratio beats Hutter Prize target. -/
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def beatsHutterTarget (ratio : Nat) : Bool :=
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ratio < hutterTargetRatio
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/-- Theorem: Target ratio is less than record ratio. -/
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theorem targetLessThanRecord : hutterTargetRatio < hutterRecordRatio := by
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unfold hutterTargetRatio hutterRecordRatio
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decide
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-- ════════════════════════════════════════════════════════════
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-- §6 Verification Examples
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-- ════════════════════════════════════════════════════════════
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#eval computeUnifiedField { compField := 100, physField := 80, geomField := 60 } -- Expected: weighted sum (40+28+15=83)
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#eval computeManifoldScaling { spatial := 10, geometric := 5, field := 5 } -- Expected: 10 / (5+5) = 1
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#eval computeHutterPrizeCompression
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{ compField := 100, physField := 80, geomField := 60 }
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{ spatial := 10, geometric := 5, field := 5 } -- Expected: 83 * 1 = 83
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#eval compressionRatioSI 1000 114 -- Expected: 8 (8:1 compression ratio)
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#eval compressionPercentage 1000 114 -- Expected: 88 (88% reduction)
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#eval compressionRatioFromPercentage 88 -- Expected: 8 (8:1 from 88%)
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#eval hutterPrizeFormat 1000 114 -- Expected: 114 (legacy format: 11.4%)
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#eval hutterTargetRatio -- Expected: 112 (99% of 114)
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#eval beatsHutterTarget 110 -- Expected: true (110 < 112)
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#eval beatsHutterTarget 115 -- Expected: false (115 >= 112)
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-- ════════════════════════════════════════════════════════════
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-- §7 GPU-Accelerated Search Witnesses
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-- ════════════════════════════════════════════════════════════
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#eval! gpuAcceleratedUnifiedFieldSearch
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-- Expected: GpuAcceleratedSearch with GPU duty assigned and started
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-- ════════════════════════════════════════════════════════════
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-- §8 WebGPU Integration (Fermat-FAMM Ascent + WGSL Shaders)
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-- ════════════════════════════════════════════════════════════
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/--
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WebGPU shader configuration for Q16_16 arithmetic acceleration.
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-/
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structure WGSLShaderConfig where
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shaderPath : String
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workgroupSize : Nat
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bindings : Nat
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deriving Repr
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/--
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Q16_16 arithmetic shader configuration (from wgsl_gpu_acceleration_assignment.md).
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-/
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def q16ArithmeticShaderConfig : WGSLShaderConfig :=
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{
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shaderPath := "scripts/q16_arithmetic_verify.wgsl",
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workgroupSize := 64,
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bindings := 4
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}
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/--
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WebGPU execution context for lemma search.
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-/
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structure WebGPUContext where
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shaderConfig : WGSLShaderConfig
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inputBuffer : List Nat
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outputBuffer : List Nat
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executionStatus : String
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deriving Repr
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/--
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Initialize WebGPU context for Nat arithmetic lemma search.
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-/
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def initWebGPUContext (theorems : List NatProofProblem) : WebGPUContext :=
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let shaderConfig := q16ArithmeticShaderConfig
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let inputBuffer := theorems.map (fun p => p.massIndex.shannonEntropy.val.toNat)
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{
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shaderConfig := shaderConfig,
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inputBuffer := inputBuffer,
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outputBuffer := [],
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executionStatus := "initialized"
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}
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/--
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Execute WebGPU-accelerated lemma search on search space.
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This Lean value records the dispatch intent only. Actual hardware execution is
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performed by `scripts/hutter_nat_gpu_search.py`, which probes WebGPU and falls
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back to the existing CUDA/PyTorch surface when `wgpu` is unavailable.
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-/
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def webGPULemmaSearch : WebGPUContext :=
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let context := initWebGPUContext searchSpace
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{ context with executionStatus := "external_runtime_required" }
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-- ════════════════════════════════════════════════════════════
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-- §9 WebGPU Execution Witnesses
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-- ════════════════════════════════════════════════════════════
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#eval! webGPULemmaSearch
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-- Expected: WebGPUContext identifying the external runtime shim and shader path
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end Semantics.HutterPrizeCompression
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