OpenAI Astra Math Reasoning Deep Dive: 3-Stage Pipeline and Lean 4 Formal Proofs

"""
Simulating Astra's multi-agent mathematical reasoning engine
Core idea: Search agent explores proof space, verification agent
checks intermediate steps, backtrack agent handles contradictions,
meta agent decides when to switch strategies
"""

import random
from typing import Any, Callable
from dataclasses import dataclass, field

# ─── Type Definitions ─────────────────────────────────────────

@dataclass
class ProofState:
    """Proof state: known facts, goal, attempted paths"""
    known_facts: set[str] = field(default_factory=set)
    goal: str = ""
    attempted_strategies: list[str] = field(default_factory=list)
    active_branches: list[list[str]] = field(default_factory=list)
    confidence: float = 0.0

class SearchAgent:
    """Search Agent: explores candidate paths in proof space"""
    def __init__(self, strategy_pool: dict[str, Callable]):
        self.strategies = strategy_pool
        self.strategy_history: list[str] = []
    
    def propose_next_step(self, state: ProofState) -> tuple[str, str]:
        """Propose next strategy based on current state"""
        available = [s for s in self.strategies 
                     if s not in state.attempted_strategies]
        if not available:
            available = list(self.strategies.keys())
        
        # Probability-weighted selection (simulating Astra's strategy evaluation)
        strategy = random.choices(
            available, 
            weights=[1.0 + len(state.attempted_strategies) * 0.1 
                     for _ in available]
        )[0]
        
        args = self.strategies[strategy](state)
        self.strategy_history.append(strategy)
        return strategy, args

class VerificationAgent:
    """Verification Agent: checks logical consistency of reasoning steps"""
    def __init__(self, axiom_system: set[str]):
        self.axioms = axiom_system
        self.logical_errors: list[str] = []
    
    def verify_step(self, state: ProofState, deduced: str) -> bool:
        """Verify that the deduced step follows from known facts"""
        if deduced in state.known_facts:
            self.logical_errors.append(f"Circular reasoning: {deduced}")
            return False
        
        conflict = any(
            self._contradicts(deduced, fact) 
            for fact in state.known_facts
        )
        if conflict:
            self.logical_errors.append(f"Contradiction: {deduced}")
            return False
        
        return True
    
    def _contradicts(self, a: str, b: str) -> bool:
        """Check if two propositions contradict each other"""
        return f"not {a}" == b or f"not {b}" == a

class BacktrackAgent:
    """Backtrack Agent: rolls back on contradiction and switches paths"""
    def __init__(self, max_retries: int = 5):
        self.retry_count = 0
        self.max_retries = max_retries
        self.failure_patterns: list[dict] = []
    
    def analyze_failure(self, state: ProofState, 
                        error: str) -> str:
        """Analyze failure reason, return suggested new strategy"""
        self.retry_count += 1
        self.failure_patterns.append({
            "state": state,
            "error": error,
            "retry": self.retry_count
        })
        
        if "global" in error.lower():
            return "switch to local method"
        elif "contradiction" in error.lower():
            return "switch to dual argument"
        else:
            return "switch to deterministic construction"

class MetaAgent:
    """Meta Agent: coordinates agents, decides overall strategy"""
    def __init__(self, agents: dict[str, Any]):
        self.agents = agents
        self.performance_log: list[dict] = []
    
    def evaluate_progress(self, state: ProofState) -> str:
        """Evaluate overall progress, decide next action"""
        if len(state.known_facts) == 0:
            return "initiate_search"
        
        error_rate = len(
            self.agents["verifier"].logical_errors
        ) / max(1, len(state.attempted_strategies))
        
        if error_rate > 0.6:
            return "switch_framework"
        elif state.confidence > 0.8:
            return "attempt_conclusion"
        else:
            return "continue_search"

# ─── Complete Reasoning Engine ───────────────────────────────

class AstraReasoningEngine:
    """Core implementation of Astra's reasoning engine"""
    def __init__(self):
        self.state = ProofState()
        
        strategies = {
            "randomized_grid": self._random_grid,
            "deterministic_median": self._deterministic_median,
            "local_mass_exclusion": self._local_mass_exclusion,
            "harmonic_measure": self._harmonic_measure,
            "dual_lp": self._dual_lp,
            "fourier_transform": self._fourier_transform,
        }
        
        self.agents = {
            "searcher": SearchAgent(strategies),
            "verifier": VerificationAgent(
                {"group_axioms", "topology_axioms", "metric_axioms"}
            ),
            "backtracker": BacktrackAgent(),
            "meta": MetaAgent({})
        }
        self.agents["meta"].agents = self.agents
    
    def _random_grid(self, state: ProofState) -> str:
        state.attempted_strategies.append("randomized_grid")
        return "attempt permutation approximation on random grid"
    
    def _deterministic_median(self, state: ProofState) -> str:
        state.attempted_strategies.append("deterministic_median")
        return "construct deterministic approximation via median operator"
    
    def _local_mass_exclusion(self, state: ProofState) -> str:
        state.attempted_strategies.append("local_mass_exclusion")
        return "construct local mass exclusion inequality"
    
    def _harmonic_measure(self, state: ProofState) -> str:
        state.attempted_strategies.append("harmonic_measure")
        return "apply harmonic measure and maximum modulus principle"
    
    def _dual_lp(self, state: ProofState) -> str:
        state.attempted_strategies.append("dual_lp")
        return "solve dual linear program for upper bound"
    
    def _fourier_transform(self, state: ProofState) -> str:
        state.attempted_strategies.append("fourier_transform")
        return "apply radial Fourier transform with Mellin reflection"
    
    def solve(self, problem: str, 
              max_iterations: int = 100) -> ProofState:
        """Main solving loop"""
        self.state.goal = problem
        
        for iteration in range(max_iterations):
            action = self.agents["meta"].evaluate_progress(self.state)
            
            if action == "attempt_conclusion":
                print(f"✓ Round {iteration}: Conclusion reached")
                break
            elif action == "switch_framework":
                print(f"↻ Round {iteration}: Switching framework")
                self.state.attempted_strategies.clear()
            
            strategy, args = self.agents["searcher"].propose_next_step(
                self.state
            )
            print(f"→ Trying strategy: {strategy} -> {args}")
            
            deduced = f"from_{strategy}: partial_result_for_{problem}"
            
            if self.agents["verifier"].verify_step(self.state, deduced):
                self.state.known_facts.add(deduced)
                self.state.confidence = min(
                    1.0, self.state.confidence + 0.15
                )
                print(f"  ✓ Verified (confidence: {self.state.confidence:.2f})")
            else:
                error = self.agents["verifier"].logical_errors[-1]
                print(f"  ✗ Failed: {error}")
                suggestion = self.agents["backtracker"].analyze_failure(
                    self.state, error
                )
                print(f"  ↻ Backtrack suggestion: {suggestion}")
        
        return self.state

# ─── Run Example ─────────────────────────────────────────────

if __name__ == "__main__":
    engine = AstraReasoningEngine()
    result = engine.solve("prove_existence_of_non_sofic_group")
    
    print(f"\n{'='*50}")
    print(f"Final State:")
    print(f"  Known facts: {len(result.known_facts)}")
    print(f"  Strategies attempted: {len(result.attempted_strategies)}")
    print(f"  Confidence: {result.confidence:.2f}")
    print(f"  Verification errors: {len(engine.agents['verifier'].logical_errors)}")

-- Example of proof and type relationship in Lean 4

-- Proposition: There exists a non-sofic group
theorem exists_non_sofic_group : ∃ (G : Type) [Group G], ¬ Sofic G :=
by
  -- Constructive proof: explicitly construct a non-sofic group
  let G := LeavittUnitGroup 2  -- Unit group of binary Leavitt algebra
  have h_non_sofic : ¬ Sofic G := non_sofic_proof G
  exact ⟨G, inferInstance, h_non_sofic⟩

-- Here "∃ (G : Type) [Group G], ¬ Sofic G" is a type
-- Its proof is constructing a term satisfying that type

# Python Type System → Lean 4 Dependent Type System Analogy

from typing import TypeVar, Generic, Protocol, runtime_checkable
from dataclasses import dataclass

# ─── Type-Level Programming Basics ──────────────────────────

# Lean: def identity {α : Type} (x : α) : α := x
T = TypeVar('T')
def identity(x: T) -> T:
    return x

# Lean: inductive Nat where | zero : Nat | succ : Nat → Nat
@dataclass
class Nat:
    """Peano axiom encoding of natural numbers"""
    pass

@dataclass
class Zero(Nat):
    """0"""
    pass

@dataclass
class Succ(Nat):
    """Successor function"""
    pred: Nat

# ─── Propositions as Types ──────────────────────────────────

@runtime_checkable
class Proposition(Protocol):
    """Proposition type"""
    def __call__(self) -> bool:
        ...

# Logical connective type signatures
class And(Generic[T, U]):
    """Conjunction: A ∧ B"""
    def __init__(self, left: T, right: U):
        self.left = left
        self.right = right

class Or(Generic[T, U]):
    """Disjunction: A ∨ B"""
    pass

class Left(Or[T, U]):
    """Left branch proof of A ∨ B"""
    def __init__(self, value: T):
        self.value = value

class Right(Or[T, U]):
    """Right branch proof of A ∨ B"""
    def __init__(self, value: U):
        self.value = value

class Implies(Generic[T, U]):
    """Implication: A → B (i.e., function type)"""
    def __init__(self, func):
        self.func = func
    
    def __call__(self, premise: T) -> U:
        return self.func(premise)

# ─── Dependent Type Example ─────────────────────────────────

# Lean: def Vector (α : Type) (n : Nat) : Type
# Vector type depends on its length
@dataclass
class Vector(Generic[T]):
    """Dependent type: length encoded in the type"""
    elements: tuple[T, ...]
    length: int
    
    def __post_init__(self):
        assert len(self.elements) == self.length

# Type-safe vector addition
def vector_add(
    v1: Vector[float], 
    v2: Vector[float]
) -> Vector[float] | None:
    """Can only add vectors of the same length"""
    if v1.length != v2.length:
        return None  # Type error!
    return Vector(
        tuple(a + b for a, b in zip(v1.elements, v2.elements)),
        v1.length
    )

# Install Lean 4 toolchain
curl -sSf https://raw.githubusercontent.com/leanprover/elan/master/elan-init.sh | sh

# Fetch Mathlib cache
lake exe cache get

# Build all proofs
lake build All

# Or build individual proofs
lake build NonSoficGroup

"""
High-dimensional sphere packing density upper bound computation
Astra determines the ceiling of the Cohn-Elkies LP method
"""

import numpy as np
from scipy import special

class SpherePackingBound:
    """Sphere packing density upper bound analyzer"""
    
    def __init__(self, dimension: int):
        self.d = dimension
        self._cached_kl_bound = None
        self._cached_ce_bound = None
    
    def kabatiansky_levenshtein_bound(self) -> float:
        """Compute KL bound (classic 1978 result)"""
        if self._cached_kl_bound is not None:
            return self._cached_kl_bound
        
        theta = np.arccos(1 / (2 * np.sqrt(2)))
        alpha = -np.log2(np.sin(theta))
        self._cached_kl_bound = alpha
        return alpha
    
    def cohn_elkies_lp_threshold(self) -> float:
        """Compute Cohn-Elkies LP ceiling exponent"""
        if self._cached_ce_bound is not None:
            return self._cached_ce_bound
        
        # Astra's asymptotic value: sqrt(e / (2*pi))
        threshold = np.sqrt(np.e / (2 * np.pi))
        alpha = -np.log2(threshold)
        self._cached_ce_bound = alpha
        return alpha
    
    def density_upper_bound(self, n: int) -> float:
        """Compute density upper bound for n-dimensional space"""
        alpha = self.cohn_elkies_lp_threshold()
        return 2 ** (-alpha * n)
    
    def compare_bounds(self, dimensions: list[int]) -> dict:
        """Compare KL bound with Astra's new bound"""
        results = {}
        for d in dimensions:
            self.d = d
            kl = self.kabatiansky_levenshtein_bound()
            ce = self.cohn_elkies_lp_threshold()
            improvement = ((ce - kl) / kl) * 100
            results[d] = {
                "kl_bound_exponent": round(kl, 6),
                "ce_bound_exponent": round(ce, 6),
                "improvement_pct": round(improvement, 4),
            }
        return results
    
    def fourier_analytic_approach(self, r: np.ndarray) -> np.ndarray:
        """Core of the Cohn-Elkies Fourier analytic method"""
        nu = self.d / 2 - 1  # Bessel function order
        
        def admissible_function(r: float) -> float:
            """Construct admissible function satisfying Cohn-Elkies conditions"""
            if r < 1e-10:
                return 1.0
            return (special.jv(nu, 2 * np.pi * r) / 
                    (2 * np.pi * r) ** nu) * np.exp(-r**2)
        
        return np.array([admissible_function(ri) for ri in r])

# Run analysis
if __name__ == "__main__":
    sp = SpherePackingBound(1000)
    
    print("Sphere Packing Density Upper Bound Improvement Analysis")
    print("=" * 60)
    print(f"KL bound exponent (1978): {sp.kabatiansky_levenshtein_bound():.6f}")
    print(f"CE LP ceiling exponent (Astra 2026): {sp.cohn_elkies_lp_threshold():.6f}")
    print(f"Improvement: {((sp.cohn_elkies_lp_threshold() - sp.kabatiansky_levenshtein_bound()) / sp.kabatiansky_levenshtein_bound() * 100):.4f}%")
    
    for d in [100, 500, 1000, 5000]:
        print(f"\nDimension d={d}:")
        print(f"  Density upper bound ≤ 2^({-sp.cohn_elkies_lp_threshold():.6f} × {d})")

// Go implementation: Core algebraic structure of non-sofic group construction
// Simulating Astra's use of the binary Leavitt algebra unit group

package main

import (
    "fmt"
    "strings"
)

// ─── Core Algebraic Structures ────────────────────────────

// LeavittAlgebra represents the binary Leavitt algebra L(2)
// Generators: e₁, e₂, f₁, f₂ satisfying:
// f_i * e_j = δ_{ij} * 1
// e₁*f₁ + e₂*f₂ = 1
type LeavittAlgebra struct {
    Generators [4]string
    Relations  []Relation
}

type Relation struct {
    LHS, RHS string
}

func NewLeavittAlgebra() *LeavittAlgebra {
    return &LeavittAlgebra{
        Generators: [4]string{"e₁", "e₂", "f₁", "f₂"},
        Relations: []Relation{
            {LHS: "f₁·e₁", RHS: "1"},
            {LHS: "f₂·e₂", RHS: "1"},
            {LHS: "f₁·e₂", RHS: "0"},
            {LHS: "f₂·e₁", RHS: "0"},
            {LHS: "e₁·f₁ + e₂·f₂", RHS: "1"},
        },
    }
}

// UnitGroup represents the unit group of the Leavitt algebra
type UnitGroup struct {
    algebra  *LeavittAlgebra
    elements map[string]*Permutation
}

// Permutation represents a finite permutation
type Permutation struct {
    n     int
    image []int
}

func NewPermutation(n int, image []int) *Permutation {
    p := &Permutation{
        n:     n,
        image: make([]int, n),
    }
    copy(p.image, image)
    return p
}

func (p *Permutation) Compose(q *Permutation) *Permutation {
    if p.n != q.n {
        panic("permutation size mismatch")
    }
    result := make([]int, p.n)
    for i := 0; i < p.n; i++ {
        result[i] = p.image[q.image[i]]
    }
    return NewPermutation(p.n, result)
}

// ThompsonV represents Thompson's group V
type ThompsonV struct {
    generators []BinaryTreeAutomorphism
}

type BinaryTreeAutomorphism struct {
    domain, codomain []string
}

func NewThompsonV() *ThompsonV {
    return &ThompsonV{
        generators: []BinaryTreeAutomorphism{
            {
                domain:  []string{"00", "01", "1"},
                codomain: []string{"01", "00", "1"},
            },
            {
                domain:  []string{"00", "010", "011", "1"},
                codomain: []string{"010", "00", "011", "1"},
            },
        },
    }
}

// NonSoficGroup constructed via amalgamated product
type NonSoficGroup struct {
    leavittUnitGroup    *UnitGroup
    thompsonV           *ThompsonV
    amalgamatedSubgroup string
}

func ConstructNonSoficGroup() *NonSoficGroup {
    algebra := NewLeavittAlgebra()
    unitGroup := &UnitGroup{
        algebra:  algebra,
        elements: make(map[string]*Permutation),
    }
    unitGroup.elements["u₁"] = NewPermutation(4, []int{1, 0, 2, 3})
    unitGroup.elements["u₂"] = NewPermutation(4, []int{0, 1, 3, 2})
    
    thompsonV := NewThompsonV()
    
    return &NonSoficGroup{
        leavittUnitGroup:    unitGroup,
        thompsonV:           thompsonV,
        amalgamatedSubgroup: "⟨u₁², u₂²⟩",
    }
}

func (nsg *NonSoficGroup) VerifyNonSofic() bool {
    fmt.Println("Verifying non-sofic property...")
    fmt.Println("Step 1: Check group is finitely presented")
    fmt.Println("  → Generators: e₁, e₂, f₁, f₂, A, B, C")
    fmt.Println("  → Relations: finite")
    
    fmt.Println("\nStep 2: Construct Kun-Thom expander graph")
    fmt.Println("  → Using group action orbit structure on binary tree")
    
    fmt.Println("\nStep 3: Prove no finite permutation approximation exists")
    fmt.Println("  → Via median argument (Astra's core innovation)")
    fmt.Println("  → Eliminating randomized grid path (Astra's self-rejected route)")
    
    fmt.Println("\nStep 4: Derive contradiction")
    fmt.Println("  → If sofic approximation existed, it would contradict")
    fmt.Println("    the expander graph property")
    fmt.Println("  → Therefore the group is not sofic")
    
    return true
}

func main() {
    fmt.Println("=" + strings.Repeat("=", 59))
    fmt.Println("  Astra Non-Sofic Group Construction -- Go Implementation")
    fmt.Println("=" + strings.Repeat("=", 59))
    
    fmt.Println("\n1. Construct binary Leavitt algebra L(2):")
    algebra := NewLeavittAlgebra()
    fmt.Printf("   Generators: %v\n", algebra.Generators)
    fmt.Printf("   Relations: %d\n", len(algebra.Relations))
    for _, r := range algebra.Relations {
        fmt.Printf("   %s = %s\n", r.LHS, r.RHS)
    }
    
    fmt.Println("\n2. Construct Leavitt unit group U(L(2)):")
    fmt.Printf("   u₁ = (0 1)\n")
    
    fmt.Println("\n3. Construct Thompson's group V:")
    thompsonV := NewThompsonV()
    fmt.Printf("   Generators: %d\n", len(thompsonV.generators))
    
    fmt.Println("\n4. Amalgamated product construction:")
    nsg := ConstructNonSoficGroup()
    result := nsg.VerifyNonSofic()
    
    fmt.Printf("\nFinal result: Non-sofic group exists = %v\n", result)
    fmt.Println("This resolves a central open question since Gromov introduced soficity in 1999.")
}

// Go implementation: Multi-agent reasoning cost and efficiency monitoring

package main

import (
    "fmt"
    "math/rand"
    "strings"
    "sync"
    "time"
)

type TokenCost struct {
    InputTokens  int64
    OutputTokens int64
    TotalCost    float64
}

type AgentMetrics struct {
    AgentName        string
    TokensConsumed   TokenCost
    StrategyAttempts int
    SuccessRate      float64
    WallClockTime    time.Duration
}

type ReasoningOrchestrator struct {
    mu        sync.Mutex
    agents    map[string]*AgentMetrics
    totalCost float64
    startTime time.Time
}

func NewReasoningOrchestrator() *ReasoningOrchestrator {
    return &ReasoningOrchestrator{
        agents:    make(map[string]*AgentMetrics),
        startTime: time.Now(),
    }
}

func (ro *ReasoningOrchestrator) RegisterAgent(name string) {
    ro.mu.Lock()
    defer ro.mu.Unlock()
    ro.agents[name] = &AgentMetrics{AgentName: name}
}

func (ro *ReasoningOrchestrator) RecordAgentAttempt(
    name string, tokens int64, success bool,
) {
    ro.mu.Lock()
    defer ro.mu.Unlock()
    
    agent, ok := ro.agents[name]
    if !ok {
        return
    }
    
    costPerToken := 30.0 / 1_000_000
    cost := float64(tokens) * costPerToken
    
    agent.TokensConsumed.TotalCost += cost
    agent.TokensConsumed.InputTokens += tokens
    agent.StrategyAttempts++
    agent.SuccessRate = (agent.SuccessRate*float64(agent.StrategyAttempts-1) +
        map[bool]float64{true: 1, false: 0}[success]) /
        float64(agent.StrategyAttempts)
    agent.WallClockTime = time.Since(ro.startTime)
    ro.totalCost += cost
}

func (ro *ReasoningOrchestrator) SimulateReasoning() {
    problems := []string{
        "Sphere Packing", "Binary Codes", "Non-Sofic Groups",
        "Connes Rigidity", "Circuit Lower Bounds", "Quantum Parallel Repetition",
        "Closest Vector Problem", "Ehrhart Volume", "Multicolor Ramsey", "Extremal Numbers",
    }
    
    agents := []string{"SearchAgent", "VerifyAgent", "BacktrackAgent", "MetaAgent"}
    for _, a := range agents {
        ro.RegisterAgent(a)
    }
    
    totalTokens := int64(0)
    
    for i, problem := range problems {
        fmt.Printf("\n▶ Problem %d/%d: %s\n", i+1, len(problems), problem)
        searchTokens := int64(0)
        
        for round := 0; round < 5+rand.Intn(10); round++ {
            tokens := int64(1000 + rand.Intn(9000))
            searchTokens += tokens
            success := rand.Float64() > 0.3
            
            ro.RecordAgentAttempt("SearchAgent", tokens, success)
            
            if success {
                verifyTokens := int64(500 + rand.Intn(2000))
                ro.RecordAgentAttempt("VerifyAgent", verifyTokens, true)
                fmt.Printf("   Round %d: success ✓ (tokens: %d)\n",
                    round, tokens+verifyTokens)
                break
            } else {
                backtrackTokens := int64(300 + rand.Intn(1000))
                ro.RecordAgentAttempt("BacktrackAgent", backtrackTokens, false)
                fmt.Printf("   Round %d: failed ✗, backtracking...\n", round)
            }
        }
        totalTokens += searchTokens
    }
    
    fmt.Printf("\n%s\n", strings.Repeat("=", 60))
    fmt.Printf("Reasoning Statistics\n")
    fmt.Printf("%s\n", strings.Repeat("=", 60))
    fmt.Printf("Total tokens: %d\n", totalTokens)
    fmt.Printf("Total cost (est.): $%.2f\n", ro.totalCost)
    fmt.Printf("Avg cost per problem: $%.2f\n", ro.totalCost/float64(len(problems)))
    
    for _, agent := range ro.agents {
        fmt.Printf("\nAgent: %s\n", agent.AgentName)
        fmt.Printf("  Attempts: %d\n", agent.StrategyAttempts)
        fmt.Printf("  Success rate: %.1f%%\n", agent.SuccessRate*100)
        fmt.Printf("  Tokens: %d\n", agent.TokensConsumed.InputTokens)
        fmt.Printf("  Cost: $%.2f\n", agent.TokensConsumed.TotalCost)
    }
}

func main() {
    orch := NewReasoningOrchestrator()
    orch.SimulateReasoning()
}

"""
Lean 4 formal proof auto-generation framework
Simulating Astra's conversion of informal arguments into Lean 4 code
"""

class Lean4Generator:
    """Convert structured mathematical arguments to Lean 4 code"""
    
    def __init__(self):
        self.imports = set()
        self.theorems = []
        self.definitions = []
    
    def generate_non_sofic_proof(self) -> str:
        """Generate Lean 4 proof framework for non-sofic group construction"""
        
        return """
import Mathlib.GroupTheory.Sofic
import Mathlib.Algebra.Algebra
import Mathlib.Algebra.Group.Units

open scoped BigOperators

/-!
# Existence of Non-Sofic Groups
Astra's construction: Amalgamated product of
the Leavitt algebra unit group and Thompson's group V
-/

/-! 
## Part 1: Binary Leavitt Algebra L(2)
-/
def LeavittAlgebra : Type := 
  Algebra (FreeAlgebra ℤ {e₁, e₂, f₁, f₂}) / 
    ⟨f₁*e₁ - 1, f₂*e₂ - 1, f₁*e₂, f₂*e₁, e₁*f₁ + e₂*f₂ - 1⟩

/-!
## Part 2: Leavitt Unit Group U(L(2))
-/
def LeavittUnitGroup : Type := 
  Units (LeavittAlgebra)

instance : Group LeavittUnitGroup := 
  Units.group

/-!
## Part 3: Thompson's Group V
-/
def ThompsonV : Type := 
  FpGroup ⟨A, B, C | A^2, B^2, C^2, (AB)^3, (AC)^3, (BC)^3⟩

/-!
## Part 4: Amalgamated Product Construction
-/
def NonSoficGroup : Type := 
  AmalgamatedProduct LeavittUnitGroup ThompsonV 
    (by
      -- Construct embedding of common subgroup
      -- Astra's reasoning: via Kun-Thom expander graph theory
      sorry)

/-!
## Part 5: Main Theorem
-/
theorem exists_non_sofic_group : ∃ (G : Type) [Group G], ¬ Sofic G := by
  refine ⟨NonSoficGroup, inferInstance, ?_⟩
  -- Non-sofic proof using median argument (deterministic method)
  sorry
"""
    
    def generate_sphere_packing_proof(self) -> str:
        """Generate Lean 4 proof framework for sphere packing bounds"""
        
        return """
import Mathlib.Analysis.Fourier
import Mathlib.MeasureTheory.Integral
import Mathlib.Analysis.SpecialFunctions.Bessel

open Real

/-!
# High-Dimensional Sphere Packing Density Upper Bound
Astra determines the ceiling of the Cohn-Elkies LP method
-/

noncomputable def radialFourierTransform 
  {d : ℕ} (f : ℝ → ℝ) (t : ℝ) : ℝ :=
  2 * π * ∫ (r : ℝ) in Set.Ioi 0, 
    f r * BesselJ ((d : ℝ)/2 - 1) (2*π*r*t) * r^(d-1) ∂ volume

structure AdmissibleFunction where
  f : ℝ → ℝ
  hf_positive : ∀ r, f r ≥ 0
  hf_fourier_positive : ∀ t ≥ 1, radialFourierTransform f t ≥ 0
  hf_normalized : f 0 = 1

theorem sphere_packing_density_upper_bound (d : ℕ) (h : AdmissibleFunction) : 
  PackingDensity d ≤ π^(d/2) / (2^d * Γ(d/2 + 1)) * h.f 0 := by
  sorry

theorem cohn_elkies_asymptotic_ceiling : 
  Filter.Tendsto (λ (d : ℕ) => 
    (LP_density_bound d) ^ (1/(d : ℝ)))
    Filter.atTop (𝓝 (Real.sqrt (Real.exp 1 / (2 * π)))) := by
  sorry
"""
    
    def compile_check(self, lean_code: str) -> bool:
        """Simulate Lean 4 compilation check"""
        print("Running lake build All ...")
        print("  ✓ Type checking passed")
        print("  ✓ Dependency resolution complete")
        print("  ✓ Termination check passed")
        print("  ✓ Compilation successful")
        return True

if __name__ == "__main__":
    generator = Lean4Generator()
    
    print("Generating non-sofic group proof...")
    proof1 = generator.generate_non_sofic_proof()
    print(f"  Generated code length: {len(proof1)} chars")
    
    print("\nGenerating sphere packing proof...")
    proof2 = generator.generate_sphere_packing_proof()
    print(f"  Generated code length: {len(proof2)} chars")
    
    print("\nVerifying all proofs...")
    generator.compile_check(proof1 + proof2)