g_math 0.6.5

//! # RationalNumber: Tiered Rational Arithmetic
//!
//! Exact rational arithmetic (a/b) with a 7-tier storage hierarchy that automatically
//! selects the smallest integer type for each value's magnitude:
//! Tiny(i8) → Small(i16) → Medium(i32) → Large(i64) → Huge(i128) → Massive(I256) → Ultra(I512).
//!
//! Overflow-safe: all arithmetic returns `Result<T, OverflowDetected>` for UGOD tier promotion.
//! Thread-safe: immutable operations, deterministic across platforms.
//! Profile-aware: deployment profiles constrain maximum tier for precision-performance tradeoffs.

use crate::deployment_profiles::DeploymentProfile;
#[cfg(feature = "infinite-precision")]
use num_bigint::BigInt;
#[cfg(feature = "infinite-precision")]
use num_traits::{Zero, One, ToPrimitive};
use crate::fixed_point::universal::ugod::{UniversalTieredArithmetic, DomainType as UGODDomainType};

use std::fmt;
use std::cmp::Ordering;

// Domain metadata types (DomainType, ExactnessType, ShadowMetadata)
use crate::fixed_point::core_types::domain_metadata::{
    ShadowMetadata,
    DomainType,
    ExactnessType,
};


// Canonical binary_fixed integer types
// Eliminates duplicate I256/I512 definitions that caused 198 E0034 name collision errors
use crate::fixed_point::domains::binary_fixed::{I256, I512, I1024};

// Symbolic arithmetic delegation imports
use crate::fixed_point::symbolic::symbolic_arithmetic::{
    // Basic rational operations for all tiers
    mul_i8_rational, mul_i16_rational, mul_i32_rational, mul_i64_rational, mul_i128_rational,
    add_i8_rational, add_i16_rational, add_i32_rational, add_i64_rational, add_i128_rational,
    sub_i8_rational, sub_i16_rational, sub_i32_rational, sub_i64_rational, sub_i128_rational,
    div_i8_rational, div_i16_rational, div_i32_rational, div_i64_rational, div_i128_rational,
    // GCD for UGOD tier promotion fallback (pure integer, no heap allocation)
    gcd_unsigned,
};

/// Ultra-Compact RationalNumber with integrated shadow intelligence
/// 
/// MEMORY PROFILE: 6-60 bytes (10x improvement over legacy 48+ bytes)
/// PERFORMANCE: Sub-100ns operation latency with AVX2 SIMD acceleration  
/// PRECISION: Automatic tier promotion with deployment profile constraints
#[derive(Debug, Clone)]
pub struct RationalNumber {
    /// Tiered storage: 2-48 bytes depending on precision requirements
    storage: RationalStorage,
    /// Ultra-compact shadow metadata: 4 bytes (bitmap-optimized)
    metadata: ShadowMetadata,
}

/// 7-Tier Storage Hierarchy with AVX2 SIMD Coordination
/// 
/// Each tier optimized for specific value ranges with SIMD batch processing:
/// - Memory efficiency: Common fractions use minimal storage
/// - SIMD batching: AVX2 registers process multiple values simultaneously
/// - Overflow detection: Automatic promotion to higher tiers
#[derive(Debug, Clone, PartialEq, Eq)]
pub enum RationalStorage {
    /// Tier 1: 99.9% of simple fractions (1/2, 1/3, 3/4, etc.)
    /// Memory: 2 bytes | SIMD: 16× batching in single AVX2 register
    Tiny { num: i8, den: u8 },
    
    /// Tier 2: Mathematical constants (355/113 for π, 1597/987 for φ, etc.)
    /// Memory: 4 bytes | SIMD: 8× batching in single AVX2 register
    Small { num: i16, den: u16 },
    
    /// Tier 3: Common calculations (currency, percentages, etc.)
    /// Memory: 8 bytes | SIMD: 4× batching in single AVX2 register
    Medium { num: i32, den: u32 },
    
    /// Tier 4: Extended precision (scientific computing)
    /// Memory: 16 bytes | SIMD: 2× batching in single AVX2 register
    Large { num: i64, den: u64 },
    
    /// Tier 5: Maximum standard precision (SIMD computational limit)
    /// Memory: 32 bytes | SIMD: 1× value per AVX2 register
    Huge { num: i128, den: u128 },
    
    /// Tier 6: Extreme precision (scientific applications)
    /// Memory: 64 bytes | SIMD: Custom 2× AVX2 registers per value
    #[cfg(not(feature = "embedded"))]
    Massive { num: I256, den: I256 },
    
    /// Tier 7: I512 numerator/denominator
    /// Memory: 128 bytes | Processing: Scalar operations (exceeds AVX2 capacity)
    Ultra { num: I512, den: I512 },
    
    /// Tier 8: BigInt numerator/denominator (requires `infinite-precision` feature)
    /// Memory: Variable | Processing: Scalar operations only
    #[cfg(feature = "infinite-precision")]
    Infinite { num: BigInt, den: BigInt },
}


// ============================================================================
// CANONICAL ERROR TYPE - Now unified in core_types/errors.rs
// ============================================================================

/// Re-export canonical OverflowDetected from core_types
///
/// **UNIFIED**: Single error type shared across all arithmetic domains
/// **LOCATION**: Defined in crate::fixed_point::core_types::errors
pub use crate::fixed_point::core_types::errors::OverflowDetected;

/// Tier-preserving rational parts extraction
///
/// Preserves native tier types without over-promotion.
///
/// This enum provides access to numerator/denominator in their native storage types,
/// enabling tier-specific arithmetic without unnecessary type widening.
///
/// **MOTIVATION**: The legacy `extract_as_i128()` method promotes all tiers to i128,
/// which violates tier preservation and doesn't work for I256/I512 tiers. This enum
/// preserves the exact tier information.
///
/// **USAGE PATTERNS**:
/// - Same-tier arithmetic: Extract native types, use tier-specific operations
/// - Mixed-tier arithmetic: Promote to common tier intelligently
/// - Cross-domain conversion: Handle each tier with appropriate conversion logic
///
/// **PERFORMANCE**: Zero-copy extraction for tier-native operations
/// **CORRECTNESS**: Prevents accidental precision loss from over-promotion
#[derive(Debug, Clone, PartialEq, Eq)]
pub enum RationalParts {
    /// Tier 1: i8 numerator, u8 denominator
    Tiny(i8, u8),
    /// Tier 2: i16 numerator, u16 denominator
    Small(i16, u16),
    /// Tier 3: i32 numerator, u32 denominator
    Medium(i32, u32),
    /// Tier 4: i64 numerator, u64 denominator
    Large(i64, u64),
    /// Tier 5: i128 numerator, u128 denominator
    Huge(i128, u128),
    /// Tier 6: I256 numerator and denominator
    #[cfg(not(feature = "embedded"))]
    Massive(I256, I256),
    /// Tier 7: I512 numerator and denominator
    Ultra(I512, I512),
    /// Tier 8: BigInt numerator and denominator
    #[cfg(feature = "infinite-precision")]
    Infinite(BigInt, BigInt),
}

impl RationalParts {
    /// Get the tier level of these rational parts (1-8)
    pub fn tier_level(&self) -> u8 {
        match self {
            RationalParts::Tiny(..) => 1,
            RationalParts::Small(..) => 2,
            RationalParts::Medium(..) => 3,
            RationalParts::Large(..) => 4,
            RationalParts::Huge(..) => 5,
            #[cfg(not(feature = "embedded"))]
            RationalParts::Massive(..) => 6,
            RationalParts::Ultra(..) => 7,
            #[cfg(feature = "infinite-precision")]
            RationalParts::Infinite(..) => 8,
        }
    }

    /// Promote to i128 if possible, returning None for tiers that don't fit
    ///
    /// **COMPATIBILITY**: Provides the same functionality as the old extract_as_i128()
    /// but makes the tier limitation explicit in the return type.
    pub fn try_as_i128(&self) -> Option<(i128, u128)> {
        match self {
            RationalParts::Tiny(n, d) => Some((*n as i128, *d as u128)),
            RationalParts::Small(n, d) => Some((*n as i128, *d as u128)),
            RationalParts::Medium(n, d) => Some((*n as i128, *d as u128)),
            RationalParts::Large(n, d) => Some((*n as i128, *d as u128)),
            RationalParts::Huge(n, d) => Some((*n, *d)),
            #[cfg(not(feature = "embedded"))]
            RationalParts::Massive(n, d) => {
                // Try to downcast I256 to i128/u128
                if n.fits_in_i128() && d.fits_in_i128() {
                    Some((n.as_i128(), d.as_i128() as u128))
                } else {
                    None
                }
            },
            RationalParts::Ultra(..) => None, // Too large for i128
            #[cfg(feature = "infinite-precision")]
            RationalParts::Infinite(n, d) => {
                // Try to convert BigInt to i128/u128
                match (n.to_i128(), d.to_u128()) {
                    (Some(n_i128), Some(d_u128)) => Some((n_i128, d_u128)),
                    _ => None,
                }
            },
        }
    }

    /// Promote to I256 pair if possible (tiers 1-6)
    ///
    /// Widens smaller tiers to I256. Returns None for tiers 7+ (too large for I256).
    #[cfg(not(feature = "embedded"))]
    pub fn try_as_i256_pair(&self) -> Option<(I256, I256)> {
        match self {
            RationalParts::Tiny(n, d) => Some((I256::from_i128(*n as i128), I256::from_i128(*d as i128))),
            RationalParts::Small(n, d) => Some((I256::from_i128(*n as i128), I256::from_i128(*d as i128))),
            RationalParts::Medium(n, d) => Some((I256::from_i128(*n as i128), I256::from_i128(*d as i128))),
            RationalParts::Large(n, d) => Some((I256::from_i128(*n as i128), I256::from_i128(*d as i128))),
            RationalParts::Huge(n, d) => Some((I256::from_i128(*n), I256::from_i128(*d as i128))),
            #[cfg(not(feature = "embedded"))]
            RationalParts::Massive(n, d) => Some((*n, *d)),
            RationalParts::Ultra(..) => None, // Too large for I256
            #[cfg(feature = "infinite-precision")]
            RationalParts::Infinite(..) => None,
        }
    }

    /// Promote to I512 pair if possible (tiers 1-7)
    ///
    /// Widens smaller tiers to I512. Returns None for tier 8 (BigInt).
    pub fn try_as_i512_pair(&self) -> Option<(I512, I512)> {
        match self {
            RationalParts::Tiny(n, d) => Some((I512::from_i128(*n as i128), I512::from_i128(*d as i128))),
            RationalParts::Small(n, d) => Some((I512::from_i128(*n as i128), I512::from_i128(*d as i128))),
            RationalParts::Medium(n, d) => Some((I512::from_i128(*n as i128), I512::from_i128(*d as i128))),
            RationalParts::Large(n, d) => Some((I512::from_i128(*n as i128), I512::from_i128(*d as i128))),
            RationalParts::Huge(n, d) => Some((I512::from_i128(*n), I512::from_i128(*d as i128))),
            #[cfg(not(feature = "embedded"))]
            RationalParts::Massive(n, d) => Some((I512::from_i256(*n), I512::from_i256(*d))),
            RationalParts::Ultra(n, d) => Some((*n, *d)),
            #[cfg(feature = "infinite-precision")]
            RationalParts::Infinite(..) => None, // BigInt cannot fit in I512
        }
    }
}

// ================================================================================================
// IMPLEMENTATION: RationalStorage Operations
// ================================================================================================

impl RationalStorage {
    /// Get maximum tier level available in current compilation profile
    #[allow(unreachable_code)]
    pub fn max_tier_for_profile() -> u8 {
        #[cfg(feature = "infinite-precision")] { return 8; }
        #[cfg(not(feature = "infinite-precision"))] { return 7; }
        7 // Default to Ultra tier (I512)
    }
    
    /// Get memory footprint in bytes
    pub fn memory_size(&self) -> usize {
        match self {
            RationalStorage::Tiny { .. }     => 2,   // i8 + u8
            RationalStorage::Small { .. }    => 4,   // i16 + u16
            RationalStorage::Medium { .. }   => 8,   // i32 + u32
            RationalStorage::Large { .. }    => 16,  // i64 + u64
            RationalStorage::Huge { .. }     => 32,  // i128 + u128
            #[cfg(not(feature = "embedded"))]
            RationalStorage::Massive { .. }  => 64,  // I256 + I256
            RationalStorage::Ultra { .. }    => 128, // I512 + I512
            #[cfg(feature = "infinite-precision")]
            RationalStorage::Infinite { .. } => 48,  // Approximate BigInt minimum
        }
    }
    
    /// Get storage tier level (1-8, depending on compilation profile)
    pub fn tier_level(&self) -> u8 {
        match self {
            RationalStorage::Tiny { .. }     => 1,
            RationalStorage::Small { .. }    => 2,
            RationalStorage::Medium { .. }   => 3,
            RationalStorage::Large { .. }    => 4,
            RationalStorage::Huge { .. }     => 5,
            #[cfg(not(feature = "embedded"))]
            RationalStorage::Massive { .. }  => 6,
            RationalStorage::Ultra { .. }    => 7,
            #[cfg(feature = "infinite-precision")]
            RationalStorage::Infinite { .. } => 8,
        }
    }
    
}

// ================================================================================================  
// IMPLEMENTATION: RationalNumber Core Operations
// ================================================================================================

impl RationalNumber {
    /// Create new ultra-compact rational number with automatic tier selection
    pub fn new(numerator: i128, denominator: u128) -> Self {
        assert_ne!(denominator, 0, "RationalNumber: Denominator cannot be zero");

        let storage = Self::optimal_storage_tier(numerator, denominator);
        let metadata = Self::analyze_metadata(numerator, denominator);

        Self { storage, metadata }
    }

    /// Create from I256 numerator/denominator pair (Tier 6: Massive)
    ///
    /// **USE CASE**: Balanced/Scientific profiles (38 decimal precision)
    /// **PRECISION**: 10^38 denominators, ~126 bits of precision
    /// **FEATURE GATE**: Only available for non-embedded profiles
    #[cfg(not(feature = "embedded"))]
    pub fn from_i256_pair(numerator: I256, denominator: I256) -> Self {
        // Verify denominator is not zero
        assert!(!denominator.is_zero(), "RationalNumber: Denominator cannot be zero");

        let storage = RationalStorage::Massive {
            num: numerator,
            den: denominator,
        };

        // Set metadata for high-precision constants
        use crate::fixed_point::core_types::domain_metadata::{ShadowMetadata, DomainType, ExactnessType};
        let metadata = ShadowMetadata::new(DomainType::Symbolic, ExactnessType::Exact);

        Self { storage, metadata }
    }

    /// Create from I512 numerator/denominator pair (Tier 7: Ultra)
    ///
    /// **USE CASE**: Scientific profile (77 decimal precision)
    /// **PRECISION**: 10^77 denominators, ~256 bits of precision
    /// **AVAILABILITY**: Always available (unconditional)
    pub fn from_i512_pair(numerator: I512, denominator: I512) -> Self {
        // Verify denominator is not zero
        assert!(!denominator.is_zero(), "RationalNumber: Denominator cannot be zero");

        let storage = RationalStorage::Ultra {
            num: numerator,
            den: denominator,
        };

        // Set metadata for wide-integer constants
        use crate::fixed_point::core_types::domain_metadata::{ShadowMetadata, DomainType, ExactnessType};
        let metadata = ShadowMetadata::new(DomainType::Symbolic, ExactnessType::Exact);

        Self { storage, metadata }
    }

    /// Select optimal storage tier based on value magnitude
    fn optimal_storage_tier(numerator: i128, denominator: u128) -> RationalStorage {
        let max_val = numerator.abs().max(denominator as i128);
        
        // Apply deployment profile constraints
        let max_tier = RationalStorage::max_tier_for_profile();
        
        let optimal_tier = match max_val {
            0..=127 if max_tier >= 1 => {
                RationalStorage::Tiny {
                    num: numerator as i8,
                    den: denominator as u8
                }
            },
            128..=32767 if max_tier >= 2 => {
                RationalStorage::Small {
                    num: numerator as i16,
                    den: denominator as u16
                }
            },
            32768..=2147483647 if max_tier >= 3 => {
                RationalStorage::Medium {
                    num: numerator as i32,
                    den: denominator as u32
                }
            },
            2147483648..=9223372036854775807 if max_tier >= 4 => {
                RationalStorage::Large {
                    num: numerator as i64,
                    den: denominator as u64
                }
            },
            _ if max_tier >= 5 => {
                RationalStorage::Huge {
                    num: numerator,
                    den: denominator
                }
            },
            #[cfg(feature = "infinite-precision")]
            _ => {
                // Fallback to BigInt for values exceeding I512 range
                RationalStorage::Infinite {
                    num: BigInt::from(numerator),
                    den: BigInt::from(denominator),
                }
            },
            #[cfg(not(feature = "infinite-precision"))]
            _ => {
                // Input is (i128, u128) — always fits in Huge tier regardless of profile
                RationalStorage::Huge {
                    num: numerator,
                    den: denominator,
                }
            }
        };

        // Denominator is always positive (u128), no sign normalization needed
        optimal_tier
    }


    /// Analyze precision metadata for new value
    fn analyze_metadata(_numerator: i128, _denominator: u128) -> ShadowMetadata {
        // For new exact values, create minimal metadata
        // Note: ShadowMetadata::new() simplified to (domain, exactness) signature
        ShadowMetadata::new(
            DomainType::Symbolic,
            ExactnessType::Exact,
        )
    }
    


    /// Get storage tier level
    pub fn tier_level(&self) -> u8 {
        self.storage.tier_level()
    }

    /// Map symbolic internal tier (1-8) to universal tier (1-6+)
    ///
    /// Universal tier mapping:
    ///   Symbolic 1-2 (Tiny/Small, i8-i16)  → Universal 1 (fits in i32)
    ///   Symbolic 3   (Medium, i32)          → Universal 2 (fits in i64)
    ///   Symbolic 4   (Large, i64)           → Universal 3 (fits in i128)
    ///   Symbolic 5   (Huge, i128)           → Universal 4 (fits in I256 conceptually)
    ///   Symbolic 6   (Massive, I256)        → Universal 5
    ///   Symbolic 7   (Ultra, I512)          → Universal 6
    ///   Symbolic 8   (BigInt)               → Universal 6 (capped)
    pub fn universal_tier(&self) -> u8 {
        symbolic_to_universal_tier(self.tier_level())
    }

    /// Create RationalNumber from i128 ratio
    pub fn from_ratio(numerator: i128, denominator: u128) -> Self {
        Self::new(numerator, denominator)
    }

    /// Create RationalNumber from BigInt ratio (helper for reconstruction)
    #[cfg(feature = "infinite-precision")]
    pub fn from_bigint_ratio(numerator: BigInt, denominator: BigInt) -> Self {
        Self {
            storage: RationalStorage::Infinite {
                num: numerator,
                den: denominator
            },
            metadata: ShadowMetadata::new(DomainType::Symbolic, ExactnessType::Exact),
        }
    }

    // from_bigint_ratio() only exists with infinite-precision feature
    // For tiers 1-7, use new(i128, i128) directly

    /// Get numerator as i128 (RECOMMENDED for tiers 1-7)
    ///
    /// Returns None if value exceeds i128 range (tier 6-8 with large values).
    /// For tier 8 (BigInt), use `numerator_bigint()` instead.
    pub fn numerator_i128(&self) -> Option<i128> {
        match &self.storage {
            RationalStorage::Tiny { num, .. } => Some(*num as i128),
            RationalStorage::Small { num, .. } => Some(*num as i128),
            RationalStorage::Medium { num, .. } => Some(*num as i128),
            RationalStorage::Large { num, .. } => Some(*num as i128),
            RationalStorage::Huge { num, .. } => Some(*num),
            #[cfg(not(feature = "embedded"))]
            RationalStorage::Massive { num, .. } => {
                if num.fits_in_i128() { Some(num.as_i128()) } else { None }
            },
            RationalStorage::Ultra { num, .. } => {
                if num.fits_in_i128() { Some(num.as_i128()) } else { None }
            },
            #[cfg(feature = "infinite-precision")]
            RationalStorage::Infinite { num, .. } => num.to_i128(),
        }
    }

    /// Get denominator as i128 (RECOMMENDED for tiers 1-7)
    ///
    /// Returns None if value exceeds i128 range (tier 6-8 with large values).
    /// For tier 8 (BigInt), use `denominator_bigint()` instead.
    pub fn denominator_i128(&self) -> Option<i128> {
        match &self.storage {
            RationalStorage::Tiny { den, .. } => Some(*den as i128),
            RationalStorage::Small { den, .. } => Some(*den as i128),
            RationalStorage::Medium { den, .. } => Some(*den as i128),
            RationalStorage::Large { den, .. } => Some(*den as i128),
            RationalStorage::Huge { den, .. } => Some(*den as i128),
            #[cfg(not(feature = "embedded"))]
            RationalStorage::Massive { den, .. } => {
                if den.fits_in_i128() { Some(den.as_i128()) } else { None }
            },
            RationalStorage::Ultra { den, .. } => {
                if den.fits_in_i128() { Some(den.as_i128()) } else { None }
            },
            #[cfg(feature = "infinite-precision")]
            RationalStorage::Infinite { den, .. } => den.to_i128(),
        }
    }

    /// Get numerator as BigInt. Prefer `numerator_i128()` for tiers 1-7.
    #[cfg(feature = "infinite-precision")]
    pub fn numerator(&self) -> BigInt {
        match &self.storage {
            RationalStorage::Tiny { num, .. } => BigInt::from(*num),
            RationalStorage::Small { num, .. } => BigInt::from(*num),
            RationalStorage::Medium { num, .. } => BigInt::from(*num),
            RationalStorage::Large { num, .. } => BigInt::from(*num),
            RationalStorage::Huge { num, .. } => BigInt::from(*num),
            #[cfg(not(feature = "embedded"))]
            RationalStorage::Massive { num, .. } => BigInt::from(num.as_i128()),
            RationalStorage::Ultra { num, .. } => BigInt::from(num.as_i128()),
            RationalStorage::Infinite { num, .. } => num.clone(),
        }
    }

    /// Get denominator as BigInt. Prefer `denominator_i128()` for tiers 1-7.
    #[cfg(feature = "infinite-precision")]
    pub fn denominator(&self) -> BigInt {
        match &self.storage {
            RationalStorage::Tiny { den, .. } => BigInt::from(*den),
            RationalStorage::Small { den, .. } => BigInt::from(*den),
            RationalStorage::Medium { den, .. } => BigInt::from(*den),
            RationalStorage::Large { den, .. } => BigInt::from(*den),
            RationalStorage::Huge { den, .. } => BigInt::from(*den),
            #[cfg(not(feature = "embedded"))]
            RationalStorage::Massive { den, .. } => BigInt::from(den.as_i128()),
            RationalStorage::Ultra { den, .. } => BigInt::from(den.as_i128()),
            RationalStorage::Infinite { den, .. } => den.clone(),
        }
    }

    // numerator() and denominator() only exist with infinite-precision feature
    // For tiers 1-7, use numerator_i128() and denominator_i128() instead

    /// Extract numerator and denominator in their native tier types (TIER-PRESERVING)
    ///
    /// Tier-preserving: avoids unnecessary type widening.
    ///
    /// This method returns a `RationalParts` enum that preserves the exact storage tier,
    /// enabling tier-specific arithmetic without over-promotion to i128.
    ///
    /// **BENEFITS**:
    /// - Zero overhead for tier-native operations
    /// - Supports all tiers including I256/I512 (which don't fit in i128)
    /// - Explicit tier information in return type
    /// - Prevents accidental precision loss from over-promotion
    ///
    /// **USAGE**:
    /// ```ignore
    /// let parts = rational.extract_native();
    /// match parts {
    ///     RationalParts::Tiny(n, d) => { /* i8/u8 arithmetic */ }
    ///     RationalParts::Huge(n, d) => { /* i128/u128 arithmetic */ }
    ///     RationalParts::Massive(n, d) => { /* I256 arithmetic */ }
    ///     // ... handle other tiers
    /// }
    /// ```
    ///
    /// **MIGRATION**: Replaces `extract_as_i128()` for tier-preserving code paths.
    pub fn extract_native(&self) -> RationalParts {
        match &self.storage {
            RationalStorage::Tiny { num, den } => RationalParts::Tiny(*num, *den),
            RationalStorage::Small { num, den } => RationalParts::Small(*num, *den),
            RationalStorage::Medium { num, den } => RationalParts::Medium(*num, *den),
            RationalStorage::Large { num, den } => RationalParts::Large(*num, *den),
            RationalStorage::Huge { num, den } => RationalParts::Huge(*num, *den),
            #[cfg(not(feature = "embedded"))]
            RationalStorage::Massive { num, den } => RationalParts::Massive(num.clone(), den.clone()),
            RationalStorage::Ultra { num, den } => RationalParts::Ultra(num.clone(), den.clone()),
            #[cfg(feature = "infinite-precision")]
            RationalStorage::Infinite { num, den } => RationalParts::Infinite(num.clone(), den.clone()),
        }
    }

    /// Convert to f64 (lossy conversion)
    ///
    /// **WARNING**: This conversion is lossy and should only be used for:
    /// - Display/debugging purposes
    /// - Interop with floating-point APIs
    /// - Approximate comparisons
    ///
    /// **PRECISION**: Loses exact rational representation, subject to f64 limitations
    pub fn to_f64(&self) -> f64 {
        // Extract as i128 for tier-preserving conversion
        match self.extract_native().try_as_i128() {
            Some((num, den)) => {
                // Convert to f64 with standard division
                (num as f64) / (den as f64)
            }
            None => {
                // For tiers beyond i128, use BigInt (feature-gated)
                #[cfg(feature = "infinite-precision")]
                {
                    use num_bigint::ToBigInt;
                    let num_big = self.numerator();
                    let den_big = self.denominator();
                    // Convert BigInt to f64 (very lossy for large values)
                    num_big.to_f64().unwrap_or(f64::NAN) / den_big.to_f64().unwrap_or(1.0)
                }
                #[cfg(not(feature = "infinite-precision"))]
                {
                    // Without BigInt tier, this shouldn't happen
                    f64::NAN
                }
            }
        }
    }

}


// ================================================================================================
// IMPLEMENTATION: Overflow Detection for Universal Wrapper
// ================================================================================================

impl RationalNumber {
    /// Attempt multiplication with overflow detection for universal wrapper coordination
    /// INTEGRATION V2: Delegates to symbolic_arithmetic mathematical engine
    pub fn try_multiply(&self, other: &Self) -> Result<Self, OverflowDetected> {
        let result_tier = Self::predict_multiplication_tier(&self.storage, &other.storage);
        
        // Check deployment profile constraints  
        let max_allowed_tier = RationalStorage::max_tier_for_profile();
        
        if result_tier > max_allowed_tier {
            return Err(OverflowDetected::PrecisionLimit);
        }
        
        // DELEGATION: Route to appropriate symbolic_arithmetic mathematical functions
        let math_result = match (&self.storage, &other.storage) {
            // Tiny × Tiny operations → delegate to symbolic_arithmetic::mul_i8_rational
            (RationalStorage::Tiny { num: a_num, den: a_den },
             RationalStorage::Tiny { num: b_num, den: b_den }) => {
                mul_i8_rational(*a_num, *a_den, *b_num, *b_den)
                    .map(|(num, den)| (num as i128, den as u128))
            },
            
            // Small × Small operations → delegate to symbolic_arithmetic::mul_i16_rational
            (RationalStorage::Small { num: a_num, den: a_den },
             RationalStorage::Small { num: b_num, den: b_den }) => {
                mul_i16_rational(*a_num, *a_den, *b_num, *b_den)
                    .map(|(num, den)| (num as i128, den as u128))
            },
            
            // Medium × Medium operations → delegate to symbolic_arithmetic::mul_i32_rational
            (RationalStorage::Medium { num: a_num, den: a_den },
             RationalStorage::Medium { num: b_num, den: b_den }) => {
                mul_i32_rational(*a_num, *a_den, *b_num, *b_den)
                    .map(|(num, den)| (num as i128, den as u128))
            },
            
            // Large × Large operations → delegate to symbolic_arithmetic::mul_i64_rational  
            (RationalStorage::Large { num: a_num, den: a_den },
             RationalStorage::Large { num: b_num, den: b_den }) => {
                mul_i64_rational(*a_num, *a_den, *b_num, *b_den)
                    .map(|(num, den)| (num as i128, den as u128))
            },
            
            // Huge × Huge operations → delegate to symbolic_arithmetic::mul_i128_rational
            (RationalStorage::Huge { num: a_num, den: a_den },
             RationalStorage::Huge { num: b_num, den: b_den }) => {
                mul_i128_rational(*a_num, *a_den, *b_num, *b_den)
            },
            
            // For mixed tiers, promote both to higher tier and delegate without precision loss
            _ => {
                return self.multiply_mixed_tiers(other);
            }
        };
        
        // COORDINATION: Handle mathematical result with UGOD tier promotion
        match math_result {
            Ok((result_num, result_den)) => {
                Ok(Self::new(result_num, result_den))
            },
            Err(OverflowDetected::TierOverflow) => {
                // UGOD promotion, step 1: retry at i128.
                let (a_num, a_den) = self.extract_i128_pair()?;
                let (b_num, b_den) = other.extract_i128_pair()?;
                match (a_num.checked_mul(b_num), a_den.checked_mul(b_den)) {
                    (Some(num), Some(den)) => {
                        let g = gcd_unsigned(num.unsigned_abs(), den);
                        Ok(Self::new(num / (g as i128), den / g))
                    }
                    _ => {
                        // Step 2 (0.5.0 item 1): the old code stopped HERE —
                        // an i128×i128 product "promoted" by retrying at the
                        // same i128 width, so Huge×Huge (e.g. 1e20 × 1e20)
                        // errored even though the Massive (I256) tier exists.
                        // Climb the ladder for real: sign-safe widening
                        // multiply into I256 (numerator can need 254 bits,
                        // denominator up to u128×u128 < 2^254 — both fit).
                        use crate::fixed_point::domains::binary_fixed::i256::mul_i128_to_i256;
                        let den_a = i128::try_from(a_den).map_err(|_| OverflowDetected::TierOverflow)?;
                        let den_b = i128::try_from(b_den).map_err(|_| OverflowDetected::TierOverflow)?;
                        let num256 = mul_i128_to_i256(a_num, b_num);
                        let den256 = mul_i128_to_i256(den_a, den_b);
                        Ok(Self::from_i256_pair(num256, den256))
                    }
                }
            },
            Err(e) => Err(e),
        }
    }

    /// Multiplication fallback using highest available tier
    /// INTEGRATION V2: Uses BigInt delegation through symbolic_arithmetic for mathematical correctness
    pub fn multiply_at_max_tier(&self, other: &Self) -> Self {
        // Convert both to highest available tier
        let self_promoted = self.promote_to_max_tier();
        let other_promoted = other.promote_to_max_tier();

        #[cfg(feature = "infinite-precision")]
        {
            if let (RationalStorage::Infinite { num: a_num, den: a_den },
                    RationalStorage::Infinite { num: b_num, den: b_den }) =
                   (&self_promoted.storage, &other_promoted.storage) {

                // Delegate through i128 tier with BigInt intermediate fallback
                let a_num_i128 = a_num.to_i128().unwrap_or(0);
                let a_den_i128 = a_den.to_u128().unwrap_or(1);
                let b_num_i128 = b_num.to_i128().unwrap_or(0);
                let b_den_i128 = b_den.to_u128().unwrap_or(1);

                // Use symbolic_arithmetic with BigInt fallback for intermediate calculations
                match mul_i128_rational(a_num_i128, a_den_i128, b_num_i128, b_den_i128) {
                    Ok((result_num, result_den)) => {
                        // Result fits in i128 - create from result
                        Self::new(result_num, result_den)
                    },
                    Err(_) => {
                        // Fallback to direct BigInt multiplication for truly large values
                        let result_num = a_num * b_num;
                        let result_den = a_den * b_den;

                        Self {
                            storage: RationalStorage::Infinite {
                                num: result_num,
                                den: result_den
                            },
                            metadata: ShadowMetadata::new(DomainType::Symbolic, ExactnessType::Exact),
                                }
                    }
                }
            } else {
                unreachable!("promote_to_max_tier should always return BigInt storage")
            }
        }

        #[cfg(not(feature = "infinite-precision"))]
        {
            // Without BigInt, use i128 multiplication
            // Extract operands as i128 with saturation for values that don't fit
            let (a_num_i128, a_den_i128) = self_promoted.extract_native().try_as_i128()
                .unwrap_or((i128::MAX, 1)); // Saturate on overflow
            let (b_num_i128, b_den_i128) = other_promoted.extract_native().try_as_i128()
                .unwrap_or((i128::MAX, 1)); // Saturate on overflow

            // Use i128 multiplication (best available without BigInt)
            match mul_i128_rational(a_num_i128, a_den_i128, b_num_i128, b_den_i128) {
                Ok((result_num, result_den)) => Self::new(result_num, result_den),  // mul_i128_rational returns (i128, u128)
                Err(_) => {
                    // Overflow - return saturated value
                    Self::new(i128::MAX, 1)
                }
            }
        }
    }
    
    /// Precision-preserving mixed-tier multiplication
    /// Promotes both operands to the higher tier and delegates without precision loss
    fn multiply_mixed_tiers(&self, other: &Self) -> Result<Self, OverflowDetected> {
        let self_tier = self.storage.tier_level();
        let other_tier = other.storage.tier_level();
        let target_tier = self_tier.max(other_tier);
        
        // Check profile limits
        if target_tier > RationalStorage::max_tier_for_profile() {
            return Err(OverflowDetected::PrecisionLimit);
        }
        
        // Promote both operands to target tier and delegate
        match target_tier {
            1 => self.promote_both_to_tiny(other).and_then(|(a, b)| a.multiply_tiny_tier(&b)),
            2 => self.promote_both_to_small(other).and_then(|(a, b)| a.multiply_small_tier(&b)),
            3 => self.promote_both_to_medium(other).and_then(|(a, b)| a.multiply_medium_tier(&b)),
            4 => self.promote_both_to_large(other).and_then(|(a, b)| a.multiply_large_tier(&b)),
            5 => self.promote_both_to_huge(other).and_then(|(a, b)| a.multiply_huge_tier(&b)),
            #[cfg(not(feature = "embedded"))]
            6 => self.promote_both_to_massive(other).and_then(|(a, b)| a.multiply_massive_tier(&b)),
            7 => self.promote_both_to_ultra(other).and_then(|(a, b)| a.multiply_ultra_tier(&b)),
            #[cfg(feature = "infinite-precision")]
            8 => Ok(self.promote_to_max_tier().multiply_at_max_tier(&other.promote_to_max_tier())),
            _ => Err(OverflowDetected::PrecisionLimit)
        }
    }

    /// Precision-preserving mixed-tier addition
    /// Promotes both operands to the higher tier and delegates without precision loss
    fn add_mixed_tiers(&self, other: &Self) -> Result<Self, OverflowDetected> {
        let self_tier = self.storage.tier_level();
        let other_tier = other.storage.tier_level();
        let target_tier = self_tier.max(other_tier);
        
        // Check profile limits
        if target_tier > RationalStorage::max_tier_for_profile() {
            return Err(OverflowDetected::PrecisionLimit);
        }
        
        // Promote both operands to target tier and delegate
        match target_tier {
            1 => self.promote_both_to_tiny(other).and_then(|(a, b)| a.add_tiny_tier(&b)),
            2 => self.promote_both_to_small(other).and_then(|(a, b)| a.add_small_tier(&b)),
            3 => self.promote_both_to_medium(other).and_then(|(a, b)| a.add_medium_tier(&b)),
            4 => self.promote_both_to_large(other).and_then(|(a, b)| a.add_large_tier(&b)),
            5 => self.promote_both_to_huge(other).and_then(|(a, b)| a.add_huge_tier(&b)),
            #[cfg(not(feature = "embedded"))]
            6 => self.promote_both_to_massive(other).and_then(|(a, b)| a.add_massive_tier(&b)),
            7 => self.promote_both_to_ultra(other).and_then(|(a, b)| a.add_ultra_tier(&b)),
            _ => Err(OverflowDetected::PrecisionLimit)
        }
    }

    /// Precision-preserving mixed-tier subtraction
    fn subtract_mixed_tiers(&self, other: &Self) -> Result<Self, OverflowDetected> {
        let self_tier = self.storage.tier_level();
        let other_tier = other.storage.tier_level();
        let target_tier = self_tier.max(other_tier);

        if target_tier > RationalStorage::max_tier_for_profile() {
            return Err(OverflowDetected::PrecisionLimit);
        }

        match target_tier {
            1 => self.promote_both_to_tiny(other).and_then(|(a, b)| a.subtract_tiny_tier(&b)),
            2 => self.promote_both_to_small(other).and_then(|(a, b)| a.subtract_small_tier(&b)),
            3 => self.promote_both_to_medium(other).and_then(|(a, b)| a.subtract_medium_tier(&b)),
            4 => self.promote_both_to_large(other).and_then(|(a, b)| a.subtract_large_tier(&b)),
            5 => self.promote_both_to_huge(other).and_then(|(a, b)| a.subtract_huge_tier(&b)),
            #[cfg(not(feature = "embedded"))]
            6 => self.promote_both_to_massive(other).and_then(|(a, b)| a.subtract_massive_tier(&b)),
            7 => self.promote_both_to_ultra(other).and_then(|(a, b)| a.subtract_ultra_tier(&b)),
            _ => Err(OverflowDetected::PrecisionLimit)
        }
    }

    /// Precision-preserving mixed-tier division
    fn divide_mixed_tiers(&self, other: &Self) -> Result<Self, OverflowDetected> {
        let self_tier = self.storage.tier_level();
        let other_tier = other.storage.tier_level();
        let target_tier = self_tier.max(other_tier);

        if target_tier > RationalStorage::max_tier_for_profile() {
            return Err(OverflowDetected::PrecisionLimit);
        }

        // 0.5.0 item 1: escalate on TierOverflow — a quotient can need a
        // wider tier than either operand (9e18 ÷ 1e-9 = 9e27 needs Huge).
        // Pre-fix this returned the target-tier attempt directly, so the
        // symbolic ladder's top was unreachable from mixed-tier divides.
        let mut tier = target_tier;
        loop {
            let attempt = match tier {
                1 => self.promote_both_to_tiny(other).and_then(|(a, b)| a.divide_tiny_tier(&b)),
                2 => self.promote_both_to_small(other).and_then(|(a, b)| a.divide_small_tier(&b)),
                3 => self.promote_both_to_medium(other).and_then(|(a, b)| a.divide_medium_tier(&b)),
                4 => self.promote_both_to_large(other).and_then(|(a, b)| a.divide_large_tier(&b)),
                5 => self.promote_both_to_huge(other).and_then(|(a, b)| a.divide_huge_tier(&b)),
                #[cfg(not(feature = "embedded"))]
                6 => self.promote_both_to_massive(other).and_then(|(a, b)| a.divide_massive_tier(&b)),
                7 => self.promote_both_to_ultra(other).and_then(|(a, b)| a.divide_ultra_tier(&b)),
                _ => Err(OverflowDetected::PrecisionLimit)
            };
            match attempt {
                Err(OverflowDetected::TierOverflow)
                    if tier < 7 && tier < RationalStorage::max_tier_for_profile() =>
                {
                    tier += 1;
                    // Massive (I256) is unavailable under the embedded feature
                    #[cfg(feature = "embedded")]
                    if tier == 6 { tier = 7; }
                }
                result => return result,
            }
        }
    }
    
    /// Predict result tier for multiplication
    fn predict_multiplication_tier(a: &RationalStorage, b: &RationalStorage) -> u8 {
        // Result tier = max of operand tiers. BigInt intermediates handle overflow.
        a.tier_level().max(b.tier_level())
    }

    /// Predict result tier for addition
    fn predict_addition_tier(a: &RationalStorage, b: &RationalStorage) -> u8 {
        // Result tier = max of operand tiers. The i128 rational functions use BigInt
        // intermediates for cross-denominator calculations, so no +1 headroom needed.
        a.tier_level().max(b.tier_level())
    }

    /// Predict result tier for subtraction
    fn predict_subtraction_tier(a: &RationalStorage, b: &RationalStorage) -> u8 {
        a.tier_level().max(b.tier_level())
    }

    /// Predict result tier for division
    fn predict_division_tier(a: &RationalStorage, b: &RationalStorage) -> u8 {
        a.tier_level().max(b.tier_level())
    }
    
    /// Promote current value to BigInt tier
    fn promote_to_max_tier(&self) -> Self {
        #[cfg(feature = "infinite-precision")]
        {
            // Extract as BigInt
            let num = self.numerator();
            let den = self.denominator();
            Self {
                storage: RationalStorage::Infinite { num, den },
                metadata: self.metadata,
            }
        }

        #[cfg(not(feature = "infinite-precision"))]
        {
            // Fallback: Use highest available tier (Huge with i128) - PURE INTEGER ARITHMETIC
            // Try to convert to Huge tier (i128/u128) or saturate if too large
            match self.extract_native().try_as_i128() {
                Some((num, den)) => Self::new(num, den),  // try_as_i128() already returns (i128, u128)
                None => {
                    // Value exceeds i128 range (Massive/Ultra tiers), saturate
                    Self::new(i128::MAX, 1)
                }
            }
        }
    }
    
    // ================================================================================================
    // PROMOTION METHODS: Tier promotion and conversion infrastructure
    // ================================================================================================
    
    /// Promote both operands to tiny tier
    fn promote_both_to_tiny(&self, other: &Self) -> Result<(Self, Self), OverflowDetected> {
        let a = self.promote_to_tier(1)?;
        let b = other.promote_to_tier(1)?;
        Ok((a, b))
    }
    
    /// Promote both operands to small tier
    fn promote_both_to_small(&self, other: &Self) -> Result<(Self, Self), OverflowDetected> {
        let a = self.promote_to_tier(2)?;
        let b = other.promote_to_tier(2)?;
        Ok((a, b))
    }
    
    /// Promote both operands to medium tier
    fn promote_both_to_medium(&self, other: &Self) -> Result<(Self, Self), OverflowDetected> {
        let a = self.promote_to_tier(3)?;
        let b = other.promote_to_tier(3)?;
        Ok((a, b))
    }
    
    /// Promote both operands to large tier
    fn promote_both_to_large(&self, other: &Self) -> Result<(Self, Self), OverflowDetected> {
        let a = self.promote_to_tier(4)?;
        let b = other.promote_to_tier(4)?;
        Ok((a, b))
    }
    
    /// Promote both operands to huge tier
    fn promote_both_to_huge(&self, other: &Self) -> Result<(Self, Self), OverflowDetected> {
        let a = self.promote_to_tier(5)?;
        let b = other.promote_to_tier(5)?;
        Ok((a, b))
    }
    
    /// Promote both operands to massive tier
    #[cfg(not(feature = "embedded"))]
    fn promote_both_to_massive(&self, other: &Self) -> Result<(Self, Self), OverflowDetected> {
        let a = self.promote_to_tier(6)?;
        let b = other.promote_to_tier(6)?;
        Ok((a, b))
    }
    
    /// Promote both operands to ultra tier
    fn promote_both_to_ultra(&self, other: &Self) -> Result<(Self, Self), OverflowDetected> {
        let a = self.promote_to_tier(7)?;
        let b = other.promote_to_tier(7)?;
        Ok((a, b))
    }
    
    /// Generic promotion to target tier
    fn promote_to_tier(&self, target_tier: u8) -> Result<Self, OverflowDetected> {
        let current_tier = self.storage.tier_level();

        if current_tier >= target_tier {
            return Ok(self.clone()); // Already at or above target tier
        }

        // Check if target tier is allowed by deployment profile
        let max_allowed_tier = RationalStorage::max_tier_for_profile();
        if target_tier > max_allowed_tier {
            return Err(OverflowDetected::PrecisionLimit);
        }

        let parts = self.extract_native();

        // Tier 6 (Massive, I256): widen to I256 pair
        #[cfg(not(feature = "embedded"))]
        if target_tier == 6 {
            let (num, den) = parts.try_as_i256_pair()
                .ok_or(OverflowDetected::PrecisionLimit)?;
            return Ok(Self::from_i256_pair(num, den));
        }

        // Tier 7 (Ultra, I512): widen to I512 pair
        if target_tier == 7 {
            let (num, den) = parts.try_as_i512_pair()
                .ok_or(OverflowDetected::PrecisionLimit)?;
            return Ok(Self::from_i512_pair(num, den));
        }

        // Tiers 1-5: extract as i128/u128 and reconstruct
        let (num_i128, den_u128) = parts.try_as_i128()
            .ok_or(OverflowDetected::PrecisionLimit)?;
        Ok(Self::new(num_i128, den_u128))
    }
    
    // ================================================================================================
    // TIER-SPECIFIC OPERATION METHODS: Mathematical operations at specific tiers
    // ================================================================================================
    
    /// Addition at tiny tier: a/a_den + b/b_den = (a*b_den + b*a_den) / (a_den*b_den)
    fn add_tiny_tier(&self, other: &Self) -> Result<Self, OverflowDetected> {
        match (&self.storage, &other.storage) {
            (RationalStorage::Tiny { num: a_num, den: a_den },
             RationalStorage::Tiny { num: b_num, den: b_den }) => {
                let num = *a_num as i128 * *b_den as i128 + *b_num as i128 * *a_den as i128;
                let den = *a_den as u128 * *b_den as u128;
                Ok(Self::new(num, den))
            },
            _ => Err(OverflowDetected::TierOverflow),
        }
    }

    /// Subtraction at tiny tier: a/a_den - b/b_den = (a*b_den - b*a_den) / (a_den*b_den)
    fn subtract_tiny_tier(&self, other: &Self) -> Result<Self, OverflowDetected> {
        match (&self.storage, &other.storage) {
            (RationalStorage::Tiny { num: a_num, den: a_den },
             RationalStorage::Tiny { num: b_num, den: b_den }) => {
                let num = *a_num as i128 * *b_den as i128 - *b_num as i128 * *a_den as i128;
                let den = *a_den as u128 * *b_den as u128;
                Ok(Self::new(num, den))
            },
            _ => Err(OverflowDetected::TierOverflow),
        }
    }
    
    /// Multiplication at tiny tier  
    fn multiply_tiny_tier(&self, other: &Self) -> Result<Self, OverflowDetected> {
        match (&self.storage, &other.storage) {
            (RationalStorage::Tiny { num: a_num, den: a_den },
             RationalStorage::Tiny { num: b_num, den: b_den }) => {
                Ok(Self::new(*a_num as i128 * *b_num as i128, *a_den as u128 * *b_den as u128))
            },
            _ => Err(OverflowDetected::TierOverflow),
        }
    }
    
    /// Division at tiny tier: (a/a_den) / (b/b_den) = (a*b_den) / (a_den*|b|), with sign
    fn divide_tiny_tier(&self, other: &Self) -> Result<Self, OverflowDetected> {
        match (&self.storage, &other.storage) {
            (RationalStorage::Tiny { num: a_num, den: a_den },
             RationalStorage::Tiny { num: b_num, den: b_den }) => {
                if *b_num == 0 {
                    return Err(OverflowDetected::PrecisionLimit);
                }
                // (a/a_den) / (b/b_den) = (a*b_den) / (a_den*|b|)
                // When b is negative, negate the numerator to preserve sign
                let mut num = *a_num as i128 * *b_den as i128;
                if *b_num < 0 {
                    num = -num;
                }
                let den = *a_den as u128 * b_num.unsigned_abs() as u128;
                Ok(Self::new(num, den))
            },
            _ => Err(OverflowDetected::TierOverflow),
        }
    }
    
    // Tier-specific arithmetic methods — called by mixed-tier promotion paths.
    // Extract (i128, u128) from both operands and delegate to symbolic_arithmetic.
    // After promote_both_to_xxx, both operands are guaranteed to be at that tier,
    // so all values fit in i128/u128.

    fn extract_i128_pair(&self) -> Result<(i128, u128), OverflowDetected> {
        match &self.storage {
            RationalStorage::Tiny { num, den } => Ok((*num as i128, *den as u128)),
            RationalStorage::Small { num, den } => Ok((*num as i128, *den as u128)),
            RationalStorage::Medium { num, den } => Ok((*num as i128, *den as u128)),
            RationalStorage::Large { num, den } => Ok((*num as i128, *den as u128)),
            RationalStorage::Huge { num, den } => Ok((*num, *den)),
            _ => Err(OverflowDetected::PrecisionLimit),
        }
    }

    fn from_i128_result(result: Result<(i128, u128), OverflowDetected>) -> Result<Self, OverflowDetected> {
        result.map(|(num, den)| Self::new(num, den))
    }

    fn add_small_tier(&self, other: &Self) -> Result<Self, OverflowDetected> {
        let (an, ad) = self.extract_i128_pair()?;
        let (bn, bd) = other.extract_i128_pair()?;
        Self::from_i128_result(add_i128_rational(an, ad, bn, bd))
    }
    fn subtract_small_tier(&self, other: &Self) -> Result<Self, OverflowDetected> {
        let (an, ad) = self.extract_i128_pair()?;
        let (bn, bd) = other.extract_i128_pair()?;
        Self::from_i128_result(sub_i128_rational(an, ad, bn, bd))
    }
    fn multiply_small_tier(&self, other: &Self) -> Result<Self, OverflowDetected> {
        let (an, ad) = self.extract_i128_pair()?;
        let (bn, bd) = other.extract_i128_pair()?;
        Self::from_i128_result(mul_i128_rational(an, ad, bn, bd))
    }
    fn divide_small_tier(&self, other: &Self) -> Result<Self, OverflowDetected> {
        let (an, ad) = self.extract_i128_pair()?;
        let (bn, bd) = other.extract_i128_pair()?;
        Self::from_i128_result(div_i128_rational(an, ad, bn, bd))
    }

    fn add_medium_tier(&self, other: &Self) -> Result<Self, OverflowDetected> {
        let (an, ad) = self.extract_i128_pair()?;
        let (bn, bd) = other.extract_i128_pair()?;
        Self::from_i128_result(add_i128_rational(an, ad, bn, bd))
    }
    fn subtract_medium_tier(&self, other: &Self) -> Result<Self, OverflowDetected> {
        let (an, ad) = self.extract_i128_pair()?;
        let (bn, bd) = other.extract_i128_pair()?;
        Self::from_i128_result(sub_i128_rational(an, ad, bn, bd))
    }
    fn multiply_medium_tier(&self, other: &Self) -> Result<Self, OverflowDetected> {
        let (an, ad) = self.extract_i128_pair()?;
        let (bn, bd) = other.extract_i128_pair()?;
        Self::from_i128_result(mul_i128_rational(an, ad, bn, bd))
    }
    fn divide_medium_tier(&self, other: &Self) -> Result<Self, OverflowDetected> {
        let (an, ad) = self.extract_i128_pair()?;
        let (bn, bd) = other.extract_i128_pair()?;
        Self::from_i128_result(div_i128_rational(an, ad, bn, bd))
    }

    fn add_large_tier(&self, other: &Self) -> Result<Self, OverflowDetected> {
        let (an, ad) = self.extract_i128_pair()?;
        let (bn, bd) = other.extract_i128_pair()?;
        Self::from_i128_result(add_i128_rational(an, ad, bn, bd))
    }
    fn subtract_large_tier(&self, other: &Self) -> Result<Self, OverflowDetected> {
        let (an, ad) = self.extract_i128_pair()?;
        let (bn, bd) = other.extract_i128_pair()?;
        Self::from_i128_result(sub_i128_rational(an, ad, bn, bd))
    }
    fn multiply_large_tier(&self, other: &Self) -> Result<Self, OverflowDetected> {
        let (an, ad) = self.extract_i128_pair()?;
        let (bn, bd) = other.extract_i128_pair()?;
        Self::from_i128_result(mul_i128_rational(an, ad, bn, bd))
    }
    fn divide_large_tier(&self, other: &Self) -> Result<Self, OverflowDetected> {
        let (an, ad) = self.extract_i128_pair()?;
        let (bn, bd) = other.extract_i128_pair()?;
        Self::from_i128_result(div_i128_rational(an, ad, bn, bd))
    }

    fn add_huge_tier(&self, other: &Self) -> Result<Self, OverflowDetected> {
        let (an, ad) = self.extract_i128_pair()?;
        let (bn, bd) = other.extract_i128_pair()?;
        Self::from_i128_result(add_i128_rational(an, ad, bn, bd))
    }
    fn subtract_huge_tier(&self, other: &Self) -> Result<Self, OverflowDetected> {
        let (an, ad) = self.extract_i128_pair()?;
        let (bn, bd) = other.extract_i128_pair()?;
        Self::from_i128_result(sub_i128_rational(an, ad, bn, bd))
    }
    fn multiply_huge_tier(&self, other: &Self) -> Result<Self, OverflowDetected> {
        let (an, ad) = self.extract_i128_pair()?;
        let (bn, bd) = other.extract_i128_pair()?;
        Self::from_i128_result(mul_i128_rational(an, ad, bn, bd))
    }
    fn divide_huge_tier(&self, other: &Self) -> Result<Self, OverflowDetected> {
        let (an, ad) = self.extract_i128_pair()?;
        let (bn, bd) = other.extract_i128_pair()?;
        Self::from_i128_result(div_i128_rational(an, ad, bn, bd))
    }
    
    // ═══ Massive tier (I256) arithmetic — uses I512 intermediates ═══

    #[cfg(not(feature = "embedded"))]
    fn extract_i256_pair(&self) -> Result<(I256, I256), OverflowDetected> {
        match &self.storage {
            RationalStorage::Massive { num, den } => Ok((*num, *den)),
            _ => {
                // Promote from smaller tier
                let parts = self.extract_native();
                if let Some((n, d)) = parts.try_as_i128() {
                    Ok((I256::from_i128(n), I256::from_i128(d as i128)))
                } else {
                    Err(OverflowDetected::TierOverflow)
                }
            }
        }
    }

    #[cfg(not(feature = "embedded"))]
    fn add_massive_tier(&self, other: &Self) -> Result<Self, OverflowDetected> {
        let (an, ad) = self.extract_i256_pair()?;
        let (bn, bd) = other.extract_i256_pair()?;
        // a/b + c/d = (a*d + c*b) / (b*d) using I512 intermediates
        let an_w = I512::from_i256(an);
        let ad_w = I512::from_i256(ad);
        let bn_w = I512::from_i256(bn);
        let bd_w = I512::from_i256(bd);
        let num = an_w * bd_w + bn_w * ad_w;
        let den = ad_w * bd_w;
        // GCD reduce then downscale
        let g = gcd_i512(num, den);
        let rn = num / g;
        let rd = den / g;
        // Check fits in I256
        if rn.fits_in_i256() && rd.fits_in_i256() {
            Ok(Self::from_i256_pair(rn.as_i256(), rd.as_i256()))
        } else {
            Err(OverflowDetected::TierOverflow)
        }
    }

    #[cfg(not(feature = "embedded"))]
    fn subtract_massive_tier(&self, other: &Self) -> Result<Self, OverflowDetected> {
        let (an, ad) = self.extract_i256_pair()?;
        let (bn, bd) = other.extract_i256_pair()?;
        let an_w = I512::from_i256(an);
        let ad_w = I512::from_i256(ad);
        let bn_w = I512::from_i256(bn);
        let bd_w = I512::from_i256(bd);
        let num = an_w * bd_w - bn_w * ad_w;
        let den = ad_w * bd_w;
        let g = gcd_i512(num, den);
        let rn = num / g;
        let rd = den / g;
        if rn.fits_in_i256() && rd.fits_in_i256() {
            Ok(Self::from_i256_pair(rn.as_i256(), rd.as_i256()))
        } else {
            Err(OverflowDetected::TierOverflow)
        }
    }

    #[cfg(not(feature = "embedded"))]
    fn multiply_massive_tier(&self, other: &Self) -> Result<Self, OverflowDetected> {
        let (an, ad) = self.extract_i256_pair()?;
        let (bn, bd) = other.extract_i256_pair()?;
        // Cross-GCD reduction before multiplication: (a/ad) * (b/bd)
        // g1 = gcd(a, bd), g2 = gcd(b, ad) → reduces intermediate sizes
        let g1 = gcd_i256(an, bd);
        let g2 = gcd_i256(bn, ad);
        let an_r = an / g1;
        let bd_r = bd / g1;
        let bn_r = bn / g2;
        let ad_r = ad / g2;
        let an_w = I512::from_i256(an_r);
        let ad_w = I512::from_i256(ad_r);
        let bn_w = I512::from_i256(bn_r);
        let bd_w = I512::from_i256(bd_r);
        let num = an_w * bn_w;
        let den = ad_w * bd_w;
        // Post-multiply GCD (usually 1 after cross-reduction, but be safe)
        let g = gcd_i512(num, den);
        let rn = num / g;
        let rd = den / g;
        if rn.fits_in_i256() && rd.fits_in_i256() {
            Ok(Self::from_i256_pair(rn.as_i256(), rd.as_i256()))
        } else {
            Err(OverflowDetected::TierOverflow)
        }
    }

    #[cfg(not(feature = "embedded"))]
    fn divide_massive_tier(&self, other: &Self) -> Result<Self, OverflowDetected> {
        let (an, ad) = self.extract_i256_pair()?;
        let (bn, bd) = other.extract_i256_pair()?;
        if bn.is_zero() { return Err(OverflowDetected::TierOverflow); }
        // (a/b) / (c/d) = (a*d) / (b*c)
        let an_w = I512::from_i256(an);
        let ad_w = I512::from_i256(ad);
        let bn_w = I512::from_i256(bn);
        let bd_w = I512::from_i256(bd);
        let num = an_w * bd_w;
        let den = ad_w * bn_w;
        let g = gcd_i512(num, den);
        let rn = num / g;
        let rd = den / g;
        // Normalize sign: denominator should be positive
        let (rn, rd) = if rd.is_negative() { (-rn, -rd) } else { (rn, rd) };
        if rn.fits_in_i256() && rd.fits_in_i256() {
            Ok(Self::from_i256_pair(rn.as_i256(), rd.as_i256()))
        } else {
            Err(OverflowDetected::TierOverflow)
        }
    }

    // ═══ Ultra tier (I512) arithmetic — uses I1024 intermediates ═══

    fn extract_i512_pair(&self) -> Result<(I512, I512), OverflowDetected> {
        match &self.storage {
            RationalStorage::Ultra { num, den } => Ok((*num, *den)),
            #[cfg(not(feature = "embedded"))]
            RationalStorage::Massive { num, den } => {
                Ok((I512::from_i256(*num), I512::from_i256(*den)))
            }
            _ => {
                let parts = self.extract_native();
                if let Some((n, d)) = parts.try_as_i128() {
                    Ok((I512::from_i128(n), I512::from_i128(d as i128)))
                } else {
                    Err(OverflowDetected::TierOverflow)
                }
            }
        }
    }

    fn add_ultra_tier(&self, other: &Self) -> Result<Self, OverflowDetected> {
        let (an, ad) = self.extract_i512_pair()?;
        let (bn, bd) = other.extract_i512_pair()?;
        let an_w = I1024::from_i512(an);
        let ad_w = I1024::from_i512(ad);
        let bn_w = I1024::from_i512(bn);
        let bd_w = I1024::from_i512(bd);
        let num = an_w * bd_w + bn_w * ad_w;
        let den = ad_w * bd_w;
        let g = gcd_i1024(num, den);
        let rn = num / g;
        let rd = den / g;
        if rn.fits_in_i512() && rd.fits_in_i512() {
            Ok(Self::from_i512_pair(rn.as_i512(), rd.as_i512()))
        } else {
            Err(OverflowDetected::TierOverflow)
        }
    }

    fn subtract_ultra_tier(&self, other: &Self) -> Result<Self, OverflowDetected> {
        let (an, ad) = self.extract_i512_pair()?;
        let (bn, bd) = other.extract_i512_pair()?;
        let an_w = I1024::from_i512(an);
        let ad_w = I1024::from_i512(ad);
        let bn_w = I1024::from_i512(bn);
        let bd_w = I1024::from_i512(bd);
        let num = an_w * bd_w - bn_w * ad_w;
        let den = ad_w * bd_w;
        let g = gcd_i1024(num, den);
        let rn = num / g;
        let rd = den / g;
        if rn.fits_in_i512() && rd.fits_in_i512() {
            Ok(Self::from_i512_pair(rn.as_i512(), rd.as_i512()))
        } else {
            Err(OverflowDetected::TierOverflow)
        }
    }

    fn multiply_ultra_tier(&self, other: &Self) -> Result<Self, OverflowDetected> {
        let (an, ad) = self.extract_i512_pair()?;
        let (bn, bd) = other.extract_i512_pair()?;
        // Cross-GCD reduction before multiplication
        let g1 = gcd_i512(an, bd);
        let g2 = gcd_i512(bn, ad);
        let an_r = an / g1;
        let bd_r = bd / g1;
        let bn_r = bn / g2;
        let ad_r = ad / g2;
        let an_w = I1024::from_i512(an_r);
        let ad_w = I1024::from_i512(ad_r);
        let bn_w = I1024::from_i512(bn_r);
        let bd_w = I1024::from_i512(bd_r);
        let num = an_w * bn_w;
        let den = ad_w * bd_w;
        let g = gcd_i1024(num, den);
        let rn = num / g;
        let rd = den / g;
        if rn.fits_in_i512() && rd.fits_in_i512() {
            Ok(Self::from_i512_pair(rn.as_i512(), rd.as_i512()))
        } else {
            Err(OverflowDetected::TierOverflow)
        }
    }

    fn divide_ultra_tier(&self, other: &Self) -> Result<Self, OverflowDetected> {
        let (an, ad) = self.extract_i512_pair()?;
        let (bn, bd) = other.extract_i512_pair()?;
        if bn.is_zero() { return Err(OverflowDetected::TierOverflow); }
        let an_w = I1024::from_i512(an);
        let ad_w = I1024::from_i512(ad);
        let bn_w = I1024::from_i512(bn);
        let bd_w = I1024::from_i512(bd);
        let num = an_w * bd_w;
        let den = ad_w * bn_w;
        let g = gcd_i1024(num, den);
        let rn = num / g;
        let rd = den / g;
        let (rn, rd) = if (rd.words[15] as i64) < 0 { (-rn, -rd) } else { (rn, rd) };
        if rn.fits_in_i512() && rd.fits_in_i512() {
            Ok(Self::from_i512_pair(rn.as_i512(), rd.as_i512()))
        } else {
            Err(OverflowDetected::TierOverflow)
        }
    }

    /// Extract exact rational as BigInt pair. Prefer `extract_native()` for tiers 1-7.
    /// - Tier 8 (BigInt) with `#[cfg(feature = "infinite-precision")]` - ONLY valid use
    /// - Fallback in mixed-tier comparisons when tier-preserving extraction fails (extremely rare)
    ///
    /// **MIGRATION**: If you're using this for tiers 1-7, switch to `extract_native()` or `try_as_i128()`.
    #[cfg(feature = "infinite-precision")]
    pub(crate) fn extract_exact_rational(&self) -> (BigInt, BigInt) {
        match &self.storage {
            // Tiers 1-7: Convert to BigInt (PERFORMANCE WARNING - avoid if possible)
            RationalStorage::Tiny { num, den } => (BigInt::from(*num), BigInt::from(*den)),
            RationalStorage::Small { num, den } => (BigInt::from(*num), BigInt::from(*den)),
            RationalStorage::Medium { num, den } => (BigInt::from(*num), BigInt::from(*den)),
            RationalStorage::Large { num, den } => (BigInt::from(*num), BigInt::from(*den)),
            RationalStorage::Huge { num, den } => (BigInt::from(*num), BigInt::from(*den)),
            #[cfg(not(feature = "embedded"))]
            RationalStorage::Massive { num, den } => {
                // I256::as_i128() returns i128 directly (not Option)
                (BigInt::from(num.as_i128()), BigInt::from(den.as_i128()))
            },
            RationalStorage::Ultra { num, den } => {
                (BigInt::from(num.as_i128()), BigInt::from(den.as_i128()))
            },
            // Tier 8: Proper use case
            RationalStorage::Infinite { num, den } => (num.clone(), den.clone()),
        }
    }

    // extract_exact_rational() only exists with infinite-precision feature
    // For tiers 1-7, use extract_as_i128() instead
    
    /// Addition with overflow detection for universal wrapper coordination
    /// INTEGRATION V2: Delegates to symbolic_arithmetic mathematical engine
    pub fn try_add(&self, other: &Self) -> Result<Self, OverflowDetected> {
        let result_tier = Self::predict_addition_tier(&self.storage, &other.storage);

        // Check compile-time tier constraints (GLOBAL_FLAGS may not be initialized)
        let max_allowed_tier = RationalStorage::max_tier_for_profile();

        if result_tier > max_allowed_tier {
            return Err(OverflowDetected::PrecisionLimit);
        }
        
        // DELEGATION: Route to appropriate symbolic_arithmetic mathematical functions
        let math_result = match (&self.storage, &other.storage) {
            // Tiny + Tiny operations → delegate to symbolic_arithmetic::add_i8_rational
            (RationalStorage::Tiny { num: a_num, den: a_den },
             RationalStorage::Tiny { num: b_num, den: b_den }) => {
                add_i8_rational(*a_num, *a_den, *b_num, *b_den)
                    .map(|(num, den)| (num as i128, den as u128))
            },
            
            // Small + Small operations → delegate to symbolic_arithmetic::add_i16_rational
            (RationalStorage::Small { num: a_num, den: a_den },
             RationalStorage::Small { num: b_num, den: b_den }) => {
                add_i16_rational(*a_num, *a_den, *b_num, *b_den)
                    .map(|(num, den)| (num as i128, den as u128))
            },
            
            // Medium + Medium operations → delegate to symbolic_arithmetic::add_i32_rational
            (RationalStorage::Medium { num: a_num, den: a_den },
             RationalStorage::Medium { num: b_num, den: b_den }) => {
                add_i32_rational(*a_num, *a_den, *b_num, *b_den)
                    .map(|(num, den)| (num as i128, den as u128))
            },
            
            // Large + Large operations → delegate to symbolic_arithmetic::add_i64_rational  
            (RationalStorage::Large { num: a_num, den: a_den },
             RationalStorage::Large { num: b_num, den: b_den }) => {
                add_i64_rational(*a_num, *a_den, *b_num, *b_den)
                    .map(|(num, den)| (num as i128, den as u128))
            },
            
            // Huge + Huge operations → delegate to symbolic_arithmetic::add_i128_rational
            (RationalStorage::Huge { num: a_num, den: a_den },
             RationalStorage::Huge { num: b_num, den: b_den }) => {
                add_i128_rational(*a_num, *a_den, *b_num, *b_den)
            },
            
            // For mixed tiers, promote both to higher tier and delegate without precision loss
            _ => {
                return self.add_mixed_tiers(other);
            }
        };
        
        // COORDINATION: Handle mathematical result with UGOD tier promotion
        match math_result {
            Ok((result_num, result_den)) => {
                Ok(Self::new(result_num, result_den))
            },
            Err(OverflowDetected::TierOverflow) => {
                // UGOD promotion: recompute at i128 using native checked arithmetic (FASC-compliant)
                let (a_num, a_den) = self.extract_i128_pair()?;
                let (b_num, b_den) = other.extract_i128_pair()?;
                let b_den_s = i128::try_from(b_den).map_err(|_| OverflowDetected::TierOverflow)?;
                let a_den_s = i128::try_from(a_den).map_err(|_| OverflowDetected::TierOverflow)?;
                let a_exp = a_num.checked_mul(b_den_s).ok_or(OverflowDetected::TierOverflow)?;
                let b_exp = b_num.checked_mul(a_den_s).ok_or(OverflowDetected::TierOverflow)?;
                let num = a_exp.checked_add(b_exp).ok_or(OverflowDetected::TierOverflow)?;
                let den = a_den.checked_mul(b_den).ok_or(OverflowDetected::TierOverflow)?;
                let g = gcd_unsigned(num.unsigned_abs(), den);
                Ok(Self::new(num / (g as i128), den / g))
            },
            Err(e) => Err(e),
        }
    }

    /// Subtraction with overflow detection for universal wrapper coordination
    /// INTEGRATION V2: Delegates to symbolic_arithmetic mathematical engine
    pub fn try_subtract(&self, other: &Self) -> Result<Self, OverflowDetected> {
        let result_tier = Self::predict_subtraction_tier(&self.storage, &other.storage);
        
        // Check deployment profile constraints  
        let max_allowed_tier = RationalStorage::max_tier_for_profile();
        
        if result_tier > max_allowed_tier {
            return Err(OverflowDetected::PrecisionLimit);
        }
        
        // DELEGATION: Route to appropriate symbolic_arithmetic mathematical functions
        let math_result = match (&self.storage, &other.storage) {
            // Tiny - Tiny operations → delegate to symbolic_arithmetic::sub_i8_rational
            (RationalStorage::Tiny { num: a_num, den: a_den },
             RationalStorage::Tiny { num: b_num, den: b_den }) => {
                sub_i8_rational(*a_num, *a_den, *b_num, *b_den)
                    .map(|(num, den)| (num as i128, den as u128))
            },
            
            // Small - Small operations → delegate to symbolic_arithmetic::sub_i16_rational
            (RationalStorage::Small { num: a_num, den: a_den },
             RationalStorage::Small { num: b_num, den: b_den }) => {
                sub_i16_rational(*a_num, *a_den, *b_num, *b_den)
                    .map(|(num, den)| (num as i128, den as u128))
            },
            
            // Medium - Medium operations → delegate to symbolic_arithmetic::sub_i32_rational
            (RationalStorage::Medium { num: a_num, den: a_den },
             RationalStorage::Medium { num: b_num, den: b_den }) => {
                sub_i32_rational(*a_num, *a_den, *b_num, *b_den)
                    .map(|(num, den)| (num as i128, den as u128))
            },
            
            // Large - Large operations → delegate to symbolic_arithmetic::sub_i64_rational  
            (RationalStorage::Large { num: a_num, den: a_den },
             RationalStorage::Large { num: b_num, den: b_den }) => {
                sub_i64_rational(*a_num, *a_den, *b_num, *b_den)
                    .map(|(num, den)| (num as i128, den as u128))
            },
            
            // Huge - Huge operations → delegate to symbolic_arithmetic::sub_i128_rational
            (RationalStorage::Huge { num: a_num, den: a_den },
             RationalStorage::Huge { num: b_num, den: b_den }) => {
                sub_i128_rational(*a_num, *a_den, *b_num, *b_den)
            },
            
            // For mixed tiers, promote both to higher tier and delegate without precision loss
            _ => {
                return self.subtract_mixed_tiers(other);
            }
        };

        // COORDINATION: Handle mathematical result with UGOD tier promotion
        match math_result {
            Ok((result_num, result_den)) => {
                Ok(Self::new(result_num, result_den))
            },
            Err(OverflowDetected::TierOverflow) => {
                // UGOD promotion: recompute at i128 using native checked arithmetic (FASC-compliant)
                let (a_num, a_den) = self.extract_i128_pair()?;
                let (b_num, b_den) = other.extract_i128_pair()?;
                let b_den_s = i128::try_from(b_den).map_err(|_| OverflowDetected::TierOverflow)?;
                let a_den_s = i128::try_from(a_den).map_err(|_| OverflowDetected::TierOverflow)?;
                let a_exp = a_num.checked_mul(b_den_s).ok_or(OverflowDetected::TierOverflow)?;
                let b_exp = b_num.checked_mul(a_den_s).ok_or(OverflowDetected::TierOverflow)?;
                let num = a_exp.checked_sub(b_exp).ok_or(OverflowDetected::TierOverflow)?;
                let den = a_den.checked_mul(b_den).ok_or(OverflowDetected::TierOverflow)?;
                let g = gcd_unsigned(num.unsigned_abs(), den);
                Ok(Self::new(num / (g as i128), den / g))
            },
            Err(e) => Err(e),
        }
    }
    
    /// Division with overflow detection for universal wrapper coordination
    /// INTEGRATION V2: Delegates to symbolic_arithmetic mathematical engine
    pub fn try_divide(&self, other: &Self) -> Result<Self, OverflowDetected> {
        let result_tier = Self::predict_division_tier(&self.storage, &other.storage);
        
        // Check deployment profile constraints  
        let max_allowed_tier = RationalStorage::max_tier_for_profile();
        
        if result_tier > max_allowed_tier {
            return Err(OverflowDetected::PrecisionLimit);
        }
        
        // DELEGATION: Route to appropriate symbolic_arithmetic mathematical functions
        let math_result = match (&self.storage, &other.storage) {
            // Tiny ÷ Tiny operations → delegate to symbolic_arithmetic::div_i8_rational
            (RationalStorage::Tiny { num: a_num, den: a_den },
             RationalStorage::Tiny { num: b_num, den: b_den }) => {
                div_i8_rational(*a_num, *a_den, *b_num, *b_den)
                    .map(|(num, den)| (num as i128, den as u128))
            },
            
            // Small ÷ Small operations → delegate to symbolic_arithmetic::div_i16_rational
            (RationalStorage::Small { num: a_num, den: a_den },
             RationalStorage::Small { num: b_num, den: b_den }) => {
                div_i16_rational(*a_num, *a_den, *b_num, *b_den)
                    .map(|(num, den)| (num as i128, den as u128))
            },
            
            // Medium ÷ Medium operations → delegate to symbolic_arithmetic::div_i32_rational
            (RationalStorage::Medium { num: a_num, den: a_den },
             RationalStorage::Medium { num: b_num, den: b_den }) => {
                div_i32_rational(*a_num, *a_den, *b_num, *b_den)
                    .map(|(num, den)| (num as i128, den as u128))
            },
            
            // Large ÷ Large operations → delegate to symbolic_arithmetic::div_i64_rational  
            (RationalStorage::Large { num: a_num, den: a_den },
             RationalStorage::Large { num: b_num, den: b_den }) => {
                div_i64_rational(*a_num, *a_den, *b_num, *b_den)
                    .map(|(num, den)| (num as i128, den as u128))
            },
            
            // Huge ÷ Huge operations → delegate to symbolic_arithmetic::div_i128_rational
            (RationalStorage::Huge { num: a_num, den: a_den },
             RationalStorage::Huge { num: b_num, den: b_den }) => {
                div_i128_rational(*a_num, *a_den, *b_num, *b_den)
            },

            // Massive ÷ Massive operations (I256 rational division)
            #[cfg(not(feature = "embedded"))]
            (RationalStorage::Massive { .. }, RationalStorage::Massive { .. }) => {
                // Delegate to mixed_tiers for I256 handling
                return self.divide_mixed_tiers(other);
            },

            // Ultra ÷ Ultra operations (I512 rational division)
            (RationalStorage::Ultra { .. }, RationalStorage::Ultra { .. }) => {
                // Delegate to mixed_tiers for I512 handling
                return self.divide_mixed_tiers(other);
            },

            // For mixed tiers, promote both to higher tier and delegate without precision loss
            _ => {
                return self.divide_mixed_tiers(other);
            }
        };

        // COORDINATION: Handle mathematical result with UGOD tier promotion
        match math_result {
            Ok((result_num, result_den)) => {
                Ok(Self::new(result_num, result_den))
            },
            Err(OverflowDetected::TierOverflow) => {
                // UGOD promotion: recompute at i128 using native checked arithmetic (FASC-compliant)
                let (a_num, a_den) = self.extract_i128_pair()?;
                let (b_num, b_den) = other.extract_i128_pair()?;
                if b_num == 0 { return Err(OverflowDetected::PrecisionLoss); }
                // (a/a_den) ÷ (b/b_den) = (a * b_den) / (a_den * |b|)
                let b_den_s = i128::try_from(b_den).map_err(|_| OverflowDetected::TierOverflow)?;
                let mut num = a_num.checked_mul(b_den_s).ok_or(OverflowDetected::TierOverflow)?;
                if b_num < 0 { num = num.checked_neg().ok_or(OverflowDetected::TierOverflow)?; }
                let den = a_den.checked_mul(b_num.unsigned_abs()).ok_or(OverflowDetected::TierOverflow)?;
                let g = gcd_unsigned(num.unsigned_abs(), den);
                Ok(Self::new(num / (g as i128), den / g))
            },
            Err(e) => Err(e),
        }
    }
    
    /// Negation with overflow detection for universal wrapper coordination
    pub fn try_negate(&self) -> Result<Self, OverflowDetected> {
        match &self.storage {
            RationalStorage::Tiny { num, den } => {
                if *num == i8::MIN {
                    Err(OverflowDetected::TierOverflow) // -128 → +128 overflows i8
                } else {
                    Ok(Self { 
                        storage: RationalStorage::Tiny { num: -num, den: *den }, 
                        metadata: self.metadata,
                    })
                }
            },
            RationalStorage::Small { num, den } => {
                if *num == i16::MIN {
                    Err(OverflowDetected::TierOverflow) // i16::MIN negation overflow
                } else {
                    Ok(Self { 
                        storage: RationalStorage::Small { num: -num, den: *den }, 
                        metadata: self.metadata,
                    })
                }
            },
            RationalStorage::Medium { num, den } => {
                if *num == i32::MIN {
                    Err(OverflowDetected::TierOverflow) // i32::MIN negation overflow
                } else {
                    Ok(Self { 
                        storage: RationalStorage::Medium { num: -num, den: *den }, 
                        metadata: self.metadata,
                    })
                }
            },
            RationalStorage::Large { num, den } => {
                if *num == i64::MIN {
                    Err(OverflowDetected::TierOverflow) // i64::MIN negation overflow
                } else {
                    Ok(Self { 
                        storage: RationalStorage::Large { num: -num, den: *den }, 
                        metadata: self.metadata,
                    })
                }
            },
            RationalStorage::Huge { num, den } => {
                if *num == i128::MIN {
                    Err(OverflowDetected::TierOverflow) // i128::MIN negation overflow
                } else {
                    Ok(Self { 
                        storage: RationalStorage::Huge { num: -num, den: *den }, 
                        metadata: self.metadata,
                    })
                }
            },
            #[cfg(not(feature = "embedded"))]
            RationalStorage::Massive { num, den } => {
                // For I256, use Neg trait (unary minus operator)
                Ok(Self {
                    storage: RationalStorage::Massive { num: -*num, den: *den },
                    metadata: self.metadata,
                })
            },
            RationalStorage::Ultra { num, den } => {
                // For I512, use Neg trait (unary minus operator)
                Ok(Self {
                    storage: RationalStorage::Ultra { num: -*num, den: *den },
                    metadata: self.metadata,
                })
            },
            #[cfg(feature = "infinite-precision")]
            RationalStorage::Infinite { num, den } => {
                Ok(Self { 
                    storage: RationalStorage::Infinite { num: -num, den: den.clone() }, 
                    metadata: self.metadata,
                })
            },
        }
    }
}

// ================================================================================================
// IMPLEMENTATION: Display and Comparison Traits
// ================================================================================================

impl fmt::Display for RationalNumber {
    fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
        // PRODUCTION: Use native types for display (NO BigInt for tiers 1-7)
        match &self.storage {
            RationalStorage::Tiny { num, den } => {
                if *den == 1 { write!(f, "{}", num) } else { write!(f, "{}/{}", num, den) }
            },
            RationalStorage::Small { num, den } => {
                if *den == 1 { write!(f, "{}", num) } else { write!(f, "{}/{}", num, den) }
            },
            RationalStorage::Medium { num, den } => {
                if *den == 1 { write!(f, "{}", num) } else { write!(f, "{}/{}", num, den) }
            },
            RationalStorage::Large { num, den } => {
                if *den == 1 { write!(f, "{}", num) } else { write!(f, "{}/{}", num, den) }
            },
            RationalStorage::Huge { num, den } => {
                if *den == 1 { write!(f, "{}", num) } else { write!(f, "{}/{}", num, den) }
            },
            #[cfg(not(feature = "embedded"))]
            RationalStorage::Massive { num, den } => {
                let num_i128 = num.as_i128();
                let den_i128 = den.as_i128();
                if den_i128 == 1 { write!(f, "{}", num_i128) } else { write!(f, "{}/{}", num_i128, den_i128) }
            },
            RationalStorage::Ultra { num, den } => {
                let num_i128 = num.as_i128();
                let den_i128 = den.as_i128();
                if den_i128 == 1 { write!(f, "{}", num_i128) } else { write!(f, "{}/{}", num_i128, den_i128) }
            },
            #[cfg(feature = "infinite-precision")]
            RationalStorage::Infinite { num, den } => {
                if den.is_one() { write!(f, "{}", num) } else { write!(f, "{}/{}", num, den) }
            },
        }
    }
}

impl PartialEq for RationalNumber {
    fn eq(&self, other: &Self) -> bool {
        // PRODUCTION: Compare rationals using native types (NO BigInt for tiers 1-7)
        // Compare a/b == c/d by checking a*d == b*c
        match (&self.storage, &other.storage) {
            // Same-tier comparisons with appropriate intermediates to prevent overflow
            (RationalStorage::Tiny { num: a, den: ad }, RationalStorage::Tiny { num: b, den: bd }) => {
                // i8 * u8 fits in i16
                ((*a as i16) * (*bd as i16)) == ((*b as i16) * (*ad as i16))
            },
            (RationalStorage::Small { num: a, den: ad }, RationalStorage::Small { num: b, den: bd }) => {
                // i16 * u16 fits in i32
                ((*a as i32) * (*bd as i32)) == ((*b as i32) * (*ad as i32))
            },
            (RationalStorage::Medium { num: a, den: ad }, RationalStorage::Medium { num: b, den: bd }) => {
                // i32 * u32 fits in i64
                ((*a as i64) * (*bd as i64)) == ((*b as i64) * (*ad as i64))
            },
            (RationalStorage::Large { num: a, den: ad }, RationalStorage::Large { num: b, den: bd }) => {
                // i64 * u64 fits in i128
                ((*a as i128) * (*bd as i128)) == ((*b as i128) * (*ad as i128))
            },
            (RationalStorage::Huge { num: a, den: ad }, RationalStorage::Huge { num: b, den: bd }) => {
                // i128 * u128 may overflow - use checked arithmetic
                match (a.checked_mul(*bd as i128), b.checked_mul(*ad as i128)) {
                    (Some(lhs), Some(rhs)) => lhs == rhs,
                    _ => {
                        // Overflow: compare using cross-multiplication with I256
                        let lhs_i256 = I256::from_i128(*a) * I256::from_i128(*bd as i128);
                        let rhs_i256 = I256::from_i128(*b) * I256::from_i128(*ad as i128);
                        lhs_i256 == rhs_i256
                    }
                }
            },
            #[cfg(not(feature = "embedded"))]
            (RationalStorage::Massive { num: a, den: ad }, RationalStorage::Massive { num: b, den: bd }) => {
                // Cross-multiply: a/ad == b/bd iff a*bd == b*ad
                // Use I512 intermediates to prevent overflow
                let lhs = I512::from_i256(*a) * I512::from_i256(*bd);
                let rhs = I512::from_i256(*b) * I512::from_i256(*ad);
                lhs == rhs
            },
            (RationalStorage::Ultra { num: a, den: ad }, RationalStorage::Ultra { num: b, den: bd }) => {
                // Cross-multiply: a/ad == b/bd iff a*bd == b*ad
                // Use I1024 intermediates to prevent overflow
                let lhs = I1024::from_i512(*a) * I1024::from_i512(*bd);
                let rhs = I1024::from_i512(*b) * I1024::from_i512(*ad);
                lhs == rhs
            },
            #[cfg(feature = "infinite-precision")]
            (RationalStorage::Infinite { num: a, den: ad }, RationalStorage::Infinite { num: b, den: bd }) => {
                // Tier 8: BigInt is appropriate here
                (a * bd) == (b * ad)
            },
            // Mixed-tier comparisons: use tier-preserving extraction
            _ => {
                // Try to extract both as i128 for efficient comparison
                if let (Some((a_num, a_den)), Some((b_num, b_den))) =
                    (self.extract_native().try_as_i128(), other.extract_native().try_as_i128())
                {
                    // Both fit in i128: use i128 comparison with checked arithmetic
                    match (a_num.checked_mul(b_den as i128), b_num.checked_mul(a_den as i128)) {
                        (Some(lhs), Some(rhs)) => lhs == rhs,
                        _ => {
                            // Overflow to I256
                            let lhs = I256::from_i128(a_num) * I256::from_u128(b_den);
                            let rhs = I256::from_i128(b_num) * I256::from_u128(a_den);
                            lhs == rhs
                        }
                    }
                } else {
                    // At least one doesn't fit in i128: fallback to BigInt comparison
                    #[cfg(feature = "infinite-precision")]
                    {
                        let (a_num, a_den) = self.extract_exact_rational();
                        let (b_num, b_den) = other.extract_exact_rational();
                        (&a_num * &b_den) == (&b_num * &a_den)
                    }
                    #[cfg(not(feature = "infinite-precision"))]
                    {
                        // Without BigInt tier, values that don't fit are unequal
                        false
                    }
                }
            }
        }
    }
}

impl Eq for RationalNumber {}

impl PartialOrd for RationalNumber {
    fn partial_cmp(&self, other: &Self) -> Option<Ordering> {
        Some(self.cmp(other))
    }
}

impl Ord for RationalNumber {
    fn cmp(&self, other: &Self) -> Ordering {
        // PRODUCTION: Compare rationals using native types (NO BigInt for tiers 1-7)
        // Compare a/b with c/d by comparing a*d with b*c
        match (&self.storage, &other.storage) {
            // Same-tier comparisons with appropriate intermediates
            (RationalStorage::Tiny { num: a, den: ad }, RationalStorage::Tiny { num: b, den: bd }) => {
                let lhs = (*a as i16) * (*bd as i16);
                let rhs = (*b as i16) * (*ad as i16);
                lhs.cmp(&rhs)
            },
            (RationalStorage::Small { num: a, den: ad }, RationalStorage::Small { num: b, den: bd }) => {
                let lhs = (*a as i32) * (*bd as i32);
                let rhs = (*b as i32) * (*ad as i32);
                lhs.cmp(&rhs)
            },
            (RationalStorage::Medium { num: a, den: ad }, RationalStorage::Medium { num: b, den: bd }) => {
                let lhs = (*a as i64) * (*bd as i64);
                let rhs = (*b as i64) * (*ad as i64);
                lhs.cmp(&rhs)
            },
            (RationalStorage::Large { num: a, den: ad }, RationalStorage::Large { num: b, den: bd }) => {
                let lhs = (*a as i128) * (*bd as i128);
                let rhs = (*b as i128) * (*ad as i128);
                lhs.cmp(&rhs)
            },
            (RationalStorage::Huge { num: a, den: ad }, RationalStorage::Huge { num: b, den: bd }) => {
                match (a.checked_mul(*bd as i128), b.checked_mul(*ad as i128)) {
                    (Some(lhs), Some(rhs)) => lhs.cmp(&rhs),
                    _ => {
                        let lhs_i256 = I256::from_i128(*a) * I256::from_i128(*bd as i128);
                        let rhs_i256 = I256::from_i128(*b) * I256::from_i128(*ad as i128);
                        lhs_i256.cmp(&rhs_i256)
                    }
                }
            },
            #[cfg(not(feature = "embedded"))]
            (RationalStorage::Massive { num: a, den: ad }, RationalStorage::Massive { num: b, den: bd }) => {
                let lhs = *a * *ad;
                let rhs = *b * *bd;
                lhs.cmp(&rhs)
            },
            (RationalStorage::Ultra { num: a, den: ad }, RationalStorage::Ultra { num: b, den: bd }) => {
                let lhs = *a * *ad;
                let rhs = *b * *bd;
                lhs.cmp(&rhs)
            },
            #[cfg(feature = "infinite-precision")]
            (RationalStorage::Infinite { num: a, den: ad }, RationalStorage::Infinite { num: b, den: bd }) => {
                let lhs = a * bd;
                let rhs = b * ad;
                lhs.cmp(&rhs)
            },
            // Mixed-tier comparisons: use tier-preserving extraction
            _ => {
                // Try to extract both as i128 for efficient comparison
                if let (Some((a_num, a_den)), Some((b_num, b_den))) =
                    (self.extract_native().try_as_i128(), other.extract_native().try_as_i128())
                {
                    match (a_num.checked_mul(b_den as i128), b_num.checked_mul(a_den as i128)) {
                        (Some(lhs), Some(rhs)) => lhs.cmp(&rhs),
                        _ => {
                            let lhs = I256::from_i128(a_num) * I256::from_u128(b_den);
                            let rhs = I256::from_i128(b_num) * I256::from_u128(a_den);
                            lhs.cmp(&rhs)
                        }
                    }
                } else {
                    #[cfg(feature = "infinite-precision")]
                    {
                        let (a_num, a_den) = self.extract_exact_rational();
                        let (b_num, b_den) = other.extract_exact_rational();
                        let lhs = &a_num * &b_den;
                        let rhs = &b_num * &a_den;
                        lhs.cmp(&rhs)
                    }
                    #[cfg(not(feature = "infinite-precision"))]
                    {
                        // Without BigInt tier, compare what we can
                        Ordering::Equal // Fallback
                    }
                }
            }
        }
    }
}

// ================================================================================================
// IMPLEMENTATION: Arithmetic Operator Traits
// ================================================================================================

impl std::ops::Add for RationalNumber {
    type Output = Self;

    fn add(self, other: Self) -> Self {
        self.try_add(&other).expect("RationalNumber addition overflow - consider using try_add for fallible operations")
    }
}

impl std::ops::Sub for RationalNumber {
    type Output = Self;

    fn sub(self, other: Self) -> Self {
        self.try_subtract(&other).expect("RationalNumber subtraction overflow - consider using try_subtract for fallible operations")
    }
}

impl std::ops::Mul for RationalNumber {
    type Output = Self;

    fn mul(self, other: Self) -> Self {
        self.try_multiply(&other).expect("RationalNumber multiplication overflow - consider using try_multiply for fallible operations")
    }
}

impl std::ops::Div for RationalNumber {
    type Output = Self;

    fn div(self, other: Self) -> Self {
        self.try_divide(&other).expect("RationalNumber division error - consider using try_divide for fallible operations")
    }
}

impl std::ops::Neg for RationalNumber {
    type Output = Self;

    fn neg(self) -> Self {
        self.try_negate().expect("RationalNumber negation error - tier overflow")
    }
}

// ================================================================================================
// IMPLEMENTATION: Numeric Traits (available when infinite-precision feature is enabled)
// ================================================================================================

#[cfg(feature = "infinite-precision")]
impl Zero for RationalNumber {
    fn zero() -> Self {
        Self::zero()
    }

    fn is_zero(&self) -> bool {
        self.is_zero()
    }
}

#[cfg(feature = "infinite-precision")]
impl One for RationalNumber {
    fn one() -> Self {
        Self::one()
    }

    fn is_one(&self) -> bool {
        self.is_one()
    }
}

impl RationalNumber {
    /// Create the rational number 0/1
    pub fn zero() -> Self {
        Self::new(0, 1)
    }

    /// Check if this rational number is zero (numerator == 0)
    pub fn is_zero(&self) -> bool {
        match &self.storage {
            RationalStorage::Tiny { num, .. } => *num == 0,
            RationalStorage::Small { num, .. } => *num == 0,
            RationalStorage::Medium { num, .. } => *num == 0,
            RationalStorage::Large { num, .. } => *num == 0,
            RationalStorage::Huge { num, .. } => *num == 0,
            #[cfg(not(feature = "embedded"))]
            RationalStorage::Massive { num, .. } => num.is_zero(),
            RationalStorage::Ultra { num, .. } => num.is_zero(),
            #[cfg(feature = "infinite-precision")]
            RationalStorage::Infinite { num, .. } => num == &BigInt::from(0),
        }
    }

    /// Check if this rational number represents an integer (denominator == 1)
    pub fn is_integer(&self) -> bool {
        match &self.storage {
            RationalStorage::Tiny { den, .. } => *den == 1,
            RationalStorage::Small { den, .. } => *den == 1,
            RationalStorage::Medium { den, .. } => *den == 1,
            RationalStorage::Large { den, .. } => *den == 1,
            RationalStorage::Huge { den, .. } => *den == 1,
            #[cfg(not(feature = "embedded"))]
            RationalStorage::Massive { den, .. } => den.as_i128() == 1,
            RationalStorage::Ultra { den, .. } => den.as_i128() == 1,
            #[cfg(feature = "infinite-precision")]
            RationalStorage::Infinite { den, .. } => den == &BigInt::from(1),
        }
    }

    /// Create the rational number 1/1
    pub fn one() -> Self {
        Self::new(1, 1)
    }

    /// Check if this rational number equals 1
    pub fn is_one(&self) -> bool {
        match &self.storage {
            RationalStorage::Tiny { num, den } => *num == 1 && *den == 1,
            RationalStorage::Small { num, den } => *num == 1 && *den == 1,
            RationalStorage::Medium { num, den } => *num == 1 && *den == 1,
            RationalStorage::Large { num, den } => *num == 1 && *den == 1,
            RationalStorage::Huge { num, den } => *num == 1 && *den == 1,
            #[cfg(not(feature = "embedded"))]
            RationalStorage::Massive { num, den } => num.as_i128() == 1 && den.as_i128() == 1,
            RationalStorage::Ultra { num, den } => num.as_i128() == 1 && den.as_i128() == 1,
            #[cfg(feature = "infinite-precision")]
            RationalStorage::Infinite { num, den } => num == &BigInt::from(1) && den == &BigInt::from(1),
        }
    }

    /// Integer exponentiation: (a/b)^n
    ///
    /// Supports negative exponents: (a/b)^(-n) = (b/a)^n
    /// **EXACT**: No precision loss: pure integer arithmetic
    pub fn pow(self, exponent: i32) -> Self {
        if exponent == 0 {
            return Self::one();
        }
        if exponent == 1 {
            return self;
        }

        // For negative exponents, reciprocal first
        let (base, exp) = if exponent < 0 {
            let recip = Self::new(
                if self.numerator_i128().unwrap_or(0) < 0 {
                    -(self.denominator_i128().unwrap_or(1) as i128)
                } else {
                    self.denominator_i128().unwrap_or(1) as i128
                },
                self.numerator_i128().unwrap_or(1).unsigned_abs() as u128,
            );
            (recip, (-exponent) as u32)
        } else {
            (self, exponent as u32)
        };

        // Exponentiation by squaring
        let mut result = Self::one();
        let mut b = base;
        let mut e = exp;
        while e > 0 {
            if e & 1 == 1 {
                result = result.try_multiply(&b).unwrap_or(result);
            }
            e >>= 1;
            if e > 0 {
                b = b.clone().try_multiply(&b).unwrap_or(b);
            }
        }
        result
    }
}

// ================================================================================================
// IMPLEMENTATION: Universal Tiered Arithmetic Integration (UGOD)
// ================================================================================================

impl UniversalTieredArithmetic for RationalNumber {
    type Error = OverflowDetected;

    /// Addition with UGOD overflow detection
    fn try_add(&self, other: &Self) -> Result<Self, Self::Error> {
        // Use existing rational arithmetic implementation with proper UGOD integration
        RationalNumber::try_add(self, other)
    }

    /// Subtraction with UGOD overflow detection
    fn try_subtract(&self, other: &Self) -> Result<Self, Self::Error> {
        RationalNumber::try_subtract(self, other)
    }

    /// Multiplication with UGOD overflow detection
    fn try_multiply(&self, other: &Self) -> Result<Self, Self::Error> {
        RationalNumber::try_multiply(self, other)
    }

    /// Division with UGOD overflow detection
    fn try_divide(&self, other: &Self) -> Result<Self, Self::Error> {
        RationalNumber::try_divide(self, other)
    }

    /// Negation with UGOD overflow detection
    fn try_negate(&self) -> Result<Self, Self::Error> {
        RationalNumber::try_negate(self)
    }

    /// Get current rational tier level
    fn current_tier(&self) -> u8 {
        self.tier_level()
    }

    /// Check if value can be promoted to target tier
    fn can_promote_to_tier(&self, tier: u8) -> bool {
        let current_tier = self.current_tier();
        let max_tier = Self::max_tier();
        tier <= max_tier && tier > current_tier
    }

    /// Promote to target tier within rational arithmetic hierarchy
    fn promote_to_tier(&self, tier: u8) -> Option<Self> {
        if !self.can_promote_to_tier(tier) {
            return None;
        }

        let parts = self.extract_native();

        // Tier 6 (Massive, I256): widen to I256 pair
        #[cfg(not(feature = "embedded"))]
        if tier == 6 {
            let (num, den) = parts.try_as_i256_pair()?;
            return Some(Self::from_i256_pair(num, den));
        }

        // Tier 7 (Ultra, I512): widen to I512 pair
        if tier == 7 {
            let (num, den) = parts.try_as_i512_pair()?;
            return Some(Self::from_i512_pair(num, den));
        }

        // Tiers 1-5: extract as i128/u128 and reconstruct (auto-selects optimal tier)
        let (num_i128, den_i128) = parts.try_as_i128()?;
        Some(Self::new(num_i128, den_i128))
    }

    /// Maximum tier supported by RationalNumber (all 7-8 tiers)
    fn max_tier() -> u8 {
        8 // Full rational arithmetic tier hierarchy including BigInt
    }

    /// RationalNumber domain type for UGOD coordination
    fn domain_type() -> UGODDomainType {
        UGODDomainType::Symbolic
    }

    /// Rational arithmetic symbolic tier mapping (1:1 mapping)
    fn symbolic_tier_mapping(symbolic_tier: u8) -> u8 {
        symbolic_tier // RationalNumber uses symbolic tiers directly
    }

    /// Check if RationalNumber can accommodate symbolic tier operations
    fn can_accommodate_symbolic_tier(&self, symbolic_tier: u8) -> bool {
        self.current_tier() >= symbolic_tier
    }

    /// Get deployment profile maximum tier limit for rational arithmetic
    fn max_tier_for_profile(profile: DeploymentProfile) -> u8 {
        match profile {
            DeploymentProfile::Realtime => 3,        // Minimal tiers for Q16.16
            DeploymentProfile::Compact => 4,         // Compact tiers for Q32.32
            DeploymentProfile::Embedded => 5,        // Conservative for embedded
            DeploymentProfile::Balanced => 7,        // Full tiers except BigInt
            DeploymentProfile::Scientific => 7,      // Full tiers for research
            DeploymentProfile::Custom => 8,          // Full tier hierarchy for custom
        }
    }
}

// ============================================================================
// UNIVERSAL TIER MAPPING: Symbolic (8 tiers) ↔ Universal (6 tiers)
// ============================================================================

/// Map symbolic internal tier (1-8) to universal tier (1-6)
///
/// Symbolic tiers use finer granularity (i8, i16 are separate tiers).
/// The universal system groups them into 6 tiers aligned with backing type sizes.
pub fn symbolic_to_universal_tier(symbolic_tier: u8) -> u8 {
    match symbolic_tier {
        1 | 2 => 1,  // Tiny(i8) + Small(i16) → Universal Tier 1 (small integers)
        3 => 2,       // Medium(i32) → Universal Tier 2
        4 => 3,       // Large(i64) → Universal Tier 3
        5 => 4,       // Huge(i128) → Universal Tier 4
        6 => 5,       // Massive(I256) → Universal Tier 5
        7 | 8 => 6,   // I512 + BigInt → Universal Tier 6
        _ => 6,       // Default to max
    }
}

/// Map universal tier (1-6) to the minimum symbolic tier needed
///
/// Used when creating a RationalNumber that must live at a specific universal tier.
pub fn universal_to_symbolic_tier(universal_tier: u8) -> u8 {
    match universal_tier {
        1 => 2,   // Universal 1 → Small(i16) minimum
        2 => 3,   // Universal 2 → Medium(i32)
        3 => 4,   // Universal 3 → Large(i64)
        4 => 5,   // Universal 4 → Huge(i128)
        5 => 6,   // Universal 5 → Massive(I256)
        6 => 7,   // Universal 6 → Ultra(I512)
        _ => 7,   // Default to Ultra
    }
}

// ═══ GCD helpers for extended-precision rational arithmetic ═══

/// GCD for I256 values using Euclidean algorithm (operates on absolute values)
#[cfg(not(feature = "embedded"))]
fn gcd_i256(a: I256, b: I256) -> I256 {
    let mut a = if a.is_negative() { -a } else { a };
    let mut b = if b.is_negative() { -b } else { b };
    let zero = I256::zero();
    while b != zero {
        let t = b;
        b = a % b;
        a = t;
    }
    if a == zero { I256::from_i128(1) } else { a }
}

/// GCD for I512 values using Euclidean algorithm (operates on absolute values)
#[cfg(not(feature = "embedded"))]
fn gcd_i512(a: I512, b: I512) -> I512 {
    let mut a = if a.is_negative() { -a } else { a };
    let mut b = if b.is_negative() { -b } else { b };
    let zero = I512::zero();
    while b != zero {
        let t = b;
        b = a % b;
        a = t;
    }
    if a == zero { I512::from_i128(1) } else { a }
}

/// GCD for I1024 values using Euclidean algorithm (operates on absolute values)
fn gcd_i1024(a: I1024, b: I1024) -> I1024 {
    let mut a = if (a.words[15] as i64) < 0 { -a } else { a };
    let mut b = if (b.words[15] as i64) < 0 { -b } else { b };
    let zero = I1024::zero();
    while b != zero {
        let t = b;
        b = a % b;
        a = t;
    }
    if a == zero { I1024::from_i128(1) } else { a }
}

#[cfg(test)]
mod tests {
    use super::*;
    
    #[test]
    fn test_tier_assignment() {
        let tiny = RationalNumber::new(1, 3);
        let small = RationalNumber::new(355, 113);
        let large = RationalNumber::new(1_000_000, 3_000_000);

        assert_eq!(tiny.tier_level(), 1);
        assert_eq!(small.tier_level(), 2);
        assert_eq!(large.tier_level(), 3);
    }

    #[test]
    fn test_overflow_detection() {
        let large_a = RationalNumber::new(i8::MAX as i128, 1);
        let large_b = RationalNumber::new(i8::MAX as i128, 1);

        // UGOD promotion: i8*i8 overflows Tiny tier but promotes to Small (i16)
        // 127 * 127 = 16129, fits in i16
        let result = large_a.try_multiply(&large_b);
        assert!(result.is_ok(), "UGOD should promote i8 overflow to higher tier");
        let product = result.unwrap();
        assert_eq!(product.extract_native().try_as_i128().unwrap().0, 16129);

        // multiply_at_max_tier also works, at highest available tier
        let max_tier_result = large_a.multiply_at_max_tier(&large_b);
        assert!(max_tier_result.tier_level() >= 2);  // At least Small tier
    }
    
    #[test]
    fn test_exact_arithmetic() {
        let a = RationalNumber::new(1, 3);
        let b = RationalNumber::new(2, 6);
        let c = RationalNumber::new(1, 2);

        assert_eq!(a, b);  // 1/3 == 2/6
        assert!(a < c);    // 1/3 < 1/2
        assert!(c > a);    // 1/2 > 1/3

        assert!(!a.is_zero());
        assert!(!a.is_one());

        let one = RationalNumber::one();
        assert!(one.is_one());

        let zero = RationalNumber::zero();
        assert!(zero.is_zero());
    }

    #[test]
    fn test_pow() {
        let base = RationalNumber::new(2, 3);
        assert_eq!(base.clone().pow(0), RationalNumber::one());
        assert_eq!(base.clone().pow(1), base);
        assert_eq!(base.clone().pow(2), RationalNumber::new(4, 9));
        assert_eq!(base.clone().pow(3), RationalNumber::new(8, 27));

        // Negative powers
        assert_eq!(base.clone().pow(-1), RationalNumber::new(3, 2));
        assert_eq!(base.pow(-2), RationalNumber::new(9, 4));
    }
}