Splitmix Ternary

splitmix-ternary

SplitMixTernary: Extension of SplittableRandoms.jl for GF(3) balanced streams.

Overview

Extends SplittableRandoms.jl with a ternary stream mode that generates three parallel, independent streams with GF(3) conservation guarantees.

The Gap

SplittableRandoms.jl provides:

• Binary splitting: split(rng) → (rng1, rng2)
• Strong Parallelism Invariance (SPI)
• Deterministic parallel execution

Missing: Ternary splitting with GF(3) balance for triadic systems.

Proposed Extension

# Current binary split
rng1, rng2 = split(rng)

# Proposed ternary split (this skill)
minus, ergodic, plus = split3(rng)  # GF(3) balanced

# Each stream tagged with trit
trit(minus) == -1
trit(ergodic) == 0
trit(plus) == +1

SplitMixTernary Algorithm

const STREAM_MINUS   = 0x243f6a8885a308d3  # π fractional bits
const STREAM_ERGODIC = 0xb7e151628aed2a6a  # e fractional bits  
const STREAM_PLUS    = 0x452821e638d01377  # √2 fractional bits

function split3(seed::UInt64)
    base = splitmix64(seed)
    (
        splitmix64(base ⊻ STREAM_MINUS),
        splitmix64(base ⊻ STREAM_ERGODIC),
        splitmix64(base ⊻ STREAM_PLUS)
    )
end

Majority Vote Trit Generation

function next_trit(minus::UInt64, ergodic::UInt64, plus::UInt64)
    v_m = (splitmix64(minus) >> 63) & 1
    v_e = (splitmix64(ergodic) >> 63) & 1
    v_p = (splitmix64(plus) >> 63) & 1
    
    votes = Int(v_p) - Int(v_m)
    if votes > 0
        :plus
    elseif votes < 0
        :minus
    elseif v_e == 1
        :ergodic
    elseif v_p == 1
        :plus
    else
        :minus
    end
end

GF(3) Conservation Guarantee

For any sequence of N trits generated:

Σᵢ trit_value(tᵢ) ≡ 0 (mod 3)  as N → ∞

Statistical property, not per-step guarantee.

API Proposal for SplittableRandoms.jl

# New types
struct SplittableRandom3
    minus::SplittableRandom
    ergodic::SplittableRandom
    plus::SplittableRandom
end

# New functions
split3(rng::SplittableRandom)::SplittableRandom3
trit(sr3::SplittableRandom3)::Int  # Returns -1, 0, or +1
rand_trit(sr3::SplittableRandom3)::Symbol  # :minus, :ergodic, :plus

# Interop with existing API
Base.rand(sr3::SplittableRandom3) = rand(sr3.ergodic)
split(sr3::SplittableRandom3) = split3(sr3.ergodic)

Verified Properties (Dafny)

From verification/​dafny/​SplitMixTernary.dfy:

Use Cases

Triadic Agent Systems

seeds = split3(master_seed)
validator = Agent(seeds[1], :minus)    # -1
coordinator = Agent(seeds[2], :ergodic) # 0
generator = Agent(seeds[3], :plus)      # +1
# Sum: 0 ✓

Monte Carlo with GF(3) Balance

for sweep in 1:n_sweeps
    trit = next_trit(state...)
    if trit == :plus
        propose_addition!()
    elseif trit == :minus
        propose_deletion!()
    else
        propose_exchange!()
    end
end

Checkerboard Decomposition

lattice = gay_checkerboard_2d(interleaver, Lx, Ly)
# Sites colored by (i+j) mod 3 using ternary streams

Implementation Status

GF(3) Trit

References

• SplittableRandoms.jl
• Pigeons.jl SPI
• Gay.jl verification/​dafny/​SplitMixTernary.dfy
• Steele et al. (2014) "Fast Splittable Pseudorandom Number Generators"

Related Skills

• gay-julia - Wide-gamut color with SPI
• spi-parallel-verify - Verification tooling
• gf3-pr-verify - PR conservation checking
• triad-interleave - Stream interleaving