Goldbach Checkpoint Verifier (4×10¹⁸ Scale)

High-performance Goldbach verification engine for extremely large ranges with:

• Checkpoint / Resume support
• QHot witness reuse (~99.9% hit rate)
• Anchor primes (bounded search)
• MR fallback
• Independent reconstruction checker

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What this does

Verifies that every even number N in a given range satisfies:

N = p + q   (p and q are prime)

This is computational verification, not a formal proof.

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Core Ideas

QHot Cache

• Reuses recent q values
• ~99.9% hit rate → most evens resolved instantly

Anchor Primes

• Only small primes p ≤ limit are tested
• Reduces search drastically

MR Fallback

• Used when QHot fails (~0.08%)
• Produces Miller–Rabin verified q values

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Example Run

Range: [4e18 .. 4e18 + 1e10]
Total evens: 5,000,000,001
QHot hits:   4,995,840,973
Sieve hits:  4,159,028
Missing:     0
Bad:         0

QHot ≈ 99.9168%
Sieve ≈ 0.0832%

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Repository Structure

src/
  goldbach_final.cpp
  goldbach_sieve_q_dump.cpp

tools/
  reconstruct_sieve_q_checker.cpp

data/
  checkpoints/
  samples/
  

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Build

g++ -O3 -march=native -std=gnu++17 -fopenmp src/goldbach_final.cpp -o goldbach

g++ -O3 -march=native -std=gnu++17 -fopenmp src/goldbach_sieve_q_dump.cpp -o goldbach_sieve_q_dump

g++ -O3 -march=native -std=gnu++17 tools/reconstruct_sieve_q_checker.cpp -o reconstruct_checker

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Run Verifier

./goldbach <start> <end> <block_bits> <threads> [p_anchor_limit] [sample_limit] [Qhot size]

Example:

./goldbach 4000000000000000000 4000010000000000000 24 12 100000 10000 256

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Resume from Checkpoint

./goldbach  0 0 24 12 100000 10000 256 --resume

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Sieve-Q Dump Mode

./goldbach_sieve_q_dump 4000000000000000000 4000000000001000000 24 1 100000 10000 --dump-sieve-q sieve_q_witness.csv

Output:

# adjusted_start=...
# adjusted_end=...
# p_anchor_limit=...
# qhot_size=1024
N,p,q

• Only fallback (~0.08%) stored
• Each q is MR-confirmed
• QHot hits are not stored

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Reconstruction Checker

./reconstruct_checker sieve_q_witness.csv

• Reconstructs all pairs using QHot + sieve data
• Validates:
  • p + q = N
  • p is anchor prime
  • q is prime (MR)

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Current Status

• Verified: up to 500 trillion evens (checkpointed)
• Misses: 0
• Reconstruction check: PASS

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Key Insight

Instead of storing all pairs (petabytes):

Store ~0.08% sieve witnesses → reconstruct 100% pairs

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Notes

• Not a formal mathematical proof
• Full output intentionally avoided (too large)
• Designed for scaling beyond 10^18

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License

MIT