Goldbach Midpoint Transitions and Twin–Cousin Rescue: A Computational Study to One Billion

Abstract

We study Goldbach representations about a moving pivot, distinguishing the loss of a

direct continuation from the loss of all representations. For pivots congruent to 1 modulo

3, direct extinction has an exact primality criterion. Dirichlet’s theorem then supplies

infinitely many such pivots. We conjecture that each extinction is rescued by a twin pair

and a prime pair at distance four, and show that this conjecture implies binary Goldbach

and infinitely many twin primes. An equivalent four-prime convolution leads to a stronger

residue-specific conjecture. An odd-only Eratosthenes sieve and a complete fallback search

test every pivot through 109: among 284,220,751 residue-specific targets, no failure occurs,

and no extinction pivot occurs in the other residue classes. A stricter right-hand orientation

already fails at 277. We give exact algorithms, finite counts, code, and the logical limits of

these observations; neither Goldbach nor the twin-prime conjecture is proved.

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BibTeX RIS

BibTeX

@misc{bilokon2026goldbach,
  title = {{Goldbach Midpoint Transitions and Twin–Cousin Rescue: A Computational Study to One Billion}},
  author = {Paul Alexander Bilokon},
  howpublished = {Zenodo},
  year = {2026},
  month = oct,
  doi = {10.5281/zenodo.23410422},
  url = {https://zenodo.org/records/23410422},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Goldbach Midpoint Transitions and Twin–Cousin Rescue: A Computational Study to One Billion
AU  - Paul Alexander Bilokon
PY  - 2026
DA  - 2026/10/16
PB  - Zenodo
DO  - 10.5281/zenodo.23410422
UR  - https://zenodo.org/records/23410422
AB  - We study Goldbach representations about a moving pivot, distinguishing the loss of a  direct continuation from the loss of all representations. For pivots congruent to 1 modulo  3, direct extinction has an exact primality criterion. Dirichlet’s theorem then supplies  infinitely many such pivots. We conjecture that each extinction is rescued by a twin pair  and a prime pair at distance four, and show that this conjecture implies binary Goldbach  and infinitely many twin primes. An equivalent four-prime convolution leads to a stronger  residue-specific conjecture. An odd-only Eratosthenes sieve and a complete fallback search  test every pivot through 109: among 284,220,751 residue-specific targets, no failure occurs,  and no extinction pivot occurs in the other residue classes. A stricter right-hand orientation  already fails at 277. We give exact algorithms, finite counts, code, and the logical limits of  these observations; neither Goldbach nor the twin-prime conjecture is proved.
ER  -

Abstract

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