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PreprintCC BY 4.0Zenodo

Sexy-Prime Bridges Across Goldbach Transitions: Exact Convolutions, Modular Extinction, Local Sieve Dynamics, and Conditional Covering Criteria

  • Paul Alexander Bilokon1
  1. 1Imperial College London
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Abstract

We investigate a proposed rescue mechanism for Goldbach representations at indices

n≡1pmod3 where the exceptional Goldbach summand $3$ is unavailable.

The proposed rescue uses a prime pair (q,q+6) and a complementary prime r,

so that q+r=2n−4 and (q+6)+r=2n+2. Writing n=3t+1 reduces the

problem to positivity of the discrete convolution

R(t)=∑a=1t−1f(a)f(a+1)f(t−a), where

f(a)=onePP(6a−1). We give exact gcd identities, the associated

singular series and its uniform positivity, elementary rough-number bounds,

finite local-covering certificates, and infinite families of prescribed

extinction and recovery. A conditional covariance criterion isolates the

additional additive information required to prove universal rescue.

Exact modular convolutions verify R(t)>0 for 2≤ t≤106.

The universal rescue assertion and the infinitude of sexy-prime pairs

remain open; the note does not claim a solution of either problem.

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Cite

BibTeXRIS
BibTeX
@misc{bilokon2026sexyprime,
  title = {{Sexy-Prime Bridges Across Goldbach Transitions: Exact Convolutions, Modular Extinction, Local Sieve Dynamics, and Conditional Covering Criteria}},
  author = {Paul Alexander Bilokon},
  howpublished = {Zenodo},
  year = {2026},
  month = oct,
  doi = {10.5281/zenodo.23463823},
  url = {https://zenodo.org/records/23463823},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Sexy-Prime Bridges Across Goldbach Transitions: Exact Convolutions, Modular Extinction, Local Sieve Dynamics, and Conditional Covering Criteria
AU  - Paul Alexander Bilokon
PY  - 2026
DA  - 2026/10/19
PB  - Zenodo
DO  - 10.5281/zenodo.23463823
UR  - https://zenodo.org/records/23463823
AB  - We investigate a proposed rescue mechanism for Goldbach representations at indices  n≡1pmod3 where the exceptional Goldbach summand $3$ is unavailable.  The proposed rescue uses a prime pair (q,q+6) and a complementary prime r,  so that q+r=2n−4 and (q+6)+r=2n+2. Writing n=3t+1 reduces the  problem to positivity of the discrete convolution  R(t)=∑_a=1^t−1f(a)f(a+1)f(t−a), where  f(a)=one_PP(6a−1). We give exact gcd identities, the associated  singular series and its uniform positivity, elementary rough-number bounds,  finite local-covering certificates, and infinite families of prescribed  extinction and recovery. A conditional covariance criterion isolates the  additional additive information required to prove universal rescue.  Exact modular convolutions verify R(t)>0 for 2≤ t≤10^6.  The universal rescue assertion and the infinitude of sexy-prime pairs  remain open; the note does not claim a solution of either problem.
ER  -