Goldbach Intersection Geometry and Prime-Pair Correlations: Exact Identities, Crossed Bridges, Modular Selection, Singular Series, and Computational Evidence
- 1Imperial College London
Abstract
For each even integer 2n, let S(n)={p∈PP:2n−p∈PP} be the set of prime summands of its Goldbach representations, and let Ik(n)=S(n)∩ S(n−k). These intersections encode prime pairs of every even separation. We develop exact correspondences, a zero-or-many theorem for nonempty intersection rows, consequences of finite and infinite fixed-gap prime-pair classes, and counterexamples showing why gap recurrence and neighboring-row overlaps alone cannot prove Goldbach. We then study crossed intersections Bh(n) and prove that they are precisely fixed-gap autocorrelations of S(n). On the residue class n≡1pmod3 where the summand 3 is unavailable, modulo-3 selection restricts nonexceptional crossed bridges to gaps divisible by six. For gaps $6,12,18$ we derive their four-prime singular series and local modulation modulo 5 and 7, prove exact overlap exclusions and a pair-count conservation law, and provide reproducible computations through n≤106. There are 258,814 tested extinction indices and no simultaneous zero of the three crossed-bridge counts. The numerical and heuristic observations are explicitly separated from theorems: universal short-gap rescue, the twin-prime conjecture, and the binary Goldbach conjecture remain unproved.
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BibTeX
@misc{bilokon2026goldbach,
title = {{Goldbach Intersection Geometry and Prime-Pair Correlations: Exact Identities, Crossed Bridges, Modular Selection, Singular Series, and Computational Evidence}},
author = {Paul Alexander Bilokon},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23463891},
url = {https://zenodo.org/records/23463891},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Goldbach Intersection Geometry and Prime-Pair Correlations: Exact Identities, Crossed Bridges, Modular Selection, Singular Series, and Computational Evidence
AU - Paul Alexander Bilokon
PY - 2026
DA - 2026/10/19
PB - Zenodo
DO - 10.5281/zenodo.23463891
UR - https://zenodo.org/records/23463891
AB - For each even integer 2n, let S(n)={p∈PP:2n−p∈PP} be the set of prime summands of its Goldbach representations, and let I_k(n)=S(n)∩ S(n−k). These intersections encode prime pairs of every even separation. We develop exact correspondences, a zero-or-many theorem for nonempty intersection rows, consequences of finite and infinite fixed-gap prime-pair classes, and counterexamples showing why gap recurrence and neighboring-row overlaps alone cannot prove Goldbach. We then study crossed intersections B_h(n) and prove that they are precisely fixed-gap autocorrelations of S(n). On the residue class n≡1pmod3 where the summand 3 is unavailable, modulo-3 selection restricts nonexceptional crossed bridges to gaps divisible by six. For gaps $6,12,18$ we derive their four-prime singular series and local modulation modulo 5 and 7, prove exact overlap exclusions and a pair-count conservation law, and provide reproducible computations through n≤10^6. There are 258,814 tested extinction indices and no simultaneous zero of the three crossed-bridge counts. The numerical and heuristic observations are explicitly separated from theorems: universal short-gap rescue, the twin-prime conjecture, and the binary Goldbach conjecture remain unproved.
ER -