The Ternary Conductor Boundary: Why Conductor Rigidity Is Specific to the Binary Goldbach Problem
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| Format: | Recurso digital |
| Langue: | anglais |
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2026
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| _version_ | 1866901616715628544 |
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| author | Chen, Ruqing |
| author_facet | Chen, Ruqing |
| contents | <p>We investigate whether the conductor rigidity framework for the binary Goldbach conjecture (genus 2, GSp(4)) extends to the ternary problem p₁+p₂+p₃=N (genus 3, GSp(6)). By computing the true discriminant of the genus-3 hyperelliptic Frey curve, we show that the algebraic identity p²−q²=(p−q)N which underpins binary conductor rigidity has no ternary analogue: N does not appear as an independent factor in the genus-3 discriminant, entering only through partial sums (N−pₖ). Consequently, the Band Shifting Law ceases to hold (R²=0.0002 against the static conduit variable ξ). The ternary conductor decomposes into summand, difference, and partial-sum contributions whose interplay is governed by prime-factor statistics rather than algebraic structure. The PPP–CCC gap shrinks from a large stable separation (binary) to a marginal offset of 0.10±0.07 (ternary). These results precisely delineate the applicability boundary of conductor rigidity: the theory is native to genus 2, where the factorisation p²−q²=(p−q)N embeds N as a universal geometric invariant of the Frey family.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18727994 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Ternary Conductor Boundary: Why Conductor Rigidity Is Specific to the Binary Goldbach Problem Chen, Ruqing Goldbach conjecture conductor rigidity ternary Goldbach applicability boundary GSp(6) genus 3 Frey curve discriminant radical function algebraic number theory Mathematics Number Theory Algebraic Geometry <p>We investigate whether the conductor rigidity framework for the binary Goldbach conjecture (genus 2, GSp(4)) extends to the ternary problem p₁+p₂+p₃=N (genus 3, GSp(6)). By computing the true discriminant of the genus-3 hyperelliptic Frey curve, we show that the algebraic identity p²−q²=(p−q)N which underpins binary conductor rigidity has no ternary analogue: N does not appear as an independent factor in the genus-3 discriminant, entering only through partial sums (N−pₖ). Consequently, the Band Shifting Law ceases to hold (R²=0.0002 against the static conduit variable ξ). The ternary conductor decomposes into summand, difference, and partial-sum contributions whose interplay is governed by prime-factor statistics rather than algebraic structure. The PPP–CCC gap shrinks from a large stable separation (binary) to a marginal offset of 0.10±0.07 (ternary). These results precisely delineate the applicability boundary of conductor rigidity: the theory is native to genus 2, where the factorisation p²−q²=(p−q)N embeds N as a universal geometric invariant of the Frey family.</p> |
| title | The Ternary Conductor Boundary: Why Conductor Rigidity Is Specific to the Binary Goldbach Problem |
| topic | Goldbach conjecture conductor rigidity ternary Goldbach applicability boundary GSp(6) genus 3 Frey curve discriminant radical function algebraic number theory Mathematics Number Theory Algebraic Geometry |
| url | https://doi.org/10.5281/zenodo.18727994 |