The Ternary Conductor Boundary: Why Conductor Rigidity Is Specific to the Binary Goldbach Problem

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.