Frey Triples with Rational 3-Isogeny: A Six-Linear Radical Bound

We study a special subfamily of primitive positive triples $a+b=c$, namely those for which the full-$2$-torsion Frey curve Ea,b,c y²=x(x-a)(x+b) admits a rational cyclic $3$-isogeny. A direct elimination of the $3$-division polynomial gives a genus-zero parametrization by coprime integers $m,n$. Up to the standard symmetries of the $abc$ equation and a normalization factor in \1,3\, the three terms are built from the six linear forms m, n, $m-n$, $m+n$, $n-2m$, and $m-2n$. Every prime at least $5$ occurs in at most one of these forms. Splitting the six forms into two additive blocks, one block has prime-to-$6$ radical at most R1/2, where R=rad(abc). Inside that block, one of the three largest prime factors is at most R1/6. Combining this allocation with Pasten's largest-prime-factor subexponential $abc$ theorem yields the unconditional estimate log c R1/6 exp(κ √(log R) log₂^* R) = R1/6+o(1) throughout the rational-$3$-isogeny Frey subfamily. We also record a near-cusp R^ε improvement obtained from the refined Pasten-Sepulveda-Manzo theorem and an effective finite-support consequence of Zywina's Runge theorem. The exponent $1/6$ is explicitly identified as the limit of the present two-block/three-factor pigeonhole mechanism; no uniform below-$1/6$ claim is made here.