Pure-Even Multi-k Families and the Arithmetic Core of Resonant Algebra

We construct an infinite family of pure-even Bring–Jerrard quintics over Q, organised by a single rational invariant k=β/α. For every fixed k≠0 the one-parameter family

x5+α(m)x+β(m),α(m)=256m2−3125k4256,β(m)=k⋅α(m)x^5+\alpha(m)x+\beta(m),\qquad \alpha(m)=256m^2-\frac{3125k^4}{256},\quad \beta(m)=k\cdot\alpha(m)x5+α(m)x+β(m),α(m)=256m2−2563125k4,β(m)=k⋅α(m)

has identically square discriminant. Irreducible specialisations therefore have even Galois group; extensive computation yields Gal=A5 with high frequency (and D5 on a minority of fibres). An enlarged search produces approximately sixty A5 seeds that fall into sixteen ratio classes, ten of which are multi-seed. Explicit rational paths in the two-parameter envelope join seeds of distinct ratios while remaining pure-even. A discriminant-height asymptotic is proved. Independent verification confirms the flagship seed x5−55x+88 (Gal=A5) and a rigid geometric control of opposite parity (Gal=S5).

The coefficient lattice is the ternary / resonant lattice arising in an exploratory framework (HQCC / “9 Maths of Unification”). All theorems stated below are arithmetic and do not depend on any physical interpretation of that lattice. Geometric multi-k fusion (a Nielsen-labelled Hurwitz family recovering several catalogue ratios) remains open.