Child-selection determinants and unstable cores in sequential distributive futile cycles

Vassena and Stadler localized instability in reaction networks to unstable cores: minimal child-selection matrices with an eigenvalue of positive real part. For the sequential distributive dual futile cycle they found exactly three unstable cores, all unstable-positive, all with determinant of modulus one, and conjectured that the same picture persists for the $n$-site cycle built by adding phosphorylation steps. We settle this conjecture by separating its two assertions. The determinant assertion holds in a class much larger than the futile cycles: every child-selection matrix of an elementary enzyme-conversion system, singular or not, has determinant $0$ or $\pm1$. The structural assertion fails in two ways. For every $n\ge3$ the cycle contains a fixed six-species minimal unstable-negative core, and for every $n\ge2$ it contains a minimal unstable-positive core of dimension $2n+1$ that specializes at $n=2$ to one of the three dual-cycle cores and remains minimally unstable under every positive column scaling; the literal dimensions of minimal positive cores are therefore unbounded. A five-species restriction of the negative witness is strictly stable at unit scaling yet is a $D$-unstable-negative core, so ordinary and scaling-invariant minimality differ within one network. For the positive family we derive a scalar return equation that identifies the spectral abscissa for arbitrary rates and losses, giving exact rate comparisons and a sharp loss threshold; at unit rates the unstable eigenvalue is $W(n-1)/(2(n-1))+O(W(n-1)^2/(n-1)^2)\sim\log n/(2n)$, so the instability weakens as the cores grow. An explicit saturating-kinetic model of the complete three-step network realizes an unstable positive equilibrium with exactly three eigenvalues in the open right half-plane. The determinant theorem, both core families and the scaling statements are verified in Lean 4; the rate-dependence, asymptotic and kinetic results have conventional proofs with exact rational certificates that a supplied script replays.