A discrete curvature measure for flux balance analysis predicts transcriptional regulation in Escherichia coli

When nutrients change, cells reroute metabolism through alternativepathways, and control must act at the switching points. Flux balanceanalysis (FBA) predicts metabolic states by solving linearoptimization problems; as nutrient and enzyme capacities vary, theoptimal response is piecewise linear, and ordinary curvature vanishesalmost everywhere. Here we show that the true curvature of theoptimal flux map is not a function but a discrete, matrix-valuedmeasure concentrated on the boundaries where the active constraintset switches; in genome-scale \emph{E.~coli} models, $93.4$--$100.0\%$of the second-order response concentrates at these transitions.Integrating the measure along physiological paths assigns each enzymea parameter-free rerouting burden, $\kmu$. In carbon-starved\emph{E.~coli}, $\kmu$ predicts transcriptional induction across$424$ genes ($r = +0.395$, $p = 2.6 \times 10^{-17}$); induced genessit in operons of global carbon and energy regulons, and thererouting mass concentrates on the fork metabolites of central carbonmetabolism. The association survives five tie-breaking protocols andmultiple stress axes, yet vanishes at the protein layer($r = -0.083$, $366$ genes, matched proteomics), consistent withcells transcribing standby capacity for rerouting while bufferingtranslation.Cyclic-perturbation memory --- $66\%$ of closed cycles fail torevert, with path-dependent drift scaling linearly in loop size(slope $1.00$) where smooth systems scale quadratically --- must livein fast post-translational state. Double-knockout epistasis mirrorsactive-set boundary overlaps ($\rho_S = 0.865$). The frameworkunites linear programming, discrete geometry, and transcriptionalregulation into a predictive foundation for metabolic systemsbiology.