Chemical Medium-Range Order Enables Stoichiometric Rigidity

Maxwell counting predicts an isostatic threshold at $\langle r\rangle = 2.4$ for covalent network glasses, but which structural correlations actually produce rigidity near this point is still unclear. In this work, we test four candidates: enthalpic stress, chemical defects, geometric interlocking, and medium-range order (MRO). We use a locally tree-like configuration model as a zero-MRO baseline and apply perturbations to test each candidate. We find that (i) enthalpic stress delays rigidity rather than enabling it; (ii) chemical defects require fractions ($\sim$40%) far above experimental values ($\sim$16% in GeSe$_2$); (iii) geometric linking density does not govern the threshold location, which is instead set by loop-induced redundancy; and (iv) only phenomenological MRO proxies recover rigidity at experimentally accessible strengths. Consequently, chalcogenide intermediate-phase data and amorphous SiO$_2$ ring statistics positively implicate chemical MRO, while DNA spatial networks independently rule out pure geometric entanglement. We conclude that rigidity near the Maxwell threshold requires chemistry-specific correlations beyond pure connectivity.