A Joint Survival Certificate for the Substituted Chrysene Network: The L^1 Transformation Geometry, Peak-Gain Well-Posedness, and a Bandwidth Floor

Abstract

The transfer geometry of the substituted chrysene network certifies robust stability and invariance of the admissible domain by two quantities that no estimate relates: the worst-case norm controls the first, a peak gain the second, and no function of the former bounds the latter, so the survival claim has stood as a conjunction of separately certified halves. We discharge it as a single certificate. The impulse response of a composite realization is identified with the integrated return amplitude of the active region, and its spectral density with the coupling-weighted density of states of the second fibration seen through the effective resolvent. Finiteness of the peak gain is then an L^1 decay condition strictly stronger than embedded-mode exclusion: the latter excludes poles, the former additionally constrains edge and van Hove regularity, and under a one-sided square-root edge hypothesis the return amplitude decays like t^(-3/2). On the resulting domain the unitary-orbit distance taken in L^1 is a pseudometric with attained infimum whose path closure dominates the H-infinity transformation metric. Its zero set is again transfer isomorphism, as it must be for any unitarily invariant norm; what changes is that its domain is strictly smaller and its infinite locus strictly larger, which costs the connectivity trichotomy of the prior paper. Under a uniform constraint-margin hypothesis, one ball in this metric certifies robust stability at a given environment level and invariance of the admissible domain at a given disturbance amplitude, for the same forms, and propagates along a transformation route. Finally the integrated return amplitude admits a lower bound of pi divided by the width of the composite spectrum, and hence an upper bound on the certifiable disturbance amplitude set by the fibration rather than by the design; both natural evasions of that bound are closed by the well-posedness criterion itself. The joint certificate is shown to be strictly more conservative than the prior robust-stability radius, which it does not supersede, and the two gains are shown not to be co-optimized. Twelve non-implications, four new targets of which two are load-bearing, three corrections to the prior paper including one to a hypothesis its invariance theorem rests on, a seventeen-item defect ledger, and an explicit witness solved in closed form are stated.