Higher Categorical Coherence Breakdown and the Dynamical Central Charge: Conceptual and Experimental Pathways via the Fractional Quantum Hall Effect

The central charge occupies a unique role in conformal field theory, simultaneously serving as a measure of degrees of freedom, as the determinant of Casimir energy through modular transformations, and as an obstruction to the naive extension of the Witt algebra. The Virasoro central extension itself is rigid: it fixes c as a label of a given conformal field theory. In this work, we propose that higher categorical coherence—the pentagon and hexagon constraints governing fusion and braiding data, one level above the cocycle responsible for the Virasoro extension—supplies an additional, physically controllable handle. We show that controlled deformations of this higher coherence (higher categorical coherence breakdown, HCCB), implemented consistently through anomaly inflow, shift the effective central charge read out by anomaly-sensitive observables in quantized steps, opening the possibility of treating the measured central charge not as a fixed label but as an experimentally addressable piecewise-quantized quantity. We then focus on the fractional quantum Hall effect (FQHE), where the chiral central charge $c_{-}$ directly governs the quantized thermal Hall conductance. After reviewing the role of edge conformal field theories and current bounds on thermal transport, we propose experimental modifications—such as engineering multi-component edge states, coupling to non-Abelian quasiparticles, or introducing controlled categorical perturbations—that could render higher coherence breakdown detectable as shifts in the effective central charge. Two further elements complete the program. First, we show that within the consistent framework, all route- and bracketing-dependent observables vanish identically (route blindness), so that the pentagon and hexagon interferometers and thermal Y-junction networks we design operate as precision null tests of the modular-functor axioms themselves—the axioms stating that anyonic amplitudes are determined by the topology of a process rather than by the bookkeeping route used to compose it. Second, we show that a quantized remnant of route sensitivity survives in exactly one consistent form: the holonomy of closed cycles of categorical controls, realizing a central-charge pump for which the integer count per cycle is a family invariant beyond any static stacking description. The resulting framework provides both a conceptual reinterpretation of the central charge as a higher obstruction in categorical terms and a concrete experimental route for probing its dynamical behavior. Beyond the quantum Hall setting, these ideas suggest a broader program: anomalies, topological phases, and even string worldsheet central charges may admit reinterpretation through higher coherence. We conclude by outlining a research agenda in which categorical methods yield new experimental observables, potentially transforming the interplay between mathematics, condensed matter physics, and high-energy theory.