The Regulatory-Channel Equivalence A Formal Mapping Between GTRS Regulatory Capacity and BCST Channel Capacity

AbstractThis paper investigates the formal mathematical relationship between the regulatory capacityparameter rho(x) of the General Theory of Regulated Stability (GTRS) and the aggregatecross-scale channel capacity Cagg of Bounded Communication Systems Theory (BCST).Three questions are addressed: (Q1) whether a well-defined transformation T exists such thatrho can be expressed as a function of channel capacity variables; (Q2) whether the GTRSstability threshold rho = 1 corresponds to the BCST coherence threshold Cagg = Cmin; and(Q3) whether the four GTRS failure modes are preserved under the mapping. Under statedassumptions, we construct a logarithmic coordinate transformation relating rho to Cagg. Thetransformation is formally a rescaling of the GTRS regulatory ratio in information-theoreticterms; its value lies in revealing structural features — compressive saturation, bottleneckidentification, and failure mode signatures — that are obscured in the linear ratiorepresentation. Threshold correspondence holds under specified conditions. Three of fourfailure modes map with distinct signatures; the fourth (structural bifurcation) defines themapping’s boundary. Connections to Geometric Bifurcation Theory (Fisher information atcritical points) are noted as consistency checks, not independent confirmations. An analogical(not formal) link to Pesin’s theorem is discussed with explicit acknowledgment thatbiological systems violate the theorem’s conditions. The precise boundaries of thecorrespondence are identified, including timescale constraints arising frommemory-dependence in the regulatory capacity variable.