Transverse Symmetry Breaking and Bounded-Slope Compensation in a Piecewise-Smooth Pressure–Capacity Model

We extend a two-dimensional piecewise-smooth model of standardization pressure and intermediate-layer capacity into a three-dimensional system by introducing a transverse coupling variable representing the production–consumption differential. The extended system acquires a double-zero eigenvalue at the critical fold point. Through an explicit evaluation of the cubic characteristic polynomial's constant coefficient, we prove that, within a sufficiently small neighborhood of the critical state–parameter point on the same smooth branch, every non-diagonal equilibrium has at least one unstable direction and is therefore linearly unstable. This is a strictly local statement: it does not extend across switching manifolds, and it does not exclude distant non-diagonal equilibria. We then propose a bounded-slope compensation path. By exact algebraic reduction, we show that the non-diagonal exclusion problem along this proposed path reduces to a set of certification protocols. The algebraic reduction is rigorous; the global exclusion of non-diagonal equilibria along the proposed compensation path is conditional on the interval-arithmetic certification of Protocols A–D, which we specify explicitly for independent computational execution. The proposed path is one modeling choice among several possible bounded-slope paths, not a unique derived control law. We conclude by identifying open problems on network-topological intermediate-layer robustness.