Explicit Admissible Bound for the Hourglass Budget

This component establishes the analytic ceiling required to complete the rigidity-based confinement program. It derives an explicit upper bound on the allowable growth of the global hourglass rigidity budget, defining the total “credit” available to the system under the analytic constraints of the Riemann function. The section formally defines the global hourglass budget as an integral of the coercive rigidity density over the truncated critical strip and introduces a flow-aligned weight function that compensates for the anisotropic expansion of triangular averaging domains. This weighting ensures scale integrability near the apex while preserving sensitivity to vertical propagation of instability. Using the functional equation symmetry and known zero-density behavior, the analysis proves that when all nontrivial zeros lie on the critical line, the total rigidity budget grows at most at the rate of the zero-counting function The resulting bound defines a precise admissible “credit limit” proportional to the number of critical-line zeros. The component then demonstrates the structural consequence of this bound: while zeros located on the neutral manifold exactly saturate the available budget, any off-critical-line singularity induces an asymmetric hourglass flow whose footprint grows linearly with height. The coercive rigidity floor prevents cancellation of this contribution, forcing an excess cost that inevitably exceeds the admissible ceiling. This result supplies the quantitative foundation for the Budget Lemma and provides the analytic counterpart to the Footprint Lower Bound. Together, they establish the global solvency constraint under which the Riemann Hypothesis emerges as a stability law: the analytic universe of the function cannot support a singularity outside the critical line without violating its fundamental rigidity budget.