qSR: A Reproducible Dimensionless Structural Self-Regulation Constant for a 9+1 Closure Class

We introduce (q_{\mathrm{SR}}), a dimensionless structural self-regulation factor associated with a defined 9+1 closure class. The construction does not claim the first discovery of the algebraic equation [q^{10}+q-1=0] or of its positive root. Instead, it assigns the root a structural interpretation as the inverse-Perron return factor of a normalized closure system consisting of nine internal closure directions and one additional degree of structural continuation. For the corresponding weighted return operator, unit full-loop normalization yields the characteristic equation [\lambda^{10}-\lambda^9-1=0,\qquad q_{\mathrm{SR}}=\lambda^{-1},] and therefore 0.8350790427235590476\ldots] The result is invariant under redistribution of internal gains that preserves the normalized loop product, with such realizations belonging to the same spectral equivalence class. Numerical experiments over large randomized ensembles reproduce the predicted value to machine precision, while perturbations of the closure conditions produce systematic deviations. The same algebraic closure also appears in independently formulated recurrence, renewal, symbolic-dynamical, and age-structured systems, suggesting that the relevant object is not the numerical constant alone but a broader return-closure structure. Nonlinear simulations further indicate that the 9+1 organization can persist beyond the exact normalized class even when the scalar return factor changes. The present work therefore proposes (q_{\mathrm{SR}}) as an exact invariant of a specified structural universality class, rather than as an empirically established universal constant of nature. A reproducible computational and falsification protocol is provided for independent verification and future cross-domain testing.