M25d Generalised Operational Number Systems: Degrees of Freedom

This short paper develops new degrees of freedom within the g-ONS programme, introducing higher-etage Hermit units, generalized TrigCore families, and the base manifold of TetraCore structures. The work defines Upsilon_HC, the slog-shelf Hermit unit, as the TetraCore analogue of the loaded tau tau_HC from the K-shelf. It satisfies the NC equation (Upsilon - 1)*slog_B(Upsilon) = i*pi/2, the direct slog-level analogue of the tau relation (tau - 1)*ln(tau) = i*pi/2. The monograph then derives three unified TrigCore families. Ordinary TrigCore generates circular trigonometry, imaginary-angle continuation produces HyperbolicTrigCore, and the new HyperTrigCore is activated by Upsilon_HC in the TetraCore exactly as tau_HC activates TrigCore in the K-shelf. The general principle is that each HC Etage possesses its own Hermit unit and corresponding trigonometric family. A major structural contribution is the interpretation of the base B as a true manifold coordinate. While the Symmetric Core is base-free, TetraCore and all higher Etages depend intrinsically on the tetration base through the Koenigs multiplier lambda_B. The admissible base interval B in (1, e^(1/e)) therefore becomes a geometric parameter space of TetraCore theories. The monograph concludes with the Degrees-of-Freedom Principle: every new Etage introduces one additional continuous degree of freedom. Base B appears at the TetraCore level, configuration C at the PentaCore level, and further higher-dimensional freedoms emerge in successive operational layers. A final Propagation Conjecture proposes that all higher Etage structures arise through systematic propagation of lower-level branch and TrigCore mechanisms.