M30c Operational Geometry (v0.9)

M30c is the geometric companion to M30a and M30b: where M30a supplies the Koenigs-phase / Wronskian engine and M30b supplies the cross-rank Lie groupoid, M30c reads the same machinery as a theory of operational geometries. Its central principle is that a geometry is not primarily a metric imposed on a passive space, but the visible form of a native compositional law in its correct Abel, Koenigs, logarithmic, or spectral coordinate. The monograph defines an operational geometry as a triple G_R = (X_R, Op_R, A_R) where X_R is a rank-shelf, Op_R is its native composition, and A_R is the flattening coordinate satisfying A_R(Op_R(a,b)) = A_R(a) + A_R(b). The pullback metric is then ds_R^2 = |dA_R|^2, so each rank produces its own native geometry. Euclidean geometry appears at R = 1, the hyperbolic line at R = 2, and rank R = 3 already gives a new non-classical double-logarithmic geometry. A major new structure is the six-knob phase diagram of operational geometry: (R, T, T_hyp, kappa, theta, B). Here R selects the rank, T is elliptic imaginary rank, T_hyp is hyperbolic rank, kappa is branch monodromy, theta is the TrigCore knob, and B is the tetrational base. M30c shows that geometries are not isolated species but knob configurations: Euclidean, hyperbolic, spherical, Minkowski, AGM, Heun, ISHE, branch, star, critical-interface, and sporadic-stabilizer geometries all become entries in one operational atlas. The key physical-geometric result is the bicomplex extension. By adjoining both the elliptic unit i with i^2 = -1 and the hyperbolic unit j withj^2 = +1, M30c obtains a rank-extended metric whose unique time-like direction is the hyperbolic rank T_hyp. Thus Lorentzian signature is not postulated: it is forced by the hyperbolic twin of the Koenigs phase. At rank 2, the elliptic and hyperbolic Hermit generators produce the Lorentz algebra. The six generators J_x, J_y, J_z, K_x, K_y, K_z satisfy the commutation relations ofsl(2,C) ~= so(3,1), so the proper Lorentz algebra emerges from the rank-2 bicomplex operational structure. M30c also lifts this picture toward rank 3, where Wronskian corrections become the operational analogue of spacetime curvature. M30c also gives a geometric interpretation of the Wick rotation: it is the knob exchangeT <-> T_hyp between elliptic Koenigs phase and hyperbolic rank. In this reading, Euclidean and Lorentzian QFT are not unrelated formalisms but two knob-related operational geometries. The Riemann-critical geometry is recast as a null-locus statement. The critical line Re(s) = 1/2 is identified with the projection of a null direction of the rank-extended Lorentzian metric, and the RH route is restated geometrically: zeta zeros should be lightlike under the operational metric. This is presented as the geometric counterpart of M30a’s commutator condition [G_R, V] = 0.