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 GR = (XR, OpR, AR) where XR is a rank-shelf, OpR is its native composition, and AR is the flattening coordinate satisfying AR (OpR (a, b) ) = AR (a) + AR (b). The pullback metric is then dsR² = |dAR|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, Tₕyp, kappa, theta, B). Here R selects the rank, T is elliptic imaginary rank, Tₕyp 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 i² = -1 and the hyperbolic unit j withj² = +1, M30c obtains a rank-extended metric whose unique time-like direction is the hyperbolic rank Tₕyp. 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 Jₓ, Jᵧ, Jᵦ, Kₓ, Kᵧ, Kᵦ 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 Tₕyp 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 GR, V = 0.