A Lorentzian Sector Bundle over the Geometric-Time Direction Sphere

This upload contains a self-contained kinematic paper developing the Lorentzian sector-bundle layer of the multi-geometric-time Omniverse formulation. The paper studies an ambient vector space with three ordinary spatial directions and a positive K = ξ − 3 dimensional geometric-time block. Its central distinction is between the scalar clock parameter t, the ordinary Minkowski time coordinate x⁰ = ct, the ambient geometric-time coordinates, the selected geometric-time carrier Ĝ, and the effective geometric-time coordinate G₀. A selected unit geometric-time direction Ĝ ∈ 𝒢 determines the effective coordinate by projection, G₀ = Ĝ · 𝐗_GT, while the externally observed sector is obtained by imposing the projection-locking condition Q_Ĝ 𝐗_GT = 0, which suppresses geometric-time components orthogonal to Ĝ. On each locked sector, after the clock convention G₀ = ct is imposed, the induced interval reduces to the ordinary (1,3)-signature Minkowski interval. The standard Lorentz group then appears as the subgroup of the ambient isometry group O(K,3) acting internally on the selected Lorentzian sector 𝓔_Ĝ = span{Ĝ, x̂₁, x̂₂, x̂₃} and trivially on the orthogonal geometric-time complement Ĝ⊥. The central result is that when K ≥ 2, the ambient kinematics do not select a unique effective geometric-time carrier. Instead, the possible oriented carriers form the sphere Sᴷ⁻¹, equivalently O(K)/O(K−1), and each point of this sphere determines a locked Lorentzian sector. The resulting object is a Lorentzian sector bundle over the geometric-time direction sphere. This sector bundle is what the paper calls the kinematic Omniverse. The manuscript also clarifies the geometric-time invariance postulate, according to which the geometric-time velocity vector 𝐯_GT has invariant magnitude c. This common propagation along the selected geometric-time carrier is treated as part of the ambient kinematic description, but it is not internally observed as relative spatial motion because observers, rods, clocks, detectors, and matter systems locked to the same sector share the same carrier drift, 𝐯_GT = cĜ. In this framework, the invariant vacuum speed of light is recovered as a null-sector corollary of the geometric-time carrier normalization, rather than being introduced as an unrelated postulate. With orientation retained, antipodal directions Ĝ and −Ĝ define opposite effective geometric-time orientations. This is not a reversal of scalar-time evolution: an antiverse-sector has the opposite orientation along the effective geometric-time carrier while still admitting a forward-increasing scalar clock within its own sector. After quotienting by the antipodal identification Ĝ ∼ −Ĝ, the unoriented sector base is real projective space, ℝℙᴷ⁻¹. The revised version also includes a more detailed discussion of action-level geometric-time locking. In particular, it explains how the projection-locking condition Q_Ĝ 𝐗_GT = 0 can be imposed at the level of the action by a transverse Dirac delta constraint supported on the selected geometric-time carrier. This action-level formulation makes the associated geometric-time permutation covariance transparent: before a carrier is selected, the raw geometric-time coordinate labels can be treated symmetrically, while a selected locked sector retains only the residual symmetry preserving its chosen carrier. This manuscript is deliberately limited to the kinematic layer. It does not assume compactification, field equations, matter content, inter-sector dynamics, physical occupation of all sectors, or observational accessibility of multiple sectors. Its purpose is to isolate the projection, locking, Lorentz-reduction, sector-bundle, geometric-time invariance, action-level locking, permutation covariance, and universe–antiverse orientation structure needed before any later dynamical or field-theoretic extension. The work is a narrow, self-contained extraction from a broader monograph-level framework, but it is written so that the kinematic construction can be read independently.