Part 10: Geometric Origin of Mass Hierarchy from Hierarchical Inertia in Discrete H4 Lattices

Overview Building upon the geometric definition of inertial mass introduced in Part 9, we investigate how discrete H4-derived geometry within the Origin Geometry framework may generate inertial hierarchy before particle identity, gauge dynamics, Yukawa couplings, or symmetry-breaking mechanisms are introduced. In Part 9, mass was interpreted as the geometric cost required to sustain metric deformation of the underlying substrate. The present work asks a subsequent question: if mass is geometric deformation cost, why should different excitation classes experience that cost so differently? Sector-Dependent Geometric Participation We propose that inertial hierarchy arises from sector-dependent geometric participation. The geometric inertia of an excitation is stratified into distinct regimes based on its geometric footprint: High Inertia: Excitations that engage four-dimensional bulk metric deformation carry large geometric inertia. Low Inertia: Excitations that remain boundary-supported, phase-like, or approximately isometric carry much smaller inertia. Intermediate Inertia: Projection-mediated excitations occupy an intermediate regime in which geometric participation is attenuated by dimensional reduction or boundary–bulk projection. The Hierarchical Inertia Operator To formalize this distinction, we introduce a reduced-sector Hierarchical Inertia Operator. The construction begins with a graph-theoretic stiffness proxy, such as the graph Laplacian, but supplements it with a diagonal participation-weight operator defined only after a projection, boundary selection, or excitation-sector decomposition has been specified. This caveat is essential: the full 600-cell is highly symmetric and vertex-transitive, so a vertex weight defined only from the full H4 action would generally be constant and could not generate inertial stratification. Hierarchy appears only after the relevant physical or geometric sector decomposition reduces the full symmetry and separates bulk, projection, and boundary degrees of freedom. Kinematic Spectral Probe and Falsifiability The resulting operator is not claimed to be a fundamental Hamiltonian or a completed dynamical law. It is a falsifiable kinematic spectral probe of whether reduced H4-derived geometry supports class-resolved inertial bands. The central prediction is that, once sector labels are fixed independently of the spectrum, the operator should exhibit robust separation between low-inertia boundary/phase-like modes, intermediate projection-mediated modes, and high-inertia bulk metric modes. Failure of this class-resolved separation under appropriate perturbation and control tests would falsify the hierarchical inertia hypothesis. Scope and Limitations This Part does not compute physical particle masses. It does not identify fermions, bosons, or Standard Model families. It does not replace the Higgs mechanism or fit experimental values. It establishes a geometric precondition for mass hierarchy: distinct inertial scales can arise because different excitation classes require different degrees of metric participation in the underlying discrete substrate.