Volume II (T): The Geometry of Force -- Platonic Archetypes, the Tiling Audit, Force-Body Duality, and the Stride-10 Protocol

The second pillar of the N.E.A. framework: Force-Body Duality. We prove that the gauge groups of the four fundamental forces are topological invariants of the cycle spaces of the three admissible geometric carriers in 3D Euclidean space. Through a systematic Platonic solids audit---classification, tiling uniqueness, and bandwidth cost screening---we identify the sole physical survivors: the cube C8 (spatial skeleton), the regular tetrahedron K4 (minimal closed volume anchor), and the regular octahedron (the addressing framework for causal directional locking, occupying the earliest logical layer of the protocol stack).

We then conduct the **Tiling Audit**—a new chapter proving that the tiling capability of a Platonic solid determines both the range and the effective strength of the force it carries. Tiling success (C8) produces long-range, power-law forces with rent diluted across the infinite translational symmetry group. Tiling failure (K4, octahedron) produces short-range, exponentially confined forces with rent trapped entirely within the local structure. The Strong Force and Gravity are both 2→3 operations constructing 3D volume, distinguished solely by the tiling fate of their carriers: Gravity succeeds (C8 tiles space, rent diluted, force vanishingly weak, becomes the background curvature of spacetime), the Strong Force fails (K4 cannot tile, rent trapped, force confined within nucleons, becomes localized matter). Mass is failed space.

The gauge group dimension is determined by the topological phase of the operation on the carrier. For fully locked volumes, dim g = N^2 - 1, where N is the cycle number of the locked simplex. K4 has cycle number m-n+1 = 3 → SU(3), dimension 8. For partially locked or open-boundary operations, the gauge group dimension equals the effective number of independent phase degrees of freedom surviving after constraints. The regular octahedron, under causal directional locking, leaves 6-3=3 independent phase rotation generators, which spontaneously close into SU(2), dimension 3. The C8 intra-cell boundary is a 1D open edge, whose sole phase degree of freedom is U(1). Gravity is the inter-cell deficit stitching across the C8 tiling gaps, with no independent carrier and no independent gauge group; spin-2 emerges as the tensorial fingerprint of the stitching operation.

The Weak Force is clarified as the 0→1 Addressing Protocol—the establishment of the first directed causal identity along the undirected skeletal edges that the Being Tax and global conservation jointly mandate (Volume I). The octahedron serves as the directional addressing framework; its six vertices lock a 3D direction for the nascent 1D chain. The octahedron cannot tile 3D space, explaining why the Weak Force is both short-range and transient—the addressing protocol retires once the Stride-10 closure is achieved, with W/Z bosons existing only as handshake headers.

We establish the Stride-10 protocol—the minimal logical addressing width of 10 causal edges for non-aliased causal direction resolution in 3D discrete lattices, verified invariant across lattice sizes—and complete the physical realization of space: the C8 tiling scaffold dynamically refreshed by the photonic Stride-1 phase synchronization pulse. The spectral band diagram maps the protocol domains of the four forces across 35 orders of magnitude. Gravity is shown to be protocol non-profitable below the atomic scale (~10^{-10} m), a fact that partially mitigates the 10^{120} vacuum energy catastrophe by a factor of 10^{100}.

Two fundamental topological constants are extracted: U_EM = 0.4π (Theorem, double-verified) and U_weak = 10√3 (Strong Evidence, Stride-10 saturation). The √3 = ΔU_internal + U_EM identity reveals the topological origin of electroweak unification. Numerical evidence from the tiling audit—IPR ratios, spectral distribution comparisons, and force-range propagator analysis—provides direct quantitative support for the tiling-force correspondence. These constants and the tiling framework provide the foundational basis for the zero-parameter settlement of all 19 Standard Model constants in Volume III.