DCLXVI LC Mechanical Engineering & Research

A weighted Voronoi cell at site p_i contains points x satisfying |x − p_i|² − w_i ≤ |x − p_j|² − w_j for every j. Pairwise boundaries are planes, and the cells partition space with disjoint interiors. Applying this to host and interstice centers gives a precise family of fully packed constructions to investigate.

Site positions, weights, nonempty-cell conditions and face topology must still be specified. This construction does not automatically retain four physical faces for T and eight for O; the literal-face mass rule must be checked against the resulting cells. The 1S:2T:1O inventory is exact bulk accounting, while a four-whole-piece repeating tile needs an explicit assignment. An HCP primitive translation cell contains two host sites.

Let I count occupied T–O face contacts and F count exposed triangular apertures. Counting faces gives F = 4T + 8O − 2I. Since every O has at most eight T neighbors, I ≤ 8O. At q = −e:

The electron shell achieves I = 8 and F = 24. For touching host spheres of radius R, each aperture has area a = (√3 − π/2)R². Define Ω_T = 3 arccos(1/3) − π and Ω_O = 4 arcsin(1/3). The curved host-wall areas per cavity are s_T = 4Ω_TR² and s_O = 6Ω_OR². All three areas are positive.

Equality in this admissible class requires the unique nine-cell shell. Thus a proposed energy E = γA with γ > 0 selects that shell as its unique connected −e minimum. A is the full Euclidean boundary of the occupied cavity union, including the curved host interfaces. A counts neither a convex hull nor a relativistic event horizon. If LSM assigns a different energy to these interfaces, includes spectral terms, or permits lattice deformation, that energy requires its own minimization.

For reproduction, V_T = [4/(3√2) − 4Ω_T/3]R³ and V_O = [8√2/3 − 2Ω_O]R³. These are the curved void volumes, not face-count mass units.

In ideal HCP, an O has six T and two O face-neighbors; a T has three O and one T face-neighbor. The FCC proof above therefore does not automatically apply to HCP. A fully regular octah with eight regular face-sharing tetrah is an FCC-compatible local cluster. Embedding that full arrangement in an otherwise HCP substrate would require a local stacking change or another explicitly constructed defect. The current gallery already uses FCC, and is preserved.

Connectedness is a restriction, not a derived confinement law. Six mutually separated T cavities have total charge −e and area 6s_T + 24a, less than the electron shell by 2s_T + s_O. They lie outside the theorem. If charge alone is conserved and such fractional fragments are allowed, the surface functional alone does not ensure electron stability. If T and O inventories are separately conserved, that particular comparison is inadmissible, and the allowed moves must say so. A physical completion must supply confinement/conservation rules and test all permitted fragmentation channels.

The bulk neutral ratio gives 2q_T + q_O = 0. Declaring the eight-face shell to carry −e then gives q_T = −e/6 and q_O = +e/3. This explains the charge normalization conditional on that identification; it is not an independent derivation of charge from geometry alone. Spin ½, fermionic statistics, inertial mass, propagation and measured electromagnetic behavior also require a dynamical construction.

For any +e T/O inventory, Q = 2O − T = 6, so F = 16O − 24. A neutral T/O core adds a multiple of 16. Relative to the 40-face electron, the bare additive ratios therefore lie on a grid with spacing 2/5. The measured ratio is not on that grid. Subdividing all faces equally leaves the ratios unchanged.

The next physical rule must account for binding, excitation, host coupling, or a different relation between face inventory and rest energy. The microscopic counts can remain integers. The additional contribution must follow from shared local rules rather than a separate adjustment for each particle.

For the selected cage plus the unchanged node, choose zero electron binding correction as the calibration reference and introduce one integer proton binding count B. Then M_e = 40K and M_p = 73,448K − B.

An exhaustive positive-integer search for the smallest K matching the CODATA ratio within one quoted standard uncertainty gives K = 346 and B = 655.

This is an observation-constrained arithmetic fit. Neither 346 nor 655 has yet been derived from the geometry or an update rule. It demonstrates compatibility with one measured ratio, not a fit to the full particle spectrum. This quantum is about 36.9 eV/c², too coarse for the observed light-neutrino sector under a universal positive integer rest-mass rule. It is a search candidate, not the final fundamental scale.

The 2T + 1O interstitial inventory has bare mass 16 face units, about 204.4 keV/c². Its identification with observed light neutrinos needs an additional rest-energy, binding or propagation mechanism: KATRIN's 2025 result bounds the effective electron-neutrino mass below 0.45 eV/c² at 90% confidence. Adding a positive host mass does not resolve that discrepancy.

Integer coordinates can yield exact square-root distances. An octahedron's edge/diagonal ratio is invariant and irrational; changing units cannot make both lengths integer multiples of one common nonzero length. Integer microscopic inventories and update rules do not require every derived ratio or average to be an integer.

The gravitational constant and electromagnetic coupling remain dynamical constraints. This convention does not derive G or the fine-structure constant. The next task is a common local integer rule for binding, excitation and host coupling that reproduces the electron, a proton and light-neutrino behavior with the same parameters.