A Discrete-Lattice Kaluza–Klein Variant with Integer-Rung Dimensional Indexing and Base-3 Matter Condensation

This is a theoretical physics paper, which describes original research that proposes to extend known frameworks. Keywords: Kaluza-Klein theory, discrete spacetime, Planck-scale lattice, extra dimensions, hierarchy problem, mass spectrum, particle masses, base-3 number theory, emergent Lorentz invariance, quantum gravity phenomenology, Newton's constant derivation, speed of light derivation, dimensional ladder, fine-structure constant, Treo ModelSubject field: Physical Sciences → Physics → Theoretical PhysicsAbstract: We propose a tuning-free discrete-lattice extension of the Kaluza–Klein tradition. The substrate is a cubic lattice whose nodes each host two paired primitive entities: a 1-dimensional vibrating object of Planck length ℓP (the Treo) and a 5-dimensional curled structure (the Void). Every Treo executes a fixed integer number S = 1. 855 × 10⁴³ of oscillation cycles per Planck time, taken as a primitive substrate parameter alongside the lattice spacing ℓP. From these two primitives the framework derives (i) an integer dimensional ladder Ud = mP · S^ ( (d−3) /2) with a fixed ratio √S ≈ 4. 307 × 10²¹ between rungs, replacing the tuneable warp factor of Randall–Sundrum-type models; (ii) the speed of light c = S·ℓP, now a derived quantity whose constancy and isotropy emerge from substrate uniformity rather than being postulated; and (iii) Newton's constant G = c³/ (S·mP) as a consequence of Planck-unit algebra combined with c = S·ℓP. Matter condensation is governed by a base-3 admissibility rule arising naturally from the cubic lattice's discrete translational symmetry. A five-factor factorisation of the electron closure constant approaches the Planck identity mP/ (mₑ·√S) to better than 10⁻⁵, with a named 0. 088% residual flagged as an open sub-problem. The mass formula m = n·Mᵤ· (1 − k·α), with Mᵤ = mₑ/ (2α) = 35. 013 MeV/c², selects 11 of 26 standard particles at clean integer k values; a perturbation-null Monte Carlo returns a comparable rate on randomly scaled masses, so the formula should be read as a classification scheme rather than a standalone predictor — the genuine predictive domain is the hadron range below 2 GeV. We execute steps (i) – (iii) of the Newton's third-law derivation target from discrete lattice translational symmetry and show as a by-product that the bound sector is gapless by axiom. The Einstein-equation target and the Stage-B stability criterion for base-3 condensation remain open. The cubic-lattice Lorentz-invariance status is identified as the principal standing technical concern.