A Scherk–Schwarz Origin for a Least-Common-Multiple Mass Spectrum

We observe that a Scherk–Schwarz compactification on M_4 \times S^1 with a \mathbb{Z}_{210} Wilson line generates a mass spectrum whose ratios are fixed by least common multiples of the primes \{2,3,5,7\}. States neutral under the \mathbb{Z}_p subgroups for p outside a subset S carry minimal charge q_S = 210/\mathrm{lcm}(S), acquire twist \theta_S = 1/\mathrm{lcm}(S), and have lightest Kaluza–Klein mass proportional to 1/\mathrm{lcm}(S). The resulting ratios are exactly 1:2:3:5:7, with no adjustable parameter. What This Paper Does The paper presents a self-contained phenomenological analysis of the \mathbb{Z}_{210} Scherk–Schwarz construction. The setup is standard: one extra dimension compactified on a circle, a Wilson line generating \mathbb{Z}_{210}, and twisted boundary conditions for fields carrying charge under the discrete group. The mass spectrum is derived as a theorem: for states neutral under exactly one \mathbb{Z}_p factor, the lightest masses are proportional to 1/\mathrm{lcm}(S), giving the ratios 1:2:3:5:7. The paper then works through the consequences: · Compactification scale: Fixing the lightest state at 172.5 GeV gives 1/R = 36.2 TeV, with light states at 172.5, 345, 517.5, 862.5, 1207.5 GeV, followed by a gap to the Kaluza–Klein tower near 35 TeV.· Selection rules: Charge conservation enforces q_1 + q_2 \equiv q_3 \pmod{210}. All couplings are fixed by charge arithmetic. If Standard Model fields are \mathbb{Z}_{210}-neutral, the lightest twisted state is absolutely stable.· Threshold degeneracy and bulk mass: With zero bulk mass, every kinematically allowed decay sits exactly at threshold. A non-zero bulk mass makes the Q-values negative, forbidding all cascade decays. The ratios 1:2:3:5:7 hold exactly only for m_0 = 0; requiring them accurate to 1\% bounds m_0 \lesssim 28 GeV.· Dark matter exclusion: A stable thermal relic annihilates through the bulk U(1) mediator, overclosing the universe by a factor \sim 10^6. Hypercharge and Higgs-portal remedies are excluded by direct detection by six and two orders of magnitude respectively. Therefore, \mathbb{Z}_{210} must be broken; the states decay and no relic survives.· Production and detectability: If the states are Standard Model singlets, production and decay are governed by the same operator, giving \sigma \times c\tau \simeq 3 \times 10^{-6} fb·cm. This means a displaced vertex and an observable rate are mutually exclusive—singlet states cannot be produced. The states must carry Standard Model gauge charge.· Colour charge exclusion: Colour-triplet scalars would be pair-produced at cross sections excluded by existing long-lived-particle searches. All five states would be ruled out.· Electroweak charge surviving window: Drell–Yan pair production estimates give event counts at HL-LHC: 172.5 and 345 GeV are excluded; 517.5 GeV sits at the current sensitivity boundary; 862.5 GeV is open and testable; 1207.5 GeV is likely out of reach. Why This Matters The Standard Model contains roughly two dozen parameters whose values are measured rather than predicted. This construction reduces one sector of the spectrum to a theorem: given a discrete symmetry, the mass ratios are fixed by least common multiples, with no free parameters. The phenomenological consequences are forced rather than chosen. The paper is explicit about what is derived and what is assumed: the Scherk–Schwarz geometry, the Wilson line, the \mathbb{Z}_{210} group, and the charge restriction on the light matter content are assumptions. The mass ratios, selection rules, threshold degeneracy, the relation \sigma \times c\tau = \text{const}, and the exclusion of singlet and coloured states are derived. The base scale is not derived; it is identified with 172.5 GeV for normalization. The construction is falsifiable: long-lived states with masses in the exact ratios 1:2:3:5:7; decays displaced, not prompt; no prompt resonant diboson decays at these masses; no direct-detection signal; a Kaluza–Klein tower near 36 TeV; and a bulk mass below \approx 28 GeV. Two long-lived states found at a ratio outside \{1,2,3,5,7\} refute the construction. A prompt resonance at any of these masses refutes it. A direct-detection signal attributable to a state at the base scale refutes it. The \mathbb{Z}_{210} Wilson line construction does not explain why the four primes \{2,3,5,7\} were selected. That origin is heuristic—a synchronization requirement in earlier work. The paper states this honestly. The construction explains what follows from those four primes, not why they are the ones that appear. Keywords: Scherk–Schwarz compactification, Wilson line, \mathbb{Z}_{210}, least common multiple, Kaluza–Klein spectrum, long-lived particles, electroweak charge, collider phenomenology, dark matter exclusion, falsifiable predictions