A Discrete Wilson Line Origin for a Least-Common-Multiple Mass Spectrum

This paper presents a five-dimensional theory on M_4 \times S^1 carrying a discrete \mathbb{Z}_{210} holonomy that generates a Kaluza–Klein spectrum whose mass ratios are fixed by least common multiples of the primes \{2,3,5,7\}. A state neutral under the \mathbb{Z}_p subgroups for p outside a subset S has minimal charge q_S = 210 / \mathrm{lcm}(S) and lightest mass proportional to 1/\mathrm{lcm}(S). With one field per subgroup, the charges are precisely the sixteen square-free divisors of 210, and the mass ratios are those divisors, with no adjustable parameter. What the Paper Does The construction begins with a U(1) gauge field on M_4 \times S^1. A scalar of charge 210 acquires a VEV, breaking U(1) \to \mathbb{Z}_{210}. The surviving flat connections are classified by \pi_1(S^1) \to \mathbb{Z}_{210}, giving a discrete Wilson line. A field of charge q then has a Kaluza–Klein spectrum m_{q,n}^2 = m_0^2 + (n + q/210)^2/R^2. Taking k=1 and m_0=0, the lightest mode for |q| < 105 has mass m_q = q/(210R). The key result is that every subgroup of \mathbb{Z}_{210} contributes a state, and the charges run over the square-free divisors of 210: 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210. The absence of states at q=4,8,9,\dots is as much a prediction as the presence of those listed. What This Paper Does That Is Novel The mass ratios are fixed by least common multiples with no free parameter. The compactification scale is set by identifying the lightest state with the top quark mass: 1/R = 210 \times 172.5 GeV = 36.2 TeV. The spectrum includes six states below 1.5 TeV: 172.5, 345, 517.5, 862.5, 1035 and 1207.5 GeV. The paper then derives the phenomenology: · The vacuum structure is topological: Because \mathbb{Z}_{210} descends from a spontaneously broken U(1), the holonomy is classified by \pi_1(S^1) \to \mathbb{Z}_{210} and cannot relax, provided the breaking scale v exceeds the compactification scale. Distinct k label superselection sectors separated by barriers, with no continuous direction along which to relax.· Bulk supersymmetry breaking is decoupled: The bulk U(1) boson is made heavy by the charge-210 VEV, suppressing gauge transmission of brane supersymmetry breaking into the bulk. Gravity mediation bounds \sqrt{F} \lesssim 8 \times 10^9 GeV.· Radiative corrections close cascade decays: A positive radiative mass shift \delta m^2 > 0 makes all Q-values negative, so cascade decays are forbidden and the c\tau \propto 1/q^3 lifetime law holds. The sign is load-bearing: if cascades opened they would proceed at gauge strength, destroying the prediction.· The charge assignment is not free: Triplet and higher representations generate a radiative bulk mass exceeding the bound that the mass ratios themselves impose. Only singlet and doublet assignments survive. The radiative correction also shifts the mass ratios below their integer values by a calculable amount: \sim 0.2\% for a singlet, \sim 0.8\% for a doublet. Measuring two masses at the percent level distinguishes which assignment is realised.· The states cannot be dark matter: Hypercharge and Higgs-portal options are excluded by direct detection. The lightest twisted state overcloses the universe by \sim 10^6 if stable.· The states cannot be Standard Model singlets: Production and decay share one operator with \sigma \times c\tau \simeq 3 \times 10^{-6} fb cm. A displaced vertex and an observable rate are mutually exclusive.· Colour is excluded: Long-lived-particle searches exclude coloured states.· What survives: An electroweak-charged scenario in which the two lightest states (172.5 and 345 GeV) are excluded, one state (517.5 GeV) sits at current sensitivity, and two states (862.5 and 1035 GeV) are testable at the HL-LHC. Why This Matters The mass spectrum is not fitted — it is derived from the group structure. The lifetimes are not independent — they are locked to masses by c\tau \propto 1/q^3 with no free parameter. The breaking scale \Lambda is bounded to two decades by requiring resolvable displaced vertices. The charge assignment is identified by the size of the radiative mass shift. Measuring any two states fixes \Lambda twice over; every remaining lifetime is then predicted with no freedom. This is a stronger test than the mass ratios alone. What Remains Open · The origin of \mathbb{Z}_{210} is assumed (the periods \{2,3,5,7\} arose from a heuristic synchronisation requirement in earlier unpublished work).· One field per subgroup of \mathbb{Z}_{210} is assumed, not derived.· The base scale is not derived; identification of the lightest state with the top quark mass is a normalisation.· Which of the two allowed charge assignments is realised is not determined by the construction; the two are distinguished by the size of the mass-ratio deviation. Keywords: discrete Wilson line, Kaluza–Klein spectrum, least common multiple, square-free divisors, \mathbb{Z}_{210} holonomy, Scherk–Schwarz compactification, displaced vertices, long-lived particles, electroweak-charged scalars, HL-LHC