The soliton candidate (retired) on The Broken-Born program

Obituary: superheavy conjugate solitons

Mon, 01 Jan 0001 00:00:00 +0000

Superheavy conjugate solitons were the third dark-matter candidate of version 2.0 of the checkpoint: magnetic monopoles of a hidden gauge force, with a benchmark mass of 3×10¹³ GeV in the final version, stable because their field configuration cannot unwind and dark because no gauge force of ordinary matter acts on them. The proposal was written up as a standalone manuscript, #14, and reviewed blind three times. The third review re-ran every number in code and found that the count of monopoles formed in the early universe had been normalized far too low. Counted correctly, the benchmark mass is overproduced by a factor of 2.1×10³ to 1.2×10⁴, and the only version that survives is a much lighter one, which is an existing published scenario that does not depend on the program. The candidate was retired on 2026-10-19 and stays dead.

Topological dark matter in the literature

Mon, 01 Jan 0001 00:00:00 +0000

Topological dark matter is the established proposal that dark matter consists of topological defects, such as magnetic monopoles, left behind when a symmetry of a hidden sector breaks in the early universe. The defects are stable because their field configuration cannot unwind, and how many survive is set by how many form at the transition and how many then annihilate in pairs. Murayama and Shu found that such defects could be the dark matter at masses of 1–10 PeV, noting that Kibble’s simple count underestimates how many form; Liveoak, Maharana and Wells showed in 2026 that hidden monopoles overclose the universe if the symmetry breaks above about 100 PeV, unless a later epoch dilutes them. This low-scale scenario is what remains of the program’s retired soliton candidate once its abundance is counted correctly. Nothing in it depends on Born reciprocity, so the program claims compatibility with it and nothing more.

Obituary: the hidden SU(2) → U(1) proposal

Mon, 01 Jan 0001 00:00:00 +0000

The hidden SU(2) → U(1) proposal was the particle content the soliton manuscript #14 assumed for its dark sector: a hidden copy of a simple gauge force, SU(2)_d, whose Higgs field breaks it down to a single unbroken U(1)_d. The breaking leaves heavy charged W′± bosons, a massless hidden photon, and ’t Hooft–Polyakov monopoles that are stable for topological reasons, which were the dark-matter candidate. A light hidden fermion ψ was added because without it the W′± would overclose the universe. #14 identified this sector with the momentum-polarized half of the program’s matter, but nothing numerical in the proposal depended on that identification. As a model it is a standard hidden-sector monopole scenario already in the literature; as the program’s own proposal it died with the soliton candidate on 2026-10-19.

Obituary: the coupling α_d ≈ 34

Mon, 01 Jan 0001 00:00:00 +0000

α_d ≈ 34 was the strength the soliton candidate assigned to the interaction between its hidden monopoles: 1/(4α), the reciprocal of four times the fine-structure constant α ≈ 1/137 of ordinary electromagnetism. The program tried three ways to fix that value. The first applied Dirac’s charge-quantization condition using our photon’s charge, which contradicted the claim that the monopoles do not couple to our photon. The second had Born reciprocity act on gauge couplings as the modular S transformation of electric–magnetic duality; a referee called it numerology, and it was set aside. The third assumed that the smallest hidden charge equals the electron’s; it was motivated but not derived, and it fell with the candidate. The number itself shifts by a factor of four with the convention used. What survives is a general fact: a weakly coupled force always has strongly coupled monopoles.

Obituary: the Kibble normalization

Mon, 01 Jan 0001 00:00:00 +0000

The Kibble normalization is the number p in the estimate of how many monopoles a symmetry-breaking phase transition in the early universe leaves behind: the yield n/s = p (T_c/M_Pl)³, the number of monopoles per unit entropy, where T_c is the transition temperature and M_Pl the Planck mass. Version 3 of the soliton manuscript #14 used p ≈ 0.023 and treated p = 1 as an upper end. The round-3 referee showed that 0.023 is what the formula gives when the expansion rate of the universe is written without its numerical prefactor; with the prefactor, one monopole per Hubble volume gives p ≈ 104. Because causality requires at least about that many, the value is a minimum, not a maximum. The correction moved the mass at which the monopoles would make up all the dark matter from v3’s (0.9–3.7)×10¹³ GeV down to 2.9–4.4×10¹² GeV, and the candidate was retired.

Overproduction at the benchmark

Mon, 01 Jan 0001 00:00:00 +0000

Overproduction at the benchmark is the corrected abundance result that retired the soliton candidate. If one monopole forms per Hubble volume when the hidden symmetry breaks, the least that causality allows, the monopoles match the observed dark-matter density only for a mass M* of 2.9–4.4×10¹² GeV. Heavier monopoles are overproduced in proportion to the fourth power of their mass, because a heavier monopole forms at a higher temperature, where more of them form per unit entropy. At the benchmark of #14 v3, 3×10¹³ GeV, they would exceed the observed density 2.1×10³ to 1.2×10⁴ times, and realistic production at the transition makes the excess larger still. The result stands and is checked by the program’s verification gate. It is why a superheavy monopole could survive only with an arbitrary dilution factor, which would remove its mass prediction.

Obituary: the √N topological floor (chat-only)

Mon, 01 Jan 0001 00:00:00 +0000

The √N topological floor was an argument, made only in the claude.ai chat that preceded this repository, about how many of the soliton candidate’s monopoles could survive annihilation in the early universe. A monopole and an antimonopole that meet can annihilate, and at the hidden sector’s strong magnetic coupling, about 34, that annihilation might have been fast enough to deplete them. The floor argument held that the Kibble mechanism leaves a net topological charge that annihilation cannot remove, so the surviving abundance would lie between a Kibble “ceiling” and that floor. It was never written into any document. In round 3 the referee showed that Kibble’s count is a minimum, not a ceiling, and that annihilation at this coupling has already been worked out for hidden sectors; with the soliton candidate retired, the planned calculation was dropped.

W′ overclosure: the minimal sector is excluded

Mon, 01 Jan 0001 00:00:00 +0000

W′ overclosure is the result that the simplest version of the soliton candidate’s hidden sector — SU(2)_d broken to U(1)_d by an adjoint Higgs field and nothing else — is excluded by its own heavy charged gauge bosons, the W′±. With no lighter particle carrying hidden charge, the W′± cannot decay. The hot hidden plasma in which the monopoles form also fills the universe with W′±, and after they stop annihilating, their density would exceed the observed dark matter 10¹⁶ to 10¹⁷ times. #14 v3 therefore added a light hidden fermion ψ for the W′± to decay into. The calculation was done at the withdrawn benchmark mass, but the statement that the minimal sector is excluded survived the candidate’s retirement and is machine-checked.

Obituary: the ψ relic window

Mon, 01 Jan 0001 00:00:00 +0000

The ψ relic window was the range of masses #14 v3 allowed for ψ, the light hidden fermion it added so that the heavy hidden W′± bosons could decay. ψ is itself stable and left over from the early universe, so its mass sets how much of the dark matter it makes up: v3 wanted ψ light enough to be a minor component, so that the monopoles dominate, and heavy enough to have annihilated before nucleosynthesis. The round-3 review found both edges wrong. The upper edge was too high because v3 had left out the antiparticle and the second member of the doublet, a factor of 4, and had bracketed an exactly known annihilation cross-section over a factor of ten; the lower edge was too low because MeV-scale ψ would still be freezing out during nucleosynthesis. The window died with these corrections, and the candidate it served was retired.

The hidden sector's thermal history

Mon, 01 Jan 0001 00:00:00 +0000

The hidden sector’s thermal history is the record of how the soliton candidate’s hidden sector would have evolved as the universe cooled, and of the constraints that evolution had to meet. Its massless hidden photon, together with any light hidden fermion ψ, is extra radiation, which measurements of the cosmic microwave background limit through the effective number of neutrino species, N_eff. A hidden sector as hot as ours would add about 0.86 to N_eff, which is excluded, so #14 required the hidden sector to start colder, by a temperature ratio ξ below 0.77 — corrected in round 3 to 0.760 — yet computed its relic abundances as if ξ were 1. A first-order hidden phase transition would also have left gravitational waves at about 10⁵–10⁷ Hz. These are requirements and predictions of a retired candidate, kept as record.

Benchmark-conditional arithmetic

Mon, 01 Jan 0001 00:00:00 +0000

Benchmark-conditional arithmetic is the set of numbers that #14 v3 computed at its benchmark monopole mass of 3×10¹³ GeV and that remain correct as arithmetic, although the benchmark itself is withdrawn. At that mass, monopoles making up all the dark matter would be spaced about 420 m apart near the Sun; they would scatter off one another so weakly, σ_T/M ~ 10⁻²⁶ cm²/g, that they would behave as collisionless cold dark matter; and if they carried magnetic charge under our own electromagnetism, their flux would exceed the Parker bound by 7.5 orders of magnitude. That last, counterfactual number is why the hidden sector has to be separate from ours. The round-3 referee recomputed each value from independently chosen constants and confirmed it.

Kinetic mixing between sectors

Mon, 01 Jan 0001 00:00:00 +0000

Kinetic mixing is a coupling between the field strengths of two U(1) gauge forces, such as our photon and a hidden photon, through a term proportional to ε times the product of the two field strengths. When ε is not zero, particles charged under the hidden force acquire a small electric charge under ours, so kinetic mixing is the standard route by which a hidden sector can leak into the visible one. Three literature results framed the soliton candidate’s hidden sector: gravity alone produces mixing only at six loops or more, and none if hidden charge conjugation is an exact symmetry; while the hidden photon stays massless, mixing does not make hidden monopoles magnetic under our electromagnetism; and what leakage remains would give the hidden fermion ψ, and through the Witten effect the monopoles, a small visible electric charge. The program closed these channels by taking its postulate P4 strictly.

Obituary: Planck-suppressed adjoint mixing

Mon, 01 Jan 0001 00:00:00 +0000

Planck-suppressed adjoint mixing was #14 v3’s estimate of how strongly its hidden photon would mix with ours. The hidden Higgs field Φ permits a gauge-invariant interaction Φ^a F^a_μν B^μν / M_*, which links the hidden field strength to the field strength of our hypercharge; once Φ takes its vacuum value v, it produces a kinetic mixing ε ~ v/M_*. v3 assumed the operator appears with a coefficient of order one at the Planck scale, which gives ε ~ 10⁻⁷–10⁻⁶. The round-3 review found no source for that assumption: every paper that derives the operator obtains it at one loop from particles charged under both forces, which the manuscript’s own sector separation excludes, and the estimate those papers give is three orders of magnitude smaller. The estimate is dead, and the program then took sector separation strictly.