The soliton candidate (retired)

The soliton candidate was the proposal that dark matter consists of superheavy magnetic monopoles of a hidden gauge force: configurations of a hidden field that wrap the space of its vacuum states and cannot unwind, so they would be stable without any further assumption. Version 2.0 of the checkpoint listed them as a third dark-matter candidate, and the idea was written up as a standalone manuscript, #14, for blind review. The third review recomputed every number with executed code and found that the estimate of how many monopoles the early universe forms had been too low by a factor of a hundred to several thousand; counted correctly, superheavy monopoles are made in far too great a number. The candidate was retired on 2026-10-19. This subsection records what was proposed, each component that failed and why, the results that survive, and the published scenario that the surviving low-mass version duplicates.

Stands: overproduction at the benchmark and W′ overclosure: the minimal sector is excluded, both [V] · Record, conditional on the withdrawn benchmark: the hidden sector’s thermal history and benchmark-conditional arithmetic · Died: superheavy conjugate solitons, the hidden SU(2) → U(1) proposal, the coupling α_d ≈ 34, the Kibble normalization, the √N topological floor (chat-only), the ψ relic window, Planck-suppressed adjoint mixing · Superseded: no page of its own; folded sections on reciprocity acting as modular S, F2, which coupling sets the soliton mass, the convention and running spread, the worklog’s “point, not a window” and the handoff’s “current best” candidate · Open: none here; two open items raised during the episode sit with their subjects, stability at equal Planck masses and the two dual cones · Conjectures: popsci-only, folded: the favoured corner of the three-way question

Map#

The record of the episode is the manuscript #14, notes/knot-dark-matter-standalone.md, in three versions, with checkpoint #11 §12 and the round-3 adjudication as its current summary and verify/soliton_obituary.py as the check on its numbers (the soliton-obituary script); flag 33 in review/wiki-build-flags.md records which of the record’s numbers the script does not check.

  • Version 1 called the objects “knots” and fixed their coupling by Dirac quantization against our photon. It is not in the repository. Round 1 of blind review graded it “B / medium” and found the coupling argument self-contradictory; among its valid hits, folded into v2, was the renaming of the “knots” as “π₂ monopole-class” (HANDOFF.md §2, §6).
  • Version 2, “Dark Matter as Superheavy Magnetic Solitons of a Spontaneously Broken Phase-Space Duality”, set the hidden coupling by “the reciprocity map, acting as the modular S transformation”, to “alpha_d = 1/(4 alpha) ~ 34”, took the Kibble normalization at “p = O(1)” and used a benchmark “M = 1e16 GeV”. It survives verbatim inside review/skeptic-review-console.html, the tool used for rounds 1 and 2. Round 2 graded it “B near B/C / medium” and left a pending list of seven items (HANDOFF.md §6).
  • Version 3 (commit 5dbaa82), “Dark Matter as Superheavy Monopoles of a Hidden Gauge Sector Selected by a Spontaneously Broken Phase-Space Duality”, was cut from that list in review/v3-worklog.md, a chunked log of the round-2 items, new findings and Paul’s decisions, closed after round 3 (“This worklog is closed.”). v3 placed the monopole mass “at about 1e13 to 4e13 GeV with no dilution required”. Round 3, a blind review with code execution, graded it C on 2026-10-19 (review round 3). It was withdrawn the same day, moved from papers/ to notes/, and given a banner: “Kept as the audit trail; not for circulation.”

Of the thirteen pages below, seven are obituaries, two are results that stand [V], two are record and two are established literature (prior art and kinetic mixing). They open with the death and the published scenario it reduces to; the components follow, from the hidden sector and its coupling, through the abundance and the particles it forced, to the mixing with ordinary electromagnetism.

  • Obituary: superheavy conjugate solitons comes first, because the death is the subject of the section. It sets out what #11 v2.0 and the three versions of #14 claimed, how the candidate was tested in three blind reviews, why the corrected Kibble count overproduces the benchmark 2.1×10³–1.2×10⁴ times, the thirteen round-3 rulings and the scoring of the pre-registered prediction, Decision 2’s three routes and the retirement, what survives, and #14’s own self-assessment. The handoff’s framing of the solitons as its current best candidate is folded in as superseded, and the popular articles’ favoured corner as popsci-only.
  • Topological dark matter in the literature is the established scenario the corrected candidate reduces to: defects of a hidden sector as relics, set by formation and annihilation; Murayama–Shu’s hidden monopoles at 1–10 PeV and Liveoak–Maharana–Wells’s overclosure above about 100 PeV [28S, 29S]; annihilation at strong magnetic coupling as Preskill’s solved case; the uncited prior art; and why the program claims only that it is “merely compatible”.
  • Obituary: the hidden SU(2) → U(1) proposal is the sector #14 postulated: an adjoint Higgs field breaking SU(2)_d to U(1)_d, leaving heavy W′±, a massless hidden photon, stable monopoles, and a light fermion ψ. The page explains each piece, why a mirror copy of the Standard Model has no stable monopoles, how v2’s unnamed group became v3’s sector, and why the model, standard and already published, is dead as the program’s own.
  • Obituary: the coupling α_d ≈ 34 starts from the Dirac relation α_m = 1/α_W′ between a hidden sector’s electric and magnetic couplings and records the three attempts to fix the value: Dirac quantization against our photon (round 1’s valid kill), reciprocity acting as modular S (superseded), and hypothesis H (dead with the candidate), with the worklog’s F2 reframing and the factor-of-four convention spread folded in as superseded.
  • Obituary: the Kibble normalization is the error that killed the candidate. It derives the yield n/s = p (T_c/M_Pl)³ for one defect per Hubble volume, shows that v3’s p ≈ 0.023 drops the Friedmann prefactor (1.66√g*)³ where the correct causal minimum is p = 104.31, explains why the count is a floor and why the reduced-Planck “factor of 3” was an artefact, and records why v3’s own check passed.
  • Overproduction at the benchmark is the corrected result, which stands [V]: the model and inputs behind it, the relation for the closing mass as the gate codes it, the abundance closing at M* = 2.9–4.4×10¹² GeV and the 3×10¹³ GeV benchmark overproduced 2.1×10³–1.2×10⁴ times, the colder-sector and Kibble–Zurek figures that only add to it, and the worklog’s earlier “point, not a window” as superseded.
  • Obituary: the √N topological floor (chat-only) is a bound on the abundance that existed only in the claude.ai chat, between a Kibble “ceiling” and a net-charge floor. Round-3 Decision 3 dropped it because its premise, the Kibble count as a ceiling, is inverted, and because with the candidate retired it had no candidate left to serve; the strong-coupling annihilation it addressed had already been worked out for hidden sectors (ruling 10).
  • W′ overclosure: the minimal sector is excluded is a result that survives the candidate [V]: with no light charged fermion the stable, thermal W′± overclose the universe by 10¹⁶–10¹⁷, which forced the addition of ψ. The page gives the gated freeze-out inputs, the annihilation coefficient by detailed balance and the colder-sector check.
  • Obituary: the ψ relic window is the mass range v3 allowed for ψ. Both edges were wrong: the upper by a miscount of states and an exact cross-section bracketed by a factor of ten, the lower because MeV-scale ψ would freeze out during nucleosynthesis.
  • The hidden sector’s thermal history is record: the dark radiation that requires a colder hidden sector, ξ < 0.760, its inconsistent use in v3, what a colder sector would need, and the gravitational-wave band a first-order hidden transition would leave, which died with the candidate.
  • Benchmark-conditional arithmetic is record: numbers that remain correct as arithmetic at the withdrawn benchmark — densities and spacing, the self-interaction σ_T/M ~ 10⁻²⁶ cm²/g, and the counterfactual flux that shows why the hidden sector must be separate from ours.
  • Kinetic mixing between sectors is the established literature on how a hidden photon can mix with ours: gravity-induced mixing and its protection, no visible magnetic flux while U(1)_d is unbroken, and the leakage channels that strict sector separation closes.
  • Obituary: Planck-suppressed adjoint mixing comes last: v3’s dimension-5 estimate of the mixing, whose order-one coefficient at the Planck scale had no source. With it gone, the program took postulate P4 strictly.

Boundaries#