Conjugate monopoles as ontology

Conjugate monopoles are hypothetical magnetic monopoles, or other topological solitons, belonging to the program’s momentum-polarized matter sector rather than to ours. The checkpoint keeps one statement about them: if they exist, they gravitate, because every kind of matter in the program couples to the same geometry, so their existence could not be without consequence. Their use as the dark matter itself, the superheavy soliton candidate, was retired on 2026-10-19 after its abundance arithmetic failed review. What remains is a statement about what such objects would do, not a claim that they exist.

Status: Alive — [D] — #11 §11.1 calls the ontology “alive and consequence-bearing”. It is an argument from postulate P4, with no calculation or script behind it. The dark-matter candidacy it fed is retired (#11 §12).

The statement#

#11 §11.1, “The monopole audit, revised [D]”, after killing the explanatory half of founding supposition 5:

As ontology: alive and consequence-bearing — by P4, conjugate-sector solitons gravitate; “true but irrelevant” is the empty set (a correction supplied by P.R.). Their candidacy as the dark matter itself is retired in v2.1 (§12).

The italic opening of §11 gives the same point in plain words: “if such objects exist, they must matter, because everything gravitates”. The founding-supposition audit records the split: “as explanation, dead (wrong duality; inflation suffices); as ontology, alive and consequence-bearing (P.R.’s “true-but-irrelevant is measure zero”); the knot candidate it fed is retired (§12)” (#11 App. A item 5; founding suppositions and their fates; Obituary: conjugate monopoles explain the null result).

What the objects would be#

A topological soliton is a particle-like configuration of fields that wraps the space of a theory’s vacuum states and cannot be smoothly unwrapped, so it is stable without any conserved charge put in by hand [T]. A magnetic monopole of an SU(2) gauge theory broken to U(1) is the standard example (the ’t Hooft–Polyakov monopole). “Conjugate-sector solitons” are such objects made of the p-polarized matter fields χ rather than of ours.

The checkpoint does not specify the gauge structure of the p-polarized sector, so it does not say which solitons it has. The one concrete proposal was #14’s hidden SU(2) broken to U(1), which is dead as a program claim (Obituary: the hidden SU(2) → U(1) proposal). The ontology on this page is conditional: if the sector has such objects, the statement applies to them.

Why they would gravitate#

The argument is a single step from postulate P4 (#11 §4; P4: polarized matter and strict sector separation):

Matter comes in x-polarized (Standard Model) and p-polarized (χ) representations; both couple universally to H; gauge forces do not cross polarizations.

H is the positive generalized metric on phase space (postulate P1). On the vacuum slice it has the form H = diag(g, g⁻¹), with g the spacetime metric (the compatibility theorem; #11 §5), so universal coupling to H includes coupling to the metric of ordinary gravity. #11 §8 says the same of the χ particles: “χ propagates on g (equivalence principle; Bullet-Cluster-safe)” (the equivalence principle; the Bullet Cluster). An object built from p-polarized fields carries energy, and under P4 that energy sources the gravitational field and responds to it like any other.

The last clause of P4 fixes what they would not do. Since v2.1, P4 is taken strictly: “no kinetic mixing between the sectors beyond what gravity induces, which vanishes if hidden charge conjugation is exact and otherwise first appears at six loops, ε ≲ 10⁻¹³ [27S]” (#11 §4; round-3 Decision 1, round-3 decisions; kinetic mixing between sectors). A conjugate monopole would carry no charge under our photon and would not interact with our matter through any gauge force. Gravity is the coupling the postulates guarantee.

“True but irrelevant” is empty#

A hypothesis is true but irrelevant when the objects it posits exist and have no effect on anything that could be observed. P.R.’s correction is that P4 rules this case out for the conjugate sector: anything that exists there gravitates, and so has consequences in principle.

Two wordings, one status (unresolved)#

#11 states the correction in two forms. §11.1 calls “true but irrelevant” “the empty set (a correction supplied by P.R.)”. App. A item 5 records P.R.’s phrase as “true-but-irrelevant is measure zero”. The first says there is no such case, the second that such cases are negligible. The checkpoint does not reconcile the two, and the status, alive, is the same under either.

#11 does not list the consequences beyond gravitation. The statement is a constraint on what the sector’s objects can be, not a prediction of a signal.

The candidacy, retired#

The ontology once had a specific candidate attached.

  • v2.0 listed “topological solitons of a p-local gauge sector (§11) as a third candidate, stable by topological charge”, and claimed they “slip through the Debt-3 loophole (§9), since topology, not collective dynamics, would localize them in momentum” (#11 §12, ‘What was claimed’).
  • The handoff tied the surviving half of the split to that candidate: “As ontology they’re alive (the soliton candidate)”, and named as its “Current best DM candidate: superheavy (~10¹³–10¹⁷ GeV) topological solitons (π₂ monopole-class) of a separate, strongly coupled dual gauge sector” (HANDOFF.md §2; DM is dark matter; the handoff).
  • Round 3 found the candidate’s abundance misnormalized, and on 2026-10-19 it was retired as a program-native proposal (round-3 Decision 2; Obituary: superheavy conjugate solitons). #11 §11.1 now reads: “Their candidacy as the dark matter itself is retired in v2.1 (§12).”
  • What stands is the root CLAUDE.md’s summary: “The framework is merely compatible with known PeV-scale topological dark matter (prior art)” (topological dark matter in the literature).

The handoff’s coupling of the ontology to the candidate is resolved by precedence: the ontology is alive in #11 §11.1, and the candidacy is retired (conflicts between sources). The two remaining dark-matter candidates are χ particles and a condensate of the momentum graviton (#11 §12).

The laboratory shadow [T]#

#11 §11.1 ends the audit with an analog in condensed matter: “Their laboratory shadow is real: Weyl semimetals host catalogued families of momentum-space Berry-curvature monopoles [16†] — right geometry, different flux.”

In a crystal, an electron’s quantum state in each energy band depends on its crystal momentum, and the way the state’s phase turns as the momentum changes defines a field over momentum space, the Berry curvature [T]. In a Weyl semimetal two bands touch at isolated points in momentum space, and each touching point is a source or sink of Berry curvature with a quantized total flux: a monopole in momentum space [T]. In the wiki’s reading, this is the sense of “right geometry”: the monopole sits in momentum space, the space toward which the program’s conjugate sector is polarized. “Different flux” is the other half: Berry curvature is a property of the electron states, not a magnetic field.

Reference [16]†: S.-Y. Xu et al., “Discovery of a Weyl fermion semimetal,” Science 349, 613 (2015); B. Q. Lv et al., Phys. Rev. X 5, 031013 (2015). Both are cited from model memory and await the † literature battery. Prediction row 8 names these monopoles as “the only shadow” of a permanent null for monopoles of our sector (row 8).

Localization in momentum space#

The founding image of the program was dark matter that clumps in momentum space, and it died: collective momentum-space clumping is “nowhere mandatory” (#11 §9; Obituary: velocity clumping, the founding image; the net theorem). One logical gap was left for single topological objects, whose localization would come from topology rather than from collective dynamics. #11 §12 lists it among the survivors of the soliton obituary: “The Debt-3 loophole — topological momentum-space localization of individual objects — remains untested, but has no candidate” (Open: the topological loophole). App. A item 3 agrees: “topological p-localization of individual solitons untested, and its candidate is retired (§12)”.

#11 §14 item 8 words it differently: “The soliton-spectrum and p-localization items are closed by the §12 obituary”. The checkpoint does not reconcile “closed” with “untested” (open problem 8).

What the statement does not claim#

  • It does not claim that conjugate monopoles exist.
  • It does not make them the dark matter; that candidacy is retired.
  • It does not explain why no monopole of our sector has been found; that explanation is dead.
  • It does not give them any coupling to our matter other than gravity.

Sources#

  • papers/broken-born-checkpoint-v2.md §11.1 — the statement [D], the retired candidacy and the laboratory shadow; §11 italic opening; App. A items 3 and 5 — the loophole and the split with “measure zero”; §4 — P4, strict; §5 — H = diag(g, g⁻¹); §8 — χ propagates on g; §12 — the candidate as claimed in v2.0, its retirement and the loophole; §13 — row 8; §14 item 8 — “closed”; ref [16]†.
  • CLAUDE.md ‘Since the handoff’ — the soliton candidate dead; the framework merely compatible with prior art.
  • review/round3-adjudication.md ‘Decisions’ 1, 2 — strict sector separation; the candidate retired.
  • HANDOFF.md §2 — the earlier tie to the soliton candidate (superseded).