The X-condensate candidate

The X-condensate candidate is the proposal that dark matter is a coherent, spatially uniform oscillation of X, the program’s momentum graviton: the massive spin-2 field that the broken reciprocity adds to gravity. If it holds, dark matter and the hidden half of the geometry are one field rather than two separate things. The idea was logged during the Debt-1 work as a direction deliberately left unopened, became one of three dark-matter candidates in version 2.0 of the checkpoint, and has been one of two since the third was retired. The record holds its profile and its history, not a calculation; like the χ-particle candidate, its abundance is unexplained.

Status: Alive — one of two candidates; #11 §12 tags the pair “[D — framework; each candidate O]”. The field it is made of is derived ([D + T], #11 §7.1); that the dark matter is made of it is open. Its lineage, bigravity dark matter, is cited from model memory [23†]; flag 20 in review/wiki-build-flags.md suspects that #11’s entry for it pairs one paper’s title with another’s arXiv number. Nothing on this page is machine-checked.

The candidate#

Checkpoint #11 §12 lists two candidates for dark matter, each with its origin, what makes it stable, and the observable that would single it out. The second row:

DM there is dark matter. The tag “[D — framework; each candidate O]” says that the framework producing the candidates is derived and each candidate is open. The other row is χ particles, and “The 5.4 : 1 abundance question (polarization-genesis) applies to both” (#11 §12).

What X is#

The program’s geometry lives in one field on phase space, the modular field Z = X + iY. Its imaginary part is the spacetime metric, Im Z = Y = g. Its real part is “Re Z = X, a new symmetric tensor: the momentum graviton. It is exactly the fluctuation that breaks η-compatibility — i.e., the local wobble of the emergent polarization” (#5 §C). X is the field that measures how the split of phase space into position and momentum varies from place to place.

Debt 1 established what kind of field X is (why X must be massive). In #11’s summary [D + T]: X is shift-symmetric and so “never auxiliary”; its local symmetry makes it “a second linearized-diff gauge field”, that is, a second graviton; a second massless graviton is excluded by the multi-graviton no-go [8†]; and “The polarization choice spontaneously breaks the dual diffs; X eats the four Goldstones and emerges massive, M_X = the polarization-breaking scale” (#11 §7.1; the gravitational Higgs mechanism). X is therefore a massive spin-2 field with five polarizations, governed by Hassan–Rosen bimetric gravity restricted to the palindromic branch.

The properties that matter for a dark-matter candidate are in #11 §7.5 [D]:

• Mass. “Five βₙ → two parameters (M_X ∝ 2(β₁+β₂); one cubic shape)”. No document gives a value for M_X. Calc note #8 §3, the source of the count, writes the proportionality for the squared Fierz–Pauli mass, “m_FP² ∝ β₁ + 2β₂ + β₃ = 2(β₁+β₂), i.e. M_X”; the difference is recorded in review/wiki-build-flags.md item 5, and this page quotes both as written (spectrum, parameters and reciprocity parity).
• Parity. The massless graviton is “reciprocity-even” and X is “reciprocity-odd”; “Reciprocity is a selection rule on vertices” (#8 §3; on the parities under #11 §6’s Z → −Z⁻¹, see flag 39 in review/wiki-build-flags.md).
• Couplings to ordinary matter. “SM couplings gravitational-strength and Yukawa-cut at 1/M_X” (SM is the Standard Model; Yukawa forces). With equal effective Planck masses, “SM matter — coupled to g — sees both eigenstates with comparable strength; safety is by range, not decoupling” (#8 §3; safety by range).

What “condensate” means here#

Calc note #6 states the idea in one sentence: “homogeneous oscillations of a massive spin-2 X are themselves a dark-matter candidate (cf. bigravity dark matter)” (#6 §4). A scalar field that oscillates uniformly through space at a frequency set by its mass behaves, averaged over the oscillation, as pressureless matter, its energy diluting with the cube of the scale factor like cold dark matter [T]; for a light scalar this is the misalignment mechanism. The uniform mode of a spin-2 field is a tensor and singles out directions in space; that case and its literature are on bigravity dark matter. Such a state is coherent: the field moves in phase everywhere, like a single standing wave, rather than as a gas of independent quanta [T].

The program’s record does not go further. No document computes how the oscillation would be produced in the early universe, its amplitude, or its energy density.

The three handles#

The last column of the table names three distinguishing features. #11 states them and does not derive them; what the record says about each:

• “unifies dark sector with geometry”. Calc note #6: “BBT may not need χ as a separate field — X-condensate dark matter would unify the geometric sector with the dark sector” (#6 §4; BBT is the Broken-Born theory). #9 phrased it as “potentially eliminating χ and unifying the dark sector with the geometry” (#9 §10 item 7). If the candidate holds, the dark matter is a state of the geometric field X, not a separate matter field.
• “coherent-field signatures”. Not specified further in any document. Row 11 relies on “coherence signatures”, with rows 1–3, to separate particles from a condensate.
• “knee shape wave-like”. #11 gives no derivation for this entry. In the predictions table the wave-like tail of the self-interaction knee is “(v*/σ)⁴”, derived for wave χ on the wave island (#11 §9; row 3). No document derives a knee for a condensate of X or relates it to wave χ.

The stability entry, “m_X-protected”, is likewise stated without an argument. No document computes a lifetime for the oscillation.

Its tests#

• Row 7: a permanent null. “Permanent null (both dark-sector candidates SM-decoupled; P4 strict)”, killed by “Any confirmed non-gravitational DM–SM coupling” (#11 §13). X does couple to ordinary matter, at gravitational strength and with a range cut off at 1/M_X (#11 §7.5); the null concerns couplings other than gravity.
• Row 11: particles or condensate. “Particles vs condensate, separable via rows 1–3 + coherence signatures”, marked “Diagnostic” (#11 §13). How to make the separation is open: “Discriminating the two remaining dark-sector candidates (§12) [O]” (#11 §14 item 9).

What is open#

• Abundance. “The 5.4 : 1 abundance question (polarization-genesis) applies to both” (#11 §12; Open: polarization-genesis and the 5.4 : 1 ratio).
• Telling it from χ. #11 §14 item 9, above.
• Cosmological stability of X. Reciprocity forces equal effective Planck masses, which “rules out the M_f ≪ M_g escape” in which a heavy spin-2 field has been shown linearly stable at late times (bimetric cosmology and its instabilities); whether “a large mass alone is enough at M_f = M_g is open” (review/v3-worklog.md, item R7; Open: stability at equal Planck masses). The question concerns X itself, and through it this candidate.
• No calculation yet. Production, amplitude, lifetime and the coherence signatures are all uncomputed.

Not the same condensate#

The word “condensate” appears twice in the dark-matter record. On the wave island of Debt 3, very light χ can condense into Bose stars (wave kinetics and Bose-star formation). Those are condensates of the matter field χ. The X-condensate is a state of the geometric field X. No document relates the two.

History#

• A door left closed (calc note #6). “A new door, deliberately left closed today: homogeneous oscillations of a massive spin-2 X are themselves a dark-matter candidate (cf. bigravity dark matter). BBT may not need χ as a separate field — X-condensate dark matter would unify the geometric sector with the dark sector. Logged as an open direction, not pursued. [O]” (#6 §4). #6 §5 lists “X-condensate dark matter (§4)” among what is “Also now open”.
• The lab notebook. Addendum v0.4 records “A new open direction: massive-X condensate as the dark matter itself (bigravity-DM-style), potentially unifying the dark sector with the geometry.”
• #9. Open problem 7: “X-condensate as the dark matter itself (bigravity-DM-style), potentially eliminating χ and unifying the dark sector with the geometry [O — direction, deliberately unopened]” (#9 §10). Its row 7 already covers it: “χ (or X-condensate) has no non-gravitational SM coupling” (#9 §9).
• #11 v2.0 and the handoff. The candidate became one row of a three-way choice with χ particles and superheavy solitons. The handoff, HANDOFF.md §2, lists “Dark-sector trilemma discrimination: particles vs X-condensate vs solitons” as open item 8.
• #11 v2.1. After round 3, the soliton candidate was retired on 2026-10-19 and “the trilemma becomes two program-native candidates (gated χ particles; the X-condensate)” (review/round3-adjudication.md, Decision 2; Obituary: superheavy conjugate solitons).
• The popular articles. Both predate the retirement and describe the three-way question. #10 lists “a condensate of the twin-graviton field itself” (‘Dividends and daydreams’); #12 writes “a condensate of the massive geometric field itself, in which case dark matter and the hidden geometry are one substance” (‘Dark matter as the other polarization’). Neither is a source for the candidate’s status (the popular articles).

Sources#

• papers/broken-born-checkpoint-v2.md §12 — the candidate profile and its tag; §7.1 — X forced massive; §7.5 — mass, parity and couplings; §9 — the wave-island tail; §13 — rows 3, 7 and 11; §14 item 9 — discriminating the two candidates
• notes/debt1a-ii-X-classification.md §4 — the door “deliberately left closed”; §5 — listed as open
• notes/debt1a-geometric-core.md §C — X as Re Z, “the local wobble of the emergent polarization”
• notes/debt1a-iv-beta-coefficients.md §3 — m_FP², equal Planck masses, safety by range
• notes/broken-born-theory.md ‘Addendum (v0.4)’ — the open direction
• notes/broken-born-program-v1.md §10 — open problem 7 of #9; §9 — row 7 of #9
• HANDOFF.md §2 — open item 8, the trilemma (superseded)
• review/round3-adjudication.md ‘Decisions’ 2 — three candidates become two
• review/v3-worklog.md ‘Round-2 pending list’ — R7, stability at equal Planck masses
• review/wiki-build-flags.md item 5 — M_X versus m_FP²; item 20 — the title and arXiv number of reference [23]
• popsci/angle-of-the-mirror.md ‘Dividends and daydreams’; popsci/universes-other-half.md ‘Dark matter as the other polarization’ — the three-way framing