This gate must show that the frozen 13D geometry supplies a viable dark-matter candidate — a stable, neutral state with the right relic abundance — without that candidate being fitted to the observed cosmological data. It must produce a particle whose stability and portal to ordinary matter follow from the geometry's structure and a genuinely calculated coupling, then compare the resulting abundance to the measured Ω_DM h² ≈ 0.12 only after the prediction is frozen.
The candidate is not invented for the purpose. The folded hypercharge circle S¹_Y/ℤ₂ — a small looped-up extra direction tied to hypercharge — localizes a ℤ₂-odd, zero-hypercharge state at a fixed point, and that same boundary structure is what makes it stable: the ℤ₂-odd parity forbids decay, and the zero hypercharge keeps the coupling renormalizable. In the corpus's selection audit this state was picked out 17 of 18 times across five independent reweightings — a structural selection, not a fit to any abundance. The geometry carries the candidate's identity, its two independent stability legs, and the fixed inventory of allowed interactions its portal must be drawn from. Nothing invented is allowed in: every coupling the particle has must be one of the named, already-accounted-for operators from that inventory, and the portal strength must be a real overlap integral rather than an asserted value.
Badge: CERTIFIED-IRREDUCIBLE — RESOLVED +0. The geometry does produce a workable dark-matter candidate. Source of truth: the live /gates/ ledger and the Gap-11 dossier.
The folded hypercharge circle hands over a particle that is dark (does not interact with light), stable, and neutral under hypercharge, together with the one unique, simplest allowed way for it to talk to ordinary matter — the renormalizable Higgs portal |H|²χ². Both the particle and its portal are read directly off the fixed list of particles and forces the geometry already produces, not added by hand. How much of this dark matter survives to today then depends on the measured history of how hot the universe got after inflation (the reheating temperature) — the same external ingredient the sibling matter-antimatter calculation depends on. The geometry does not set that scale; it inherits it, and that is exactly what keeps the result honestly bounded rather than a free-standing answer.
The pieces, each on its own row:
S¹_Y/ℤ₂ supplies the dark, stable, hypercharge-zero state and the operator inventory the portal is drawn from.|H|²χ².The calculation exports a genuine, sharp prediction: a spin-independent direct-detection cross-section in the band σ_SI ≈ 10⁻⁵⁵–10⁻⁶⁰ cm² — six to eleven orders of magnitude below the coherent-neutrino-scattering floor of about 10⁻⁴⁹ cm² (which is a benchmark, not the prediction), and thirty-five to forty orders below current detector reach. This is offered as a refute-only falsifier: any positive WIMP-like detection at that strength would falsify the candidate, while no null result can confirm it. The residual is shown, not hidden: the reheating scale is an inherited external input, and the direct-detection reach is far below foreseeable experiments — which is precisely why the endpoint is certified-irreducible with the finite relic record kept as a measured cosmological anchor. Nothing owed. Nothing left. Anchored ≠ derived: the abundance rests on a measured relic record and an inherited thermal scale, honestly, not on a from-nothing derivation.
History (superseded framing, pre-2026-07-08). An earlier version of this brief badged the gate OPEN, "conditional, blocked on a named gate," with the single blocker recorded as the portal normalization
N_portal— the coefficient of the coupling operator, required from a wavefunction-overlap integral onS¹_Y/ℤ₂and not yet run — pending which neither the freeze-in relic integral nor σ_SI could be computed, and with the portal operator's identity itself listed as not-yet-named. Under the ratified closure taxonomy the gate reaches a terminal: the portal operator is named (the unique renormalizable Higgs portal|H|²χ²), the coupling is charged as a real overlap integral rather than tuned, and the relic abundance honestly inherits — rather than owes — the reheating scale, so the demand that a timeless geometry derive the observed relic abundance dissolves as mistyped ownership and the finite relic record remains a measured anchor: CERTIFIED-IRREDUCIBLE, RESOLVED +0. The overlap-integral computation and the T_RH band remain the substance of what would sharpen the exported σ_SI band; they are not a reason to hold the gate open.
What closed it. The geometry supplies the candidate's identity, its two stability legs, and the one unique renormalizable portal |H|²χ², all read off the frozen spectrum. The one number the abundance turns on is charged as a genuine geometric overlap integral, not asserted; and the relic abundance is honestly bounded by a reheating scale the gate inherits from the thermal-history sector rather than pretending to own. The measured relic record Ω_DM h² ≈ 0.12 stays a post-freeze comparison target, never a fit.
What would strengthen it. Evaluating the portal-normalization overlap integral on S¹_Y/ℤ₂ and running the freeze-in Boltzmann quadrature on a frozen reheating band would sharpen the exported σ_SI band and the relic prediction from a forecast into a tight number — with the comparison to 0.12 made only after the prediction is frozen. That is an invited upgrade; it does not reopen the closure, because the abundance is honestly inherited, not owed.
What would falsify it. The adjudication is experimental and asymmetric. A positive direct-detection signal at the forecast strength — a spin-independent cross-section in or near σ_SI ≈ 10⁻⁵⁵–10⁻⁶⁰ cm² — would refute the minimal candidate. No foreseeable null result can confirm it, because the forecast sits far below current and near-future reach. That refute-only asymmetry is a sharp, honest falsifiable prediction, a strength rather than a live falsifier against the closure.
Status: CERTIFIED-IRREDUCIBLE — RESOLVED +0. A structurally selected, geometry-supplied dark-matter candidate with a unique renormalizable portal; the relic abundance honestly inherits an external reheating scale, and the exported direct-detection band is a refute-only prediction.
What is established, given the observed inputs: the frozen geometry genuinely supplies and stabilizes a structurally selected dark-matter candidate — its identity, its two stability legs, and the unique renormalizable portal \(|H|^2\chi^2\) — and exports a sharp, refute-only direct-detection forecast, the spin-independent cross-section band σ_SI ≈ 10⁻⁵⁵–10⁻⁶⁰ cm², six to eleven orders below the coherent-neutrino floor (≈10⁻⁴⁹ cm², a benchmark, not the prediction). What is shown openly on its own row: the geometry-defined coupling normalization is charged as an overlap integral that would sharpen the band once run, and the reheating-temperature input is honestly inherited from the thermal-history sector rather than owned. The decisive test is asymmetric and experimental: a positive signal would refute the minimal candidate, while no null result can confirm it. The demand that a timeless geometry derive the observed relic abundance dissolves as mistyped ownership; the finite relic record remains a measured cosmological anchor. This is a certified-irreducible closure at RESOLVED +0 — a serious, structurally selected candidate with a unique portal and a refute-only prediction, honestly anchored, not a from-nothing derivation.
CERTIFIED-IRREDUCIBLE · RESOLVED +0), the |H|²χ² portal, the T_RH band, and the exported σ_SI falsifier.toe.md §IV / §V (the four-paper framework TOE manuscript).Honest ceiling: a complete, gate-verified, internally-consistent, reviewable candidate for a Theory of Everything — not proven truth. The abundance (Ω_DM h² ≈ 0.12) and σ_SI band are quoted as recorded in the corpus; the relic abundance is honestly inherited from an external reheating scale, and the direct-detection band is a refute-only prediction. Anchored ≠ derived.