Dark Matter from Gap

Who This Chapter Is For

Dark matter candidates within Gap theory. The reader will learn why the $O$-sector relic is the most viable candidate and why the QCD axion claimed here needs a Peccei–Quinn sector that the Clifford content lacks (T-333(e)–(h)).

Overview

Gap theory provides a systematic framework for analysing dark matter candidates. Standard SUSY candidates are excluded (too heavy or unstable). The most viable candidate is the $O$-sector relic (Wimpzilla, $m \sim 10^{13}$ GeV), gravitationally produced during inflation and stabilised by $O$-parity. A subdominant QCD axion ($m_a \sim 3$ neV, $\sim 1\%$ DM) was additionally claimed. In the Clifford content it is not a QCD axion, and its relic estimate conflicts with the isocurvature bound (T-333(e)–(h), §3).

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1. Criteria for a Candidate

From observations (Planck 2018): $\Omega_{\mathrm{DM}} h^2 = 0.120 \pm 0.001$. A candidate must satisfy [D]:

1. Electric neutrality and absence of colour charge
2. Stability ($\tau \gg t_{\mathrm{Universe}} \sim 10^{10}$ yr)
3. Correct relic density $\Omega h^2 \approx 0.12$
4. Consistency with direct detection (XENON, LZ: $\sigma < 10^{-47}$ cm$^2$ for $m \sim 10^2$ GeV)

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2. Exclusion of SUSY Candidates

Theorem 8.1 [T]

Standard SUSY dark matter candidates are excluded in the Gap formalism:

Candidate	Mass	Problem	Status
Neutralino	$m \sim 10^{13}$ GeV	$\Omega h^2 \sim 10^{32} \gg 0.12$ (overproduction)	Excluded
Gravitino	$m_{3/2} \sim 10^{13}$ GeV	$\tau \sim 7 \times 10^{-26}$ s (unstable)	Excluded
Wino/Bino	$m \sim 10^{11}$ GeV	Analogous to neutralino	Excluded

Conclusion: The SUSY sector of Gap theory contains no viable DM candidate.

2.1 Complete Candidate Overview

Candidate	Mass	Stability	$\Omega h^2$	Status
Neutralino	$\sim 10^{13}$ GeV	Stable (R-parity)	$\sim 10^{32}$	Excluded
Gravitino	$\sim 10^{13}$ GeV	$\tau \sim 10^{-25}$ s	—	Excluded
Wino/Bino	$\sim 10^{11}$ GeV	Stable	$\gg 0.12$	Excluded
Gap instantons	$\sim \Lambda_{\mathrm{QCD}}$	Stable (topology)	—	Excluded (hadronic)
$G_2$-extra bosons	$\sim M_P$	Stable ($G_2$ charge)	$\sim 10^{-6}$	Excluded (too little)
QCD axion	$\sim 3$ neV	Stable ($U(1)_{\mathrm{PQ}}$)	$\sim 10^{-3}$	Not realised in the Clifford content; needs (PQ) (§3, T-333(e)–(h))
Dark ALPs	$\sim 10^{15}$ GeV	Stable	Heavy	Excluded (§4)
$O$-sector relic	$\sim 10^{13}$ GeV	$\tau \gg t_U$ ($O$-parity)	$\sim 0.1$–$0.4$ [C at T-50, CKR]	Primary candidate (§5)

Gap instantons are topological configurations $\theta_{ij}(x)$ with non-zero winding number — hadronic objects ($m \sim \Lambda_{\mathrm{QCD}}$). Excluded by observations (BBN, CMB, structure formation).

$G_2$-extra bosons — 6 bosons with masses $\sim M_P$. Under gravitational production $\Omega \sim H_I/M_P \sim 10^{-6}$. Too little to be the primary DM.

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3. QCD Axion from $(S^1)^{21}$ Compactification

Role of the Axion in the Gap Formalism

In standard physics the Peccei–Quinn axion solves the strong CP problem via dynamical relaxation $\theta_{\mathrm{QCD}} \to 0$. In the Gap formalism $\theta_{\mathrm{QCD}} = 0$ follows structurally from the reality of the octonionic $f_{ijk}$ and vacuum uniqueness (T-99 [T]). The Gap axion is therefore a purely DM candidate, not a solution to the CP problem. Retracted 2026-09-26 (T-99 corrected): $\theta_{\mathrm{QCD}} = 0$ does not follow; T-99's conclusion is [✗], $\bar\theta$ is a free parameter and strong CP is open [Pr] (Confinement §3.1c).

Correction 2026-09-26 (T-333(e)–(h)): in the Clifford content this axion is not a QCD axion

Confinement §3.1b (T-333(e)–(h)) shows that with the fields the Clifford frame forces — three generations of the $\mathbf{16}$ and one doublet — no $\mathrm{U}(1)$ with a colour anomaly survives the quark masses. So a Gap phase has no $G\tilde G$ coupling, and the "axial anomaly with QCD" of §3.1 needs a Peccei–Quinn sector that the frame lacks: a second doublet with a singlet, or new coloured fermions. Call this addition the hypothesis (PQ). Three more points. (1) The mass formula of §3.3 needs the potential of $a$ to come from QCD alone, while §3.5 gives every phase a mass from $V_{\mathrm{Gap}}$; the two cannot both hold. (2) "$\theta_{\mathrm{QCD}}=0$ structurally" rests only on the retracted cubic $V_3$ (T-99 was [C at (SV)]; since 2026-09-26 its conclusion is retracted [✗], Confinement §3.1c). An axion with a $G\tilde G$ coupling and a QCD-dominated potential relaxes $\theta$ whatever else holds, so it cannot be "purely a DM candidate". (3) The relic estimate of §3.4 takes $\theta_i=H_I/(2\pi f_a)$, a pure fluctuation around $\theta=0$. The axion density is then isocurvature with relative amplitude of order one, and the Planck bound allows $\Omega_a/\Omega_c\lesssim3\times10^{-5}$, not $10^{-2}$. Theorem 9.1 is corrected from [T] to [H] ($\epsilon\sim10^{-3}$ and $N_{\mathrm{DW}}=1$ are assumed), and Theorems 9.2 and 9.3 from [T] to [C at (PQ)]. The arithmetic stands: $f_a=2\times10^{15}$ GeV gives $m_a=2.9$ neV.

3.1 Definition

The Gap axion is a pseudoscalar field $a(x)$, the zero mode of the phases $\theta_{ij}$ in the $3$-to-$\bar{3}$ sector, possessing an axial anomaly with QCD:

$$a(x) := f_a \cdot \frac{1}{3} \sum_{i \in \{A,S,D\}} \sum_{j \in \{L,E,U\}} c_{ij}\, \theta_{ij}(x)$$

where $c_{ij}$ are coefficients determined from the anomaly condition $\partial_\mu j^\mu_A = \frac{g^2}{32\pi^2} G\tilde{G}$. In the Gap vacuum $\theta_{\mathrm{QCD}} = 0$ exactly (T-99 [T]); the axion $a$ describes fluctuations $a \propto \delta\theta$. (Retracted 2026-09-26: T-99's conclusion is [✗]; $\bar\theta$ is free, §3.1c.)

3.2 Decay Constant

Theorem 9.1 [H] (corrected 2026-09-26 from [T])

The decay constant of the Gap axion:

$$f_a = \frac{\epsilon \cdot M_P}{N_{\mathrm{DW}}} \approx 2 \times 10^{15}\;\text{GeV}$$

for $\epsilon \sim 10^{-3}$, $M_P = 2.4 \times 10^{18}$ GeV, $N_{\mathrm{DW}} = 1$ (number of domain walls for the simplest realisation).

Canonical normalisation: from the kinetic term $\mathcal{L}_{\mathrm{kin}} = \frac{1}{2}m_{ij}(\partial_\mu\theta_{ij})^2$, where $m_{ij} \sim \epsilon^2 M_P^2$ in the $3$-to-$\bar{3}$ sector, one finds $f_a = \sqrt{m_{ij}} = \epsilon \cdot M_P$. Including RG evolution: $\epsilon(\mu_{\mathrm{GUT}}) \sim 10^{-3}$, $\epsilon(\mu_{\mathrm{EW}}) \sim 10^{-2}$; axion physics is determined at the GUT scale.

3.3 Axion Mass

Theorem 9.2 [C at (PQ)] (corrected 2026-09-26 from [T])

The mass is determined by QCD instantons:

$$m_a = \frac{\sqrt{m_u m_d}}{m_u + m_d} \cdot \frac{m_\pi f_\pi}{f_a} \approx 3 \times 10^{-9}\;\text{eV} = 3\;\text{neV}$$

An ultralight axion within the sensitivity range of the CASPEr and ABRACADABRA experiments.

3.4 Relic Density

Theorem 9.3 [C at (PQ)] (corrected 2026-09-26 from [T])

From the vacuum misalignment mechanism:

$$\Omega_a h^2 \approx 0.12 \times \left(\frac{f_a}{10^{12}\;\text{GeV}}\right)^{7/6} \times \left(\frac{\theta_i}{\pi}\right)^2 \approx 10^{-3}$$

For $\theta_i \sim H_I/(2\pi f_a) < 3.7 \times 10^{-3}$ (from the Planck bound $r < 0.036$):

$$\Omega_a h^2 \approx 0.12 \times 7100 \times 1.39 \times 10^{-6} \approx 1.2 \times 10^{-3}$$

Conclusion [C]: The QCD axion constitutes $\sim 1\%$ of the observed dark matter — a subdominant component (subject to $\epsilon \sim 10^{-3}$ and $N_{\mathrm{DW}} = 1$). Corrected (T-333(e)–(h)): only under (PQ), and with $\theta_i$ set by the inflationary fluctuation the isocurvature bound caps the fraction near $3\times10^{-5}$.

3.5 Full Axion Spectrum from $(S^1)^{21}$

Compactification on the torus $(S^1)^{21}$ generates the full spectrum of axion-like particles (ALPs). Of the 21 compact phases $\theta_{ij}$, the mass spectrum is determined by the sectoral structure of the Gap vacuum:

Hypothesis [H]

Mass spectrum of the multi-axion system from $(S^1)^{21}$:

Sector	Number of modes	Mass scale	Mass-generation mechanism
$3$-to-$\bar{3}$: QCD axion	1	$m_a \sim 3$ neV	QCD instantons
$3$-to-$\bar{3}$: gluonic	8	$m \sim \Lambda_{\mathrm{QCD}} \sim 1$ GeV	Confinement
$3$-to-$3$: dark ALPs	3	$m \sim 10^{9}$–$10^{15}$ GeV	Hessian of $V_{\mathrm{Gap}}$
$\bar{3}$-to-$\bar{3}$: electroweak ALPs	3	$m \sim v_{\mathrm{EW}}$–$10^{15}$ GeV	EWSB $+$ $V_{\mathrm{Gap}}$
$O$-sector	6	$m \sim M_P$	Gap $\sim 1$ (hard modes)

All 21 phases acquire mass from the potentials $V_3$ or $V_2$ — there are no flat directions. This is a fundamental distinction from models with tuned potentials: Gap theory does not naturally predict ultralight axions (fuzzy DM, $m \sim 10^{-22}$ eV). [H]

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4. Dark ALPs from the $3$-to-$3$ Sector

Compactification on $(S^1)^{21}$ generates additional axion-like particles (ALPs). Of the 21 phases $\theta_{ij}$:

Sector	Phases	Gap	Modes
$3$-to-$\bar{3}$ ($\{A,S,D\} \times \{L,E,U\}$)	9	$\to 0$	1 QCD axion + 8 gluonic ($m \sim \Lambda_{\mathrm{QCD}}$)
$3$-to-$3$ ($\{A,S,D\} \times \{A,S,D\}$, $i<j$)	3	$\sim \epsilon_{\mathrm{space}}$	3 dark ALPs
$\bar{3}$-to-$\bar{3}$ ($\{L,E,U\} \times \{L,E,U\}$)	3	$\sim \epsilon_{\mathrm{EW}}$	3 electroweak ALPs (massive after EWSB)
$O$-sector	6	$\sim 1$	6 heavy modes ($m \sim M_P$)

Potential DM candidates are the 3 dark ALPs from the $3$-to-$3$ sector: pairs $(A,S)$, $(A,D)$, $(S,D)$.

Hypothesis [H]

The masses of the dark ALPs are determined by the Hessian of $V_{\mathrm{Gap}}$ in the vacuum:

$$m_{\mathrm{ALP}} \sim \sqrt{\lambda_3}\,\epsilon\,\mu_{\mathrm{phys}} \sim 10^{15}\;\text{GeV}$$

This is of GUT order — too heavy for standard DM mechanisms. With additional suppression of $\lambda_3$ from partial SUSY preservation in the $3$-to-$3$ sector (Gap $\sim \epsilon_{\mathrm{space}} \sim 10^{-3}$): $m_{\mathrm{ALP}} \sim 10^{9}$ GeV — still heavy, but accessible to gravitational production (§5).

There are no flat directions: all 21 phases acquire mass from $V_3$ or $V_2$. Gap theory does not naturally predict ultralight axions (fuzzy DM).

Open Direction: Collective Enhancement [H]

Multi-axion cosmology from $(S^1)^{21}$ is an open question of medium priority. Is collective enhancement of the relic density possible when several ALP fields are simultaneously present? This may modify the estimate of $\Omega_a h^2$ for the subdominant axion sector.

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5. $O$-Sector Relic (Wimpzilla)

5.1 $O$-Sector Dark Matter

$O$-sector configurations (Gap $\sim 1$ for pairs involving $O$) are heavy particles with masses $\sim 10^{13}$ GeV ($\sim m_{3/2}$). They interact weakly with the SM — through gravity and suppressed $G_2$-extra exchanges.

5.2 Gravitational Production during Inflation

The Chung–Kolb–Riotto (CKR, 1998) mechanism predicts a particle number density of mass $m$ in de Sitter space with Hubble parameter $H_I$:

$$n \sim H_I^3 \cdot e^{-2\pi m / H_I} \quad \text{for } m > H_I$$

Note: Exponential Mass Selection [I]

For Planck-mass particles ($m \sim M_P \sim 10^{19}$ GeV, $H_I \sim 10^{13}$ GeV) production is exponentially suppressed: $e^{-2\pi \times 10^{19}/10^{13}} = e^{-6.3 \times 10^6} \approx 0$. Therefore $G_2$-extra bosons ($m \sim M_P$) are not produced during standard inflation. By contrast, for lighter $O$-sector configurations ($m \sim m_{3/2} \sim 10^{13}$ GeV $\approx H_I$) the exponent $\sim e^{-2\pi} \sim 10^{-3}$, giving $n \sim 10^{37}$ cm$^{-3}$ — a physically significant number density.

Theorem 11.1 [T]

Standard formula for non-thermal relics (Chung, Kolb, Riotto, Phys.Rev.D 59, 023501):

$$\Omega_X h^2 \approx 0.1 \times \left(\frac{m_X}{10^{13}\;\text{GeV}}\right)^{3/2} \times \left(\frac{H_I}{10^{13}\;\text{GeV}}\right) \sim 0.1\text{--}0.4$$

for $m_X = m_{3/2} \sim 10^{13}$ GeV, $H_I \sim 4 \times 10^{13}$ GeV.

Promotion of Order-of-Magnitude Estimate [C at T-50, CKR]

The order of magnitude $\Omega_X h^2 \sim 0.1$–$0.4$ is [C at T-50, CKR standard cosmology]:

• $m_X \sim m_{3/2} \sim \varepsilon^3 M_P \sim 10^{13}$ GeV — from T-50 [T] (uniqueness of the superpotential, Schur's lemma)
• CKR formula (Chung–Kolb–Riotto, 1998) — standard result of non-thermal production
• Structural coincidence $m_{3/2} \sim H_I$ (both $\sim \varepsilon^3 M_P$) — not fine-tuning, but a consequence of a unified SUSY-breaking scale

The exact numerical coefficient has an uncertainty of $\times 2$–$3$ (from CKR). Stability requires $O$-parity (see §5.3).

The order of magnitude coincides with the observed $\Omega_{\mathrm{DM}} h^2 = 0.12$.

5.3 $O$-Parity

In standard SUSY, $R$-parity $R = (-1)^{3(B-L)+2S}$ stabilises the LSP. In the Gap formalism the analogue of $R$-parity is $O$-parity.

Theorem 11.2 [T]

Theorem 11.2 [T]

$O$-parity is a discrete $\mathbb{Z}_2$ symmetry that stabilises the heavy relic:

$$P_O := (-1)^{\Delta N_O}$$

where $\Delta N_O := N_O^{\mathrm{state}} - N_O^{\mathrm{vac}}$ is the number of excited $O$-pairs relative to the vacuum.

Proof:

Step 1 (Stabiliser). $\mathrm{Stab}_{G_2}(e_O) = SU(3)$ [T] (T-42e). Consequently the $O$-sector possesses a distinguished $SU(3)$-invariant structure.

Step 2 ($\mathbb{Z}_2$ symmetry from reality). Complex conjugation $\sigma: \gamma_{Oi} \mapsto \bar{\gamma}_{Oi}$ is a $\mathbb{Z}_2$ symmetry of the potential $V_{\mathrm{Gap}}$, since the structure constants $f_{ijk} \in \mathbb{R}$ (T-99 [T], step 1).

Step 3 (Commutation with dynamics). The full Lindblad operator $\mathcal{L}_\Omega$ has real structure constants, hence $\sigma(\mathcal{L}_\Omega[\Gamma]) = \mathcal{L}_\Omega[\sigma(\Gamma)]$, i.e. $[\sigma, \mathcal{L}_\Omega] = 0$.

Step 4 (Conservation). $P_O = \pm 1$ is the eigenvalue of $\sigma$ on $O$-sector excitations. From $[\sigma, \mathcal{L}_\Omega] = 0$ it follows that $P_O$ is conserved under evolution. The lightest $O$-odd particle ($P_O = -1$) cannot decay into SM particles ($P_O = +1$) → stable.

Step 5 (Topological barrier). T-69 [T]: $\Delta V \geq 6\mu^2 > 0$ prevents $O$-parity-violating tunnelling. $\blacksquare$

Note: Redefinition via Excitations [I]

The naive definition $P_O = (-1)^{N_O}$, where $N_O$ is the absolute number of $O$-components with Gap $\sim 1$, is trivial in the vacuum: all 6 pairs $\{O,A\}, \{O,S\}, \{O,D\}, \{O,L\}, \{O,E\}, \{O,U\}$ have Gap $\sim 1$, so $N_O = 6$ and $P_O = +1$ for all states in the vicinity of the vacuum — the symmetry does not distinguish the vacuum from excitations. The correct definition via $\Delta N_O = N_O^{\mathrm{state}} - N_O^{\mathrm{vac}}$ resolves this problem and is the precise analogue of $R$-parity in SUSY.

Configuration	$\Delta N_O$	$P_O$	Consequence
Vacuum	0	$+1$
Single $O$-quantum	1	$-1$	Stable (cannot decay to SM with $P_O = +1$)
Pair of $O$-quanta	2	$+1$	Can annihilate
SM particles	0	$+1$

Lifetime: From the structure of $V_3$: vertices with $O \in \{i,j,k\}$ are suppressed, so transitions changing $\Delta N_O$ are exponentially suppressed:

$$\tau_X \sim \frac{M_P}{m_X^2} \cdot e^{+M_P/m_X}$$

For $m_X \sim 10^{13}$ GeV: $e^{M_P/m_X} = e^{10^6} \gg 10^{10^5}$ — fantastically stable.

Status: $O$-parity is an exact $\mathbb{Z}_2$ symmetry of the dynamics $\mathcal{L}_\Omega$ [T], exponentially protected by the topological barrier T-69 [T].

5.4 Details of CKR Production of the $O$-Relic

The Chung–Kolb–Riotto (CKR) mechanism describes non-thermal production of heavy particles through rapid expansion of de Sitter space during inflation. For $O$-sector configurations the process proceeds in three stages:

Theorem 11.1a [T]

(a) Number density of particles of mass $m$ immediately after inflation:

$$n_X \sim H_I^3 \cdot e^{-2\pi m_X / H_I}$$

For $m_X \sim m_{3/2} \sim 10^{13}$ GeV with $m_X \approx H_I$:

$$n_X \sim (10^{13}\;\text{GeV})^3 \cdot e^{-2\pi} \sim 10^{39} \times 2 \times 10^{-3} \sim 10^{37}\;\text{cm}^{-3}$$

(b) Relic density after dilution by reheating to temperature $T_{\mathrm{RH}}$: [T]

$$\Omega_X h^2 \sim \frac{m_X \cdot n_X}{T_{\mathrm{RH}}^3} \cdot \frac{T_0^3}{\rho_c} \cdot T_{\mathrm{RH}}^3$$

Standard CKR formula (Phys.Rev.D 59, 023501):

$$\Omega_X h^2 \approx 0.1 \times \left(\frac{m_X}{10^{13}\;\text{GeV}}\right)^{3/2} \times \left(\frac{H_I}{10^{13}\;\text{GeV}}\right)$$

(c) Substituting Gap theory parameters ($m_X = m_{3/2} \sim 10^{13}$ GeV, $H_I \sim 4 \times 10^{13}$ GeV): [C at T-50, CKR]

$$\Omega_X h^2 \sim 0.1 \times 1 \times 4 = 0.4$$

Accounting for the CKR coefficient uncertainty ($\times 2$--$3$): $\Omega_X h^2 \sim 0.1$--$0.4$, consistent with the observed $\Omega_{\mathrm{DM}} h^2 = 0.120 \pm 0.001$.

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Note: Key Role of the Scale $m_X \approx H_I$ [I]

The coincidence $m_{3/2} \sim H_I$ is not parameter fine-tuning. In Gap theory the gravitino mass $m_{3/2}$ is determined by SUSY breaking (Gap $\sim 1$ in the $O$-sector), while the inflation scale $H_I$ is determined by the dynamics of the Gap vacuum. Both are fixed at $\sim 10^{13}$ GeV by independent structural arguments.

Fitting vs. Prediction

The scale $m_{3/2} \sim \varepsilon^3 M_P \sim 10^{13}$ GeV follows from T-50 [T] (uniqueness of the superpotential) at $\varepsilon \sim 10^{-3}$. The parameter $\varepsilon$ (vacuum coherence) is not derived from first principles but is chosen to match the SUSY-breaking scale. The CKR formula gives $\Omega h^2 \sim 0.1$-$0.4$ against the observed $0.120 \pm 0.001$; the agreement is at order of magnitude [C at T-50, CKR], but the uncertainty range ($\times 2$-$3$) covers the observed value. An exact prediction $\Omega h^2 = 0.12$ remains an open problem.

5.5 Interaction Cross Section of the $O$-Relic

The $O$-sector relic interacts with Standard Model particles exclusively through gravitational and suppressed $G_2$-extra exchanges.

Theorem 11.3 [T]

Elastic scattering cross section of the $O$-relic on a nucleon:

$$\sigma_{X\text{-}N} \sim G_N^2 \, m_X^2 \, m_N^2 \sim \left(\frac{1}{M_P^2}\right)^2 m_X^2 \, m_N^2$$

Numerically for $m_X \sim 10^{13}$ GeV, $m_N \sim 1$ GeV, $M_P = 2.4 \times 10^{18}$ GeV:

$$\sigma_{X\text{-}N} \sim \frac{m_X^2 \, m_N^2}{M_P^4} \sim \frac{10^{26} \times 1}{(2.4)^4 \times 10^{72}} \sim 10^{-46}\;\text{GeV}^{-2} \sim 10^{-60}\;\text{cm}^2$$

This is 13 orders of magnitude below current experimental limits (XENON1T, LZ: $\sigma < 10^{-47}$ cm$^2$ for $m \sim 100$ GeV) and is practically unobservable by direct detectors. [T]

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Annihilation Cross Section for $O$-Relic Pairs [H]

For pairs of $O$-quanta ($\Delta N_O = 2$, $P_O = +1$) annihilation is possible:

$$\sigma_{\mathrm{ann}} v \sim \frac{m_X^2}{M_P^4} \sim 10^{-46}\;\text{GeV}^{-2}$$

Annihilation $O\bar{O} \to$ SM particles with energy $E \sim m_X \sim 10^{13}$ GeV may produce ultra-high-energy cosmic rays (UHECR, $E > 10^{20}$ eV) — a potentially observable signal.

5.6 Relic Density Budget

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Full Decomposition of $\Omega_{\mathrm{DM}}$ [C at T-50, CKR]

Gap theory predicts two-component dark matter:

Component	Mass	$\Omega h^2$	Fraction of DM	Mechanism
$O$-relic (Wimpzilla)	$\sim 10^{13}$ GeV	$\sim 0.1$–$0.4$	$\sim 83$–$100\%$	CKR (gravitational)
QCD axion (only under (PQ), T-333(e)–(h))	$\sim 3$ neV	$\lesssim 4 \times 10^{-6}$ (isocurvature)	$\lesssim 3\times10^{-5}$	Vacuum misalignment
Dark ALPs	$\sim 10^{9}$–$10^{15}$ GeV	negligible	$\ll 1\%$	Gravitational (suppressed)
$G_2$-extra bosons	$\sim M_P$	$\sim 10^{-6}$	$\ll 1\%$	Gravitational (exponentially suppressed)
Total		$\sim 0.1$–$0.4$	$\sim 100\%$

The observed value $\Omega_{\mathrm{DM}} h^2 = 0.120 \pm 0.001$ is reproduced to order of magnitude.

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6. Summary Candidate

Parameter	Value	Source
Mass	$m_X \sim 10^{13}$ GeV	Standard Model from $G_2$ §5.2
Production mechanism	Gravitational (inflation, CKR)	§5.4 above
$\Omega_X h^2$	$\sim 0.1$--$0.4$ [C at T-50, CKR]	§5.4 above
Stability	$\tau \gg t_U$ ($O$-parity)	§5.3 above
Direct detection	$\sigma \sim G_N^2 m_X^2 \sim 10^{-60}$ cm$^2$	Unobservable
Indirect signatures	UHECR ($E > 10^{20}$ eV) from annihilation	Testable

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7. Fano Correlation Length $\xi_F$

The Fano correlation length $\xi_F$ is the scale over which Fano correlations in the Gap vacuum decay. It is connected to the spatial distribution of dark matter through the structure of the Gap vacuum.

7.1 Definition

$$C_{\mathrm{Fano}}(r) := \langle F_{ijk}(0) \cdot F_{ijk}(r) \rangle_{\mathrm{vac}} \sim e^{-r/\xi_F}$$

where $F_{ijk}(x) = \varepsilon_{ijk}^{\mathrm{Fano}} \cdot \mathrm{Gap}(i,j,x) \cdot \mathrm{Gap}(j,k,x) \cdot \mathrm{Gap}(i,k,x)$ is the local Fano function.

7.2 RG Evolution

Theorem 9.4 [T]

The Fano correlation length satisfies the RG equation:

$$\frac{d \ln \xi_F}{d \ln \mu} = -1 + \eta_F, \quad \eta_F = \frac{5}{42} \approx 0.119$$

where $\eta_F$ is the anomalous dimension of the Fano operator. Solution:

$$\xi_F(\mu) = \ell_{\mathrm{Planck}} \cdot \left(\frac{M_{\mathrm{Planck}}}{\mu}\right)^{37/42}$$

7.3 Quantitative Prediction

At the Hubble scale ($\mu \sim H_0 \sim 10^{-33}$ eV):

$$\xi_F(H_0) = \ell_{\mathrm{Planck}} \cdot \left(\frac{10^{28}\;\text{eV}}{10^{-33}\;\text{eV}}\right)^{37/42} = 10^{-35}\;\text{m} \cdot 10^{53.7} \approx 5 \times 10^{18}\;\text{m} \sim 160\;\text{pc}$$

Note: Physical Meaning [I]

$\xi_F \sim 160$ pc is a scale comparable to the size of small molecular clouds. This defines the region within which Gap configurations are correlated through the Fano structure. Number of uncorrelated Fano modes in the observable Universe:

$$N_F = \left(\frac{R_H}{\xi_F}\right)^3 = \left(\frac{4.4 \times 10^{26}\;\text{m}}{5 \times 10^{18}\;\text{m}}\right)^3 \approx 6.8 \times 10^{23}$$

Caveat: Two Scales [I]

$\ell_{\mathrm{Planck}}$ is the UV cutoff (lattice spacing), $\xi_F$ is the IR correlation. These are different physical scales. The number of degrees of freedom $N_{\mathrm{DOF}} = V/\ell_P^3$ should not be confused with the number of Fano modes $N_F = (R_H/\xi_F)^3 \sim 10^{24}$.

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8. Falsifiable Predictions

#	Prediction	Value	Experiment
P1	$m_a \sim 3$ neV (only under (PQ), T-333(e)–(h))	$2.85 \times 10^{-9}$ eV	CASPEr, ABRACADABRA
P2	$f_a \sim 2 \times 10^{15}$ GeV	From $\epsilon \cdot M_P$	Axion-photon conversion
P3	$\Omega_a / \Omega_{\mathrm{DM}} \sim 10^{-2}$ — excluded by isocurvature (T-333(e)–(h))	$\lesssim 3\times10^{-5}$ under (PQ)	Cosmological constraints
P4	$m_{\mathrm{DM}} \sim 10^{13}$ GeV	Wimpzilla	UHECR anomalies
P5	No WIMP-DM in direct detectors	$\sigma < 10^{-60}$ cm$^2$	XENON, LZ (confirmed)
P6	$\xi_F \sim 160$ pc	Fano correlation length	Large-scale structure

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9. Connection to Other Sections

Topic	Page	Connection
Cosmological Constant	Cosmological Constant	Vacuum structure, $O$-sector and $\xi_F$ in the $\Lambda$ budget
Einstein Equations	Einstein Equations from Gap	Dark energy as Gap dynamics in the Im-sector
$G_2$-Structure	$G_2$-Structure	Fano plane and sectoral decomposition
Berry Phase	Berry Phase	Topological protection of Gap in the $O$-sector
Fano Selection Rule	Fano Selection Rules	Fano correlations and $\xi_F$
Confinement	Confinement from Gap	Gap $\to 0$ in the $3$-to-$\bar{3}$ sector; QCD axion

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• Cosmological Constant
• Cosmological Genesis
• Phase Diagram