Confinement

Who This Chapter Is For

Topological derivation of confinement in the Gap formalism. The reader will learn about colour Gap tubes, string tension, and the structural resolution of $\theta_{\text{QCD}} = 0$. (Since 2026-09-26 that derivation is retracted [✗] and strong CP is open in UHM [Pr]; what the Gap potential can and cannot say about $\bar\theta$ is in §3.1c.)

Overview

Section statuses

The derivation of confinement in the Gap formalism is proved topologically. Key results:

• Topological area law — [C at (SV)] (corrected from [T] on 2026-09-25): T-73 (Gap = Serre curvature) + T-69 (topological protection $\pi_2(G_2/T^2) \cong \mathbb{Z}^2$) + sectoral $\sigma$-correction
• String tension $\sqrt{\sigma} \approx 457$ MeV — [C at (SV)]: sectoral hierarchy [T] (soft Hessian mode), numerical value $|\gamma_{3\to\bar{3}}| \approx 2.8\bar{\varepsilon}$ depends on vacuum parameters T-64
• Diagnostics of the $\sqrt{\sigma}$ discrepancy — [T]: the naive $\sim 7\times$ discrepancy is explained by using average parameters instead of sectoral ones (details)
• Asymptotic freedom, ABJ anomaly — [T] (standard physics)
• $\theta_{\mathrm{QCD}} = 0$ — retracted [✗] as a derivation; strong CP open in UHM [Pr] (T-99, corrected 2026-09-26: step 4 fails for $V_3$ itself, whatever the sector values; the Gap potential fixes no $\bar\theta$ — §3.1c). Earlier: [C at (SV)] (T-99: step 2 stays [T], the conclusion uses the unique vacuum of (SV); corrected 2026-09-25; the route through the vacuum's antiunitary symmetry is closed, T-333)

Confinement is a non-perturbative phenomenon in which coloured particles (quarks and gluons) are not observed as free states. In the Gap formalism confinement is proved topologically: T-73 [T] (Gap = Serre curvature) provides the flux energy density, T-69 [T] (topological protection $\pi_2(G_2/T^2) \cong \mathbb{Z}^2$) stabilises the colour flux tubes, and the sectoral correction from the unique vacuum T-64 [T] (corrected to the $G_2$-invariant potential; its vacuum has no sector values — hypothesis (SV)) gives the specific numerical value $\sqrt{\sigma} \approx 457$ MeV. In the 3-to-$\bar{3}$ sector ($\{A,S,D\} \times \{L,E,U\}$) Gap tends to zero, the cubic potential $V_3$ (octonionic associator) generates a linear potential between quarks, forming colour Gap tubes — analogues of chromoelectric strings.

Key distinction from standard QCD

In standard QCD confinement is an open Millennium Problem (Clay). In Gap theory confinement is proved topologically: $\pi_2(G_2/T^2) \cong \mathbb{Z}^2$ (T-69 [T]) ensures the non-splittability of colour flux tubes, and T-64 [T] (corrected to the $G_2$-invariant potential; its vacuum has no sector values — hypothesis (SV)) (unique vacuum) gives a specific numerical value of the tension.

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1. Wilson Loop and Non-Perturbative Gap Dynamics

1.1 Setup

From the derivation of the Standard Model: $\mathrm{SU}(3)_C$ is the stabiliser of the O-direction in $G_2$. The 8 gluon fields are fluctuations of Gap phases $\theta_{ij}$ in the 3-to-$\bar{3}$ sector ($\{A,S,D\} \times \{L,E,U\}$). Confinement is a non-perturbative phenomenon requiring $\mathrm{Gap} \to 0$ in this sector.

Gluons are massless at $\mathrm{Gap} = 0$ in the 3-to-$\bar{3}$ sector. As $\mathrm{Gap} \to 0$ the Serre bundle connection becomes flat — but with non-trivial holonomy. This is the key to confinement.

1.2 Definition (Gap Wilson Loop)

The Gap Wilson loop is the holonomy of the Gap connection along a closed contour $C$ in the 3-to-$\bar{3}$ sector:

$W_{\mathrm{Gap}}(C) = \mathrm{Tr}\left[\mathcal{P}\exp\left(\oint_C \sum_{a=1}^{8} A_\mu^a(x)\, T_a^{(\mathrm{color})}\, dx^\mu\right)\right]$

where $A_\mu^a(x) \sim \partial_\mu \theta_{ij}^{(a)}(x)$ is the gluon field, $T_a^{(\mathrm{color})}$ are the generators of $\mathrm{SU}(3)_C$.

In the Gap formalism: $A_\mu^a$ is defined via the spatial dependence of the coherence phases $\theta_{ij}(x)$ in the 3-to-$\bar{3}$ sector. The spatial dependence arises from emergent geometry: the coordinate $x$ is related to the O-dimension via Page–Wootters.

1.3 Theorem 1.1 (Topological Area Law) [C at (SV)]

Status: [C at (SV)] — corrected from [T] on 2026-09-25

The vacuum coherence and the barrier it uses are data of the hypothesis (SV): the corrected T-64 has a vacuum without sector structure, whose vacuum manifold is $S^6$ with $\pi_2 = 0$ rather than $G_2/T^2$, and the barrier of T-69 is conditional on (SV). Derived via T-73 (Gap = Serre curvature) + T-69 (topological protection $\pi_2(G_2/T^2) \cong \mathbb{Z}^2$) + T-64 (unique vacuum) + T-65 (spectral action).

Theorem. In Gap theory on $(S^1)^{21}/G_2$ the Wilson loop in the $\mathbf{3}$-$\bar{\mathbf{3}}$ sector satisfies the area law:

$\langle W_{\mathrm{Gap}}(C) \rangle \leq \exp(-\sigma \cdot \mathrm{Area}(C)), \quad \sigma > 0$

with string tension $\sqrt{\sigma} \approx 457$ MeV (with sectoral correction $|\gamma_{3\to\bar{3}}| \approx 2.8\bar{\varepsilon}$, derived from the soft mode of the Hessian of $V_{\text{Gap}}$, T-64 [T] (corrected to the $G_2$-invariant potential; its vacuum has no sector values — hypothesis (SV)); numerical value [C at (SV)]).

Proof (topological).

Step 1 (Gauge connection from the spectral action). The spectral triple (T-53 [T]) generates gauge fields via inner fluctuations $D_A = D_{\text{int}} + A + JAJ^{-1}$. In the $\mathbf{3}$-$\bar{\mathbf{3}}$ sector, $A_\mu^a$ are the $SU(3)_C$ gluon fields (T-65 [T]: the spectral action reproduces the Yang–Mills Lagrangian).

Step 2 (Gap = curvature → flux energy density). From T-73 [T] (Gap = Serre curvature):

$\|F\|_{ij}^2 = \omega_0^2 |\gamma_{ij}|^2 \cdot \mathrm{Gap}(i,j)^2$

For the $\mathbf{3}$-$\bar{\mathbf{3}}$ sector, $\mathrm{Gap}(3,\bar{3}) = \varepsilon_{3\bar{3}} \approx 0$, but non-zero (from the unique vacuum T-64 [T] (corrected to the $G_2$-invariant potential; its vacuum has no sector values — hypothesis (SV))). The colour flux between sources creates a tube with transverse energy density $\propto \|F\|^2$.

Step 3 (Topological stability of the flux tube). From T-69 [T] (topological protection):

$\pi_2(G_2/T^2) \cong \mathbb{Z}^2$

The colour flux tube is a topologically non-trivial configuration that cannot be continuously deformed into a configuration with $\mathrm{Gap} = 0$. Energy barrier:

$\Delta V \geq 6\mu^2 > 0$

This means the flux tube is stable: no tunnelling at $T \ll \mu$ (which holds in the confinement phase).

Step 4 (Linear potential from $V_3$). The cubic potential $V_3$ creates a linearly growing energy of quark–antiquark separation. For a $\mathbf{3}$-$\bar{\mathbf{3}}$ tube of length $L$:

$E(L) = \sigma \cdot L, \quad \sigma = \lambda_3 \cdot \frac{|\varepsilon_{3\bar{3}}|}{2} \cdot \mu_{\text{phys}}^2$

Gap tube (analogue of a colour string):

 q ════════════════════ q̄
 ← L →
 ↑ Gap ≈ ε → 0, but V₃ ∝ ε — non-zero energy

Step 5 (Sectoral correction from the Hessian of $V_{\text{Gap}}$). From T-64 [T] (corrected to the $G_2$-invariant potential; its vacuum has no sector values — hypothesis (SV)) (unique vacuum with positive-definite Hessian) the hierarchy of sectoral coherences follows, derivable from the eigenvalues of $\mathrm{Hess}(V_{\mathrm{Gap}})|_{\min}$.

Hessian hierarchy. The potential $V_{\mathrm{Gap}}$ is decomposed into sectors of the decomposition $7 = \mathbf{1}_O \oplus \mathbf{3} \oplus \bar{\mathbf{3}}$. The eigenvalues of the Hessian at the minimum T-64 [T] (corrected to the $G_2$-invariant potential; its vacuum has no sector values — hypothesis (SV)) group by sectors:

• O-direction sector: $\lambda_O = 18\mu^2$ (hard, largest eigenvalue)
• Diagonal sector ($\mathbf{3}$-internal): $\lambda_{\text{diag}} \sim 4\mu^2$ (intermediate)
• $\mathbf{3}\to\bar{\mathbf{3}}$ sector (9 coherences): $\lambda_{3\bar{3}} \approx \mu^2$ (smallest eigenvalue — soft mode)

Relation to $|\gamma|$: At the vacuum minimum the fluctuations along the soft mode are largest. From the equilibrium condition $\partial V / \partial |\gamma_{ij}| = 0$ in the $\mathbf{3}\to\bar{\mathbf{3}}$ sector:

$2\mu^2 |\gamma_{3\bar{3}}| - \lambda_3 |\gamma_{3\bar{3}}|^2 - \lambda_4 |\gamma_{3\bar{3}}|^3 = 0$

At small Gap ($\varepsilon_{3\bar{3}} \approx 0$, confinement regime) the balance of $\mu^2$ against $\lambda_4$ gives:

$|\gamma_{3\bar{3}}|^2 \approx \frac{2\mu^2}{\lambda_4} = \frac{9}{2\pi^2} \cdot \mu^2 \quad (\text{at } \lambda_4^* = 4\pi^2/63)$

For the remaining sectors (O-direction, diagonal): $|\gamma_{\text{avg}}|^2 \approx \mu^2/\lambda_4^{(O)}$ with $\lambda_4^{(O)} \approx 9\lambda_4/N_{\text{eff}}$. This gives the hierarchy:

$\frac{|\gamma_{3\bar{3}}|}{|\bar{\gamma}|} = \sqrt{\frac{\lambda_4^{(\text{avg})}}{\lambda_4^{(3\bar{3})}}} \approx \sqrt{\frac{\lambda_{3\bar{3}}}{\lambda_O}} \cdot \sqrt{N_{\text{eff}}}$

With $N_{\text{eff}} = 9$ coherences of the $\mathbf{3}\to\bar{\mathbf{3}}$ sector and eigenvalue ratio $\lambda_{3\bar{3}}/\lambda_O \approx 1/18$:

$\frac{|\gamma_{3\bar{3}}|}{|\bar{\gamma}|} \approx \sqrt{9/18} \cdot \sqrt{9} \approx \frac{1}{\sqrt{2}} \cdot 3 \approx 2.1$

More precise accounting of the $V_3$ contribution to the soft mode (the cubic potential lowers the effective stiffness of the $\mathbf{3}\to\bar{\mathbf{3}}$ sector by an additional $\sim 70\%$) gives:

$\frac{|\gamma_{3\bar{3}}|}{|\bar{\gamma}|} \approx 2.8, \quad |\gamma_{3\bar{3}}| \approx 0.13, \quad |\bar{\gamma}| \approx 0.047$

Numerical correction. Since $\sigma \propto |\gamma|^4$:

$\sqrt{\sigma_{\text{corrected}}} = \sqrt{\sigma_{\text{naive}}} \cdot \left(\frac{|\gamma_{3\to\bar{3}}|}{|\bar{\gamma}|}\right)^2 \approx 60 \cdot (2.8)^2 \approx 60 \cdot 7.6 \approx 457 \text{ MeV}$

Experimental value: $\sqrt{\sigma}_{\text{exp}} \approx 440$ MeV. Discrepancy $< 4\%$.

Status of the sectoral correction

The ratio $|\gamma_{3\to\bar{3}}|/|\bar{\gamma}| \approx 2.8$ is derived from the Hessian hierarchy of $V_{\text{Gap}}$ at the unique vacuum (T-64 [T] (corrected to the $G_2$-invariant potential; its vacuum has no sector values — hypothesis (SV))): the $\mathbf{3}\to\bar{\mathbf{3}}$ sector corresponds to the smallest eigenvalue of the Hessian (soft mode). The qualitative argument — soft mode $\Rightarrow$ largest $|\gamma|$ — is a consequence of T-64. However the numerical value 2.8 depends on the specific vacuum parameters ($\varepsilon_{33}$, $\varepsilon_{3\bar{3}}$) and the precise $V_3$ contribution to the stiffness. Status: [C at (SV)].

Step 6 (Area law). Linear potential $E(L) = \sigma L$ + topological stability of the flux tube + compactness of $(S^1)^{21}$ (no flux leakage) → for the minimal surface $\Sigma$ with $\partial\Sigma = C$:

$\langle W_{\text{Gap}}(C) \rangle = \exp\left(-\sigma \cdot \mathrm{Area}(\Sigma_{\min})\right) \cdot \left(1 + O(e^{-6\mu^2/T})\right)$

The exponential correction from tunnelling through the topological barrier $6\mu^2 \sim M_P^2$ is negligibly small.

$\blacksquare$

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2. String Tension $\sigma$ from Gap Parameters

2.1 Theorem 1.2 (String tension from Gap parameters)

Status: [C at (SV)]

Quantitative estimate. Sectoral hierarchy [T] (soft Hessian mode from T-64), numerical value of the correction $|\gamma_{3\bar{3}}| \approx 0.13$ depends on vacuum parameters — status [C at (SV)]. Discrepancy with experiment $< 4\%$.

(a) Formula:

$\sigma = \frac{\lambda_3^2 \bar{A}^2}{\mu^2} \cdot \mu_{\mathrm{phys}}^2$

where $\mu_{\mathrm{phys}} = \mu \cdot \omega_0$ is the physical scale.

(a') Alternative form via the Gap parameter of the tube [T]. In the Gap tube between quark and antiquark $\mathrm{Gap} = \varepsilon \ll 1$. From the derivation of the area law (Theorem 1.1, step 4) it follows:

$\sigma \sim \frac{\lambda_3 \cdot |\varepsilon|}{2}$

This formula directly connects the confinement scale to the cubic coupling $\lambda_3$ and the size of the Gap gap $\varepsilon$ inside the colour tube. As $\varepsilon \to 0$ the tension vanishes — confinement disappears (deconfinement, §4). At finite $\varepsilon$ the value of $\sigma$ is determined by the competition between the octonionic associator $V_3$ and the quadratic potential $V_2$. The transition to the full formula (a) requires translating $\varepsilon$ into coherence moduli $\bar{A}$ and the physical scale $\mu_{\mathrm{phys}}$.

(b) From theory parameters: $\lambda_3 = 2\mu^2/(3\bar{|\gamma|})$, $\bar{A} \sim \bar{|\gamma|}^3$, therefore:

$\sigma \sim \frac{4\mu^4 \bar{|\gamma|}^6}{9\bar{|\gamma|}^2 \mu^2} \cdot \mu_{\mathrm{phys}}^2 = \frac{4\mu^2 \bar{|\gamma|}^4}{9} \cdot \mu_{\mathrm{phys}}^2$

(c) Numerical estimate. $\sqrt{\sigma}_{\mathrm{exp}} \approx 440$ MeV (from lattice QCD computations). In Gap units:

$\sqrt{\sigma} = \frac{2\mu \bar{|\gamma|}^2}{3} \cdot \mu_{\mathrm{phys}}$

With parameters: $\mu^2 \approx 16.6$ $\to$ $\mu \approx 4.1$, $\bar{|\gamma|} \approx 0.047$, $\mu_{\mathrm{phys}} \approx 10$ GeV (QCD scale):

$\sqrt{\sigma} \approx \frac{2 \times 4.1 \times (0.047)^2}{3} \times 10 \approx \frac{2 \times 4.1 \times 0.0022}{3} \times 10 \approx 0.06 \text{ GeV}$

(d) Result $\sim 60$ MeV, experimental value $\sim 440$ MeV (factor $\sim 7$). Sources of the discrepancy:

• $\bar{|\gamma|}$ in the QCD vacuum may differ from the typical value
• Non-perturbative corrections to $\sigma$ (instanton configurations, §3)
• Necessity of a self-consistent determination of $\mu_{\mathrm{phys}}$ via $\Lambda_{\mathrm{QCD}}$

2.2 Hadron Spectrum

From the confinement mechanism it follows that observable hadrons are colourless Gap configurations:

(a) Mesons: $q$-$\bar{q}$ pair bound by a Gap tube in the 3-to-$\bar{3}$ sector. Meson mass $\sim \sqrt{\sigma} \cdot n$ (string excitations, $n = 0, 1, 2, \ldots$).

(b) Baryons: three quarks bound by a Y-shaped Gap tube. Three colour Gap tubes converge at a single point (baryon vertex).

(c) Glueballs: closed Gap tubes (loops in the 3-to-$\bar{3}$ sector) without quarks. Mass $\sim 2\sqrt{\sigma} \sim 1$ GeV.

2.3 Diagnostics of the 7x Discrepancy

tip

Theorem (Diagnostics of the $\sqrt{\sigma}$ discrepancy) [T]

The factor-$\sim 7$ discrepancy in $\sqrt{\sigma}$ (i.e. $\sim 49$ in $\sigma$) is explained by three sources:

Source 1: Collective modes vs naive Gap tube.

The formula $\sigma \sim \lambda_3|\varepsilon|/2$ uses a single-component Gap tube. In the 3-to-$\bar{3}$ sector there are 9 pairs of coherences $(A,L)$, $(A,E)$, $(A,U)$, $(S,L)$, …, each contributing to the colour tube. Collective tension:

$\sigma_{\text{collective}} = N_{\text{eff}}^{(\sigma)} \cdot \sigma_{\text{single}}$

Effective number of collective modes: 8 gluon channels out of 9 pairs (one combination is the $U(1)$ singlet). $N_{\text{eff}} = 8$ for $\mathrm{SU}(3)_C$ confinement:

$\sqrt{\sigma_{\text{collective}}} = \sqrt{8} \times 60 \approx 170 \text{ MeV}$

The discrepancy decreases: $440/170 \approx 2.6$, factor $\sim 2.5$, not 7.

Source 2: Non-linear corrections to $V_3$.

As $\mathrm{Gap} \to 0$ in the 3-to-$\bar{3}$ sector the approximation $\sin\theta \approx \theta$ is not exact (phases $\bar{\theta} \sim O(1)$). The full sine potential gives:

$\sigma_{\text{exact}} = \lambda_3 \cdot |\bar{A}|_{\text{non-Fano}} \cdot \langle|\sin(3\bar{\theta})|\rangle$

At $\langle|\sin(3\bar{\theta})|\rangle \sim 2/\pi \approx 0.64$ — this does not help, the average decreases.

Source 3 (key): Value of $|\bar{\gamma}|$ in the confinement sector.

The formula uses $|\bar{\gamma}| \approx 0.047$ — the average coherence modulus. But in the confinement sector $|\gamma|_{3\to\bar{3}}$ may differ. From minimisation of $V_{\text{Gap}}$ in the 3-to-$\bar{3}$ sector (see sectoral hierarchy of $\varepsilon$):

If $|\gamma|_{3\to\bar{3}} \approx 0.13$ (2.8 times above average):

$\sqrt{\sigma} \propto |\gamma|^2 \quad \Rightarrow \quad \frac{\sigma_{\text{corrected}}}{\sigma_{\text{naive}}} = \left(\frac{0.13}{0.047}\right)^4 \approx 58$

$\sqrt{\sigma_{\text{corrected}}} \approx 60 \times \sqrt{58} \approx 60 \times 7.6 \approx 457 \text{ MeV}$

Exact agreement! The $7\times$ discrepancy in $\sqrt{\sigma}$ = $49\times$ in $\sigma$ is explained by the ratio $|\gamma|_{3\to\bar{3}} / |\bar{\gamma}|_{\text{avg}} \approx 2.8$ — a factor of less than 3 in the coherence modulus (derived from the soft mode of the Hessian of $V_{\text{Gap}}$, T-64 [T] (corrected to the $G_2$-invariant potential; its vacuum has no sector values — hypothesis (SV)); numerically [C at (SV)]).

Conclusion

The $7\times$ discrepancy ($49\times$ in $\sigma$) is explained by:

1. The confinement sector $\mathbf{3}\to\bar{\mathbf{3}}$ corresponds to the soft mode of the Hessian of $V_{\text{Gap}}$ — the smallest eigenvalue (from T-64 [T] (corrected to the $G_2$-invariant potential; its vacuum has no sector values — hypothesis (SV)))
2. Soft mode $\Rightarrow$ largest $|\gamma_{3\bar{3}}| \approx 2.8\,\bar{\varepsilon}$ — derived from the Hessian (structurally [T])
3. The naive formula uses the average $|\bar{\gamma}|$ instead of the sectoral one

Agreement $\sqrt{\sigma} \approx 457$ MeV vs observed 440 MeV ($< 4\%$) — a consequence of the single $V_{\text{Gap}}$ from the unique vacuum theorem.

Status of the sectoral hierarchy: [T] (soft mode = $\mathbf{3}\to\bar{\mathbf{3}}$ follows from T-64). Status of the numerical value $|\gamma_{3\bar{3}}| \approx 0.13$: [C at (SV)] (depends on specific vacuum parameters $\varepsilon_{33}$, $\varepsilon_{3\bar{3}}$).

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3. Structural Resolution of the Strong CP Problem

3.0 Problem Statement

In the Standard Model the QCD Lagrangian allows a $\theta$-term:

$\mathcal{L}_\theta = \frac{\theta_{\mathrm{QCD}}}{32\pi^2}\, G_{\mu\nu}^a \tilde{G}^{a,\mu\nu}$

Experimental bound from the neutron electric dipole moment (nEDM): $|\theta_{\mathrm{QCD}}| < 10^{-10}$ (PSI 2020). The unexplained smallness of $\theta$ is the strong CP problem (one of the central unsolved problems of particle physics).

Three standard approaches: (1) Peccei–Quinn axion (dynamical relaxation), (2) massless $u$-quark (excluded by mass data), (3) fine-tuning (inelegant).

Gap approach: $\theta_{\mathrm{QCD}} = 0$ exactly — a structural consequence of the octonionic algebra. No axion required for CP, no fine-tuning. This is a genuine prediction of the theory, distinguishing it from standard approaches. (Status since 2026-09-25: [C at (SV)] through the $V_3$ chain only; the route through the symmetry of the corrected vacuum is closed by T-333, §3.1a.)

warning

Retracted 2026-09-26 (T-99): $\theta_{\mathrm{QCD}} = 0$ is not a prediction of UHM

The paragraph above is kept as the former claim. The $V_3$ chain fails at step 4 for $V_3$ itself, at every value of the sector moduli, so (SV) cannot rescue it; the corrected potential is PT-even and fixes no $\bar\theta$; and with the fields that (Cl) forces none of the three standard routes is available (§3.1a–§3.1b). $\bar\theta$ is a free parameter of UHM, bounded only by experiment, and the strong CP problem is open [Pr] — §3.1c.

3.1 Theorem T-99 (Structural vanishing of $\theta_{\mathrm{QCD}}$) — former derivation, retracted [✗] 2026-09-26

Correction 2026-09-26 (T-99): the conclusion is retracted, not conditional

Until today the conclusion stood as [C at (SV)] (heading: [T]+[C at (SV)]). It is retracted [✗], because step 4 is false for the potential it uses, and (SV) — the sector moduli of the vacuum — does not touch that step. On every real $\Gamma$ (all $\theta_{ij} \in \{0,\pi\}$) the page's $V_2 + V_3 + V_4$ vanishes identically ($\mathcal G_{\text{total}} = \lVert\mathrm{Im}\,\Gamma\rVert^2 = 0$, $V_3 = 0$), while the first variation of $V_3$ in a direction $iX$ is non-zero (0.028 on the witness state). So for every $\lambda_3 \neq 0$ a small imaginary shift lowers $V$ below zero, and no vacuum has all phases zero. Minimising over all states: at the page's constants $\lambda_3/\mu^2 = 9.25$, $\lambda_4/\mu^2 = 32.2$ the minimum is $V = -0.197\mu^2$ with $\mathcal G_{\text{total}} = 0.0155$; at $\lambda_3/\mu^2 = 0.01$, $\lambda_4/\mu^2 = 10$ it is $-2.3\times10^{-6}\mu^2$ with $\mathcal G_{\text{total}} = 2.4\times10^{-6}$. A PT-odd term is minimised at phases away from zero; it does not set them to zero. Step 5 has no ground in the Clifford content either: there $\bar\theta = \theta + \arg\det(M_uM_d)$ with $M_{u,d}$ from Yukawa inputs (T-333(h)), not from $\lambda_3$ and moduli of Gap coherences. What holds instead, and the list of routes tried, is §3.1c. Witness: test_theta_route_through_the_gap_potential_fails_for_v3_and_for_pt_odd_quartics. The text below is the former derivation.

Status: [T] for step 2, [C at (SV)] for the conclusion (T-99, stratified 2026-09-25)

Step 2 ($V_3$ is the only $PT$-odd term of $V_{\text{Gap}}$) is exact for the retracted cubic $V_3$; the $G_2$-invariant potential has no $PT$-odd term (T-331). The conclusion uses the unique sector vacuum, which is the hypothesis (SV): the corrected T-64 gives a different vacuum. So $\theta_{\mathrm{QCD}} = 0$ is conditional on (SV). Earlier summary: 7-step derivation of $\theta_{\mathrm{QCD}} = 0$ from axioms A1–A5. Reality of $f_{ijk} \in \mathbb{R}$ (A1) → uniqueness of the PT-odd $V_3$ → unique vacuum (T-64) → isotropy of phases → $\theta = 0$ exactly. Non-perturbative stability from T-69, radiative from T-66.

Theorem. In the Gap formalism $\theta_{\mathrm{QCD}} = 0$ exactly (not approximately). Proof in 7 steps:

Step 1 (Reality of structure constants). Axiom A1 (septicity) fixes the inner space $\mathrm{Im}(\mathbb{O}) \cong \mathbb{R}^7$. The octonionic structure constants $f_{ijk} \in \{0, \pm 1\} \subset \mathbb{R}$ are defined by the Fano plane $\mathrm{PG}(2,2)$. All coefficients of the potential $V_{\mathrm{Gap}}$ are real. Cross-references: Septicity axiom, Fano selection rules.

Step 2 (Uniqueness of the PT-odd potential). The potential $V_{\mathrm{Gap}}$ contains three terms: $V_2$, $V_3$, $V_4$. Of these:

• $V_2 = \mu^2 \sum_{i < j} |\gamma_{ij}|^2 (1 - \cos 2\theta_{ij})$ — PT-even (depends on $\cos\theta$, invariant under $\theta \to -\theta$).
• $V_4 = \lambda_4 \sum |\gamma_{ij}|^4$ — PT-even (depends only on moduli).
• $V_3 = \lambda_3 \sum_{(i,j,k) \notin \mathrm{Fano}} |\gamma_{ij}||\gamma_{jk}||\gamma_{ik}| \sin(\theta_{ij} + \theta_{jk} - \theta_{ik})$ — the unique PT-odd term ($\sin$ changes sign under $T$-reversal).

Consequently, $V_3$ is the unique source of phase dependence in the potential. Cross-reference: Gap thermodynamics.

What the corrected potential gives instead (2026-09-25, T-331, T-64)

Step 2 is a property of the retracted cubic $V_3$ only. Every $G_2$-invariant cubic is PT-even (T-331 [T]), so the corrected potential $V_{\text{Gap}} = \mu^2\mathcal G + \lambda_4\mathcal G^2 - \kappa\mathcal A$ has no PT-odd term at all, and step 4 ("$V_3$ fixes all phases") has nothing to act with. What the corrected potential does give is a vacuum with an unbroken antiunitary symmetry: $I/7$ is PT-invariant ([T] for $0 < \kappa \le \mu^2/48$), and the colour-invariant vacuum $\Gamma_v$ of the Gap phase is invariant under $g_v\circ\mathrm{PT}$ with $g_v \in G_2$, $g_vv = -v$ ([T], T-64). Turning this into $\theta_{\mathrm{QCD}} = 0$ needs one more step, and it is the precise obstruction: the antiunitary symmetry must be identified with CP of the colour sector, and $\theta_{\mathrm{QCD}}$ of step 5 is the phase of $\det(M_uM_d)$, which requires the quark mass matrices — the Yukawa structure, which is open (standard model, Theorem 2.6(f)). Until it is closed, $\theta_{\mathrm{QCD}} = 0$ stays [C at (SV)].

Resolved negatively (T-333, §3.1a): with the Yukawa couplings classified (T-332), no lift of this antiunitary symmetry to the fermions can give $\bar\theta = 0$ while keeping $m_t \neq m_b$ and the observed CKM phase. The route through the vacuum symmetry is closed [✗]. $\theta_{\mathrm{QCD}} = 0$ keeps only the $V_3$ chain of steps 3–5, [C at (SV)], and the $G_2$-invariant potential no longer contains $V_3$.

Step 3 (Uniqueness of the vacuum). From T-64 [T] (corrected to the $G_2$-invariant potential; its vacuum has no sector values — hypothesis (SV)) (global minimisation of $V_{\mathrm{Gap}}$): $G_2$-orbital reduction $21D \to 5D$ leads to a unique global minimum with positive-definite Hessian ($\mathrm{Hess}(V_{\mathrm{Gap}})|_{\min} > 0$). The vacuum is uniquely determined.

Step 4 (Isotropy of phases at the minimum). At the minimum of $V_{\mathrm{Gap}}$:

• From $V_2$: $\sin^2\theta_{ij}$ is minimised at $\theta_{ij} = 0$ or $\pi$ for all $(i,j) \in 3\text{-to-}\bar{3}$.
• From $V_3$: for Fano triplets $\sin(\theta_{ij} + \theta_{jk} - \theta_{ik})$ is minimised at $\theta_{ij} = \theta_{jk} = \theta_{ik} = 0$ (not $\pi$, which increases $V_3$).
• Hessian: eigenvalue $\lambda_1 = 18\mu^2 > 0$ confirms that $\theta_{ij} = 0 \;\forall (i,j) \in 3\text{-to-}\bar{3}$ is a stable minimum.

Conclusion: all phases vanish in the vacuum.

Refuted 2026-09-26 [✗]: $\sin$ is not minimised at $0$; real states are not even stationary points of $V_2 + V_3 + V_4$ when $\lambda_3 \neq 0$, and the vacuum has $\mathcal G_{\text{total}} > 0$ (box at the head of §3.1).

Step 5 (Vanishing of $\theta_{\mathrm{QCD}}$). The parameter $\theta_{\mathrm{QCD}}$ in the Gap formalism:

$\theta_{\mathrm{QCD}} = \arg\left(\det(M_u \cdot M_d)\right) = \arg\left(\lambda_3^2 \cdot \prod_{(i,j) \in 3\text{-to-}\bar{3}} |\gamma_{ij}|\right)$

From steps 1–4: $\lambda_3 \in \mathbb{R}$ (step 1), $|\gamma_{ij}| \in \mathbb{R}_+$ (moduli are real), all phases $\theta_{ij} = 0$ (step 4). Consequently, the argument of the product of real positive numbers is identically zero:

$\theta_{\mathrm{QCD}} = 0 \quad \text{(exactly, not approximately)}$

Retracted 2026-09-26 [✗]: the formula identifies the quark mass matrices with $\lambda_3$ and the moduli of Gap coherences; with the fields that (Cl) forces the quark masses come from Yukawa couplings whose phases are inputs (T-333(h)), and the $\theta$ of the gauge action enters $\bar\theta$ on its own.

Step 6 (Non-perturbative stability). From T-69 [T] (topological protection): $\pi_2(G_2/T^2) \cong \mathbb{Z}^2$ guarantees topological stability of the vacuum. Energy barrier:

$\Delta V \geq 6\mu^2 > 0$

Instanton configurations (§3.3) do not violate the isotropy of phases: they rearrange the windings $\theta_{ij}$ with the vacuum fixed at $\theta_{ij} = 0$. The topological charge $\mathbb{Z}_2$ forbids a continuous deformation to $\theta \neq 0$.

Step 7 (Radiative stability). From T-66 (UV finiteness: field-space [T], order-by-order [C]): radiative corrections preserve $G_2$-symmetry. The coefficient $\lambda_3$ runs under RG but remains real (RG preserves the reality of coefficients of a real potential). Phase isotropy $\theta_{ij} = 0$ is a property of the minimum, not violated by loop corrections.

$\blacksquare$

3.1a The vacuum's antiunitary symmetry on the fermions (T-333)

tip

Status: Theorem 3.1a is [T] as mathematics and [C at (Cl)] in UHM; the comparison with the data uses the measured $m_t/m_b$ and $J$

The corrected vacuum keeps an antiunitary symmetry: $\mathrm{PT}$ at $I/7$, $\Theta_v = g_v\circ\mathrm{PT}$ on the orbit $S^6$. The box after step 2 asked whether this symmetry acts as CP on quarks once the Yukawa structure is fixed. It does not, in any lift that the data allow. Registry row T-333; checks in website/scripts/check_core_numbers.py.

Theorem 3.1a (T-333).

(a) The lifts [T]. On $\mathbb C^7$, $\mathrm{PT}$ is complex conjugation, which on $\mathcal S = \mathbb C\otimes\mathbb O$ is $J = \gamma_8$, a Clifford generator in the charged directions of the Higgs plane. Take $g_v \in G_2$ of order 2, the identity on a quaternionic line not through $e_O$ and $-1$ on its complement, so $g_ve_O = -e_O$. Then $\Theta_v = g_vJ$ keeps the vacuum $\Gamma_v$ of T-64, and $\mathrm{PT}$ alone does not. On $\mathcal S_{\mathbb C} = \mathcal S\otimes\mathbb C'$ each has a $\mathbb C'$-linear lift ($\otimes 1$) and a $\mathbb C'$-antilinear lift ($\otimes K'$):

lift	field unit $\omega$	halves $V_L$, $V_R$	on the vector $\mathbb R^{10}$	type
$J\otimes1$	commutes	kept	rotation, det $+1$	element of $\mathrm{Spin}(10)$
$\Theta_v\otimes1$	commutes	exchanged; $\mathfrak{su}(2)_L\to\mathfrak{su}(2)_R$	rotation, det $+1$; fixes $\gamma_{10}$, $iL_{e_O}$	element of $\mathrm{Spin}(10)$, left–right exchange
$J\otimes K'$	anticommutes	exchanged	reflection, det $-1$	P-type
$\Theta_v\otimes K'$	anticommutes	kept; normalises $\mathfrak g_{\mathrm{SM}}$	reflection; fixes $iL_{e_O}$, reverses $\gamma_{10}$	CP-type

(b) The linear lifts do not act on $\theta$ [T]. $J\otimes1$ and $\Theta_v\otimes1$ are unitary internal transformations of the $\mathbf{16}$ in the connected group $\mathrm{Spin}(10)$. They leave the $\theta$-term unchanged, so they cannot set $\theta = 0$. The canonical lift of an operator on $\mathcal S$ is the $\mathbb C'$-linear one, so the canonical action of the vacuum's symmetry on fermions is a gauge-group element, not CP.

(c) Exchanging lifts force $m_t = m_b$ [T]. $J\otimes K'$ and $\Theta_v\otimes1$ map $\mathfrak g_{\mathrm{SM}}$ to its mirror. Together with $\mathfrak g_{\mathrm{SM}}$ the mirror generates the left–right algebra $\mathfrak{su}(3)\oplus\mathfrak{su}(2)_L\oplus\mathfrak{su}(2)_R\oplus\mathfrak u(1)_{B-L}$ (dimension 15). A Yukawa coupling invariant under $G_{\mathrm{SM}}$ and under such a lift is therefore left–right equivariant. With the one real doublet, T-332(b) then gives $\lvert m_u\rvert = \lvert m_d\rvert$ and $\lvert m_\nu\rvert = \lvert m_e\rvert$, refuted by $y_t/y_b \approx 68$.

(d) The CP-type lift forces $J = 0$ [T]. Suppose $\Theta_v\otimes K'$, combined with any unitary action on the families (a generalised CP), is an unbroken symmetry of the quark Yukawa couplings. Then every CP-odd weak-basis invariant vanishes: $\bar\theta$, and also the Jarlskog invariant (Bernabéu, Branco and Gronau, Phys. Lett. B 169, 243 (1986)). The measured value is $J = (3.12^{+0.13}_{-0.12})\times10^{-5}$ (PDG 2024, CKM review, §12).

So under (Cl) with one doublet, no lift of the vacuum's antiunitary symmetry makes $\bar\theta = 0$ while keeping $m_t \neq m_b$ and $J \neq 0$. Fixing the Yukawa structure resolves the obstruction named after step 2, and negatively: the route "vacuum symmetry → $\theta = 0$" is closed [✗]. Three routes survive, each as a hypothesis [H]. (i) Left–right parity with a complex bidoublet (two doublets, against T-296), Hermitian Yukawa matrices and relatively real vacuum values. Hermitian mass matrices have a real determinant and can still carry $J \neq 0$ (Babu and Mohapatra, Phys. Rev. D 41, 1286 (1990)). (ii) Spontaneous CP violation of Nelson–Barr type, which the corpus does not contain. (iii) A Peccei–Quinn axion, with the Gap axion of §3.2 then required to relax $\theta$. With the corrected potential PT-even (T-331), the Gap sector has no source of CP violation at all. The CKM phase must be an input of the Yukawa sector, so §3.3(b) no longer holds for the corrected potential.

Proof. (a) The table is computed: each lift is tested against $\omega$ and against the volume $\omega_4$ of the colour-free plane, and conjugation by it is expanded in the ten Clifford vectors. The rotation and its determinant are read off from that expansion. The images of $\mathfrak{su}(2)_L$ and $\mathfrak g_{\mathrm{SM}}$ are computed. (b) Elements of a connected gauge group preserve $\int G\tilde G$. (c) The Lie closure is computed. If $Y$ is equivariant under $G$ and under $T$, it is equivariant under $TGT^{-1}$ and hence under the group they generate. (d) This is the cited theorem; it is checked on random generalised-CP-invariant Yukawa matrices, $\lvert J\rvert < 10^{-12}$. Hermitian ones give $\mathrm{Im}\det M_uM_d = 0$ with $\lvert J\rvert > 10^{-3}$. $\blacksquare$

Witnesses: test_vacuum_antiunitary_lifts_are_gauge_parity_or_cp, test_an_unbroken_cp_or_lr_symmetry_contradicts_the_quark_data.

3.1b Peccei–Quinn and Nelson–Barr in the Clifford content (T-333, continued)

tip

Status: Theorem 3.1b is [T] as mathematics and [C at (Cl)] in UHM; with the fields that (Cl) forces, $\bar\theta$ is a free parameter and the strong CP problem is open in UHM [Pr]

T-333 left three routes to $\bar\theta=0$: left–right parity with two doublets, Nelson–Barr, and an axion. This section tests the last two against the fields that the Clifford frame forces — three generations of the $\mathbf{16}$ (T-329) and one doublet in the colour-free plane — and checks the axion of the dark-matter page. Registry row T-333, items (e)–(h); checks in website/scripts/check_core_numbers.py.

Theorem 3.1b (T-333(e)–(h)).

(e) No Peccei–Quinn symmetry [T]. Let $Y_u$ and $Y_d$ be the $3\times3$ matrices of $Q\tilde Hu^c$ and $QHd^c$, with $\det Y_u\neq0$ and $\det Y_d\neq0$. Every phase rotation of $Q_i$, $u^c_i$, $d^c_i$ and $H$ that keeps all their non-zero entries has zero colour anomaly, $\sum_i(2q_{Q_i}+q_{u_i}+q_{d_i})=0$. All six quarks are massive ($m_u\approx2.2$ MeV, PDG), so the Clifford content with one doublet has no Peccei–Quinn symmetry and no axion. In one generation this is T-332(i): a coupling with $\lvert\beta\rvert\neq\lvert\alpha\rvert$ keeps only hypercharge, $B$ and $L$. A colour-anomalous phase appears only at $\beta=\pm\alpha$, where a whole charge type is massless. That is the massless-quark solution, which the data exclude.

(f) What an axion needs [T for the statement]. A colour-anomalous $\mathrm{U}(1)$ that survives the quark masses needs new fields of one of two kinds. The first kind is a second doublet, so that $Q\tilde H_uu^c$ and $QH_dd^c$ carry independent phases (Peccei and Quinn, Phys. Rev. Lett. 38, 1440 (1977)). With $f_a=v$ this is the Weinberg–Wilczek axion (Phys. Rev. Lett. 40, 223 and 279 (1978)), long excluded. The invisible version adds a singlet with $f_a\gg v$ (Dine, Fischler and Srednicki, Phys. Lett. B 104, 199 (1981); Zhitnitsky, Sov. J. Nucl. Phys. 31, 260 (1980)). The second kind is new coloured fermions whose mass comes from a singlet (Kim, Phys. Rev. Lett. 43, 103 (1979); Shifman, Vainshtein and Zakharov, Nucl. Phys. B 166, 493 (1980)). The first kind is the complex bidoublet of T-332(c), against T-296. The second is not in $\mathcal S_{\mathbb C}$: its 32 real components are one chiral $\mathbf{16}$, forced by T-329, and the hypercharges of its coloured states are not closed under a change of sign.

(g) The Gap axion is not a QCD axion in this content [T for the implication]. The dark-matter page defines the axion as a zero mode of Gap phases "possessing an axial anomaly with QCD". A coupling to $G\tilde G$ through an anomaly is the anomaly of a fermion current, so by (e) no Gap phase acquires one in the Clifford content. The mass formula $m_a\propto m_\pi f_\pi/f_a$ also needs the potential of $a$ to come from QCD alone. The same page (§3.5) gives all 21 phases a mass from $V_{\mathrm{Gap}}$, with no flat direction. A QCD axion needs the non-QCD part of its potential below about $10^{-10}\chi_{\mathrm{top}}$, where $\chi_{\mathrm{top}}^{1/4}\approx75.6$ MeV (Borsanyi et al., Nature 539, 69 (2016)). The arithmetic of the page is right: $f_a=2\times10^{15}$ GeV gives $m_a=2.9$ neV. Its relic estimate takes $\theta_i=H_I/(2\pi f_a)$, a pure inflationary fluctuation around $\theta=0$. The axion density is then an isocurvature mode with relative amplitude of order one. The Planck limit on uncorrelated dark-matter isocurvature (Planck Collaboration, Astron. Astrophys. 641, A10 (2020)), taken as $\beta_{\mathrm{iso}}<0.038$, allows $\Omega_a/\Omega_c\lesssim3\times10^{-5}$ with the relative power $4/N$ at $N=60$ e-folds, not $10^{-2}$.

(h) No spontaneous CP violation, so no Nelson–Barr [T]. Nelson–Barr (Nelson, Phys. Lett. B 136, 387 (1984); Barr, Phys. Rev. Lett. 53, 329 (1984)) keeps CP exact in the Lagrangian, so that $\theta=0$ there. It breaks CP only by complex vacuum values of heavy singlets that couple the light quarks to vector-like heavy quarks. The Clifford content has none of this. (1) It has no vector-like quark, by (f). (2) The Gap vacuum does not break CP. It is invariant under the CP-type lift $\Theta_v\otimes K'$ (T-333), which fixes $iL_{e_O}$ and reverses $\gamma_{10}$. Hypercharge rotates the neutral plane of the Higgs (T-332(h)) and acts on $\mathcal S$ as $(B-L)/2$, which commutes with $\Gamma_v$. So a hypercharge rotation composed with $\Theta_v\otimes K'$ fixes both $\Gamma_v$ and any neutral Higgs vacuum value. CP is broken spontaneously only when no generalised CP leaves the vacuum invariant (Branco, Lavoura and Silva, CP Violation, Oxford University Press (1999)). If CP were exact in the Lagrangian it would therefore stay unbroken, and T-333(d) would give $J=0$, against $J=3.12\times10^{-5}$. So CP must be broken explicitly in the Yukawa couplings, and then nothing protects $\theta$.

So under (Cl) none of the three routes of T-333 is open without a field that the frame does not contain. Parity needs a second doublet. Nelson–Barr needs vector-like quarks and a CP-breaking singlet. An axion needs a second doublet and a singlet, or new coloured fermions. With the forced content $\bar\theta$ is a free parameter of the Yukawa sector, and the strong CP problem is open in UHM [Pr]. Each extension is falsifiable: a charged Higgs (parity, DFSZ), a vector-like quark (Nelson–Barr, KSVZ), or an axion signal.

Proof. (e) If $\det Y_u\neq0$, some permutation $\sigma$ has $(Y_u)_{i\sigma(i)}\neq0$ for every $i$. Invariance of those entries gives $q_{Q_i}-q_H+q_{u_{\sigma(i)}}=0$, so $\sum q_u=-\sum q_Q+3q_H$. In the same way $\sum q_d=-\sum q_Q-3q_H$, and the sum gives the claim. It is checked on 300 random supports with non-singular matrices. (f) The hypercharges of the coloured states of $\mathcal S_{\mathbb C}$ are computed. (g) The numbers use $m_u=2.16$ MeV, $m_d=4.67$ MeV, $m_\pi=135$ MeV, $f_\pi=92$ MeV, $A_s=2.1\times10^{-9}$ and $r<0.036$, which give $H_I/(2\pi f_a)=3.7\times10^{-3}$ as on the dark-matter page. (h) $\Gamma_v$, extended to $\mathcal S$, commutes with $(B-L)/2$ (computed). The rest is T-333 and the cited criterion. $\blacksquare$

Witness: test_no_peccei_quinn_symmetry_in_the_clifford_content.

3.1c What the Gap potential says about $\bar\theta$ (T-99, corrected)

tip

Status: Theorem 3.1c is [T] as mathematics and [C at (Cl)] in UHM; $\theta_{\mathrm{QCD}} = 0$ is not derived, strong CP is open [Pr] (2026-09-26)

This replaces the conclusion of T-99. T-99 set out to show that the Gap sector makes $\bar\theta$ vanish. The true statement is weaker and exact: the Gap sector is CP-neutral, and no term of a $G_2$-invariant Gap potential can make $\bar\theta$ vanish. Registry row T-99; checks in website/scripts/check_core_numbers.py.

Theorem 3.1c (T-99, corrected).

(a) The corrected Gap sector is CP-neutral [T]. Every $G_2$-invariant polynomial of degree $\le 3$ on $\mathrm{Herm}(\mathbb C^7)$ is PT-even (T-331), so $V_{\text{Gap}} = \mu^2\mathcal G + \lambda_4\mathcal G^2 - \kappa\mathcal A$ is PT-invariant. Each of its vacua keeps an antiunitary symmetry: $\mathrm{PT}$ at $I/7$, $g_v\circ\mathrm{PT}$ on the orbit $S^6$ (T-64). The Gap sector carries no CP-odd phase of its own.

(b) CP-neutrality does not reach $\bar\theta$ [T]. The canonical lift of that symmetry to the fermions is an element of $\mathrm{Spin}(10)$ and leaves the $\theta$-term unchanged; the lifts that act as P or CP contradict $m_t \neq m_b$ or $J \neq 0$ (§3.1a). With one doublet and three $\mathbf{16}$ CP must be broken explicitly in the Yukawa couplings, and then nothing protects $\bar\theta = \theta + \arg\det(M_uM_d)$ (§3.1b).

(c) A PT-odd Gap term is a source of phases, not a guard [T]. The first PT-odd $G_2$-invariants appear in degree 4. There are three (T-331), of types $SX_{\mathbf 7}^2X_{\mathbf{14}}$, $SX_{\mathbf 7}X_{\mathbf{14}}^2$ and $S^3X_{\mathbf 7}$, where $S$ is the traceless part of $\mathrm{Re}\,\Gamma$ and $X = \mathrm{Im}\,\Gamma$. The last one is $Q_{\rm odd} = \langle \varphi\cdot X, \varphi\cdot(NS)\rangle$ with $N_{pq} = \varphi_{pab}S_{ac}S_{bd}\varphi_{qcd}$. A potential containing any PT-odd term is invariant under no $g\circ\mathrm{PT}$ with $g \in G_2$, since $f(g\bar\Gamma g^{-1}) = f(\bar\Gamma) = -f(\Gamma)$. $Q_{\rm odd}$ is linear in $X$; on a real state with $S \neq 0$ its first variation in $\mathrm{Im}\,\Gamma$ is non-zero (0.0019 on the witness state). So $\mu^2\mathcal G + \lambda_4\mathcal G^2 + \varepsilon Q_{\rm odd}$ goes below zero off the real states (minimum $-0.0042\mu^2$ with $\mathcal G = 0.0034$ at $\varepsilon = 0.5\mu^2$, $\lambda_4 = 10\mu^2$). The cubic $V_3$ of the former derivation behaves the same way (box at the head of §3.1).

(d) Consequence. In UHM with the fields that (Cl) forces, $\bar\theta$ is a free parameter, fixed by no axiom, no Gap potential and no symmetry of the vacuum. UHM predicts no value of the neutron EDM; the bound $\lvert\bar\theta\rvert \lesssim 10^{-10}$ is an input. The strong CP problem is open in UHM [Pr].

Routes tried before the retraction.

route	outcome	where
$V_3$ chain of T-99 under (SV)	step 4 false for every $\lambda_3 \neq 0$; (SV) fixes moduli, not phases	box at the head of §3.1
antiunitary symmetry of the corrected vacuum	the linear lift is a gauge element; P- or CP-type lifts give $m_t = m_b$ or $J = 0$	§3.1a, T-333(a)–(d)
PT-odd $G_2$-invariant quartics	break every $g\circ\mathrm{PT}$; move the vacuum off the real states	(c) above
Peccei–Quinn axion	no colour-anomalous $\mathrm U(1)$ with all quarks massive; needs a second doublet or new coloured fermions	§3.1b(e)–(g)
Nelson–Barr	no vector-like quark, no spontaneous CP violation	§3.1b(h)
left–right parity	needs a complex bidoublet (two doublets, against T-296)	§3.1a, T-332(c)
massless $u$ quark	$m_u = 2.16$ MeV (PDG)	§3.1b(e)

Proof. (a) is T-331 with T-64. (b) is T-333. (c): invariance under $G_2$ and the sign under PT are computed on random states; the first variation at a real state is linear in $X$ and is evaluated directly, and a shift $R \to R + itX$ with the sign of $t$ opposite to it lowers $V$ below its value $0$ on the real states. (d) follows from (a)–(c) and T-333(e)–(h). $\blacksquare$

Witness: test_theta_route_through_the_gap_potential_fails_for_v3_and_for_pt_odd_quartics.

3.2 Corollary: Axion without PQ Mechanism

Reinterpretation of the axion's role

In standard physics the Peccei–Quinn axion solves the strong CP problem via dynamical relaxation $\theta \to 0$. In the Gap formalism $\theta_{\mathrm{QCD}} = 0$ follows structurally (T-99), so an axion is not needed for CP. Its role is purely as a DM candidate. Conditional (2026-09-25): this holds only through the $V_3$ chain of T-99, [C at (SV)]. The route through the vacuum's antiunitary symmetry is closed (T-333), and a Peccei–Quinn axion is one of the three routes left open. Update (T-333(e)–(h), §3.1b): in the Clifford content no $\mathrm{U}(1)$ with a colour anomaly survives the quark masses, so the Gap axion has no $G\tilde G$ coupling there. It is not a QCD axion and relaxes nothing. The table below describes it only on the hypothesis (PQ) of an added Peccei–Quinn sector. Update 2026-09-26 (T-99 corrected, §3.1c): the $V_3$ chain is retracted [✗] as well, so the premise of this box — "$\theta_{\mathrm{QCD}} = 0$ follows structurally" — no longer holds; strong CP is open [Pr].

The Gap axion (§3.4, definition in dark matter, §3.1) — a pseudoscalar field $a(x)$, the zero mode of phases $\theta_{ij}$ in the 3-to-$\bar{3}$ sector — exists as a particle (Goldstone boson from the $(S^1)^{21}$ compactification). But its role is fundamentally different:

	Standard axion	Gap axion
Solves strong CP?	Yes (dynamical relaxation)	No (T-99: $\theta = 0$ structurally — retracted 2026-09-26; strong CP open [Pr], §3.1c)
DM candidate?	Yes ($\sim 100\%$ at $f_a \sim 10^{12}$ GeV)	Yes, subdominant ($\sim 1\%$ DM)
Mass	$m_a \sim 10^{-5}$ eV	$m_a \sim 3$ neV (from $f_a \sim 2 \times 10^{15}$ GeV)
$f_a$	Free parameter	Fixed: $f_a = \varepsilon \cdot M_P$

Cross-reference: dark matter from Gap, §3.

3.3 Corollary: Dual Role of $V_3$

The cubic potential $V_3$ (octonionic associator) plays a dual role:

(a) Cause of $\theta_{\mathrm{QCD}} = 0$ (as argued with the retracted cubic; see the box after step 2 of T-99). $V_3$ is the unique PT-odd term of the potential. At the minimum of $V_{\mathrm{Gap}}$ it fixes all phases to $\theta_{ij} = 0$, making $\theta_{\mathrm{QCD}} = 0$ a structural result (T-99, steps 2 and 4). Retracted [✗] 2026-09-26: a PT-odd term does not fix the phases at zero; the vacuum of $V_2 + V_3 + V_4$ has $\mathcal G_{\text{total}} > 0$ (§3.1c).

(b) Unique source of CP violation in CKM. (For the retracted $V_3$ only: the $G_2$-invariant potential is PT-even (T-331) and contains no source of CP violation, so the CKM phase is an input of the Yukawa sector, T-333.) The same $V_3$ generates complex phases in the Yukawa matrices $Y^u$, $Y^d$ via generation mixing, giving a non-zero phase $\delta_{\mathrm{CP}} \neq 0$ in the CKM matrix.

This explains the CP paradox: why strong CP violation is zero ($\theta_{\mathrm{QCD}} = 0$), while weak CP violation is non-zero ($\delta_{\mathrm{CP}} \approx 64.6°$). Answer: $V_3$ sets the vacuum phases to zero ($\theta_{ij} = 0$), but generates inter-generational phases via loop corrections. Cross-reference: CKM matrix, §4.

3.4 Gap Instantons and the $\theta$-Vacuum

(a) Topology: $\pi_3(\mathrm{SU}(3)) = \mathbb{Z}$. An instanton is a map $S^3 \to \mathrm{SU}(3)$ with non-zero winding number $n$.

(b) Gap instanton. In Gap language: an instanton is a configuration $\theta_{ij}(x)$ in the 3-to-$\bar{3}$ sector in which all 8 phases complete a full rotation from 0 to $2\pi$ upon traversal of a three-dimensional sphere in spatial coordinates.

(c) Instanton action:

$S_{\mathrm{inst}} = \frac{8\pi^2}{g_s^2} = \frac{8\pi^2}{4\pi\,\alpha_s} = \frac{2\pi}{\alpha_s}$

In Gap parameters: $\alpha_s = g_s^2/(4\pi)$ is determined via the Gap coupling constant in the 3-to-$\bar{3}$ sector. From the relation $g_s \sim 1/\sqrt{\lambda_4 \cdot N_{\mathrm{eff}}}$:

$\alpha_s(\mu) = \frac{\lambda_4(\mu)}{4\pi \cdot 9}$

where 9 is the number of coherences in the 3-to-$\bar{3}$ sector.

(d) $\theta$-vacuum. The full vacuum is a superposition of instanton sectors:

$|\theta\rangle = \sum_{n=-\infty}^{\infty} e^{in\theta} |n\rangle$

From T-99 (step 5): $\theta_{\mathrm{QCD}} = 0$ exactly, so the physical vacuum = $|0\rangle$ — the unique instanton sector without a phase factor. Retracted 2026-09-26: step 5 is retracted [✗]; the physical vacuum is $|\bar\theta\rangle$ with $\bar\theta$ a free parameter (§3.1c).

---

4. Deconfinement and Phase Transition

4.1 Theorem 2.1 (Deconfinement as a Gap Phase Transition)

Statuses of §4

Polyakov loop as order parameter — [T] (from the $\mathbb{Z}_3$ centre of $\mathrm{SU}(3)_C$ [T-42e]). Critical temperature $T_c \sim 170$ MeV — [C at (SV)] (depends on vacuum parameters). Crossover with dynamical quarks — [H] (qualitative model).

As $T_{\mathrm{eff}}$ rises above the critical value $T_{\mathrm{deconf}}$ the system undergoes a phase transition from the confinement phase to the deconfinement phase:

(a) Confinement phase ($T < T_{\mathrm{deconf}}$):

• $\mathrm{Gap} \to 0$ in the 3-to-$\bar{3}$ sector
• Area law
• Linear potential $V(L) = \sigma \cdot L$
• Quarks confined in colourless hadrons

(b) Deconfinement phase ($T > T_{\mathrm{deconf}}$):

• $\mathrm{Gap} > 0$ in the 3-to-$\bar{3}$ sector (thermal fluctuations break isotropy)
• Perimeter law: $W(C) \sim \exp(-\mu \cdot P(C))$
• Potential screened: $V(L) = \sigma \cdot L \cdot \exp(-L/\lambda_D)$
• Free quarks and gluons

(c) Critical temperature:

$T_{\mathrm{deconf}} = T_c^{(3\bar{3})} = \frac{\mu^2_{3\bar{3}}}{\Gamma_2 / \kappa_0 \cdot k_B \ln 9}$

from the Gap-theory phase diagram restricted to the 3-to-$\bar{3}$ sector ($N_{\mathrm{eff}} = 9$, not 21).

(d) Prediction. For 3-to-$\bar{3}$: $N_{\mathrm{eff}} = 9$, $\mu^2 \approx 16.6$ in Gap units. Translation to physical units via $\Lambda_{\mathrm{QCD}}$:

$T_{\mathrm{deconf}} \sim \Lambda_{\mathrm{QCD}} \sim 170 \text{ MeV}$

— consistent with lattice QCD computations ($T_c \approx 150\text{--}170$ MeV for the crossover transition).

4.2 Order Parameter of Deconfinement (Polyakov Loop)

The confinement–deconfinement phase transition is characterised by an order parameter — the Polyakov loop $\langle P \rangle$:

$P = \frac{1}{N_c}\mathrm{Tr}\left[\mathcal{P}\exp\left(i\oint_0^{1/T} A_0^a T_a \, d\tau\right)\right]$

In the Gap formalism $A_0^a \sim \partial_\tau \theta_{ij}^{(a)}$, and the Polyakov loop measures the holonomy of the Gap connection along the temporally compactified coordinate $\tau \in [0, 1/T]$.

Theorem (Polyakov loop as order parameter) [T]

The Polyakov loop $\langle P \rangle$ is the order parameter of deconfinement for pure $\mathrm{SU}(3)_C$. Proof: $\mathrm{SU}(3)_C = \mathrm{Stab}_{G_2}(e_O)$ [T-42e [T]]. The centre $Z(\mathrm{SU}(3)) = \mathbb{Z}_3$ acts on the Polyakov loop as $P \mapsto e^{2\pi i k/3} P$, $k=0,1,2$. In the confinement phase $\mathbb{Z}_3$-symmetry is exact → $\langle P \rangle = 0$ (the unique $\mathbb{Z}_3$-invariant value). Deconfinement = spontaneous breaking of $\mathbb{Z}_3$ → $\langle P \rangle \neq 0$. This is the standard result (Svetitsky–Yaffe, 1982), applied to $\mathrm{SU}(3)_C$ derived from the $G_2$-structure. $\blacksquare$

(a) At $T < T_c$: $\langle P \rangle = 0$ — the centre $\mathbb{Z}_3$-symmetry of $\mathrm{SU}(3)_C$ is unbroken. The Gap phases $\theta_{ij}$ average to zero upon traversal of the thermal circle. The free energy of a single quark is infinite: $F_q = -T\ln\langle P \rangle \to \infty$.

(b) At $T > T_c$: $\langle P \rangle \neq 0$ — the centre $\mathbb{Z}_3$-symmetry is spontaneously broken. Thermal fluctuations break the isotropy of the Gap vacuum in the 3-to-$\bar{3}$ sector, Gap acquires a non-zero value, and the holonomy becomes non-trivial. The quark free energy is finite.

(c) Critical temperature [C at (SV)]. The formula for $T_c$ (§4.1) depends on the vacuum parameters T-64 [T] (corrected to the $G_2$-invariant potential; its vacuum has no sector values — hypothesis (SV)); qualitatively $T_c \sim \Lambda_{\mathrm{QCD}} \sim 170$ MeV.

(d) Nature of the transition [H]. For pure $\mathrm{SU}(3)$ (without dynamical quarks) the transition is first order — $\langle P \rangle$ undergoes a jump. With $N_f = 2+1$ dynamical quarks the transition broadens into a crossover. In the Gap formalism: dynamical quarks are fermionic Gap configurations, their presence explicitly breaks $\mathbb{Z}_3$-symmetry ($\langle P \rangle \neq 0$ already at $T < T_c$), turning the phase transition into an analytic crossover.

Computational problem C18: finite-temperature Gap lattice. Realisable as MVP-12 in SYNARC.

(d) Quark–gluon plasma (QGP). At $T \gg T_c$ the system enters the quark–gluon plasma phase, where:

• $\mathrm{Gap}(\text{3-to-}\bar{3}) \sim O(1)$ — colour degrees of freedom are deconfined
• QGP pressure: $p \approx \frac{\pi^2}{90}\left(2(N_c^2-1) + \frac{7}{2}N_c N_f\right)T^4$ — ideal Stefan–Boltzmann gas
• Corrections $\sim \alpha_s(T)$ are computed by standard perturbative RG (see Gap renormalisation group)

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5. Asymptotic Freedom

Asymptotic freedom — the decrease of the coupling constant $\alpha_s$ with increasing energy — is a fundamental property of $\mathrm{SU}(3)_C$, ensuring the transition from confinement (IR) to free quarks (UV). In the Gap formalism asymptotic freedom follows from the general RG structure: the beta function of $\lambda_4$ in the 3-to-$\bar{3}$ sector, restricted to $N_{\mathrm{eff}} = 9$ coherences, reproduces the standard one-loop QCD result.

5.1 Theorem 3.1 (Running Coupling Constant)

Status: Theorem [T]

The $\mathrm{SU}(3)_C$ coupling constant in the Gap formalism runs under RG according to the standard formula.

(a) One-loop beta function for $\alpha_s$ in the 3-to-$\bar{3}$ sector:

$\beta_{\alpha_s} = -\frac{\alpha_s^2}{2\pi}\left(\frac{11}{3}N_c - \frac{2}{3}N_f\right)$

In the Gap formalism: $N_c = 3$ (number of colours $= \dim(\text{3-sector})$), $N_f$ — number of active fermion generations.

(b) Sign: for $N_f < 33/2 = 16.5$ (satisfied for the SM with $N_f = 6$): $\beta_{\alpha_s} < 0$ $\to$ asymptotic freedom. At lower energy (larger distance) $\alpha_s$ grows $\to$ confinement.

(c) Relation to Gap parameters:

$\alpha_s(\mu) = \frac{\lambda_4(\mu)}{4\pi \cdot 9} = \frac{4\pi^2/63}{4\pi \cdot 9} \cdot \left(1 + \beta_{\lambda_4} \ln(\mu/\Lambda)\right)^{-1} = \frac{\pi}{567}\cdot\left(1 + \beta_{\lambda_4} \ln(\mu/\Lambda)\right)^{-1}$

using the Wilson–Fisher value $\lambda_4^* = 4\pi^2/63$.

(d) $\Lambda_{\mathrm{QCD}}$ from Gap:

$\Lambda_{\mathrm{QCD}} = \mu_{\mathrm{phys}} \cdot \exp\left(-\frac{2\pi}{(11 - 2N_f/3)\, \alpha_s(\mu_{\mathrm{phys}})}\right)$

5.1a Relation to the Gap RG Flow [T]

The running coupling constant $\alpha_s$ is a special case of the RG flow of $V_{\mathrm{Gap}}$ parameters. The correspondence is established as follows:

(a) General one-loop $\beta$-function for $\lambda_4$ (see Gap renormalisation group, §2):

$\beta_{\lambda_4} = -\epsilon\lambda_4 + \frac{(N+8)}{6}\frac{\lambda_4^2}{8\pi^2}$

Upon restriction to the 3-to-$\bar{3}$ sector: $N = N_{\mathrm{eff}} = 9$. The relation $\alpha_s = \lambda_4/(4\pi \cdot 9)$ and substitution of $\epsilon = 0$ (physical $d=4$ dimensions) give the standard QCD beta with the correct coefficient.

(b) The Wilson–Fisher fixed point $\lambda_4^* = 4\pi^2/63$ (from RG analysis) determines the value of $\alpha_s$ at the confinement scale:

$\alpha_s^* = \frac{\lambda_4^*}{4\pi \cdot 9} = \frac{4\pi^2}{63 \cdot 36\pi} = \frac{\pi}{567} \approx 0.0055$

This value corresponds to the deep perturbative regime. Under RG flow to the IR ($\mu \to \Lambda_{\mathrm{QCD}}$) the coupling grows to $\alpha_s \sim 1$, signalling confinement.

(c) Two-loop corrections (see RG flow, §3) modify the running of $\alpha_s$ at intermediate energies. RG suppression of $\lambda_3$ in the flow from $\mu_{\mathrm{Planck}}$ to $\mu_{\mathrm{EW}}$ (factor $\sim 10^{-14.5}$) is critical for quantitative predictions of CKM mixing angles and the $\Lambda$ budget.

5.2 Corollary (Running of Quark Masses)

Quark masses (defined via the Higgs coupling) run under RG:

$m_q(\mu) = m_q(\mu_0) \cdot \left(\frac{\alpha_s(\mu)}{\alpha_s(\mu_0)}\right)^{12/(33 - 2N_f)}$

The anomalous mass dimension $\gamma_m = 12/(33 - 2N_f)$ is the standard QCD result. In the Gap formalism: $12 = 4 \cdot 3$, where 4 is the number of components of the quark doublet $Q_L$ in one colour, 3 is the number of colours. The agreement is ensured by the fact that Gap theory in the 3-to-$\bar{3}$ sector reduces to standard QCD.

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6. ABJ Axial Anomaly from Cliff(7)

The Adler–Bell–Jackiw (ABJ, 1969) axial anomaly — quantum violation of the classical conservation of the axial current — is reproduced in the Gap formalism via the Clifford algebra $\mathrm{Cliff}(7)$ underlying the 7-dimensional internal structure.

6.1 Axial Current in the Gap Formalism [T]

Status: Theorem [T]

The axial current and its anomaly are fully reproduced from the $\mathrm{Cliff}(7)$-structure of Gap fermions.

(a) The chiral operator in the Gap formalism is defined via $\mathrm{Cliff}(7)$-elements:

$\gamma_5 = i\,\Gamma_O\,\Gamma_A\,\Gamma_S\,\Gamma_D$

where $\Gamma_X$ are generators of $\mathrm{Cliff}(7)$ associated with the 7 coherence dimensions. Axial current:

$j_5^\mu = \sum_{\mathrm{fermions}} \bar{\chi}\,\gamma^\mu\,\gamma_5\,\chi = n_L^\mu - n_R^\mu$

where $n_L$ is the number of configurations with $\mathrm{Gap}(E,U) = 0$ (left-handed), $n_R$ — with $\mathrm{Gap}(E,U) \neq 0$ (right-handed).

(b) Classical conservation: in the absence of gauge fields chirality is conserved ($\partial_\mu j_5^\mu = 0$). In Gap language: $\mathrm{Gap}(E,U) = 0$ cannot spontaneously become $\mathrm{Gap}(E,U) \neq 0$ without interaction.

6.2 Quantum Anomaly from the Index Theorem [T]

(a) Dirac operator on Gap space:

$D_{\mathrm{Gap}} = \sum_{\mu=0}^{3}\gamma^\mu D_\mu, \qquad D_\mu = \partial_\mu + A_\mu^a T_a$

where $A_\mu^a$ is the Gap gauge field (as in §1.2).

(b) Dirac index (Atiyah–Singer theorem):

$\mathrm{ind}(D) = n_+ - n_- = \frac{1}{32\pi^2}\int d^4x\, F_{\mu\nu}^a\,\tilde{F}^{a,\mu\nu}$

where $n_\pm$ are the numbers of zero modes with positive/negative chirality, $\tilde{F}^{\mu\nu} = \frac{1}{2}\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$ is the dual tensor.

(c) Anomalous divergence of the axial current:

$\partial_\mu j_5^\mu = \frac{N_f \cdot g_s^2}{16\pi^2}\, G_{\mu\nu}^a\,\tilde{G}^{a,\mu\nu}$

The coefficient $N_f = 3$ is the number of fermion generations. In the Gap formalism: $g_s^2/(16\pi^2) = \alpha_s/(4\pi)$, where $\alpha_s = \lambda_4/(4\pi \cdot 9)$ (from §5.1).

(d) Role of $\mathrm{Cliff}(7)$ [T]. The standard proof of the anomaly (Fujikawa, 1979) is based on the non-invariance of the path integral measure. Adaptation to the Gap formalism: replacing the ordinary Dirac operator by the Gap-Dirac operator does not change the topological nature of the anomaly. The coefficient is determined by the structure of the Clifford algebra; for the physical subspace $\mathrm{Cliff}(1,3) \subset \mathrm{Cliff}(7)$ the result coincides with the standard one. Key point: $\gamma_5$ is defined via four of the seven generators of $\mathrm{Cliff}(7)$ ($O, A, S, D$), and its anticommutation with $D_{\mathrm{Gap}}$ guarantees the existence of a chiral symmetry, broken at the quantum level.

6.3 Decay $\pi^0 \to \gamma\gamma$ [T]

The decay of the neutral pion is the classical confirmation of the ABJ anomaly and the number of colours $N_c = 3$.

(a) Amplitude:

$\mathcal{A}(\pi^0 \to \gamma\gamma) = \frac{\alpha\, N_c}{2\pi\, f_\pi}\,\epsilon_{\mu\nu\rho\sigma}\,\epsilon_1^\mu\, k_1^\nu\, \epsilon_2^\rho\, k_2^\sigma$

where $N_c = 3 = \dim(\{A,S,D\})$ is the number of colours from the Gap structure, $f_\pi \approx 93$ MeV is the pion decay constant.

(b) Lifetime:

$\tau(\pi^0) = \frac{64\pi}{\left(\alpha N_c / (\pi f_\pi)\right)^2 m_\pi^3} \approx 8.4 \times 10^{-17}\;\text{s}$

Observed value: $(8.5 \pm 0.5) \times 10^{-17}$ s. Exact agreement — confirms $N_c = 3$ from the $G_2$ decomposition.

(c) Interpretation in the Gap formalism. $\pi^0$ is a superposition of quark–antiquark Gap configurations $(u\bar{u} - d\bar{d})/\sqrt{2}$. The decay $\pi^0 \to \gamma\gamma$ is a rearrangement of the Gap profile: from a configuration with $\mathrm{Gap}(\text{3-to-}\bar{3}) \neq 0$ (quark pair) to a configuration with $\mathrm{Gap} = 0$ (photons — massless, colourless). The anomaly ensures non-conservation of the axial current, permitting this transition.

6.4 Anomalous Ward Identities [T]

From the ABJ anomaly the modified Ward identities for axial vertices follow:

$q_\mu\,\Gamma_5^{\mu,ab}(p,q) = 2m\,\Gamma_5^{ab}(p,q) + \frac{\alpha_s}{2\pi}\,\delta^{ab}\,\epsilon_{\mu\nu\rho\sigma}\,p^\mu q^\nu\epsilon_1^\rho\epsilon_2^\sigma$

The second term is the anomalous contribution, absent classically. In the Gap formalism this term arises from the non-trivial topology of the space of Gap configurations: $\pi_3(\mathrm{SU}(3)) = \mathbb{Z}$ generates instanton configurations (§3) that connect the axial anomaly with the $\theta$-vacuum.

6.5 Cancellation of Gauge Anomalies (T-175b) [T] for the Standard-Model content; [C at (FE)] as a UHM result

Theorem (Cancellation of gauge anomalies) [T] for the representation content of Step 3; [C at (FE)] as a UHM result

For the one-generation fermion content of Step 3 — the Standard-Model fermions, which UHM imports with Connes' $H_F$ — the $\mathrm{SU}(3)_C \times \mathrm{SU}(2)_L \times \mathrm{U}(1)_Y$ gauge anomalies cancel completely. (Until 2026-09-25 the statement read: "The UHM spectral triple (T-53) with unimodularity guarantees complete cancellation"; that derivation is retracted in Step 2 below.)

$\mathrm{tr}(T^a \{T^b, T^c\}) = 0 \quad \text{for all gauge generators}$

Frame of the premise (2026-09-26). (FE) is the premise of the axis frame. In the Clifford frame it is replaced by (Cl₀), and there the cancellation is no longer imported: the generation $\mathcal S_{\mathbb C}$ is the $\mathbf{16}$ of $\mathrm{Spin}(10)$, which has no cubic invariant, so every anomaly vanishes (T-329(e), [T] as mathematics, [C at (Cl)] in UHM; Standard Model §2.6, Premises of UHM).

Proof.

Step 1 (Unimodularity = anomaly cancellation). Alvarez, Gracia-Bondia, Martin (Phys. Lett. B364, 1995) proved: in the NCG model of the Standard Model the unimodularity condition $\det(u)|_{\mathcal{H}_{\text{int}}} = 1$ is strictly equivalent to the cancellation of gauge anomalies (in the absence of right-handed neutrinos; with right-handed neutrinos — also true with automatic adjustment of hypercharges).

Step 2 (UHM satisfies unimodularity) — retracted [✗] (2026-09-25). Former text: "The spectral triple T-53 has $A_{\text{int}} = \mathbb{C} \oplus M_3(\mathbb{C}) \oplus M_3(\mathbb{C})$, real structure $J$ (KO-dim 6) and is Morita-equivalent to the Connes algebra $\mathbb{C} \oplus \mathbb{H} \oplus M_3(\mathbb{C})$ (T-175a). The unitary group $U(A_{\text{int}}) = U(1) \times U(3) \times U(3)$ after unimodularity gives:

$SU(A_{\text{int}}) = \{u : \det(u)|_{\mathcal{H}_{\text{int}}} = 1\} \to \mathrm{SU}(3)_C \times \mathrm{SU}(2)_L \times \mathrm{U}(1)_Y\text{."}$

Three of its inputs fail: no real structure of KO-dimension 6 exists on $\mathbb{C}^7$ — its $\chi = \pm 1$ eigenspaces would need equal dimension, and 7 is odd (spacetime, Step 6); the Morita equivalence T-175a is retracted (the centres $\mathbb{C}^3$ and $\mathbb{C}\oplus\mathbb{R}\oplus\mathbb{C}$ differ); and unimodularity cannot produce an $\mathrm{SU}(2)$ from $U(1) \times U(3) \times U(3)$ — the condition $\det u = 1$ is a single constraint that cuts the rank from 7 to 6 and leaves $\mathrm{U}(1)^{2} \times \mathrm{SU}(3)^{2}$ up to finite quotients, with no $\mathrm{SU}(2)$ factor at all. The anomaly cancellation of Step 1 is a theorem about Connes' model with its imported $H_F$; UHM inherits it only together with that import, with the electroweak group [C at (FE)] (Standard Model).

Step 3 (Explicit verification). The UHM fermion representation (from the sectoral decomposition + HE) for one generation:

Fermion	$(\mathrm{SU}(3)_C, \mathrm{SU}(2)_L, Y)$	Multiplicity
$Q_L$	$(3, 2, +1/6)$	6
$u_R$	$(3, 1, +2/3)$	3
$d_R$	$(3, 1, -1/3)$	3
$L_L$	$(1, 2, -1/2)$	2
$e_R$	$(1, 1, -1)$	1

Verification of all 5 cancellation conditions ($N_g = 3$ generations factor out):

• $\mathrm{tr}(Y) = 6 \cdot \frac{1}{6} + 3 \cdot \frac{2}{3} + 3 \cdot (-\frac{1}{3}) + 2 \cdot (-\frac{1}{2}) + 1 \cdot (-1) = 1 + 2 - 1 - 1 - 1 = 0$ $\checkmark$
• $\mathrm{tr}(Y^3) = 6 \cdot (\frac{1}{6})^3 + 3 \cdot (-\frac{2}{3})^3 + 3 \cdot (\frac{1}{3})^3 + 2 \cdot (-\frac{1}{2})^3 + 1\cdot(1)^3 = \frac{1}{36} - \frac{8}{9} + \frac{1}{9} - \frac{1}{4} + 1 = 0$ $\checkmark$ (all fields written as left-handed Weyl: the right-handed $u,d,e$ enter via their conjugates $Y\to-Y$; the earlier line used unconjugated $Y$ and summed to $-4/9$)
• $\mathrm{SU}(3)^2 \times \mathrm{U}(1)_Y$: $2 \cdot \frac{1}{6} - \frac{2}{3} + \frac{1}{3} = 0$ $\checkmark$
• $\mathrm{SU}(2)^2 \times \mathrm{U}(1)_Y$: $3 \cdot \frac{1}{6} + (-\frac{1}{2}) = 0$ $\checkmark$
• Gravitational $\mathrm{tr}(Y) = 0$ — coincides with the first. $\checkmark$

All anomaly coefficients vanish. $\blacksquare$

Relation to the ABJ anomaly

Sections 6.1–6.4 prove the chiral ABJ anomaly ($\partial_\mu j_5^\mu \neq 0$) — the correct anomaly that must exist. T-175b proves the cancellation of gauge anomalies ($\mathrm{tr}(T^a\{T^b,T^c\}) = 0$) — the consistency condition that must be satisfied. Both results are consistent: the chiral anomaly breaks a global symmetry, the gauge anomalies are cancelled for the local symmetry.

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7. Complete Picture of Confinement in the Gap Formalism

7.1 Diagram

 UV (high energies) IR (low energies)
 Gap(3-to-3̄) ~ O(1) Gap(3-to-3̄) → 0
 αs ≪ 1 αs ~ 1
 ─────────────────────────────────────────────────→
 Free quarks Confinement
 Perimeter law W(C) Area law W(C)
 V(L) → const V(L) = σ·L

 ←── Asymptotic freedom ───→
 ←── RG: βα < 0 ───────────────→

7.2 Self-Consistency

Confinement in Gap theory is self-consistent:

1. $\mathrm{SU}(3)_C$ arises from $G_2$ as the stabiliser of the O-direction [T]
2. 8 gluons are fluctuations of Gap phases in the 3-to-$\bar{3}$ sector [T]
3. $\mathrm{Gap} \to 0$ in this sector creates the conditions for confinement [T]
4. $V_3$ generates a linear potential (area law) [T] (topological proof); string tension $\sigma \sim \lambda_3|\varepsilon|/2$ [T]
5. String tension expressed via Gap parameters [C at (SV)] (naive discrepancy $\sim 7\times$; diagnostics: sectoral correction from the soft Hessian mode $\to$ $\sim 457$ MeV; hierarchy [T], numerical value [C at (SV)])
6. $\theta_{\mathrm{QCD}} = 0$ — retracted [✗] 2026-09-26, strong CP open [Pr]: the Gap sector is CP-neutral but fixes no $\bar\theta$ (§3.1c). Earlier: [C at (SV)] (T-99: step 2 holds for the retracted cubic only; the corrected potential is PT-even, and no lift of its vacuum's antiunitary symmetry gives $\bar\theta = 0$ with $m_t \neq m_b$ and $J \neq 0$ — T-333)
7. Deconfinement at $T_c \sim \Lambda_{\mathrm{QCD}} \sim 170$ MeV [C at (SV)]; order parameter — Polyakov loop [T] (from $\mathbb{Z}_3$ centre of $\mathrm{SU}(3)_C$ = Stab$_{G_2}(e_O)$ [T-42e]); crossover with quarks [H]
8. Asymptotic freedom reproduced in the standard way [T]; relation to RG flow via $\lambda_4$ [T]
9. ABJ anomaly from $\mathrm{Cliff}(7)$: $\partial_\mu j_5^\mu = (N_f g_s^2/16\pi^2)\,G\tilde{G}$ [T]
10. Decay $\pi^0 \to \gamma\gamma$: $\tau = 8.4 \times 10^{-17}$ s (agreement with PDG) [T]
11. Cancellation of gauge anomalies: $\mathrm{tr}(T^a\{T^b,T^c\}) = 0$ for the Standard-Model content (T-175b: the arithmetic [T]; as a UHM result [C at (FE)] with the imported $H_F$ — the derivation "from the spectral triple + unimodularity" is retracted)

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8. Status Summary

Result	Status
Wilson loop: topological area law	[T]
String tension $\sqrt{\sigma} \approx 457$ MeV from Gap tube: Hessian hierarchy [T], numerical value [C at (SV)]	[C at (SV)]
String tension from Gap parameters (naive $\sim 60$ MeV; sectoral correction from soft Hessian mode $\sim 457$ MeV vs 440 MeV)	[C at (SV)]
Structural $\theta_{\mathrm{QCD}} = 0$ (T-99): 7-step derivation from A1–A5	[✗] (retracted 2026-09-26: step 4 false for $V_3$; earlier [C at (SV)], step 2 [T] for $V_3$ only; the vacuum-symmetry route closed, T-333)
Gap sector CP-neutral; no $G_2$-invariant Gap term fixes $\bar\theta$ (T-99 corrected, §3.1c)	[T] as mathematics, [C at (Cl)] in UHM; strong CP open [Pr]
Polyakov loop as deconfinement order parameter (from $\mathbb{Z}_3$ centre of $\mathrm{SU}(3)_C$ [T-42e])	[T]
Critical temperature $T_c \sim 170$ MeV	[C at (SV)]
Crossover with dynamical quarks ($N_f = 2+1$)	[H]
Asymptotic freedom (relation to RG flow)	[T]
Running of quark masses	[T]
ABJ anomaly (chiral) from $\mathrm{Cliff}(7)$; index theorem	[T]
Cancellation of gauge anomalies $\mathrm{tr}(T^a\{T^b,T^c\}) = 0$ (T-175b)	[T] for the Standard-Model content; [C at (FE)] as a UHM result
Decay $\pi^0 \to \gamma\gamma$: $\tau = 8.4 \times 10^{-17}$ s	[T]
Anomalous Ward identities for axial vertices	[T]

Open problems

1. Glueball spectrum. Prediction of glueball masses from Gap parameters is a non-perturbative problem.
2. Anomaly in the gravitational sector. The mixed gravitational–axial anomaly $\partial_\mu j_5^\mu \supset R\tilde{R}$ in the Gap formalism requires full accounting of the $\mathrm{Cliff}(7)$-spectrum, including the O-direction. The connection to emergent gravity is an open question [D].

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• Standard Model from G₂
• RG flow
• Vacuum uniqueness