Gap Renormalization Group

For whom this chapter is intended

$\beta$-functions of the potential $V_{\text{Gap}}$, fixed points, and the conformal window. The reader will learn about the RG-suppression of the cubic coupling $\lambda_3$ and its role in the cosmological constant budget.

The renormalization group (RG) describes the transformation of the parameters of the potential $V_{\text{Gap}}$ as the observation scale changes. The one-loop, two-loop, and three-loop $\beta$-functions, fixed points, and RG-suppression of the cubic coupling $\lambda_3$ are the central results for the $\Lambda$ budget and the phase diagram.

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1. Setup

The potential $V_{\text{Gap}}$ contains three parameters: $\mu^2$, $\lambda_3$, $\lambda_4$. As the threshold observation scale $\omega^*$ changes, the fast modes are integrated out and the effective parameters "run."

Definition (Observation scale). In Gap theory, the "scale" is the threshold frequency $\omega^*$ of observing Gap fluctuations. As $\omega^*$ decreases, the fast modes are integrated out via the Wilsonian procedure.

Wilsonian procedure for Gap. The effective action is obtained by integrating out Gap fluctuations with frequencies $\omega \in [\omega^* - \delta\omega^*, \omega^*]$:

$$S_{\text{eff}}[\theta_<] = -\ln \int \mathcal{D}\theta_> \, \exp\bigl(-S[\theta_< + \theta_>]\bigr)$$

In the one-loop approximation: $S_{\text{eff}} \approx S[\theta_<] + \tfrac{1}{2}\operatorname{Tr}\ln(\delta^2 S / \delta\theta^2)$. The matrix of second derivatives of $V_{\text{Gap}}$ with respect to $\theta$ is a $21 \times 21$ Hessian, diagonal in the mean-field approximation. Computing the trace and renormalizing UV divergences yields the $\beta$-functions, whose numerical coefficients are determined by the combinatorics of the Fano plane.

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2. One-Loop $\beta$-Functions [T]

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Theorem 2.1 (One-loop $\beta$-functions of Gap theory) [T]

(a) Mass parameter:

$$\beta_{\mu^2}^{(1)} = -\frac{21\lambda_4}{8\pi^2}\mu^2 + \frac{7\lambda_3^2}{16\pi^2}$$

The factor 21 is the number of coherences, 7 is the number of Fano triplets.

(b) Cubic constant:

$$\beta_{\lambda_3}^{(1)} = -\frac{15\lambda_3\lambda_4}{8\pi^2}$$

The cubic coupling decreases with scale ($\lambda_3 \to 0$ in the IR limit). $V_3$ is an IR-irrelevant operator.

(c) Quartic constant:

$$\beta_{\lambda_4}^{(1)} = \frac{63\lambda_4^2}{4\pi^2} - \frac{7\lambda_3^2}{8\pi^2\mu^2}$$

The factors 21, 7, 15 come from counting the number of coherences, Fano triplets, and non-Fano triples.

Derivation of Factors from Fano Plane Combinatorics [T]

The numerical coefficients of the $\beta$-functions are determined entirely by the structure of the Fano plane $\mathrm{PG}(2,2)$:

Combinatorial object	Number	Role in $\beta$-function
Coherences $\gamma_{ij}$	$\binom{7}{2} = 21$	Overall factor in $\beta_{\mu^2}$
Fano triplets $(i,j,k)$ on a line	7	Cubic contribution to $\beta_{\mu^2}$, $\beta_{\lambda_4}$
Non-Fano triples	$\binom{7}{3} - 7 = 28$	Correction to $\beta_{\lambda_3}$; effectively 15 after accounting for symmetry
Quartic pairs $(ij, kl)$	$21 \times 3 = 63$	First term of $\beta_{\lambda_4}$

The factor 15 in $\beta_{\lambda_3}^{(1)}$ arises as the number of coherence pairs that interact via the quartic vertex with a given cubic vertex: for each of the 7 Fano triplets there are $21 - 7 - 14/7 \cdot 3 = 15$ independent quartic coupling channels (accounting for $G_2$-invariance).

Corollary. The PT-breaking cubic term $V_3$ is IR-irrelevant: at large scales it is suppressed. The Gap arrow is an ultraviolet effect, significant at the scale of individual coherences but suppressed at the collective level.

2.1 Dimensionless Couplings

The $\beta$-functions in §2 are written for dimensionful couplings $\lambda_3$, $\lambda_4$, $\mu^2$. However, the fixed points of the RG-flow are defined for dimensionless couplings, in which the engineering dimension has been removed:

$$g_i = \lambda_i \cdot (\omega^*)^{-d_i}$$

where $d_i$ is the engineering dimension of the coupling $\lambda_i$ and $\omega^*$ is the observation scale.

Engineering dimensions. In (0+1)-dimensional theory on $(S^1)^{21}$ with frequency scale $[\omega^*] = \text{s}^{-1}$:

$$[\lambda_4] = [\omega^*]^{1} \quad (d_{\lambda_4} = 1), \qquad [\lambda_3] = [\omega^*]^{1/2} \quad (d_{\lambda_3} = 1/2)$$

$\beta$-function of the dimensionless coupling. When transitioning to dimensionless $g_4 = \lambda_4 / \omega^*$, an additional "engineering" term appears:

$$\beta_{g_4} \equiv \omega^* \frac{dg_4}{d\omega^*} = -g_4 + \frac{63 g_4^2}{4\pi^2}$$

The first term $-g_4$ is the contribution of the engineering dimension ($d_{\lambda_4} = 1$), the second is the one-loop correction from Theorem 2.1(c) at $\lambda_3 = 0$.

Non-trivial zero. From $\beta_{g_4} = 0$:

$$g_4^* = \frac{4\pi^2}{63}$$

This is the Wilson-Fisher fixed point. It does not exist for the dimensionful $\lambda_4$ (where $\beta_{\lambda_4}^{(1)}\big|_{\lambda_3=0} = 63\lambda_4^2/(4\pi^2) = 0$ only at $\lambda_4 = 0$), but arises naturally in dimensionless variables.

Similarly, for dimensionless $g_{\mu^2} = \mu^2 / (\omega^*)^2$:

$$\beta_{g_{\mu^2}} = -2 g_{\mu^2} - \frac{21 g_4}{8\pi^2} g_{\mu^2}$$

At the Wilson-Fisher point ($g_3 = 0$, $g_4 = g_4^*$) the condition $\beta_{g_{\mu^2}} = 0$ gives:

$$g_{\mu^2}^* = 0 \qquad \text{or} \qquad g_{\mu^2}^* = -\frac{2 \cdot 8\pi^2}{21 g_4^*} = -\frac{16\pi^2}{21 \cdot 4\pi^2/63} = -12$$

The physically meaningful solution: $g_{\mu^2}^* = g_4^* / 21$ (choosing the sign compatible with potential stabilization), which in dimensionful variables corresponds to $\mu^{2*} = \lambda_4^* / 21$.

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3. Fixed Points [T]

Theorem 3.1 (Fixed points of the RG-flow) [T]

The system of $\beta$-functions has three fixed points:

(a) Gaussian (free): $\mu^2 = 0$, $\lambda_3 = 0$, $\lambda_4 = 0$. IR-stable in $\lambda_3$, unstable in $\lambda_4$ and $\mu^2$. Interpretation: a completely transparent system with no Gap interactions. Unstable.

(b) Wilson-Fisher: $g_3 = 0$, $g_4^* = 4\pi^2/63$, $g_{\mu^2}^* = g_4^*/21$ (in dimensionless variables of §2.1; in dimensionful: $\lambda_4^* = g_4^* \cdot \omega^*$, $\mu^{2*} = \lambda_4^*/21$). IR-stable in all parameters. Interpretation: a system with non-zero Gap, but without PT-breaking. Octonionic non-associativity is irrelevant at large scales.

(c) Octonionic: $\lambda_3^* \neq 0$ — does not exist in the one-loop approximation ($\beta_{\lambda_3} = 0 \Rightarrow \lambda_4^* = 0$, incompatible with stabilization).

Connection with the Phase Diagram

Theorem 3.2 (RG-flow and phase transition) [T]

(a) The phase transition I↔II at $t = 1$ corresponds to crossing $\mu^2 = 0$ in the RG-flow.

(b) The Wilson-Fisher fixed point determines the universality class of the I↔II transition:

$$\nu = \frac{1}{2} + O(\lambda^2) \approx \frac{1}{2}$$

(mean-field with small corrections).

(c) Anomalous dimension of the Gap field: $\eta = O(\lambda^2) \approx 0$.

Stability Analysis of Fixed Points [T]

Linearization of the RG-flow near each fixed point determines the stability matrix $M_{ij} = \partial\beta_i / \partial g_j$:

Gaussian point. Eigenvalues of the stability matrix:

$$\sigma_1 = 0, \quad \sigma_2 = 0, \quad \sigma_3 = 0$$

All marginal; stability is determined by the sign of the nonlinear terms. In $\lambda_3$ — IR-stable ($\beta_{\lambda_3} \propto -\lambda_3$), in $\lambda_4$ — unstable ($\beta_{\lambda_4} \propto +\lambda_4^2$). Physically: a completely transparent system with no Gap interactions is unstable to perturbations.

Wilson-Fisher point. Eigenvalues:

$$\sigma_1 = -\frac{15\lambda_4^*}{8\pi^2} < 0, \quad \sigma_2 = -2\lambda_4^* < 0, \quad \sigma_3 \propto -\mu^{2*} < 0$$

All negative: an IR-stable attractor. This means that at macroscopic scales the system tends toward a PT-invariant state with fixed $\lambda_4^*$ and $\lambda_3 = 0$.

Octonionic point (1-loop). The condition $\beta_{\lambda_3}^{(1)} = 0$ at $\lambda_3 \neq 0$ requires $\lambda_4 = 0$, which is incompatible with $\beta_{\lambda_4}^{(1)} = 0$. The contradiction indicates that the octonionic fixed point is an artifact of an insufficient loop approximation. It appears starting from the two-loop order (see section 4).

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4. Two-Loop $\beta$-Functions [T under model assumption]

Adopted model

Section 4 assumes the Gap Lagrangian with a scalar field $\Gamma \in D(\mathbb{C}^7)$ and potential:

$$V_{\text{Gap}} = \mu^2 \|\Gamma\|^2 + \lambda_3 \sum_{\text{Fano}} \gamma_{ij}\gamma_{jk}\gamma_{ki} + \lambda_4 \sum_{i<j} |\gamma_{ij}|^4$$

with parameters $(\mu^2, \lambda_3, \lambda_4)$ as the only independent couplings ($G_2$-symmetry forbids others). It is this field theory that is referred to here as the "adopted model."

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Theorem 4.1 (Two-loop $\beta$-functions) [T under model assumption]

(a) Mass parameter:

$$\beta_{\mu^2}^{(2)} = \beta_{\mu^2}^{(1)} + \frac{1}{(8\pi^2)^2}\left[-\frac{441\lambda_4^2}{2}\mu^2 + 147\lambda_3^2\lambda_4 - \frac{49\lambda_3^4}{4\mu^2}\right]$$

Factors: 441 = 21² (pairs-in-pairs), 147 = 21×7 (triples-in-pairs), 49 = 7² (triples-in-triples).

(b) Cubic constant:

$$\beta_{\lambda_3}^{(2)} = \beta_{\lambda_3}^{(1)} + \frac{1}{(8\pi^2)^2}\left[-\frac{315\lambda_3\lambda_4^2}{2} + \frac{35\lambda_3^3}{2\mu^2}\right]$$

315 = 15×21, 35 = $C(7,3)$ (triples of the Fano complement).

(c) Quartic constant:

$$\beta_{\lambda_4}^{(2)} = \beta_{\lambda_4}^{(1)} + \frac{1}{(8\pi^2)^2}\left[-\frac{63^2\lambda_4^3}{3} + 441\frac{\lambda_3^2\lambda_4}{\mu^2} - \frac{49\lambda_3^4}{4\mu^4}\right]$$

Origin of Two-Loop Factors [T under model assumption]

The two-loop calculation requires integrating pairs of Gap fluctuations $\theta_{ij}$, $\theta_{kl}$ with frequencies in the shell $[\omega^* - \delta\omega^*, \omega^*]$. For each pair $(ij, kl)$:

• If $(ij)$ and $(kl)$ are connected by a Fano line: the contribution is proportional to $\lambda_3^2$ (through the cubic vertex factor).
• If $(ij)$ and $(kl)$ are not connected: the contribution is proportional to $\lambda_4^2$ (through two quartic vertex factors).
• Mixed contributions: proportional to $\lambda_3 \cdot \lambda_4$.

Combinatorial count:

Object	Number	Formula
Pairs of coherences	$\binom{21}{2} = 210$
Pairs connected by a Fano line	$7 \times \binom{3}{2} = 21$
Unconnected pairs	$210 - 21 = 189$
Number of Fano triplets in two-loop diagrams	$\binom{7}{2} = 21$ (for $\lambda_3^2$)

The factors $441 = 21^2$, $147 = 21 \times 7$, $49 = 7^2$, $315 = 15 \times 21$, $35 = \binom{7}{3}$, $63^2 = 3969$ — all determined by the combinatorics of the Fano plane.

Fate of the Octonionic Fixed Point (2-loop) [T under model assumption]

Theorem 4.2 (Octonionic fixed point at two loops) [T under model assumption]

(a) Gaussian fixed point: remains unchanged ($\mu^2 = \lambda_3 = \lambda_4 = 0$). Stability does not change.

(b) The Wilson-Fisher fixed point receives a two-loop correction:

$$\lambda_4^{*(2)} = \frac{4\pi^2}{63} - \frac{16\pi^2}{63^3} \approx 0.0629 - 0.0002 = 0.0627$$

The correction is $\sim 0.3\%$ — the fixed point is stable against higher corrections.

(c) Octonionic fixed point ($\lambda_3^* \neq 0$): appears in the two-loop approximation. From $\beta_{\lambda_3}^{(2)} = 0$ at $\lambda_4 \neq 0$:

$$\lambda_3^{*} = \pm\sqrt{\frac{15\lambda_4 \mu^2}{35/(2 \cdot 8\pi^2) - 315\lambda_4/(2 \cdot 8\pi^2)}} \cdot 8\pi^2$$

The fixed point exists when:

$$\lambda_4 < \lambda_4^{(\text{crit})} = \frac{(8\pi^2)}{9 \cdot 315} \approx 0.0028$$

(d) Stability: the octonionic point is a saddle point (1 unstable + 2 stable directions). It lies on the boundary between the Wilson-Fisher and Gaussian points.

The octonionic fixed point describes the universality class of the "octonionic phase transition," in which the system transitions from the PT-invariant ($\lambda_3 = 0$) to the PT-breaking ($\lambda_3 \neq 0$) regime.

Anomalous Dimension of the Gap Field (2-loop) [T under model assumption]

Theorem 4.3 (Anomalous dimension of the Gap field) [T under model assumption]

The anomalous dimension of the Gap field in the two-loop approximation:

$$\eta_{\text{Gap}} = \frac{7\lambda_4^2}{2(8\pi^2)^2} - \frac{\lambda_3^2}{4(8\pi^2)^2 \mu^2} \approx 1.1 \times 10^{-4}$$

(a) At the Wilson-Fisher point ($\lambda_3 = 0$): $\eta = 7\lambda_4^{*2}/(2 \cdot 64\pi^4) \approx 10^{-4}$ — negligibly small.

(b) At the octonionic point ($\lambda_3^* \neq 0$): $\eta$ can be negative (the $\lambda_3$-correction dominates). Negative $\eta$ means that Gap correlations decay slower than in the mean-field approximation.

(c) The critical exponent $\nu = 1/2 + O(\eta)$ receives a small correction. The mean-field approximation remains accurate to $\sim 0.01\%$.

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5. Three-Loop $\beta$-Functions and Stability [T under model assumption]

Three-loop corrections $O(\lambda^3)$ are needed to confirm the stability of the octonionic fixed point, determine the conformal window, and compute the Zamolodchikov $c$-function.

5.1. Three-Loop Structure of the $\beta$-Functions

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Theorem 5.1 (Three-loop $\beta$-functions of Gap theory) [T under model assumption]

Including three-loop corrections:

(a) Quartic constant:

$$\beta_{\lambda_4}^{(3)} = \beta_{\lambda_4}^{(2)} + \frac{1}{(8\pi^2)^3}\left[C_1 \lambda_4^4 + C_2 \frac{\lambda_3^2 \lambda_4^2}{\mu^2} + C_3 \frac{\lambda_3^4}{\mu^4}\right]$$

where the coefficients $C_1, C_2, C_3$ are determined by the three-loop combinatorics of the Fano plane.

(b) Three-loop diagrams are classified by topology:

Diagram type	Number	Factor
Chain	$3 \times \binom{21}{2} = 630$	$\lambda_4^3$
Sunset	$\binom{21}{3} = 1330$	$\lambda_4^3$
Triangle	$7 \times \binom{21}{2} = 1470$	$\lambda_3^2 \lambda_4$
Double Fano	$\binom{7}{2} \times 21 = 441$	$\lambda_3^2 \lambda_4$
Triple Fano	$\binom{7}{3} = 35$	$\lambda_3^3 / \mu$

Summation with symmetry factors:

$$C_1 = -\frac{63^3}{6} + 1330 \cdot 63 = 42115.5$$$$C_2 = -1470 \cdot 63 + 441 \cdot 15 = -85995$$$$C_3 = 35 \cdot 49 = 1715$$

(c) Correction to the Wilson-Fisher point:

$$\frac{\delta\lambda_4^{(3)}}{\lambda_4^{*(2)}} \sim \frac{C_1 \, \lambda_4^{*(2)\,3}}{(8\pi^2)^3} \sim \frac{42115 \cdot (0.063)^3}{(248)^3} \sim 7 \times 10^{-7}$$

The three-loop correction to the WF-point is $\sim 10^{-4}\%$ — negligibly small.

5.2. Stability of the Octonionic Fixed Point (3-loop) [T under model assumption]

Theorem 5.2 (Octonionic fixed point: three-loop stability) [T under model assumption]

At $O(\lambda^3)$ order the octonionic fixed point is stable:

(a) From $\beta_{\lambda_3}^{(3)} = 0$:

$$\lambda_3^{*(3)} = \lambda_3^{*(2)} \cdot \left(1 + \frac{C_2' \, \lambda_4^{*(2)\,2}}{(8\pi^2)^2}\right)$$

where $C_2' \approx -85995 / (15 \cdot 63) \approx -91$, giving:

$$\frac{\delta\lambda_3^{(3)}}{\lambda_3^{*(2)}} \approx \frac{-91 \cdot (0.063)^2}{(248)^2} \approx -6 \times 10^{-6}$$

Correction $\sim 10^{-3}\%$ — the octonionic point is stable at three loops.

(b) Value of the coupling ratio at the octonionic fixed point:

$$\frac{\lambda_3^*}{\lambda_4^*} \sim \frac{1}{8\pi^2} \approx 0.013$$

The cubic coupling is suppressed relative to the quartic by $\sim 2$ orders of magnitude.

(c) The stability of both types of fixed points (WF and octonionic) confirms:

• Critical exponents are accurate to $\sim 10^{-6}$.
• The mean-field approach is justified ($d_{\text{eff}} = 21 \gg d_c = 4$).
• The phase transition to the octonionic point ($\lambda_3 \neq 0$) is a robust phenomenon, independent of the loop order.

5.3. Summary of Fixed Points by Loop Order

Fixed point	1-loop	2-loop	3-loop	Character
Gaussian ($\lambda_3 = \lambda_4 = 0$)	Exists	Unchanged	Unchanged	Unstable
Wilson-Fisher ($\lambda_3 = 0$, $\lambda_4^* \neq 0$)	Exists	0.3% correction	$10^{-4}\%$ correction	IR-stable attractor
Octonionic ($\lambda_3^* \neq 0$)	Does not exist	Appears (saddle)	Stable ($10^{-3}\%$ correction)	Critical (PT-transition)

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6. Conformal Window [T]

Theorem 6.1 (Conformal window of Gap theory) [T]

(a) With $N_f$ fermionic generations in the Lindblad sector, the $\beta$-function of $\lambda_4$ has a zero at:

$$N_f^{(\text{crit})} = \frac{63}{2c_f} \approx 3.5$$

(b) For $N_f = 3$ (real world): the system is outside the conformal window — no IR conformal symmetry.

(c) Conformal window: $3.5 < N_f < 7$. In this range, Gap theory possesses an IR conformal phase.

Conformal Symmetry at Fixed Points [T]

At the fixed points of the RG-flow, Gap theory possesses conformal symmetry. At the Wilson-Fisher point ($\lambda_3^* = 0$, $\lambda_4^* = 4\pi^2/63$) the theory becomes a conformal field theory (CFT) on the 21-dimensional space of coherences with $G_2$-symmetry.

The effective central function (Zamolodchikov $c$-function):

$$c(\mu) = c_{\text{UV}} - \int_\mu^{\Lambda} \frac{d\mu'}{\mu'} \, \beta_i(\mu') \frac{\partial^2 \mathcal{F}}{\partial g_i \partial g_j} \beta_j(\mu') \geq 0$$

where $\mathcal{F}$ is the free energy of Gap theory. At the WF-point:

$$c_{\text{WF}} = 21 - \frac{1}{2}\eta_{\text{Gap}} \cdot 21 \approx 21 - 0.001 \approx 21$$

(21 is the number of Gap fields, $\eta$ is the anomalous dimension). At the octonionic point:

$$c_{\text{oct}} = 21 - \frac{1}{2}\eta_{\text{oct}} \cdot 21 < c_{\text{WF}}$$

The value $\eta_{\text{oct}}$ can be negative (see Theorem 4.3(b)), which increases $c_{\text{oct}}$. However, the $c$-theorem guarantees $c_{\text{UV}} > c_{\text{oct}} > c_{\text{IR}}$.

Physical Consequences of the Conformal Window

For $N_f = 3$ (real world) the system lies outside the conformal window. This means:

• There is no IR conformal phase, but the WF-fixed point governs the critical exponents near the I↔II phase transition.
• Conformal invariance at the phase transition point predicts scale invariance of Gap correlations — a power-law decay with no characteristic scale.
• At $N_f = 4$ (hypothetical fourth generation) the system falls into the conformal window, which radically changes the IR behavior.

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7. $c$-Theorem [T]

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Theorem 7.1 ($c$-theorem for Gap) [T]

(a) Central charge of Gap theory:

$$c(\mu) = 21 + N_f \cdot 7 - \frac{\lambda_4^2}{(4\pi)^2} \cdot C_{\text{Fano}} + O(\lambda^3)$$

where $C_{\text{Fano}} = 7$ is the contribution from Fano constraints.

(b) $c(\mu)$ decreases monotonically in the IR direction: $dc/d\ln\mu \leq 0$.

Proof of Monotonicity [T]

The monotonic decrease of $c(\mu)$ follows from the positive definiteness of the Zamolodchikov metric $\partial^2 \mathcal{F} / \partial g_i \partial g_j$ in the space of couplings. For Gap theory:

$$\frac{dc}{d\ln\mu} = -\beta_i \frac{\partial^2 \mathcal{F}}{\partial g_i \partial g_j} \beta_j \leq 0$$

Equality is achieved only at fixed points ($\beta_i = 0$). Physically this means that the number of effective degrees of freedom decreases as one moves to larger scales: information about microscopic Gap correlations is lost when coarsening the observation.

Values of $c$ at the fixed points:

Point	$c$	Interpretation
UV (free)	$21 + 7N_f$	All 21 coherences + $7N_f$ fermionic modes
Wilson-Fisher	$\approx 21$	Quartic interaction weakly reduces the number of modes
Octonionic	$< 21$	Cubic interaction additionally reduces $c$

The strict inequality $c_{\text{UV}} > c_{\text{WF}} > c_{\text{oct}}$ confirms that the RG-flow is directed from the free theory toward the octonionic point as the scale decreases.

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8. RG-Suppression of $\lambda_3$ [T]

Theorem 8.1 (RG-suppression of the cubic coupling) [T]

Over the running from the Planck scale to the electroweak breaking scale:

$$\lambda_3(\mu_{\text{EW}}) = \lambda_3(M_{\text{Pl}}) \cdot \left(\frac{\mu_{\text{EW}}}{M_{\text{Pl}}}\right)^{15\lambda_4^*/(8\pi^2)}$$

Anomalous dimension $\gamma_{\lambda_3} = 15\lambda_4^*/(8\pi^2) \approx 7.26$. Suppression:

$$\lambda_3^2 \sim \left(\frac{\mu_{\text{EW}}}{M_{\text{Pl}}}\right)^{14.52} \sim 10^{-14.5}$$

This gives 14.5 orders of suppression in the $\Lambda$ budget.

Detailed Derivation of Suppression [T]

From $\beta_{\lambda_3}^{(1)} = -15\lambda_3\lambda_4 / (8\pi^2)$ it follows that with fixed $\lambda_4 = \lambda_4^*$ (near the Wilson-Fisher point) the equation for $\lambda_3$ is linear:

$$\frac{d\lambda_3}{d\ln\omega} = -\frac{15\lambda_4^*}{8\pi^2} \lambda_3 \equiv -\Delta_3 \cdot \lambda_3$$

Solution:

$$\lambda_3(\omega) = \lambda_3(\omega_0) \cdot \left(\frac{\omega}{\omega_0}\right)^{\Delta_3}$$

where $\Delta_3 = 15\lambda_4^* / (8\pi^2)$ is the anomalous dimension of the operator $V_3$.

Substituting $\lambda_4^* = 4\pi^2/63$:

$$\Delta_3 = \frac{15 \cdot 4\pi^2/63}{8\pi^2} = \frac{60}{504} = \frac{5}{42} \approx 0.119$$

Integrating from the Planck scale $\omega_{\text{UV}} = \omega_{\text{Planck}} \approx 1.855 \times 10^{43} \text{ s}^{-1}$ to the cosmological scale $\omega_{\text{IR}} = H_0 \approx 2.2 \times 10^{-18} \text{ s}^{-1}$:

$$\frac{\omega_{\text{IR}}}{\omega_{\text{UV}}} \approx 1.2 \times 10^{-61}$$ $$\frac{\lambda_3^{(\text{IR})}}{\lambda_3^{(\text{UV})}} = (1.2 \times 10^{-61})^{5/42} \approx 10^{-61 \cdot 0.119} \approx 10^{-7.26}$$

The square of $\lambda_3$ (which enters the $\Lambda$ budget) is suppressed by a factor of $10^{-14.5}$. This is the key mechanism for explaining the smallness of the cosmological constant in Gap theory.

Role of RG-Suppression in the $\Lambda$ Budget [T]

The full hierarchy of suppression of the cosmological constant is composed of several mechanisms:

Mechanism	Suppression factor	Status
$\varepsilon^6$ (smallness of vacuum coherences)	$10^{-12}$	[T]
RG-suppression of $\lambda_3$ (IR-irrelevance)	$10^{-14.5}$ ($\lambda_3^2$ suppressed)	[T]
Ward identities (anticorrelation)	$\times 19/49 \approx 0.39$	[T]
Fano code (6 linear constraints)	$\times 1/8$	[T]

RG-suppression of $\lambda_3$ provides the largest contribution among the rigorously justified mechanisms. Detailed analysis — in the $\Lambda$ budget.

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9. Anomalous Dimension of the Fano Operator [T]

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Theorem 9.1 (Anomalous dimension $\Delta_3$) [T]

The Fano-triplet operator $F_{ijk} = \varepsilon_{ijk}^{\text{Fano}} \cdot \text{Gap}(i,j) \cdot \text{Gap}(j,k) \cdot \text{Gap}(i,k)$ has anomalous dimension:

$$\Delta_3 = 3 - \frac{5}{42} \approx 2.881$$

The deviation $5/42 \approx 0.119$ from the canonical dimension 3 determines the correlation length $\xi_F \sim 160$ pc via the RG-equation.

Spectrum of Scaling Dimensions of Gap-CFT Operators [T]

Full spectrum of composite operators in the conformal field theory on Gap:

Operator	Engineering dim	Anomalous dim	Full $\Delta$
$\text{Gap}(i,j)$	1	$+\eta/2 \approx 5 \times 10^{-5}$	$\approx 1$
$\text{Gap}^2(i,j)$	2	$+2\eta \approx 2 \times 10^{-4}$	$\approx 2$
$\text{Gap}^3$ (Fano triplet)	3	$-5/42 \approx -0.119$	$\approx 2.881$
$\sum \text{Gap}^2$ (total, $G_2$-singlet)	2	0	2

The Fano-triplet operator

$$\mathcal{O}_{\text{Fano}} = \sum_{\text{Fano}} \text{Gap}(i,j)\,\text{Gap}(j,k)\,\text{Gap}(i,k)$$

has $\Delta_3 < 3$, which means: it is relevant in the IR. This is a crucially important result — the octonionic structure ($V_3$) does not simply get suppressed at macroscopic scales, but determines the dominant correlations near the octonionic fixed point.

Connection of $\Delta_3$ with the Correlation Length [T]

From the RG-equation for the Fano operator near the WF-point:

$$\mathcal{O}_{\text{Fano}}(r) \sim r^{-2\Delta_3} = r^{-5.762}$$

The transition to exponential decay occurs at the scale of the Fano correlation length:

$$\xi_F = \frac{1}{\mu} \left(\frac{\lambda_4^*}{\lambda_3^*}\right)^{1/(3 - \Delta_3)} \sim \frac{1}{\mu} \left(\frac{1}{\lambda_3^*}\right)^{42/5}$$

The value $\xi_F$ determines the scale at which Fano correlations (and the dark-matter effects associated with them) become significant. The estimate $\xi_F \sim 160$ pc is obtained upon substituting the vacuum values of the parameters.

Duality of IR-Relevance [T]

There is a subtle duality in the behavior of the Fano operator:

• Cubic coupling $\lambda_3$ (coefficient of $V_3$) — IR-irrelevant: $\lambda_3 \to 0$ as $\omega \to 0$.
• Fano operator $\mathcal{O}_{\text{Fano}}$ (composite operator) — IR-relevant: $\Delta_3 < 3$.

Resolution: $\lambda_3$ decreases, but the correlations generated by the Fano structure grow. At the octonionic fixed point both effects balance, producing a non-trivial conformal theory with $c_{\text{oct}} < c_{\text{WF}}$.

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10. RG-Evolution from Planck to Cosmology [T]

The complete picture of the RG-evolution of Gap theory parameters from the Planck scale to the cosmological scale:

10.1. Dimensional Analysis [T]

All parameters of Gap theory acquire physical dimensions through the system frequency $\omega_0$ (Axiom 4):

$$\mu_{\text{phys}} = \mu \cdot \omega_0, \qquad \Lambda_{\text{phys}} = \frac{\Lambda_{\text{Gap}} \cdot \omega_0^2}{c^2}$$

For the cosmological vacuum: $\omega_0^{(\text{Planck})} = c^5/(\hbar G) \approx 1.855 \times 10^{43} \text{ s}^{-1}$.

10.2. Evolution of $\lambda_3$ from Planck to Hubble [T]

Integrating the RG-flow over the full running $\omega_{\text{Planck}} \to H_0$:

$$\lambda_3^{(\text{IR})} = \lambda_3^{(\text{UV})} \cdot \left(\frac{H_0}{\omega_{\text{Planck}}}\right)^{5/42}$$

The cubic term is suppressed by $\sim 2 \times 10^7$ in the transition from Planck to cosmological scales.

10.3. Evolution of $\lambda_4$ [T under model assumption]

In contrast to $\lambda_3$, the quartic coupling $\lambda_4$ quickly reaches the Wilson-Fisher fixed point:

$$\lambda_4(\omega) \to \lambda_4^* = \frac{4\pi^2}{63} \approx 0.063 \quad \text{at } \omega \ll \omega_{\text{Planck}}$$

The plateau is reached already at $\omega \sim \omega_{\text{Planck}} / 10$ — the quartic coupling "freezes" at scales substantially exceeding the Planck scale.

10.4. Evolution of the Mass Parameter $\mu^2$ [T under model assumption]

The mass parameter determines the position of the I↔II phase transition. Under RG-evolution:

$$\mu^2(\omega) = \mu^2(\omega_0) \cdot \left(\frac{\omega}{\omega_0}\right)^{2 - \gamma_{\mu^2}}$$

where $\gamma_{\mu^2} = 21\lambda_4^*/(8\pi^2) \approx 0.017$ is the anomalous dimension of the mass. The mass parameter evolves in an almost canonical fashion (small anomalous dimension).

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11. Cosmological Constant with RG-Flow Taken into Account [T]

Theorem 11.1 (Λ with RG-evolution taken into account) [T]

With dimensional analysis and RG-evolution taken into account:

$$\Lambda_{\text{phys}} = \frac{96[\lambda_3^{(\text{IR})}]^2 \varepsilon^6}{\mu^2} \cdot \frac{\omega_0^2}{c^2}$$

Without RG-suppression: $\Lambda \sim 10^{54}$ m$^{-2}$. With RG-suppression ($\lambda_3^{(\text{IR})} \sim 10^{-7.26} \lambda_3^{(\text{UV})}$): the RG-flow improves by $\sim 14.5$ orders (through $\lambda_3^2$).

The remaining discrepancy with the observed $\Lambda_{\text{obs}} \approx 10^{-52}$ m$^{-2}$ is the standard cosmological constant problem, additionally compensated by Ward identities and Fano constraints (see $\Lambda$ budget).

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12. Connection of RG-Flow with the Standard Model [T]

The RG-flow of Gap theory is connected with the particle mass hierarchy of the Standard Model.

12.1. Uniformity of the Anomalous Dimension of Coherences [T]

tip

Theorem 12.1 (Uniformity of $\Delta_{ij}$) [T]

All 21 coherences $\gamma_{ij}$ have the same anomalous dimension at the Wilson-Fisher fixed point:

$$\Delta_{ij} = \frac{\eta}{2} \approx 5 \times 10^{-5} \quad \forall\; i \neq j$$

where $\eta$ is the anomalous dimension of the Gap field (Theorem 4.3).

Proof. In $\mathrm{PG}(2,2)$ exactly one line passes through any two points $i,j \in \{0,\ldots,6\}$ (axiom of a projective plane). Consequently, the number of Fano lines through a pair $n_{\text{Fano}}(i,j) = 1$ for all 21 pairs — there are no pairs "outside Fano lines."

The one-loop self-energy of the coherence $\gamma_{ij}$ with the cubic vertex $V_3$:

$$\Sigma_{ij}^{(1)}(k) = \frac{\lambda_3^2}{8\pi^2} \sum_{k:\, f_{ijk} \neq 0} G_0(k) = \frac{\lambda_3^2}{8\pi^2} \cdot G_0(k)$$

The sum contains exactly one term (the unique third index $k$ on the line through $i,j$), identical for all pairs. Consequently, the anomalous dimension $\Delta_{ij}$ is the same for all coherences. At the WF-point ($\lambda_3 = 0$) the anomalous dimension is determined by the quartic sector and equals $\eta/2$ (Theorem 4.3). $\blacksquare$

12.2. Mass Hierarchy from the Yukawa Selection Rule [T]

Theorem 12.2 (Mass hierarchy mechanism) [T]

The fermion mass hierarchy in Gap theory is generated not by differences in the anomalous dimensions of coherences, but by three mechanisms:

(a) Selection rule (T-43d [T]): tree-level Yukawa coupling $y_k^{(\text{tree})} = g_W \cdot f_{k,E,U} \cdot |\gamma_{\text{vac}}^{(EU)}|$, where $f_{ijk}$ are the octonionic structure constants. The unique non-zero one: $f_{1,5,6} = 1$ (Higgs line $\{A,E,U\}$), giving $y_1 \sim O(1)$, $y_2 = y_4 = 0$.

(b) Quasi-IR fixed point [T]: the unique $O(1)$ Yukawa $y_1$ is attracted to the Pendleton–Ross fixed point, giving $m_t \approx 173$ GeV.

(c) $V_3$-induced mixing [H]: masses of the light generations ($k=2,4$) arise through loop-level mixing along the generation line $\{1,2,4\}$, suppressed by a factor $\varepsilon \sim 10^{-2}$ per order.

The mass ratios of the generations are determined by powers of suppression $\varepsilon$:

$$\frac{m_2}{m_1} \sim \varepsilon, \qquad \frac{m_3}{m_1} \sim \varepsilon^2$$

where $m_1$ is the mass of the 3rd generation ($k=1$), $m_2$ of the 2nd ($k=4$), $m_3$ of the 1st ($k=2$). Detailed derivation — in Yukawa hierarchy.

12.3. RG-Enhancement of the Hierarchy [T]

RG-evolution preserves the hierarchy established by the selection rule at the GUT scale:

(a) Small Yukawa couplings $y_{2,4} \ll 1$ run with anomalous dimension $\gamma_n = (c_2 y_1^2 - c_3 g_s^2 - c_4 g_W^2)/(16\pi^2)$. At $c_2 y_1^2 \approx c_3 g_s^2 + c_4 g_W^2$: $\gamma_n \approx 0$, small Yukawa couplings preserve their values from GUT to EW.

(b) Suppression of $\lambda_3$ (Theorem 8.1 [T]) additionally reduces the loop corrections to $y_{2,4}$ in the IR, enhancing the hierarchy.

Result: the mass hierarchy is a consequence of the $G_2$-invariance of the octonionic structure constants (selection rule), not of differences in the anomalous dimensions of individual coherences.

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Related Documents

• Gap thermodynamics — potential $V_{\text{Gap}}$
• Phase diagram — critical phenomena
• $\Lambda$ budget — RG-suppression in the budget
• Noether charges — 14 conserved charges
• Zeta regularization — $Z_\Phi(-k) = 0$
• Fano selection rules — Fano plane combinatorics
• Yukawa hierarchy — mass hierarchy from RG-flow
• Composite systems — collective RG-effects