13 Explicit G2 Metric

The key discovery: the standard G2 form \(\varphi _0\) scaled by \(c = (65/32)^{1/14}\) is the exact analytical solution satisfying GIFT constraints.

13.1 The Standard G2 3-form

Definition 13.1 Associative 3-form

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The standard G2 3-form on \(\mathbb {R}^7\):

\[ \varphi _0 = e^{123} + e^{145} + e^{167} + e^{246} - e^{257} - e^{347} - e^{356} \]

In 0-indexed notation:

\[ \varphi _0 = e^{012} + e^{034} + e^{056} + e^{135} - e^{146} - e^{236} - e^{245} \]

Theorem 13.2 Seven Terms

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\(\varphi _0\) has exactly 7 non-zero terms (of 35 independent components).

Definition 13.3 Signs Pattern

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The signs are: \([+1, +1, +1, +1, -1, -1, -1]\)

13.2 Linear Index Representation

Definition 13.4 C(7,3) Components

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A 3-form on \(\mathbb {R}^7\) has \(\binom {7}{3} = 35\) independent components, indexed lexicographically: \((0,1,2) \mapsto 0\), \((0,1,3) \mapsto 1\), etc.

Theorem 13.5 Non-zero Indices

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The 7 non-zero indices are: \(\{ 0, 9, 14, 20, 23, 27, 28\} \)

Index	Triple	Sign
0	\((0,1,2)\)	\(+1\)
9	\((0,3,4)\)	\(+1\)
14	\((0,5,6)\)	\(+1\)
20	\((1,3,5)\)	\(+1\)
23	\((1,4,6)\)	\(-1\)
27	\((2,3,6)\)	\(-1\)
28	\((2,4,5)\)	\(-1\)

Theorem 13.6 Sparsity

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Only \(7/35 = 20\% \) of components are non-zero.

13.3 The GIFT Scale Factor

Definition 13.7 Scale Factor

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To achieve \(\det (g) = 65/32\), we scale \(\varphi _0\) by:

\[ c = \left(\frac{65}{32}\right)^{1/14} \approx 1.0543 \]

Theorem 13.8 Scaling Derivation

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For \(\varphi = c \cdot \varphi _0\) and metric \(g_{ij} = \frac{1}{6}\sum _{k,l} \varphi _{ikl}\varphi _{jkl}\):

Standard \(\varphi _0\) gives \(g = I_7\), so \(\det (g) = 1\)
Scaling \(\varphi \mapsto c \cdot \varphi \) gives \(g \mapsto c^2 \cdot g\)
Therefore \(\det (g) \mapsto c^{14} \cdot \det (g)\)
Setting \(c^{14} = 65/32\) yields \(\det (g) = 65/32\)

13.4 The Explicit Metric

Theorem 13.9 Scaled Identity Metric

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The induced metric is:

\[ g = c^2 \cdot I_7 = \left(\frac{65}{32}\right)^{1/7} \cdot I_7 \approx 1.1115 \cdot I_7 \]

Explicitly:

\[ g_{ij} = \begin{cases} (65/32)^{1/7} & \text{if } i = j \\ 0 & \text{otherwise} \end{cases} \]

Theorem 13.10 Determinant Verification

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\(\det (g) = \left[(65/32)^{1/7}\right]^7 = 65/32 = 2.03125\) exactly.

13.5 Torsion Vanishes

Theorem 13.11 Zero Torsion

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For a constant 3-form \(\varphi (x) = \varphi _0\):

\(d\varphi = 0\) (exterior derivative of constant)
\(d{*}\varphi = 0\) (same reasoning)

Therefore \(T = 0\) exactly.

Theorem 13.12 Joyce Satisfied

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\(\| T\| = 0 {\lt} 0.0288 = \varepsilon _{\text{Joyce}}\) with infinite margin.

13.6 Summary

Theorem 13.13 Analytical G2 Metric

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The canonical GIFT G2 metric on \(K_7\) is given by:

3-form (\(35\) components, \(7\) non-zero):

\[ \varphi _i = \begin{cases} +c & i \in \{ 0, 9, 14, 20\} \\ -c & i \in \{ 23, 27, 28\} \\ 0 & \text{otherwise} \end{cases} \]

where \(c = (65/32)^{1/14}\).

Metric (\(7 \times 7\) diagonal):

\[ g = (65/32)^{1/7} \cdot I_7 \]

Properties:

\(\det (g) = 65/32\) (exact)
\(\| T\| = 0\) (torsion-free)
\(\mathrm{Hol}(g) = G_2\) (by construction)

Remark 13.14 Simplicity

This is the simplest possible G2 structure satisfying GIFT constraints. The solution is a constant 3-form with only 7 non-zero components and a diagonal metric. No PINN training or Fourier analysis is required—the standard G2 form is the answer.

Remark 13.15 G2 vs Fano

The G2 3-form indices are different from Fano plane lines:

\begin{align*} \text{G2 3-form:} & \quad (0,1,2), (0,3,4), (0,5,6), (1,3,5), (1,4,6), (2,3,6), (2,4,5) \\ \text{Fano lines:} & \quad (0,1,3), (1,2,4), (2,3,5), (3,4,6), (4,5,0), (5,6,1), (6,0,2) \end{align*}

Both have 7 terms but represent different structures (3-form vs cross-product).