Dependencies

\(u \times u = 0\)

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.

A vector \(v \in \mathbb {R}^8\) has all half-integer coordinates if \(v_i \in \mathbb {Z}+ \frac{1}{2}\) for all \(i\).

A vector \(v \in \mathbb {R}^8\) has all integer coordinates if \(v_i \in \mathbb {Z}\) for all \(i\).

\(\alpha ^{-1}_{\mathrm{alg}} = \frac{\mathrm{dim}(E_8) + \mathrm{rank}(E_8)}{2} = \frac{248 + 8}{2} = 128\)

\(\alpha ^{-1}_{\mathrm{bulk}} = \frac{H^*}{D_{\mathrm{bulk}}} = \frac{99}{11} = 9\)

\(\mathrm{dim}(G_2) - p_2 = 14 - 2 = 12\)

\(b_2= \binom {7}{2}\) (pairs of imaginary octonion units)

\(b_3= 3 \cdot b_2+ \mathrm{dim}(G_2)\)

\(|2I| = 120\)

\(h(E_8) = 30 = \) icosahedron edges

For \(u, v \in \mathbb {R}^7\), the cross product is:

The PINN parameterizes perturbations via the 14-dimensional \(\mathfrak {g}_2\) adjoint:

Only 14 functions are learned (not 35).

Determinant error: \(|\det (g) - 65/32| {\lt} 10^{-6}\)

\(\mathrm{dim}(G_2) = 14\)

\(\mathrm{dim}(\mathrm{SO}(n)) = \frac{n(n-1)}{2}\)

The \(E_8\) lattice consists of all \(v \in \mathbb {R}^8\) satisfying either:

1. All coordinates are integers with even sum, or

2. All coordinates are half-integers with even sum

For a root \(\alpha \) with \(\langle \alpha , \alpha \rangle = 2\), the Weyl reflection is:

The structure constants \(\varepsilon _{ijk}\) for the 7D cross product:

• \(\varepsilon _{ijk} = +1\) for \((i,j,k)\) a cyclic permutation of a Fano line

• \(\varepsilon _{ijk} = -1\) for anticyclic permutations

• \(\varepsilon _{ijk} = 0\) otherwise

\(\displaystyle \sum _k \varepsilon _{ijk} \varepsilon _{lmk}\)

The Fano plane has 7 lines (cyclic triples):

\(F_0 = 0, F_1 = 1, F_{n+2} = F_n + F_{n+1}\)

The trained PINN is evaluated on a grid and FFT identifies dominant modes. Coefficients are rationalized to \(\mathbb {Q}\) within tolerance \(10^{-8}\).

\(\mathrm{fund}(E_7) = 56\)

The geometric part encodes K\(_7\) topology:

\(H^*= b_2+ b_3+ 1 = 99\)

\(\{ 1, 2, 3, 7, 11, 19, 43, 67, 163\} \)

The octonions \(\mathbb {O}\) have 7 imaginary units: \(|\mathrm{Im}(\mathbb {O})| = 7\)

\(j(\tau ) = q^{-1} + 744 + 196884q + \ldots \)

\(\varepsilon _0 = 0.0288\) (scaled: 288)

The denominator is \(b_2= 21\).

The numerator is \(\mathrm{dim}(G_2) = 14\).

\(\lambda _H^2\) numerator: \(\mathrm{dim}(G_2) + 3 = 17\)

\(L_0 = 2, L_1 = 1, L_{n+2} = L_n + L_{n+1}\)

\(m_\mu /m_e\) base: \(\mathrm{dim}(J_3(\mathbb {O})) = 27\) (exceptional Jordan algebra)

The smallest faithful representation: \(196883\)

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

In 0-indexed notation:

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

The standard associative 3-form \(\varphi _0 = \sum _{ijk} \varepsilon _{ijk}\, dx^i \wedge dx^j \wedge dx^k\) where \(\varepsilon _{ijk}\) are the Fano plane structure constants, normalized for \(\det (g) = 65/32\).

\(\| T(\varphi _0)\| = 0.00141\) (scaled: 141)

The 7D correction to the Kronecker formula:

Let \(\mathbb {R}^8\) denote the 8-dimensional Euclidean space with standard inner product.

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

The chiral spinor of \(\mathrm{SO}(16)\) has dimension \(2^8/2 = 128\).

The spinorial part: \(2^{|\mathrm{Im}(\mathbb {O})|} = 128\)

The standard basis vectors \(e_i\) for \(i \in \{ 0, \ldots , 7\} \) satisfy \(\langle e_i, e_j \rangle = \delta _{ij}\).

A vector \(v\) has even sum if \(\sum _{i=0}^{7} v_i \in 2\mathbb {Z}\).

Direct GIFT constants that are prime: \(\{ 2, 3, 5, 7, 11, 13, 17, 19, 31, 61\} \)

PINN torsion bound: \(\| T\| {\lt} 0.001\)

The denominator is \(b_3+ \mathrm{dim}(G_2) = 77 + 14 = 91\).

The numerator is \(b_2= 21\).

\(\psi _{ijlm} = -\psi _{ljim}\) (verified for all \(7^4 = 2401\) index combinations)

\(\displaystyle \sum _{i,j,l,m} \psi _{ijlm} \, u_i u_l v_j v_m = 0\)

For integers \(n_0, \ldots , n_7\): \(\left(\sum _i n_i^2\right) \mod 2 = \left(\sum _i n_i\right) \mod 2\)

\(\alpha ^{-1}_{\mathrm{base}} = 128 + 9 = 137\)

With torsion correction:

(experimental: \(137.035999...\), deviation: 0.002%)

\(\alpha _s = \frac{\sqrt{2}}{12}\), where \(12 = \mathrm{dim}(G_2) - p_2\)

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

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

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

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

Properties:

• \(\det (g) = 65/32\) (exact)

• \(\| T\| = 0\) (torsion-free)

• \(\mathrm{Hol}(g) = G_2\) (by construction)

\(b_2= 21\)

\(b_3= 77\)

\(b_3= b_2+ \mathrm{fund}(E_7) = 21 + 56 = 77\)

\(b_3= 77 = L_4 \times L_5 = 7 \times 11\)

\(30 = p_2 \times N_{\mathrm{gen}} \times W = 2 \times 3 \times 5\)

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

The G2 constraint reduces parameters from 35 to 14: \(35 - 14 = 21 = b_2\)

\(\mathrm{dim}(\mathrm{SO}(16)) = \frac{16 \times 15}{2} = 120\)

\(\mathrm{dim}(\mathrm{SO}(7)) = \frac{7 \times 6}{2} = 21 = b_2\)

For \(v, w \in E_8\): \(\langle v, w \rangle \in \mathbb {Z}\)

\(240 = 2 \times |2I| = \mathrm{rank}(E_8) \times h(E_8)\)

For \(v \in E_8\): \(\| v \| ^2 \in 2\mathbb {Z}\)

For \(v, w \in E_8\): \(v - w \in E_8\)

For all \(i, j, k\): \(\varepsilon _{ijk} = -\varepsilon _{jik}\)

Icosahedron Euler characteristic: \(V + F - E = 12 + 20 - 30 = 2 = p_2\)

The Fano plane has exactly 7 lines.

\(F_{12} = 144 = (\mathrm{dim}(G_2) - p_2)^2 = 12^2\)

\(F_3 = 2 = p_2\) (Pontryagin class)

\(F_6 = 8 = \mathrm{rank}(E_8)\)

\(F_8 = 21 = b_2\)

Complete embedding \(F_3\) through \(F_{12}\) in GIFT constants.

\(\mathrm{fund}(E_7) = 2 \cdot b_2+ \mathrm{dim}(G_2) = 42 + 14 = 56\)

\(u \times v = -v \times u\)

The cross product is bilinear:

Physical interpretation:

• 120 = topology + holonomy + Cartan \(\to \) gauge bosons

• 128 = \(2^7\) from octonions \(\to \) fermions

\(b_2+ b_3+ \mathrm{dim}(G_2) + \mathrm{rank}(E_8) = 120 = \mathrm{dim}(\mathrm{SO}(16))\)

\(163 = \mathrm{dim}(E_8) - \mathrm{rank}(E_8) - b_3= 248 - 8 - 77\)

All 9 Heegner numbers have GIFT expressions.

For \(v, w \in \mathbb {R}^8\): \(\langle v, w \rangle = \sum _{i=0}^{7} v_i w_i\)

\(744 = N_{\mathrm{gen}} \times \mathrm{dim}(E_8) = 3 \times 248\)

\(744 = \mathrm{dim}(E_8) + \mathrm{dim}(E_8\times E_8) = 248 + 496\)

All conditions verified: topological (\(b_2= 21\), \(b_3= 77\)), analytic (contraction mapping), existence.

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

The PINN torsion is well below Joyce threshold: \(0.001 {\lt} 0.0288\)

\(\exists \varphi : K_7\to \Omega ^3\), torsion-free \(G_2\) structure.

\(L_4 = 7 = \mathrm{dim}(K_7)\)

\(L_5 = 11 = D_{\mathrm{bulk}}\) (M-theory dimension)

\(m_\mu /m_e \approx 27^\phi \) where \(\phi = (1 + \sqrt{5})/2\) is the golden ratio.

\(3477 = 3 \times 19 \times 61 = N_{\mathrm{gen}} \times p_8 \times \kappa _T^{-1}\)

\(20 \times \| T\| _{\text{PINN}} {\lt} \varepsilon _{\text{Joyce}}\)

The induced metric is:

Explicitly:

\(47, 59, 71\) form an AP with common difference \(12 = \mathrm{dim}(G_2) - p_2\)

\(196883 = 47 \times 59 \times 71\)

\(196883 = L_8 \times (b_3- L_6) \times (b_3- 6)\)

The spectral index \(n_s = \zeta (11)/\zeta (5)\) uses:

• \(11 = D_{\mathrm{bulk}}\) (M-theory dimension)

• \(5 = \) Weyl factor

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

For \(v \in \mathbb {R}^8\): \(\| v \| ^2 = \sum _{i=0}^{7} v_i^2\)

\(\Omega _{DE} = \ln (2) \times \frac{98}{99} = \ln (2) \times \frac{H^*- 1}{H^*}\)

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

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

\(\| T(\varphi _0)\| {\lt} \varepsilon _0\) (20\(\times \) safety margin)

For \(\alpha , v \in E_8\) with \(\langle \alpha , \alpha \rangle = 2\): \(s_\alpha (v) \in E_8\)

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

1. Standard \(\varphi _0\) gives \(g = I_7\), so \(\det (g) = 1\)

2. Scaling \(\varphi \mapsto c \cdot \varphi \) gives \(g \mapsto c^2 \cdot g\)

3. Therefore \(\det (g) \mapsto c^{14} \cdot \det (g)\)

4. Setting \(c^{14} = 65/32\) yields \(\det (g) = 65/32\)

\(2^{|\mathrm{Im}(\mathbb {O})|} = 2^7 = 128\)

The spinorial part equals the \(\mathrm{SO}(16)\) spinor dimension.

For all \(i, j \in \{ 0, \ldots , 7\} \):

The target value \(65/32\) lies in the certified interval for \(\det (g)\).

Every element of tier1_primes is prime.

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.

\(\frac{3}{13} = 0.230769\ldots \) vs experimental \(0.23122 \pm 0.00004\) (deviation: 0.19%).