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.

The adaptive cutoff exponent:

determined by the normalization constraint \(\alpha = 1\) uniformly across height windows.

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)

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

\(|2I| = 120\)

\(c_1 = v_0^2\) (from Cheeger inequality)

\(c_2 = \frac{16 v_1}{1 - v_1}\) (from Rayleigh quotient)

The cosine-squared mollifier kernel:

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

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

Cross-section of TCS cylindrical end with dimension and Betti numbers.

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

Only 14 functions are learned (not 35).

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

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

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

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

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 sequence \(\gamma _n\) (\(n \in \mathbb {N}^+\)) of positive imaginary parts of non-trivial zeros of the Riemann zeta function: \(\zeta (\tfrac {1}{2} + i\gamma _n) = 0\), \(\gamma _n {\gt} 0\).

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: \(|\mathrm{Im}(\mathbb {O})| = 7\)

A multiple \(7k\) is matched if \(\exists \, n : |\gamma _n - 7k| / (7k) {\lt} 0.5\% \).

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

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

Standard Kovalev construction with \(\dim = 5\) and \(b_0 = 1, b_1 = 1, b_2 = 22, b_3 = 22, b_4 = 23, b_5 = 1\).

The denominator is \(b_2= 21\).

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

\(L_0 = \frac{2 v_0}{h_0}\) (neck dominates for \(L {\gt} L_0\))

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

The spectral parameter arising from the functional equation \(\zeta (s) = \zeta (1-s)\):

At \(s = \tfrac {1}{2} + i\gamma \): \(s(1-s) = \tfrac {1}{4} + \gamma ^2\).

The multiset of lengths of closed geodesics on a compact Riemannian manifold.

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

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

The smallest faithful representation: \(196883\)

\(G_2\)-holonomy manifolds admit exactly \(h = 1\) parallel spinor (Berger classification; Joyce 2000, Table 10.1).

\(\mathrm{SU}(3)\)-holonomy manifolds (Calabi-Yau 3-folds) admit \(h = 2\) parallel spinors.

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 \(\det (g) = 65/32\).

The physical mass gap numerator: \(\mathrm{dim}(G_2) - h = 14 - 1 = 13\).

The physical mass gap ratio: \(\lambda _1 = 13/99\).

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

For prime \(p\) and power \(m\):

The 7D correction to the Kronecker formula:

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

\(S_w(T) = S_w(T; \theta (T))\) using the adaptive cutoff.

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

Conjecture: \(\exists \, n_1, n_2, n_{20}, n_{60}, n_{107}\) such that \(\lambda _{n_i}\) matches \(C_i^2\) within specified precision.

A GIFT constant \(C\) has a spectral match at index \(n\) with precision \(p\) if:

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 \(\sum _{i=0}^{7} v_i \in 2\mathbb {Z}\).

The six hypotheses (H1)–(H6) for spectral bounds:

1. Volume normalization: \(\mathrm{Vol}(K) = 1\)

2. Bounded neck volume: \(v_0 \leq \mathrm{Vol}(N) \leq v_1\)

3. Product metric: \(g|_N = dt^2 + g_Y\)

4. Block Cheeger bound: \(h(M_i \setminus N) \geq h_0\)

5. Balanced blocks: \(\mathrm{Vol}(M_i) \in [1/4, 3/4]\)

6. Neck minimality: \(\mathrm{Area}(\Gamma ) \geq \mathrm{Area}(Y)\)

A TCS manifold \(K = M_1 \cup _N M_2\) consists of:

• Two asymptotically cylindrical manifolds \(M_1, M_2\) (building blocks)

• A cylindrical neck \(N \cong Y \times [0, L]\) with cross-section \(Y\)

• Neck length parameter \(L {\gt} 0\)

• Volume normalization: \(\mathrm{Vol}(K) = 1\)

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

\(163 \equiv 2 \pmod{7}\)

\(248 \equiv 3 \pmod{7}\)

The five correspondences hold simultaneously.

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

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)

\(b_2= 3 \cdot \mathrm{dim}(K_7)\)

\(b_2= 21\)

\(b_3= 11 \cdot \mathrm{dim}(K_7)\)

\(b_3= 77\)

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

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

\(14/99 - 13/99 = 1/99 = h/H^*\). The correction is exactly the parallel spinor count divided by the total cohomological dimension.

For TCS manifold with neck length \(L\): \(\exists c {\gt} 0\) such that no eigenvalues in \((0, c/L)\). Reference: Crowley-Goette-Nordstrom 2024, Inventiones Math.

Over the first 100,000 zeros: \(\max _n |N_{\text{approx}}(T_n) - n| = 0.111 {\lt} 0.5\) (safety factor \(4.52\times \)).

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

\(\frac{7}{2\pi } \in (1.11, 1.12)\), ensuring high density of zeros near each multiple of 7.

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

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

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

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

\(\mathrm{dim}(G_2) = 2 \cdot \mathrm{dim}(K_7)\)

Every element of direct_primes is prime.

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:

The 107th zeta zero (\(248.1020\ldots \)) is within \(0.04\% \) of \(\mathrm{dim}(E_8) = 248\).

The first zeta zero (\(14.1347\ldots \)) is within \(0.96\% \) of \(\mathrm{dim}(G_2) = 14\).

The 20th zeta zero (\(77.1448\ldots \)) is within \(0.19\% \) of \(b_3= 77\).

The second zeta zero (\(21.0220\ldots \)) is within \(0.10\% \) of \(b_2= 21\).

The 60th zeta zero (\(163.0307\ldots \)) is within \(0.02\% \) of the largest Heegner number \(163\).

The sequence \((\gamma _n)\) is strictly increasing.

\(\gamma _n {\gt} 0\) for all \(n\).

\(w(0) = 1\) and \(0 {\lt} 13/99 {\lt} 1\): the spectral gap falls within the compact support of the cosine-squared kernel.

Physical interpretation:

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

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

For \(K_7\) constructed via TCS, primitive closed geodesic lengths satisfy \(\ell _\gamma (p) = c \cdot \log (p)\) for primes \(p\) and a geometric constant \(c {\gt} 0\). Status: Conjectural (Category E).

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

Under the geodesic-prime correspondence, the geometric side of the trace formula is proportional to \(S_w(T)\) from the MollifiedSum module. Status: Conjectural (Category E).

The GIFT prediction \(\lambda _1 = 14/99 = \dim (G_2)/H^*\) is consistent with TCS bounds: \(1/100 {\lt} 14/99 {\lt} 1/4\).

Complete GIFT framework: 290+ certified relations spanning E\(_8\) lattice, G\(_2\) holonomy, physical predictions, spectral theory, number-theoretic correspondences, and the Selberg bridge.

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

\(163 = |\mathrm{Roots}(E_8)| - b_3= 240 - 77\).

All 9 Heegner numbers have GIFT expressions.

\(H^*= 14 \cdot \mathrm{dim}(K_7) + 1\)

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

The spectral gap scales as \(\lambda _1 = \Theta (1/L^2)\) as \(L \to \infty \).

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

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

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: \(0.001 {\lt} 0.0288\)

For K3 \(\times \) S\(^1\):

• 2-forms: \(2(b_1 + b_2) = 2(1 + 22) = 46\)

• 3-forms: \(2(b_2 + b_3) = 2(22 + 22) = 88\)

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

\(w(x) \leq 1\) for all \(x \in \mathbb {R}\).

\(w(x) \geq 0\) for all \(x \in \mathbb {R}\).

\(w(x) = 0\) for all \(x \geq 1\).

\(w(0) = 1\).

The standard prime power cutoff \(K = 3\) equals the number of fermion generations \(N_{\mathrm{gen}} = 3\).

The spectral parameters \((\lambda _n)\) are strictly increasing.

\(\lambda _n {\gt} 0\) for all \(n\).

For TCS family \((M_T, g_T)\) with cross-section \(X\):

Reference: Langlais 2024, Comm. Math. Phys.

98.0% of zeros are localized to their correct inter-Gram interval. The 2.0% failure rate is predicted (within 8%) by a simplified GUE model.

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

\(L_5 = 11 = D_{\mathrm{bulk}}\) (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:

Complete verification of the constructive core: kernel properties + sum well-definedness + standard \(K = 3\).

\(47, 59, 71\) 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 \(v \in \mathbb {R}^8\): \(\| v \| ^2 = \sum _{i=0}^{7} v_i^2\)

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

\(99^2 - 50 \times 14^2 = 1\) and \(14 - 1 = 13\): the Pell equation connects bare topological constants, while the spinor correction shifts \(14 \to 13\).

\(99^2 - 50 \times 14^2 = 1\). The bare topological constants \((14, 99)\) satisfy a Pell equation with discriminant \(50 = 2 \times 5^2\).

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

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

Master certificate: 11 properties verified, zero axioms.

\(\gcd (13, 99) = 1\). The ratio \(13/99\) is in lowest terms.

\(13/99 = (\mathrm{dim}(G_2) - h) / H^*\) where all quantities are topological.

\(\mathrm{dim}(G_2) - h = 14 - 1 = 13\).

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

The adaptive formula explains 93.9% of the variance in zero corrections \(\delta _n\) (\(R^2 = 0.939\), out-of-sample \(R^2 = 0.919\)).

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

\(S_w(T; \theta )\) is a finite sum and therefore always a well-defined real number. Unlike the formal Dirichlet series \(-\zeta '/\zeta (s)\) which diverges on \(\mathrm{Re}(s) = \frac{1}{2}\), no convergence issues arise.

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 \(\det (g) = 65/32\)

Master certificate combining MollifiedSum properties, Spectral properties, and cross-module identities.

\(\mathrm{dim}(G_2) - h = 13 = \mathrm{rank}(E_8) + \text{Weyl} = \alpha _{\mathrm{sum}}\).

\((13/99) \times 99 = 13 = \mathrm{dim}(G_2) - h\).

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

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

Reference squared values: \(14^2 = 196\), \(21^2 = 441\), \(77^2 = 5929\), \(163^2 = 26569\), \(248^2 = 61504\).

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

For \(\mathrm{SU}(3)\) holonomy: \(\mathrm{dim}(\mathrm{SU}(3)) - h = 8 - 2 = 6\). Numerically validated: \(\lambda _1 \times H^*= 5.996\) on \(T^6/\mathbb {Z}_3\).

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

Complete verification of TCS spectral bounds and GIFT prediction compatibility.

For TCS manifold \(K\) with neck length \(L {\gt} L_0\) satisfying (H1)–(H6):

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.

For a compact Riemannian manifold \(M\) and a suitable test function \(h\):

Reference: Selberg (1956); Duistermaat-Guillemin (1975), Invent. Math. 29:39–79.

For typical TCS with \(v_0 = v_1 = 1/2\), \(h_0 = 1\):

• \(c_1 = 1/4\)

• \(c_2 = 16\)

• \(L_0 = 1\)

• Bound ratio: \(c_2/c_1 = 64\)

This ratio appears in both the GIFT mass gap (\(\lambda _1 = 14/99\)) and the first zeta zero (\(\gamma _1 \approx 14\)).

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

Complete verification: all five primary correspondences hold and the structural identities \(14 = 2 \times 7\), \(21 = 3 \times 7\), \(77 = 11 \times 7\) are confirmed.