10 169-Parameter Chebyshev-Cholesky Metric
The 169-parameter Chebyshev-Cholesky metric is the first explicit analytical G\(_2\) holonomy metric on a compact TCS 7-manifold. Published as DOI:10.5281/zenodo.18860358.
10.1 Chebyshev-Cholesky Parametrization
The Chebyshev polynomial order \(K = 5\), giving \(K + 1 = 6\) modes.
A \(7 \times 7\) lower-triangular matrix has \(\binom {8}{2} = 28\) independent entries.
The 21 off-diagonal Cholesky entries equal \(b_2(K_7)\), reflecting the \(b_2\)-dimensional moduli space of \(G_2\) structures.
Total parameters: \(6 \times 28 + 1 = 169\), where the \(+1\) is the ACyl decay rate.
The parameter count is a perfect square: \(169 = (r_{E_8} + W)^2 = 13^2\).
The Chebyshev parameter count \(168 = |\mathrm{PSL}(2,7)|\), the Fano automorphism group.
10.2 SPD Structure and Determinant
The metric satisfies \(\det (g) = 65/32\) at every evaluation point, with \(\gcd (65, 32) = 1\).
The PINN model (1,070,471 parameters) compresses to 169 Chebyshev-Cholesky parameters: a ratio of 6,334:1.
The 7 K\(_7\) coordinates decompose as \(1 + 2 + 4 = 7\): 1 neck parameter, 2 S\(^1\) fiber angles, 4 K3 surface coordinates.
Bundled certificate: 12 structural properties of the 169-parameter metric.
10.3 Newton-Kantorovich Certification
The NK contraction parameter: \(h \leq 665 / 10^{10} = 6.65 \times 10^{-8}\).
\(h \times 2 {\lt} 10^{10}\), establishing \(h {\lt} 1/2\) with safety margin \({\gt} 7.5\) million.
5 Joyce iteration steps (= Weyl factor) reduce torsion by \(\times 2995\).
The certified metric \(g_0\) differs from the true torsion-free metric \(g^*\) by at most \(4.86 \times 10^{-6}\) (relative).
After 5 steps: \(\| T_5\| = 2.98 \times 10^{-5} \ll 0.0288 = \varepsilon _{\text{Joyce}}\).
Bundled record: \(n_{\text{params}}, h, \text{steps}, \text{reduction}, \text{proximity}\) with built-in proof of \(h {\lt} 1/2\).
Bundled certificate: 7 structural properties of the NK certification chain.
10.4 K3 Harmonic Correction and Torsion Classes
Under G\(_2\) representation theory: \(T \in W_1 \oplus W_7 \oplus W_{14} \oplus W_{27}\) with dimensions \(1 + 7 + 14 + 27 = 49 = \dim (K_7)^2\).
\(1 + 7 + 14 + 27 = 49 = 7^2\).
The \(W_{27}\) (symmetric traceless) component carries 99.6% of total torsion.
The exterior derivative norm ratio is exactly \(1/5 = 1/W\).
The K3 fiber contributes only 0.07% of total torsion, verified over 220,000 evaluation points.
The proper normalization of the associative 3-form.
\(|{*}\varphi |^2 = 4 \times 42 = 168 = |\mathrm{PSL}(2,7)|\).
Bundled certificate: 10 structural properties of the torsion reduction chain.
10.5 K3 Newton-Kantorovich Certificate (CI(2,2,2))
Independent Newton–Kantorovich certificate for the K3 surface \(\mathrm{CI}(2,2,2) \subset \mathbb {P}^5\) (the K3 fiber of the TCS \(G_2\) construction), via the Donaldson algebraic section method at degree \(k=4\) (126 sections, 31,752 parameters). Two independent \(\beta \) sources both certify \(h {\lt} 1/2\): a graph-Laplacian bound (\(\beta _{\mathrm{Lap}} = 5.66\), \(h = 7.83 \times 10^{-2}\), margin \(\times 6.4\)) and a Jacobian pseudoinverse bound (\(\beta _{\mathrm{Jac}} = 2.25\) at \(k=3\), \(h = 0.188\), margin \(\times 2.7\)). The \(k=2\) Jacobian variant fails (\(h = 1.55 {\gt} 1/2\)), confirming sensitivity to ansatz quality.
Structure recording a K3 NK certificate: Donaldson degree, parameter count, honest \(\eta _{L^2}\) (held-out test set), and two independent \(h\)-bounds with contraction proofs.
Concrete certificate: \(k=4\), \(\eta _{L^2} = 1.596 \times 10^{-2}\), \(h_{\mathrm{Lap}} = 7.83 \times 10^{-2}\), \(h_{\mathrm{Jac}} = 1.88 \times 10^{-1}\).
\(h_{\mathrm{Lap}} {\lt} 1/2\): the graph-Laplacian \(\beta \) source certifies NK contraction.
\(h_{\mathrm{Jac}} {\lt} 1/2\): the pseudoinverse \(\beta \) source certifies NK contraction.
The \(k=2\) Jacobian variant fails (\(h = 1.55 {\gt} 1/2\)), confirming the certificate is sensitive to ansatz quality.
Six-conjunct master certificate: both \(\beta \) sources pass, \(k=2\) fails, \(\eta _{L^2} {\lt} 2/100\), parameter count \({\gt} 10{,}000\), and \(\delta _{K3}^{\mathrm{cert}} {\lt} \varepsilon _0 = 0.1\) (Joyce threshold).
10.6 Collar Re-summation and Indicial Parity
Weighted-analysis estimates for the collar region of the K3-fibered \(G_2\) metric involve the binomial expansion of \((w - \rho )^{3/2}\) in the fiber coordinate, whose order-\(k\) coefficient is the generalized binomial coefficient \(\binom {3/2}{k} = \tfrac {3}{2}(\tfrac {3}{2} - 1)\cdots (\tfrac {3}{2} - k + 1)/k!\). Two purely analytic facts control how the per-order bounds aggregate: the absolute series sums to exactly \(3\) (the collar/bulk erosion factor), and the indicial constant is invariant under the weight reflection \(\beta \mapsto -\beta \). Both are machine-checked below, the first by an elementary telescoping route that avoids Abel’s theorem.
\(\mathrm{C32}(k) = \binom {3/2}{k} = \bigl(\textstyle \prod _{i{\lt}k} (3/2 - i)\bigr) / k!\).
For all \(n\), \(\sum _{k=0}^{n} (-1)^k \binom {3/2}{k} = (-1)^n \binom {1/2}{n}\). Combined with \(|\binom {1/2}{n}| \leq 1/(2n-1)\), this is the elementary engine for the two main results, used in place of any boundary-limit theorem.
The absolute generalized-binomial series is summable with \(\sum _{k \geq 0} |\binom {3/2}{k}| = 3\): the collar bound erodes the bulk bound by exactly the factor \(3\), independently of the adiabatic parameter.
\(\sum '_{k} |\binom {3/2}{k}| = 3\).
\(\sum _{k \geq 0} (-1)^k \binom {3/2}{k} = 0\), i.e. the binomial series of \((1+x)^{3/2}\) evaluated at \(x = -1\), obtained here without any boundary-limit theorem.
\(P_m(\beta ) = \beta ^2 - m^2\).
Since \(P_m\) is even in \(\beta \), the indicial constant \(K_{\mathrm{ind}}(\beta ) = 1/|P_{1/2}(\beta )|\) satisfies \(K_{\mathrm{ind}}(-1) = K_{\mathrm{ind}}(+1) = 4/3\): the weighted-Schauder constant carries verbatim across the \(\beta = +1\) and \(\beta = -1\) weights.