23 Spectral Scaling on the TCS Neck

The neck \(N \cong [0,L] \times Y\) of the TCS manifold \(K_7\) supports a waveguide spectrum. The longitudinal Neumann eigenvalues \(\lambda _n = n^2 \pi ^2/L^2\) connect to GIFT topological constants through remarkable arithmetic identities: the eigenvalue sum \(1^2+2^2+3^2 = \mathrm{dim}(G_2)\), the sub-gap mode count equals \(N_{\mathrm{gen}}\), and the neck length ratio \(L^2/\pi ^2 = H^*/\mathrm{dim}(G_2)\) satisfies a Pell equation.

23.1 Neumann Eigenvalue Hierarchy

Definition 23.1 Neumann eigenvalue ratio

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The \(n\)-th Neumann eigenvalue ratio \(\lambda _n/\lambda _1 = n^2\), from the 1D problem \(-f'' = \lambda f\) on \([0,L]\) with \(f'(0) = f'(L) = 0\).

Theorem 23.2 Second spectral gap \(= N_{\mathrm{gen}}\)

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The gap between the first two excited modes is \(\lambda _2 - \lambda _1 = 3 = N_{\mathrm{gen}}\).

Theorem 23.3 Third spectral gap \(=\) Weyl factor

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The gap \(\lambda _3 - \lambda _2 = 5 = W\).

Theorem 23.4 Eigenvalue sum \(= \mathrm{dim}(G_2)\)

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\(1^2 + 2^2 + 3^2 = 14 = \mathrm{dim}(G_2)\): the holonomy dimension emerges as a sum of eigenvalue ratios.

Theorem 23.5 Eigenvalue sum \(= h(E_8)\)

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\(1^2 + 2^2 + 3^2 + 4^2 = 30 = h(E_8)\): the Coxeter number of \(E_8\) emerges at the fourth level.

23.2 Neck Length Ratio and Selection Constant

Theorem 23.6 Neck ratio from GIFT constants

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\(L^2/\pi ^2 = H^*/\mathrm{dim}(G_2) = 99/14\).

Theorem 23.7 Selection constant structure

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\(\mathrm{dim}(G_2) = p_2 \times \mathrm{dim}(K_7) = 2 \times 7\): the Pontryagin-manifold factorization of the holonomy dimension.

Theorem 23.8 \(\kappa \times H^*= L^2/\pi ^2\)

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\((1/14) \times 99 = 99/14\): the rational skeleton of \(L^2 = \kappa H^*\).

23.3 Division Algorithm Connection

Theorem 23.9 Euclidean division of \(H^*\) by \(\mathrm{dim}(G_2)\)

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\(H^*= \mathrm{dim}(K_7) \times \mathrm{dim}(G_2) + 1\), i.e., \(99 = 7 \times 14 + 1\). The quotient is \(\mathrm{dim}(K_7)\) and the remainder is 1 (the parallel spinor count for \(G_2\) holonomy from Berger’s classification).

Theorem 23.10 Remainder \(=\) spinor correction

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\(H^*\bmod \mathrm{dim}(G_2) = 1\), and \((\mathrm{dim}(G_2) - 1)/H^*= 13/99\): the physical spectral gap.

23.4 Sub-Gap Mode Counting

Theorem 23.11 Sub-gap threshold \(= \mathrm{dim}(K_7)\)

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\(\lfloor L^2/\pi ^2 \rfloor = \lfloor 99/14 \rfloor = 7 = \mathrm{dim}(K_7)\).

Theorem 23.12 Sub-gap mode count \(= N_{\mathrm{gen}}\)

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Modes with \(n^2 \le 7\): \(n = 0, 1, 2\) (since \(3^2 = 9 {\gt} 7\)). The count is \(3 = N_{\mathrm{gen}}\): the three fermion generations correspond to the three longitudinal waveguide modes below the cross-section gap.

23.5 Second Eigenvalue and \(E_7\)

Theorem 23.13 \(\lambda _2 \times H^*= \mathrm{dim}(\mathrm{fund.}\; E_7)\)

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\(4 \times \mathrm{dim}(G_2) = 56 = \mathrm{dim}(\mathrm{fund.}\; E_7) = b_3- b_2\).

Theorem 23.14 \(56 = C(\mathrm{rank}(E_8), N_{\mathrm{gen}})\)

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\(\mathrm{dim}(\mathrm{fund.}\; E_7) = \binom {8}{3}\): a binomial coefficient involving the rank of \(E_8\) and \(N_{\mathrm{gen}}\).

23.6 Pell Equation

Theorem 23.15 Pell equation for \((H^*, \mathrm{dim}(G_2))\)

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\(99^2 - 50 \times 14^2 = 1\). The discriminant \(50 = p_2 \times W^2 = 2 \times 5^2\) connects to the Pontryagin class and Weyl factor.

23.7 Spectral-Topological Dictionary

Theorem 23.16 Spectral ladder

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\(\lambda _1 H^*= 14\), \(\lambda _2 H^*= 56\), gap \(= 42 = 2b_2\).

23.8 Master Certificate

Theorem 23.17 Spectral Scaling Certificate

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All 12 conjuncts of the spectral scaling certificate are proven.