Let \[ S_*=\{1,15,18,22,27\}. \] Exact enumeration gives \[ \max_{|S|=5}\tau(G_S)= 2708274425686971438523646018073797918175286202944373217379948692092215150697249050994964490229261. \] Multiplication by 4 modulo 101, followed by replacing residues above 50 with their negatives, sends \(S_*\) to \[ \{4,7,13,29,41\}. \] Thus the sampled incumbent in the candidate record already attained the maximum.

The group \((\mathbf Z/101\mathbf Z)^\times/\{\pm1\}\) has order 50 and acts on the five-element step sets. The \(\binom{50}{5}=2{,}118{,}760\) sets split into 42,376 orbits: 42,375 orbits of size 50 and one orbit of size 10. The exact search finds one maximizing orbit. The next orbit is represented by \(\{7,15,16,18,28\}\), with tree count \[ 2705790677807473137809613684625587492780256938436447544415672932035143210140285775448737562551029. \] Every step is nonzero modulo the prime 101, so each graph in the search is connected.

For \(\zeta=e^{2\pi i/101}\), the nonzero Laplacian eigenvalues are \[ \lambda_k=10-\sum_{s\in S}(\zeta^{ks}+\zeta^{-ks}), \qquad 1\leq k\leq100. \] Kirchhoff's theorem gives \[ \tau(G_S)=\frac1{101}\prod_{k=1}^{100}\lambda_k. \] Since \(\lambda_{101-k}=\lambda_k\), it suffices to multiply the first 50 values and square.

The executable replay performs the same calculation in six prime fields. Each modulus is congruent to 1 modulo 101 and comes with an explicit element of order 101. It reconstructs \(\tau(G_S)\) by the Chinese remainder theorem. The combined modulus has 366 bits and exceeds \(10^{110}\). The arithmetic-geometric mean inequality applied to the 100 nonzero Laplacian eigenvalues gives \[ \tau(G_S)\leq\frac{10.1^{100}}{101}<10^{99}. \] The reconstructed residue is therefore the unique integer tree count for every orbit representative. This exact comparison avoids any ordering decision based on floating-point logarithms.

Wang and Yang derive a spectral formula for spanning trees of circulant graphs. Zhang, Yong, and Golin develop Chebyshev formulas for fixed and varying jumps. Mednykh and Mednykh give further exact formulas, the square-form arithmetic theorem used as a check here, and Mahler-measure asymptotics.

Lonc, Parol, and Wojciechowski study the maximum number of spanning trees among \(2k\)-regular circulants as the number of vertices tends to infinity. Their result addresses asymptotic growth for fixed degree. The sources audited do not tabulate the exact order-101, degree-10 optimum or the two maximizing step-set representatives stated in this fixture.

Targeted searches using the order, degree, winning step sets, and the 97-digit count found no matching publication. This is a focused prior-art check, so the originality of the finite result remains unverified.

The program stores each step set as a 50-bit integer. For a first unseen set, it generates all 50 signed-unit images, marks the orbit seen, and uses the smallest bit mask as representative. Assertions fix the complete orbit distribution before accepting the optimum.

For each representative, six finite-field Fourier tables supply the 50 paired Laplacian eigenvalues. Native integer modular arithmetic produces six exact residues, and incremental Chinese remaindering reconstructs the tree count. The product of the moduli is larger than the universal spectral upper bound, so no two possible counts share a residue vector.

A SHA-256 stream covers the representative mask, its orbit size, and its reconstructed 48-byte little-endian tree count. Its digest is `422da911117db6d7008b3683660583752b1152e49a15522c52f9267d664b4771`. The stable seven-line output has SHA-256 digest `3936bd3c064fae74cbcc3e4147ab4db608e7f558ad84e02e1cd345d3ff56643d`.

The replay constructs the integer Laplacian directly from the canonical step set and deletes vertex 100. Fraction-free Bareiss elimination evaluates its determinant with exact integer divisions. Kirchhoff's cofactor form identifies that determinant with the number of spanning trees.

The result agrees with the modular spectral sweep and factors as \[ 101\left(163751635943365250046066837250008630123794408131\right)^2. \] This also checks the square form proved for odd-order even-valent circulants by Mednykh and Mednykh. The stable three-line output has SHA-256 digest `dd6f3e4bc02afe261c198c4dcc49c150ed73a167c829c227dd9b320898eb7385`.