Write \(\gamma\) for the discrete-time spectral gap under the convention in the problem. The certified interval is \[ \frac{1}{170005193383307227693056} \leq \gamma \leq \frac{48265367094829605687889}{36363194510128970638045284} =0.00132731372325897517\ldots. \] The lower endpoint equals \(1/(36\,2^{72})\). The upper endpoint is an exact Rayleigh quotient for total magnetization \(M=\sum_v\sigma_v\). This trial function lies in the spin-flip-odd, translation-invariant symmetry sector. Its projected transition eigenvalue is \[ \frac{36314929143034141032357395}{36363194510128970638045284}. \]

For the upper bound, reversibility and the heat-bath projection identity give \[ \langle M,(I-P)M\rangle_\pi =\mathbb E_\pi\!\left[\operatorname{Var}(\sigma_0\mid\sigma_{V\setminus\{0\}})\right]. \] The exact transfer calculation finds \[ \sum_\sigma 2^{a(\sigma)}M(\sigma)^2 =273793464546853425980576256, \] where \(a(\sigma)\) is the number of agreeing edges. With common conditional-variance scale \(7225=25\cdot17^2\), it also finds \[ \sum_\sigma 2^{a(\sigma)}\,7225\operatorname{Var}(\sigma_0\mid\sigma_{V\setminus\{0\}}) =2625635969958730549421161600. \] Their quotient gives the stated upper endpoint.

For the lower bound, compare with independent random-scan heat bath under the uniform measure. If \(m\) and \(M_0\) are the minimum and maximum density ratios of the Ising measure against uniform measure, edge conductances give \(\mathcal E_P\geq m\mathcal E_0\), while \(\operatorname{Var}_\pi f\leq M_0\operatorname{Var}_0f\). The independent chain has gap \(1/36\). The Gibbs weights are proportional to \(2^{a(\sigma)}\), and the bipartite torus realizes both \(a=0\) and \(a=72\). Hence \(m/M_0=2^{-72}\), proving the lower endpoint.

The calculation supplies rigorous bounds. Determining the exact second eigenvalue still requires a certificate that covers every symmetry sector.

Holley and Stroock develop functional inequalities and perturbation methods for stochastic Ising models. Martinelli's Saint-Flour lectures give a broad account of reversible Glauber dynamics, finite-volume spectral gaps, and temperature regimes. Lubetzky and Sly study heat-bath Glauber dynamics on square tori throughout the strong-spatial-mixing regime and relate finite-torus gaps to the infinite-volume gap.

These sources support the chain convention and the use of Poincare comparison. They do not tabulate the second eigenvalue for \(C_6\mathbin{\square}C_6\) at \(\beta=(\log2)/2\). Targeted searches for the exact size, the five rational conditional probabilities, and a finite-volume characteristic polynomial also found no such certificate. This is a status audit rather than a proof of novelty.

A full resolution needs a decomposition of the transition operator under spin flip and the automorphism group of the torus, followed by an exact characteristic polynomial or rational eigenvalue enclosures for every block. The present one-dimensional Ritz calculation covers one invariant trial subspace and supplies the upper endpoint recorded here.

For a configuration \(\sigma\), its Gibbs weight differs by a constant factor from \(2^{a(\sigma)}\), where \(a(\sigma)\) counts agreeing edges. Encode each row by a six-bit integer. The integer transfer matrix is \[ T_{rs}=2^{h(s)+v(r,s)}, \] where \(h(s)\) counts horizontal agreements in row \(s\), and \(v(r,s)\) counts vertical agreements. Thus \(\operatorname{tr}(T^6)\) contracts all \(2^{36}\) configurations. Its value is \[ 674103569746667088362626. \]

The magnetization second moment is obtained from the six cyclic row separations. The one-site conditional variance depends on the four-neighbor field magnitude. Its three values are \(1\), \(16/25\), and \(64/289\). A three-row insertion followed by \(T^4\) gives its exact weighted sum. The quotient is the Ritz value of \(I-P\) on the span of total magnetization, so it bounds the full gap above without assuming that the leading odd mode is magnetization.

Entries are serialized row-major. Each nonnegative integer is written as a four-byte big-endian length followed by its shortest big-endian byte string. The transfer hash is `c9bdfa7af6e49df8c031fbe5def55787bff5beaf45aa1b3775f6f7f3b1f07897`; the \(T^6\) hash is `61ed667ffcb10c3467853248b81cd65acb909f6b8035d9672e1662ceb28ac732`.