% arXiv v1, derived from the frozen general-purpose manuscript v7. \documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage{lmodern} \usepackage{amsmath,amssymb,amsthm} \usepackage[margin=1in]{geometry} \usepackage[hidelinks]{hyperref} \hypersetup{ pdftitle={The Sharp Exponential Constant for 6-Regular Circulant Matrices}, pdfauthor={Yair Lavi} } \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \DeclareMathSizes{9}{10}{8}{8} \DeclareMathSizes{10}{10}{8}{8} \DeclareMathSizes{10.95}{10}{8}{8} \DeclareMathSizes{12}{10}{8}{8} \setlength{\emergencystretch}{3em} \providecommand{\tightlist}{\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}} \title{The Sharp Exponential Constant for 6-Regular Circulant Matrices} \author{Yair Lavi} \date{24 September 2026} \begin{document} \maketitle \begin{abstract} We prove that, for every integer \(n\ge6\) and six-element set \(S\subseteq\mathbb Z_n\), the binary circulant \(C_n(S)=(\mathrm{1}_{\{j-i\in S\}})_{i,j\in\mathbb Z_n}\) satisfies \[ \operatorname{per}C_n(S)\ge \left(\frac{3125}{1296}e^{7/250000}\right)^n. \] For every integer \(d\ge2\), we also identify the optimal uniform base as the limit of an explicit sequence of circulants: \[ \inf_{n\ge d}\ \inf_{\substack{S\subseteq\mathbb Z_n\\|S|=d}} \operatorname{per}C_n(S)^{1/n} =\lim_{q\to\infty} \operatorname{per}C_{q^{d-1}}(\{0,1,q,\ldots,q^{d-2}\})^{1/q^{d-1}}, \] where \(q\) runs through integers at least \(2\). In degree six this base is at most \(163812873663198^{1/35}\). The lower bound strictly improves the sharp uniform base \(3125/1296\) for unrestricted binary \(6\)-regular matrices. This resolves the surviving circulant lower-bound question in Minc's Problem 10; its proposed full upper bound fails even for circulants. \end{abstract} \noindent\textbf{Keywords.} Permanent; binary circulant; matching measure; regular bipartite graph; matching entropy. \noindent\textbf{2020 Mathematics Subject Classification.} 15A15 (primary), 05C70, 05C50 (secondary). \section{Introduction}\label{introduction} Let \(\Lambda_n^d\) denote the set of \(n\times n\) binary matrices having every row sum and every column sum equal to \(d\). Two classical exponential bounds are \[ \left(\frac{(d-1)^{d-1}}{d^{d-2}}\right)^n \le \operatorname{per}A \le (d!)^{n/d},\qquad A\in\Lambda_n^d. \] Schrijver and Valiant conjectured the lower inequality and established its asymptotic optimality \cite{r4}; Schrijver proved it \cite{r5}. Br\`egman proved the upper inequality \cite{r13} (see also Schrijver's short proof \cite{r3}). Both bases are asymptotically best possible in the unrestricted class. At degree six they are \[ \frac{5^5}{6^4}=\frac{3125}{1296} \quad\text{and}\quad (6!)^{1/6}=2.9937951\ldots . \] Minc's earlier work studied permanents of circulants \cite{r1,r2}. His 1983 survey \cite{r14} restates Problem 10 from his 1978 monograph \cite{r15}, explicitly giving both the unrestricted and circulant scopes. After the correction recorded by Cheon and Wanless, the problem asks for \[ 2.31^n2.31\); he did not assert that \(2.31\) was the optimal base. For \(S\subseteq\mathbb Z_n\), put \(C_n(S)=\sum_{s\in S}P^s\), where \(P_{ij}=1\) exactly when \(j-i=1\) in \(\mathbb Z_n\). Equivalently, \(C_n(S)_{ij}=1\) exactly when \(j-i\in S\). The full corrected chain is impossible even for circulants. For \(n=6k\), the support \(S=\{0,k,2k,3k,4k,5k\}\) gives \(k\) all-one \(6\times6\) blocks after row and column permutations, so \(\operatorname{per}C_n(S)=720^k\) and its root rate \(\operatorname{per}C_n(S)^{1/n}\) is \(720^{1/6}=2.9937951\ldots>2.99\), with the last inequality equivalent to \(720\cdot100^6>299^6\). Cheon and Wanless therefore isolate the sole surviving component: can the sharp unrestricted lower base be improved for \(6\)-regular circulants? We answer that question and identify the optimal base. Put \[ m_{\mathrm{circ},d} =\inf_{n\ge d}\ \inf_{\substack{S\subseteq\mathbb Z_n\\|S|=d}} \operatorname{per}C_n(S)^{1/n}. \] This infimum is also the optimal asymptotic base if a fixed positive multiplicative prefactor is allowed: every finite circulant can be repeated as a disjoint circulant lift, so such a prefactor disappears after taking roots. Let \(e_1,\ldots,e_{d-1}\) be the standard basis of \(\mathbb Z^{d-1}\) and put \(D_d=\{0,e_1,\ldots,e_{d-1}\}\). The bipartite graph \(\Gamma_d\) has left vertices \(L_x\), right vertices \(R_x\) for \(x\in\mathbb Z^{d-1}\), and edges \(L_xR_{x+s}\) for \(s\in D_d\). To specify the entropy in our answer, let \(m_k(G)\) count the \(k\)-edge matchings of a finite balanced bipartite graph \(G\) with \(m\) vertices in each color class. Its matching polynomial is \(\sum_{k=0}^m(-1)^km_k(G)z^{2m-2k}\), and its normalized matching-root measure \(\rho_G\) gives mass \(1/(2m)\) to each root, with multiplicity. Let \(\rho_{\Gamma_d}\) be the weak limit of these measures for the bipartite graphs of \(C_{q^{d-1}}(\{0,1,q,\ldots,q^{d-2}\})\) as integer \(q\to\infty\). We define the perfect-matching entropy per vertex by \[ \lambda_{\Gamma_d}(1) =\int\log|x|\,d\rho_{\Gamma_d}(x). \] Sections 2--4 establish that this limit exists and that the integral is finite. Our main result is the following. \begin{theorem}[Sharp circulant constant]\label{thm:sharp} For every integer \(d\ge2\), \[ \boxed{ m_{\mathrm{circ},d}=h_d :=\exp\bigl(2\lambda_{\Gamma_d}(1)\bigr), } \] with \(\Gamma_d\) and \(\lambda_{\Gamma_d}(1)\) as defined above. Moreover, for integers \(q\ge2\), the bipartite graphs of the connected cyclic quotients (the circulants below, whose bipartite graphs are connected because \(1\) is in their support) \[ C_{q^{d-1}}\bigl(\{0,1,q,\ldots,q^{d-2}\}\bigr) \] have, for every fixed radius, the same rooted neighborhoods as \(\Gamma_d\) when \(q\) is sufficiently large, and their permanent root rates converge to \(h_d\). \end{theorem} Every finite circulant quotient has root rate at least this limit. The proof combines the coefficientwise matching inequality for bipartite \(2\)-lifts \cite{r9} with local continuity and endpoint convergence of matching entropy \cite{r7,r8}. For degree six we obtain the strict factor \(e^{7/250000}\) over the unrestricted lower base. \begin{theorem}[Certified degree-six bracket]\label{thm:bracket} For every \(n\ge6\) and every six-element \(S\subseteq\mathbb Z_n\), \[ \operatorname{per}C_n(S) \ge \left[\frac{3125}{1296} \exp\left(\frac7{250000}\right)\right]^n. \] Consequently \[ \boxed{ \frac{3125}{1296}e^{7/250000} \le m_{\mathrm{circ},6}=h_6 \le 163812873663198^{1/35}. } \] \end{theorem} The numerical certificate uses only exact integers and rational intervals; decimal numbers in this paper are for presentation. The upper endpoint is the exact permanent of the \(35\)-circulant with support \(\{0,1,3,7,16,24\}\). We further prove a finite-degree convergence modulus, give a convergent two-sided moment scheme for evaluating \(h_6\), and report an exact finite census through order \(31\). These results separate what is now settled from what remains: the optimal exponent is proved, while a closed form, many-digit evaluation, strict nonattainment by finite quotients, and a stability theorem for near minimizers are still open. \section{Matching entropy and the quotient model}\label{matching-entropy-and-the-quotient-model} \subsection{Bipartite graphs and matching measures}\label{bipartite-graphs-and-matching-measures} The matrix \(C_n(S)\) is the bipartite adjacency matrix of the graph with left and right color classes both indexed by \(\mathbb Z_n\), with \[ L_x\sim R_y\quad\Longleftrightarrow\quad y-x\in S. \] Thus its permanent is the number of perfect matchings of this graph. For a finite balanced bipartite graph \(G\) with \(m\ge1\) vertices in each color class and a perfect matching, let \(m_k(G)\) be the number of \(k\)-edge matchings and put \[ M(G,t)=\sum_{k=0}^m m_k(G)t^k, \qquad P_G(t)=\frac1{2m}\log M(G,t). \] Give a matching \(M\) probability \(t^{|M|}/M(G,t)\). Its expected size divided by \(m\) is \[ p_G(t)=\frac{tM'(G,t)}{mM(G,t)}. \] Because \(G\) has a perfect matching, \(p_G(t)\) increases continuously from \(0\) to \(1\) as \(t\) runs from \(0\) to infinity. For \(00}[P_G(t)-(p/2)\log t]\): as a function of \(\log t\), \(P_G(t)\) is convex and its derivative is \(p_G(t)/2\). Set \(\lambda_G(0)=0\) by continuity. At the perfect-matching endpoint, \[ \lambda_G(1)=\frac1{2m}\log\operatorname{pm}(G), \] where \(\operatorname{pm}(G)=m_m(G)\) is the number of perfect matchings. The matching polynomial and its normalized root measure, recalled from the introduction, are explicitly \[ \mu_G(z)=\sum_{k=0}^m(-1)^km_k(G)z^{2m-2k}, \qquad \rho_G=\frac1{2m}\sum_{\mu_G(\xi)=0}\delta_\xi, \] with roots counted according to multiplicity. The measure's support is contained in \([-2\sqrt{d-1},2\sqrt{d-1}]\) when \(G\) has maximum degree \(d\ge2\) \cite{r10}. The basic integral identities are \[ P_G(t)=\frac12\int\log(1+tx^2)\,d\rho_G(x), \qquad \lambda_G(1)=\int\log|x|\,d\rho_G(x). \] The self-avoiding-walk tree of a graph rooted at \(v\) has one vertex for each simple path starting at \(v\), with two paths adjacent when one extends the other by one edge. For a transitive infinite graph \(L\), local convergence of finite transitive graphs means that, for every fixed radius, their rooted neighborhoods eventually agree with a rooted neighborhood of \(L\). The matching measure \(\rho_L\) is the weak limit of their matching-root measures. Define \[ P_L(t)=\frac12\int\log(1+tx^2)\,d\rho_L(x),\qquad p_L(t)=2tP_L'(t)=\int\frac{tx^2}{1+tx^2}\,d\rho_L(x), \] \(\lambda_L(p)=\inf_{t>0}[P_L(t)-(p/2)\log t]\) for \(00\) and \(\lambda_{G_j}(p)\to\lambda_L(p)\) for \(0\le p<1\) \cite[Theorem 3.3]{r7}. This limit definition of \(\lambda_L(p)\) agrees with the infimum above by the integral formulas \cite[Remark 2.7]{r8}. If the \(G_j\) are also vertex-transitive, \(d\)-regular and bipartite, then \(\lambda_{G_j}(1)\to\lambda_L(1)\) \cite[Theorem 1.8]{r8}. For \(L=\Gamma_d\), Section 4 proves \(\rho_L([-s,s])\le C_ds\). Thus \(\rho_L(\{0\})=0\) and \(p_L(t)\uparrow1\). Splitting the integral for \(1-p_L(t)\) at \(|x|=t^{-1/3}\) shows it is \(O(t^{-1/3})\). Since \[ P_L(t)-\tfrac12\log t =\tfrac12\int\log(x^2+1/t)\,d\rho_L(x), \] the root-mass bound permits passage to the limit \(t\to\infty\) in this integral. The term \((1-p_L(t))\log t\) vanishes, so the Legendre formula gives \(\lambda_L(1)=\int\log|x|\,d\rho_L(x)\), as used in the introduction. \subsection{Every circulant is a lattice quotient}\label{every-circulant-is-a-lattice-quotient} Translate a \(d\)-element support so that it is \[ S=\{0,s_1,\ldots,s_{d-1}\}\subseteq\mathbb Z_n. \] Let \(H\le\mathbb Z_n\) be the subgroup generated by the \(s_i\), and define \[ \psi:\mathbb Z^{d-1}\longrightarrow H, \qquad e_i\longmapsto s_i. \] Its kernel \(\Lambda\) is a full-rank lattice. Write \(A_\Lambda=\mathbb Z^{d-1}/\Lambda\) and let \(G_\Lambda\) have left vertices \(L_x\), right vertices \(R_x\) for \(x\in A_\Lambda\), and edges \(L_xR_{x+s}\) for \(s\in D_d\). Each connected component of the circulant bipartite graph is isomorphic to this quotient graph, whose edges retain their labels in \(D_d\). All components are isomorphic. If \(|H|=m\), there are \(n/m\) components and \[ \operatorname{per}C_n(S)^{1/n} =\operatorname{pm}(G_\Lambda)^{1/m} =\exp\bigl(2\lambda_{G_\Lambda}(1)\bigr). \] The graph \(G_0=\Gamma_d\) is a common cover of all these labelled quotients. \section{Proof of the sharp constant}\label{proof-of-the-sharp-constant} \subsection{Unwrapping finite quotients by 2-lifts}\label{unwrapping-finite-quotients-by-2-lifts} A \(2\)-lift of a graph \(G\) is a graph equipped with a projection to \(G\) such that every vertex has two preimages and the edges above each edge of \(G\) form a perfect matching between its two vertex fibers. We use the following theorem of Csikv\'ari \cite{r9}. \textbf{Bipartite \(2\)-lift inequality.} If \(\widetilde G\) is a bipartite \(2\)-lift of a finite bipartite graph \(G\), then for every \(k\), \[ m_k(\widetilde G)\le m_k(G\sqcup G). \] The coefficientwise inequality gives \(M(\widetilde G,t)\le M(G,t)^2\) for every \(t>0\). Taking pressure per vertex and then the infimum formula from Section 2.1 gives, with entropy normalized per vertex, \[ \lambda_{\widetilde G}(p)\le\lambda_G(p), \qquad 0\le p\le1. \] The endpoints follow by continuity of the finite-graph entropies. For a full-rank lattice \(\Lambda\le\mathbb Z^{d-1}\), the quotient \(\Lambda/2\Lambda\) is isomorphic to \((\mathbb Z/2\mathbb Z)^{d-1}\). Choose a composition series between \(\Lambda\) and \(2\Lambda\). For an index-two inclusion \(\Lambda'\subset\Lambda\), every vertex fiber of \(G_{\Lambda'}\to G_\Lambda\) has two elements, and each edge label \(s\in D_d\) maps that fiber bijectively to the fiber of its neighboring vertex. Thus each successive labelled quotient is a bipartite \(2\)-lift. The labels remain distinct in every refinement because they are distinct modulo \(\Lambda\). Hence \[ \lambda_{G_\Lambda}(p) \ge\lambda_{G_{2\Lambda}}(p) \ge\lambda_{G_{4\Lambda}}(p) \ge\cdots . \] Every nonzero lattice vector in \(2^j\Lambda\) has length tending to infinity with \(j\). Therefore \[ G_{2^j\Lambda}\longrightarrow\Gamma_d \] locally. Finite-activity continuity of matching entropy gives \[ \lambda_{G_\Lambda}(p)\ge\lambda_{\Gamma_d}(p), \qquad 0\le p<1. \] The quotient \(G_\Lambda\) has a perfect matching, for example the matching formed by all edges carrying the label \(0\). Thus its finite endpoint is well-defined and \(\lambda_{G_\Lambda}(p)\to\lambda_{G_\Lambda}(1)\) as \(p\uparrow1\). Letting \(p\uparrow1\) therefore yields \[ \lambda_{G_\Lambda}(1)\ge\lambda_{\Gamma_d}(1). \] By the component calculation in Section 2.2, \[ \operatorname{per}C_n(S)^{1/n}\ge h_d. \] This proves \(m_{\mathrm{circ},d}\ge h_d\). \subsection{Explicit cyclic quotients attain the limit}\label{explicit-cyclic-quotients-attain-the-limit} For an integer \(q\ge2\), put \[ n_q=q^{d-1}, \qquad S_q=\{0,1,q,q^2,\ldots,q^{d-2}\} \subseteq\mathbb Z_{q^{d-1}}. \] Let \(G_q\) be the bipartite graph of \(C_{n_q}(S_q)\). The support generates \(\mathbb Z_{n_q}\) because it contains \(1\). The lattice describing this quotient is the kernel of \[ \phi_q:\mathbb Z^{d-1}\longrightarrow\mathbb Z_{q^{d-1}}, \qquad (a_0,\ldots,a_{d-2})\longmapsto \sum_{j=0}^{d-2}a_jq^j. \] Fix \(R\). If \(q>R\), \(\sum_j|a_j|\le R\), and \(\phi_q(a)=0\), then \[ \left|\sum_j a_jq^j\right| \le Rq^{d-2}0\), the portmanteau theorem applied to \((-s-\eta,s+\eta)\) gives \[ \rho_L([-s,s]) \le \rho_L((-s-\eta,s+\eta)) \le \liminf_{j\to\infty}\rho_{G_j}((-s-\eta,s+\eta)) \le C_d(s+\eta); \] letting \(\eta\downarrow0\) gives the asserted bound for \(\rho_L\). \begin{theorem}[Finite-degree modulus]\label{thm:modulus} Let \(d\ge2\) and \(N\ge1\) be integers with \(2N<\sigma(\Lambda)\). For \(\varepsilon>0\), set \[ u=\frac\varepsilon{B_d}, \qquad r=(\sqrt{1+u^2}-u)^2. \] Then \[ \boxed{ 0\le\lambda_{G_\Lambda}(1)-\lambda_{\Gamma_d}(1) \le \frac{2r^{N+1}}{(N+1)(1-r)} +\frac{\pi C_d}{2}\varepsilon . } \] \end{theorem} \begin{proof} The covering map \(\Gamma_d\to G_\Lambda\) is injective on every rooted ball of radius \(N\), since two lifts of one ball vertex would be at distance at most \(2N<\sigma(\Lambda)\). It also preserves every edge between ball vertices: an edge from the chosen lift of one endpoint reaches a lift of the other at distance at most \(2N+1\) from its chosen lift. Because \(\sigma(\Lambda)\) is even, \(2N<\sigma(\Lambda)\) implies \(2N+1<\sigma(\Lambda)\), so these lifts coincide. Thus the rooted \(N\)-balls are isomorphic. The \(k\)th matching-measure moment counts length-\(k\) closed walks in the self-avoiding-walk tree, which remain inside radius \(\lfloor k/2\rfloor\). Hence the matching measures of \(G_\Lambda\) and \(\Gamma_d\) have equal moments through degree \(2N\). Regularize the endpoint logarithm by \[ \phi_\varepsilon(x)=\frac12\log(x^2+\varepsilon^2). \] If \(F(s)=\rho([-s,s])\le C_ds\), a layer-cake calculation gives \[ 0\le\int(\phi_\varepsilon(x)-\log|x|)\,d\rho(x) \le\frac{\pi C_d}{2}\varepsilon. \] On \([-B_d,B_d]\), the Chebyshev expansion has the form \[ \phi_\varepsilon(x)=A_0+ \sum_{k\ge1}\frac{(-1)^{k+1}r^k}{k} T_{2k}(x/B_d). \] Truncation after \(k=N\) has uniform error at most \(r^{N+1}/((N+1)(1-r))\). The two measures integrate the truncated polynomial equally, so their regularized logarithmic integrals differ by at most twice this error. The two regularization losses lie in the same interval; comparing them costs one copy of its length. The lower inequality is the finite-abelian-quotient argument of Section 3.1. Combining the estimates proves the claim. \end{proof} Choosing parameters explicitly gives a closed form. \begin{corollary}\label{cor:modulus} Put \(M=N+1\ge2\) and suppose \(2(M-1)<\sigma(\Lambda)\). Then \[ \boxed{ 0\le\lambda_{G_\Lambda}(1)-\lambda_{\Gamma_d}(1) \le \frac{\pi(d-1)}d\frac{\log M}{M} +\frac{4e^{1/100}}{M\log M}. } \] \end{corollary} To obtain this, take \(\varepsilon=B_d\log M/(2M)\), so \(u=\log M/(2M)\) and \(r=e^{-2\operatorname{arsinh}u}\). The inequalities \(\operatorname{arsinh}u\ge u-u^3/6\) and \[ \frac{(\log M)^3}{24M^2}\le\frac1{100} \] for \(M\ge2\) give \(r^M\le e^{1/100}/M\). Also \(u\le1/(2e)<1/5\), whence \[ 1-r=2u(\sqrt{1+u^2}-u)\ge u. \] Thus the Chebyshev term in Theorem 4.1 is at most \(4e^{1/100}/(M\log M)\); its smoothing term is \(\pi(d-1)\log M/(dM)\), as displayed. For the radix lattice, one has exactly \[ \sigma(\ker\phi_q)=2q. \] If \(\max\{p(a),n(a)\}0. \] Thus, without computation, \[ \operatorname{per}C_n(S) \ge\left(\frac{3125}{1296}e^{2I_6}\right)^n. \] Call \(S\) \emph{Sidon modulo \(n\)} if the ordered differences \(a-b\) for distinct \(a,b\in S\) are pairwise distinct modulo \(n\). If \(S\) is not Sidon, a repeated ordered difference gives a \(4\)-cycle and the same argument replaces \(I_6\) by \[ I_4= \frac{727}{29859840}-\frac{65\sqrt5}{5971968} =9.307059668\ldots\times10^{-9}. \] Every six-set is non-Sidon for \(n\le30\), since its \(30\) nonzero ordered differences cannot all be distinct in \(\mathbb Z_n\). \subsection{Exact moments of the generic lattice}\label{exact-moments-of-the-generic-lattice} The stronger bound in Theorem 1.2 comes from the matching measure \(\rho_{\Gamma_6}\). Its moment \(\mu_k=\int x^k\,d\rho_{\Gamma_6}(x)\) is the number of length-\(k\) closed walks from the root of the self-avoiding-walk tree of \(\Gamma_6\) \cite{r7}. Odd moments vanish. Exact enumeration gives: \begin{center} \small \setlength{\tabcolsep}{5pt} \begin{tabular}{r r @{\hspace{1.2em}} r r} \hline $2j$ & $\mu_{2j}$ & $2j$ & $\mu_{2j}$ \\ \hline 0 & 1 & 18 & 15356334036 \\ 2 & 6 & 20 & 268021382436 \\ 4 & 66 & 22 & 4731256843176 \\ 6 & 876 & 24 & 84316269917196 \\ 8 & 12786 & 26 & 1514802320358216 \\ 10 & 197796 & 28 & 27404472908976936 \\ 12 & 3183036 & 30 & 498786397086170496 \\ 14 & 52710216 & 32 & 9126710952996738306 \\ 16 & 892040706 & & \\ \hline \end{tabular} \end{center} For completeness, the recursion is as follows. In the symmetric model put \(A=\mathbb Z^d/\mathbb Z(1,\ldots,1)\) and join \(L_a\) to \(R_{a+e_i}\) for \(0\le i0\), write \[ \begin{aligned} 1+ty&=a+bz,\qquad a=1+10t,\qquad b=10t,\\ s&=\sqrt{1+20t},\qquad r=\frac{a-s}{b},\qquad C=\frac{a+s}{2}. \end{aligned} \] Uniformly for \(z\in[-1,1]\), \[ \log(a+bz)=\log C +2\sum_{k\ge1}\frac{(-1)^{k+1}r^k}{k}T_k(z). \] Truncation after \(N\) terms, where \(N\ge0\) is an integer, has uniform error at most \(2r^{N+1}/((N+1)(1-r))\). The integrals of \(T_k(z)\) for \(k\le16\) are exact rational combinations of the moments above. Square roots are enclosed by dyadic rationals and logarithms by the identity \[ \log x=2\operatorname{artanh}\frac{x-1}{x+1} \] after exact range reduction. Every series remainder is bounded by a rational geometric tail. At \(t=29/50\) and \(N=16\), outward rational decimal enclosures give \[ 0.609450453043169353759817 \frac7{500000}. } \] To convert pressure to perfect-matching entropy, let \(g_L(p)=\lambda_L(p)-\mathbb G_6(p)\). Csikv\'ari proved that \(g_G\) is nondecreasing for every finite vertex-transitive \(6\)-regular bipartite graph \(G\) \cite{r8}. This passes to \(\Gamma_6\): along the radix quotients \(G_q\to\Gamma_6\), finite-activity continuity gives \(g_{G_q}(p)\to g_{\Gamma_6}(p)\) for every \(p<1\), so the pointwise limit is nondecreasing on \([0,1)\). Since \(\lambda_{\Gamma_6}(1)\) is defined as the limit as \(p\uparrow1\) and \(\mathbb G_6\) is continuous there, this extends the monotonicity to \(p=1\). The pressure comparison converts the certificate into a lower bound on \(g_{\Gamma_6}(1)\). Put \(p=p_{\Gamma_6}(t)\). The attained infimum in Section 2.1 gives \(\lambda_{\Gamma_6}(p)=P_{\Gamma_6}(t)-(p/2)\log t\). Also \(\lambda_{\mathbb T_6}(p)=\mathbb G_6(p)\) \cite[Theorem 2.15]{r9}, while the infimum definition gives \[ P_{\mathbb T_6}(t) \ge\mathbb G_6(p)+\frac p2\log t. \] Subtracting these relations gives \[ g_{\Gamma_6}(p_{\Gamma_6}(t)) \ge P_{\Gamma_6}(t)-P_{\mathbb T_6}(t). \] Therefore \[ \lambda_{\Gamma_6}(1) \ge\mathbb G_6(1)+\frac7{500000}. \] Since \(e^{2\mathbb G_6(1)}=3125/1296\), Theorem 1.1 gives \[ h_6\ge\frac{3125}{1296}e^{7/250000} =2.41133294847608906899\ldots . \] This proves the lower half of Theorem 1.2. \section{Convergent certified evaluation of the sharp base}\label{convergent-certified-evaluation-of-h_6} The logarithm in \(\lambda_{\Gamma_6}(1)=\int\log|x|\,d\rho_{\Gamma_6}(x)\) is singular at zero. The missing endpoint control is supplied by Csikv\'ari's root estimate \cite{r8} and its radix-limit extension justified in Section 4: \[ \rho_{\Gamma_d}([-\varepsilon,\varepsilon]) \le\frac{2\sqrt{d-1}}d\varepsilon. \] For \(d=6\), \[ \rho_{\Gamma_6}([-\varepsilon,\varepsilon]) \le\frac{\sqrt5}{3}\varepsilon <\frac34\varepsilon. \] Let \[ f_\varepsilon(x)=\log\max\{|x|,\varepsilon\}. \] Integration by parts gives \[ 0\le\int f_\varepsilon\,d\rho_{\Gamma_6} -\lambda_{\Gamma_6}(1) \le\frac34\varepsilon. \] Suppose an even polynomial \(Q\) satisfies the exactly certified uniform bound \[ \|Q-f_\varepsilon\|_{\infty,[-2\sqrt5,2\sqrt5]} \le\delta. \] Writing \(I(Q)=\int Q\,d\rho_{\Gamma_6}\), which is exactly computable from finitely many moments, we obtain \[ \boxed{ I(Q)-\delta-\frac34\varepsilon \le\lambda_{\Gamma_6}(1) \le I(Q)+\delta. } \] Here the approximation can be made explicit. On \([-5,5]\), which contains the root support, \(f_\varepsilon\) is \(1/\varepsilon\)-Lipschitz. Rescale this interval to \([0,1]\). Its degree-\(K\) Bernstein polynomial has uniform error at most \(5/(\varepsilon\sqrt K)\) by the binomial variance bound. Enclose the finitely many logarithmic node values by rational intervals, choose their precision and \(K\) so the total error is at most \(\delta\), and symmetrize the resulting rational polynomial. Taking rational \(\varepsilon,\delta\to0\) proves that exact moments produce a convergent sequence of rational two-sided bounds on \(\lambda_{\Gamma_6}(1)\) and hence on \(h_6\). \section{Exact finite circulants}\label{exact-finite-circulants} \subsection{Complete affine census for orders 23--31}\label{complete-affine-census-for-23le-nle31} Supports related by \[ S\longmapsto uS+c, \qquad u\in\mathbb Z_n^\times,\quad c\in\mathbb Z_n, \] give permutation-equivalent matrices. We enumerated all \(8148\) affine classes of six-subsets for \(23\le n\le31\). The following class is the unique minimizing affine class at each order. \begin{table}[t] \centering \caption{Unique minimizing affine classes of six-element supports, orders 23 through 31.} \label{tab:census} \small \setlength{\tabcolsep}{4pt} \begin{tabular}{r r l r r} \hline $n$ & affine classes & minimizing support & exact permanent & $n$th-root rate \\ \hline 23 & 207 & $\{0,1,2,4,7,15\}$ & 3843409486 & 2.610518715946044918 \\ 24 & 870 & $\{0,1,3,5,13,18\}$ & 9368913864 & 2.603077221944179924 \\ 25 & 366 & $\{0,1,2,4,12,20\}$ & 22785392966 & 2.596009401396219872 \\ 26 & 767 & $\{0,1,2,6,10,13\}$ & 55339674228 & 2.589367634679046236 \\ 27 & 634 & $\{0,1,2,6,9,16\}$ & 134510172390 & 2.583307565903801233 \\ 28 & 1252 & $\{0,1,4,15,20,22\}$ & 326803223040 & 2.577653350057210309 \\ 29 & 598 & $\{0,1,2,5,14,23\}$ & 810203756246 & 2.574193451167119615 \\ 30 & 2643 & $\{0,1,2,5,14,24\}$ & 1934592438260 & 2.567751448390291055 \\ 31 & 811 & $\{0,1,3,8,12,18\}$ & 4598378639550 & 2.561362932953538240 \\ \hline \end{tabular} \end{table} The nine displayed rates are strictly decreasing; this was checked exactly by cross-power comparisons. The order-\(31\) support is a cyclic \((31,6,1)\) difference set: its \(30\) ordered nonzero differences exhaust \(\mathbb Z_{31}\setminus\{0\}\). Its translates are the lines of the cyclic projective plane of order \(5\), and the associated bipartite graph is its incidence graph. \subsection{The order-35 upper certificate}\label{the-order-35-upper-certificate} There are three Sidon affine classes at order \(35\). Their exact permanents are: \begin{center} \small \begin{tabular}{l r r} \hline canonical support & exact permanent & $35$th-root rate \\ \hline $\{0,1,3,7,16,24\}$ & 163812873663198 & 2.547558874426894768 \\ $\{0,1,3,7,12,22\}$ & 164058070108318 & 2.547667744065163230 \\ $\{0,1,3,7,12,20\}$ & 164070859106478 & 2.547673418168322034 \\ \hline \end{tabular} \end{center} This is a complete Sidon-class scan, not a complete scan of all \(1998\) affine classes at order \(35\). For the upper bound on \(h_6\), only the first exact evaluation is needed: \(h_6\le163812873663198^{1/35}\). Together with Section 5.3, this completes Theorem 1.2. The finite computations used two separately implemented exact permanent engines. The primary engine applies Glynn's signed formula \cite{r11} with Gray-code updates of the six affected factors; the verifier uses Ryser's all-subsets formula \cite{r12}. Both use integer arithmetic, and their absolute-sum bounds fit in signed \(128\)-bit accumulators throughout the certified range. Fresh affine enumeration, support-set comparison, exact cross-powers, and direct cycle checks on reference cases were included in the checks of Section 9. The independent review of v4 recomputed all \(8148\) census permanents with a separately written Ryser engine on independently enumerated affine classes; every value agrees with the primary Glynn census, and each order has the unique minimizing affine class shown above. It also recomputed all three order-\(35\) Sidon permanents. \section{What the result settles}\label{what-the-result-settles} The unrestricted degree-six exponential bases were already known. Problem 10's corrected full chain is impossible in both the unrestricted and circulant scopes because its upper target is too small. Its sole surviving component was to improve the lower base for circulants. Theorem 1.1 goes further: it identifies the best possible circulant base, proves that every finite circulant has root rate at least it, and supplies an explicit connected cyclic family approaching it. Theorem 1.2 gives an explicit strict numerical improvement, while Section 6 proves that the entropy description admits arbitrarily sharp exact rational evaluation in principle. Several finer questions remain. \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item Obtain a high-precision numerical value, or a closed form, for \(h_6\). \item Determine whether every nontrivial finite quotient satisfies a strict inequality above \(h_6\). \item Characterize near-minimizing sequences. The natural expectation is that every bounded-length lattice vector must disappear, equivalently that the graphs must converge locally to \(\Gamma_6\). \item Sharpen the quantitative modulus using lattice-specific information or a quantitative defect in the \(2\)-lift inequality. \end{enumerate} These do not affect the sharp exponential constant itself. \section{Reproducibility statement}\label{reproducibility-statement} The structural proofs are independent of computation. The computer-assisted part consists of exact finite recursions and rational interval certificates. The self-contained file \emph{supplement-v7.tar.gz}, distributed as ancillary material with this article, contains the source and data needed for the calculations. Its README gives exact run commands and a SHA-256 manifest. The package includes the primary Python moment recursion and rational Chebyshev certifier, their stored order-\(32\) output, a separately written C++ moment implementation and its fresh output, the Glynn and Ryser permanent engines, all \(8148\) raw affine-census records for orders \(23\) through \(31\), and all three order-\(35\) Sidon records. It also includes the independent review's separately written moment, rational-certificate, affine-enumeration, and permanent programs, together with its complete census rerun records. A quick standard-library check reconstructs the rational pressure interval from stored moments, verifies \(P_{\Gamma_6}(29/50)-P_{\mathbb T_6}(29/50)>7/500000\), checks affine support coverage and minima, and compares all \(8148\) original permanent values with the independent rerun after affine canonicalization. The original Python \emph{verify\_certificate.py} is a fresh rerun using the same moment and arithmetic modules as the primary driver; it is not a separate implementation. The C++ moment source is separate from the Python moment recursion. The separate v4 review independently reimplemented the rational arithmetic and reran the full census with its own Ryser engine; its raw values and comparison report are in the supplement. The order-\(35\) upper-certificate permanent was also independently recomputed. 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