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We have $\\Im t,\\Im s,\\Im\\delta>0$, $\\Im\\tau<0$ and $|\\rho|<1$. The last follows from $N(\\epsilon)=1$ and $0<2-7^{1/3}<1$ at the real embedding. The numbers $1,t,s$ are rationally independent: a rational linear relation becomes a polynomial of degree at most two in $\\beta$, contradicting its irreducible cubic minimal polynomial.\n\\begin{figure}[htbp]\\centering\\includegraphics[width=.96\\linewidth]{figures/01_periods.png}\\caption{The chosen cubic embedding and its actual periods. The right panel has equal aspect; rounded drawing coordinates illustrate the exact definitions in \\eqref{eq:fix","\\tau&=(\\beta^2+2\\beta)/15,&t&=-\\tau,&s&=-(\\beta+2)/15,&\n \\delta&=(\\tau+2s+4)/7.\n\\end{aligned}\n\\end{equation}\nThe real cube root $7^{1/3}$ is positive; this fixes the lower complex embedding of $\\beta$. We have $\\Im t,\\Im s,\\Im\\delta>0$, $\\Im\\tau<0$ and $|\\rho|<1$. The last follows from $N(\\epsilon)=1$ and $0<2-7^{1/3}<1$ at the real embedding. The numbers $1,t,s$ are rationally independent: a rational linear relation becomes a polynomial of degree at most two in $\\beta$, contradicting its irreducible cubic minimal polynomial.\n\\begin{figure}[htbp]\\centering\\includegraphics[width=.96\\linewidth]{figures/01_periods.png}\\caption{The chosen cubic embedding and its actual periods. The right panel has equal aspect; rounded drawing coordinates illustrate the exact definitions in \\eqref{eq:fixed-data}.}\\label{fig:periods}\\end{figure}","au+2s+4)/7.\n\\end{aligned}\n\\end{equation}\nThe real cube root $7^{1/3}$ is positive; this fixes the lower complex embedding of $\\beta$. We have $\\Im t,\\Im s,\\Im\\delta>0$, $\\Im\\tau<0$ and $|\\rho|<1$. The last follows from $N(\\epsilon)=1$ and $0<2-7^{1/3}<1$ at the real embedding. The numbers $1,t,s$ are rationally independent: a rational linear relation becomes a polynomial of degree at most two in $\\beta$, contradicting its irreducible cubic minimal polynomial.\n\\begin{figure}[htbp]\\centering\\includegraphics[width=.96\\linewidth]{figures/01_periods.png}\\caption{The chosen cubic embedding and its actual periods. The right panel has equal aspect; rounded drawing coordinates illustrate the exact definitions in \\eqref{eq:fixed-data}.}\\label{fig:periods}\\end{figure}","For positive imaginary periods use the additive product convention\n\\begin{align}\n \\theta_0(z;a)&=\\prod_{n\\ge0}(1-e^{2\\pi i(z+na)})(1-e^{2\\pi i((n+1)a-z)}),\\label{eq:theta-product}\\\\\n \\Gamma_{\\rm ell}(z;a,b)&=\\prod_{j,k\\ge0}\n \\frac{1-e^{2\\pi i((j+1)a+(k+1)b-z)}}{1-e^{2\\pi i(ja+kb+z)}}.\\label{eq:gamma-product}\n\\end{align}\nThe negative-first-period continuation is\n\\[\n \\Gamma_{\\rm ell}(z;-t,s)=\\Gamma_{\\rm ell}(z+t;t,s)^{-1}.\n\\]\nConsequently the main scalar function is\n\\[\n F(z)=\\frac{\\Gamma_{\\rm ell}(z;\\tau,s)^5}{\\Gamma_{\\rm ell}(5z;5\\tau,5s)}\n =\\frac{\\Gamma_{\\rm ell}(5(z+t);5t,5s)}{\\Gamma_{\\rm ell}(z+t;t,s)^5},\n \\qquad U_\\Gamma=F(1/3).\n\\]\nThis convention agrees with the checked primary authors' copy of Bergeron--Charollois--Garc\\'ia \\cite{BCG}. All analytic identities at removable or polar points mean meromorphic identities. Pointwise divisions require nonzero denominators.","\\Gamma_{\\rm ell}(z;a,b)&=\\prod_{j,k\\ge0}\n \\frac{1-e^{2\\pi i((j+1)a+(k+1)b-z)}}{1-e^{2\\pi i(ja+kb+z)}}.\\label{eq:gamma-product}\n\\end{align}\nThe negative-first-period continuation is\n\\[\n \\Gamma_{\\rm ell}(z;-t,s)=\\Gamma_{\\rm ell}(z+t;t,s)^{-1}.\n\\]\nConsequently the main scalar function is\n\\[\n F(z)=\\frac{\\Gamma_{\\rm ell}(z;\\tau,s)^5}{\\Gamma_{\\rm ell}(5z;5\\tau,5s)}\n =\\frac{\\Gamma_{\\rm ell}(5(z+t);5t,5s)}{\\Gamma_{\\rm ell}(z+t;t,s)^5},\n \\qquad U_\\Gamma=F(1/3).\n\\]\nThis convention agrees with the checked primary authors' copy of Bergeron--Charollois--Garc\\'ia \\cite{BCG}. All analytic identities at removable or polar points mean meromorphic identities. Pointwise divisions require nonzero denominators.","Put $q_a(z)=\\theta_0(z;a)^5/\\theta_0(5z;5a)$. Direct product identities and the inherited modular normalization give\n\\begin{equation}\\label{eq:F-identities}\n\\begin{aligned}\n F(z+1)&=F(z),&\\frac{F(z+\\tau)}{F(z)}&=q_s(z),&\n \\frac{F(z+s)}{F(z)}&=q_t(z)^{-1},","Put $q_a(z)=\\theta_0(z;a)^5/\\theta_0(5z;5a)$. Direct product identities and the inherited modular normalization give\n\\begin{equation}\\label{eq:F-identities}\n\\begin{aligned}\n F(z+1)&=F(z),&\\frac{F(z+\\tau)}{F(z)}&=q_s(z),&\n \\frac{F(z+s)}{F(z)}&=q_t(z)^{-1},\\\\\n q_s(z)&=q_t(\\rho z),&F(0)&=\\epsilon^2,&F(1/3)F(2/3)&=1.\n\\end{aligned}\n\\end{equation}\nThe two elliptic lattices are homothetic, $\\rho\\Lambda_s=\\Lambda_t$, with $\\Lambda_a=\\mathbb Z+\\mathbb Za$. This is not an endomorphism of the single elliptic lattice $\\Lambda_t$. Section~\\ref{sec:theta} proves the exact rational fifth-torsion normalization and explains the covariance without omitting its scalar.","q_s(z)&=q_t(\\rho z),&F(0)&=\\epsilon^2,&F(1/3)F(2/3)&=1.\n\\end{aligned}\n\\end{equation}\nThe two elliptic lattices are homothetic, $\\rho\\Lambda_s=\\Lambda_t$, with $\\Lambda_a=\\mathbb Z+\\mathbb Za$. This is not an endomorphism of the single elliptic lattice $\\Lambda_t$. Section~\\ref{sec:theta} proves the exact rational fifth-torsion normalization and explains the covariance without omitting its scalar.","We use $F_0=\\mathbb Q(\\omega)$ for the imaginary-quadratic field, to separate it from the scalar gamma function $F$. The eta quotient is $\\mathcal H(a)=\\eta(5a)/\\eta(a)^5$; the integer step function $H(n)$ used in the divisor formula is a different object. The symbol $R_0$ is the scalar anchor, while $R_*$ is a counting cutoff radius. In the arithmetic block $\\varrho$ is a ray character and a radical lift, as explicitly indicated there; it is distinct from the contracting period multiplier $\\rho$.","gamma function $F$. The eta quotient is $\\mathcal H(a)=\\eta(5a)/\\eta(a)^5$; the integer step function $H(n)$ used in the divisor formula is a different object. The symbol $R_0$ is the scalar anchor, while $R_*$ is a counting cutoff radius. In the arithmetic block $\\varrho$ is a ray character and a radical lift, as explicitly indicated there; it is distinct from the contracting period multiplier $\\rho$.","\\subsection{The arithmetic candidate is fixed independently}\nLet $J=2+3\\omega$ and choose the Gauss branch\n\\[\n X=\\beta^{-1}\\sum_{a=1}^{6}\\chi_7(a)e^{6\\pi ia/7},\\qquad\n (\\chi_7(1),\\ldots,\\chi_7(6))=(1,\\omega,\\omega^2,\\omega^2,\\omega,1),\\quad X^3=J.\n\\]\nThen $E=\\mathbb Q(\\omega,\\beta,X)$ has degree eighteen. The retained arithmetic construction defines\n\\begin{align*}\n \\nu&=(X+\\beta/X-\\beta^2)/3,&v&=-\\beta^2\\nu-2\\nu^2,","\\subsection{The arithmetic candidate is fixed independently}\nLet $J=2+3\\omega$ and choose the Gauss branch\n\\[\n X=\\beta^{-1}\\sum_{a=1}^{6}\\chi_7(a)e^{6\\pi ia/7},\\qquad\n (\\chi_7(1),\\ldots,\\chi_7(6))=(1,\\omega,\\omega^2,\\omega^2,\\omega,1),\\quad X^3=J.\n\\]\nThen $E=\\mathbb Q(\\omega,\\beta,X)$ has degree eighteen. The retained arithmetic construction defines\n\\begin{align*}\n \\nu&=(X+\\beta/X-\\beta^2)/3,&v&=-\\beta^2\\nu-2\\nu^2,\\\\\n Y&=(4\\beta^2+\\beta-14)v^2+(11\\beta-23)v-\\beta^2+3\\beta-3,\\\\\n Z&=\\frac{(-2\\beta^2+7\\beta-8)v^2+(-10\\beta^2+11\\beta+17)v}{3}\n       +7-\\beta-\\beta^2,\\\\\n U_{\\rm alg}&=\\frac{Y-(1+2\\omega)Z}{2}.\n\\end{align*}\nThese formulas choose no coefficient or branch by fitting $U_\\Gamma$. The original desired identity is $U_\\Gamma=U_{\\rm alg}$. The established modulus comparison is $|U_\\Gamma|=|U_{\\rm alg}|$. The remaining exact phase comparison is stated in Section~\\ref{sec:remaining}; neither numerical proximity nor the analytic reconstruction supplies it.","U_{\\rm alg}&=\\frac{Y-(1+2\\omega)Z}{2}.\n\\end{align*}\nThese formulas choose no coefficient or branch by fitting $U_\\Gamma$. The original desired identity is $U_\\Gamma=U_{\\rm alg}$. The established modulus comparison is $|U_\\Gamma|=|U_{\\rm alg}|$. The remaining exact phase comparison is stated in Section~\\ref{sec:remaining}; neither numerical proximity nor the analytic reconstruction supplies it.","its bytes with the inaccessible v2 artifact. Product/continuation conventions, Theorem~3.2 and the fixed example in Section~5.2 are the controlling uses.\n\\bibitem{Milne} J.~S.~Milne, \\emph{Class Field Theory}, version~4.03, August~6, 2020. Cubic reciprocity and Artin induction in Chapter~VIII. \\url{https://www.jmilne.org/math/CourseNotes/CFTc.pdf}.\n\\bibitem{Cogdell} J.~W.~Cogdell, \\emph{On Artin $L$-functions}, author's PDF, Section~2.2, local inertia-invariant Euler factors and induction compatibility. \\url{https://math.osu.edu/~cogdell/artin-www.pdf}.\n\\bibitem{Sutherland} A.~V.~Sutherland, \\emph{18.785 Lecture~19: The analytic class number formula}, November~15, 2016. Residue formula and completed zeta functional equation. MIT course notes. \\url{https://math.mit.edu/classes/18.785/2016fa/LectureNotes19.pdf}.\n\\bibitem{DLMFTheta} NIST Digital Library of Mathematical Functions, Section~21.","Stark modulus & Conductor $3J$, character normalization, Epstein derivative and unit norm comparison & Modulus equality does not determine complex phase","\\subsection{The scalar missing from a closed finite orbit}\nThe earlier index-23 and index-10 orbit decompositions are $529=1+528$ and $100=1+3+8\\cdot12$. They are exact finite permutation data. For values $z_j$ on a cycle of length $d$, replacing every value by $cz_j$ leaves all adjacent ratios unchanged, including their product one, but multiplies the cycle product by $c^d$. An anchor on another fixed-point orbit does not remove this freedom. Thus orbit closure alone does not evaluate a primitive analytic sector or fix its scalar. The modulus proof instead uses Artin induction, the regulator and normalized CM Green products; the complex phase still requires an additional comparison.","\\subsection{Three geometries must stay distinct}\nThe scalar plane carries the meromorphic correction and its canonical divisor product. The intermediate cover $X=\\mathbb C^3/\\mathbb Z^3$ carries periodic multipliers and a degree-seven principal cover. The compact CM threefold has lattice $\\mathbb Z[\\omega]^3$ and degree 49 for the same matrix. Finite projective representations or oriented torsion sectors can encode cocycle data in the compact model. They do not alone provide a scalar meromorphic section with the divisor of $R$, a periodic relative multiplier or a based arithmetic identification. The final regularizer resolves the first of these scalar-plane requirements while leaving the others explicit.","duct. The intermediate cover $X=\\mathbb C^3/\\mathbb Z^3$ carries periodic multipliers and a degree-seven principal cover. The compact CM threefold has lattice $\\mathbb Z[\\omega]^3$ and degree 49 for the same matrix. Finite projective representations or oriented torsion sectors can encode cocycle data in the compact model. They do not alone provide a scalar meromorphic section with the divisor of $R$, a periodic relative multiplier or a based arithmetic identification. The final regularizer resolves the first of these scalar-plane requirements while leaving the others explicit.","\\subsection{Exactly normalized elliptic norms}\nOn $E_t=\\mathbb C/\\Lambda_t$ define $N_nf(w)=\\prod_{nz=w}f(z)$ for integers $n\\ge1$, with all $n^2$ preimages counted. The product is independent of the representative lift. Root-of-unity averaging and vertical residue splitting in \\eqref{eq:theta-product} give\n\\begin{equation}\\label{eq:qnorm}\n N_nq_t(w)=\\begin{cases}q_t(w),&5\\nmid n,","\\subsection{Exactly normalized elliptic norms}\nOn $E_t=\\mathbb C/\\Lambda_t$ define $N_nf(w)=\\prod_{nz=w}f(z)$ for integers $n\\ge1$, with all $n^2$ preimages counted. The product is independent of the representative lift. Root-of-unity averaging and vertical residue splitting in \\eqref{eq:theta-product} give\n\\begin{equation}\\label{eq:qnorm}\n N_nq_t(w)=\\begin{cases}q_t(w),&5\\nmid n,\\\\1,&5\\mid n.\\end{cases}\n\\end{equation}\nIn the prime-to-five case the two product identities are\n\\[\n \\prod_{a=0}^{n-1}\\theta_0(z+a/n;t)=\\theta_0(nz;nt),\\qquad\n \\prod_{b=0}^{n-1}\\theta_0(nz+bt;nt)=\\theta_0(nz;t).\n\\]\nMultiplication by five permutes the real residues in the smoothing denominator. When $5\\mid n$, real averaging retains only frequencies already deleted by fifth smoothing. This proves the scalar one in both cases, rather than a norm formula up to a root of unity.","1,&5\\mid n.\\end{cases}\n\\end{equation}\nIn the prime-to-five case the two product identities are\n\\[\n \\prod_{a=0}^{n-1}\\theta_0(z+a/n;t)=\\theta_0(nz;nt),\\qquad\n \\prod_{b=0}^{n-1}\\theta_0(nz+bt;nt)=\\theta_0(nz;t).\n\\]\nMultiplication by five permutes the real residues in the smoothing denominator. When $5\\mid n$, real averaging retains only frequencies already deleted by fifth smoothing. This proves the scalar one in both cases, rather than a norm formula up to a root of unity.","The point $S=s\\bmod\\Lambda_t$ has infinite order by rational independence. Put\n\\[\n \\psi(w)=\\frac{q_t(w)^2}{q_t(2w)q_t(2w+s)}.\n\\]\n\\begin{proposition}[All finite norms of the return kernel]\\label{prop:allnorms}\nFor every integer $n\\ge1$,\n\\[\n N_n\\psi(w)=\n \\begin{cases}\n 1,&5\\mid n,","The point $S=s\\bmod\\Lambda_t$ has infinite order by rational independence. Put\n\\[\n \\psi(w)=\\frac{q_t(w)^2}{q_t(2w)q_t(2w+s)}.\n\\]\n\\begin{proposition}[All finite norms of the return kernel]\\label{prop:allnorms}\nFor every integer $n\\ge1$,\n\\[\n N_n\\psi(w)=\n \\begin{cases}\n 1,&5\\mid n,\\\\[2pt]\n \\displaystyle\\frac{q_t(w)^2}{q_t(2w)q_t(2w+ns)},&5\\nmid n,\\ n\\text{ odd},\\\\[7pt]\n \\displaystyle\\frac1{q_t(w)^2q_t(w+ks)^4},&5\\nmid n,\\ n=2k.\n \\end{cases}\n\\]\n\\end{proposition}\n\\begin{proof}\nFor odd $n$, doubling permutes $E_t[n]$. For $n=2k$, it maps $E_t[n]$ to $E_t[k]$ with four lifts per point, so each denominator norm is a fourth power. Apply \\eqref{eq:qnorm} and cancel the numerator. If $5\\mid n$ then all these norms are one, including the even case since $5\\mid k$.\n\\end{proof}\nFor $n=2k+1$ prime to five, $B_k(w)=\\prod_{j=0}^{k-1}q_t(2w+(2j+1)s)$ gives\n\\[\n N_{2k+1}\\psi(w)=\\psi(w)\\frac{B_k(w)}{B_k(w+s)}.\n\\]\nFor $n=2k$ prime to five, $C_k(w)=\\prod_{j=0}^{k-1}q_t(w+js)^4$ gives\n\\[\n N_{2k}\\psi(w)=q_t(w)^{-6}\\frac{C_k(w)}{C_k(w+s)}.\n\\]\nFinite products therefore retain explicit endpoint multipliers. Clo…","[7pt]\n \\displaystyle\\frac1{q_t(w)^2q_t(w+ks)^4},&5\\nmid n,\\ n=2k.\n \\end{cases}\n\\]\n\\end{proposition}\n\\begin{proof}\nFor odd $n$, doubling permutes $E_t[n]$. For $n=2k$, it maps $E_t[n]$ to $E_t[k]$ with four lifts per point, so each denominator norm is a fourth power. Apply \\eqref{eq:qnorm} and cancel the numerator. If $5\\mid n$ then all these norms are one, including the even case since $5\\mid k$.\n\\end{proof}\nFor $n=2k+1$ prime to five, $B_k(w)=\\prod_{j=0}^{k-1}q_t(2w+(2j+1)s)$ gives\n\\[\n N_{2k+1}\\psi(w)=\\psi(w)\\frac{B_k(w)}{B_k(w+s)}.\n\\]\nFor $n=2k$ prime to five, $C_k(w)=\\prod_{j=0}^{k-1}q_t(w+js)^4$ gives\n\\[\n N_{2k}\\psi(w)=q_t(w)^{-6}\\frac{C_k(w)}{C_k(w+s)}.\n\\]\nFinite products therefore retain explicit endpoint multipliers. Closing a permutation of labels does not remove them or fix the scalar normalization of a","[7pt]\n \\displaystyle\\frac1{q_t(w)^2q_t(w+ks)^4},&5\\nmid n,\\ n=2k.\n \\end{cases}\n\\]\n\\end{proposition}\n\\begin{proof}\nFor odd $n$, doubling permutes $E_t[n]$. For $n=2k$, it maps $E_t[n]$ to $E_t[k]$ with four lifts per point, so each denominator norm is a fourth power. Apply \\eqref{eq:qnorm} and cancel the numerator. If $5\\mid n$ then all these norms are one, including the even case since $5\\mid k$.\n\\end{proof}\nFor $n=2k+1$ prime to five, $B_k(w)=\\prod_{j=0}^{k-1}q_t(2w+(2j+1)s)$ gives\n\\[\n N_{2k+1}\\psi(w)=\\psi(w)\\frac{B_k(w)}{B_k(w+s)}.\n\\]\nFor $n=2k$ prime to five, $C_k(w)=\\prod_{j=0}^{k-1}q_t(w+js)^4$ gives\n\\[\n N_{2k}\\psi(w)=q_t(w)^{-6}\\frac{C_k(w)}{C_k(w+s)}.\n\\]\nFinite products therefore retain explicit endpoint multipliers. Closing a permutation of labels does not remove them or fix the scalar normalization of a separate orbit.","yle\\frac1{q_t(w)^2q_t(w+ks)^4},&5\\nmid n,\\ n=2k.\n \\end{cases}\n\\]\n\\end{proposition}\n\\begin{proof}\nFor odd $n$, doubling permutes $E_t[n]$. For $n=2k$, it maps $E_t[n]$ to $E_t[k]$ with four lifts per point, so each denominator norm is a fourth power. Apply \\eqref{eq:qnorm} and cancel the numerator. If $5\\mid n$ then all these norms are one, including the even case since $5\\mid k$.\n\\end{proof}\nFor $n=2k+1$ prime to five, $B_k(w)=\\prod_{j=0}^{k-1}q_t(2w+(2j+1)s)$ gives\n\\[\n N_{2k+1}\\psi(w)=\\psi(w)\\frac{B_k(w)}{B_k(w+s)}.\n\\]\nFor $n=2k$ prime to five, $C_k(w)=\\prod_{j=0}^{k-1}q_t(w+js)^4$ gives\n\\[\n N_{2k}\\psi(w)=q_t(w)^{-6}\\frac{C_k(w)}{C_k(w+s)}.\n\\]\nFinite products therefore retain explicit endpoint multipliers. Closing a permutation of labels does not remove them or fix the scalar normalization of a separate orbit.","ks)^4},&5\\nmid n,\\ n=2k.\n \\end{cases}\n\\]\n\\end{proposition}\n\\begin{proof}\nFor odd $n$, doubling permutes $E_t[n]$. For $n=2k$, it maps $E_t[n]$ to $E_t[k]$ with four lifts per point, so each denominator norm is a fourth power. Apply \\eqref{eq:qnorm} and cancel the numerator. If $5\\mid n$ then all these norms are one, including the even case since $5\\mid k$.\n\\end{proof}\nFor $n=2k+1$ prime to five, $B_k(w)=\\prod_{j=0}^{k-1}q_t(2w+(2j+1)s)$ gives\n\\[\n N_{2k+1}\\psi(w)=\\psi(w)\\frac{B_k(w)}{B_k(w+s)}.\n\\]\nFor $n=2k$ prime to five, $C_k(w)=\\prod_{j=0}^{k-1}q_t(w+js)^4$ gives\n\\[\n N_{2k}\\psi(w)=q_t(w)^{-6}\\frac{C_k(w)}{C_k(w+s)}.\n\\]\nFinite products therefore retain explicit endpoint multipliers. Closing a permutation of labels does not remove them or fix the scalar normalization of a separate orbit.","position}\n\\begin{proof}\nFor odd $n$, doubling permutes $E_t[n]$. For $n=2k$, it maps $E_t[n]$ to $E_t[k]$ with four lifts per point, so each denominator norm is a fourth power. Apply \\eqref{eq:qnorm} and cancel the numerator. If $5\\mid n$ then all these norms are one, including the even case since $5\\mid k$.\n\\end{proof}\nFor $n=2k+1$ prime to five, $B_k(w)=\\prod_{j=0}^{k-1}q_t(2w+(2j+1)s)$ gives\n\\[\n N_{2k+1}\\psi(w)=\\psi(w)\\frac{B_k(w)}{B_k(w+s)}.\n\\]\nFor $n=2k$ prime to five, $C_k(w)=\\prod_{j=0}^{k-1}q_t(w+js)^4$ gives\n\\[\n N_{2k}\\psi(w)=q_t(w)^{-6}\\frac{C_k(w)}{C_k(w+s)}.\n\\]\nFinite products therefore retain explicit endpoint multipliers. Closing a permutation of labels does not remove them or fix the scalar normalization of a separate orbit.","t[n]$. For $n=2k$, it maps $E_t[n]$ to $E_t[k]$ with four lifts per point, so each denominator norm is a fourth power. Apply \\eqref{eq:qnorm} and cancel the numerator. If $5\\mid n$ then all these norms are one, including the even case since $5\\mid k$.\n\\end{proof}\nFor $n=2k+1$ prime to five, $B_k(w)=\\prod_{j=0}^{k-1}q_t(2w+(2j+1)s)$ gives\n\\[\n N_{2k+1}\\psi(w)=\\psi(w)\\frac{B_k(w)}{B_k(w+s)}.\n\\]\nFor $n=2k$ prime to five, $C_k(w)=\\prod_{j=0}^{k-1}q_t(w+js)^4$ gives\n\\[\n N_{2k}\\psi(w)=q_t(w)^{-6}\\frac{C_k(w)}{C_k(w+s)}.\n\\]\nFinite products therefore retain explicit endpoint multipliers. Closing a permutation of labels does not remove them or fix the scalar normalization of a separate orbit.","maps $E_t[n]$ to $E_t[k]$ with four lifts per point, so each denominator norm is a fourth power. Apply \\eqref{eq:qnorm} and cancel the numerator. If $5\\mid n$ then all these norms are one, including the even case since $5\\mid k$.\n\\end{proof}\nFor $n=2k+1$ prime to five, $B_k(w)=\\prod_{j=0}^{k-1}q_t(2w+(2j+1)s)$ gives\n\\[\n N_{2k+1}\\psi(w)=\\psi(w)\\frac{B_k(w)}{B_k(w+s)}.\n\\]\nFor $n=2k$ prime to five, $C_k(w)=\\prod_{j=0}^{k-1}q_t(w+js)^4$ gives\n\\[\n N_{2k}\\psi(w)=q_t(w)^{-6}\\frac{C_k(w)}{C_k(w+s)}.\n\\]\nFinite products therefore retain explicit endpoint multipliers. Closing a permutation of labels does not remove them or fix the scalar normalization of a separate orbit.","point, so each denominator norm is a fourth power. Apply \\eqref{eq:qnorm} and cancel the numerator. If $5\\mid n$ then all these norms are one, including the even case since $5\\mid k$.\n\\end{proof}\nFor $n=2k+1$ prime to five, $B_k(w)=\\prod_{j=0}^{k-1}q_t(2w+(2j+1)s)$ gives\n\\[\n N_{2k+1}\\psi(w)=\\psi(w)\\frac{B_k(w)}{B_k(w+s)}.\n\\]\nFor $n=2k$ prime to five, $C_k(w)=\\prod_{j=0}^{k-1}q_t(w+js)^4$ gives\n\\[\n N_{2k}\\psi(w)=q_t(w)^{-6}\\frac{C_k(w)}{C_k(w+s)}.\n\\]\nFinite products therefore retain explicit endpoint multipliers. Closing a permutation of labels does not remove them or fix the scalar normalization of a separate orbit.","\\subsection{The exact information lost by doubling}\nQuotient nonzero meromorphic elliptic functions by constants and translation coboundaries $A(w+s)/A(w)$. Set $e=[q_t]$, $p=[\\psi]$ in additive notation. These generate a free $\\mathbb Z^2$: the divisor charges on the disjoint infinite translation orbits of $0$ and $-S/2$ are respectively $(4,4)$ and $(0,-4)$. A finite divisor coboundary has charge zero on every orbit by telescoping, so a relation $ae+bp=0$ forces $b=0$ then $a=0$.","uotient nonzero meromorphic elliptic functions by constants and translation coboundaries $A(w+s)/A(w)$. Set $e=[q_t]$, $p=[\\psi]$ in additive notation. These generate a free $\\mathbb Z^2$: the divisor charges on the disjoint infinite translation orbits of $0$ and $-S/2$ are respectively $(4,4)$ and $(0,-4)$. A finite divisor coboundary has charge zero on every orbit by telescoping, so a relation $ae+bp=0$ forces $b=0$ then $a=0$.","eromorphic elliptic functions by constants and translation coboundaries $A(w+s)/A(w)$. Set $e=[q_t]$, $p=[\\psi]$ in additive notation. These generate a free $\\mathbb Z^2$: the divisor charges on the disjoint infinite translation orbits of $0$ and $-S/2$ are respectively $(4,4)$ and $(0,-4)$. A finite divisor coboundary has charge zero on every orbit by telescoping, so a relation $ae+bp=0$ forces $b=0$ then $a=0$.","elliptic functions by constants and translation coboundaries $A(w+s)/A(w)$. Set $e=[q_t]$, $p=[\\psi]$ in additive notation. These generate a free $\\mathbb Z^2$: the divisor charges on the disjoint infinite translation orbits of $0$ and $-S/2$ are respectively $(4,4)$ and $(0,-4)$. A finite divisor coboundary has charge zero on every orbit by telescoping, so a relation $ae+bp=0$ forces $b=0$ then $a=0$.","The norm matrices on this free subgroup are\n\\[\n N_n=I\\quad(n\\text{ odd},5\\nmid n),\\qquad\n N_n=\\begin{pmatrix}1&-6\\\\0&0\\end{pmatrix}\\quad(n\\text{ even},5\\nmid n),\\qquad\n N_n=0\\quad(5\\mid n).\n\\]\nThe middle matrix is a split integral projector with image $\\mathbb Z(1,0)$ and kernel $\\mathbb Z(6,1)$. Its primitive kernel class $6e+p$ is nonzero, as its two charges are $28,-4$. Thus repeated full doubling cannot recover the deleted component. The residue-ten matrix law is multiplicative; its finite coefficient statement belongs to the inherited formal core, whereas the application of divisors to analytic functions is an ordinary argument.","0&0\\end{pmatrix}\\quad(n\\text{ even},5\\nmid n),\\qquad\n N_n=0\\quad(5\\mid n).\n\\]\nThe middle matrix is a split integral projector with image $\\mathbb Z(1,0)$ and kernel $\\mathbb Z(6,1)$. Its primitive kernel class $6e+p$ is nonzero, as its two charges are $28,-4$. Thus repeated full doubling cannot recover the deleted component. The residue-ten matrix law is multiplicative; its finite coefficient statement belongs to the inherited formal core, whereas the application of divisors to analytic functions is an ordinary argument.","\\subsection{The fifth-isogeny differential retains its constant}\nLet $a=\\wp_t(1/5)$, $b=\\wp_t(2/5)$ and $L_t=(q_t'/q_t)'$. The enlarged lattice is $\\Lambda_t^+=\\tfrac15\\mathbb Z+\\mathbb Zt$, with $\\wp_+(z)=25\\wp_{5t}(5z)$. The degree-five quotient satisfies\n\\begin{equation}\\label{eq:isogeny-differential}\n L_t(z)=\\wp_+(z)-5\\wp_t(z)+2(a+b).\n\\end{equation}\nAt zero $L_t$ has principal part $-4/z^2$; at the four nonzero kernel points it has $+1/(z-P_j)^2$, with no simple-pole terms. Thus $L_t-\\wp_++5\\wp_t$ is a constant elliptic function. The canonical formula gives $q_t(z)=(a-b)^","\\subsection{The fifth-isogeny differential retains its constant}\nLet $a=\\wp_t(1/5)$, $b=\\wp_t(2/5)$ and $L_t=(q_t'/q_t)'$. The enlarged lattice is $\\Lambda_t^+=\\tfrac15\\mathbb Z+\\mathbb Zt$, with $\\wp_+(z)=25\\wp_{5t}(5z)$. The degree-five quotient satisfies\n\\begin{equation}\\label{eq:isogeny-differential}\n L_t(z)=\\wp_+(z)-5\\wp_t(z)+2(a+b).\n\\end{equation}\nAt zero $L_t$ has principal part $-4/z^2$; at the four nonzero kernel points it has $+1/(z-P_j)^2$, with no simple-pole terms. Thus $L_t-\\wp_++5\\wp_t$ is a constant elliptic function. The canonical formula gives $q_t(z)=(a-b)^2z^4[1+(a+b)z^2+","\\subsection{The fifth-isogeny differential retains its constant}\nLet $a=\\wp_t(1/5)$, $b=\\wp_t(2/5)$ and $L_t=(q_t'/q_t)'$. The enlarged lattice is $\\Lambda_t^+=\\tfrac15\\mathbb Z+\\mathbb Zt$, with $\\wp_+(z)=25\\wp_{5t}(5z)$. The degree-five quotient satisfies\n\\begin{equation}\\label{eq:isogeny-differential}\n L_t(z)=\\wp_+(z)-5\\wp_t(z)+2(a+b).\n\\end{equation}\nAt zero $L_t$ has principal part $-4/z^2$; at the four nonzero kernel points it has $+1/(z-P_j)^2$, with no simple-pole terms. Thus $L_t-\\wp_++5\\wp_t$ is a constant elliptic function. The canonical formula gives $q_t(z)=(a-b)^2z^4[1+(a+b)z^2+O(z^4)]$, fixing that","\\subsection{The fifth-isogeny differential retains its constant}\nLet $a=\\wp_t(1/5)$, $b=\\wp_t(2/5)$ and $L_t=(q_t'/q_t)'$. The enlarged lattice is $\\Lambda_t^+=\\tfrac15\\mathbb Z+\\mathbb Zt$, with $\\wp_+(z)=25\\wp_{5t}(5z)$. The degree-five quotient satisfies\n\\begin{equation}\\label{eq:isogeny-differential}\n L_t(z)=\\wp_+(z)-5\\wp_t(z)+2(a+b).\n\\end{equation}\nAt zero $L_t$ has principal part $-4/z^2$; at the four nonzero kernel points it has $+1/(z-P_j)^2$, with no simple-pole terms. Thus $L_t-\\wp_++5\\wp_t$ is a constant elliptic function. The canonical formula gives $q_t(z)=(a-b)^2z^4[1+(a+b)z^2+O(z^4)]$, fixing that constant as $2(a+b)$. The degree-five quotient is equivalently the p","\\subsection{The fifth-isogeny differential retains its constant}\nLet $a=\\wp_t(1/5)$, $b=\\wp_t(2/5)$ and $L_t=(q_t'/q_t)'$. The enlarged lattice is $\\Lambda_t^+=\\tfrac15\\mathbb Z+\\mathbb Zt$, with $\\wp_+(z)=25\\wp_{5t}(5z)$. The degree-five quotient satisfies\n\\begin{equation}\\label{eq:isogeny-differential}\n L_t(z)=\\wp_+(z)-5\\wp_t(z)+2(a+b).\n\\end{equation}\nAt zero $L_t$ has principal part $-4/z^2$; at the four nonzero kernel points it has $+1/(z-P_j)^2$, with no simple-pole terms. Thus $L_t-\\wp_++5\\wp_t$ is a constant elliptic function. The canonical formula gives $q_t(z)=(a-b)^2z^4[1+(a+b)z^2+O(z^4)]$, fixing that constant as $2(a+b)$. The degree-five quotient is equivalently the paired sum\n\\[\n \\wp_+(z)=\\wp_t(z)+","\\subsection{The fifth-isogeny differential retains its constant}\nLet $a=\\wp_t(1/5)$, $b=\\wp_t(2/5)$ and $L_t=(q_t'/q_t)'$. The enlarged lattice is $\\Lambda_t^+=\\tfrac15\\mathbb Z+\\mathbb Zt$, with $\\wp_+(z)=25\\wp_{5t}(5z)$. The degree-five quotient satisfies\n\\begin{equation}\\label{eq:isogeny-differential}\n L_t(z)=\\wp_+(z)-5\\wp_t(z)+2(a+b).\n\\end{equation}\nAt zero $L_t$ has principal part $-4/z^2$; at the four nonzero kernel points it has $+1/(z-P_j)^2$, with no simple-pole terms. Thus $L_t-\\wp_++5\\wp_t$ is a constant elliptic function. The canonical formula gives $q_t(z)=(a-b)^2z^4[1+(a+b)z^2+O(z^4)]$, fixing that constant as $2(a+b)$. The degree-five quotient is equivalently the paired sum\n\\[\n \\wp_+(z)=\\wp_t(z)+\\sum_{j=1}^{4}[\\wp_t(z+j/5)-\\wp_t(j/5)].\n\\]\nThe analytic identity determines a rational isogeny map once elliptic coordin","\\subsection{The fifth-isogeny differential retains its constant}\nLet $a=\\wp_t(1/5)$, $b=\\wp_t(2/5)$ and $L_t=(q_t'/q_t)'$. The enlarged lattice is $\\Lambda_t^+=\\tfrac15\\mathbb Z+\\mathbb Zt$, with $\\wp_+(z)=25\\wp_{5t}(5z)$. The degree-five quotient satisfies\n\\begin{equation}\\label{eq:isogeny-differential}\n L_t(z)=\\wp_+(z)-5\\wp_t(z)+2(a+b).\n\\end{equation}\nAt zero $L_t$ has principal part $-4/z^2$; at the four nonzero kernel points it has $+1/(z-P_j)^2$, with no simple-pole terms. Thus $L_t-\\wp_++5\\wp_t$ is a constant elliptic function. The canonical formula gives $q_t(z)=(a-b)^2z^4[1+(a+b)z^2+O(z^4)]$, fixing that constant as $2(a+b)$. The degree-five quotient is equivalently the paired sum\n\\[\n \\wp_+(z)=\\wp_t(z)+\\sum_{j=1}^{4}[\\wp_t(z+j/5)-\\wp_t(j/5)].\n\\]\nThe analytic identity determines a rational isogeny map once elliptic coordinates are chosen. It does no","\\subsection{The fifth-isogeny differential retains its constant}\nLet $a=\\wp_t(1/5)$, $b=\\wp_t(2/5)$ and $L_t=(q_t'/q_t)'$. The enlarged lattice is $\\Lambda_t^+=\\tfrac15\\mathbb Z+\\mathbb Zt$, with $\\wp_+(z)=25\\wp_{5t}(5z)$. The degree-five quotient satisfies\n\\begin{equation}\\label{eq:isogeny-differential}\n L_t(z)=\\wp_+(z)-5\\wp_t(z)+2(a+b).\n\\end{equation}\nAt zero $L_t$ has principal part $-4/z^2$; at the four nonzero kernel points it has $+1/(z-P_j)^2$, with no simple-pole terms. Thus $L_t-\\wp_++5\\wp_t$ is a constant elliptic function. The canonical formula gives $q_t(z)=(a-b)^2z^4[1+(a+b)z^2+O(z^4)]$, fixing that constant as $2(a+b)$. The degree-five quotient is equivalently the paired sum\n\\[\n \\wp_+(z)=\\wp_t(z)+\\sum_{j=1}^{4}[\\wp_t(z+j/5)-\\wp_t(j/5)].\n\\]\nThe analytic identity determines a rational isogeny map once elliptic coordinates are chosen. It does not assert that the period-dep","l retains its constant}\nLet $a=\\wp_t(1/5)$, $b=\\wp_t(2/5)$ and $L_t=(q_t'/q_t)'$. The enlarged lattice is $\\Lambda_t^+=\\tfrac15\\mathbb Z+\\mathbb Zt$, with $\\wp_+(z)=25\\wp_{5t}(5z)$. The degree-five quotient satisfies\n\\begin{equation}\\label{eq:isogeny-differential}\n L_t(z)=\\wp_+(z)-5\\wp_t(z)+2(a+b).\n\\end{equation}\nAt zero $L_t$ has principal part $-4/z^2$; at the four nonzero kernel points it has $+1/(z-P_j)^2$, with no simple-pole terms. Thus $L_t-\\wp_++5\\wp_t$ is a constant elliptic function. The canonical formula gives $q_t(z)=(a-b)^2z^4[1+(a+b)z^2+O(z^4)]$, fixing that constant as $2(a+b)$. The degree-five quotient is equivalently the paired sum\n\\[\n \\wp_+(z)=\\wp_t(z)+\\sum_{j=1}^{4}[\\wp_t(z+j/5)-\\wp_t(j/5)].\n\\]\nThe analytic identity determines a rational isogeny map once elliptic coordinates are chosen. It does not assert that the period-dependent parameter is an algebraic number or identify this","wp_t(2/5)$ and $L_t=(q_t'/q_t)'$. The enlarged lattice is $\\Lambda_t^+=\\tfrac15\\mathbb Z+\\mathbb Zt$, with $\\wp_+(z)=25\\wp_{5t}(5z)$. The degree-five quotient satisfies\n\\begin{equation}\\label{eq:isogeny-differential}\n L_t(z)=\\wp_+(z)-5\\wp_t(z)+2(a+b).\n\\end{equation}\nAt zero $L_t$ has principal part $-4/z^2$; at the four nonzero kernel points it has $+1/(z-P_j)^2$, with no simple-pole terms. Thus $L_t-\\wp_++5\\wp_t$ is a constant elliptic function. The canonical formula gives $q_t(z)=(a-b)^2z^4[1+(a+b)z^2+O(z^4)]$, fixing that constant as $2(a+b)$. The degree-five quotient is equivalently the paired sum\n\\[\n \\wp_+(z)=\\wp_t(z)+\\sum_{j=1}^{4}[\\wp_t(z+j/5)-\\wp_t(j/5)].\n\\]\nThe analytic identity determines a rational isogeny map once elliptic coordinates are chosen. It does not assert that the period-dependent parameter is an algebraic number or identify this cubic-period curve with the separate equianharmonic","mathbb Z+\\mathbb Zt$, with $\\wp_+(z)=25\\wp_{5t}(5z)$. The degree-five quotient satisfies\n\\begin{equation}\\label{eq:isogeny-differential}\n L_t(z)=\\wp_+(z)-5\\wp_t(z)+2(a+b).\n\\end{equation}\nAt zero $L_t$ has principal part $-4/z^2$; at the four nonzero kernel points it has $+1/(z-P_j)^2$, with no simple-pole terms. Thus $L_t-\\wp_++5\\wp_t$ is a constant elliptic function. The canonical formula gives $q_t(z)=(a-b)^2z^4[1+(a+b)z^2+O(z^4)]$, fixing that constant as $2(a+b)$. The degree-five quotient is equivalently the paired sum\n\\[\n \\wp_+(z)=\\wp_t(z)+\\sum_{j=1}^{4}[\\wp_t(z+j/5)-\\wp_t(j/5)].\n\\]\nThe analytic identity determines a rational isogeny map once elliptic coordinates are chosen. It does not assert that the period-dependent parameter is an algebraic number or identify this cubic-period curve with the separate equianharmonic CM curve.","-five quotient satisfies\n\\begin{equation}\\label{eq:isogeny-differential}\n L_t(z)=\\wp_+(z)-5\\wp_t(z)+2(a+b).\n\\end{equation}\nAt zero $L_t$ has principal part $-4/z^2$; at the four nonzero kernel points it has $+1/(z-P_j)^2$, with no simple-pole terms. Thus $L_t-\\wp_++5\\wp_t$ is a constant elliptic function. The canonical formula gives $q_t(z)=(a-b)^2z^4[1+(a+b)z^2+O(z^4)]$, fixing that constant as $2(a+b)$. The degree-five quotient is equivalently the paired sum\n\\[\n \\wp_+(z)=\\wp_t(z)+\\sum_{j=1}^{4}[\\wp_t(z+j/5)-\\wp_t(j/5)].\n\\]\nThe analytic identity determines a rational isogeny map once elliptic coordinates are chosen. It does not assert that the period-dependent parameter is an algebraic number or identify this cubic-period curve with the separate equianharmonic CM curve.","tial}\n L_t(z)=\\wp_+(z)-5\\wp_t(z)+2(a+b).\n\\end{equation}\nAt zero $L_t$ has principal part $-4/z^2$; at the four nonzero kernel points it has $+1/(z-P_j)^2$, with no simple-pole terms. Thus $L_t-\\wp_++5\\wp_t$ is a constant elliptic function. The canonical formula gives $q_t(z)=(a-b)^2z^4[1+(a+b)z^2+O(z^4)]$, fixing that constant as $2(a+b)$. The degree-five quotient is equivalently the paired sum\n\\[\n \\wp_+(z)=\\wp_t(z)+\\sum_{j=1}^{4}[\\wp_t(z+j/5)-\\wp_t(j/5)].\n\\]\nThe analytic identity determines a rational isogeny map once elliptic coordinates are chosen. It does not assert that the period-dependent parameter is an algebraic number or identify this cubic-period curve with the separate equianharmonic CM curve.","\\[\n \\omega^2+\\omega+1=0,\\quad J=2+3\\omega,\\quad\n \\beta^3=7,\\quad \\xi^3=J.\n\\]","Use the fixed embedding \\(\\omega=e^{2\\pi i/3}\\),\n\\(\\beta=7^{1/3}\\omega^2\\), and the retained Gauss branch","\\[\n \\xi=\\beta^{-1}\\sum_{k=1}^6\\chi_7(k)e^{6\\pi i k/7},\n \\qquad (\\chi_7(1),\\ldots,\\chi_7(6))=(1,\\omega,\\omega^2,\\omega^2,\\omega,1).\n\\]","\\[\n E=\\mathbb Q(\\omega,\\beta,\\xi),\\quad F_0=\\mathbb Q(\\omega),\n \\qquad \\operatorname{Gal}(E/F_0)=C_3^2.\n\\]","The actual arithmetic candidate \\(U=U_{\\rm alg}\\) is defined by the\nretained rational formulas in \\nolinkurl{prior/ARITHMETIC_INPUT.md}. This\nblock starts from those formulas. No gamma value is used to find or fit\ncoefficients.","\\[\n a_0=(1+\\omega)\\xi^2-\\xi+1,\\quad\n a_1=-\\xi^2+(1+\\omega)\\xi+1,\\quad\n a_2=-\\omega\\xi^2-\\omega\\xi+1.\n\\]","\\[\n a_0a_1a_2=1,\\qquad a_j^3-3a_j^2+6\\omega a_j-1=0.\n\\]","\\[\n \\epsilon_j=2-\\omega^j\\beta,\\qquad \\epsilon=\\epsilon_0,\n \\qquad \\epsilon_0\\epsilon_1\\epsilon_2=1.\n\\]","In the fixed embedding \\(\\epsilon_1=2-7^{1/3}>0\\), and\n\\(\\epsilon_2=\\overline\\epsilon\\).","\\[\n \\iota(\\beta)=\\beta,\\quad \\iota(\\omega)=\\omega^2,\\quad \\iota(\\xi)=\\beta/\\xi,\n\\]","whereas actual complex conjugation \\(\\kappa\\) is","\\[\n \\kappa(\\beta)=\\omega^2\\beta,\\quad \\kappa(\\omega)=\\omega^2,\n \\quad \\kappa(\\xi)=\\omega\\beta/\\xi.\n\\]","\\[\n \\boxed{\\kappa(a_j)=\\iota(a_{j+1})},\\qquad j\\pmod3. \\tag{Q1}\n\\]","All displayed algebraic identities are checked in an 18-dimensional\nrational quotient implementation. Construction of \\(U\\), the cubic\ncoordinate, and the \\(a_j\\) is also independently repeated by SymPy\nGroebner reduction over \\(\\mathbb Q\\).","Let \\(\\gamma_{ij}\\) fix \\(\\omega\\) and send","\\[\n \\beta\\mapsto\\omega^i\\beta,\\qquad \\xi\\mapsto\\omega^j\\xi.\n\\]","There are four order-three subgroups of \\(C_3^2\\), generated by","\\[\n (1,0),\\ (0,1),\\ (1,1),\\ (1,2).\n\\]","Every nonidentity group element lies in exactly one of these four\nsubgroups; the identity lies in all four. Hence, for any nonzero\n\\(z\\in E\\),","\\[\n \\boxed{\n \\prod_{H\\text{ a line}}N_{E/E^H}(z)\n   =z^3N_{E/F_0}(z).\n } \\tag{Q2}\n\\]","\\[\n \\mathcal A=a_0a_1^3=\\frac{a_1^2}{a_2},\\qquad\n \\mathcal B=a_0^2a_1^4=\\left(\\frac{a_1}{a_2}\\right)^2.\n\\]","For the actual \\(U\\), exact field arithmetic gives:","\\midrule\\noalign{}\n\\endhead\n\\bottomrule\\noalign{}\n\\endlastfoot\n(1,0) & \\(F_0(\\xi)\\) & \\(\\omega\\mathcal A\\)","(0,1) & \\(F_0(\\beta)\\) & \\(\\omega\\epsilon_2/\\epsilon_1\\)","(1,1) & \\(F_0(\\beta/\\xi)\\) & \\(\\omega/\\iota(\\mathcal A)\\)","(1,2) & \\(F_0(\\beta\\xi)\\) & \\(\\omega\\)","Also \\(N_{E/F_0}(U)=1\\). Thus (Q2) proves","\\[\n \\boxed{\n U_{\\rm alg}^3=\\omega\\frac{\\epsilon_2}{\\epsilon_1}\n                 \\frac{\\mathcal A}{\\iota(\\mathcal A)}.\n } \\tag{Q3}\n\\]","The same calculation for the inherited pre-smoothing unit \\(W\\) gives\nline norms","\\[\n -\\mathcal B,\\quad -\\epsilon_2/\\epsilon_1,\\quad\n -1/\\iota(\\mathcal B),\\quad -1,\n\\]","and total norm \\(-1\\). Consequently","\\[\n \\boxed{W^3=-\\frac{\\epsilon_2}{\\epsilon_1}\n                   \\frac{\\mathcal B}{\\iota(\\mathcal B)}.} \\tag{Q4}\n\\]","The normalized polynomial \\(U_{21}(\\beta,\\xi,\\omega)=21U\\) is explicitly\ndefined in \\texttt{H12CMNormCertificate.lean}. It is the normal form of\nthe inherited candidate, as checked in both rational implementations.\nAlso","\\[\n \\mathcal A=4(1+\\omega)\\xi^2-(6\\omega+4)\\xi+1,\n\\]","\\[\n 7\\iota(\\mathcal A)\n =-8\\beta^2\\omega\\xi-12\\beta^2\\xi\n    +6\\beta\\omega\\xi^2+16\\beta\\xi^2+7.\n\\]","\\[\n U_{21}^3(2-\\omega\\beta)(7\\iota\\mathcal A)\n =64827\\,\\omega[2+(1+\\omega)\\beta]\\mathcal A,\n\\]","using only \\(\\beta^3=7\\), \\(\\xi^3=2+3\\omega\\), and\n\\(\\omega^2+\\omega+1=0\\). Here \\(64827=21^3\\cdot7\\). Five small\npolynomial reductions replace one resource-heavy monolithic certificate.\nThe field corollary gives (Q3) with explicit nonzero denominator\nhypotheses.","The geometric field interpretation and the conversion from the original\nrational Gauss formula to \\(U_{21}\\) are not silently assumed as axioms\nin Lean: the formal statement is about that explicit polynomial\npresentation, while the conversion is verified by the two separate exact\narithmetic implementations.","\\[\n \\iota(\\mathcal A)=\\frac{\\kappa(a_0)^2}{\\kappa(a_1)},\n\\]","and \\(a_0a_1a_2=1\\) therefore gives","\\[\n \\frac{\\mathcal A}{\\iota\\mathcal A}\n  =\\frac{a_1^2\\overline{a_1}^{\\,3}\\overline{a_2}^{\\,2}}{a_2}.\n\\]","Moreover \\(\\epsilon_2/\\epsilon_1=\\epsilon\\overline\\epsilon^{\\,2}\\).\nEquation (Q3) becomes","\\[\n U_{\\rm alg}^3=\n \\omega\\epsilon\\overline\\epsilon^{\\,2}\n \\frac{a_1^2\\overline{a_1}^{\\,3}\\overline{a_2}^{\\,2}}{a_2}. \\tag{Q5}\n\\]","\\[\n \\boxed{C=\\frac{\\epsilon a_1a_2^3}{\\omega}}\n\\]","\\[\n R_+=|\\epsilon|^4|a_1|^6|a_2|^4.\n\\]","\\[\n \\boxed{C U_{\\rm alg}^3=R_+>0.} \\tag{Q6}\n\\]","The positivity uses actual complex conjugation, not \\(\\iota\\). The\npolynomial factor rearrangement and positivity implication are checked\nin Lean, with their hypotheses exposed.","\\[\n C\\approx13.80641265880838653-46.68563220754137422i,\n\\]","\\[\n R_+\\approx3619549.58981670163340845644692077486836844.\n\\]","\\[\n |U_\\Gamma|^3=|\\epsilon^3a_2a_1^5|.\n\\]","Deleting absolute-value signs would assert a false identity even for\n\\(U_{\\rm alg}\\). In fact","\\[\n \\boxed{\n \\frac{U_{\\rm alg}^3}{\\epsilon^3a_2a_1^5}\n =\\omega\\left(\\frac{\\overline\\epsilon}{\\epsilon}\\right)^2\n       \\left(\\frac{\\overline{a_1}}{a_1}\\right)^3\n       \\left(\\frac{\\overline{a_2}}{a_2}\\right)^2.\n } \\tag{Q7}\n\\]","The right side has modulus one but is not in \\(\\mu_6\\): the rational\nquotient calculation proves its sixth power differs from one. Its value\nis approximately","\\[\n 0.2625544763772951843-0.9649171710225931849i.\n\\]","We do not claim here that it is not a root of unity of any order. The\nverified negative statement is precisely that it is not a sixth root of\nunity. This correction is different from the still-unidentified\ndiscrepancy \\(U_\\Gamma/U_{\\rm alg}\\).","Use the inherited exact modulus theorem \\(|U_\\Gamma|=|U_{\\rm alg}|\\). It\nis not re-proved in the corresponding source checkpoint. From (Q6),","\\[\n |C U_\\Gamma^3|=R_+.\n\\]","\\[\n 3.6\\cdot10^6<\\Re(C U_\\Gamma^3)<3.7\\cdot10^6.\n\\]","\\[\n \\boxed{\\Im(C U_\\Gamma^3)=0} \\tag{Q8}\n\\]","would imply \\(C U_\\Gamma^3=R_+\\), then \\(U_\\Gamma^3=U_{\\rm alg}^3\\). The\nretained historical checkpoint distance bound \\(|U_\\Gamma/U_{\\rm alg}-1|<1/2\\)\nwould select \\(U_\\Gamma=U_{\\rm alg}\\).","\\[\n \\overline{C U_\\Gamma^3}=C U_\\Gamma^3,\n \\qquad\n \\left(\\frac{U_\\Gamma}{\\overline{U_\\Gamma}}\\right)^3\n   =\\frac{\\overline C}{C}. \\tag{Q9}\n\\]","\\[\n \\overline{F_{t,s}(z)}=F_{-\\bar t,-\\bar s}(-\\bar z).\n\\]","\\[\n F_{-\\bar t,-\\bar s}(1/3)=\\overline{U_\\Gamma}^{\\,-1}.\n\\]","\\subsection{The seven-point product}\nUse the fixed periods of Section~\\ref{sec:setting}. Define\n\\[\n S_5(z;a,b)=\\frac{\\Gamma_{\\rm ell}(z;a,b)^5}\n {\\Gamma_{\\rm ell}(5z;5a,5b)},\\qquad\n F(z)=S_5(z+t;t,s)^{-1}.\n\\]\nAll identities below are meromorphic identities; their quotients at apparent exceptional points mean meromorphic continuation. Let\n\\[\n \\Lambda_5=\\mathbb Z\\tau+\\mathbb Zs+\\tfrac15\\mathbb Z,\n \\qquad \\delta=\\frac{\\tau+2s+4}{7}.\n\\]\nThe period identities, obtained from $\\beta^3=7$, are\n\\begin{equation}\\label{eq:period-identities}\n\\begin{aligned}\n \\beta\\tau&=2\\tau+4s+1,&\\beta s&=-\\tau,&\\beta&=-15s-2,","sec:setting}. Define\n\\[\n S_5(z;a,b)=\\frac{\\Gamma_{\\rm ell}(z;a,b)^5}\n {\\Gamma_{\\rm ell}(5z;5a,5b)},\\qquad\n F(z)=S_5(z+t;t,s)^{-1}.\n\\]\nAll identities below are meromorphic identities; their quotients at apparent exceptional points mean meromorphic continuation. Let\n\\[\n \\Lambda_5=\\mathbb Z\\tau+\\mathbb Zs+\\tfrac15\\mathbb Z,\n \\qquad \\delta=\\frac{\\tau+2s+4}{7}.\n\\]\nThe period identities, obtained from $\\beta^3=7$, are\n\\begin{equation}\\label{eq:period-identities}\n\\begin{aligned}\n \\beta\\tau&=2\\tau+4s+1,&\\beta s&=-\\tau,&\\beta&=-15s-2,",",b)=\\frac{\\Gamma_{\\rm ell}(z;a,b)^5}\n {\\Gamma_{\\rm ell}(5z;5a,5b)},\\qquad\n F(z)=S_5(z+t;t,s)^{-1}.\n\\]\nAll identities below are meromorphic identities; their quotients at apparent exceptional points mean meromorphic continuation. Let\n\\[\n \\Lambda_5=\\mathbb Z\\tau+\\mathbb Zs+\\tfrac15\\mathbb Z,\n \\qquad \\delta=\\frac{\\tau+2s+4}{7}.\n\\]\nThe period identities, obtained from $\\beta^3=7$, are\n\\begin{equation}\\label{eq:period-identities}\n\\begin{aligned}\n \\beta\\tau&=2\\tau+4s+1,&\\beta s&=-\\tau,&\\beta&=-15s-2,\\\\\n \\beta\\delta&=-8s-1,&7\\delta&=\\tau+2s+4.\n\\end{aligned}\n\\end{equation}\nThe seven-point product and its correction are\n\\begin{equation}\\label{eq:p7}\n P_7(z)=\\prod_{k=0}^{6}F(z+k\\delta)\n =\\frac{S_5(z;s,\\delta)}{S_5(z;t,\\delta)S_5(z+s;t,\\delta)},\n \\qquad R(z)=\\frac{P_7(z)}{F(\\beta z)}.\n\\end{equation}\nThe product identity is an actual gamma multiplication identity, not the assertion $P_7(z)=F(\\beta z)$. For completeness, its elementary product proof is retained from v0.12: rewrite $t=2s+4-7\\delta$, regard periods modulo integers, and perform real root averaging and vertical residue splitting in t…","\\beta\\delta&=-8s-1,&7\\delta&=\\tau+2s+4.\n\\end{aligned}\n\\end{equation}\nThe seven-point product and its correction are\n\\begin{equation}\\label{eq:p7}\n P_7(z)=\\prod_{k=0}^{6}F(z+k\\delta)\n =\\frac{S_5(z;s,\\delta)}{S_5(z;t,\\delta)S_5(z+s;t,\\delta)},\n \\qquad R(z)=\\frac{P_7(z)}{F(\\beta z)}.\n\\end{equation}\nThe product identity is an actual gamma multiplication identity, not the assertion $P_7(z)=F(\\beta z)$. For completeness, its elementary product proof is retained from v0.12: rewrite $t=2s+4-7\\delta$, regard periods modulo integers, and perform real root averaging and vertical residue splitting in the defining gamma products. The resulting split factors are precisely $S_5(z;s,\\delta)$, $S_5(z;t,\\delta)^{-1}$ and $S_5(z+s;t,\\delta)^{-1}$. The full source derivation and its positive-period convention are reproduced in Appendix~\\ref{sec:seven-product-proof}. No scalar inferred from a numerical sample is inserted.","int product and its correction are\n\\begin{equation}\\label{eq:p7}\n P_7(z)=\\prod_{k=0}^{6}F(z+k\\delta)\n =\\frac{S_5(z;s,\\delta)}{S_5(z;t,\\delta)S_5(z+s;t,\\delta)},\n \\qquad R(z)=\\frac{P_7(z)}{F(\\beta z)}.\n\\end{equation}\nThe product identity is an actual gamma multiplication identity, not the assertion $P_7(z)=F(\\beta z)$. For completeness, its elementary product proof is retained from v0.12: rewrite $t=2s+4-7\\delta$, regard periods modulo integers, and perform real root averaging and vertical residue splitting in the defining gamma products. The resulting split factors are precisely $S_5(z;s,\\delta)$, $S_5(z;t,\\delta)^{-1}$ and $S_5(z+s;t,\\delta)^{-1}$. The full source derivation and its positive-period convention are reproduced in Appendix~\\ref{sec:seven-product-proof}. No scalar inferred from a numerical sample is inserted.","{F(\\beta z)}.\n\\end{equation}\nThe product identity is an actual gamma multiplication identity, not the assertion $P_7(z)=F(\\beta z)$. For completeness, its elementary product proof is retained from v0.12: rewrite $t=2s+4-7\\delta$, regard periods modulo integers, and perform real root averaging and vertical residue splitting in the defining gamma products. The resulting split factors are precisely $S_5(z;s,\\delta)$, $S_5(z;t,\\delta)^{-1}$ and $S_5(z+s;t,\\delta)^{-1}$. The full source derivation and its positive-period convention are reproduced in Appendix~\\ref{sec:seven-product-proof}. No scalar inferred from a numerical sample is inserted.","equation}\nThe product identity is an actual gamma multiplication identity, not the assertion $P_7(z)=F(\\beta z)$. For completeness, its elementary product proof is retained from v0.12: rewrite $t=2s+4-7\\delta$, regard periods modulo integers, and perform real root averaging and vertical residue splitting in the defining gamma products. The resulting split factors are precisely $S_5(z;s,\\delta)$, $S_5(z;t,\\delta)^{-1}$ and $S_5(z+s;t,\\delta)^{-1}$. The full source derivation and its positive-period convention are reproduced in Appendix~\\ref{sec:seven-product-proof}. No scalar inferred from a numerical sample is inserted.","ity is an actual gamma multiplication identity, not the assertion $P_7(z)=F(\\beta z)$. For completeness, its elementary product proof is retained from v0.12: rewrite $t=2s+4-7\\delta$, regard periods modulo integers, and perform real root averaging and vertical residue splitting in the defining gamma products. The resulting split factors are precisely $S_5(z;s,\\delta)$, $S_5(z;t,\\delta)^{-1}$ and $S_5(z+s;t,\\delta)^{-1}$. The full source derivation and its positive-period convention are reproduced in Appendix~\\ref{sec:seven-product-proof}. No scalar inferred from a numerical sample is inserted.","\\par\n\\begin{figure}[htbp]\\centering\\includegraphics[width=.92\\linewidth]{figures/04_seven_cover.png}\\caption{The exact action of the inverse unit on the seven cover labels is $k\\mapsto4k\\pmod7$. The diagram places the fixed label and two three-cycles schematically; representative transport retains its theta carries.}\\label{fig:cover}\\end{figure}\n\\par","\\subsection{Base divisor and the integral lift}\nDefine\n\\[\n H(n)=\\mathbf1_{\\{n>0\\}},\\qquad\n w(c)=\\begin{cases}4,&5\\mid c,\\\\-1,&5\\nmid c.\\end{cases}\n\\]\nThe function $H$ here is an integer step function, distinct from the eta quotient $\\mathcal H$ and the locally defined CM Green quantity $H(z)$ used elsewhere.","-1,&5\\nmid c.\\end{cases}\n\\]\nThe function $H$ here is an integer step function, distinct from the eta quotient $\\mathcal H$ and the locally defined CM Green quantity $H(z)$ used elsewhere.","\\begin{theorem}[Base half-space divisor]\\label{thm:base-divisor}\nThe divisor of $F$ is supported on $\\Lambda_5$. At its uniquely parametrized point,\n\\begin{equation}\\label{eq:base-divisor}\n \\operatorname{ord}_{a\\tau+bs+c/5}F=w(c)\\,[H(a)-H(b)].\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor positive periods $t,s$, the defining gamma product has a simple zero at $jt+ks+\\mathbb Z$ for $j,k\\ge1$, a simple pole for $j,k\\le0$, and no divisor in the mixed cones. Rational independence of $1,t,s$ prevents collisions. The argument $z+t$ has coordinates $(1-a,b)$ in the $(t,s)$ basis, giving gamma order $H(b)-H(a)$. In\n\\[\n F(z)=\\frac{\\Gamma_{\\rm ell}(5(z+t);5t,5s)}\n {\\Gamma_{\\rm ell}(z+t;t,s)^5},\n\\]\nthe numerator contributes for every integer $c$ and the denominator only when $5\\mid","\\begin{theorem}[Base half-space divisor]\\label{thm:base-divisor}\nThe divisor of $F$ is supported on $\\Lambda_5$. At its uniquely parametrized point,\n\\begin{equation}\\label{eq:base-divisor}\n \\operatorname{ord}_{a\\tau+bs+c/5}F=w(c)\\,[H(a)-H(b)].\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor positive periods $t,s$, the defining gamma product has a simple zero at $jt+ks+\\mathbb Z$ for $j,k\\ge1$, a simple pole for $j,k\\le0$, and no divisor in the mixed cones. Rational independence of $1,t,s$ prevents collisions. The argument $z+t$ has coordinates $(1-a,b)$ in the $(t,s)$ basis, giving gamma order $H(b)-H(a)$. In\n\\[\n F(z)=\\frac{\\Gamma_{\\rm ell}(5(z+t);5t,5s)}\n {\\Gamma_{\\rm ell}(z+t;t,s)^5},\n\\]\nthe numerator contributes for every integer $c$ and the denominator only when $5\\mid c$. Their difference is \\eq","\\begin{theorem}[Base half-space divisor]\\label{thm:base-divisor}\nThe divisor of $F$ is supported on $\\Lambda_5$. At its uniquely parametrized point,\n\\begin{equation}\\label{eq:base-divisor}\n \\operatorname{ord}_{a\\tau+bs+c/5}F=w(c)\\,[H(a)-H(b)].\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor positive periods $t,s$, the defining gamma product has a simple zero at $jt+ks+\\mathbb Z$ for $j,k\\ge1$, a simple pole for $j,k\\le0$, and no divisor in the mixed cones. Rational independence of $1,t,s$ prevents collisions. The argument $z+t$ has coordinates $(1-a,b)$ in the $(t,s)$ basis, giving gamma order $H(b)-H(a)$. In\n\\[\n F(z)=\\frac{\\Gamma_{\\rm ell}(5(z+t);5t,5s)}\n {\\Gamma_{\\rm ell}(z+t;t,s)^5},\n\\]\nthe numerator contributes for every integer $c$ and the denominator only when $5\\mid c$. Their difference is \\eqref{eq:base-divisor}. The same product lists all possible zeros and poles.\n\\end{proof}\nThe rank-three additive group $\\Lambda_5\\subset","\\begin{theorem}[Base half-space divisor]\\label{thm:base-divisor}\nThe divisor of $F$ is supported on $\\Lambda_5$. At its uniquely parametrized point,\n\\begin{equation}\\label{eq:base-divisor}\n \\operatorname{ord}_{a\\tau+bs+c/5}F=w(c)\\,[H(a)-H(b)].\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor positive periods $t,s$, the defining gamma product has a simple zero at $jt+ks+\\mathbb Z$ for $j,k\\ge1$, a simple pole for $j,k\\le0$, and no divisor in the mixed cones. Rational independence of $1,t,s$ prevents collisions. The argument $z+t$ has coordinates $(1-a,b)$ in the $(t,s)$ basis, giving gamma order $H(b)-H(a)$. In\n\\[\n F(z)=\\frac{\\Gamma_{\\rm ell}(5(z+t);5t,5s)}\n {\\Gamma_{\\rm ell}(z+t;t,s)^5},\n\\]\nthe numerator contributes for every integer $c$ and the denominator only when $5\\mid c$. Their difference is \\eqref{eq:base-divisor}. The same product lists all possible zeros and poles.\n\\end{proof}\nThe rank-three additive group $\\Lambda_5\\subset\\mathbb C$ is nondiscrete. This does not make the divisor dense: its","\\begin{theorem}[Base half-space divisor]\\label{thm:base-divisor}\nThe divisor of $F$ is supported on $\\Lambda_5$. At its uniquely parametrized point,\n\\begin{equation}\\label{eq:base-divisor}\n \\operatorname{ord}_{a\\tau+bs+c/5}F=w(c)\\,[H(a)-H(b)].\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor positive periods $t,s$, the defining gamma product has a simple zero at $jt+ks+\\mathbb Z$ for $j,k\\ge1$, a simple pole for $j,k\\le0$, and no divisor in the mixed cones. Rational independence of $1,t,s$ prevents collisions. The argument $z+t$ has coordinates $(1-a,b)$ in the $(t,s)$ basis, giving gamma order $H(b)-H(a)$. In\n\\[\n F(z)=\\frac{\\Gamma_{\\rm ell}(5(z+t);5t,5s)}\n {\\Gamma_{\\rm ell}(z+t;t,s)^5},\n\\]\nthe numerator contributes for every integer $c$ and the denominator only when $5\\mid c$. Their difference is \\eqref{eq:base-divisor}. The same product lists all possible zeros and poles.\n\\end{proof}\nThe rank-three additive group $\\Lambda_5\\subset\\mathbb C$ is nondiscrete. This does not make the divisor dense: its selected nonzero coefficients have locally finite support.","[Base half-space divisor]\\label{thm:base-divisor}\nThe divisor of $F$ is supported on $\\Lambda_5$. At its uniquely parametrized point,\n\\begin{equation}\\label{eq:base-divisor}\n \\operatorname{ord}_{a\\tau+bs+c/5}F=w(c)\\,[H(a)-H(b)].\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor positive periods $t,s$, the defining gamma product has a simple zero at $jt+ks+\\mathbb Z$ for $j,k\\ge1$, a simple pole for $j,k\\le0$, and no divisor in the mixed cones. Rational independence of $1,t,s$ prevents collisions. The argument $z+t$ has coordinates $(1-a,b)$ in the $(t,s)$ basis, giving gamma order $H(b)-H(a)$. In\n\\[\n F(z)=\\frac{\\Gamma_{\\rm ell}(5(z+t);5t,5s)}\n {\\Gamma_{\\rm ell}(z+t;t,s)^5},\n\\]\nthe numerator contributes for every integer $c$ and the denominator only when $5\\mid c$. Their difference is \\eqref{eq:base-divisor}. The same product lists all possible zeros and poles.\n\\end{proof}\nThe rank-three additive group $\\Lambda_5\\subset\\mathbb C$ is nondiscrete. This does not make the divisor dense: its selected nonzero coefficients have locally finite support.","supported on $\\Lambda_5$. At its uniquely parametrized point,\n\\begin{equation}\\label{eq:base-divisor}\n \\operatorname{ord}_{a\\tau+bs+c/5}F=w(c)\\,[H(a)-H(b)].\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor positive periods $t,s$, the defining gamma product has a simple zero at $jt+ks+\\mathbb Z$ for $j,k\\ge1$, a simple pole for $j,k\\le0$, and no divisor in the mixed cones. Rational independence of $1,t,s$ prevents collisions. The argument $z+t$ has coordinates $(1-a,b)$ in the $(t,s)$ basis, giving gamma order $H(b)-H(a)$. In\n\\[\n F(z)=\\frac{\\Gamma_{\\rm ell}(5(z+t);5t,5s)}\n {\\Gamma_{\\rm ell}(z+t;t,s)^5},\n\\]\nthe numerator contributes for every integer $c$ and the denominator only when $5\\mid c$. Their difference is \\eqref{eq:base-divisor}. The same product lists all possible zeros and poles.\n\\end{proof}\nThe rank-three additive group $\\Lambda_5\\subset\\mathbb C$ is nondiscrete. This does not make the divisor dense: its selected nonzero coefficients have locally finite support.","parametrized point,\n\\begin{equation}\\label{eq:base-divisor}\n \\operatorname{ord}_{a\\tau+bs+c/5}F=w(c)\\,[H(a)-H(b)].\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor positive periods $t,s$, the defining gamma product has a simple zero at $jt+ks+\\mathbb Z$ for $j,k\\ge1$, a simple pole for $j,k\\le0$, and no divisor in the mixed cones. Rational independence of $1,t,s$ prevents collisions. The argument $z+t$ has coordinates $(1-a,b)$ in the $(t,s)$ basis, giving gamma order $H(b)-H(a)$. In\n\\[\n F(z)=\\frac{\\Gamma_{\\rm ell}(5(z+t);5t,5s)}\n {\\Gamma_{\\rm ell}(z+t;t,s)^5},\n\\]\nthe numerator contributes for every integer $c$ and the denominator only when $5\\mid c$. Their difference is \\eqref{eq:base-divisor}. The same product lists all possible zeros and poles.\n\\end{proof}\nThe rank-three additive group $\\Lambda_5\\subset\\mathbb C$ is nondiscrete. This does not make the divisor dense: its selected nonzero coefficients have locally finite support.","begin{equation}\\label{eq:base-divisor}\n \\operatorname{ord}_{a\\tau+bs+c/5}F=w(c)\\,[H(a)-H(b)].\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor positive periods $t,s$, the defining gamma product has a simple zero at $jt+ks+\\mathbb Z$ for $j,k\\ge1$, a simple pole for $j,k\\le0$, and no divisor in the mixed cones. Rational independence of $1,t,s$ prevents collisions. The argument $z+t$ has coordinates $(1-a,b)$ in the $(t,s)$ basis, giving gamma order $H(b)-H(a)$. In\n\\[\n F(z)=\\frac{\\Gamma_{\\rm ell}(5(z+t);5t,5s)}\n {\\Gamma_{\\rm ell}(z+t;t,s)^5},\n\\]\nthe numerator contributes for every integer $c$ and the denominator only when $5\\mid c$. Their difference is \\eqref{eq:base-divisor}. The same product lists all possible zeros and poles.\n\\end{proof}\nThe rank-three additive group $\\Lambda_5\\subset\\mathbb C$ is nondiscrete. This does not make the divisor dense: its selected nonzero coefficients have locally finite support.","abel{eq:base-divisor}\n \\operatorname{ord}_{a\\tau+bs+c/5}F=w(c)\\,[H(a)-H(b)].\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor positive periods $t,s$, the defining gamma product has a simple zero at $jt+ks+\\mathbb Z$ for $j,k\\ge1$, a simple pole for $j,k\\le0$, and no divisor in the mixed cones. Rational independence of $1,t,s$ prevents collisions. The argument $z+t$ has coordinates $(1-a,b)$ in the $(t,s)$ basis, giving gamma order $H(b)-H(a)$. In\n\\[\n F(z)=\\frac{\\Gamma_{\\rm ell}(5(z+t);5t,5s)}\n {\\Gamma_{\\rm ell}(z+t;t,s)^5},\n\\]\nthe numerator contributes for every integer $c$ and the denominator only when $5\\mid c$. Their difference is \\eqref{eq:base-divisor}. The same product lists all possible zeros and poles.\n\\end{proof}\nThe rank-three additive group $\\Lambda_5\\subset\\mathbb C$ is nondiscrete. This does not make the divisor dense: its selected nonzero coefficients have locally finite support.","me{ord}_{a\\tau+bs+c/5}F=w(c)\\,[H(a)-H(b)].\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor positive periods $t,s$, the defining gamma product has a simple zero at $jt+ks+\\mathbb Z$ for $j,k\\ge1$, a simple pole for $j,k\\le0$, and no divisor in the mixed cones. Rational independence of $1,t,s$ prevents collisions. The argument $z+t$ has coordinates $(1-a,b)$ in the $(t,s)$ basis, giving gamma order $H(b)-H(a)$. In\n\\[\n F(z)=\\frac{\\Gamma_{\\rm ell}(5(z+t);5t,5s)}\n {\\Gamma_{\\rm ell}(z+t;t,s)^5},\n\\]\nthe numerator contributes for every integer $c$ and the denominator only when $5\\mid c$. Their difference is \\eqref{eq:base-divisor}. The same product lists all possible zeros and poles.\n\\end{proof}\nThe rank-three additive group $\\Lambda_5\\subset\\mathbb C$ is nondiscrete. This does not make the divisor dense: its selected nonzero coefficients have locally finite support.","+c/5}F=w(c)\\,[H(a)-H(b)].\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor positive periods $t,s$, the defining gamma product has a simple zero at $jt+ks+\\mathbb Z$ for $j,k\\ge1$, a simple pole for $j,k\\le0$, and no divisor in the mixed cones. Rational independence of $1,t,s$ prevents collisions. The argument $z+t$ has coordinates $(1-a,b)$ in the $(t,s)$ basis, giving gamma order $H(b)-H(a)$. In\n\\[\n F(z)=\\frac{\\Gamma_{\\rm ell}(5(z+t);5t,5s)}\n {\\Gamma_{\\rm ell}(z+t;t,s)^5},\n\\]\nthe numerator contributes for every integer $c$ and the denominator only when $5\\mid c$. Their difference is \\eqref{eq:base-divisor}. The same product lists all possible zeros and poles.\n\\end{proof}\nThe rank-three additive group $\\Lambda_5\\subset\\mathbb C$ is nondiscrete. This does not make the divisor dense: its selected nonzero coefficients have locally finite support.","as a simple zero at $jt+ks+\\mathbb Z$ for $j,k\\ge1$, a simple pole for $j,k\\le0$, and no divisor in the mixed cones. Rational independence of $1,t,s$ prevents collisions. The argument $z+t$ has coordinates $(1-a,b)$ in the $(t,s)$ basis, giving gamma order $H(b)-H(a)$. In\n\\[\n F(z)=\\frac{\\Gamma_{\\rm ell}(5(z+t);5t,5s)}\n {\\Gamma_{\\rm ell}(z+t;t,s)^5},\n\\]\nthe numerator contributes for every integer $c$ and the denominator only when $5\\mid c$. Their difference is \\eqref{eq:base-divisor}. The same product lists all possible zeros and poles.\n\\end{proof}\nThe rank-three additive group $\\Lambda_5\\subset\\mathbb C$ is nondiscrete. This does not make the divisor dense: its selected nonzero coefficients have locally finite support.","Z$ for $j,k\\ge1$, a simple pole for $j,k\\le0$, and no divisor in the mixed cones. Rational independence of $1,t,s$ prevents collisions. The argument $z+t$ has coordinates $(1-a,b)$ in the $(t,s)$ basis, giving gamma order $H(b)-H(a)$. In\n\\[\n F(z)=\\frac{\\Gamma_{\\rm ell}(5(z+t);5t,5s)}\n {\\Gamma_{\\rm ell}(z+t;t,s)^5},\n\\]\nthe numerator contributes for every integer $c$ and the denominator only when $5\\mid c$. Their difference is \\eqref{eq:base-divisor}. The same product lists all possible zeros and poles.\n\\end{proof}\nThe rank-three additive group $\\Lambda_5\\subset\\mathbb C$ is nondiscrete. This does not make the divisor dense: its selected nonzero coefficients have locally finite support.","z+t$ has coordinates $(1-a,b)$ in the $(t,s)$ basis, giving gamma order $H(b)-H(a)$. In\n\\[\n F(z)=\\frac{\\Gamma_{\\rm ell}(5(z+t);5t,5s)}\n {\\Gamma_{\\rm ell}(z+t;t,s)^5},\n\\]\nthe numerator contributes for every integer $c$ and the denominator only when $5\\mid c$. Their difference is \\eqref{eq:base-divisor}. The same product lists all possible zeros and poles.\n\\end{proof}\nThe rank-three additive group $\\Lambda_5\\subset\\mathbb C$ is nondiscrete. This does not make the divisor dense: its selected nonzero coefficients have locally finite support.","Multiplication by $\\beta$ in the ordered basis $(\\tau,s,1/5)$ is\n\\[\n N_5=\\begin{pmatrix}2&-1&0","Multiplication by $\\beta$ in the ordered basis $(\\tau,s,1/5)$ is\n\\[\n N_5=\\begin{pmatrix}2&-1&0\\\\4&0&-3\\\\5&0&-2\\end{pmatrix},\\qquad\n N_5^3=7I,\\quad\\det N_5=7,\\quad\n N_5^{-1}=\\frac17\\begin{pmatrix}0&-2&3\\\\-7&-4&6\\\\0&-5&4\\end{pmatrix}.\n\\]\nIts quotient generator is $(1,2,20)/7$, representing $\\delta$. Thus\n\\[\n \\beta^{-1}\\Lambda_5=\\Lambda_5+\\mathbb Z\\delta.\n\\]\nWrite a potential singular point as\n\\begin{equation}\\label{eq:lambda}\n \\lambda(a,b,c)=\\frac{a\\tau+bs+c/5}{\\beta},\\qquad\n p=\\left\\lceil\\frac{3c-2b}{7}\\right\\rceil,\\qquad\n k=7p+2b-3c\\in\\{0,\\ldots,6\\}.\n\\end{equation}\nExactly one argument $\\lambda+k\\delta$ lies in $\\Lambda_5$. Its integral coordinates are\n\\begin{equation}\\label{eq:lift}\n (\\widehat a,\\widehat b,\\widehat c)=(p,2p-a,20p+5b-8c).\n\\end{equation}\nIndeed $N_5(\\widehat a,\\widehat b,\\widehat c)^T=(a,b-8k,c-5k)^T$, which expresses $\\beta(\\lambda+k\\delta)=\\beta\\lambda+k(-8s-1)$. The other six digits fail integrality.","0&-5&4\\end{pmatrix}.\n\\]\nIts quotient generator is $(1,2,20)/7$, representing $\\delta$. Thus\n\\[\n \\beta^{-1}\\Lambda_5=\\Lambda_5+\\mathbb Z\\delta.\n\\]\nWrite a potential singular point as\n\\begin{equation}\\label{eq:lambda}\n \\lambda(a,b,c)=\\frac{a\\tau+bs+c/5}{\\beta},\\qquad\n p=\\left\\lceil\\frac{3c-2b}{7}\\right\\rceil,\\qquad\n k=7p+2b-3c\\in\\{0,\\ldots,6\\}.\n\\end{equation}\nExactly one argument $\\lambda+k\\delta$ lies in $\\Lambda_5$. Its integral coordinates are\n\\begin{equation}\\label{eq:lift}\n (\\widehat a,\\widehat b,\\widehat c)=(p,2p-a,20p+5b-8c).\n\\end{equation}\nIndeed $N_5(\\widehat a,\\widehat b,\\widehat c)^T=(a,b-8k,c-5k)^T$, which expresses $\\beta(\\lambda+k\\delta)=\\beta\\lambda+k(-8s-1)$. The other six digits fail integrality. Also\n\\[\n \\widehat c-2c=5(4p+b-2c),\\qquad","0&-5&4\\end{pmatrix}.\n\\]\nIts quotient generator is $(1,2,20)/7$, representing $\\delta$. Thus\n\\[\n \\beta^{-1}\\Lambda_5=\\Lambda_5+\\mathbb Z\\delta.\n\\]\nWrite a potential singular point as\n\\begin{equation}\\label{eq:lambda}\n \\lambda(a,b,c)=\\frac{a\\tau+bs+c/5}{\\beta},\\qquad\n p=\\left\\lceil\\frac{3c-2b}{7}\\right\\rceil,\\qquad\n k=7p+2b-3c\\in\\{0,\\ldots,6\\}.\n\\end{equation}\nExactly one argument $\\lambda+k\\delta$ lies in $\\Lambda_5$. Its integral coordinates are\n\\begin{equation}\\label{eq:lift}\n (\\widehat a,\\widehat b,\\widehat c)=(p,2p-a,20p+5b-8c).\n\\end{equation}\nIndeed $N_5(\\widehat a,\\widehat b,\\widehat c)^T=(a,b-8k,c-5k)^T$, which expresses $\\beta(\\lambda+k\\delta)=\\beta\\lambda+k(-8s-1)$. The other six digits fail integrality. Also\n\\[\n \\widehat c-2c=5(4p+b-2c),\\qquad w(\\widehat c)=w(c).\n\\]","0&-5&4\\end{pmatrix}.\n\\]\nIts quotient generator is $(1,2,20)/7$, representing $\\delta$. Thus\n\\[\n \\beta^{-1}\\Lambda_5=\\Lambda_5+\\mathbb Z\\delta.\n\\]\nWrite a potential singular point as\n\\begin{equation}\\label{eq:lambda}\n \\lambda(a,b,c)=\\frac{a\\tau+bs+c/5}{\\beta},\\qquad\n p=\\left\\lceil\\frac{3c-2b}{7}\\right\\rceil,\\qquad\n k=7p+2b-3c\\in\\{0,\\ldots,6\\}.\n\\end{equation}\nExactly one argument $\\lambda+k\\delta$ lies in $\\Lambda_5$. Its integral coordinates are\n\\begin{equation}\\label{eq:lift}\n (\\widehat a,\\widehat b,\\widehat c)=(p,2p-a,20p+5b-8c).\n\\end{equation}\nIndeed $N_5(\\widehat a,\\widehat b,\\widehat c)^T=(a,b-8k,c-5k)^T$, which expresses $\\beta(\\lambda+k\\delta)=\\beta\\lambda+k(-8s-1)$. The other six digits fail integrality. Also\n\\[\n \\widehat c-2c=5(4p+b-2c),\\qquad w(\\widehat c)=w(c).\n\\]\nThis transports the fifth-smoothing weight exactly.","nd{pmatrix}.\n\\]\nIts quotient generator is $(1,2,20)/7$, representing $\\delta$. Thus\n\\[\n \\beta^{-1}\\Lambda_5=\\Lambda_5+\\mathbb Z\\delta.\n\\]\nWrite a potential singular point as\n\\begin{equation}\\label{eq:lambda}\n \\lambda(a,b,c)=\\frac{a\\tau+bs+c/5}{\\beta},\\qquad\n p=\\left\\lceil\\frac{3c-2b}{7}\\right\\rceil,\\qquad\n k=7p+2b-3c\\in\\{0,\\ldots,6\\}.\n\\end{equation}\nExactly one argument $\\lambda+k\\delta$ lies in $\\Lambda_5$. Its integral coordinates are\n\\begin{equation}\\label{eq:lift}\n (\\widehat a,\\widehat b,\\widehat c)=(p,2p-a,20p+5b-8c).\n\\end{equation}\nIndeed $N_5(\\widehat a,\\widehat b,\\widehat c)^T=(a,b-8k,c-5k)^T$, which expresses $\\beta(\\lambda+k\\delta)=\\beta\\lambda+k(-8s-1)$. The other six digits fail integrality. Also\n\\[\n \\widehat c-2c=5(4p+b-2c),\\qquad w(\\widehat c)=w(c).\n\\]\nThis transports the fifth-smoothing weight exactly.","epresenting $\\delta$. Thus\n\\[\n \\beta^{-1}\\Lambda_5=\\Lambda_5+\\mathbb Z\\delta.\n\\]\nWrite a potential singular point as\n\\begin{equation}\\label{eq:lambda}\n \\lambda(a,b,c)=\\frac{a\\tau+bs+c/5}{\\beta},\\qquad\n p=\\left\\lceil\\frac{3c-2b}{7}\\right\\rceil,\\qquad\n k=7p+2b-3c\\in\\{0,\\ldots,6\\}.\n\\end{equation}\nExactly one argument $\\lambda+k\\delta$ lies in $\\Lambda_5$. Its integral coordinates are\n\\begin{equation}\\label{eq:lift}\n (\\widehat a,\\widehat b,\\widehat c)=(p,2p-a,20p+5b-8c).\n\\end{equation}\nIndeed $N_5(\\widehat a,\\widehat b,\\widehat c)^T=(a,b-8k,c-5k)^T$, which expresses $\\beta(\\lambda+k\\delta)=\\beta\\lambda+k(-8s-1)$. The other six digits fail integrality. Also\n\\[\n \\widehat c-2c=5(4p+b-2c),\\qquad w(\\widehat c)=w(c).\n\\]\nThis transports the fifth-smoothing weight exactly.","ial singular point as\n\\begin{equation}\\label{eq:lambda}\n \\lambda(a,b,c)=\\frac{a\\tau+bs+c/5}{\\beta},\\qquad\n p=\\left\\lceil\\frac{3c-2b}{7}\\right\\rceil,\\qquad\n k=7p+2b-3c\\in\\{0,\\ldots,6\\}.\n\\end{equation}\nExactly one argument $\\lambda+k\\delta$ lies in $\\Lambda_5$. Its integral coordinates are\n\\begin{equation}\\label{eq:lift}\n (\\widehat a,\\widehat b,\\widehat c)=(p,2p-a,20p+5b-8c).\n\\end{equation}\nIndeed $N_5(\\widehat a,\\widehat b,\\widehat c)^T=(a,b-8k,c-5k)^T$, which expresses $\\beta(\\lambda+k\\delta)=\\beta\\lambda+k(-8s-1)$. The other six digits fail integrality. Also\n\\[\n \\widehat c-2c=5(4p+b-2c),\\qquad w(\\widehat c)=w(c).\n\\]\nThis transports the fifth-smoothing weight exactly.","ac{a\\tau+bs+c/5}{\\beta},\\qquad\n p=\\left\\lceil\\frac{3c-2b}{7}\\right\\rceil,\\qquad\n k=7p+2b-3c\\in\\{0,\\ldots,6\\}.\n\\end{equation}\nExactly one argument $\\lambda+k\\delta$ lies in $\\Lambda_5$. Its integral coordinates are\n\\begin{equation}\\label{eq:lift}\n (\\widehat a,\\widehat b,\\widehat c)=(p,2p-a,20p+5b-8c).\n\\end{equation}\nIndeed $N_5(\\widehat a,\\widehat b,\\widehat c)^T=(a,b-8k,c-5k)^T$, which expresses $\\beta(\\lambda+k\\delta)=\\beta\\lambda+k(-8s-1)$. The other six digits fail integrality. Also\n\\[\n \\widehat c-2c=5(4p+b-2c),\\qquad w(\\widehat c)=w(c).\n\\]\nThis transports the fifth-smoothing weight exactly.","n\\{0,\\ldots,6\\}.\n\\end{equation}\nExactly one argument $\\lambda+k\\delta$ lies in $\\Lambda_5$. Its integral coordinates are\n\\begin{equation}\\label{eq:lift}\n (\\widehat a,\\widehat b,\\widehat c)=(p,2p-a,20p+5b-8c).\n\\end{equation}\nIndeed $N_5(\\widehat a,\\widehat b,\\widehat c)^T=(a,b-8k,c-5k)^T$, which expresses $\\beta(\\lambda+k\\delta)=\\beta\\lambda+k(-8s-1)$. The other six digits fail integrality. Also\n\\[\n \\widehat c-2c=5(4p+b-2c),\\qquad w(\\widehat c)=w(c).\n\\]\nThis transports the fifth-smoothing weight exactly.","\\begin{theorem}[Complete ceiling divisor]\\label{thm:complete-divisor}\nThere are no zeros or poles of $R$ outside $\\beta^{-1}\\Lambda_5$. At \\eqref{eq:lambda},\n\\begin{equation}\\label{eq:complete-divisor}\n m_\\lambda:=\\operatorname{ord}_{\\lambda}R\n =w(c)\\,[H(p)-H(2p-a)-H(a)+H(b)]\\in\\{-4,-1,0,1,4\\}.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nApply Theorem~\\ref{thm:base-divisor} at the unique integral numerator lift \\eqref{eq:lift} and subtract the denominator order at $\\beta\\lambda$. The weights agree. To bound the bracket, when $a\\le0,p>0$ it is $H(b)$; when $a>0,p\\le0$ it is $H(b)-1$. In the other cases it is a difference of two zero-one values. Thus it lies in $\\{-1,0,1\\}$.\n\\end{proof}\nIn particular,\n\\[\n \\operatorname{ord}_{n\\tau}R=4\\quad(n\\ge1),\\qquad\n \\operatorname{ord}_{ns}R=-4\\quad(n\\in\\mat","\\begin{theorem}[Complete ceiling divisor]\\label{thm:complete-divisor}\nThere are no zeros or poles of $R$ outside $\\beta^{-1}\\Lambda_5$. At \\eqref{eq:lambda},\n\\begin{equation}\\label{eq:complete-divisor}\n m_\\lambda:=\\operatorname{ord}_{\\lambda}R\n =w(c)\\,[H(p)-H(2p-a)-H(a)+H(b)]\\in\\{-4,-1,0,1,4\\}.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nApply Theorem~\\ref{thm:base-divisor} at the unique integral numerator lift \\eqref{eq:lift} and subtract the denominator order at $\\beta\\lambda$. The weights agree. To bound the bracket, when $a\\le0,p>0$ it is $H(b)$; when $a>0,p\\le0$ it is $H(b)-1$. In the other cases it is a difference of two zero-one values. Thus it lies in $\\{-1,0,1\\}$.\n\\end{proof}\nIn particular,\n\\[\n \\operatorname{ord}_{n\\tau}R=4\\quad(n\\ge1),\\qquad\n \\operatorname{ord}_{ns}R=-4\\quad(n\\in\\mathbb Z\\setminus\\{0\\}),\\qquad\n \\","\\begin{theorem}[Complete ceiling divisor]\\label{thm:complete-divisor}\nThere are no zeros or poles of $R$ outside $\\beta^{-1}\\Lambda_5$. At \\eqref{eq:lambda},\n\\begin{equation}\\label{eq:complete-divisor}\n m_\\lambda:=\\operatorname{ord}_{\\lambda}R\n =w(c)\\,[H(p)-H(2p-a)-H(a)+H(b)]\\in\\{-4,-1,0,1,4\\}.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nApply Theorem~\\ref{thm:base-divisor} at the unique integral numerator lift \\eqref{eq:lift} and subtract the denominator order at $\\beta\\lambda$. The weights agree. To bound the bracket, when $a\\le0,p>0$ it is $H(b)$; when $a>0,p\\le0$ it is $H(b)-1$. In the other cases it is a difference of two zero-one values. Thus it lies in $\\{-1,0,1\\}$.\n\\end{proof}\nIn particular,\n\\[\n \\operatorname{ord}_{n\\tau}R=4\\quad(n\\ge1),\\qquad\n \\operatorname{ord}_{ns}R=-4\\quad(n\\in\\mathbb Z\\setminus\\{0\\}),\\qquad\n \\operatorname{ord}_{-n}R=4\\quad(n\\ge1).\n\\]\nFor these assertions the triples are $(2n,4n,5n)$, $(-n,0,0)$ and $(0,15n,10n)$ respectively. At zero the order is zero.","There are no zeros or poles of $R$ outside $\\beta^{-1}\\Lambda_5$. At \\eqref{eq:lambda},\n\\begin{equation}\\label{eq:complete-divisor}\n m_\\lambda:=\\operatorname{ord}_{\\lambda}R\n =w(c)\\,[H(p)-H(2p-a)-H(a)+H(b)]\\in\\{-4,-1,0,1,4\\}.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nApply Theorem~\\ref{thm:base-divisor} at the unique integral numerator lift \\eqref{eq:lift} and subtract the denominator order at $\\beta\\lambda$. The weights agree. To bound the bracket, when $a\\le0,p>0$ it is $H(b)$; when $a>0,p\\le0$ it is $H(b)-1$. In the other cases it is a difference of two zero-one values. Thus it lies in $\\{-1,0,1\\}$.\n\\end{proof}\nIn particular,\n\\[\n \\operatorname{ord}_{n\\tau}R=4\\quad(n\\ge1),\\qquad\n \\operatorname{ord}_{ns}R=-4\\quad(n\\in\\mathbb Z\\setminus\\{0\\}),\\qquad\n \\operatorname{ord}_{-n}R=4\\quad(n\\ge1).\n\\]\nFor these assertions the triples are $(2n,4n,5n)$, $(-n,0,0)$ and $(0,15n,10n)$ respectively. At zero the order is zero. Lean verifies the integer identities; the interpretation as complex orders is the ordinary product argument.","_5$. At \\eqref{eq:lambda},\n\\begin{equation}\\label{eq:complete-divisor}\n m_\\lambda:=\\operatorname{ord}_{\\lambda}R\n =w(c)\\,[H(p)-H(2p-a)-H(a)+H(b)]\\in\\{-4,-1,0,1,4\\}.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nApply Theorem~\\ref{thm:base-divisor} at the unique integral numerator lift \\eqref{eq:lift} and subtract the denominator order at $\\beta\\lambda$. The weights agree. To bound the bracket, when $a\\le0,p>0$ it is $H(b)$; when $a>0,p\\le0$ it is $H(b)-1$. In the other cases it is a difference of two zero-one values. Thus it lies in $\\{-1,0,1\\}$.\n\\end{proof}\nIn particular,\n\\[\n \\operatorname{ord}_{n\\tau}R=4\\quad(n\\ge1),\\qquad\n \\operatorname{ord}_{ns}R=-4\\quad(n\\in\\mathbb Z\\setminus\\{0\\}),\\qquad\n \\operatorname{ord}_{-n}R=4\\quad(n\\ge1).\n\\]\nFor these assertions the triples are $(2n,4n,5n)$, $(-n,0,0)$ and $(0,15n,10n)$ respectively. At zero the order is zero. Lean verifies the integer identities; the interpretation as complex orders is the ordinary product argument.","lambda},\n\\begin{equation}\\label{eq:complete-divisor}\n m_\\lambda:=\\operatorname{ord}_{\\lambda}R\n =w(c)\\,[H(p)-H(2p-a)-H(a)+H(b)]\\in\\{-4,-1,0,1,4\\}.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nApply Theorem~\\ref{thm:base-divisor} at the unique integral numerator lift \\eqref{eq:lift} and subtract the denominator order at $\\beta\\lambda$. The weights agree. To bound the bracket, when $a\\le0,p>0$ it is $H(b)$; when $a>0,p\\le0$ it is $H(b)-1$. In the other cases it is a difference of two zero-one values. Thus it lies in $\\{-1,0,1\\}$.\n\\end{proof}\nIn particular,\n\\[\n \\operatorname{ord}_{n\\tau}R=4\\quad(n\\ge1),\\qquad\n \\operatorname{ord}_{ns}R=-4\\quad(n\\in\\mathbb Z\\setminus\\{0\\}),\\qquad\n \\operatorname{ord}_{-n}R=4\\quad(n\\ge1).\n\\]\nFor these assertions the triples are $(2n,4n,5n)$, $(-n,0,0)$ and $(0,15n,10n)$ respectively. At zero the order is zero. Lean verifies the integer identities; the interpretation as complex orders is the ordinary product argument.","in{equation}\\label{eq:complete-divisor}\n m_\\lambda:=\\operatorname{ord}_{\\lambda}R\n =w(c)\\,[H(p)-H(2p-a)-H(a)+H(b)]\\in\\{-4,-1,0,1,4\\}.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nApply Theorem~\\ref{thm:base-divisor} at the unique integral numerator lift \\eqref{eq:lift} and subtract the denominator order at $\\beta\\lambda$. The weights agree. To bound the bracket, when $a\\le0,p>0$ it is $H(b)$; when $a>0,p\\le0$ it is $H(b)-1$. In the other cases it is a difference of two zero-one values. Thus it lies in $\\{-1,0,1\\}$.\n\\end{proof}\nIn particular,\n\\[\n \\operatorname{ord}_{n\\tau}R=4\\quad(n\\ge1),\\qquad\n \\operatorname{ord}_{ns}R=-4\\quad(n\\in\\mathbb Z\\setminus\\{0\\}),\\qquad\n \\operatorname{ord}_{-n}R=4\\quad(n\\ge1).\n\\]\nFor these assertions the triples are $(2n,4n,5n)$, $(-n,0,0)$ and $(0,15n,10n)$ respectively. At zero the order is zero. Lean verifies the integer identities; the interpretation as complex orders is the ordinary product argument.","{eq:complete-divisor}\n m_\\lambda:=\\operatorname{ord}_{\\lambda}R\n =w(c)\\,[H(p)-H(2p-a)-H(a)+H(b)]\\in\\{-4,-1,0,1,4\\}.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nApply Theorem~\\ref{thm:base-divisor} at the unique integral numerator lift \\eqref{eq:lift} and subtract the denominator order at $\\beta\\lambda$. The weights agree. To bound the bracket, when $a\\le0,p>0$ it is $H(b)$; when $a>0,p\\le0$ it is $H(b)-1$. In the other cases it is a difference of two zero-one values. Thus it lies in $\\{-1,0,1\\}$.\n\\end{proof}\nIn particular,\n\\[\n \\operatorname{ord}_{n\\tau}R=4\\quad(n\\ge1),\\qquad\n \\operatorname{ord}_{ns}R=-4\\quad(n\\in\\mathbb Z\\setminus\\{0\\}),\\qquad\n \\operatorname{ord}_{-n}R=4\\quad(n\\ge1).\n\\]\nFor these assertions the triples are $(2n,4n,5n)$, $(-n,0,0)$ and $(0,15n,10n)$ respectively. At zero the order is zero. Lean verifies the integer identities; the interpretation as complex orders is the ordinary product argument.","a)+H(b)]\\in\\{-4,-1,0,1,4\\}.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nApply Theorem~\\ref{thm:base-divisor} at the unique integral numerator lift \\eqref{eq:lift} and subtract the denominator order at $\\beta\\lambda$. The weights agree. To bound the bracket, when $a\\le0,p>0$ it is $H(b)$; when $a>0,p\\le0$ it is $H(b)-1$. In the other cases it is a difference of two zero-one values. Thus it lies in $\\{-1,0,1\\}$.\n\\end{proof}\nIn particular,\n\\[\n \\operatorname{ord}_{n\\tau}R=4\\quad(n\\ge1),\\qquad\n \\operatorname{ord}_{ns}R=-4\\quad(n\\in\\mathbb Z\\setminus\\{0\\}),\\qquad\n \\operatorname{ord}_{-n}R=4\\quad(n\\ge1).\n\\]\nFor these assertions the triples are $(2n,4n,5n)$, $(-n,0,0)$ and $(0,15n,10n)$ respectively. At zero the order is zero. Lean verifies the integer identities; the interpretation as complex orders is the ordinary product argument.","\\end{theorem}\n\\begin{proof}\nApply Theorem~\\ref{thm:base-divisor} at the unique integral numerator lift \\eqref{eq:lift} and subtract the denominator order at $\\beta\\lambda$. The weights agree. To bound the bracket, when $a\\le0,p>0$ it is $H(b)$; when $a>0,p\\le0$ it is $H(b)-1$. In the other cases it is a difference of two zero-one values. Thus it lies in $\\{-1,0,1\\}$.\n\\end{proof}\nIn particular,\n\\[\n \\operatorname{ord}_{n\\tau}R=4\\quad(n\\ge1),\\qquad\n \\operatorname{ord}_{ns}R=-4\\quad(n\\in\\mathbb Z\\setminus\\{0\\}),\\qquad\n \\operatorname{ord}_{-n}R=4\\quad(n\\ge1).\n\\]\nFor these assertions the triples are $(2n,4n,5n)$, $(-n,0,0)$ and $(0,15n,10n)$ respectively. At zero the order is zero. Lean verifies the integer identities; the interpretation as complex orders is the ordinary product argument.","he bracket, when $a\\le0,p>0$ it is $H(b)$; when $a>0,p\\le0$ it is $H(b)-1$. In the other cases it is a difference of two zero-one values. Thus it lies in $\\{-1,0,1\\}$.\n\\end{proof}\nIn particular,\n\\[\n \\operatorname{ord}_{n\\tau}R=4\\quad(n\\ge1),\\qquad\n \\operatorname{ord}_{ns}R=-4\\quad(n\\in\\mathbb Z\\setminus\\{0\\}),\\qquad\n \\operatorname{ord}_{-n}R=4\\quad(n\\ge1).\n\\]\nFor these assertions the triples are $(2n,4n,5n)$, $(-n,0,0)$ and $(0,15n,10n)$ respectively. At zero the order is zero. Lean verifies the integer identities; the interpretation as complex orders is the ordinary product argument.","en $a\\le0,p>0$ it is $H(b)$; when $a>0,p\\le0$ it is $H(b)-1$. In the other cases it is a difference of two zero-one values. Thus it lies in $\\{-1,0,1\\}$.\n\\end{proof}\nIn particular,\n\\[\n \\operatorname{ord}_{n\\tau}R=4\\quad(n\\ge1),\\qquad\n \\operatorname{ord}_{ns}R=-4\\quad(n\\in\\mathbb Z\\setminus\\{0\\}),\\qquad\n \\operatorname{ord}_{-n}R=4\\quad(n\\ge1).\n\\]\nFor these assertions the triples are $(2n,4n,5n)$, $(-n,0,0)$ and $(0,15n,10n)$ respectively. At zero the order is zero. Lean verifies the integer identities; the interpretation as complex orders is the ordinary product argument.","it is $H(b)$; when $a>0,p\\le0$ it is $H(b)-1$. In the other cases it is a difference of two zero-one values. Thus it lies in $\\{-1,0,1\\}$.\n\\end{proof}\nIn particular,\n\\[\n \\operatorname{ord}_{n\\tau}R=4\\quad(n\\ge1),\\qquad\n \\operatorname{ord}_{ns}R=-4\\quad(n\\in\\mathbb Z\\setminus\\{0\\}),\\qquad\n \\operatorname{ord}_{-n}R=4\\quad(n\\ge1).\n\\]\nFor these assertions the triples are $(2n,4n,5n)$, $(-n,0,0)$ and $(0,15n,10n)$ respectively. At zero the order is zero. Lean verifies the integer identities; the interpretation as complex orders is the ordinary product argument.","\\subsection{All transition multipliers and the scalar anchor}\n\\begin{theorem}[Theta transitions]\\label{thm:transitions}\nThe following hold meromorphically:\n\\begin{align}\n K(z):=\\frac{R(z+1)}{R(z)}&=\\prod_{r=1}^{15}q_t(\\beta z-rs)^{-1},\\label{eq:K}\\\\\n \\frac{R(z+s)}{R(z)}&=\\frac{q_s(\\beta z-\\tau)}{\\prod_{k=0}^{6}q_t(z+k\\delta)},\\label{eq:Rs}\\\\\n \\frac{R(z+\\tau)}{R(z)}&=\n \\frac{\\prod_{k=0}^{6}q_s(z+k\\delta)\\prod_{r=0}^{3}q_t(\\beta z+rs)}\n {q_s(\\beta z)q_s(\\beta z+\\tau)},\\label{eq:Rtau}\\\\\n \\frac{R(z+\\delta)}{R(z)}&=\n \\frac{q_s(z)}{q_t(z)q_t(z+s)\\prod_{r=1}^{8}q_t(\\beta z-rs)}.\\label{eq:Rdelta}\n\\end{align}\n\\end{theorem}\n\\begin{proof}\nUse $F(z+1)=F(z)$, $F(z+\\tau)/F(z)=q_s(z)$ and $F(z+s)/F(z)=q_t(z)^{-1}$. For \\eqref{eq:K}, $P_7$ is period one and $\\beta=-15s-2$, producing fifteen negative $s$ shifts in the denominator. For \\eqref{eq:Rs} use $\\beta s=-\\tau$. For \\eqref{eq:Rtau}, $\\beta\\tau=2\\tau+4s+1$ gives two $q_s$ factors and four reciprocal $q_t$ factors; the $2\\tau$ shift disappears from $q_t$ by ellipticity. Finally $P_7(z+\\delta)/P_7(z)=F(z+7\\delta)/F(z)$ and the last two identities…","\\frac{R(z+\\delta)}{R(z)}&=\n \\frac{q_s(z)}{q_t(z)q_t(z+s)\\prod_{r=1}^{8}q_t(\\beta z-rs)}.\\label{eq:Rdelta}\n\\end{align}\n\\end{theorem}\n\\begin{proof}\nUse $F(z+1)=F(z)$, $F(z+\\tau)/F(z)=q_s(z)$ and $F(z+s)/F(z)=q_t(z)^{-1}$. For \\eqref{eq:K}, $P_7$ is period one and $\\beta=-15s-2$, producing fifteen negative $s$ shifts in the denominator. For \\eqref{eq:Rs} use $\\beta s=-\\tau$. For \\eqref{eq:Rtau}, $\\beta\\tau=2\\tau+4s+1$ gives two $q_s$ factors and four reciprocal $q_t$ factors; the $2\\tau$ shift disappears from $q_t$ by ellipticity. Finally $P_7(z+\\delta)/P_7(z)=F(z+7\\delta)/F(z)$ and the last two identities of \\eqref{eq:period-identities} give \\eqref{eq:Rdelta}.\n\\end{proof}\nUnder $z\\mapsto z+1$ the triple changes to $(a,b-15,c-10)$, leaving $p$ and $w(c)$ unchanged. Formula~\\eqref{eq:complete-divisor} gives\n\\[\n \\operatorname{ord}_{z+1}R-\\operatorname{ord","\\frac{R(z+\\delta)}{R(z)}&=\n \\frac{q_s(z)}{q_t(z)q_t(z+s)\\prod_{r=1}^{8}q_t(\\beta z-rs)}.\\label{eq:Rdelta}\n\\end{align}\n\\end{theorem}\n\\begin{proof}\nUse $F(z+1)=F(z)$, $F(z+\\tau)/F(z)=q_s(z)$ and $F(z+s)/F(z)=q_t(z)^{-1}$. For \\eqref{eq:K}, $P_7$ is period one and $\\beta=-15s-2$, producing fifteen negative $s$ shifts in the denominator. For \\eqref{eq:Rs} use $\\beta s=-\\tau$. For \\eqref{eq:Rtau}, $\\beta\\tau=2\\tau+4s+1$ gives two $q_s$ factors and four reciprocal $q_t$ factors; the $2\\tau$ shift disappears from $q_t$ by ellipticity. Finally $P_7(z+\\delta)/P_7(z)=F(z+7\\delta)/F(z)$ and the last two identities of \\eqref{eq:period-identities} give \\eqref{eq:Rdelta}.\n\\end{proof}\nUnder $z\\mapsto z+1$ the triple changes to $(a,b-15,c-10)$, leaving $p$ and $w(c)$ unchanged. Formula~\\eqref{eq:complete-divisor} gives\n\\[\n \\operatorname{ord}_{z+1}R-\\operatorname{ord}_zR\n =-w(c)\\mathbf1_{\\{1","\\frac{R(z+\\delta)}{R(z)}&=\n \\frac{q_s(z)}{q_t(z)q_t(z+s)\\prod_{r=1}^{8}q_t(\\beta z-rs)}.\\label{eq:Rdelta}\n\\end{align}\n\\end{theorem}\n\\begin{proof}\nUse $F(z+1)=F(z)$, $F(z+\\tau)/F(z)=q_s(z)$ and $F(z+s)/F(z)=q_t(z)^{-1}$. For \\eqref{eq:K}, $P_7$ is period one and $\\beta=-15s-2$, producing fifteen negative $s$ shifts in the denominator. For \\eqref{eq:Rs} use $\\beta s=-\\tau$. For \\eqref{eq:Rtau}, $\\beta\\tau=2\\tau+4s+1$ gives two $q_s$ factors and four reciprocal $q_t$ factors; the $2\\tau$ shift disappears from $q_t$ by ellipticity. Finally $P_7(z+\\delta)/P_7(z)=F(z+7\\delta)/F(z)$ and the last two identities of \\eqref{eq:period-identities} give \\eqref{eq:Rdelta}.\n\\end{proof}\nUnder $z\\mapsto z+1$ the triple changes to $(a,b-15,c-10)$, leaving $p$ and $w(c)$ unchanged. Formula~\\eqref{eq:complete-divisor} gives\n\\[\n \\operatorname{ord}_{z+1}R-\\operatorname{ord}_zR\n =-w(c)\\mathbf1_{\\{1\\le b\\le15\\}},\n\\]\nthe divisor","\\frac{R(z+\\delta)}{R(z)}&=\n \\frac{q_s(z)}{q_t(z)q_t(z+s)\\prod_{r=1}^{8}q_t(\\beta z-rs)}.\\label{eq:Rdelta}\n\\end{align}\n\\end{theorem}\n\\begin{proof}\nUse $F(z+1)=F(z)$, $F(z+\\tau)/F(z)=q_s(z)$ and $F(z+s)/F(z)=q_t(z)^{-1}$. For \\eqref{eq:K}, $P_7$ is period one and $\\beta=-15s-2$, producing fifteen negative $s$ shifts in the denominator. For \\eqref{eq:Rs} use $\\beta s=-\\tau$. For \\eqref{eq:Rtau}, $\\beta\\tau=2\\tau+4s+1$ gives two $q_s$ factors and four reciprocal $q_t$ factors; the $2\\tau$ shift disappears from $q_t$ by ellipticity. Finally $P_7(z+\\delta)/P_7(z)=F(z+7\\delta)/F(z)$ and the last two identities of \\eqref{eq:period-identities} give \\eqref{eq:Rdelta}.\n\\end{proof}\nUnder $z\\mapsto z+1$ the triple changes to $(a,b-15,c-10)$, leaving $p$ and $w(c)$ unchanged. Formula~\\eqref{eq:complete-divisor} gives\n\\[\n \\operatorname{ord}_{z+1}R-\\operatorname{ord}_zR\n =-w(c)\\mathbf1_{\\{1\\le b\\le15\\}},\n\\]\nthe divisor of \\eqref{eq:K}. Under $z","\\frac{R(z+\\delta)}{R(z)}&=\n \\frac{q_s(z)}{q_t(z)q_t(z+s)\\prod_{r=1}^{8}q_t(\\beta z-rs)}.\\label{eq:Rdelta}\n\\end{align}\n\\end{theorem}\n\\begin{proof}\nUse $F(z+1)=F(z)$, $F(z+\\tau)/F(z)=q_s(z)$ and $F(z+s)/F(z)=q_t(z)^{-1}$. For \\eqref{eq:K}, $P_7$ is period one and $\\beta=-15s-2$, producing fifteen negative $s$ shifts in the denominator. For \\eqref{eq:Rs} use $\\beta s=-\\tau$. For \\eqref{eq:Rtau}, $\\beta\\tau=2\\tau+4s+1$ gives two $q_s$ factors and four reciprocal $q_t$ factors; the $2\\tau$ shift disappears from $q_t$ by ellipticity. Finally $P_7(z+\\delta)/P_7(z)=F(z+7\\delta)/F(z)$ and the last two identities of \\eqref{eq:period-identities} give \\eqref{eq:Rdelta}.\n\\end{proof}\nUnder $z\\mapsto z+1$ the triple changes to $(a,b-15,c-10)$, leaving $p$ and $w(c)$ unchanged. Formula~\\eqref{eq:complete-divisor} gives\n\\[\n \\operatorname{ord}_{z+1}R-\\operatorname{ord}_zR\n =-w(c)\\mathbf1_{\\{1\\le b\\le15\\}},\n\\]\nthe divisor of \\eqref{eq:K}. Under $z\\mapsto z+\\delta$ the triple beco","\\frac{R(z+\\delta)}{R(z)}&=\n \\frac{q_s(z)}{q_t(z)q_t(z+s)\\prod_{r=1}^{8}q_t(\\beta z-rs)}.\\label{eq:Rdelta}\n\\end{align}\n\\end{theorem}\n\\begin{proof}\nUse $F(z+1)=F(z)$, $F(z+\\tau)/F(z)=q_s(z)$ and $F(z+s)/F(z)=q_t(z)^{-1}$. For \\eqref{eq:K}, $P_7$ is period one and $\\beta=-15s-2$, producing fifteen negative $s$ shifts in the denominator. For \\eqref{eq:Rs} use $\\beta s=-\\tau$. For \\eqref{eq:Rtau}, $\\beta\\tau=2\\tau+4s+1$ gives two $q_s$ factors and four reciprocal $q_t$ factors; the $2\\tau$ shift disappears from $q_t$ by ellipticity. Finally $P_7(z+\\delta)/P_7(z)=F(z+7\\delta)/F(z)$ and the last two identities of \\eqref{eq:period-identities} give \\eqref{eq:Rdelta}.\n\\end{proof}\nUnder $z\\mapsto z+1$ the triple changes to $(a,b-15,c-10)$, leaving $p$ and $w(c)$ unchanged. Formula~\\eqref{eq:complete-divisor} gives\n\\[\n \\operatorname{ord}_{z+1}R-\\operatorname{ord}_zR\n =-w(c)\\mathbf1_{\\{1\\le b\\le15\\}},\n\\]\nthe divisor of \\eqref{eq:K}. Under $z\\mapsto z+\\delta$ the triple becomes $(a,b-8,c-5)$; its digit dec","\\frac{R(z+\\delta)}{R(z)}&=\n \\frac{q_s(z)}{q_t(z)q_t(z+s)\\prod_{r=1}^{8}q_t(\\beta z-rs)}.\\label{eq:Rdelta}\n\\end{align}\n\\end{theorem}\n\\begin{proof}\nUse $F(z+1)=F(z)$, $F(z+\\tau)/F(z)=q_s(z)$ and $F(z+s)/F(z)=q_t(z)^{-1}$. For \\eqref{eq:K}, $P_7$ is period one and $\\beta=-15s-2$, producing fifteen negative $s$ shifts in the denominator. For \\eqref{eq:Rs} use $\\beta s=-\\tau$. For \\eqref{eq:Rtau}, $\\beta\\tau=2\\tau+4s+1$ gives two $q_s$ factors and four reciprocal $q_t$ factors; the $2\\tau$ shift disappears from $q_t$ by ellipticity. Finally $P_7(z+\\delta)/P_7(z)=F(z+7\\delta)/F(z)$ and the last two identities of \\eqref{eq:period-identities} give \\eqref{eq:Rdelta}.\n\\end{proof}\nUnder $z\\mapsto z+1$ the triple changes to $(a,b-15,c-10)$, leaving $p$ and $w(c)$ unchanged. Formula~\\eqref{eq:complete-divisor} gives\n\\[\n \\operatorname{ord}_{z+1}R-\\operatorname{ord}_zR\n =-w(c)\\mathbf1_{\\{1\\le b\\le15\\}},\n\\]\nthe divisor of \\eqref{eq:K}. Under $z\\mapsto z+\\delta$ the triple becomes $(a,b-8,c-5)$; its digit decreases by one modulo seven, and its pivot increases exactly when","\\frac{R(z+\\delta)}{R(z)}&=\n \\frac{q_s(z)}{q_t(z)q_t(z+s)\\prod_{r=1}^{8}q_t(\\beta z-rs)}.\\label{eq:Rdelta}\n\\end{align}\n\\end{theorem}\n\\begin{proof}\nUse $F(z+1)=F(z)$, $F(z+\\tau)/F(z)=q_s(z)$ and $F(z+s)/F(z)=q_t(z)^{-1}$. For \\eqref{eq:K}, $P_7$ is period one and $\\beta=-15s-2$, producing fifteen negative $s$ shifts in the denominator. For \\eqref{eq:Rs} use $\\beta s=-\\tau$. For \\eqref{eq:Rtau}, $\\beta\\tau=2\\tau+4s+1$ gives two $q_s$ factors and four reciprocal $q_t$ factors; the $2\\tau$ shift disappears from $q_t$ by ellipticity. Finally $P_7(z+\\delta)/P_7(z)=F(z+7\\delta)/F(z)$ and the last two identities of \\eqref{eq:period-identities} give \\eqref{eq:Rdelta}.\n\\end{proof}\nUnder $z\\mapsto z+1$ the triple changes to $(a,b-15,c-10)$, leaving $p$ and $w(c)$ unchanged. Formula~\\eqref{eq:complete-divisor} gives\n\\[\n \\operatorname{ord}_{z+1}R-\\operatorname{ord}_zR\n =-w(c)\\mathbf1_{\\{1\\le b\\le15\\}},\n\\]\nthe divisor of \\eqref{eq:K}. Under $z\\mapsto z+\\delta$ the triple becomes $(a,b-8,c-5)$; its digit decreases by one modulo seven, and its pivot increases exactly when the old digit is zero. These…","rac{q_s(z)}{q_t(z)q_t(z+s)\\prod_{r=1}^{8}q_t(\\beta z-rs)}.\\label{eq:Rdelta}\n\\end{align}\n\\end{theorem}\n\\begin{proof}\nUse $F(z+1)=F(z)$, $F(z+\\tau)/F(z)=q_s(z)$ and $F(z+s)/F(z)=q_t(z)^{-1}$. For \\eqref{eq:K}, $P_7$ is period one and $\\beta=-15s-2$, producing fifteen negative $s$ shifts in the denominator. For \\eqref{eq:Rs} use $\\beta s=-\\tau$. For \\eqref{eq:Rtau}, $\\beta\\tau=2\\tau+4s+1$ gives two $q_s$ factors and four reciprocal $q_t$ factors; the $2\\tau$ shift disappears from $q_t$ by ellipticity. Finally $P_7(z+\\delta)/P_7(z)=F(z+7\\delta)/F(z)$ and the last two identities of \\eqref{eq:period-identities} give \\eqref{eq:Rdelta}.\n\\end{proof}\nUnder $z\\mapsto z+1$ the triple changes to $(a,b-15,c-10)$, leaving $p$ and $w(c)$ unchanged. Formula~\\eqref{eq:complete-divisor} gives\n\\[\n \\operatorname{ord}_{z+1}R-\\operatorname{ord}_zR\n =-w(c)\\mathbf1_{\\{1\\le b\\le15\\}},\n\\]\nthe divisor of \\eqref{eq:K}. Under $z\\mapsto z+\\delta$ the triple becomes $(a,b-8,c-5)$; its digit decreases by one modulo seven, and its pivot increases exactly when the old digit is zero. These universal carry statements in…","=1}^{8}q_t(\\beta z-rs)}.\\label{eq:Rdelta}\n\\end{align}\n\\end{theorem}\n\\begin{proof}\nUse $F(z+1)=F(z)$, $F(z+\\tau)/F(z)=q_s(z)$ and $F(z+s)/F(z)=q_t(z)^{-1}$. For \\eqref{eq:K}, $P_7$ is period one and $\\beta=-15s-2$, producing fifteen negative $s$ shifts in the denominator. For \\eqref{eq:Rs} use $\\beta s=-\\tau$. For \\eqref{eq:Rtau}, $\\beta\\tau=2\\tau+4s+1$ gives two $q_s$ factors and four reciprocal $q_t$ factors; the $2\\tau$ shift disappears from $q_t$ by ellipticity. Finally $P_7(z+\\delta)/P_7(z)=F(z+7\\delta)/F(z)$ and the last two identities of \\eqref{eq:period-identities} give \\eqref{eq:Rdelta}.\n\\end{proof}\nUnder $z\\mapsto z+1$ the triple changes to $(a,b-15,c-10)$, leaving $p$ and $w(c)$ unchanged. Formula~\\eqref{eq:complete-divisor} gives\n\\[\n \\operatorname{ord}_{z+1}R-\\operatorname{ord}_zR\n =-w(c)\\mathbf1_{\\{1\\le b\\le15\\}},\n\\]\nthe divisor of \\eqref{eq:K}. Under $z\\mapsto z+\\delta$ the triple becomes $(a,b-8,c-5)$; its digit decreases by one modulo seven, and its pivot increases exactly when the old digit is zero. These universal carry statements include negative coordinates.","rs)}.\\label{eq:Rdelta}\n\\end{align}\n\\end{theorem}\n\\begin{proof}\nUse $F(z+1)=F(z)$, $F(z+\\tau)/F(z)=q_s(z)$ and $F(z+s)/F(z)=q_t(z)^{-1}$. For \\eqref{eq:K}, $P_7$ is period one and $\\beta=-15s-2$, producing fifteen negative $s$ shifts in the denominator. For \\eqref{eq:Rs} use $\\beta s=-\\tau$. For \\eqref{eq:Rtau}, $\\beta\\tau=2\\tau+4s+1$ gives two $q_s$ factors and four reciprocal $q_t$ factors; the $2\\tau$ shift disappears from $q_t$ by ellipticity. Finally $P_7(z+\\delta)/P_7(z)=F(z+7\\delta)/F(z)$ and the last two identities of \\eqref{eq:period-identities} give \\eqref{eq:Rdelta}.\n\\end{proof}\nUnder $z\\mapsto z+1$ the triple changes to $(a,b-15,c-10)$, leaving $p$ and $w(c)$ unchanged. Formula~\\eqref{eq:complete-divisor} gives\n\\[\n \\operatorname{ord}_{z+1}R-\\operatorname{ord}_zR\n =-w(c)\\mathbf1_{\\{1\\le b\\le15\\}},\n\\]\nthe divisor of \\eqref{eq:K}. Under $z\\mapsto z+\\delta$ the triple becomes $(a,b-8,c-5)$; its digit decreases by one modulo seven, and its pivot increases exactly when the old digit is zero. These universal carry statements include negative coordinates.","ign}\n\\end{theorem}\n\\begin{proof}\nUse $F(z+1)=F(z)$, $F(z+\\tau)/F(z)=q_s(z)$ and $F(z+s)/F(z)=q_t(z)^{-1}$. For \\eqref{eq:K}, $P_7$ is period one and $\\beta=-15s-2$, producing fifteen negative $s$ shifts in the denominator. For \\eqref{eq:Rs} use $\\beta s=-\\tau$. For \\eqref{eq:Rtau}, $\\beta\\tau=2\\tau+4s+1$ gives two $q_s$ factors and four reciprocal $q_t$ factors; the $2\\tau$ shift disappears from $q_t$ by ellipticity. Finally $P_7(z+\\delta)/P_7(z)=F(z+7\\delta)/F(z)$ and the last two identities of \\eqref{eq:period-identities} give \\eqref{eq:Rdelta}.\n\\end{proof}\nUnder $z\\mapsto z+1$ the triple changes to $(a,b-15,c-10)$, leaving $p$ and $w(c)$ unchanged. Formula~\\eqref{eq:complete-divisor} gives\n\\[\n \\operatorname{ord}_{z+1}R-\\operatorname{ord}_zR\n =-w(c)\\mathbf1_{\\{1\\le b\\le15\\}},\n\\]\nthe divisor of \\eqref{eq:K}. Under $z\\mapsto z+\\delta$ the triple becomes $(a,b-8,c-5)$; its digit decreases by one modulo seven, and its pivot increases exactly when the old digit is zero. These universal carry statements include negative coordinates.","f}\nUse $F(z+1)=F(z)$, $F(z+\\tau)/F(z)=q_s(z)$ and $F(z+s)/F(z)=q_t(z)^{-1}$. For \\eqref{eq:K}, $P_7$ is period one and $\\beta=-15s-2$, producing fifteen negative $s$ shifts in the denominator. For \\eqref{eq:Rs} use $\\beta s=-\\tau$. For \\eqref{eq:Rtau}, $\\beta\\tau=2\\tau+4s+1$ gives two $q_s$ factors and four reciprocal $q_t$ factors; the $2\\tau$ shift disappears from $q_t$ by ellipticity. Finally $P_7(z+\\delta)/P_7(z)=F(z+7\\delta)/F(z)$ and the last two identities of \\eqref{eq:period-identities} give \\eqref{eq:Rdelta}.\n\\end{proof}\nUnder $z\\mapsto z+1$ the triple changes to $(a,b-15,c-10)$, leaving $p$ and $w(c)$ unchanged. Formula~\\eqref{eq:complete-divisor} gives\n\\[\n \\operatorname{ord}_{z+1}R-\\operatorname{ord}_zR\n =-w(c)\\mathbf1_{\\{1\\le b\\le15\\}},\n\\]\nthe divisor of \\eqref{eq:K}. Under $z\\mapsto z+\\delta$ the triple becomes $(a,b-8,c-5)$; its digit decreases by one modulo seven, and its pivot increases exactly when the old digit is zero. These universal carry statements include negative coordinates.","g fifteen negative $s$ shifts in the denominator. For \\eqref{eq:Rs} use $\\beta s=-\\tau$. For \\eqref{eq:Rtau}, $\\beta\\tau=2\\tau+4s+1$ gives two $q_s$ factors and four reciprocal $q_t$ factors; the $2\\tau$ shift disappears from $q_t$ by ellipticity. Finally $P_7(z+\\delta)/P_7(z)=F(z+7\\delta)/F(z)$ and the last two identities of \\eqref{eq:period-identities} give \\eqref{eq:Rdelta}.\n\\end{proof}\nUnder $z\\mapsto z+1$ the triple changes to $(a,b-15,c-10)$, leaving $p$ and $w(c)$ unchanged. Formula~\\eqref{eq:complete-divisor} gives\n\\[\n \\operatorname{ord}_{z+1}R-\\operatorname{ord}_zR\n =-w(c)\\mathbf1_{\\{1\\le b\\le15\\}},\n\\]\nthe divisor of \\eqref{eq:K}. Under $z\\mapsto z+\\delta$ the triple becomes $(a,b-8,c-5)$; its digit decreases by one modulo seven, and its pivot increases exactly when the old digit is zero. These universal carry statements include negative coordinates.","denominator. For \\eqref{eq:Rs} use $\\beta s=-\\tau$. For \\eqref{eq:Rtau}, $\\beta\\tau=2\\tau+4s+1$ gives two $q_s$ factors and four reciprocal $q_t$ factors; the $2\\tau$ shift disappears from $q_t$ by ellipticity. Finally $P_7(z+\\delta)/P_7(z)=F(z+7\\delta)/F(z)$ and the last two identities of \\eqref{eq:period-identities} give \\eqref{eq:Rdelta}.\n\\end{proof}\nUnder $z\\mapsto z+1$ the triple changes to $(a,b-15,c-10)$, leaving $p$ and $w(c)$ unchanged. Formula~\\eqref{eq:complete-divisor} gives\n\\[\n \\operatorname{ord}_{z+1}R-\\operatorname{ord}_zR\n =-w(c)\\mathbf1_{\\{1\\le b\\le15\\}},\n\\]\nthe divisor of \\eqref{eq:K}. Under $z\\mapsto z+\\delta$ the triple becomes $(a,b-8,c-5)$; its digit decreases by one modulo seven, and its pivot increases exactly when the old digit is zero. These universal carry statements include negative coordinates.","q:Rs} use $\\beta s=-\\tau$. For \\eqref{eq:Rtau}, $\\beta\\tau=2\\tau+4s+1$ gives two $q_s$ factors and four reciprocal $q_t$ factors; the $2\\tau$ shift disappears from $q_t$ by ellipticity. Finally $P_7(z+\\delta)/P_7(z)=F(z+7\\delta)/F(z)$ and the last two identities of \\eqref{eq:period-identities} give \\eqref{eq:Rdelta}.\n\\end{proof}\nUnder $z\\mapsto z+1$ the triple changes to $(a,b-15,c-10)$, leaving $p$ and $w(c)$ unchanged. Formula~\\eqref{eq:complete-divisor} gives\n\\[\n \\operatorname{ord}_{z+1}R-\\operatorname{ord}_zR\n =-w(c)\\mathbf1_{\\{1\\le b\\le15\\}},\n\\]\nthe divisor of \\eqref{eq:K}. Under $z\\mapsto z+\\delta$ the triple becomes $(a,b-8,c-5)$; its digit decreases by one modulo seven, and its pivot increases exactly when the old digit is zero. These universal carry statements include negative coordinates.","e $\\beta s=-\\tau$. For \\eqref{eq:Rtau}, $\\beta\\tau=2\\tau+4s+1$ gives two $q_s$ factors and four reciprocal $q_t$ factors; the $2\\tau$ shift disappears from $q_t$ by ellipticity. Finally $P_7(z+\\delta)/P_7(z)=F(z+7\\delta)/F(z)$ and the last two identities of \\eqref{eq:period-identities} give \\eqref{eq:Rdelta}.\n\\end{proof}\nUnder $z\\mapsto z+1$ the triple changes to $(a,b-15,c-10)$, leaving $p$ and $w(c)$ unchanged. Formula~\\eqref{eq:complete-divisor} gives\n\\[\n \\operatorname{ord}_{z+1}R-\\operatorname{ord}_zR\n =-w(c)\\mathbf1_{\\{1\\le b\\le15\\}},\n\\]\nthe divisor of \\eqref{eq:K}. Under $z\\mapsto z+\\delta$ the triple becomes $(a,b-8,c-5)$; its digit decreases by one modulo seven, and its pivot increases exactly when the old digit is zero. These universal carry statements include negative coordinates.","$ gives two $q_s$ factors and four reciprocal $q_t$ factors; the $2\\tau$ shift disappears from $q_t$ by ellipticity. Finally $P_7(z+\\delta)/P_7(z)=F(z+7\\delta)/F(z)$ and the last two identities of \\eqref{eq:period-identities} give \\eqref{eq:Rdelta}.\n\\end{proof}\nUnder $z\\mapsto z+1$ the triple changes to $(a,b-15,c-10)$, leaving $p$ and $w(c)$ unchanged. Formula~\\eqref{eq:complete-divisor} gives\n\\[\n \\operatorname{ord}_{z+1}R-\\operatorname{ord}_zR\n =-w(c)\\mathbf1_{\\{1\\le b\\le15\\}},\n\\]\nthe divisor of \\eqref{eq:K}. Under $z\\mapsto z+\\delta$ the triple becomes $(a,b-8,c-5)$; its digit decreases by one modulo seven, and its pivot increases exactly when the old digit is zero. These universal carry statements include negative coordinates.","$P_7(z+\\delta)/P_7(z)=F(z+7\\delta)/F(z)$ and the last two identities of \\eqref{eq:period-identities} give \\eqref{eq:Rdelta}.\n\\end{proof}\nUnder $z\\mapsto z+1$ the triple changes to $(a,b-15,c-10)$, leaving $p$ and $w(c)$ unchanged. Formula~\\eqref{eq:complete-divisor} gives\n\\[\n \\operatorname{ord}_{z+1}R-\\operatorname{ord}_zR\n =-w(c)\\mathbf1_{\\{1\\le b\\le15\\}},\n\\]\nthe divisor of \\eqref{eq:K}. Under $z\\mapsto z+\\delta$ the triple becomes $(a,b-8,c-5)$; its digit decreases by one modulo seven, and its pivot increases exactly when the old digit is zero. These universal carry statements include negative coordinates.",")$ and the last two identities of \\eqref{eq:period-identities} give \\eqref{eq:Rdelta}.\n\\end{proof}\nUnder $z\\mapsto z+1$ the triple changes to $(a,b-15,c-10)$, leaving $p$ and $w(c)$ unchanged. Formula~\\eqref{eq:complete-divisor} gives\n\\[\n \\operatorname{ord}_{z+1}R-\\operatorname{ord}_zR\n =-w(c)\\mathbf1_{\\{1\\le b\\le15\\}},\n\\]\nthe divisor of \\eqref{eq:K}. Under $z\\mapsto z+\\delta$ the triple becomes $(a,b-8,c-5)$; its digit decreases by one modulo seven, and its pivot increases exactly when the old digit is zero. These universal carry statements include negative coordinates.","\\begin{theorem}[Three-theta anchor]\\label{thm:anchor}\n$R$ is regular and nonzero at zero, and\n\\begin{equation}\\label{eq:R0}\n R_0:=R(0)=\\prod_{j=0}^{2}q_t(u_j)^{-1},\\qquad\n u_j=\\frac{t+\\delta}{2}+j\\delta.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nLet $\\mathcal H(a)=\\eta(5a)/\\eta(a)^5$. Direct product cancellation gives\n\\[\n S_5(z;a,b)\\sim\\frac5{(2\\pi)^4}\\mathcal H(a)\\mathcal H(b)z^{-4}.\n\\]\nThe ratio in \\eqref{eq:p7} therefore has value $P_7(0)=[\\mathcal H(s)/\\mathcal H(t)]/S_5(s;t,\\delta)$. Since $F(0)=\\mathcal H(s)/\\mathcal H(t)=\\epsilon^2$, the eta factors cancel and $R_0=S_5(s;t,\\delta)^{-1}$. The period relation is $s=(t+\\delta)/2+3\\delta-2$. At the central point, $S_5((t+\\delta)/2;t,\\delta)=1$ by termwise coincidence of the gamma numerator an","\\begin{theorem}[Three-theta anchor]\\label{thm:anchor}\n$R$ is regular and nonzero at zero, and\n\\begin{equation}\\label{eq:R0}\n R_0:=R(0)=\\prod_{j=0}^{2}q_t(u_j)^{-1},\\qquad\n u_j=\\frac{t+\\delta}{2}+j\\delta.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nLet $\\mathcal H(a)=\\eta(5a)/\\eta(a)^5$. Direct product cancellation gives\n\\[\n S_5(z;a,b)\\sim\\frac5{(2\\pi)^4}\\mathcal H(a)\\mathcal H(b)z^{-4}.\n\\]\nThe ratio in \\eqref{eq:p7} therefore has value $P_7(0)=[\\mathcal H(s)/\\mathcal H(t)]/S_5(s;t,\\delta)$. Since $F(0)=\\mathcal H(s)/\\mathcal H(t)=\\epsilon^2$, the eta factors cancel and $R_0=S_5(s;t,\\delta)^{-1}$. The period relation is $s=(t+\\delta)/2+3\\delta-2$. At the central point, $S_5((t+\\delta)/2;t,\\delta)=1$ by termwise coincidence of the gamma numerator and denominator factors, at both scales. Three positive $\\delta$ shifts now give \\eqref{eq:R0}. The three arguments have $s$ coefficients $1/7,3/7,5/7","\\begin{theorem}[Three-theta anchor]\\label{thm:anchor}\n$R$ is regular and nonzero at zero, and\n\\begin{equation}\\label{eq:R0}\n R_0:=R(0)=\\prod_{j=0}^{2}q_t(u_j)^{-1},\\qquad\n u_j=\\frac{t+\\delta}{2}+j\\delta.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nLet $\\mathcal H(a)=\\eta(5a)/\\eta(a)^5$. Direct product cancellation gives\n\\[\n S_5(z;a,b)\\sim\\frac5{(2\\pi)^4}\\mathcal H(a)\\mathcal H(b)z^{-4}.\n\\]\nThe ratio in \\eqref{eq:p7} therefore has value $P_7(0)=[\\mathcal H(s)/\\mathcal H(t)]/S_5(s;t,\\delta)$. Since $F(0)=\\mathcal H(s)/\\mathcal H(t)=\\epsilon^2$, the eta factors cancel and $R_0=S_5(s;t,\\delta)^{-1}$. The period relation is $s=(t+\\delta)/2+3\\delta-2$. At the central point, $S_5((t+\\delta)/2;t,\\delta)=1$ by termwise coincidence of the gamma numerator and denominator factors, at both scales. Three positive $\\delta$ shifts now give \\eqref{eq:R0}. The three arguments have $s$ coefficients $1/7,3/7,5/7$ relative to $(1,t,s)$ and thus avoid the divisor of $q_t$.\n\\end{proof}\nA nonze","\\begin{theorem}[Three-theta anchor]\\label{thm:anchor}\n$R$ is regular and nonzero at zero, and\n\\begin{equation}\\label{eq:R0}\n R_0:=R(0)=\\prod_{j=0}^{2}q_t(u_j)^{-1},\\qquad\n u_j=\\frac{t+\\delta}{2}+j\\delta.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nLet $\\mathcal H(a)=\\eta(5a)/\\eta(a)^5$. Direct product cancellation gives\n\\[\n S_5(z;a,b)\\sim\\frac5{(2\\pi)^4}\\mathcal H(a)\\mathcal H(b)z^{-4}.\n\\]\nThe ratio in \\eqref{eq:p7} therefore has value $P_7(0)=[\\mathcal H(s)/\\mathcal H(t)]/S_5(s;t,\\delta)$. Since $F(0)=\\mathcal H(s)/\\mathcal H(t)=\\epsilon^2$, the eta factors cancel and $R_0=S_5(s;t,\\delta)^{-1}$. The period relation is $s=(t+\\delta)/2+3\\delta-2$. At the central point, $S_5((t+\\delta)/2;t,\\delta)=1$ by termwise coincidence of the gamma numerator and denominator factors, at both scales. Three positive $\\delta$ shifts now give \\eqref{eq:R0}. The three arguments have $s$ coefficients $1/7,3/7,5/7$ relative to $(1,t,s)$ and thus avoid the divisor of $q_t$.\n\\end{proof}\nA nonzero meromorphic function with transitions \\eqref{eq:K}, \\eqref{eq:Rs}, \\eqref{eq:Rdelta} and anchor \\eqref","l{thm:anchor}\n$R$ is regular and nonzero at zero, and\n\\begin{equation}\\label{eq:R0}\n R_0:=R(0)=\\prod_{j=0}^{2}q_t(u_j)^{-1},\\qquad\n u_j=\\frac{t+\\delta}{2}+j\\delta.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nLet $\\mathcal H(a)=\\eta(5a)/\\eta(a)^5$. Direct product cancellation gives\n\\[\n S_5(z;a,b)\\sim\\frac5{(2\\pi)^4}\\mathcal H(a)\\mathcal H(b)z^{-4}.\n\\]\nThe ratio in \\eqref{eq:p7} therefore has value $P_7(0)=[\\mathcal H(s)/\\mathcal H(t)]/S_5(s;t,\\delta)$. Since $F(0)=\\mathcal H(s)/\\mathcal H(t)=\\epsilon^2$, the eta factors cancel and $R_0=S_5(s;t,\\delta)^{-1}$. The period relation is $s=(t+\\delta)/2+3\\delta-2$. At the central point, $S_5((t+\\delta)/2;t,\\delta)=1$ by termwise coincidence of the gamma numerator and denominator factors, at both scales. Three positive $\\delta$ shifts now give \\eqref{eq:R0}. The three arguments have $s$ coefficients $1/7,3/7,5/7$ relative to $(1,t,s)$ and thus avoid the divisor of $q_t$.\n\\end{proof}\nA nonzero meromorphic function with transitions \\eqref{eq:K}, \\eqref{eq:Rs}, \\eqref{eq:Rdelta} and anchor \\eqref{eq:R0} equals $R$. Its ratio with $R$ has periods …","quation}\\label{eq:R0}\n R_0:=R(0)=\\prod_{j=0}^{2}q_t(u_j)^{-1},\\qquad\n u_j=\\frac{t+\\delta}{2}+j\\delta.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nLet $\\mathcal H(a)=\\eta(5a)/\\eta(a)^5$. Direct product cancellation gives\n\\[\n S_5(z;a,b)\\sim\\frac5{(2\\pi)^4}\\mathcal H(a)\\mathcal H(b)z^{-4}.\n\\]\nThe ratio in \\eqref{eq:p7} therefore has value $P_7(0)=[\\mathcal H(s)/\\mathcal H(t)]/S_5(s;t,\\delta)$. Since $F(0)=\\mathcal H(s)/\\mathcal H(t)=\\epsilon^2$, the eta factors cancel and $R_0=S_5(s;t,\\delta)^{-1}$. The period relation is $s=(t+\\delta)/2+3\\delta-2$. At the central point, $S_5((t+\\delta)/2;t,\\delta)=1$ by termwise coincidence of the gamma numerator and denominator factors, at both scales. Three positive $\\delta$ shifts now give \\eqref{eq:R0}. The three arguments have $s$ coefficients $1/7,3/7,5/7$ relative to $(1,t,s)$ and thus avoid the divisor of $q_t$.\n\\end{proof}\nA nonzero meromorphic function with transitions \\eqref{eq:K}, \\eqref{eq:Rs}, \\eqref{eq:Rdelta} and anchor \\eqref{eq:R0} equals $R$. Its ratio with $R$ has periods $1,s,\\delta$. Their rational independence makes this rank-thre…","\\frac{t+\\delta}{2}+j\\delta.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nLet $\\mathcal H(a)=\\eta(5a)/\\eta(a)^5$. Direct product cancellation gives\n\\[\n S_5(z;a,b)\\sim\\frac5{(2\\pi)^4}\\mathcal H(a)\\mathcal H(b)z^{-4}.\n\\]\nThe ratio in \\eqref{eq:p7} therefore has value $P_7(0)=[\\mathcal H(s)/\\mathcal H(t)]/S_5(s;t,\\delta)$. Since $F(0)=\\mathcal H(s)/\\mathcal H(t)=\\epsilon^2$, the eta factors cancel and $R_0=S_5(s;t,\\delta)^{-1}$. The period relation is $s=(t+\\delta)/2+3\\delta-2$. At the central point, $S_5((t+\\delta)/2;t,\\delta)=1$ by termwise coincidence of the gamma numerator and denominator factors, at both scales. Three positive $\\delta$ shifts now give \\eqref{eq:R0}. The three arguments have $s$ coefficients $1/7,3/7,5/7$ relative to $(1,t,s)$ and thus avoid the divisor of $q_t$.\n\\end{proof}\nA nonzero meromorphic function with transitions \\eqref{eq:K}, \\eqref{eq:Rs}, \\eqref{eq:Rdelta} and anchor \\eqref{eq:R0} equals $R$. Its ratio with $R$ has periods $1,s,\\delta$. Their rational independence makes this rank-three subgroup nondiscrete. Arbitrarily small nonzero periods force the deriva…","orem}\n\\begin{proof}\nLet $\\mathcal H(a)=\\eta(5a)/\\eta(a)^5$. Direct product cancellation gives\n\\[\n S_5(z;a,b)\\sim\\frac5{(2\\pi)^4}\\mathcal H(a)\\mathcal H(b)z^{-4}.\n\\]\nThe ratio in \\eqref{eq:p7} therefore has value $P_7(0)=[\\mathcal H(s)/\\mathcal H(t)]/S_5(s;t,\\delta)$. Since $F(0)=\\mathcal H(s)/\\mathcal H(t)=\\epsilon^2$, the eta factors cancel and $R_0=S_5(s;t,\\delta)^{-1}$. The period relation is $s=(t+\\delta)/2+3\\delta-2$. At the central point, $S_5((t+\\delta)/2;t,\\delta)=1$ by termwise coincidence of the gamma numerator and denominator factors, at both scales. Three positive $\\delta$ shifts now give \\eqref{eq:R0}. The three arguments have $s$ coefficients $1/7,3/7,5/7$ relative to $(1,t,s)$ and thus avoid the divisor of $q_t$.\n\\end{proof}\nA nonzero meromorphic function with transitions \\eqref{eq:K}, \\eqref{eq:Rs}, \\eqref{eq:Rdelta} and anchor \\eqref{eq:R0} equals $R$. Its ratio with $R$ has periods $1,s,\\delta$. Their rational independence makes this rank-three subgroup nondiscrete. Arbitrarily small nonzero periods force the derivative to vanish at every regular point; hence the ra…","ta(a)^5$. Direct product cancellation gives\n\\[\n S_5(z;a,b)\\sim\\frac5{(2\\pi)^4}\\mathcal H(a)\\mathcal H(b)z^{-4}.\n\\]\nThe ratio in \\eqref{eq:p7} therefore has value $P_7(0)=[\\mathcal H(s)/\\mathcal H(t)]/S_5(s;t,\\delta)$. Since $F(0)=\\mathcal H(s)/\\mathcal H(t)=\\epsilon^2$, the eta factors cancel and $R_0=S_5(s;t,\\delta)^{-1}$. The period relation is $s=(t+\\delta)/2+3\\delta-2$. At the central point, $S_5((t+\\delta)/2;t,\\delta)=1$ by termwise coincidence of the gamma numerator and denominator factors, at both scales. Three positive $\\delta$ shifts now give \\eqref{eq:R0}. The three arguments have $s$ coefficients $1/7,3/7,5/7$ relative to $(1,t,s)$ and thus avoid the divisor of $q_t$.\n\\end{proof}\nA nonzero meromorphic function with transitions \\eqref{eq:K}, \\eqref{eq:Rs}, \\eqref{eq:Rdelta} and anchor \\eqref{eq:R0} equals $R$. Its ratio with $R$ has periods $1,s,\\delta$. Their rational independence makes this rank-three subgroup nondiscrete. Arbitrarily small nonzero periods force the derivative to vanish at every regular point; hence the ratio is constant, and its value at zero is one. Thi…","{eq:p7} therefore has value $P_7(0)=[\\mathcal H(s)/\\mathcal H(t)]/S_5(s;t,\\delta)$. Since $F(0)=\\mathcal H(s)/\\mathcal H(t)=\\epsilon^2$, the eta factors cancel and $R_0=S_5(s;t,\\delta)^{-1}$. The period relation is $s=(t+\\delta)/2+3\\delta-2$. At the central point, $S_5((t+\\delta)/2;t,\\delta)=1$ by termwise coincidence of the gamma numerator and denominator factors, at both scales. Three positive $\\delta$ shifts now give \\eqref{eq:R0}. The three arguments have $s$ coefficients $1/7,3/7,5/7$ relative to $(1,t,s)$ and thus avoid the divisor of $q_t$.\n\\end{proof}\nA nonzero meromorphic function with transitions \\eqref{eq:K}, \\eqref{eq:Rs}, \\eqref{eq:Rdelta} and anchor \\eqref{eq:R0} equals $R$. Its ratio with $R$ has periods $1,s,\\delta$. Their rational independence makes this rank-three subgroup nondiscrete. Arbitrarily small nonzero periods force the derivative to vanish at every regular point; hence the ratio is constant, and its value at zero is one. This is analytic uniqueness, not an arithmetic evaluation at $1/3$.","/S_5(s;t,\\delta)$. Since $F(0)=\\mathcal H(s)/\\mathcal H(t)=\\epsilon^2$, the eta factors cancel and $R_0=S_5(s;t,\\delta)^{-1}$. The period relation is $s=(t+\\delta)/2+3\\delta-2$. At the central point, $S_5((t+\\delta)/2;t,\\delta)=1$ by termwise coincidence of the gamma numerator and denominator factors, at both scales. Three positive $\\delta$ shifts now give \\eqref{eq:R0}. The three arguments have $s$ coefficients $1/7,3/7,5/7$ relative to $(1,t,s)$ and thus avoid the divisor of $q_t$.\n\\end{proof}\nA nonzero meromorphic function with transitions \\eqref{eq:K}, \\eqref{eq:Rs}, \\eqref{eq:Rdelta} and anchor \\eqref{eq:R0} equals $R$. Its ratio with $R$ has periods $1,s,\\delta$. Their rational independence makes this rank-three subgroup nondiscrete. Arbitrarily small nonzero periods force the derivative to vanish at every regular point; hence the ratio is constant, and its value at zero is one. This is analytic uniqueness, not an arithmetic evaluation at $1/3$.",". Since $F(0)=\\mathcal H(s)/\\mathcal H(t)=\\epsilon^2$, the eta factors cancel and $R_0=S_5(s;t,\\delta)^{-1}$. The period relation is $s=(t+\\delta)/2+3\\delta-2$. At the central point, $S_5((t+\\delta)/2;t,\\delta)=1$ by termwise coincidence of the gamma numerator and denominator factors, at both scales. Three positive $\\delta$ shifts now give \\eqref{eq:R0}. The three arguments have $s$ coefficients $1/7,3/7,5/7$ relative to $(1,t,s)$ and thus avoid the divisor of $q_t$.\n\\end{proof}\nA nonzero meromorphic function with transitions \\eqref{eq:K}, \\eqref{eq:Rs}, \\eqref{eq:Rdelta} and anchor \\eqref{eq:R0} equals $R$. Its ratio with $R$ has periods $1,s,\\delta$. Their rational independence makes this rank-three subgroup nondiscrete. Arbitrarily small nonzero periods force the derivative to vanish at every regular point; hence the ratio is constant, and its value at zero is one. This is analytic uniqueness, not an arithmetic evaluation at $1/3$.",")/\\mathcal H(t)=\\epsilon^2$, the eta factors cancel and $R_0=S_5(s;t,\\delta)^{-1}$. The period relation is $s=(t+\\delta)/2+3\\delta-2$. At the central point, $S_5((t+\\delta)/2;t,\\delta)=1$ by termwise coincidence of the gamma numerator and denominator factors, at both scales. Three positive $\\delta$ shifts now give \\eqref{eq:R0}. The three arguments have $s$ coefficients $1/7,3/7,5/7$ relative to $(1,t,s)$ and thus avoid the divisor of $q_t$.\n\\end{proof}\nA nonzero meromorphic function with transitions \\eqref{eq:K}, \\eqref{eq:Rs}, \\eqref{eq:Rdelta} and anchor \\eqref{eq:R0} equals $R$. Its ratio with $R$ has periods $1,s,\\delta$. Their rational independence makes this rank-three subgroup nondiscrete. Arbitrarily small nonzero periods force the derivative to vanish at every regular point; hence the ratio is constant, and its value at zero is one. This is analytic uniqueness, not an arithmetic evaluation at $1/3$.","tors cancel and $R_0=S_5(s;t,\\delta)^{-1}$. The period relation is $s=(t+\\delta)/2+3\\delta-2$. At the central point, $S_5((t+\\delta)/2;t,\\delta)=1$ by termwise coincidence of the gamma numerator and denominator factors, at both scales. Three positive $\\delta$ shifts now give \\eqref{eq:R0}. The three arguments have $s$ coefficients $1/7,3/7,5/7$ relative to $(1,t,s)$ and thus avoid the divisor of $q_t$.\n\\end{proof}\nA nonzero meromorphic function with transitions \\eqref{eq:K}, \\eqref{eq:Rs}, \\eqref{eq:Rdelta} and anchor \\eqref{eq:R0} equals $R$. Its ratio with $R$ has periods $1,s,\\delta$. Their rational independence makes this rank-three subgroup nondiscrete. Arbitrarily small nonzero periods force the derivative to vanish at every regular point; hence the ratio is constant, and its value at zero is one. This is analytic uniqueness, not an arithmetic evaluation at $1/3$.","$ by termwise coincidence of the gamma numerator and denominator factors, at both scales. Three positive $\\delta$ shifts now give \\eqref{eq:R0}. The three arguments have $s$ coefficients $1/7,3/7,5/7$ relative to $(1,t,s)$ and thus avoid the divisor of $q_t$.\n\\end{proof}\nA nonzero meromorphic function with transitions \\eqref{eq:K}, \\eqref{eq:Rs}, \\eqref{eq:Rdelta} and anchor \\eqref{eq:R0} equals $R$. Its ratio with $R$ has periods $1,s,\\delta$. Their rational independence makes this rank-three subgroup nondiscrete. Arbitrarily small nonzero periods force the derivative to vanish at every regular point; hence the ratio is constant, and its value at zero is one. This is analytic uniqueness, not an arithmetic evaluation at $1/3$.","dence of the gamma numerator and denominator factors, at both scales. Three positive $\\delta$ shifts now give \\eqref{eq:R0}. The three arguments have $s$ coefficients $1/7,3/7,5/7$ relative to $(1,t,s)$ and thus avoid the divisor of $q_t$.\n\\end{proof}\nA nonzero meromorphic function with transitions \\eqref{eq:K}, \\eqref{eq:Rs}, \\eqref{eq:Rdelta} and anchor \\eqref{eq:R0} equals $R$. Its ratio with $R$ has periods $1,s,\\delta$. Their rational independence makes this rank-three subgroup nondiscrete. Arbitrarily small nonzero periods force the derivative to vanish at every regular point; hence the ratio is constant, and its value at zero is one. This is analytic uniqueness, not an arithmetic evaluation at $1/3$.","ma numerator and denominator factors, at both scales. Three positive $\\delta$ shifts now give \\eqref{eq:R0}. The three arguments have $s$ coefficients $1/7,3/7,5/7$ relative to $(1,t,s)$ and thus avoid the divisor of $q_t$.\n\\end{proof}\nA nonzero meromorphic function with transitions \\eqref{eq:K}, \\eqref{eq:Rs}, \\eqref{eq:Rdelta} and anchor \\eqref{eq:R0} equals $R$. Its ratio with $R$ has periods $1,s,\\delta$. Their rational independence makes this rank-three subgroup nondiscrete. Arbitrarily small nonzero periods force the derivative to vanish at every regular point; hence the ratio is constant, and its value at zero is one. This is analytic uniqueness, not an arithmetic evaluation at $1/3$.","}, \\eqref{eq:Rs}, \\eqref{eq:Rdelta} and anchor \\eqref{eq:R0} equals $R$. Its ratio with $R$ has periods $1,s,\\delta$. Their rational independence makes this rank-three subgroup nondiscrete. Arbitrarily small nonzero periods force the derivative to vanish at every regular point; hence the ratio is constant, and its value at zero is one. This is analytic uniqueness, not an arithmetic evaluation at $1/3$.","\\subsection{The conjugation divisor cannot be discarded}\nPut $R^\\#(z)=\\overline{R(\\overline z)}$. The cubic fields $\\mathbb Q(\\beta)$ and $\\mathbb Q(\\overline\\beta)$ are distinct: equality would put $\\beta/\\overline\\beta=\\omega$ in a cubic field. Their intersection is $\\mathbb Q$. The support of $\\operatorname{div}R$ lies in the first, and that of $R^\\#$ in the second. Since $\\tau$ is irrational (indeed $\\beta=2-1/(4+15\\tau)$),\n\\[\n \\operatorname{ord}_{\\tau}(R/R^\\#)=4.\n\\]\nThus this naive conjugation quotient is not a holomorphic unit. A constant polarization, root of unity or nowhere-zero","\\subsection{The conjugation divisor cannot be discarded}\nPut $R^\\#(z)=\\overline{R(\\overline z)}$. The cubic fields $\\mathbb Q(\\beta)$ and $\\mathbb Q(\\overline\\beta)$ are distinct: equality would put $\\beta/\\overline\\beta=\\omega$ in a cubic field. Their intersection is $\\mathbb Q$. The support of $\\operatorname{div}R$ lies in the first, and that of $R^\\#$ in the second. Since $\\tau$ is irrational (indeed $\\beta=2-1/(4+15\\tau)$),\n\\[\n \\operatorname{ord}_{\\tau}(R/R^\\#)=4.\n\\]\nThus this naive conjugation quotient is not a holomorphic unit. A constant polarization, root of unity or nowhere-zero holomorphic gauge cannot change this","\\subsection{The conjugation divisor cannot be discarded}\nPut $R^\\#(z)=\\overline{R(\\overline z)}$. The cubic fields $\\mathbb Q(\\beta)$ and $\\mathbb Q(\\overline\\beta)$ are distinct: equality would put $\\beta/\\overline\\beta=\\omega$ in a cubic field. Their intersection is $\\mathbb Q$. The support of $\\operatorname{div}R$ lies in the first, and that of $R^\\#$ in the second. Since $\\tau$ is irrational (indeed $\\beta=2-1/(4+15\\tau)$),\n\\[\n \\operatorname{ord}_{\\tau}(R/R^\\#)=4.\n\\]\nThus this naive conjugation quotient is not a holomorphic unit. A constant polarization, root of unity or nowhere-zero holomorphic gauge cannot change this order. A relative scalar model m","\\subsection{The conjugation divisor cannot be discarded}\nPut $R^\\#(z)=\\overline{R(\\overline z)}$. The cubic fields $\\mathbb Q(\\beta)$ and $\\mathbb Q(\\overline\\beta)$ are distinct: equality would put $\\beta/\\overline\\beta=\\omega$ in a cubic field. Their intersection is $\\mathbb Q$. The support of $\\operatorname{div}R$ lies in the first, and that of $R^\\#$ in the second. Since $\\tau$ is irrational (indeed $\\beta=2-1/(4+15\\tau)$),\n\\[\n \\operatorname{ord}_{\\tau}(R/R^\\#)=4.\n\\]\nThus this naive conjugation quotient is not a holomorphic unit. A constant polarization, root of unity or nowhere-zero holomorphic gauge cannot change this order. A relative scalar model must match the complete divisor, beginning with order four at $\\","\\subsection{The conjugation divisor cannot be discarded}\nPut $R^\\#(z)=\\overline{R(\\overline z)}$. The cubic fields $\\mathbb Q(\\beta)$ and $\\mathbb Q(\\overline\\beta)$ are distinct: equality would put $\\beta/\\overline\\beta=\\omega$ in a cubic field. Their intersection is $\\mathbb Q$. The support of $\\operatorname{div}R$ lies in the first, and that of $R^\\#$ in the second. Since $\\tau$ is irrational (indeed $\\beta=2-1/(4+15\\tau)$),\n\\[\n \\operatorname{ord}_{\\tau}(R/R^\\#)=4.\n\\]\nThus this naive conjugation quotient is not a holomorphic unit. A constant polarization, root of unity or nowhere-zero holomorphic gauge cannot change this order. A relative scalar model must match the complete divisor, beginning with order four at $\\tau$; agreement at this one point alone is insuffici","\\subsection{The conjugation divisor cannot be discarded}\nPut $R^\\#(z)=\\overline{R(\\overline z)}$. The cubic fields $\\mathbb Q(\\beta)$ and $\\mathbb Q(\\overline\\beta)$ are distinct: equality would put $\\beta/\\overline\\beta=\\omega$ in a cubic field. Their intersection is $\\mathbb Q$. The support of $\\operatorname{div}R$ lies in the first, and that of $R^\\#$ in the second. Since $\\tau$ is irrational (indeed $\\beta=2-1/(4+15\\tau)$),\n\\[\n \\operatorname{ord}_{\\tau}(R/R^\\#)=4.\n\\]\nThus this naive conjugation quotient is not a holomorphic unit. A constant polarization, root of unity or nowhere-zero holomorphic gauge cannot change this order. A relative scalar model must match the complete divisor, beginning with order four at $\\tau$; agreement at this one point alone is insufficient. Divisor matching on $\\mathbb C$ a","\\subsection{The conjugation divisor cannot be discarded}\nPut $R^\\#(z)=\\overline{R(\\overline z)}$. The cubic fields $\\mathbb Q(\\beta)$ and $\\mathbb Q(\\overline\\beta)$ are distinct: equality would put $\\beta/\\overline\\beta=\\omega$ in a cubic field. Their intersection is $\\mathbb Q$. The support of $\\operatorname{div}R$ lies in the first, and that of $R^\\#$ in the second. Since $\\tau$ is irrational (indeed $\\beta=2-1/(4+15\\tau)$),\n\\[\n \\operatorname{ord}_{\\tau}(R/R^\\#)=4.\n\\]\nThus this naive conjugation quotient is not a holomorphic unit. A constant polarization, root of unity or nowhere-zero holomorphic gauge cannot change this order. A relative scalar model must match the complete divisor, beginning with order four at $\\tau$; agreement at this one point alone is insufficient. Divisor matching on $\\mathbb C$ also does not supply torus periodicity","\\subsection{The conjugation divisor cannot be discarded}\nPut $R^\\#(z)=\\overline{R(\\overline z)}$. The cubic fields $\\mathbb Q(\\beta)$ and $\\mathbb Q(\\overline\\beta)$ are distinct: equality would put $\\beta/\\overline\\beta=\\omega$ in a cubic field. Their intersection is $\\mathbb Q$. The support of $\\operatorname{div}R$ lies in the first, and that of $R^\\#$ in the second. Since $\\tau$ is irrational (indeed $\\beta=2-1/(4+15\\tau)$),\n\\[\n \\operatorname{ord}_{\\tau}(R/R^\\#)=4.\n\\]\nThus this naive conjugation quotient is not a holomorphic unit. A constant polarization, root of unity or nowhere-zero holomorphic gauge cannot change this order. A relative scalar model must match the complete divisor, beginning with order four at $\\tau$; agreement at this one point alone is insufficient. Divisor matching on $\\mathbb C$ also does not supply torus periodicity or a principal-norm cobounda","ion{The conjugation divisor cannot be discarded}\nPut $R^\\#(z)=\\overline{R(\\overline z)}$. The cubic fields $\\mathbb Q(\\beta)$ and $\\mathbb Q(\\overline\\beta)$ are distinct: equality would put $\\beta/\\overline\\beta=\\omega$ in a cubic field. Their intersection is $\\mathbb Q$. The support of $\\operatorname{div}R$ lies in the first, and that of $R^\\#$ in the second. Since $\\tau$ is irrational (indeed $\\beta=2-1/(4+15\\tau)$),\n\\[\n \\operatorname{ord}_{\\tau}(R/R^\\#)=4.\n\\]\nThus this naive conjugation quotient is not a holomorphic unit. A constant polarization, root of unity or nowhere-zero holomorphic gauge cannot change this order. A relative scalar model must match the complete divisor, beginning with order four at $\\tau$; agreement at this one point alone is insufficient. Divisor matching on $\\mathbb C$ also does not supply torus periodicity or a principal-norm coboundary. The special-value reality problem at the","r cannot be discarded}\nPut $R^\\#(z)=\\overline{R(\\overline z)}$. The cubic fields $\\mathbb Q(\\beta)$ and $\\mathbb Q(\\overline\\beta)$ are distinct: equality would put $\\beta/\\overline\\beta=\\omega$ in a cubic field. Their intersection is $\\mathbb Q$. The support of $\\operatorname{div}R$ lies in the first, and that of $R^\\#$ in the second. Since $\\tau$ is irrational (indeed $\\beta=2-1/(4+15\\tau)$),\n\\[\n \\operatorname{ord}_{\\tau}(R/R^\\#)=4.\n\\]\nThus this naive conjugation quotient is not a holomorphic unit. A constant polarization, root of unity or nowhere-zero holomorphic gauge cannot change this order. A relative scalar model must match the complete divisor, beginning with order four at $\\tau$; agreement at this one point alone is insufficient. Divisor matching on $\\mathbb C$ also does not supply torus periodicity or a principal-norm coboundary. The special-value reality problem at the rational point $1/3$ remains separate.","the first, and that of $R^\\#$ in the second. Since $\\tau$ is irrational (indeed $\\beta=2-1/(4+15\\tau)$),\n\\[\n \\operatorname{ord}_{\\tau}(R/R^\\#)=4.\n\\]\nThus this naive conjugation quotient is not a holomorphic unit. A constant polarization, root of unity or nowhere-zero holomorphic gauge cannot change this order. A relative scalar model must match the complete divisor, beginning with order four at $\\tau$; agreement at this one point alone is insufficient. Divisor matching on $\\mathbb C$ also does not supply torus periodicity or a principal-norm coboundary. The special-value reality problem at the rational point $1/3$ remains separate.","d $\\beta=2-1/(4+15\\tau)$),\n\\[\n \\operatorname{ord}_{\\tau}(R/R^\\#)=4.\n\\]\nThus this naive conjugation quotient is not a holomorphic unit. A constant polarization, root of unity or nowhere-zero holomorphic gauge cannot change this order. A relative scalar model must match the complete divisor, beginning with order four at $\\tau$; agreement at this one point alone is insufficient. Divisor matching on $\\mathbb C$ also does not supply torus periodicity or a principal-norm coboundary. The special-value reality problem at the rational point $1/3$ remains separate.","nstant polarization, root of unity or nowhere-zero holomorphic gauge cannot change this order. A relative scalar model must match the complete divisor, beginning with order four at $\\tau$; agreement at this one point alone is insufficient. Divisor matching on $\\mathbb C$ also does not supply torus periodicity or a principal-norm coboundary. The special-value reality problem at the rational point $1/3$ remains separate.","\\section{Full seven-product series proof}\\label{sec:seven-product-proof}\nFor positive imaginary periods $t,s,d$ with $t+7d=2s+4$, put $P=e^{2\\pi it}$, $Q=e^{2\\pi is}$, $D=e^{2\\pi id}$ and $Z=e^{2\\pi iz}$. Then $PD^7=Q^2$. In a common convergence region the fifth-smoothed logarithms are\n\\begin{align*}\n \\log S_5(z;a,b)&=5\\sum_{\\substack{n\\ge1","\\section{Full seven-product series proof}\\label{sec:seven-product-proof}\nFor positive imaginary periods $t,s,d$ with $t+7d=2s+4$, put $P=e^{2\\pi it}$, $Q=e^{2\\pi is}$, $D=e^{2\\pi id}$ and $Z=e^{2\\pi iz}$. Then $PD^7=Q^2$. In a common convergence region the fifth-smoothed logarithms are\n\\begin{align*}\n \\log S_5(z;a,b)&=5\\sum_{\\substack{n\\ge1\\\\5\\nmid n}}\n \\frac{Z^n-(A_0B_0/Z)^n}{n(1-A_0^n)(1-B_0^n)},\\\\\n \\log F_{t,s}(z)&=-5\\sum_{\\substack{n\\ge1\\\\5\\nmid n}}\n \\frac{P^nZ^n-Q^nZ^{-n}}{n(1-P^n)(1-Q^n)},\n\\end{align*}\nwhere $A_0=e^{2\\pi ia},B_0=e^{2\\pi ib}$ in the first line. At one surviving frequency suppress the index $n$. Summing seven shifts multiplies the two coefficients by $\\sum_{k=0}^{6}D^k$ and $\\sum_{k=0}^{6}D^{-k}$. The exact identities are\n\\begin{align*}\n \\frac{P\\sum_{k=0}^{6}D^k}{(1-P)(1-Q)}\n &=\\frac{1+Q}{(1-P)(1-D)}-\\frac1{(1-Q)(1-D)},","5\\nmid n}}\n \\frac{P^nZ^n-Q^nZ^{-n}}{n(1-P^n)(1-Q^n)},\n\\end{align*}\nwhere $A_0=e^{2\\pi ia},B_0=e^{2\\pi ib}$ in the first line. At one surviving frequency suppress the index $n$. Summing seven shifts multiplies the two coefficients by $\\sum_{k=0}^{6}D^k$ and $\\sum_{k=0}^{6}D^{-k}$. The exact identities are\n\\begin{align*}\n \\frac{P\\sum_{k=0}^{6}D^k}{(1-P)(1-Q)}\n &=\\frac{1+Q}{(1-P)(1-D)}-\\frac1{(1-Q)(1-D)},","5\\nmid n}}\n \\frac{P^nZ^n-Q^nZ^{-n}}{n(1-P^n)(1-Q^n)},\n\\end{align*}\nwhere $A_0=e^{2\\pi ia},B_0=e^{2\\pi ib}$ in the first line. At one surviving frequency suppress the index $n$. Summing seven shifts multiplies the two coefficients by $\\sum_{k=0}^{6}D^k$ and $\\sum_{k=0}^{6}D^{-k}$. The exact identities are\n\\begin{align*}\n \\frac{P\\sum_{k=0}^{6}D^k}{(1-P)(1-Q)}\n &=\\frac{1+Q}{(1-P)(1-D)}-\\frac1{(1-Q)(1-D)},\\\\\n \\frac{Q\\sum_{k=0}^{6}D^{-k}}{(1-P)(1-Q)}\n &=\\frac{D}{1-D}\\left[-\\frac{Q}{1-Q}\n +\\frac{P(1+1/Q)}{1-P}\\right].\n\\end{align*}\nThey follow from $PD^7=Q^2$ and the two rearrangements\n\\[\n P-Q^2=(1+Q)(1-Q)-(1-P)\n =P(1+Q)(1-Q)-Q^2(1-P).\n\\]\nThese coefficients match exactly the logarithm of $S_5(z;s,d)/[S_5(z;t,d)S_5(z+s;t,d)]$. A common region exists: choose $\\Im s>6\\Im d$ and $0<\\Im z<\\Im s-6\\Im d$. Then $\\Im t=2\\Im s-7\\Im d>0$ and every series is absolutely convergent. Exponentiating proves the identity there. The positive-parameter cone $\\Im d>0,\\ 2\\Im s>7\\Im d$ is connected; meromorphic continuation in parameters and argument proves \\eqref{eq:p7} throughout. The real shift four contribute…","\\frac{Q\\sum_{k=0}^{6}D^{-k}}{(1-P)(1-Q)}\n &=\\frac{D}{1-D}\\left[-\\frac{Q}{1-Q}\n +\\frac{P(1+1/Q)}{1-P}\\right].\n\\end{align*}\nThey follow from $PD^7=Q^2$ and the two rearrangements\n\\[\n P-Q^2=(1+Q)(1-Q)-(1-P)\n =P(1+Q)(1-Q)-Q^2(1-P).\n\\]\nThese coefficients match exactly the logarithm of $S_5(z;s,d)/[S_5(z;t,d)S_5(z+s;t,d)]$. A common region exists: choose $\\Im s>6\\Im d$ and $0<\\Im z<\\Im s-6\\Im d$. Then $\\Im t=2\\Im s-7\\Im d>0$ and every series is absolutely convergent. Exponentiating proves the identity there. The positive-parameter cone $\\Im d>0,\\ 2\\Im s>7\\Im d$ is connected; meromorphic continuation in parameters and argument proves \\eqref{eq:p7} throughout. The real shift four contributes no extra exponential.","m $PD^7=Q^2$ and the two rearrangements\n\\[\n P-Q^2=(1+Q)(1-Q)-(1-P)\n =P(1+Q)(1-Q)-Q^2(1-P).\n\\]\nThese coefficients match exactly the logarithm of $S_5(z;s,d)/[S_5(z;t,d)S_5(z+s;t,d)]$. A common region exists: choose $\\Im s>6\\Im d$ and $0<\\Im z<\\Im s-6\\Im d$. Then $\\Im t=2\\Im s-7\\Im d>0$ and every series is absolutely convergent. Exponentiating proves the identity there. The positive-parameter cone $\\Im d>0,\\ 2\\Im s>7\\Im d$ is connected; meromorphic continuation in parameters and argument proves \\eqref{eq:p7} throughout. The real shift four contributes no extra exponential.","The raw identity $P_7(z)=F(\\beta z)$ is false. At $z=\\tau=-t$, $F$ has a zero of order four while the other six shifted factors are nonzero: their $t$ coefficients have nonintegral residues $-k/7$. The denominator argument of positive-period $F(\\beta\\tau)$ has mixed-sign coordinates $(-1,4)$, since $\\beta\\tau=2\\tau+4s+1$, and is regular and nonzero. Thus $\\operatorname{ord}_{\\tau}R=4$. An overall scalar cannot repair a nonzero divisor order.","\\section{Cubic divisor growth and canonical reconstruction}\\label{sec:reconstruction}\n\\noindent\\textit{Source anchor D: v0.14. This section gives the analytic arguments in rendered mathematical notation, incorporating the fresh audit's precise finite-theta grammar and the distinct cutoff symbol $R_*$.}","\\subsection{Explicit three-dimensional families}\nFor integers $n\\ge1$, $1\\le i,j,k\\le n$, define\n\\begin{equation}\\label{eq:families}\n Z_{i,j,k}^{(n)}=\\lambda(-i,j,5(n+k)),\\qquad\n P_{i,j,k,r}^{(n)}=\\lambda(-i,j,5(n+k)-r),\\quad 1\\le r\\le4.\n\\end{equation}\nEvery such triple has $a<0$, $b>0$, $3c>2b$. Hence $p>0$ and the bracket in \\eqref{eq:complete-divisor} is one. The $Z$ points have order four and the $P$ points have order minus one. Rational independence makes the parameter map injective. The pole indices are distinct because $5(k-k')=r-r'$ forces both sides to vanish; the zero and pole families are disjoint by $c$ modulo five.\n\\begin{theorem}[Two-sided cubic growth]\\label{thm:cubic-growth}\nLet $n_+(r)$ and $n_-(r)$ count the zeros and poles of $R$ in $|z|\\l","\\subsection{Explicit three-dimensional families}\nFor integers $n\\ge1$, $1\\le i,j,k\\le n$, define\n\\begin{equation}\\label{eq:families}\n Z_{i,j,k}^{(n)}=\\lambda(-i,j,5(n+k)),\\qquad\n P_{i,j,k,r}^{(n)}=\\lambda(-i,j,5(n+k)-r),\\quad 1\\le r\\le4.\n\\end{equation}\nEvery such triple has $a<0$, $b>0$, $3c>2b$. Hence $p>0$ and the bracket in \\eqref{eq:complete-divisor} is one. The $Z$ points have order four and the $P$ points have order minus one. Rational independence makes the parameter map injective. The pole indices are distinct because $5(k-k')=r-r'$ forces both sides to vanish; the zero and pole families are disjoint by $c$ modulo five.\n\\begin{theorem}[Two-sided cubic growth]\\label{thm:cubic-growth}\nLet $n_+(r)$ and $n_-(r)$ count the zeros and poles of $R$ in $|z|\\le r$, with multipli","\\subsection{Explicit three-dimensional families}\nFor integers $n\\ge1$, $1\\le i,j,k\\le n$, define\n\\begin{equation}\\label{eq:families}\n Z_{i,j,k}^{(n)}=\\lambda(-i,j,5(n+k)),\\qquad\n P_{i,j,k,r}^{(n)}=\\lambda(-i,j,5(n+k)-r),\\quad 1\\le r\\le4.\n\\end{equation}\nEvery such triple has $a<0$, $b>0$, $3c>2b$. Hence $p>0$ and the bracket in \\eqref{eq:complete-divisor} is one. The $Z$ points have order four and the $P$ points have order minus one. Rational independence makes the parameter map injective. The pole indices are distinct because $5(k-k')=r-r'$ forces both sides to vanish; the zero and pole families are disjoint by $c$ modulo five.\n\\begin{theorem}[Two-sided cubic growth]\\label{thm:cubic-growth}\nLet $n_+(r)$ and $n_-(r)$ count the zeros and poles of $R$ in $|z|\\le r$, with multiplicity. Then\n\\[\n n_+(r)=\\Theta(r^3),\\qquad n_-(r)=\\Theta(r^3),\\qquad\n n_D(r):=n_+(r)+n_-(r)=\\Theta(r^3).\n\\]\nIn particular, for $K_0=(|\\tau|+|s|+2)/|\\beta","\\subsection{Explicit three-dimensional families}\nFor integers $n\\ge1$, $1\\le i,j,k\\le n$, define\n\\begin{equation}\\label{eq:families}\n Z_{i,j,k}^{(n)}=\\lambda(-i,j,5(n+k)),\\qquad\n P_{i,j,k,r}^{(n)}=\\lambda(-i,j,5(n+k)-r),\\quad 1\\le r\\le4.\n\\end{equation}\nEvery such triple has $a<0$, $b>0$, $3c>2b$. Hence $p>0$ and the bracket in \\eqref{eq:complete-divisor} is one. The $Z$ points have order four and the $P$ points have order minus one. Rational independence makes the parameter map injective. The pole indices are distinct because $5(k-k')=r-r'$ forces both sides to vanish; the zero and pole families are disjoint by $c$ modulo five.\n\\begin{theorem}[Two-sided cubic growth]\\label{thm:cubic-growth}\nLet $n_+(r)$ and $n_-(r)$ count the zeros and poles of $R$ in $|z|\\le r$, with multiplicity. Then\n\\[\n n_+(r)=\\Theta(r^3),\\qquad n_-(r)=\\Theta(r^3),\\qquad\n n_D(r):=n_+(r)+n_-(r)=\\Theta(r^3).\n\\]\nIn particular, for $K_0=(|\\tau|+|s|+2)/|\\beta|$,\n\\[\n n_+(K_0n)\\ge4n^3,\\qquad n_-(K_0n)","\\subsection{Explicit three-dimensional families}\nFor integers $n\\ge1$, $1\\le i,j,k\\le n$, define\n\\begin{equation}\\label{eq:families}\n Z_{i,j,k}^{(n)}=\\lambda(-i,j,5(n+k)),\\qquad\n P_{i,j,k,r}^{(n)}=\\lambda(-i,j,5(n+k)-r),\\quad 1\\le r\\le4.\n\\end{equation}\nEvery such triple has $a<0$, $b>0$, $3c>2b$. Hence $p>0$ and the bracket in \\eqref{eq:complete-divisor} is one. The $Z$ points have order four and the $P$ points have order minus one. Rational independence makes the parameter map injective. The pole indices are distinct because $5(k-k')=r-r'$ forces both sides to vanish; the zero and pole families are disjoint by $c$ modulo five.\n\\begin{theorem}[Two-sided cubic growth]\\label{thm:cubic-growth}\nLet $n_+(r)$ and $n_-(r)$ count the zeros and poles of $R$ in $|z|\\le r$, with multiplicity. Then\n\\[\n n_+(r)=\\Theta(r^3),\\qquad n_-(r)=\\Theta(r^3),\\qquad\n n_D(r):=n_+(r)+n_-(r)=\\Theta(r^3).\n\\]\nIn particular, for $K_0=(|\\tau|+|s|+2)/|\\beta|$,\n\\[\n n_+(K_0n)\\ge4n^3,\\qquad n_-(K_0n)\\ge4n^3","\\subsection{Explicit three-dimensional families}\nFor integers $n\\ge1$, $1\\le i,j,k\\le n$, define\n\\begin{equation}\\label{eq:families}\n Z_{i,j,k}^{(n)}=\\lambda(-i,j,5(n+k)),\\qquad\n P_{i,j,k,r}^{(n)}=\\lambda(-i,j,5(n+k)-r),\\quad 1\\le r\\le4.\n\\end{equation}\nEvery such triple has $a<0$, $b>0$, $3c>2b$. Hence $p>0$ and the bracket in \\eqref{eq:complete-divisor} is one. The $Z$ points have order four and the $P$ points have order minus one. Rational independence makes the parameter map injective. The pole indices are distinct because $5(k-k')=r-r'$ forces both sides to vanish; the zero and pole families are disjoint by $c$ modulo five.\n\\begin{theorem}[Two-sided cubic growth]\\label{thm:cubic-growth}\nLet $n_+(r)$ and $n_-(r)$ count the zeros and poles of $R$ in $|z|\\le r$, with multiplicity. Then\n\\[\n n_+(r)=\\Theta(r^3),\\qquad n_-(r)=\\Theta(r^3),\\qquad\n n_D(r):=n_+(r)+n_-(r)=\\Theta(r^3).\n\\]\nIn particular, for $K_0=(|\\tau|+|s|+2)/|\\beta|$,\n\\[\n n_+(K_0n)\\ge4n^3,\\qquad n_-(K_0n)\\ge4n^3.\n\\]\nThes","\\subsection{Explicit three-dimensional families}\nFor integers $n\\ge1$, $1\\le i,j,k\\le n$, define\n\\begin{equation}\\label{eq:families}\n Z_{i,j,k}^{(n)}=\\lambda(-i,j,5(n+k)),\\qquad\n P_{i,j,k,r}^{(n)}=\\lambda(-i,j,5(n+k)-r),\\quad 1\\le r\\le4.\n\\end{equation}\nEvery such triple has $a<0$, $b>0$, $3c>2b$. Hence $p>0$ and the bracket in \\eqref{eq:complete-divisor} is one. The $Z$ points have order four and the $P$ points have order minus one. Rational independence makes the parameter map injective. The pole indices are distinct because $5(k-k')=r-r'$ forces both sides to vanish; the zero and pole families are disjoint by $c$ modulo five.\n\\begin{theorem}[Two-sided cubic growth]\\label{thm:cubic-growth}\nLet $n_+(r)$ and $n_-(r)$ count the zeros and poles of $R$ in $|z|\\le r$, with multiplicity. Then\n\\[\n n_+(r)=\\Theta(r^3),\\qquad n_-(r)=\\Theta(r^3),\\qquad\n n_D(r):=n_+(r)+n_-(r)=\\Theta(r^3).\n\\]\nIn particular, for $K_0=(|\\tau|+|s|+2)/|\\beta|$,\n\\[\n n_+(K_0n)\\ge4n^3,\\qquad n_-(K_0n)\\ge4n^3.\n\\]\nThese are unsigne","\\subsection{Explicit three-dimensional families}\nFor integers $n\\ge1$, $1\\le i,j,k\\le n$, define\n\\begin{equation}\\label{eq:families}\n Z_{i,j,k}^{(n)}=\\lambda(-i,j,5(n+k)),\\qquad\n P_{i,j,k,r}^{(n)}=\\lambda(-i,j,5(n+k)-r),\\quad 1\\le r\\le4.\n\\end{equation}\nEvery such triple has $a<0$, $b>0$, $3c>2b$. Hence $p>0$ and the bracket in \\eqref{eq:complete-divisor} is one. The $Z$ points have order four and the $P$ points have order minus one. Rational independence makes the parameter map injective. The pole indices are distinct because $5(k-k')=r-r'$ forces both sides to vanish; the zero and pole families are disjoint by $c$ modulo five.\n\\begin{theorem}[Two-sided cubic growth]\\label{thm:cubic-growth}\nLet $n_+(r)$ and $n_-(r)$ count the zeros and poles of $R$ in $|z|\\le r$, with multiplicity. Then\n\\[\n n_+(r)=\\Theta(r^3),\\qquad n_-(r)=\\Theta(r^3),\\qquad\n n_D(r):=n_+(r)+n_-(r)=\\Theta(r^3).\n\\]\nIn particular, for $K_0=(|\\tau|+|s|+2)/|\\beta|$,\n\\[\n n_+(K_0n)\\ge4n^3,\\qquad n_-(K_0n)\\ge4n^3.\n\\]\nThese are unsigned counts; no cubic lower bound for a signed zero-minus-pole mas","ection{Explicit three-dimensional families}\nFor integers $n\\ge1$, $1\\le i,j,k\\le n$, define\n\\begin{equation}\\label{eq:families}\n Z_{i,j,k}^{(n)}=\\lambda(-i,j,5(n+k)),\\qquad\n P_{i,j,k,r}^{(n)}=\\lambda(-i,j,5(n+k)-r),\\quad 1\\le r\\le4.\n\\end{equation}\nEvery such triple has $a<0$, $b>0$, $3c>2b$. Hence $p>0$ and the bracket in \\eqref{eq:complete-divisor} is one. The $Z$ points have order four and the $P$ points have order minus one. Rational independence makes the parameter map injective. The pole indices are distinct because $5(k-k')=r-r'$ forces both sides to vanish; the zero and pole families are disjoint by $c$ modulo five.\n\\begin{theorem}[Two-sided cubic growth]\\label{thm:cubic-growth}\nLet $n_+(r)$ and $n_-(r)$ count the zeros and poles of $R$ in $|z|\\le r$, with multiplicity. Then\n\\[\n n_+(r)=\\Theta(r^3),\\qquad n_-(r)=\\Theta(r^3),\\qquad\n n_D(r):=n_+(r)+n_-(r)=\\Theta(r^3).\n\\]\nIn particular, for $K_0=(|\\tau|+|s|+2)/|\\beta|$,\n\\[\n n_+(K_0n)\\ge4n^3,\\qquad n_-(K_0n)\\ge4n^3.\n\\]\nThese are unsigned counts; no cubic lower bound for a signed zero-minus-pole mass is asserted.\n\\end{theorem}\n\\begi…","Z_{i,j,k}^{(n)}=\\lambda(-i,j,5(n+k)),\\qquad\n P_{i,j,k,r}^{(n)}=\\lambda(-i,j,5(n+k)-r),\\quad 1\\le r\\le4.\n\\end{equation}\nEvery such triple has $a<0$, $b>0$, $3c>2b$. Hence $p>0$ and the bracket in \\eqref{eq:complete-divisor} is one. The $Z$ points have order four and the $P$ points have order minus one. Rational independence makes the parameter map injective. The pole indices are distinct because $5(k-k')=r-r'$ forces both sides to vanish; the zero and pole families are disjoint by $c$ modulo five.\n\\begin{theorem}[Two-sided cubic growth]\\label{thm:cubic-growth}\nLet $n_+(r)$ and $n_-(r)$ count the zeros and poles of $R$ in $|z|\\le r$, with multiplicity. Then\n\\[\n n_+(r)=\\Theta(r^3),\\qquad n_-(r)=\\Theta(r^3),\\qquad\n n_D(r):=n_+(r)+n_-(r)=\\Theta(r^3).\n\\]\nIn particular, for $K_0=(|\\tau|+|s|+2)/|\\beta|$,\n\\[\n n_+(K_0n)\\ge4n^3,\\qquad n_-(K_0n)\\ge4n^3.\n\\]\nThese are unsigned counts; no cubic lower bound for a signed zero-minus-pole mass is asserted.\n\\end{theorem}\n\\begin{proof}\nIn \\eqref{eq:families}, $|a|,b\\le n$ and $0<c\\le10n$. The triangle inequality places every displayed point in $|z|\\le K_…","\\quad 1\\le r\\le4.\n\\end{equation}\nEvery such triple has $a<0$, $b>0$, $3c>2b$. Hence $p>0$ and the bracket in \\eqref{eq:complete-divisor} is one. The $Z$ points have order four and the $P$ points have order minus one. Rational independence makes the parameter map injective. The pole indices are distinct because $5(k-k')=r-r'$ forces both sides to vanish; the zero and pole families are disjoint by $c$ modulo five.\n\\begin{theorem}[Two-sided cubic growth]\\label{thm:cubic-growth}\nLet $n_+(r)$ and $n_-(r)$ count the zeros and poles of $R$ in $|z|\\le r$, with multiplicity. Then\n\\[\n n_+(r)=\\Theta(r^3),\\qquad n_-(r)=\\Theta(r^3),\\qquad\n n_D(r):=n_+(r)+n_-(r)=\\Theta(r^3).\n\\]\nIn particular, for $K_0=(|\\tau|+|s|+2)/|\\beta|$,\n\\[\n n_+(K_0n)\\ge4n^3,\\qquad n_-(K_0n)\\ge4n^3.\n\\]\nThese are unsigned counts; no cubic lower bound for a signed zero-minus-pole mass is asserted.\n\\end{theorem}\n\\begin{proof}\nIn \\eqref{eq:families}, $|a|,b\\le n$ and $0<c\\le10n$. The triangle inequality places every displayed point in $|z|\\le K_0n$. There are $n^3$ distinct zero points of multiplicity four and $4n^3$ distinct sim…","p>0$ and the bracket in \\eqref{eq:complete-divisor} is one. The $Z$ points have order four and the $P$ points have order minus one. Rational independence makes the parameter map injective. The pole indices are distinct because $5(k-k')=r-r'$ forces both sides to vanish; the zero and pole families are disjoint by $c$ modulo five.\n\\begin{theorem}[Two-sided cubic growth]\\label{thm:cubic-growth}\nLet $n_+(r)$ and $n_-(r)$ count the zeros and poles of $R$ in $|z|\\le r$, with multiplicity. Then\n\\[\n n_+(r)=\\Theta(r^3),\\qquad n_-(r)=\\Theta(r^3),\\qquad\n n_D(r):=n_+(r)+n_-(r)=\\Theta(r^3).\n\\]\nIn particular, for $K_0=(|\\tau|+|s|+2)/|\\beta|$,\n\\[\n n_+(K_0n)\\ge4n^3,\\qquad n_-(K_0n)\\ge4n^3.\n\\]\nThese are unsigned counts; no cubic lower bound for a signed zero-minus-pole mass is asserted.\n\\end{theorem}\n\\begin{proof}\nIn \\eqref{eq:families}, $|a|,b\\le n$ and $0<c\\le10n$. The triangle inequality places every displayed point in $|z|\\le K_0n$. There are $n^3$ distinct zero points of multiplicity four and $4n^3$ distinct simple poles.","bracket in \\eqref{eq:complete-divisor} is one. The $Z$ points have order four and the $P$ points have order minus one. Rational independence makes the parameter map injective. The pole indices are distinct because $5(k-k')=r-r'$ forces both sides to vanish; the zero and pole families are disjoint by $c$ modulo five.\n\\begin{theorem}[Two-sided cubic growth]\\label{thm:cubic-growth}\nLet $n_+(r)$ and $n_-(r)$ count the zeros and poles of $R$ in $|z|\\le r$, with multiplicity. Then\n\\[\n n_+(r)=\\Theta(r^3),\\qquad n_-(r)=\\Theta(r^3),\\qquad\n n_D(r):=n_+(r)+n_-(r)=\\Theta(r^3).\n\\]\nIn particular, for $K_0=(|\\tau|+|s|+2)/|\\beta|$,\n\\[\n n_+(K_0n)\\ge4n^3,\\qquad n_-(K_0n)\\ge4n^3.\n\\]\nThese are unsigned counts; no cubic lower bound for a signed zero-minus-pole mass is asserted.\n\\end{theorem}\n\\begin{proof}\nIn \\eqref{eq:families}, $|a|,b\\le n$ and $0<c\\le10n$. The triangle inequality places every displayed point in $|z|\\le K_0n$. There are $n^3$ distinct zero points of multiplicity four and $4n^3$ distinct simple poles.","is one. The $Z$ points have order four and the $P$ points have order minus one. Rational independence makes the parameter map injective. The pole indices are distinct because $5(k-k')=r-r'$ forces both sides to vanish; the zero and pole families are disjoint by $c$ modulo five.\n\\begin{theorem}[Two-sided cubic growth]\\label{thm:cubic-growth}\nLet $n_+(r)$ and $n_-(r)$ count the zeros and poles of $R$ in $|z|\\le r$, with multiplicity. Then\n\\[\n n_+(r)=\\Theta(r^3),\\qquad n_-(r)=\\Theta(r^3),\\qquad\n n_D(r):=n_+(r)+n_-(r)=\\Theta(r^3).\n\\]\nIn particular, for $K_0=(|\\tau|+|s|+2)/|\\beta|$,\n\\[\n n_+(K_0n)\\ge4n^3,\\qquad n_-(K_0n)\\ge4n^3.\n\\]\nThese are unsigned counts; no cubic lower bound for a signed zero-minus-pole mass is asserted.\n\\end{theorem}\n\\begin{proof}\nIn \\eqref{eq:families}, $|a|,b\\le n$ and $0<c\\le10n$. The triangle inequality places every displayed point in $|z|\\le K_0n$. There are $n^3$ distinct zero points of multiplicity four and $4n^3$ distinct simple poles.",". The $Z$ points have order four and the $P$ points have order minus one. Rational independence makes the parameter map injective. The pole indices are distinct because $5(k-k')=r-r'$ forces both sides to vanish; the zero and pole families are disjoint by $c$ modulo five.\n\\begin{theorem}[Two-sided cubic growth]\\label{thm:cubic-growth}\nLet $n_+(r)$ and $n_-(r)$ count the zeros and poles of $R$ in $|z|\\le r$, with multiplicity. Then\n\\[\n n_+(r)=\\Theta(r^3),\\qquad n_-(r)=\\Theta(r^3),\\qquad\n n_D(r):=n_+(r)+n_-(r)=\\Theta(r^3).\n\\]\nIn particular, for $K_0=(|\\tau|+|s|+2)/|\\beta|$,\n\\[\n n_+(K_0n)\\ge4n^3,\\qquad n_-(K_0n)\\ge4n^3.\n\\]\nThese are unsigned counts; no cubic lower bound for a signed zero-minus-pole mass is asserted.\n\\end{theorem}\n\\begin{proof}\nIn \\eqref{eq:families}, $|a|,b\\le n$ and $0<c\\le10n$. The triangle inequality places every displayed point in $|z|\\le K_0n$. There are $n^3$ distinct zero points of multiplicity four and $4n^3$ distinct simple poles.","the $P$ points have order minus one. Rational independence makes the parameter map injective. The pole indices are distinct because $5(k-k')=r-r'$ forces both sides to vanish; the zero and pole families are disjoint by $c$ modulo five.\n\\begin{theorem}[Two-sided cubic growth]\\label{thm:cubic-growth}\nLet $n_+(r)$ and $n_-(r)$ count the zeros and poles of $R$ in $|z|\\le r$, with multiplicity. Then\n\\[\n n_+(r)=\\Theta(r^3),\\qquad n_-(r)=\\Theta(r^3),\\qquad\n n_D(r):=n_+(r)+n_-(r)=\\Theta(r^3).\n\\]\nIn particular, for $K_0=(|\\tau|+|s|+2)/|\\beta|$,\n\\[\n n_+(K_0n)\\ge4n^3,\\qquad n_-(K_0n)\\ge4n^3.\n\\]\nThese are unsigned counts; no cubic lower bound for a signed zero-minus-pole mass is asserted.\n\\end{theorem}\n\\begin{proof}\nIn \\eqref{eq:families}, $|a|,b\\le n$ and $0<c\\le10n$. The triangle inequality places every displayed point in $|z|\\le K_0n$. There are $n^3$ distinct zero points of multiplicity four and $4n^3$ distinct simple poles.","e distinct because $5(k-k')=r-r'$ forces both sides to vanish; the zero and pole families are disjoint by $c$ modulo five.\n\\begin{theorem}[Two-sided cubic growth]\\label{thm:cubic-growth}\nLet $n_+(r)$ and $n_-(r)$ count the zeros and poles of $R$ in $|z|\\le r$, with multiplicity. Then\n\\[\n n_+(r)=\\Theta(r^3),\\qquad n_-(r)=\\Theta(r^3),\\qquad\n n_D(r):=n_+(r)+n_-(r)=\\Theta(r^3).\n\\]\nIn particular, for $K_0=(|\\tau|+|s|+2)/|\\beta|$,\n\\[\n n_+(K_0n)\\ge4n^3,\\qquad n_-(K_0n)\\ge4n^3.\n\\]\nThese are unsigned counts; no cubic lower bound for a signed zero-minus-pole mass is asserted.\n\\end{theorem}\n\\begin{proof}\nIn \\eqref{eq:families}, $|a|,b\\le n$ and $0<c\\le10n$. The triangle inequality places every displayed point in $|z|\\le K_0n$. There are $n^3$ distinct zero points of multiplicity four and $4n^3$ distinct simple poles.","'$ forces both sides to vanish; the zero and pole families are disjoint by $c$ modulo five.\n\\begin{theorem}[Two-sided cubic growth]\\label{thm:cubic-growth}\nLet $n_+(r)$ and $n_-(r)$ count the zeros and poles of $R$ in $|z|\\le r$, with multiplicity. Then\n\\[\n n_+(r)=\\Theta(r^3),\\qquad n_-(r)=\\Theta(r^3),\\qquad\n n_D(r):=n_+(r)+n_-(r)=\\Theta(r^3).\n\\]\nIn particular, for $K_0=(|\\tau|+|s|+2)/|\\beta|$,\n\\[\n n_+(K_0n)\\ge4n^3,\\qquad n_-(K_0n)\\ge4n^3.\n\\]\nThese are unsigned counts; no cubic lower bound for a signed zero-minus-pole mass is asserted.\n\\end{theorem}\n\\begin{proof}\nIn \\eqref{eq:families}, $|a|,b\\le n$ and $0<c\\le10n$. The triangle inequality places every displayed point in $|z|\\le K_0n$. There are $n^3$ distinct zero points of multiplicity four and $4n^3$ distinct simple poles.","os and poles of $R$ in $|z|\\le r$, with multiplicity. Then\n\\[\n n_+(r)=\\Theta(r^3),\\qquad n_-(r)=\\Theta(r^3),\\qquad\n n_D(r):=n_+(r)+n_-(r)=\\Theta(r^3).\n\\]\nIn particular, for $K_0=(|\\tau|+|s|+2)/|\\beta|$,\n\\[\n n_+(K_0n)\\ge4n^3,\\qquad n_-(K_0n)\\ge4n^3.\n\\]\nThese are unsigned counts; no cubic lower bound for a signed zero-minus-pole mass is asserted.\n\\end{theorem}\n\\begin{proof}\nIn \\eqref{eq:families}, $|a|,b\\le n$ and $0<c\\le10n$. The triangle inequality places every displayed point in $|z|\\le K_0n$. There are $n^3$ distinct zero points of multiplicity four and $4n^3$ distinct simple poles.","R$ in $|z|\\le r$, with multiplicity. Then\n\\[\n n_+(r)=\\Theta(r^3),\\qquad n_-(r)=\\Theta(r^3),\\qquad\n n_D(r):=n_+(r)+n_-(r)=\\Theta(r^3).\n\\]\nIn particular, for $K_0=(|\\tau|+|s|+2)/|\\beta|$,\n\\[\n n_+(K_0n)\\ge4n^3,\\qquad n_-(K_0n)\\ge4n^3.\n\\]\nThese are unsigned counts; no cubic lower bound for a signed zero-minus-pole mass is asserted.\n\\end{theorem}\n\\begin{proof}\nIn \\eqref{eq:families}, $|a|,b\\le n$ and $0<c\\le10n$. The triangle inequality places every displayed point in $|z|\\le K_0n$. There are $n^3$ distinct zero points of multiplicity four and $4n^3$ distinct simple poles.","ad n_-(r)=\\Theta(r^3),\\qquad\n n_D(r):=n_+(r)+n_-(r)=\\Theta(r^3).\n\\]\nIn particular, for $K_0=(|\\tau|+|s|+2)/|\\beta|$,\n\\[\n n_+(K_0n)\\ge4n^3,\\qquad n_-(K_0n)\\ge4n^3.\n\\]\nThese are unsigned counts; no cubic lower bound for a signed zero-minus-pole mass is asserted.\n\\end{theorem}\n\\begin{proof}\nIn \\eqref{eq:families}, $|a|,b\\le n$ and $0<c\\le10n$. The triangle inequality places every displayed point in $|z|\\le K_0n$. There are $n^3$ distinct zero points of multiplicity four and $4n^3$ distinct simple poles.","uad\n n_D(r):=n_+(r)+n_-(r)=\\Theta(r^3).\n\\]\nIn particular, for $K_0=(|\\tau|+|s|+2)/|\\beta|$,\n\\[\n n_+(K_0n)\\ge4n^3,\\qquad n_-(K_0n)\\ge4n^3.\n\\]\nThese are unsigned counts; no cubic lower bound for a signed zero-minus-pole mass is asserted.\n\\end{theorem}\n\\begin{proof}\nIn \\eqref{eq:families}, $|a|,b\\le n$ and $0<c\\le10n$. The triangle inequality places every displayed point in $|z|\\le K_0n$. There are $n^3$ distinct zero points of multiplicity four and $4n^3$ distinct simple poles.","ular, for $K_0=(|\\tau|+|s|+2)/|\\beta|$,\n\\[\n n_+(K_0n)\\ge4n^3,\\qquad n_-(K_0n)\\ge4n^3.\n\\]\nThese are unsigned counts; no cubic lower bound for a signed zero-minus-pole mass is asserted.\n\\end{theorem}\n\\begin{proof}\nIn \\eqref{eq:families}, $|a|,b\\le n$ and $0<c\\le10n$. The triangle inequality places every displayed point in $|z|\\le K_0n$. There are $n^3$ distinct zero points of multiplicity four and $4n^3$ distinct simple poles.","For the upper bound first consider $F$. A nonzero coefficient in \\eqref{eq:base-divisor} has $H(a)\\ne H(b)$, so the imaginary parts of $a\\tau$ and $bs$ have the same sign. Therefore\n\\[\n |\\Im(a\\tau+bs)|=|a|\\Im t+|b|\\Im s.\n\\]\nA bounded complex radius bounds $|a|$ and $|b|$ by $O(r)$; the real part then bounds $|c|$ by $O(r)$. There are $O((1+r)^3)$ triples, and absolute weights are at most four. The seven translated numerator factors and the scaled denominator in \\eqref{eq:p7} give a finite union of such cone supports and cannot increase this order. Combining the upper bound with the explicit lower families, choosing $n=\\lfloor r/K_0\\rfloor$, proves the assertion for all sufficiently large $r$.\n\\end{proof}","A bounded complex radius bounds $|a|$ and $|b|$ by $O(r)$; the real part then bounds $|c|$ by $O(r)$. There are $O((1+r)^3)$ triples, and absolute weights are at most four. The seven translated numerator factors and the scaled denominator in \\eqref{eq:p7} give a finite union of such cone supports and cannot increase this order. Combining the upper bound with the explicit lower families, choosing $n=\\lfloor r/K_0\\rfloor$, proves the assertion for all sufficiently large $r$.\n\\end{proof}","hen bounds $|c|$ by $O(r)$. There are $O((1+r)^3)$ triples, and absolute weights are at most four. The seven translated numerator factors and the scaled denominator in \\eqref{eq:p7} give a finite union of such cone supports and cannot increase this order. Combining the upper bound with the explicit lower families, choosing $n=\\lfloor r/K_0\\rfloor$, proves the assertion for all sufficiently large $r$.\n\\end{proof}","\\subsection{A worked ceiling calculation}\nAt $n=1$ the family consists of the zero $\\lambda(-1,1,10)$ and four poles $\\lambda(-1,1,c)$, $c=6,7,8,9$. The exact digit data are\n\\begin{center}\n\\begin{tabular}{rrrrr}\\toprule\n$c$&$3c-2$&$p=\\lceil(3c-2)/7\\rceil$&$k=7p+2-3c$&$m_\\lambda$","midrule\n6&16&3&5&$-1$","7&19&3&2&$-1$","8&22&4&6&$-1$","9&25&4&3&$-1$","10&28&4&0&$4$","bottomrule\n\\end{tabular}\n\\end{center}\nFor example, the zero has integral lift $(\\widehat a,\\widehat b,\\widehat c)=(4,9,5)$, so $H(4)-H(9)-H(-1)+H(1)=1$ and $w(10)=4$. The $c=6$ pole has lift $(3,7,17)$ and $w(6)=-1$. This finite example illustrates the exact global formula; it does not establish the universal statement by sampling. The same construction at $n=3$ has 27 zero points and 108 pole points, each side with total multiplicity 108.\n\\begin{figure}[htbp]\\centering\\includegraphics[width=.8\\linewidth]{figures/08_complex_families.png}\\caption{The actual $n=2$ divisor families have eight order-four zeros and 32 simple poles. Drawing coordinates are rounded; the cubic-growth proof uses the exact integer parametrization for all $n$.}\\label{fig:family}\\end{figure}","$. The $c=6$ pole has lift $(3,7,17)$ and $w(6)=-1$. This finite example illustrates the exact global formula; it does not establish the universal statement by sampling. The same construction at $n=3$ has 27 zero points and 108 pole points, each side with total multiplicity 108.\n\\begin{figure}[htbp]\\centering\\includegraphics[width=.8\\linewidth]{figures/08_complex_families.png}\\caption{The actual $n=2$ divisor families have eight order-four zeros and 32 simple poles. Drawing coordinates are rounded; the cubic-growth proof uses the exact integer parametrization for all $n$.}\\label{fig:family}\\end{figure}","ame construction at $n=3$ has 27 zero points and 108 pole points, each side with total multiplicity 108.\n\\begin{figure}[htbp]\\centering\\includegraphics[width=.8\\linewidth]{figures/08_complex_families.png}\\caption{The actual $n=2$ divisor families have eight order-four zeros and 32 simple poles. Drawing coordinates are rounded; the cubic-growth proof uses the exact integer parametrization for all $n$.}\\label{fig:family}\\end{figure}","\\subsection{The precise finite-theta obstruction}\n\\begin{theorem}[Fixed affine finite-theta class]\\label{thm:theta-obstruction}\nFix finitely many symmetric period matrices $\\Omega_\\ell$ with positive-definite imaginary parts $\\Im\\Omega_\\ell$, affine maps $z\\mapsto A_\\ell z+b_\\ell$, and finite derivatives of their Riemann theta restrictions. Let $P,Q$ be nonzero entire functions formed by finite polynomial expressions in these restrictions, derivatives and polynomials in $z$. Let $u$ be a single global nowhere-zero entire function. Every meromorphic function\n\\[\n u(z)\\frac{P(z)}{Q(z)}\n\\]\nhas zero and pole counts $O(r^2)$. Thus $R$ is not representable in this class.\n\\end{theorem}\n\\begin{proof}\nFor $Y=\\Im\\Omega>0$, the defining theta series has terms bounded by\n\\[\n \\exp\\{-\\pi n^TYn+2\\pi\\|n\\|\\,\\|\\Im(Az+b)\\|\\}.\n\\]\nCompleting the square, and absorbing half of the positive quadr","\\subsection{The precise finite-theta obstruction}\n\\begin{theorem}[Fixed affine finite-theta class]\\label{thm:theta-obstruction}\nFix finitely many symmetric period matrices $\\Omega_\\ell$ with positive-definite imaginary parts $\\Im\\Omega_\\ell$, affine maps $z\\mapsto A_\\ell z+b_\\ell$, and finite derivatives of their Riemann theta restrictions. Let $P,Q$ be nonzero entire functions formed by finite polynomial expressions in these restrictions, derivatives and polynomials in $z$. Let $u$ be a single global nowhere-zero entire function. Every meromorphic function\n\\[\n u(z)\\frac{P(z)}{Q(z)}\n\\]\nhas zero and pole counts $O(r^2)$. Thus $R$ is not representable in this class.\n\\end{theorem}\n\\begin{proof}\nFor $Y=\\Im\\Omega>0$, the defining theta series has terms bounded by\n\\[\n \\exp\\{-\\pi n^TYn+2\\pi\\|n\\|\\,\\|\\Im(Az+b)\\|\\}.\n\\]\nCompleting the square, and absorbing half of the positive quadratic form into a convergent Gaussian sum, gives $C\\exp(C","\\subsection{The precise finite-theta obstruction}\n\\begin{theorem}[Fixed affine finite-theta class]\\label{thm:theta-obstruction}\nFix finitely many symmetric period matrices $\\Omega_\\ell$ with positive-definite imaginary parts $\\Im\\Omega_\\ell$, affine maps $z\\mapsto A_\\ell z+b_\\ell$, and finite derivatives of their Riemann theta restrictions. Let $P,Q$ be nonzero entire functions formed by finite polynomial expressions in these restrictions, derivatives and polynomials in $z$. Let $u$ be a single global nowhere-zero entire function. Every meromorphic function\n\\[\n u(z)\\frac{P(z)}{Q(z)}\n\\]\nhas zero and pole counts $O(r^2)$. Thus $R$ is not representable in this class.\n\\end{theorem}\n\\begin{proof}\nFor $Y=\\Im\\Omega>0$, the defining theta series has terms bounded by\n\\[\n \\exp\\{-\\pi n^TYn+2\\pi\\|n\\|\\,\\|\\Im(Az+b)\\|\\}.\n\\]\nCompleting the square, and absorbing half of the positive quadratic form into a convergent Gaussian sum, gives $C\\exp(C'|z|^2)$ for a fixed affine restriction.","\\subsection{The precise finite-theta obstruction}\n\\begin{theorem}[Fixed affine finite-theta class]\\label{thm:theta-obstruction}\nFix finitely many symmetric period matrices $\\Omega_\\ell$ with positive-definite imaginary parts $\\Im\\Omega_\\ell$, affine maps $z\\mapsto A_\\ell z+b_\\ell$, and finite derivatives of their Riemann theta restrictions. Let $P,Q$ be nonzero entire functions formed by finite polynomial expressions in these restrictions, derivatives and polynomials in $z$. Let $u$ be a single global nowhere-zero entire function. Every meromorphic function\n\\[\n u(z)\\frac{P(z)}{Q(z)}\n\\]\nhas zero and pole counts $O(r^2)$. Thus $R$ is not representable in this class.\n\\end{theorem}\n\\begin{proof}\nFor $Y=\\Im\\Omega>0$, the defining theta series has terms bounded by\n\\[\n \\exp\\{-\\pi n^TYn+2\\pi\\|n\\|\\,\\|\\Im(Az+b)\\|\\}.\n\\]\nCompleting the square, and absorbing half of the positive quadratic form into a convergent Gaussian sum, gives $C\\exp(C'|z|^2)$ for a fixed affine restriction. Cauchy estimates on fixed-radius circles give the same form for each f","ne finite-theta class]\\label{thm:theta-obstruction}\nFix finitely many symmetric period matrices $\\Omega_\\ell$ with positive-definite imaginary parts $\\Im\\Omega_\\ell$, affine maps $z\\mapsto A_\\ell z+b_\\ell$, and finite derivatives of their Riemann theta restrictions. Let $P,Q$ be nonzero entire functions formed by finite polynomial expressions in these restrictions, derivatives and polynomials in $z$. Let $u$ be a single global nowhere-zero entire function. Every meromorphic function\n\\[\n u(z)\\frac{P(z)}{Q(z)}\n\\]\nhas zero and pole counts $O(r^2)$. Thus $R$ is not representable in this class.\n\\end{theorem}\n\\begin{proof}\nFor $Y=\\Im\\Omega>0$, the defining theta series has terms bounded by\n\\[\n \\exp\\{-\\pi n^TYn+2\\pi\\|n\\|\\,\\|\\Im(Az+b)\\|\\}.\n\\]\nCompleting the square, and absorbing half of the positive quadratic form into a convergent Gaussian sum, gives $C\\exp(C'|z|^2)$ for a fixed affine restriction. Cauchy estimates on fixed-radius circles give the same form for each fixed derivative. Finite sums, products and polynomial factors preserve this bound. Jensen's formula \\cite{Jensen} applied abo…","-theta class]\\label{thm:theta-obstruction}\nFix finitely many symmetric period matrices $\\Omega_\\ell$ with positive-definite imaginary parts $\\Im\\Omega_\\ell$, affine maps $z\\mapsto A_\\ell z+b_\\ell$, and finite derivatives of their Riemann theta restrictions. Let $P,Q$ be nonzero entire functions formed by finite polynomial expressions in these restrictions, derivatives and polynomials in $z$. Let $u$ be a single global nowhere-zero entire function. Every meromorphic function\n\\[\n u(z)\\frac{P(z)}{Q(z)}\n\\]\nhas zero and pole counts $O(r^2)$. Thus $R$ is not representable in this class.\n\\end{theorem}\n\\begin{proof}\nFor $Y=\\Im\\Omega>0$, the defining theta series has terms bounded by\n\\[\n \\exp\\{-\\pi n^TYn+2\\pi\\|n\\|\\,\\|\\Im(Az+b)\\|\\}.\n\\]\nCompleting the square, and absorbing half of the positive quadratic form into a convergent Gaussian sum, gives $C\\exp(C'|z|^2)$ for a fixed affine restriction. Cauchy estimates on fixed-radius circles give the same form for each fixed derivative. Finite sums, products and polynomial factors preserve this bound. Jensen's formula \\cite{Jensen} applied about a poin…","rices $\\Omega_\\ell$ with positive-definite imaginary parts $\\Im\\Omega_\\ell$, affine maps $z\\mapsto A_\\ell z+b_\\ell$, and finite derivatives of their Riemann theta restrictions. Let $P,Q$ be nonzero entire functions formed by finite polynomial expressions in these restrictions, derivatives and polynomials in $z$. Let $u$ be a single global nowhere-zero entire function. Every meromorphic function\n\\[\n u(z)\\frac{P(z)}{Q(z)}\n\\]\nhas zero and pole counts $O(r^2)$. Thus $R$ is not representable in this class.\n\\end{theorem}\n\\begin{proof}\nFor $Y=\\Im\\Omega>0$, the defining theta series has terms bounded by\n\\[\n \\exp\\{-\\pi n^TYn+2\\pi\\|n\\|\\,\\|\\Im(Az+b)\\|\\}.\n\\]\nCompleting the square, and absorbing half of the positive quadratic form into a convergent Gaussian sum, gives $C\\exp(C'|z|^2)$ for a fixed affine restriction. Cauchy estimates on fixed-radius circles give the same form for each fixed derivative. Finite sums, products and polynomial factors preserve this bound. Jensen's formula \\cite{Jensen} applied about a point where $P$ is nonzero, using an outer circle of twice the radius, bounds its zer…","parts $\\Im\\Omega_\\ell$, affine maps $z\\mapsto A_\\ell z+b_\\ell$, and finite derivatives of their Riemann theta restrictions. Let $P,Q$ be nonzero entire functions formed by finite polynomial expressions in these restrictions, derivatives and polynomials in $z$. Let $u$ be a single global nowhere-zero entire function. Every meromorphic function\n\\[\n u(z)\\frac{P(z)}{Q(z)}\n\\]\nhas zero and pole counts $O(r^2)$. Thus $R$ is not representable in this class.\n\\end{theorem}\n\\begin{proof}\nFor $Y=\\Im\\Omega>0$, the defining theta series has terms bounded by\n\\[\n \\exp\\{-\\pi n^TYn+2\\pi\\|n\\|\\,\\|\\Im(Az+b)\\|\\}.\n\\]\nCompleting the square, and absorbing half of the positive quadratic form into a convergent Gaussian sum, gives $C\\exp(C'|z|^2)$ for a fixed affine restriction. Cauchy estimates on fixed-radius circles give the same form for each fixed derivative. Finite sums, products and polynomial factors preserve this bound. Jensen's formula \\cite{Jensen} applied about a point where $P$ is nonzero, using an outer circle of twice the radius, bounds its zero count by $O(r^2)$; the same holds for $Q$. Cancella…","a_\\ell$, affine maps $z\\mapsto A_\\ell z+b_\\ell$, and finite derivatives of their Riemann theta restrictions. Let $P,Q$ be nonzero entire functions formed by finite polynomial expressions in these restrictions, derivatives and polynomials in $z$. Let $u$ be a single global nowhere-zero entire function. Every meromorphic function\n\\[\n u(z)\\frac{P(z)}{Q(z)}\n\\]\nhas zero and pole counts $O(r^2)$. Thus $R$ is not representable in this class.\n\\end{theorem}\n\\begin{proof}\nFor $Y=\\Im\\Omega>0$, the defining theta series has terms bounded by\n\\[\n \\exp\\{-\\pi n^TYn+2\\pi\\|n\\|\\,\\|\\Im(Az+b)\\|\\}.\n\\]\nCompleting the square, and absorbing half of the positive quadratic form into a convergent Gaussian sum, gives $C\\exp(C'|z|^2)$ for a fixed affine restriction. Cauchy estimates on fixed-radius circles give the same form for each fixed derivative. Finite sums, products and polynomial factors preserve this bound. Jensen's formula \\cite{Jensen} applied about a point where $P$ is nonzero, using an outer circle of twice the radius, bounds its zero count by $O(r^2)$; the same holds for $Q$. Cancellation only decre…","of their Riemann theta restrictions. Let $P,Q$ be nonzero entire functions formed by finite polynomial expressions in these restrictions, derivatives and polynomials in $z$. Let $u$ be a single global nowhere-zero entire function. Every meromorphic function\n\\[\n u(z)\\frac{P(z)}{Q(z)}\n\\]\nhas zero and pole counts $O(r^2)$. Thus $R$ is not representable in this class.\n\\end{theorem}\n\\begin{proof}\nFor $Y=\\Im\\Omega>0$, the defining theta series has terms bounded by\n\\[\n \\exp\\{-\\pi n^TYn+2\\pi\\|n\\|\\,\\|\\Im(Az+b)\\|\\}.\n\\]\nCompleting the square, and absorbing half of the positive quadratic form into a convergent Gaussian sum, gives $C\\exp(C'|z|^2)$ for a fixed affine restriction. Cauchy estimates on fixed-radius circles give the same form for each fixed derivative. Finite sums, products and polynomial factors preserve this bound. Jensen's formula \\cite{Jensen} applied about a point where $P$ is nonzero, using an outer circle of twice the radius, bounds its zero count by $O(r^2)$; the same holds for $Q$. Cancellation only decreases their divisor counts. The factor $u$ contributes no finite divisor.…","functions formed by finite polynomial expressions in these restrictions, derivatives and polynomials in $z$. Let $u$ be a single global nowhere-zero entire function. Every meromorphic function\n\\[\n u(z)\\frac{P(z)}{Q(z)}\n\\]\nhas zero and pole counts $O(r^2)$. Thus $R$ is not representable in this class.\n\\end{theorem}\n\\begin{proof}\nFor $Y=\\Im\\Omega>0$, the defining theta series has terms bounded by\n\\[\n \\exp\\{-\\pi n^TYn+2\\pi\\|n\\|\\,\\|\\Im(Az+b)\\|\\}.\n\\]\nCompleting the square, and absorbing half of the positive quadratic form into a convergent Gaussian sum, gives $C\\exp(C'|z|^2)$ for a fixed affine restriction. Cauchy estimates on fixed-radius circles give the same form for each fixed derivative. Finite sums, products and polynomial factors preserve this bound. Jensen's formula \\cite{Jensen} applied about a point where $P$ is nonzero, using an outer circle of twice the radius, bounds its zero count by $O(r^2)$; the same holds for $Q$. Cancellation only decreases their divisor counts. The factor $u$ contributes no finite divisor. Theorem~\\ref{thm:cubic-growth} contradicts this bound for $R$.\n\\…","on. Every meromorphic function\n\\[\n u(z)\\frac{P(z)}{Q(z)}\n\\]\nhas zero and pole counts $O(r^2)$. Thus $R$ is not representable in this class.\n\\end{theorem}\n\\begin{proof}\nFor $Y=\\Im\\Omega>0$, the defining theta series has terms bounded by\n\\[\n \\exp\\{-\\pi n^TYn+2\\pi\\|n\\|\\,\\|\\Im(Az+b)\\|\\}.\n\\]\nCompleting the square, and absorbing half of the positive quadratic form into a convergent Gaussian sum, gives $C\\exp(C'|z|^2)$ for a fixed affine restriction. Cauchy estimates on fixed-radius circles give the same form for each fixed derivative. Finite sums, products and polynomial factors preserve this bound. Jensen's formula \\cite{Jensen} applied about a point where $P$ is nonzero, using an outer circle of twice the radius, bounds its zero count by $O(r^2)$; the same holds for $Q$. Cancellation only decreases their divisor counts. The factor $u$ contributes no finite divisor. Theorem~\\ref{thm:cubic-growth} contradicts this bound for $R$.\n\\end{proof}\nThe single global unit in this grammar cannot be moved inside arbitrary sums: $1+e^{z^3}$ has cubically many zeros and would invalidate that enlarged c…","\\|n\\|\\,\\|\\Im(Az+b)\\|\\}.\n\\]\nCompleting the square, and absorbing half of the positive quadratic form into a convergent Gaussian sum, gives $C\\exp(C'|z|^2)$ for a fixed affine restriction. Cauchy estimates on fixed-radius circles give the same form for each fixed derivative. Finite sums, products and polynomial factors preserve this bound. Jensen's formula \\cite{Jensen} applied about a point where $P$ is nonzero, using an outer circle of twice the radius, bounds its zero count by $O(r^2)$; the same holds for $Q$. Cancellation only decreases their divisor counts. The factor $u$ contributes no finite divisor. Theorem~\\ref{thm:cubic-growth} contradicts this bound for $R$.\n\\end{proof}\nThe single global unit in this grammar cannot be moved inside arbitrary sums: $1+e^{z^3}$ has cubically many zeros and would invalidate that enlarged claim. The theorem concerns fixed finite dimensions and affine one-dimensional restrictions. It leaves infinite products, nonlinear arguments, varying period matrices, nontrivial gerbe transition data and special-value identities at $1/3$ outside its scope.","quadratic form into a convergent Gaussian sum, gives $C\\exp(C'|z|^2)$ for a fixed affine restriction. Cauchy estimates on fixed-radius circles give the same form for each fixed derivative. Finite sums, products and polynomial factors preserve this bound. Jensen's formula \\cite{Jensen} applied about a point where $P$ is nonzero, using an outer circle of twice the radius, bounds its zero count by $O(r^2)$; the same holds for $Q$. Cancellation only decreases their divisor counts. The factor $u$ contributes no finite divisor. Theorem~\\ref{thm:cubic-growth} contradicts this bound for $R$.\n\\end{proof}\nThe single global unit in this grammar cannot be moved inside arbitrary sums: $1+e^{z^3}$ has cubically many zeros and would invalidate that enlarged claim. The theorem concerns fixed finite dimensions and affine one-dimensional restrictions. It leaves infinite products, nonlinear arguments, varying period matrices, nontrivial gerbe transition data and special-value identities at $1/3$ outside its scope.","gent Gaussian sum, gives $C\\exp(C'|z|^2)$ for a fixed affine restriction. Cauchy estimates on fixed-radius circles give the same form for each fixed derivative. Finite sums, products and polynomial factors preserve this bound. Jensen's formula \\cite{Jensen} applied about a point where $P$ is nonzero, using an outer circle of twice the radius, bounds its zero count by $O(r^2)$; the same holds for $Q$. Cancellation only decreases their divisor counts. The factor $u$ contributes no finite divisor. Theorem~\\ref{thm:cubic-growth} contradicts this bound for $R$.\n\\end{proof}\nThe single global unit in this grammar cannot be moved inside arbitrary sums: $1+e^{z^3}$ has cubically many zeros and would invalidate that enlarged claim. The theorem concerns fixed finite dimensions and affine one-dimensional restrictions. It leaves infinite products, nonlinear arguments, varying period matrices, nontrivial gerbe transition data and special-value identities at $1/3$ outside its scope.","iction. Cauchy estimates on fixed-radius circles give the same form for each fixed derivative. Finite sums, products and polynomial factors preserve this bound. Jensen's formula \\cite{Jensen} applied about a point where $P$ is nonzero, using an outer circle of twice the radius, bounds its zero count by $O(r^2)$; the same holds for $Q$. Cancellation only decreases their divisor counts. The factor $u$ contributes no finite divisor. Theorem~\\ref{thm:cubic-growth} contradicts this bound for $R$.\n\\end{proof}\nThe single global unit in this grammar cannot be moved inside arbitrary sums: $1+e^{z^3}$ has cubically many zeros and would invalidate that enlarged claim. The theorem concerns fixed finite dimensions and affine one-dimensional restrictions. It leaves infinite products, nonlinear arguments, varying period matrices, nontrivial gerbe transition data and special-value identities at $1/3$ outside its scope.",". Finite sums, products and polynomial factors preserve this bound. Jensen's formula \\cite{Jensen} applied about a point where $P$ is nonzero, using an outer circle of twice the radius, bounds its zero count by $O(r^2)$; the same holds for $Q$. Cancellation only decreases their divisor counts. The factor $u$ contributes no finite divisor. Theorem~\\ref{thm:cubic-growth} contradicts this bound for $R$.\n\\end{proof}\nThe single global unit in this grammar cannot be moved inside arbitrary sums: $1+e^{z^3}$ has cubically many zeros and would invalidate that enlarged claim. The theorem concerns fixed finite dimensions and affine one-dimensional restrictions. It leaves infinite products, nonlinear arguments, varying period matrices, nontrivial gerbe transition data and special-value identities at $1/3$ outside its scope.","en} applied about a point where $P$ is nonzero, using an outer circle of twice the radius, bounds its zero count by $O(r^2)$; the same holds for $Q$. Cancellation only decreases their divisor counts. The factor $u$ contributes no finite divisor. Theorem~\\ref{thm:cubic-growth} contradicts this bound for $R$.\n\\end{proof}\nThe single global unit in this grammar cannot be moved inside arbitrary sums: $1+e^{z^3}$ has cubically many zeros and would invalidate that enlarged claim. The theorem concerns fixed finite dimensions and affine one-dimensional restrictions. It leaves infinite products, nonlinear arguments, varying period matrices, nontrivial gerbe transition data and special-value identities at $1/3$ outside its scope.","R$.\n\\end{proof}\nThe single global unit in this grammar cannot be moved inside arbitrary sums: $1+e^{z^3}$ has cubically many zeros and would invalidate that enlarged claim. The theorem concerns fixed finite dimensions and affine one-dimensional restrictions. It leaves infinite products, nonlinear arguments, varying period matrices, nontrivial gerbe transition data and special-value identities at $1/3$ outside its scope.","\\subsection{The genus-three divisor product}\nLet $D=\\{\\lambda:m_\\lambda\\ne0\\}$; zero is not in $D$. Define\n\\begin{equation}\\label{eq:WD}\n E_3(w)=(1-w)e^{w+w^2/2+w^3/3},\\qquad\n W_D(z)=\\prod_{\\lambda\\in D}E_3(z/\\lambda)^{m_\\lambda}.\n\\end{equation}\nNegative exponents mean the quotient of the separate positive and negative entire products.\n\\begin{theorem}[Canonical regularizer and absolute genus]\\label{thm:regularizer}\nThe products defining $W_D$ converge normally on compact sets away from their poles, and\n\\[\n \\operatorname{div}W_D=\\operatorname{div}R,\\qquad W_D(0)=1,\\qquad\n (\\log W_D)^{(j)}(0)=0\\quad(1\\le j\\le3).\n\\]\nGenus three is sufficient and minimal for this ungrouped absolute construction. Special signed groupings are not excluded.\n\\end{theorem}\n\\begin{proof}\nThe c","\\subsection{The genus-three divisor product}\nLet $D=\\{\\lambda:m_\\lambda\\ne0\\}$; zero is not in $D$. Define\n\\begin{equation}\\label{eq:WD}\n E_3(w)=(1-w)e^{w+w^2/2+w^3/3},\\qquad\n W_D(z)=\\prod_{\\lambda\\in D}E_3(z/\\lambda)^{m_\\lambda}.\n\\end{equation}\nNegative exponents mean the quotient of the separate positive and negative entire products.\n\\begin{theorem}[Canonical regularizer and absolute genus]\\label{thm:regularizer}\nThe products defining $W_D$ converge normally on compact sets away from their poles, and\n\\[\n \\operatorname{div}W_D=\\operatorname{div}R,\\qquad W_D(0)=1,\\qquad\n (\\log W_D)^{(j)}(0)=0\\quad(1\\le j\\le3).\n\\]\nGenus three is sufficient and minimal for this ungrouped absolute construction. Special signed groupings are not excluded.\n\\end{theorem}\n\\begin{proof}\nThe cubic upper bound giv","\\subsection{The genus-three divisor product}\nLet $D=\\{\\lambda:m_\\lambda\\ne0\\}$; zero is not in $D$. Define\n\\begin{equation}\\label{eq:WD}\n E_3(w)=(1-w)e^{w+w^2/2+w^3/3},\\qquad\n W_D(z)=\\prod_{\\lambda\\in D}E_3(z/\\lambda)^{m_\\lambda}.\n\\end{equation}\nNegative exponents mean the quotient of the separate positive and negative entire products.\n\\begin{theorem}[Canonical regularizer and absolute genus]\\label{thm:regularizer}\nThe products defining $W_D$ converge normally on compact sets away from their poles, and\n\\[\n \\operatorname{div}W_D=\\operatorname{div}R,\\qquad W_D(0)=1,\\qquad\n (\\log W_D)^{(j)}(0)=0\\quad(1\\le j\\le3).\n\\]\nGenus three is sufficient and minimal for this ungrouped absolute construction. Special signed groupings are not excluded.\n\\end{theorem}\n\\begin{proof}\nThe cubic upper bound gives $\\sum_D|m_\\lambda|/|\\lambda|^4<\\infty$, by partial summation or dyadic shells. For $|w|<1$,\n\\[\n \\log E_3(w)=-\\sum_{k\\ge4}\\frac{w^k}","t}\nLet $D=\\{\\lambda:m_\\lambda\\ne0\\}$; zero is not in $D$. Define\n\\begin{equation}\\label{eq:WD}\n E_3(w)=(1-w)e^{w+w^2/2+w^3/3},\\qquad\n W_D(z)=\\prod_{\\lambda\\in D}E_3(z/\\lambda)^{m_\\lambda}.\n\\end{equation}\nNegative exponents mean the quotient of the separate positive and negative entire products.\n\\begin{theorem}[Canonical regularizer and absolute genus]\\label{thm:regularizer}\nThe products defining $W_D$ converge normally on compact sets away from their poles, and\n\\[\n \\operatorname{div}W_D=\\operatorname{div}R,\\qquad W_D(0)=1,\\qquad\n (\\log W_D)^{(j)}(0)=0\\quad(1\\le j\\le3).\n\\]\nGenus three is sufficient and minimal for this ungrouped absolute construction. Special signed groupings are not excluded.\n\\end{theorem}\n\\begin{proof}\nThe cubic upper bound gives $\\sum_D|m_\\lambda|/|\\lambda|^4<\\infty$, by partial summation or dyadic shells. For $|w|<1$,\n\\[\n \\log E_3(w)=-\\sum_{k\\ge4}\\frac{w^k}{k}.\n\\]\nOn each fixed compact set the tail logarithms are thus absolutely and uniformly summable. The finitely many remaining factors supply exactly the stated orders, and each factor has its first three logarit…","gin{equation}\\label{eq:WD}\n E_3(w)=(1-w)e^{w+w^2/2+w^3/3},\\qquad\n W_D(z)=\\prod_{\\lambda\\in D}E_3(z/\\lambda)^{m_\\lambda}.\n\\end{equation}\nNegative exponents mean the quotient of the separate positive and negative entire products.\n\\begin{theorem}[Canonical regularizer and absolute genus]\\label{thm:regularizer}\nThe products defining $W_D$ converge normally on compact sets away from their poles, and\n\\[\n \\operatorname{div}W_D=\\operatorname{div}R,\\qquad W_D(0)=1,\\qquad\n (\\log W_D)^{(j)}(0)=0\\quad(1\\le j\\le3).\n\\]\nGenus three is sufficient and minimal for this ungrouped absolute construction. Special signed groupings are not excluded.\n\\end{theorem}\n\\begin{proof}\nThe cubic upper bound gives $\\sum_D|m_\\lambda|/|\\lambda|^4<\\infty$, by partial summation or dyadic shells. For $|w|<1$,\n\\[\n \\log E_3(w)=-\\sum_{k\\ge4}\\frac{w^k}{k}.\n\\]\nOn each fixed compact set the tail logarithms are thus absolutely and uniformly summable. The finitely many remaining factors supply exactly the stated orders, and each factor has its first three logarithmic jets zero at the origin. Normal convergence justifies taking th…","{thm:regularizer}\nThe products defining $W_D$ converge normally on compact sets away from their poles, and\n\\[\n \\operatorname{div}W_D=\\operatorname{div}R,\\qquad W_D(0)=1,\\qquad\n (\\log W_D)^{(j)}(0)=0\\quad(1\\le j\\le3).\n\\]\nGenus three is sufficient and minimal for this ungrouped absolute construction. Special signed groupings are not excluded.\n\\end{theorem}\n\\begin{proof}\nThe cubic upper bound gives $\\sum_D|m_\\lambda|/|\\lambda|^4<\\infty$, by partial summation or dyadic shells. For $|w|<1$,\n\\[\n \\log E_3(w)=-\\sum_{k\\ge4}\\frac{w^k}{k}.\n\\]\nOn each fixed compact set the tail logarithms are thus absolutely and uniformly summable. The finitely many remaining factors supply exactly the stated orders, and each factor has its first three logarithmic jets zero at the origin. Normal convergence justifies taking these jets termwise.","y from their poles, and\n\\[\n \\operatorname{div}W_D=\\operatorname{div}R,\\qquad W_D(0)=1,\\qquad\n (\\log W_D)^{(j)}(0)=0\\quad(1\\le j\\le3).\n\\]\nGenus three is sufficient and minimal for this ungrouped absolute construction. Special signed groupings are not excluded.\n\\end{theorem}\n\\begin{proof}\nThe cubic upper bound gives $\\sum_D|m_\\lambda|/|\\lambda|^4<\\infty$, by partial summation or dyadic shells. For $|w|<1$,\n\\[\n \\log E_3(w)=-\\sum_{k\\ge4}\\frac{w^k}{k}.\n\\]\nOn each fixed compact set the tail logarithms are thus absolutely and uniformly summable. The finitely many remaining factors supply exactly the stated orders, and each factor has its first three logarithmic jets zero at the origin. Normal convergence justifies taking these jets termwise.","ir poles, and\n\\[\n \\operatorname{div}W_D=\\operatorname{div}R,\\qquad W_D(0)=1,\\qquad\n (\\log W_D)^{(j)}(0)=0\\quad(1\\le j\\le3).\n\\]\nGenus three is sufficient and minimal for this ungrouped absolute construction. Special signed groupings are not excluded.\n\\end{theorem}\n\\begin{proof}\nThe cubic upper bound gives $\\sum_D|m_\\lambda|/|\\lambda|^4<\\infty$, by partial summation or dyadic shells. For $|w|<1$,\n\\[\n \\log E_3(w)=-\\sum_{k\\ge4}\\frac{w^k}{k}.\n\\]\nOn each fixed compact set the tail logarithms are thus absolutely and uniformly summable. The finitely many remaining factors supply exactly the stated orders, and each factor has its first three logarithmic jets zero at the origin. Normal convergence justifies taking these jets termwise.","The lower bound $n_D(r)\\ge c r^3$ yields divergence of $\\sum_D|m_\\lambda|/|\\lambda|^3$: partial summation contains $3\\int n_D(r)r^{-4}\\,dr$, which diverges logarithmically. For a fixed $z\\ne0$,\n\\[\n \\log E_2(z/\\lambda)=-\\frac{z^3}{3\\lambda^3}+O(|\\lambda|^{-4}).\n\\]\nIts weighted absolute logarithm sum diverges, because the fourth-power remainder is summable. This proves failure of ungrouped absolute genus two, without a claim about cancellations in specially grouped signed products.\n\\end{proof}\n\\begin{proposition}[Explicit analytic tail]\\label{prop:tail}\nIf $n_D(r)=\\sum_{|\\lambda|\\le r}|m_\\lambda|\\le Ar^3$ for all $r\\ge R_*$, then for $T\\ge R_*$ and $|z|\\le T/2$,\n\\begin{equation}\\label{eq:tail}\n \\sum_{|\\lambda|>T}\\frac{|m_\\la","The lower bound $n_D(r)\\ge c r^3$ yields divergence of $\\sum_D|m_\\lambda|/|\\lambda|^3$: partial summation contains $3\\int n_D(r)r^{-4}\\,dr$, which diverges logarithmically. For a fixed $z\\ne0$,\n\\[\n \\log E_2(z/\\lambda)=-\\frac{z^3}{3\\lambda^3}+O(|\\lambda|^{-4}).\n\\]\nIts weighted absolute logarithm sum diverges, because the fourth-power remainder is summable. This proves failure of ungrouped absolute genus two, without a claim about cancellations in specially grouped signed products.\n\\end{proof}\n\\begin{proposition}[Explicit analytic tail]\\label{prop:tail}\nIf $n_D(r)=\\sum_{|\\lambda|\\le r}|m_\\lambda|\\le Ar^3$ for all $r\\ge R_*$, then for $T\\ge R_*$ and $|z|\\le T/2$,\n\\begin{equation}\\label{eq:tail}\n \\sum_{|\\lambda|>T}\\frac{|m_\\lambda|}{|\\lambda|^4}\\le\\frac{4A}{T},\n \\qquad |\\log W_{","The lower bound $n_D(r)\\ge c r^3$ yields divergence of $\\sum_D|m_\\lambda|/|\\lambda|^3$: partial summation contains $3\\int n_D(r)r^{-4}\\,dr$, which diverges logarithmically. For a fixed $z\\ne0$,\n\\[\n \\log E_2(z/\\lambda)=-\\frac{z^3}{3\\lambda^3}+O(|\\lambda|^{-4}).\n\\]\nIts weighted absolute logarithm sum diverges, because the fourth-power remainder is summable. This proves failure of ungrouped absolute genus two, without a claim about cancellations in specially grouped signed products.\n\\end{proof}\n\\begin{proposition}[Explicit analytic tail]\\label{prop:tail}\nIf $n_D(r)=\\sum_{|\\lambda|\\le r}|m_\\lambda|\\le Ar^3$ for all $r\\ge R_*$, then for $T\\ge R_*$ and $|z|\\le T/2$,\n\\begin{equation}\\label{eq:tail}\n \\sum_{|\\lambda|>T}\\frac{|m_\\lambda|}{|\\lambda|^4}\\le\\frac{4A}{T},\n \\qquad |\\log W_{D,>T}(z)|\\le\\frac{2A|z|^4}{T}.\n\\end{equation}\n\\end{pr","The lower bound $n_D(r)\\ge c r^3$ yields divergence of $\\sum_D|m_\\lambda|/|\\lambda|^3$: partial summation contains $3\\int n_D(r)r^{-4}\\,dr$, which diverges logarithmically. For a fixed $z\\ne0$,\n\\[\n \\log E_2(z/\\lambda)=-\\frac{z^3}{3\\lambda^3}+O(|\\lambda|^{-4}).\n\\]\nIts weighted absolute logarithm sum diverges, because the fourth-power remainder is summable. This proves failure of ungrouped absolute genus two, without a claim about cancellations in specially grouped signed products.\n\\end{proof}\n\\begin{proposition}[Explicit analytic tail]\\label{prop:tail}\nIf $n_D(r)=\\sum_{|\\lambda|\\le r}|m_\\lambda|\\le Ar^3$ for all $r\\ge R_*$, then for $T\\ge R_*$ and $|z|\\le T/2$,\n\\begin{equation}\\label{eq:tail}\n \\sum_{|\\lambda|>T}\\frac{|m_\\lambda|}{|\\lambda|^4}\\le\\frac{4A}{T},\n \\qquad |\\log W_{D,>T}(z)|\\le\\frac{2A|z|^4}{T}.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nPartial summation gives the","The lower bound $n_D(r)\\ge c r^3$ yields divergence of $\\sum_D|m_\\lambda|/|\\lambda|^3$: partial summation contains $3\\int n_D(r)r^{-4}\\,dr$, which diverges logarithmically. For a fixed $z\\ne0$,\n\\[\n \\log E_2(z/\\lambda)=-\\frac{z^3}{3\\lambda^3}+O(|\\lambda|^{-4}).\n\\]\nIts weighted absolute logarithm sum diverges, because the fourth-power remainder is summable. This proves failure of ungrouped absolute genus two, without a claim about cancellations in specially grouped signed products.\n\\end{proof}\n\\begin{proposition}[Explicit analytic tail]\\label{prop:tail}\nIf $n_D(r)=\\sum_{|\\lambda|\\le r}|m_\\lambda|\\le Ar^3$ for all $r\\ge R_*$, then for $T\\ge R_*$ and $|z|\\le T/2$,\n\\begin{equation}\\label{eq:tail}\n \\sum_{|\\lambda|>T}\\frac{|m_\\lambda|}{|\\lambda|^4}\\le\\frac{4A}{T},\n \\qquad |\\log W_{D,>T}(z)|\\le\\frac{2A|z|^4}{T}.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nPartial summation gives the first upper bound by dropping its nonpositive boundary term and using","thmically. For a fixed $z\\ne0$,\n\\[\n \\log E_2(z/\\lambda)=-\\frac{z^3}{3\\lambda^3}+O(|\\lambda|^{-4}).\n\\]\nIts weighted absolute logarithm sum diverges, because the fourth-power remainder is summable. This proves failure of ungrouped absolute genus two, without a claim about cancellations in specially grouped signed products.\n\\end{proof}\n\\begin{proposition}[Explicit analytic tail]\\label{prop:tail}\nIf $n_D(r)=\\sum_{|\\lambda|\\le r}|m_\\lambda|\\le Ar^3$ for all $r\\ge R_*$, then for $T\\ge R_*$ and $|z|\\le T/2$,\n\\begin{equation}\\label{eq:tail}\n \\sum_{|\\lambda|>T}\\frac{|m_\\lambda|}{|\\lambda|^4}\\le\\frac{4A}{T},\n \\qquad |\\log W_{D,>T}(z)|\\le\\frac{2A|z|^4}{T}.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nPartial summation gives the first upper bound by dropping its nonpositive boundary term and using $4\\int_T^\\infty Ar^3r^{-5}\\,dr=4A/T$. For the tail factors the canonical branch satisfies $|\\log E_3(w)|\\le |w|^4/[4(1-|w|)]\\le |w|^4/2$, proving the second inequality. The condition $|z|\\le T/2$ keeps these tail factors away from zeros and fixes the logarithm through its power series.\n\\end{proof}\nTh…","frac{z^3}{3\\lambda^3}+O(|\\lambda|^{-4}).\n\\]\nIts weighted absolute logarithm sum diverges, because the fourth-power remainder is summable. This proves failure of ungrouped absolute genus two, without a claim about cancellations in specially grouped signed products.\n\\end{proof}\n\\begin{proposition}[Explicit analytic tail]\\label{prop:tail}\nIf $n_D(r)=\\sum_{|\\lambda|\\le r}|m_\\lambda|\\le Ar^3$ for all $r\\ge R_*$, then for $T\\ge R_*$ and $|z|\\le T/2$,\n\\begin{equation}\\label{eq:tail}\n \\sum_{|\\lambda|>T}\\frac{|m_\\lambda|}{|\\lambda|^4}\\le\\frac{4A}{T},\n \\qquad |\\log W_{D,>T}(z)|\\le\\frac{2A|z|^4}{T}.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nPartial summation gives the first upper bound by dropping its nonpositive boundary term and using $4\\int_T^\\infty Ar^3r^{-5}\\,dr=4A/T$. For the tail factors the canonical branch satisfies $|\\log E_3(w)|\\le |w|^4/[4(1-|w|)]\\le |w|^4/2$, proving the second inequality. The condition $|z|\\le T/2$ keeps these tail factors away from zeros and fixes the logarithm through its power series.\n\\end{proof}\nThis is an analytic bound conditional on an explicit countin…","+O(|\\lambda|^{-4}).\n\\]\nIts weighted absolute logarithm sum diverges, because the fourth-power remainder is summable. This proves failure of ungrouped absolute genus two, without a claim about cancellations in specially grouped signed products.\n\\end{proof}\n\\begin{proposition}[Explicit analytic tail]\\label{prop:tail}\nIf $n_D(r)=\\sum_{|\\lambda|\\le r}|m_\\lambda|\\le Ar^3$ for all $r\\ge R_*$, then for $T\\ge R_*$ and $|z|\\le T/2$,\n\\begin{equation}\\label{eq:tail}\n \\sum_{|\\lambda|>T}\\frac{|m_\\lambda|}{|\\lambda|^4}\\le\\frac{4A}{T},\n \\qquad |\\log W_{D,>T}(z)|\\le\\frac{2A|z|^4}{T}.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nPartial summation gives the first upper bound by dropping its nonpositive boundary term and using $4\\int_T^\\infty Ar^3r^{-5}\\,dr=4A/T$. For the tail factors the canonical branch satisfies $|\\log E_3(w)|\\le |w|^4/[4(1-|w|)]\\le |w|^4/2$, proving the second inequality. The condition $|z|\\le T/2$ keeps these tail factors away from zeros and fixes the logarithm through its power series.\n\\end{proof}\nThis is an analytic bound conditional on an explicit counting constant $A$ and cu…","4}).\n\\]\nIts weighted absolute logarithm sum diverges, because the fourth-power remainder is summable. This proves failure of ungrouped absolute genus two, without a claim about cancellations in specially grouped signed products.\n\\end{proof}\n\\begin{proposition}[Explicit analytic tail]\\label{prop:tail}\nIf $n_D(r)=\\sum_{|\\lambda|\\le r}|m_\\lambda|\\le Ar^3$ for all $r\\ge R_*$, then for $T\\ge R_*$ and $|z|\\le T/2$,\n\\begin{equation}\\label{eq:tail}\n \\sum_{|\\lambda|>T}\\frac{|m_\\lambda|}{|\\lambda|^4}\\le\\frac{4A}{T},\n \\qquad |\\log W_{D,>T}(z)|\\le\\frac{2A|z|^4}{T}.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nPartial summation gives the first upper bound by dropping its nonpositive boundary term and using $4\\int_T^\\infty Ar^3r^{-5}\\,dr=4A/T$. For the tail factors the canonical branch satisfies $|\\log E_3(w)|\\le |w|^4/[4(1-|w|)]\\le |w|^4/2$, proving the second inequality. The condition $|z|\\le T/2$ keeps these tail factors away from zeros and fixes the logarithm through its power series.\n\\end{proof}\nThis is an analytic bound conditional on an explicit counting constant $A$ and cutoff $R_*$. A b…","logarithm sum diverges, because the fourth-power remainder is summable. This proves failure of ungrouped absolute genus two, without a claim about cancellations in specially grouped signed products.\n\\end{proof}\n\\begin{proposition}[Explicit analytic tail]\\label{prop:tail}\nIf $n_D(r)=\\sum_{|\\lambda|\\le r}|m_\\lambda|\\le Ar^3$ for all $r\\ge R_*$, then for $T\\ge R_*$ and $|z|\\le T/2$,\n\\begin{equation}\\label{eq:tail}\n \\sum_{|\\lambda|>T}\\frac{|m_\\lambda|}{|\\lambda|^4}\\le\\frac{4A}{T},\n \\qquad |\\log W_{D,>T}(z)|\\le\\frac{2A|z|^4}{T}.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nPartial summation gives the first upper bound by dropping its nonpositive boundary term and using $4\\int_T^\\infty Ar^3r^{-5}\\,dr=4A/T$. For the tail factors the canonical branch satisfies $|\\log E_3(w)|\\le |w|^4/[4(1-|w|)]\\le |w|^4/2$, proving the second inequality. The condition $|z|\\le T/2$ keeps these tail factors away from zeros and fixes the logarithm through its power series.\n\\end{proof}\nThis is an analytic bound conditional on an explicit counting constant $A$ and cutoff $R_*$. A bounded integer search alone do…","n_D(r)=\\sum_{|\\lambda|\\le r}|m_\\lambda|\\le Ar^3$ for all $r\\ge R_*$, then for $T\\ge R_*$ and $|z|\\le T/2$,\n\\begin{equation}\\label{eq:tail}\n \\sum_{|\\lambda|>T}\\frac{|m_\\lambda|}{|\\lambda|^4}\\le\\frac{4A}{T},\n \\qquad |\\log W_{D,>T}(z)|\\le\\frac{2A|z|^4}{T}.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nPartial summation gives the first upper bound by dropping its nonpositive boundary term and using $4\\int_T^\\infty Ar^3r^{-5}\\,dr=4A/T$. For the tail factors the canonical branch satisfies $|\\log E_3(w)|\\le |w|^4/[4(1-|w|)]\\le |w|^4/2$, proving the second inequality. The condition $|z|\\le T/2$ keeps these tail factors away from zeros and fixes the logarithm through its power series.\n\\end{proof}\nThis is an analytic bound conditional on an explicit counting constant $A$ and cutoff $R_*$. A bounded integer search alone does not certify those infinite-support hypotheses. No evaluated numerical enclosure of $W_D(1/3)$ is claimed here.","nd $|z|\\le T/2$,\n\\begin{equation}\\label{eq:tail}\n \\sum_{|\\lambda|>T}\\frac{|m_\\lambda|}{|\\lambda|^4}\\le\\frac{4A}{T},\n \\qquad |\\log W_{D,>T}(z)|\\le\\frac{2A|z|^4}{T}.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nPartial summation gives the first upper bound by dropping its nonpositive boundary term and using $4\\int_T^\\infty Ar^3r^{-5}\\,dr=4A/T$. For the tail factors the canonical branch satisfies $|\\log E_3(w)|\\le |w|^4/[4(1-|w|)]\\le |w|^4/2$, proving the second inequality. The condition $|z|\\le T/2$ keeps these tail factors away from zeros and fixes the logarithm through its power series.\n\\end{proof}\nThis is an analytic bound conditional on an explicit counting constant $A$ and cutoff $R_*$. A bounded integer search alone does not certify those infinite-support hypotheses. No evaluated numerical enclosure of $W_D(1/3)$ is claimed here.","da|^4}\\le\\frac{4A}{T},\n \\qquad |\\log W_{D,>T}(z)|\\le\\frac{2A|z|^4}{T}.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nPartial summation gives the first upper bound by dropping its nonpositive boundary term and using $4\\int_T^\\infty Ar^3r^{-5}\\,dr=4A/T$. For the tail factors the canonical branch satisfies $|\\log E_3(w)|\\le |w|^4/[4(1-|w|)]\\le |w|^4/2$, proving the second inequality. The condition $|z|\\le T/2$ keeps these tail factors away from zeros and fixes the logarithm through its power series.\n\\end{proof}\nThis is an analytic bound conditional on an explicit counting constant $A$ and cutoff $R_*$. A bounded integer search alone does not certify those infinite-support hypotheses. No evaluated numerical enclosure of $W_D(1/3)$ is claimed here.","tive boundary term and using $4\\int_T^\\infty Ar^3r^{-5}\\,dr=4A/T$. For the tail factors the canonical branch satisfies $|\\log E_3(w)|\\le |w|^4/[4(1-|w|)]\\le |w|^4/2$, proving the second inequality. The condition $|z|\\le T/2$ keeps these tail factors away from zeros and fixes the logarithm through its power series.\n\\end{proof}\nThis is an analytic bound conditional on an explicit counting constant $A$ and cutoff $R_*$. A bounded integer search alone does not certify those infinite-support hypotheses. No evaluated numerical enclosure of $W_D(1/3)$ is claimed here.","erm and using $4\\int_T^\\infty Ar^3r^{-5}\\,dr=4A/T$. For the tail factors the canonical branch satisfies $|\\log E_3(w)|\\le |w|^4/[4(1-|w|)]\\le |w|^4/2$, proving the second inequality. The condition $|z|\\le T/2$ keeps these tail factors away from zeros and fixes the logarithm through its power series.\n\\end{proof}\nThis is an analytic bound conditional on an explicit counting constant $A$ and cutoff $R_*$. A bounded integer search alone does not certify those infinite-support hypotheses. No evaluated numerical enclosure of $W_D(1/3)$ is claimed here.","4/[4(1-|w|)]\\le |w|^4/2$, proving the second inequality. The condition $|z|\\le T/2$ keeps these tail factors away from zeros and fixes the logarithm through its power series.\n\\end{proof}\nThis is an analytic bound conditional on an explicit counting constant $A$ and cutoff $R_*$. A bounded integer search alone does not certify those infinite-support hypotheses. No evaluated numerical enclosure of $W_D(1/3)$ is claimed here.","\\subsection{The fourth logarithmic derivative fixes the remaining degree}\n\\begin{theorem}[Cubic-exponential reconstruction]\\label{thm:reconstruction}\nFor the actual correction $R$,\n\\begin{equation}\\label{eq:fourth}\n (\\log R)^{(4)}(z)=-6\\sum_{\\lambda\\in D}\\frac{m_\\lambda}{(z-\\lambda)^4},\n\\end{equation}\nlocally normally away from $D$. Consequently\n\\begin{equation}\\label{eq:cubic-reconstruction}\n R(z)=R_0\\exp(\\alpha_1z+\\alpha_2z^2+\\alpha_3z^3)W_D(z),\\qquad\n \\alpha_j=\\frac{(\\log R)^{(j)}(0)}{j!}.\n\\end{equation}\nThe logarithmic derivatives are branch independent.\n\\end{theorem}\n\\begin{proof}\nThe notation $(\\log R)^{(4)}$ means the third derivative of $R'/R$, a single-valued meromorphic function. The sine partial fractions \\cite{DLMFSine} give\n\\[\n \\frac{d^4}{dw^4}\\log(1-e^{2\\pi iw})\n =-6\\sum_{n\\in\\mathbb Z}(w-n)^{-4}.\n\\]\nThe linear exponential factor relating $1-e^{2\\pi iw","\\subsection{The fourth logarithmic derivative fixes the remaining degree}\n\\begin{theorem}[Cubic-exponential reconstruction]\\label{thm:reconstruction}\nFor the actual correction $R$,\n\\begin{equation}\\label{eq:fourth}\n (\\log R)^{(4)}(z)=-6\\sum_{\\lambda\\in D}\\frac{m_\\lambda}{(z-\\lambda)^4},\n\\end{equation}\nlocally normally away from $D$. Consequently\n\\begin{equation}\\label{eq:cubic-reconstruction}\n R(z)=R_0\\exp(\\alpha_1z+\\alpha_2z^2+\\alpha_3z^3)W_D(z),\\qquad\n \\alpha_j=\\frac{(\\log R)^{(j)}(0)}{j!}.\n\\end{equation}\nThe logarithmic derivatives are branch independent.\n\\end{theorem}\n\\begin{proof}\nThe notation $(\\log R)^{(4)}$ means the third derivative of $R'/R$, a single-valued meromorphic function. The sine partial fractions \\cite{DLMFSine} give\n\\[\n \\frac{d^4}{dw^4}\\log(1-e^{2\\pi iw})\n =-6\\sum_{n\\in\\mathbb Z}(w-n)^{-4}.\n\\]\nThe linear exponential factor relating $1-e^{2\\pi iw}$ to $\\sin\\pi w$ has zero fourth logarithmic derivative. Apply this identity to each factor of the defining d","\\subsection{The fourth logarithmic derivative fixes the remaining degree}\n\\begin{theorem}[Cubic-exponential reconstruction]\\label{thm:reconstruction}\nFor the actual correction $R$,\n\\begin{equation}\\label{eq:fourth}\n (\\log R)^{(4)}(z)=-6\\sum_{\\lambda\\in D}\\frac{m_\\lambda}{(z-\\lambda)^4},\n\\end{equation}\nlocally normally away from $D$. Consequently\n\\begin{equation}\\label{eq:cubic-reconstruction}\n R(z)=R_0\\exp(\\alpha_1z+\\alpha_2z^2+\\alpha_3z^3)W_D(z),\\qquad\n \\alpha_j=\\frac{(\\log R)^{(j)}(0)}{j!}.\n\\end{equation}\nThe logarithmic derivatives are branch independent.\n\\end{theorem}\n\\begin{proof}\nThe notation $(\\log R)^{(4)}$ means the third derivative of $R'/R$, a single-valued meromorphic function. The sine partial fractions \\cite{DLMFSine} give\n\\[\n \\frac{d^4}{dw^4}\\log(1-e^{2\\pi iw})\n =-6\\sum_{n\\in\\mathbb Z}(w-n)^{-4}.\n\\]\nThe linear exponential factor relating $1-e^{2\\pi iw}$ to $\\sin\\pi w$ has zero fourth logarithmic derivative. Apply this identity to each factor of the defining double gamma products. Their three-index cone","\\subsection{The fourth logarithmic derivative fixes the remaining degree}\n\\begin{theorem}[Cubic-exponential reconstruction]\\label{thm:reconstruction}\nFor the actual correction $R$,\n\\begin{equation}\\label{eq:fourth}\n (\\log R)^{(4)}(z)=-6\\sum_{\\lambda\\in D}\\frac{m_\\lambda}{(z-\\lambda)^4},\n\\end{equation}\nlocally normally away from $D$. Consequently\n\\begin{equation}\\label{eq:cubic-reconstruction}\n R(z)=R_0\\exp(\\alpha_1z+\\alpha_2z^2+\\alpha_3z^3)W_D(z),\\qquad\n \\alpha_j=\\frac{(\\log R)^{(j)}(0)}{j!}.\n\\end{equation}\nThe logarithmic derivatives are branch independent.\n\\end{theorem}\n\\begin{proof}\nThe notation $(\\log R)^{(4)}$ means the third derivative of $R'/R$, a single-valued meromorphic function. The sine partial fractions \\cite{DLMFSine} give\n\\[\n \\frac{d^4}{dw^4}\\log(1-e^{2\\pi iw})\n =-6\\sum_{n\\in\\mathbb Z}(w-n)^{-4}.\n\\]\nThe linear exponential factor relating $1-e^{2\\pi iw}$ to $\\sin\\pi w$ has zero fourth logarithmic derivative. Apply this identity to each factor of the defining double gamma products. Their three-index cone supports have $O(r^3)$ counting. Initially work off the union of th…","fourth}\n (\\log R)^{(4)}(z)=-6\\sum_{\\lambda\\in D}\\frac{m_\\lambda}{(z-\\lambda)^4},\n\\end{equation}\nlocally normally away from $D$. Consequently\n\\begin{equation}\\label{eq:cubic-reconstruction}\n R(z)=R_0\\exp(\\alpha_1z+\\alpha_2z^2+\\alpha_3z^3)W_D(z),\\qquad\n \\alpha_j=\\frac{(\\log R)^{(j)}(0)}{j!}.\n\\end{equation}\nThe logarithmic derivatives are branch independent.\n\\end{theorem}\n\\begin{proof}\nThe notation $(\\log R)^{(4)}$ means the third derivative of $R'/R$, a single-valued meromorphic function. The sine partial fractions \\cite{DLMFSine} give\n\\[\n \\frac{d^4}{dw^4}\\log(1-e^{2\\pi iw})\n =-6\\sum_{n\\in\\mathbb Z}(w-n)^{-4}.\n\\]\nThe linear exponential factor relating $1-e^{2\\pi iw}$ to $\\sin\\pi w$ has zero fourth logarithmic derivative. Apply this identity to each factor of the defining double gamma products. Their three-index cone supports have $O(r^3)$ counting. Initially work off the union of the individual gamma-factor divisors. The fourth-power sums are locally absolutely convergent there; rearrangement, multiplication by the argument scale $\\beta$ and cancellation of multiplicities are legitimat…","}\\frac{m_\\lambda}{(z-\\lambda)^4},\n\\end{equation}\nlocally normally away from $D$. Consequently\n\\begin{equation}\\label{eq:cubic-reconstruction}\n R(z)=R_0\\exp(\\alpha_1z+\\alpha_2z^2+\\alpha_3z^3)W_D(z),\\qquad\n \\alpha_j=\\frac{(\\log R)^{(j)}(0)}{j!}.\n\\end{equation}\nThe logarithmic derivatives are branch independent.\n\\end{theorem}\n\\begin{proof}\nThe notation $(\\log R)^{(4)}$ means the third derivative of $R'/R$, a single-valued meromorphic function. The sine partial fractions \\cite{DLMFSine} give\n\\[\n \\frac{d^4}{dw^4}\\log(1-e^{2\\pi iw})\n =-6\\sum_{n\\in\\mathbb Z}(w-n)^{-4}.\n\\]\nThe linear exponential factor relating $1-e^{2\\pi iw}$ to $\\sin\\pi w$ has zero fourth logarithmic derivative. Apply this identity to each factor of the defining double gamma products. Their three-index cone supports have $O(r^3)$ counting. Initially work off the union of the individual gamma-factor divisors. The fourth-power sums are locally absolutely convergent there; rearrangement, multiplication by the argument scale $\\beta$ and cancellation of multiplicities are legitimate. After coefficient cancellation, meromorphic …","begin{equation}\\label{eq:cubic-reconstruction}\n R(z)=R_0\\exp(\\alpha_1z+\\alpha_2z^2+\\alpha_3z^3)W_D(z),\\qquad\n \\alpha_j=\\frac{(\\log R)^{(j)}(0)}{j!}.\n\\end{equation}\nThe logarithmic derivatives are branch independent.\n\\end{theorem}\n\\begin{proof}\nThe notation $(\\log R)^{(4)}$ means the third derivative of $R'/R$, a single-valued meromorphic function. The sine partial fractions \\cite{DLMFSine} give\n\\[\n \\frac{d^4}{dw^4}\\log(1-e^{2\\pi iw})\n =-6\\sum_{n\\in\\mathbb Z}(w-n)^{-4}.\n\\]\nThe linear exponential factor relating $1-e^{2\\pi iw}$ to $\\sin\\pi w$ has zero fourth logarithmic derivative. Apply this identity to each factor of the defining double gamma products. Their three-index cone supports have $O(r^3)$ counting. Initially work off the union of the individual gamma-factor divisors. The fourth-power sums are locally absolutely convergent there; rearrangement, multiplication by the argument scale $\\beta$ and cancellation of multiplicities are legitimate. After coefficient cancellation, meromorphic continuation gives \\eqref{eq:fourth} on $\\mathbb C\\setminus D$, including points where individu…","j=\\frac{(\\log R)^{(j)}(0)}{j!}.\n\\end{equation}\nThe logarithmic derivatives are branch independent.\n\\end{theorem}\n\\begin{proof}\nThe notation $(\\log R)^{(4)}$ means the third derivative of $R'/R$, a single-valued meromorphic function. The sine partial fractions \\cite{DLMFSine} give\n\\[\n \\frac{d^4}{dw^4}\\log(1-e^{2\\pi iw})\n =-6\\sum_{n\\in\\mathbb Z}(w-n)^{-4}.\n\\]\nThe linear exponential factor relating $1-e^{2\\pi iw}$ to $\\sin\\pi w$ has zero fourth logarithmic derivative. Apply this identity to each factor of the defining double gamma products. Their three-index cone supports have $O(r^3)$ counting. Initially work off the union of the individual gamma-factor divisors. The fourth-power sums are locally absolutely convergent there; rearrangement, multiplication by the argument scale $\\beta$ and cancellation of multiplicities are legitimate. After coefficient cancellation, meromorphic continuation gives \\eqref{eq:fourth} on $\\mathbb C\\setminus D$, including points where individual singularities canceled. A conditionally convergent third-power sum is never used.","j)}(0)}{j!}.\n\\end{equation}\nThe logarithmic derivatives are branch independent.\n\\end{theorem}\n\\begin{proof}\nThe notation $(\\log R)^{(4)}$ means the third derivative of $R'/R$, a single-valued meromorphic function. The sine partial fractions \\cite{DLMFSine} give\n\\[\n \\frac{d^4}{dw^4}\\log(1-e^{2\\pi iw})\n =-6\\sum_{n\\in\\mathbb Z}(w-n)^{-4}.\n\\]\nThe linear exponential factor relating $1-e^{2\\pi iw}$ to $\\sin\\pi w$ has zero fourth logarithmic derivative. Apply this identity to each factor of the defining double gamma products. Their three-index cone supports have $O(r^3)$ counting. Initially work off the union of the individual gamma-factor divisors. The fourth-power sums are locally absolutely convergent there; rearrangement, multiplication by the argument scale $\\beta$ and cancellation of multiplicities are legitimate. After coefficient cancellation, meromorphic continuation gives \\eqref{eq:fourth} on $\\mathbb C\\setminus D$, including points where individual singularities canceled. A conditionally convergent third-power sum is never used.","e of $R'/R$, a single-valued meromorphic function. The sine partial fractions \\cite{DLMFSine} give\n\\[\n \\frac{d^4}{dw^4}\\log(1-e^{2\\pi iw})\n =-6\\sum_{n\\in\\mathbb Z}(w-n)^{-4}.\n\\]\nThe linear exponential factor relating $1-e^{2\\pi iw}$ to $\\sin\\pi w$ has zero fourth logarithmic derivative. Apply this identity to each factor of the defining double gamma products. Their three-index cone supports have $O(r^3)$ counting. Initially work off the union of the individual gamma-factor divisors. The fourth-power sums are locally absolutely convergent there; rearrangement, multiplication by the argument scale $\\beta$ and cancellation of multiplicities are legitimate. After coefficient cancellation, meromorphic continuation gives \\eqref{eq:fourth} on $\\mathbb C\\setminus D$, including points where individual singularities canceled. A conditionally convergent third-power sum is never used.","tor relating $1-e^{2\\pi iw}$ to $\\sin\\pi w$ has zero fourth logarithmic derivative. Apply this identity to each factor of the defining double gamma products. Their three-index cone supports have $O(r^3)$ counting. Initially work off the union of the individual gamma-factor divisors. The fourth-power sums are locally absolutely convergent there; rearrangement, multiplication by the argument scale $\\beta$ and cancellation of multiplicities are legitimate. After coefficient cancellation, meromorphic continuation gives \\eqref{eq:fourth} on $\\mathbb C\\setminus D$, including points where individual singularities canceled. A conditionally convergent third-power sum is never used.","amma products. Their three-index cone supports have $O(r^3)$ counting. Initially work off the union of the individual gamma-factor divisors. The fourth-power sums are locally absolutely convergent there; rearrangement, multiplication by the argument scale $\\beta$ and cancellation of multiplicities are legitimate. After coefficient cancellation, meromorphic continuation gives \\eqref{eq:fourth} on $\\mathbb C\\setminus D$, including points where individual singularities canceled. A conditionally convergent third-power sum is never used.","The canonical factor satisfies $(\\log E_3(z/\\lambda))^{(4)}=-6/(z-\\lambda)^4$, so the same identity holds for $W_D$. Its divisor agrees with that of $R$. Thus $R/W_D$ extends to a nowhere-zero entire function, and on the simply connected plane has an entire logarithm $h$. The fourth derivative of $h$ vanishes, hence $h$ is a polynomial of degree at most three. Its value at zero is fixed by $R_0$, and the vanished canonical jets identify the coefficients in \\eqref{eq:cubic-reconstruction}.\n\\end{proof}\nDivisor matching alone would give $e^{h(z)}$ for an arbitrary entire $h$. The fourth-log identity is the extra analytic statement forcing a cubic polynomial; no unproved finite-order estimate for the ratio is used.\n\\begin{figure}[htbp]\\centering\\includegraphics[width=.9","The canonical factor satisfies $(\\log E_3(z/\\lambda))^{(4)}=-6/(z-\\lambda)^4$, so the same identity holds for $W_D$. Its divisor agrees with that of $R$. Thus $R/W_D$ extends to a nowhere-zero entire function, and on the simply connected plane has an entire logarithm $h$. The fourth derivative of $h$ vanishes, hence $h$ is a polynomial of degree at most three. Its value at zero is fixed by $R_0$, and the vanished canonical jets identify the coefficients in \\eqref{eq:cubic-reconstruction}.\n\\end{proof}\nDivisor matching alone would give $e^{h(z)}$ for an arbitrary entire $h$. The fourth-log identity is the extra analytic statement forcing a cubic polynomial; no unproved finite-order estimate for the ratio is used.\n\\begin{figure}[htbp]\\centering\\includegraphics[width=.92\\linewidth]{figures/13_ambiguity.png}","The canonical factor satisfies $(\\log E_3(z/\\lambda))^{(4)}=-6/(z-\\lambda)^4$, so the same identity holds for $W_D$. Its divisor agrees with that of $R$. Thus $R/W_D$ extends to a nowhere-zero entire function, and on the simply connected plane has an entire logarithm $h$. The fourth derivative of $h$ vanishes, hence $h$ is a polynomial of degree at most three. Its value at zero is fixed by $R_0$, and the vanished canonical jets identify the coefficients in \\eqref{eq:cubic-reconstruction}.\n\\end{proof}\nDivisor matching alone would give $e^{h(z)}$ for an arbitrary entire $h$. The fourth-log identity is the extra analytic statement forcing a cubic polynomial; no unproved finite-order estimate for the ratio is used.\n\\begin{figure}[htbp]\\centering\\includegraphics[width=.92\\linewidth]{figures/13_ambiguity.png}\\caption{The logical role of the four","The canonical factor satisfies $(\\log E_3(z/\\lambda))^{(4)}=-6/(z-\\lambda)^4$, so the same identity holds for $W_D$. Its divisor agrees with that of $R$. Thus $R/W_D$ extends to a nowhere-zero entire function, and on the simply connected plane has an entire logarithm $h$. The fourth derivative of $h$ vanishes, hence $h$ is a polynomial of degree at most three. Its value at zero is fixed by $R_0$, and the vanished canonical jets identify the coefficients in \\eqref{eq:cubic-reconstruction}.\n\\end{proof}\nDivisor matching alone would give $e^{h(z)}$ for an arbitrary entire $h$. The fourth-log identity is the extra analytic statement forcing a cubic polynomial; no unproved finite-order estimate for the ratio is used.\n\\begin{figure}[htbp]\\centering\\includegraphics[width=.92\\linewidth]{figures/13_ambiguity.png}\\caption{The logical role of the fourth logarithmic","The canonical factor satisfies $(\\log E_3(z/\\lambda))^{(4)}=-6/(z-\\lambda)^4$, so the same identity holds for $W_D$. Its divisor agrees with that of $R$. Thus $R/W_D$ extends to a nowhere-zero entire function, and on the simply connected plane has an entire logarithm $h$. The fourth derivative of $h$ vanishes, hence $h$ is a polynomial of degree at most three. Its value at zero is fixed by $R_0$, and the vanished canonical jets identify the coefficients in \\eqref{eq:cubic-reconstruction}.\n\\end{proof}\nDivisor matching alone would give $e^{h(z)}$ for an arbitrary entire $h$. The fourth-log identity is the extra analytic statement forcing a cubic polynomial; no unproved finite-order estimate for the ratio is used.\n\\begin{figure}[htbp]\\centering\\includegraphics[width=.92\\linewidth]{figures/13_ambiguity.png}\\caption{The logical role of the fourth logarithmic derivative. Matching every divisor order makes the ratio a holomorphic unit; the additional analytic ide","The canonical factor satisfies $(\\log E_3(z/\\lambda))^{(4)}=-6/(z-\\lambda)^4$, so the same identity holds for $W_D$. Its divisor agrees with that of $R$. Thus $R/W_D$ extends to a nowhere-zero entire function, and on the simply connected plane has an entire logarithm $h$. The fourth derivative of $h$ vanishes, hence $h$ is a polynomial of degree at most three. Its value at zero is fixed by $R_0$, and the vanished canonical jets identify the coefficients in \\eqref{eq:cubic-reconstruction}.\n\\end{proof}\nDivisor matching alone would give $e^{h(z)}$ for an arbitrary entire $h$. The fourth-log identity is the extra analytic statement forcing a cubic polynomial; no unproved finite-order estimate for the ratio is used.\n\\begin{figure}[htbp]\\centering\\includegraphics[width=.92\\linewidth]{figures/13_ambiguity.png}\\caption{The logical role of the fourth logarithmic derivative. Matching every divisor order makes the ratio a holomorphic unit; the additional analytic identity fixes the remaining expo","The canonical factor satisfies $(\\log E_3(z/\\lambda))^{(4)}=-6/(z-\\lambda)^4$, so the same identity holds for $W_D$. Its divisor agrees with that of $R$. Thus $R/W_D$ extends to a nowhere-zero entire function, and on the simply connected plane has an entire logarithm $h$. The fourth derivative of $h$ vanishes, hence $h$ is a polynomial of degree at most three. Its value at zero is fixed by $R_0$, and the vanished canonical jets identify the coefficients in \\eqref{eq:cubic-reconstruction}.\n\\end{proof}\nDivisor matching alone would give $e^{h(z)}$ for an arbitrary entire $h$. The fourth-log identity is the extra analytic statement forcing a cubic polynomial; no unproved finite-order estimate for the ratio is used.\n\\begin{figure}[htbp]\\centering\\includegraphics[width=.92\\linewidth]{figures/13_ambiguity.png}\\caption{The logical role of the fourth logarithmic derivative. Matching every divisor order makes the ratio a holomorphic unit; the additional analytic identity fixes the remaining exponent to degree at mo","The canonical factor satisfies $(\\log E_3(z/\\lambda))^{(4)}=-6/(z-\\lambda)^4$, so the same identity holds for $W_D$. Its divisor agrees with that of $R$. Thus $R/W_D$ extends to a nowhere-zero entire function, and on the simply connected plane has an entire logarithm $h$. The fourth derivative of $h$ vanishes, hence $h$ is a polynomial of degree at most three. Its value at zero is fixed by $R_0$, and the vanished canonical jets identify the coefficients in \\eqref{eq:cubic-reconstruction}.\n\\end{proof}\nDivisor matching alone would give $e^{h(z)}$ for an arbitrary entire $h$. The fourth-log identity is the extra analytic statement forcing a cubic polynomial; no unproved finite-order estimate for the ratio is used.\n\\begin{figure}[htbp]\\centering\\includegraphics[width=.92\\linewidth]{figures/13_ambiguity.png}\\caption{The logical role of the fourth logarithmic derivative. Matching every divisor order makes the ratio a holomorphic unit; the additional analytic identity fixes the remaining exponent to degree at most three.}\\label{fig:ambiguity}\\end{figure}","that of $R$. Thus $R/W_D$ extends to a nowhere-zero entire function, and on the simply connected plane has an entire logarithm $h$. The fourth derivative of $h$ vanishes, hence $h$ is a polynomial of degree at most three. Its value at zero is fixed by $R_0$, and the vanished canonical jets identify the coefficients in \\eqref{eq:cubic-reconstruction}.\n\\end{proof}\nDivisor matching alone would give $e^{h(z)}$ for an arbitrary entire $h$. The fourth-log identity is the extra analytic statement forcing a cubic polynomial; no unproved finite-order estimate for the ratio is used.\n\\begin{figure}[htbp]\\centering\\includegraphics[width=.92\\linewidth]{figures/13_ambiguity.png}\\caption{The logical role of the fourth logarithmic derivative. Matching every divisor order makes the ratio a holomorphic unit; the additional analytic identity fixes the remaining exponent to degree at most three.}\\label{fig:ambiguity}\\end{figure}","o a nowhere-zero entire function, and on the simply connected plane has an entire logarithm $h$. The fourth derivative of $h$ vanishes, hence $h$ is a polynomial of degree at most three. Its value at zero is fixed by $R_0$, and the vanished canonical jets identify the coefficients in \\eqref{eq:cubic-reconstruction}.\n\\end{proof}\nDivisor matching alone would give $e^{h(z)}$ for an arbitrary entire $h$. The fourth-log identity is the extra analytic statement forcing a cubic polynomial; no unproved finite-order estimate for the ratio is used.\n\\begin{figure}[htbp]\\centering\\includegraphics[width=.92\\linewidth]{figures/13_ambiguity.png}\\caption{The logical role of the fourth logarithmic derivative. Matching every divisor order makes the ratio a holomorphic unit; the additional analytic identity fixes the remaining exponent to degree at most three.}\\label{fig:ambiguity}\\end{figure}","\\subsection{Reflection fixes the quadratic coefficient}\nThe product and continuation convention yield\n\\[\n F(z)F(-z)=\\frac{q_t(z)}{q_s(z)},\\qquad\n \\frac{F(w+7\\delta)}{F(w)}=\\frac{q_s(w)}{q_t(w)q_t(w+s)}.\n\\]\nPair the labels $k$ and $7-k$ in $P_7(z)P_7(-z)$. For $1\\le k\\le6$,\n\\[\n F(z+k\\delta)F(-z+(7-k)\\delta)=q_t(z+k\\delta-s)^{-1}.\n\\]\nThe unpaired $k=0$ term retains $q_t(z)/q_s(z)$. Dividing by the reflected denominator gives\n\\begin{equation}\\label{eq:reflection}\n R(z)R(-z)=T(z):=\n \\frac{q_t(z)q_s(\\beta z)}{q_s(z)q_t(\\beta z)}\n \\prod_{k=1}^{6}q_t(z+k\\delta-s)^{-1}.\n\\end{equation}\nIts apparent singularities at zero are removable, and $T(0)=R_0^2$.","coefficient}\nThe product and continuation convention yield\n\\[\n F(z)F(-z)=\\frac{q_t(z)}{q_s(z)},\\qquad\n \\frac{F(w+7\\delta)}{F(w)}=\\frac{q_s(w)}{q_t(w)q_t(w+s)}.\n\\]\nPair the labels $k$ and $7-k$ in $P_7(z)P_7(-z)$. For $1\\le k\\le6$,\n\\[\n F(z+k\\delta)F(-z+(7-k)\\delta)=q_t(z+k\\delta-s)^{-1}.\n\\]\nThe unpaired $k=0$ term retains $q_t(z)/q_s(z)$. Dividing by the reflected denominator gives\n\\begin{equation}\\label{eq:reflection}\n R(z)R(-z)=T(z):=\n \\frac{q_t(z)q_s(\\beta z)}{q_s(z)q_t(\\beta z)}\n \\prod_{k=1}^{6}q_t(z+k\\delta-s)^{-1}.\n\\end{equation}\nIts apparent singularities at zero are removable, and $T(0)=R_0^2$.","$P_7(z)P_7(-z)$. For $1\\le k\\le6$,\n\\[\n F(z+k\\delta)F(-z+(7-k)\\delta)=q_t(z+k\\delta-s)^{-1}.\n\\]\nThe unpaired $k=0$ term retains $q_t(z)/q_s(z)$. Dividing by the reflected denominator gives\n\\begin{equation}\\label{eq:reflection}\n R(z)R(-z)=T(z):=\n \\frac{q_t(z)q_s(\\beta z)}{q_s(z)q_t(\\beta z)}\n \\prod_{k=1}^{6}q_t(z+k\\delta-s)^{-1}.\n\\end{equation}\nIts apparent singularities at zero are removable, and $T(0)=R_0^2$.","Let $L_t=(q_t'/q_t)'$ and $S_t=\\wp_t(1/5)+\\wp_t(2/5)$. The normalized theta formula gives\n\\[\n L_t(z)=-4z^{-2}+2S_t+O(z^2),\\qquad S_s=\\rho^2S_t.\n\\]\nSince $7\\delta-2s=-t+4$, evenness and $t$-periodicity pair the six translated $L_t$ terms. Using $s=u_0+3\\delta-2$, their sum is $2\\sum_{j=0}^{2}L_t(u_j)$. The second logarithmic derivative of \\eqref{eq:reflection} cancels all $z^{-2}$ terms, leaving\n\\[\n (\\log T)''(0)=2(1-\\beta^2)(1-\\rho^2)S_t-2\\sum_{j=0}^{2}L_t(u_j).\n\\]\nThe left side is $4\\alpha_2$. Therefore\n\\begin{equation}\\label{eq:alpha2}\n \\boxed{\\alpha_2=\\frac12\\left[(1-\\beta^2)(1-\\rho^2)S_t-\n \\sum_{j=0}^{2}L_t(u_j)\\right].}\n\\end{equation}\nThe factor $1/2$ is fixed by reflection and is not a numerical fit.","s\n\\[\n L_t(z)=-4z^{-2}+2S_t+O(z^2),\\qquad S_s=\\rho^2S_t.\n\\]\nSince $7\\delta-2s=-t+4$, evenness and $t$-periodicity pair the six translated $L_t$ terms. Using $s=u_0+3\\delta-2$, their sum is $2\\sum_{j=0}^{2}L_t(u_j)$. The second logarithmic derivative of \\eqref{eq:reflection} cancels all $z^{-2}$ terms, leaving\n\\[\n (\\log T)''(0)=2(1-\\beta^2)(1-\\rho^2)S_t-2\\sum_{j=0}^{2}L_t(u_j).\n\\]\nThe left side is $4\\alpha_2$. Therefore\n\\begin{equation}\\label{eq:alpha2}\n \\boxed{\\alpha_2=\\frac12\\left[(1-\\beta^2)(1-\\rho^2)S_t-\n \\sum_{j=0}^{2}L_t(u_j)\\right].}\n\\end{equation}\nThe factor $1/2$ is fixed by reflection and is not a numerical fit.","ad S_s=\\rho^2S_t.\n\\]\nSince $7\\delta-2s=-t+4$, evenness and $t$-periodicity pair the six translated $L_t$ terms. Using $s=u_0+3\\delta-2$, their sum is $2\\sum_{j=0}^{2}L_t(u_j)$. The second logarithmic derivative of \\eqref{eq:reflection} cancels all $z^{-2}$ terms, leaving\n\\[\n (\\log T)''(0)=2(1-\\beta^2)(1-\\rho^2)S_t-2\\sum_{j=0}^{2}L_t(u_j).\n\\]\nThe left side is $4\\alpha_2$. Therefore\n\\begin{equation}\\label{eq:alpha2}\n \\boxed{\\alpha_2=\\frac12\\left[(1-\\beta^2)(1-\\rho^2)S_t-\n \\sum_{j=0}^{2}L_t(u_j)\\right].}\n\\end{equation}\nThe factor $1/2$ is fixed by reflection and is not a numerical fit.","$, their sum is $2\\sum_{j=0}^{2}L_t(u_j)$. The second logarithmic derivative of \\eqref{eq:reflection} cancels all $z^{-2}$ terms, leaving\n\\[\n (\\log T)''(0)=2(1-\\beta^2)(1-\\rho^2)S_t-2\\sum_{j=0}^{2}L_t(u_j).\n\\]\nThe left side is $4\\alpha_2$. Therefore\n\\begin{equation}\\label{eq:alpha2}\n \\boxed{\\alpha_2=\\frac12\\left[(1-\\beta^2)(1-\\rho^2)S_t-\n \\sum_{j=0}^{2}L_t(u_j)\\right].}\n\\end{equation}\nThe factor $1/2$ is fixed by reflection and is not a numerical fit.","\\subsection{The period-one transition fixes the cubic coefficient}\nThe points $0,1,1/3$ are regular and nonzero for $R$ and $W_D$. For $1$, \\eqref{eq:K} has nonzero factors at zero. For $1/3$, $\\beta/3=-5s-2/3$, whose rational coordinate is not a rational fifth; rational independence excludes a singular lattice point. Define\n\\begin{equation}\\label{eq:BD}\n B_D(1)=(\\log W_D)''(1)\n =\\sum_{\\lambda\\in D}m_\\lambda\\left[-\\frac1{(\\lambda-1)^2}\n +\\frac1{\\lambda^2}+\\frac2{\\lambda^3}\\right].\n\\end{equation}\nThe combined summand is $O(|\\lambda|^{-4})$, so this is an absolutely convergent sum. Its terms must remain combined; the separately divergent lower-power sums are not substituted. Taking two logarithmic derivatives of \\eqref{eq:cubic-reconstruction} at the transition \\eqref{eq:K} giv","\\subsection{The period-one transition fixes the cubic coefficient}\nThe points $0,1,1/3$ are regular and nonzero for $R$ and $W_D$. For $1$, \\eqref{eq:K} has nonzero factors at zero. For $1/3$, $\\beta/3=-5s-2/3$, whose rational coordinate is not a rational fifth; rational independence excludes a singular lattice point. Define\n\\begin{equation}\\label{eq:BD}\n B_D(1)=(\\log W_D)''(1)\n =\\sum_{\\lambda\\in D}m_\\lambda\\left[-\\frac1{(\\lambda-1)^2}\n +\\frac1{\\lambda^2}+\\frac2{\\lambda^3}\\right].\n\\end{equation}\nThe combined summand is $O(|\\lambda|^{-4})$, so this is an absolutely convergent sum. Its terms must remain combined; the separately divergent lower-power sums are not substituted. Taking two logarithmic derivatives of \\eqref{eq:cubic-reconstruction} at the transition \\eqref{eq:K} gives\n\\[\n (\\log K)''(0)=6\\alpha_3+B","\\subsection{The period-one transition fixes the cubic coefficient}\nThe points $0,1,1/3$ are regular and nonzero for $R$ and $W_D$. For $1$, \\eqref{eq:K} has nonzero factors at zero. For $1/3$, $\\beta/3=-5s-2/3$, whose rational coordinate is not a rational fifth; rational independence excludes a singular lattice point. Define\n\\begin{equation}\\label{eq:BD}\n B_D(1)=(\\log W_D)''(1)\n =\\sum_{\\lambda\\in D}m_\\lambda\\left[-\\frac1{(\\lambda-1)^2}\n +\\frac1{\\lambda^2}+\\frac2{\\lambda^3}\\right].\n\\end{equation}\nThe combined summand is $O(|\\lambda|^{-4})$, so this is an absolutely convergent sum. Its terms must remain combined; the separately divergent lower-power sums are not substituted. Taking two logarithmic derivatives of \\eqref{eq:cubic-reconstruction} at the transition \\eqref{eq:K} gives\n\\[\n (\\log K)''(0)=6\\alpha_3+B_D(1).\n\\]","\\subsection{The period-one transition fixes the cubic coefficient}\nThe points $0,1,1/3$ are regular and nonzero for $R$ and $W_D$. For $1$, \\eqref{eq:K} has nonzero factors at zero. For $1/3$, $\\beta/3=-5s-2/3$, whose rational coordinate is not a rational fifth; rational independence excludes a singular lattice point. Define\n\\begin{equation}\\label{eq:BD}\n B_D(1)=(\\log W_D)''(1)\n =\\sum_{\\lambda\\in D}m_\\lambda\\left[-\\frac1{(\\lambda-1)^2}\n +\\frac1{\\lambda^2}+\\frac2{\\lambda^3}\\right].\n\\end{equation}\nThe combined summand is $O(|\\lambda|^{-4})$, so this is an absolutely convergent sum. Its terms must remain combined; the separately divergent lower-power sums are not substituted. Taking two logarithmic derivatives of \\eqref{eq:cubic-reconstruction} at the transition \\eqref{eq:K} gives\n\\[\n (\\log K)''(0)=6\\alpha_3+B_D(1).\n\\]\nSince $L_","\\subsection{The period-one transition fixes the cubic coefficient}\nThe points $0,1,1/3$ are regular and nonzero for $R$ and $W_D$. For $1$, \\eqref{eq:K} has nonzero factors at zero. For $1/3$, $\\beta/3=-5s-2/3$, whose rational coordinate is not a rational fifth; rational independence excludes a singular lattice point. Define\n\\begin{equation}\\label{eq:BD}\n B_D(1)=(\\log W_D)''(1)\n =\\sum_{\\lambda\\in D}m_\\lambda\\left[-\\frac1{(\\lambda-1)^2}\n +\\frac1{\\lambda^2}+\\frac2{\\lambda^3}\\right].\n\\end{equation}\nThe combined summand is $O(|\\lambda|^{-4})$, so this is an absolutely convergent sum. Its terms must remain combined; the separately divergent lower-power sums are not substituted. Taking two logarithmic derivatives of \\eqref{eq:cubic-reconstruction} at the transition \\eqref{eq:K} gives\n\\[\n (\\log K)''(0)=6\\alpha_3+B_D(1).\n\\]\nSince $L_t$ is even,\n\\begin{equation}\\label{eq:alpha3}\n \\boxed","\\subsection{The period-one transition fixes the cubic coefficient}\nThe points $0,1,1/3$ are regular and nonzero for $R$ and $W_D$. For $1$, \\eqref{eq:K} has nonzero factors at zero. For $1/3$, $\\beta/3=-5s-2/3$, whose rational coordinate is not a rational fifth; rational independence excludes a singular lattice point. Define\n\\begin{equation}\\label{eq:BD}\n B_D(1)=(\\log W_D)''(1)\n =\\sum_{\\lambda\\in D}m_\\lambda\\left[-\\frac1{(\\lambda-1)^2}\n +\\frac1{\\lambda^2}+\\frac2{\\lambda^3}\\right].\n\\end{equation}\nThe combined summand is $O(|\\lambda|^{-4})$, so this is an absolutely convergent sum. Its terms must remain combined; the separately divergent lower-power sums are not substituted. Taking two logarithmic derivatives of \\eqref{eq:cubic-reconstruction} at the transition \\eqref{eq:K} gives\n\\[\n (\\log K)''(0)=6\\alpha_3+B_D(1).\n\\]\nSince $L_t$ is even,\n\\begin{equation}\\label{eq:alpha3}\n \\boxed{\\alpha_3=-\\frac16\\","\\subsection{The period-one transition fixes the cubic coefficient}\nThe points $0,1,1/3$ are regular and nonzero for $R$ and $W_D$. For $1$, \\eqref{eq:K} has nonzero factors at zero. For $1/3$, $\\beta/3=-5s-2/3$, whose rational coordinate is not a rational fifth; rational independence excludes a singular lattice point. Define\n\\begin{equation}\\label{eq:BD}\n B_D(1)=(\\log W_D)''(1)\n =\\sum_{\\lambda\\in D}m_\\lambda\\left[-\\frac1{(\\lambda-1)^2}\n +\\frac1{\\lambda^2}+\\frac2{\\lambda^3}\\right].\n\\end{equation}\nThe combined summand is $O(|\\lambda|^{-4})$, so this is an absolutely convergent sum. Its terms must remain combined; the separately divergent lower-power sums are not substituted. Taking two logarithmic derivatives of \\eqref{eq:cubic-reconstruction} at the transition \\eqref{eq:K} gives\n\\[\n (\\log K)''(0)=6\\alpha_3+B_D(1).\n\\]\nSince $L_t$ is even,\n\\begin{equation}\\label{eq:alpha3}\n \\boxed{\\alpha_3=-\\frac16\\left[\\beta^2\\sum_{r=1}^{15}L_t(rs)+B_D(1)\\right].}\n\\end{equation}\nEvaluating the period-one ratio also determines\n\\begin{equation}\\label{eq:expalpha1}\n e^{\\alpha_1}=\\frac{K(0)e^{-\\alpha_2-\\al…","_D$. For $1$, \\eqref{eq:K} has nonzero factors at zero. For $1/3$, $\\beta/3=-5s-2/3$, whose rational coordinate is not a rational fifth; rational independence excludes a singular lattice point. Define\n\\begin{equation}\\label{eq:BD}\n B_D(1)=(\\log W_D)''(1)\n =\\sum_{\\lambda\\in D}m_\\lambda\\left[-\\frac1{(\\lambda-1)^2}\n +\\frac1{\\lambda^2}+\\frac2{\\lambda^3}\\right].\n\\end{equation}\nThe combined summand is $O(|\\lambda|^{-4})$, so this is an absolutely convergent sum. Its terms must remain combined; the separately divergent lower-power sums are not substituted. Taking two logarithmic derivatives of \\eqref{eq:cubic-reconstruction} at the transition \\eqref{eq:K} gives\n\\[\n (\\log K)''(0)=6\\alpha_3+B_D(1).\n\\]\nSince $L_t$ is even,\n\\begin{equation}\\label{eq:alpha3}\n \\boxed{\\alpha_3=-\\frac16\\left[\\beta^2\\sum_{r=1}^{15}L_t(rs)+B_D(1)\\right].}\n\\end{equation}\nEvaluating the period-one ratio also determines\n\\begin{equation}\\label{eq:expalpha1}\n e^{\\alpha_1}=\\frac{K(0)e^{-\\alpha_2-\\alpha_3}}{W_D(1)}.\n\\end{equation}\nThis evaluates the exponential, without choosing an unjustified principal logarithm.","ambda\\in D}m_\\lambda\\left[-\\frac1{(\\lambda-1)^2}\n +\\frac1{\\lambda^2}+\\frac2{\\lambda^3}\\right].\n\\end{equation}\nThe combined summand is $O(|\\lambda|^{-4})$, so this is an absolutely convergent sum. Its terms must remain combined; the separately divergent lower-power sums are not substituted. Taking two logarithmic derivatives of \\eqref{eq:cubic-reconstruction} at the transition \\eqref{eq:K} gives\n\\[\n (\\log K)''(0)=6\\alpha_3+B_D(1).\n\\]\nSince $L_t$ is even,\n\\begin{equation}\\label{eq:alpha3}\n \\boxed{\\alpha_3=-\\frac16\\left[\\beta^2\\sum_{r=1}^{15}L_t(rs)+B_D(1)\\right].}\n\\end{equation}\nEvaluating the period-one ratio also determines\n\\begin{equation}\\label{eq:expalpha1}\n e^{\\alpha_1}=\\frac{K(0)e^{-\\alpha_2-\\alpha_3}}{W_D(1)}.\n\\end{equation}\nThis evaluates the exponential, without choosing an unjustified principal logarithm.",")^2}\n +\\frac1{\\lambda^2}+\\frac2{\\lambda^3}\\right].\n\\end{equation}\nThe combined summand is $O(|\\lambda|^{-4})$, so this is an absolutely convergent sum. Its terms must remain combined; the separately divergent lower-power sums are not substituted. Taking two logarithmic derivatives of \\eqref{eq:cubic-reconstruction} at the transition \\eqref{eq:K} gives\n\\[\n (\\log K)''(0)=6\\alpha_3+B_D(1).\n\\]\nSince $L_t$ is even,\n\\begin{equation}\\label{eq:alpha3}\n \\boxed{\\alpha_3=-\\frac16\\left[\\beta^2\\sum_{r=1}^{15}L_t(rs)+B_D(1)\\right].}\n\\end{equation}\nEvaluating the period-one ratio also determines\n\\begin{equation}\\label{eq:expalpha1}\n e^{\\alpha_1}=\\frac{K(0)e^{-\\alpha_2-\\alpha_3}}{W_D(1)}.\n\\end{equation}\nThis evaluates the exponential, without choosing an unjustified principal logarithm.","2{\\lambda^3}\\right].\n\\end{equation}\nThe combined summand is $O(|\\lambda|^{-4})$, so this is an absolutely convergent sum. Its terms must remain combined; the separately divergent lower-power sums are not substituted. Taking two logarithmic derivatives of \\eqref{eq:cubic-reconstruction} at the transition \\eqref{eq:K} gives\n\\[\n (\\log K)''(0)=6\\alpha_3+B_D(1).\n\\]\nSince $L_t$ is even,\n\\begin{equation}\\label{eq:alpha3}\n \\boxed{\\alpha_3=-\\frac16\\left[\\beta^2\\sum_{r=1}^{15}L_t(rs)+B_D(1)\\right].}\n\\end{equation}\nEvaluating the period-one ratio also determines\n\\begin{equation}\\label{eq:expalpha1}\n e^{\\alpha_1}=\\frac{K(0)e^{-\\alpha_2-\\alpha_3}}{W_D(1)}.\n\\end{equation}\nThis evaluates the exponential, without choosing an unjustified principal logarithm.","gent lower-power sums are not substituted. Taking two logarithmic derivatives of \\eqref{eq:cubic-reconstruction} at the transition \\eqref{eq:K} gives\n\\[\n (\\log K)''(0)=6\\alpha_3+B_D(1).\n\\]\nSince $L_t$ is even,\n\\begin{equation}\\label{eq:alpha3}\n \\boxed{\\alpha_3=-\\frac16\\left[\\beta^2\\sum_{r=1}^{15}L_t(rs)+B_D(1)\\right].}\n\\end{equation}\nEvaluating the period-one ratio also determines\n\\begin{equation}\\label{eq:expalpha1}\n e^{\\alpha_1}=\\frac{K(0)e^{-\\alpha_2-\\alpha_3}}{W_D(1)}.\n\\end{equation}\nThis evaluates the exponential, without choosing an unjustified principal logarithm.","\\begin{theorem}[Branch-free special cube]\\label{thm:cube}\nThe exact analytic reconstruction at $z=1/3$ is\n\\begin{equation}\\label{eq:cube}\n \\boxed{R(1/3)^3=R_0^3K(0)\\frac{W_D(1/3)^3}{W_D(1)}\n \\exp\\left(-\\frac23\\alpha_2-\\frac89\\alpha_3\\right).}\n\\end{equation}\nThe right side consists of specified theta values and derivatives, algebraic period data and absolutely convergent genus-three products and sums. It contains no gamma value, arbitrary normalization scalar or new root choice.\n\\end{theorem}\n\\begin{proof}\nCube \\eqref{eq:cubic-reconstruction} at $1/3$ to get $R_0^3e^{\\alpha_1+\\alpha_2/3+\\alpha_3/9}W_D(1/3)^3$, then substitute \\eqref{eq:expalpha1}. The resulting exponents are $-2\\alpha_2/3$ and $-8\\alpha_3/9$.\n\\end{proof}\nThis is an analytic representation by an infinite transcendental produc","\\begin{theorem}[Branch-free special cube]\\label{thm:cube}\nThe exact analytic reconstruction at $z=1/3$ is\n\\begin{equation}\\label{eq:cube}\n \\boxed{R(1/3)^3=R_0^3K(0)\\frac{W_D(1/3)^3}{W_D(1)}\n \\exp\\left(-\\frac23\\alpha_2-\\frac89\\alpha_3\\right).}\n\\end{equation}\nThe right side consists of specified theta values and derivatives, algebraic period data and absolutely convergent genus-three products and sums. It contains no gamma value, arbitrary normalization scalar or new root choice.\n\\end{theorem}\n\\begin{proof}\nCube \\eqref{eq:cubic-reconstruction} at $1/3$ to get $R_0^3e^{\\alpha_1+\\alpha_2/3+\\alpha_3/9}W_D(1/3)^3$, then substitute \\eqref{eq:expalpha1}. The resulting exponents are $-2\\alpha_2/3$ and $-8\\alpha_3/9$.\n\\end{proof}\nThis is an analytic representation by an infinite transcendental product. It does not identify that product with a class-field value or prove $U_\\Gamma=U_{\\rm alg}$.\n\\begin{figure}[htbp]\\centering\\includegraphics[","^3=R_0^3K(0)\\frac{W_D(1/3)^3}{W_D(1)}\n \\exp\\left(-\\frac23\\alpha_2-\\frac89\\alpha_3\\right).}\n\\end{equation}\nThe right side consists of specified theta values and derivatives, algebraic period data and absolutely convergent genus-three products and sums. It contains no gamma value, arbitrary normalization scalar or new root choice.\n\\end{theorem}\n\\begin{proof}\nCube \\eqref{eq:cubic-reconstruction} at $1/3$ to get $R_0^3e^{\\alpha_1+\\alpha_2/3+\\alpha_3/9}W_D(1/3)^3$, then substitute \\eqref{eq:expalpha1}. The resulting exponents are $-2\\alpha_2/3$ and $-8\\alpha_3/9$.\n\\end{proof}\nThis is an analytic representation by an infinite transcendental product. It does not identify that product with a class-field value or prove $U_\\Gamma=U_{\\rm alg}$.\n\\begin{figure}[htbp]\\centering\\includegraphics[width=.96\\linewidth]{figures/16_cube.png}\\caption{For $P(z)=\\alpha_1z+\\alpha_2z^2+\\alpha_3z^3$, the exact coefficients of $3P(1/3)-P(1)$ are $0,-2/3,-8/9$. Cubing removes the possible relative logarithm-lift factor. This illustrates the analytic identity \\eqref{eq:cube}, not an arithmetic evaluation.}\\label{…","frac{W_D(1/3)^3}{W_D(1)}\n \\exp\\left(-\\frac23\\alpha_2-\\frac89\\alpha_3\\right).}\n\\end{equation}\nThe right side consists of specified theta values and derivatives, algebraic period data and absolutely convergent genus-three products and sums. It contains no gamma value, arbitrary normalization scalar or new root choice.\n\\end{theorem}\n\\begin{proof}\nCube \\eqref{eq:cubic-reconstruction} at $1/3$ to get $R_0^3e^{\\alpha_1+\\alpha_2/3+\\alpha_3/9}W_D(1/3)^3$, then substitute \\eqref{eq:expalpha1}. The resulting exponents are $-2\\alpha_2/3$ and $-8\\alpha_3/9$.\n\\end{proof}\nThis is an analytic representation by an infinite transcendental product. It does not identify that product with a class-field value or prove $U_\\Gamma=U_{\\rm alg}$.\n\\begin{figure}[htbp]\\centering\\includegraphics[width=.96\\linewidth]{figures/16_cube.png}\\caption{For $P(z)=\\alpha_1z+\\alpha_2z^2+\\alpha_3z^3$, the exact coefficients of $3P(1/3)-P(1)$ are $0,-2/3,-8/9$. Cubing removes the possible relative logarithm-lift factor. This illustrates the analytic identity \\eqref{eq:cube}, not an arithmetic evaluation.}\\label{fig:cube}\\end…","specified theta values and derivatives, algebraic period data and absolutely convergent genus-three products and sums. It contains no gamma value, arbitrary normalization scalar or new root choice.\n\\end{theorem}\n\\begin{proof}\nCube \\eqref{eq:cubic-reconstruction} at $1/3$ to get $R_0^3e^{\\alpha_1+\\alpha_2/3+\\alpha_3/9}W_D(1/3)^3$, then substitute \\eqref{eq:expalpha1}. The resulting exponents are $-2\\alpha_2/3$ and $-8\\alpha_3/9$.\n\\end{proof}\nThis is an analytic representation by an infinite transcendental product. It does not identify that product with a class-field value or prove $U_\\Gamma=U_{\\rm alg}$.\n\\begin{figure}[htbp]\\centering\\includegraphics[width=.96\\linewidth]{figures/16_cube.png}\\caption{For $P(z)=\\alpha_1z+\\alpha_2z^2+\\alpha_3z^3$, the exact coefficients of $3P(1/3)-P(1)$ are $0,-2/3,-8/9$. Cubing removes the possible relative logarithm-lift factor. This illustrates the analytic identity \\eqref{eq:cube}, not an arithmetic evaluation.}\\label{fig:cube}\\end{figure}","lues and derivatives, algebraic period data and absolutely convergent genus-three products and sums. It contains no gamma value, arbitrary normalization scalar or new root choice.\n\\end{theorem}\n\\begin{proof}\nCube \\eqref{eq:cubic-reconstruction} at $1/3$ to get $R_0^3e^{\\alpha_1+\\alpha_2/3+\\alpha_3/9}W_D(1/3)^3$, then substitute \\eqref{eq:expalpha1}. The resulting exponents are $-2\\alpha_2/3$ and $-8\\alpha_3/9$.\n\\end{proof}\nThis is an analytic representation by an infinite transcendental product. It does not identify that product with a class-field value or prove $U_\\Gamma=U_{\\rm alg}$.\n\\begin{figure}[htbp]\\centering\\includegraphics[width=.96\\linewidth]{figures/16_cube.png}\\caption{For $P(z)=\\alpha_1z+\\alpha_2z^2+\\alpha_3z^3$, the exact coefficients of $3P(1/3)-P(1)$ are $0,-2/3,-8/9$. Cubing removes the possible relative logarithm-lift factor. This illustrates the analytic identity \\eqref{eq:cube}, not an arithmetic evaluation.}\\label{fig:cube}\\end{figure}","t choice.\n\\end{theorem}\n\\begin{proof}\nCube \\eqref{eq:cubic-reconstruction} at $1/3$ to get $R_0^3e^{\\alpha_1+\\alpha_2/3+\\alpha_3/9}W_D(1/3)^3$, then substitute \\eqref{eq:expalpha1}. The resulting exponents are $-2\\alpha_2/3$ and $-8\\alpha_3/9$.\n\\end{proof}\nThis is an analytic representation by an infinite transcendental product. It does not identify that product with a class-field value or prove $U_\\Gamma=U_{\\rm alg}$.\n\\begin{figure}[htbp]\\centering\\includegraphics[width=.96\\linewidth]{figures/16_cube.png}\\caption{For $P(z)=\\alpha_1z+\\alpha_2z^2+\\alpha_3z^3$, the exact coefficients of $3P(1/3)-P(1)$ are $0,-2/3,-8/9$. Cubing removes the possible relative logarithm-lift factor. This illustrates the analytic identity \\eqref{eq:cube}, not an arithmetic evaluation.}\\label{fig:cube}\\end{figure}","a_3/9}W_D(1/3)^3$, then substitute \\eqref{eq:expalpha1}. The resulting exponents are $-2\\alpha_2/3$ and $-8\\alpha_3/9$.\n\\end{proof}\nThis is an analytic representation by an infinite transcendental product. It does not identify that product with a class-field value or prove $U_\\Gamma=U_{\\rm alg}$.\n\\begin{figure}[htbp]\\centering\\includegraphics[width=.96\\linewidth]{figures/16_cube.png}\\caption{For $P(z)=\\alpha_1z+\\alpha_2z^2+\\alpha_3z^3$, the exact coefficients of $3P(1/3)-P(1)$ are $0,-2/3,-8/9$. Cubing removes the possible relative logarithm-lift factor. This illustrates the analytic identity \\eqref{eq:cube}, not an arithmetic evaluation.}\\label{fig:cube}\\end{figure}","ng exponents are $-2\\alpha_2/3$ and $-8\\alpha_3/9$.\n\\end{proof}\nThis is an analytic representation by an infinite transcendental product. It does not identify that product with a class-field value or prove $U_\\Gamma=U_{\\rm alg}$.\n\\begin{figure}[htbp]\\centering\\includegraphics[width=.96\\linewidth]{figures/16_cube.png}\\caption{For $P(z)=\\alpha_1z+\\alpha_2z^2+\\alpha_3z^3$, the exact coefficients of $3P(1/3)-P(1)$ are $0,-2/3,-8/9$. Cubing removes the possible relative logarithm-lift factor. This illustrates the analytic identity \\eqref{eq:cube}, not an arithmetic evaluation.}\\label{fig:cube}\\end{figure}","2\\alpha_2/3$ and $-8\\alpha_3/9$.\n\\end{proof}\nThis is an analytic representation by an infinite transcendental product. It does not identify that product with a class-field value or prove $U_\\Gamma=U_{\\rm alg}$.\n\\begin{figure}[htbp]\\centering\\includegraphics[width=.96\\linewidth]{figures/16_cube.png}\\caption{For $P(z)=\\alpha_1z+\\alpha_2z^2+\\alpha_3z^3$, the exact coefficients of $3P(1/3)-P(1)$ are $0,-2/3,-8/9$. Cubing removes the possible relative logarithm-lift factor. This illustrates the analytic identity \\eqref{eq:cube}, not an arithmetic evaluation.}\\label{fig:cube}\\end{figure}","\\subsection{The repaired conjugation quotient and its boundary}\nFor $f^\\#(z)=\\overline{f(\\overline z)}$, the canonical product $W_D^\\#$ has the conjugate divisor. Hence\n\\begin{equation}\\label{eq:regularized-conjugation}\n \\frac{R/R^\\#}{W_D/W_D^\\#}\n =\\frac{R_0}{\\overline{R_0}}\n \\exp\\left(\\sum_{j=1}^{3}(\\alpha_j-\\overline{\\alpha_j})z^j\\right).\n\\end{equation}\nThis is entire and nowhere zero, with modulus one on the real axis. It cancels all unmatched divisor orders by a canonical meromorphic factor. That factor is not a holomorphic-unit gauge or a finite theta section. Regularity on the scalar plane","\\subsection{The repaired conjugation quotient and its boundary}\nFor $f^\\#(z)=\\overline{f(\\overline z)}$, the canonical product $W_D^\\#$ has the conjugate divisor. Hence\n\\begin{equation}\\label{eq:regularized-conjugation}\n \\frac{R/R^\\#}{W_D/W_D^\\#}\n =\\frac{R_0}{\\overline{R_0}}\n \\exp\\left(\\sum_{j=1}^{3}(\\alpha_j-\\overline{\\alpha_j})z^j\\right).\n\\end{equation}\nThis is entire and nowhere zero, with modulus one on the real axis. It cancels all unmatched divisor orders by a canonical meromorphic factor. That factor is not a holomorphic-unit gauge or a finite theta section. Regularity on the scalar plane does not establish periodic des","\\subsection{The repaired conjugation quotient and its boundary}\nFor $f^\\#(z)=\\overline{f(\\overline z)}$, the canonical product $W_D^\\#$ has the conjugate divisor. Hence\n\\begin{equation}\\label{eq:regularized-conjugation}\n \\frac{R/R^\\#}{W_D/W_D^\\#}\n =\\frac{R_0}{\\overline{R_0}}\n \\exp\\left(\\sum_{j=1}^{3}(\\alpha_j-\\overline{\\alpha_j})z^j\\right).\n\\end{equation}\nThis is entire and nowhere zero, with modulus one on the real axis. It cancels all unmatched divisor orders by a canonical meromorphic factor. That factor is not a holomorphic-unit gauge or a finite theta section. Regularity on the scalar plane does not establish periodic descent to $\\mathbb T^3$, identify a based multiplier with $U_\\Gamma/U_{\\rm alg}$, or supply the principal-norm coboundary of Section~\\ref{sec:rigidity}. Those remain distinct obligations.","\\#}\n =\\frac{R_0}{\\overline{R_0}}\n \\exp\\left(\\sum_{j=1}^{3}(\\alpha_j-\\overline{\\alpha_j})z^j\\right).\n\\end{equation}\nThis is entire and nowhere zero, with modulus one on the real axis. It cancels all unmatched divisor orders by a canonical meromorphic factor. That factor is not a holomorphic-unit gauge or a finite theta section. Regularity on the scalar plane does not establish periodic descent to $\\mathbb T^3$, identify a based multiplier with $U_\\Gamma/U_{\\rm alg}$, or supply the principal-norm coboundary of Section~\\ref{sec:rigidity}. Those remain distinct obligations.","_{j=1}^{3}(\\alpha_j-\\overline{\\alpha_j})z^j\\right).\n\\end{equation}\nThis is entire and nowhere zero, with modulus one on the real axis. It cancels all unmatched divisor orders by a canonical meromorphic factor. That factor is not a holomorphic-unit gauge or a finite theta section. Regularity on the scalar plane does not establish periodic descent to $\\mathbb T^3$, identify a based multiplier with $U_\\Gamma/U_{\\rm alg}$, or supply the principal-norm coboundary of Section~\\ref{sec:rigidity}. Those remain distinct obligations.","\\section{The exact remaining arithmetic comparison}\\label{sec:remaining}\nThe fixed arithmetic presentation gives, with $a_0,a_1,a_2$ as in the polarized block,\n\\[\n C_{\\rm pol}=\\frac{\\epsilon a_1a_2^3}{\\omega},\\qquad\n C_{\\rm pol}U_{\\rm alg}^3=R_+:=|\\epsilon|^4|a_1|^6|a_2|^4>0.\n\\]\nThe modulus theorem gives $|C_{\\rm pol}U_\\Gamma^3|=R_+$. The exact statement still missing is\n\\begin{equation}\\label{eq:remaining-reality}\n \\Im(C_{\\rm pol}U_\\Gamma^3)=0.\n\\end{equation}\nReality together with a valid positive-real-part enclosure would give $C_{\\rm pol}U_\\Gamma^3=R_+$, hence $U_\\Gamma^3=U_{\\rm alg}^3$. A valid independent bound $|U_\\Gamma/U_{\\rm alg}-1|<1/2$ would select the root $U_\\Gamma=U_{\\rm alg}$. A numerical enclosure containing zero for the imaginary part proves no exact reality statement; equality remains unproved however","\\section{The exact remaining arithmetic comparison}\\label{sec:remaining}\nThe fixed arithmetic presentation gives, with $a_0,a_1,a_2$ as in the polarized block,\n\\[\n C_{\\rm pol}=\\frac{\\epsilon a_1a_2^3}{\\omega},\\qquad\n C_{\\rm pol}U_{\\rm alg}^3=R_+:=|\\epsilon|^4|a_1|^6|a_2|^4>0.\n\\]\nThe modulus theorem gives $|C_{\\rm pol}U_\\Gamma^3|=R_+$. The exact statement still missing is\n\\begin{equation}\\label{eq:remaining-reality}\n \\Im(C_{\\rm pol}U_\\Gamma^3)=0.\n\\end{equation}\nReality together with a valid positive-real-part enclosure would give $C_{\\rm pol}U_\\Gamma^3=R_+$, hence $U_\\Gamma^3=U_{\\rm alg}^3$. A valid independent bound $|U_\\Gamma/U_{\\rm alg}-1|<1/2$ would select the root $U_\\Gamma=U_{\\rm alg}$. A numerical enclosure containing zero for the imaginary part proves no exact reality statement; equality remains unproved however small the displayed residual.","in the polarized block,\n\\[\n C_{\\rm pol}=\\frac{\\epsilon a_1a_2^3}{\\omega},\\qquad\n C_{\\rm pol}U_{\\rm alg}^3=R_+:=|\\epsilon|^4|a_1|^6|a_2|^4>0.\n\\]\nThe modulus theorem gives $|C_{\\rm pol}U_\\Gamma^3|=R_+$. The exact statement still missing is\n\\begin{equation}\\label{eq:remaining-reality}\n \\Im(C_{\\rm pol}U_\\Gamma^3)=0.\n\\end{equation}\nReality together with a valid positive-real-part enclosure would give $C_{\\rm pol}U_\\Gamma^3=R_+$, hence $U_\\Gamma^3=U_{\\rm alg}^3$. A valid independent bound $|U_\\Gamma/U_{\\rm alg}-1|<1/2$ would select the root $U_\\Gamma=U_{\\rm alg}$. A numerical enclosure containing zero for the imaginary part proves no exact reality statement; equality remains unproved however small the displayed residual.","pol}=\\frac{\\epsilon a_1a_2^3}{\\omega},\\qquad\n C_{\\rm pol}U_{\\rm alg}^3=R_+:=|\\epsilon|^4|a_1|^6|a_2|^4>0.\n\\]\nThe modulus theorem gives $|C_{\\rm pol}U_\\Gamma^3|=R_+$. The exact statement still missing is\n\\begin{equation}\\label{eq:remaining-reality}\n \\Im(C_{\\rm pol}U_\\Gamma^3)=0.\n\\end{equation}\nReality together with a valid positive-real-part enclosure would give $C_{\\rm pol}U_\\Gamma^3=R_+$, hence $U_\\Gamma^3=U_{\\rm alg}^3$. A valid independent bound $|U_\\Gamma/U_{\\rm alg}-1|<1/2$ would select the root $U_\\Gamma=U_{\\rm alg}$. A numerical enclosure containing zero for the imaginary part proves no exact reality statement; equality remains unproved however small the displayed residual.","ol}U_{\\rm alg}^3=R_+:=|\\epsilon|^4|a_1|^6|a_2|^4>0.\n\\]\nThe modulus theorem gives $|C_{\\rm pol}U_\\Gamma^3|=R_+$. The exact statement still missing is\n\\begin{equation}\\label{eq:remaining-reality}\n \\Im(C_{\\rm pol}U_\\Gamma^3)=0.\n\\end{equation}\nReality together with a valid positive-real-part enclosure would give $C_{\\rm pol}U_\\Gamma^3=R_+$, hence $U_\\Gamma^3=U_{\\rm alg}^3$. A valid independent bound $|U_\\Gamma/U_{\\rm alg}-1|<1/2$ would select the root $U_\\Gamma=U_{\\rm alg}$. A numerical enclosure containing zero for the imaginary part proves no exact reality statement; equality remains unproved however small the displayed residual.","]\nThe modulus theorem gives $|C_{\\rm pol}U_\\Gamma^3|=R_+$. The exact statement still missing is\n\\begin{equation}\\label{eq:remaining-reality}\n \\Im(C_{\\rm pol}U_\\Gamma^3)=0.\n\\end{equation}\nReality together with a valid positive-real-part enclosure would give $C_{\\rm pol}U_\\Gamma^3=R_+$, hence $U_\\Gamma^3=U_{\\rm alg}^3$. A valid independent bound $|U_\\Gamma/U_{\\rm alg}-1|<1/2$ would select the root $U_\\Gamma=U_{\\rm alg}$. A numerical enclosure containing zero for the imaginary part proves no exact reality statement; equality remains unproved however small the displayed residual.","\\subsection{The actual two commuting actions}\nOn $X=\\mathbb C^3/\\mathbb Z^3\\simeq(\\mathbb C^*)^3$ and its real torus $\\mathbb T^3$, use\n\\[\n M=\\begin{pmatrix}0&1&0","\\subsection{The actual two commuting actions}\nOn $X=\\mathbb C^3/\\mathbb Z^3\\simeq(\\mathbb C^*)^3$ and its real torus $\\mathbb T^3$, use\n\\[\n M=\\begin{pmatrix}0&1&0\\\\-4&2&15\\\\-1&0&4\\end{pmatrix},\\quad\n N=2I-M=\\begin{pmatrix}2&-1&0\\\\4&0&-15\\\\1&0&-2\\end{pmatrix},\\quad\n A=M^T,\\quad B=N^T.\n\\]\nThe period row $(\\tau,s,1)$ satisfies $(\\tau,s,1)M=\\epsilon(\\tau,s,1)$ and $(\\tau,s,1)N=\\beta(\\tau,s,1)$. We have $MN=NM$, $\\det M=1$, $\\det N=7$ and $N^3=7I$. Thus $N$ is a degree-seven cover on $X$. On the compact lattice $\\mathbb Z[\\omega]^3$ its degree is 49; these refer to different lattices.","1&0&-2\\end{pmatrix},\\quad\n A=M^T,\\quad B=N^T.\n\\]\nThe period row $(\\tau,s,1)$ satisfies $(\\tau,s,1)M=\\epsilon(\\tau,s,1)$ and $(\\tau,s,1)N=\\beta(\\tau,s,1)$. We have $MN=NM$, $\\det M=1$, $\\det N=7$ and $N^3=7I$. Thus $N$ is a degree-seven cover on $X$. On the compact lattice $\\mathbb Z[\\omega]^3$ its degree is 49; these refer to different lattices.","The kernel on $X$ has representatives $j(1,2,4)^T/7$, $0\\le j\\le6$. The conductor point $p_3=e_2/3$ obeys $Mp_3-p_3=(0,5,1)^T$ and $Np_3-p_3=(0,-5,-1)^T$. The dual character $e_k(v)=e^{2\\pi ik^Tv}$ satisfies $e_k(Mv)=e_{Ak}(v)$. Frequency transport is distinct from the scalar return sequence $F(\\rho^m/3)$.","The cubic characteristic polynomial $(x-2)^3+7$ gives a stable real eigenvalue $a=2-7^{1/3}\\in(0,1)$ and a conjugate unstable pair of modulus $a^{-1/2}>1$. Every nonzero integer $A$-orbit escapes in both directions: otherwise an integer vector would tend to zero along positive or negative iterates. Fourier orbit sums are therefore absolutely convergent for smooth coefficients. The dual degree-seven image is exactly\n\\[\n k\\in B\\mathbb Z^3\\quad\\Longleftrightarrow\\quad\n \\ell(k):=k_0+2k_1+4k_2\\equiv0\\pmod7.\n\\]\nIndeed $k-\\ell(k)e_0=B(-k_1,k_2,-8k_2)^T$ and $B(0,-2,15)^T=7e_0$. If $\\ell(k)=7r$ its explicit preimage is $(-k_1,k_2-2r,15r-8k_2)^T$.","tic polynomial $(x-2)^3+7$ gives a stable real eigenvalue $a=2-7^{1/3}\\in(0,1)$ and a conjugate unstable pair of modulus $a^{-1/2}>1$. Every nonzero integer $A$-orbit escapes in both directions: otherwise an integer vector would tend to zero along positive or negative iterates. Fourier orbit sums are therefore absolutely convergent for smooth coefficients. The dual degree-seven image is exactly\n\\[\n k\\in B\\mathbb Z^3\\quad\\Longleftrightarrow\\quad\n \\ell(k):=k_0+2k_1+4k_2\\equiv0\\pmod7.\n\\]\nIndeed $k-\\ell(k)e_0=B(-k_1,k_2,-8k_2)^T$ and $B(0,-2,15)^T=7e_0$. If $\\ell(k)=7r$ its explicit preimage is $(-k_1,k_2-2r,15r-8k_2)^T$.","stable pair of modulus $a^{-1/2}>1$. Every nonzero integer $A$-orbit escapes in both directions: otherwise an integer vector would tend to zero along positive or negative iterates. Fourier orbit sums are therefore absolutely convergent for smooth coefficients. The dual degree-seven image is exactly\n\\[\n k\\in B\\mathbb Z^3\\quad\\Longleftrightarrow\\quad\n \\ell(k):=k_0+2k_1+4k_2\\equiv0\\pmod7.\n\\]\nIndeed $k-\\ell(k)e_0=B(-k_1,k_2,-8k_2)^T$ and $B(0,-2,15)^T=7e_0$. If $\\ell(k)=7r$ its explicit preimage is $(-k_1,k_2-2r,15r-8k_2)^T$.","ery nonzero integer $A$-orbit escapes in both directions: otherwise an integer vector would tend to zero along positive or negative iterates. Fourier orbit sums are therefore absolutely convergent for smooth coefficients. The dual degree-seven image is exactly\n\\[\n k\\in B\\mathbb Z^3\\quad\\Longleftrightarrow\\quad\n \\ell(k):=k_0+2k_1+4k_2\\equiv0\\pmod7.\n\\]\nIndeed $k-\\ell(k)e_0=B(-k_1,k_2,-8k_2)^T$ and $B(0,-2,15)^T=7e_0$. If $\\ell(k)=7r$ its explicit preimage is $(-k_1,k_2-2r,15r-8k_2)^T$.","orbit escapes in both directions: otherwise an integer vector would tend to zero along positive or negative iterates. Fourier orbit sums are therefore absolutely convergent for smooth coefficients. The dual degree-seven image is exactly\n\\[\n k\\in B\\mathbb Z^3\\quad\\Longleftrightarrow\\quad\n \\ell(k):=k_0+2k_1+4k_2\\equiv0\\pmod7.\n\\]\nIndeed $k-\\ell(k)e_0=B(-k_1,k_2,-8k_2)^T$ and $B(0,-2,15)^T=7e_0$. If $\\ell(k)=7r$ its explicit preimage is $(-k_1,k_2-2r,15r-8k_2)^T$.","rwise an integer vector would tend to zero along positive or negative iterates. Fourier orbit sums are therefore absolutely convergent for smooth coefficients. The dual degree-seven image is exactly\n\\[\n k\\in B\\mathbb Z^3\\quad\\Longleftrightarrow\\quad\n \\ell(k):=k_0+2k_1+4k_2\\equiv0\\pmod7.\n\\]\nIndeed $k-\\ell(k)e_0=B(-k_1,k_2,-8k_2)^T$ and $B(0,-2,15)^T=7e_0$. If $\\ell(k)=7r$ its explicit preimage is $(-k_1,k_2-2r,15r-8k_2)^T$.","\\subsection{The convergent transfer proof}\nFor functions on $\\mathbb T^3$, let\n\\[\n S_Nh(v)=\\sum_{\\kappa\\in\\ker N}h(v+\\kappa),\\quad\n C_Nh(v)=h(Nv),\\quad L_Nh(v)=\\sum_{Nw=v}h(w),\\quad\n \\delta_Mh=h-h\\circ M.\n\\]\nThe transfer $L_N$ is unnormalized. For $h=\\sum c_ke_k$,\n\\[\n S_Nh=\\sum_{k\\in B\\mathbb Z^3}7c_ke_k,\\qquad\n L_Nh=\\sum_k7c_{Bk}e_k,\\qquad S_Nh=(L_Nh)\\circ N,\n \\qquad L_NC_Nh=7h.\n\\]\nSince $M$ permutes $\\ker N$ and commutes with $N$, the required operators commute with composition by $M$.","ction{The convergent transfer proof}\nFor functions on $\\mathbb T^3$, let\n\\[\n S_Nh(v)=\\sum_{\\kappa\\in\\ker N}h(v+\\kappa),\\quad\n C_Nh(v)=h(Nv),\\quad L_Nh(v)=\\sum_{Nw=v}h(w),\\quad\n \\delta_Mh=h-h\\circ M.\n\\]\nThe transfer $L_N$ is unnormalized. For $h=\\sum c_ke_k$,\n\\[\n S_Nh=\\sum_{k\\in B\\mathbb Z^3}7c_ke_k,\\qquad\n L_Nh=\\sum_k7c_{Bk}e_k,\\qquad S_Nh=(L_Nh)\\circ N,\n \\qquad L_NC_Nh=7h.\n\\]\nSince $M$ permutes $\\ker N$ and commutes with $N$, the required operators commute with composition by $M$.","er proof}\nFor functions on $\\mathbb T^3$, let\n\\[\n S_Nh(v)=\\sum_{\\kappa\\in\\ker N}h(v+\\kappa),\\quad\n C_Nh(v)=h(Nv),\\quad L_Nh(v)=\\sum_{Nw=v}h(w),\\quad\n \\delta_Mh=h-h\\circ M.\n\\]\nThe transfer $L_N$ is unnormalized. For $h=\\sum c_ke_k$,\n\\[\n S_Nh=\\sum_{k\\in B\\mathbb Z^3}7c_ke_k,\\qquad\n L_Nh=\\sum_k7c_{Bk}e_k,\\qquad S_Nh=(L_Nh)\\circ N,\n \\qquad L_NC_Nh=7h.\n\\]\nSince $M$ permutes $\\ker N$ and commutes with $N$, the required operators commute with composition by $M$.",")=\\sum_{\\kappa\\in\\ker N}h(v+\\kappa),\\quad\n C_Nh(v)=h(Nv),\\quad L_Nh(v)=\\sum_{Nw=v}h(w),\\quad\n \\delta_Mh=h-h\\circ M.\n\\]\nThe transfer $L_N$ is unnormalized. For $h=\\sum c_ke_k$,\n\\[\n S_Nh=\\sum_{k\\in B\\mathbb Z^3}7c_ke_k,\\qquad\n L_Nh=\\sum_k7c_{Bk}e_k,\\qquad S_Nh=(L_Nh)\\circ N,\n \\qquad L_NC_Nh=7h.\n\\]\nSince $M$ permutes $\\ker N$ and commutes with $N$, the required operators commute with composition by $M$.","\\begin{theorem}[Principal-norm rigidity]\\label{thm:principal-rigidity}\nLet $h,r$ be smooth mean-zero complex functions on $\\mathbb T^3$. If\n\\begin{equation}\\label{eq:norm-coboundary}\n S_Nh-h\\circ N=\\delta_Mr,\n\\end{equation}\nthen $h=\\delta_Mg$ for a smooth mean-zero $g$, with convergent formula\n\\[\n g=-\\frac17\\sum_{j\\ge1}L_N^jr.\n\\]\nIf $h,r$ extend holomorphically to all of $X$, so does $g$.\n\\end{theorem}\n\\begin{proof}\nApply $L_N/7$ to \\eqref{eq:norm-coboundary}. The trace relation gives $L_Nh-h=(1/7)\\delta_M L_Nr$, and induction yields\n\\[\n L_N^nh-h=\\frac17\\delta_M\\sum_{j=1}^{n}L_N^jr.\n\\]\nBecause $B^3=7I$,\n\\[\n L_N^{3n}f=\\sum_{k\\ne0}343^nc_{7^nk}e_k\n\\]\nfor a mean-zero $f$. For any $C^q$ norm choose $s>\\max(q,3)$. Smooth Fourier coefficients have finite weighted $\\ell^1$ no","\\begin{theorem}[Principal-norm rigidity]\\label{thm:principal-rigidity}\nLet $h,r$ be smooth mean-zero complex functions on $\\mathbb T^3$. If\n\\begin{equation}\\label{eq:norm-coboundary}\n S_Nh-h\\circ N=\\delta_Mr,\n\\end{equation}\nthen $h=\\delta_Mg$ for a smooth mean-zero $g$, with convergent formula\n\\[\n g=-\\frac17\\sum_{j\\ge1}L_N^jr.\n\\]\nIf $h,r$ extend holomorphically to all of $X$, so does $g$.\n\\end{theorem}\n\\begin{proof}\nApply $L_N/7$ to \\eqref{eq:norm-coboundary}. The trace relation gives $L_Nh-h=(1/7)\\delta_M L_Nr$, and induction yields\n\\[\n L_N^nh-h=\\frac17\\delta_M\\sum_{j=1}^{n}L_N^jr.\n\\]\nBecause $B^3=7I$,\n\\[\n L_N^{3n}f=\\sum_{k\\ne0}343^nc_{7^nk}e_k\n\\]\nfor a mean-zero $f$. For any $C^q$ norm choose $s>\\max(q,3)$. Smooth Fourier coefficients have finite weighted $\\ell^1$ norms of every order, and the displayed expression is bou","\\begin{theorem}[Principal-norm rigidity]\\label{thm:principal-rigidity}\nLet $h,r$ be smooth mean-zero complex functions on $\\mathbb T^3$. If\n\\begin{equation}\\label{eq:norm-coboundary}\n S_Nh-h\\circ N=\\delta_Mr,\n\\end{equation}\nthen $h=\\delta_Mg$ for a smooth mean-zero $g$, with convergent formula\n\\[\n g=-\\frac17\\sum_{j\\ge1}L_N^jr.\n\\]\nIf $h,r$ extend holomorphically to all of $X$, so does $g$.\n\\end{theorem}\n\\begin{proof}\nApply $L_N/7$ to \\eqref{eq:norm-coboundary}. The trace relation gives $L_Nh-h=(1/7)\\delta_M L_Nr$, and induction yields\n\\[\n L_N^nh-h=\\frac17\\delta_M\\sum_{j=1}^{n}L_N^jr.\n\\]\nBecause $B^3=7I$,\n\\[\n L_N^{3n}f=\\sum_{k\\ne0}343^nc_{7^nk}e_k\n\\]\nfor a mean-zero $f$. For any $C^q$ norm choose $s>\\max(q,3)$. Smooth Fourier coefficients have finite weighted $\\ell^1$ norms of every order, and the displayed expression is bounded, up to a derivative convention constant, by\n\\[\n 7^{n(3-s)}\\sum_{k\\ne0}","\\begin{theorem}[Principal-norm rigidity]\\label{thm:principal-rigidity}\nLet $h,r$ be smooth mean-zero complex functions on $\\mathbb T^3$. If\n\\begin{equation}\\label{eq:norm-coboundary}\n S_Nh-h\\circ N=\\delta_Mr,\n\\end{equation}\nthen $h=\\delta_Mg$ for a smooth mean-zero $g$, with convergent formula\n\\[\n g=-\\frac17\\sum_{j\\ge1}L_N^jr.\n\\]\nIf $h,r$ extend holomorphically to all of $X$, so does $g$.\n\\end{theorem}\n\\begin{proof}\nApply $L_N/7$ to \\eqref{eq:norm-coboundary}. The trace relation gives $L_Nh-h=(1/7)\\delta_M L_Nr$, and induction yields\n\\[\n L_N^nh-h=\\frac17\\delta_M\\sum_{j=1}^{n}L_N^jr.\n\\]\nBecause $B^3=7I$,\n\\[\n L_N^{3n}f=\\sum_{k\\ne0}343^nc_{7^nk}e_k\n\\]\nfor a mean-zero $f$. For any $C^q$ norm choose $s>\\max(q,3)$. Smooth Fourier coefficients have finite weighted $\\ell^1$ norms of every order, and the displayed expression is bounded, up to a derivative convention constant, by\n\\[\n 7^{n(3-s)}\\sum_{k\\ne0}\\|k\\|^s|c_k|.\n\\]\nIt tends geomet","\\begin{theorem}[Principal-norm rigidity]\\label{thm:principal-rigidity}\nLet $h,r$ be smooth mean-zero complex functions on $\\mathbb T^3$. If\n\\begin{equation}\\label{eq:norm-coboundary}\n S_Nh-h\\circ N=\\delta_Mr,\n\\end{equation}\nthen $h=\\delta_Mg$ for a smooth mean-zero $g$, with convergent formula\n\\[\n g=-\\frac17\\sum_{j\\ge1}L_N^jr.\n\\]\nIf $h,r$ extend holomorphically to all of $X$, so does $g$.\n\\end{theorem}\n\\begin{proof}\nApply $L_N/7$ to \\eqref{eq:norm-coboundary}. The trace relation gives $L_Nh-h=(1/7)\\delta_M L_Nr$, and induction yields\n\\[\n L_N^nh-h=\\frac17\\delta_M\\sum_{j=1}^{n}L_N^jr.\n\\]\nBecause $B^3=7I$,\n\\[\n L_N^{3n}f=\\sum_{k\\ne0}343^nc_{7^nk}e_k\n\\]\nfor a mean-zero $f$. For any $C^q$ norm choose $s>\\max(q,3)$. Smooth Fourier coefficients have finite weighted $\\ell^1$ norms of every order, and the displayed expression is bounded, up to a derivative convention constant, by\n\\[\n 7^{n(3-s)}\\sum_{k\\ne0}\\|k\\|^s|c_k|.\n\\]\nIt tends geometrically to zero and is summ","\\begin{theorem}[Principal-norm rigidity]\\label{thm:principal-rigidity}\nLet $h,r$ be smooth mean-zero complex functions on $\\mathbb T^3$. If\n\\begin{equation}\\label{eq:norm-coboundary}\n S_Nh-h\\circ N=\\delta_Mr,\n\\end{equation}\nthen $h=\\delta_Mg$ for a smooth mean-zero $g$, with convergent formula\n\\[\n g=-\\frac17\\sum_{j\\ge1}L_N^jr.\n\\]\nIf $h,r$ extend holomorphically to all of $X$, so does $g$.\n\\end{theorem}\n\\begin{proof}\nApply $L_N/7$ to \\eqref{eq:norm-coboundary}. The trace relation gives $L_Nh-h=(1/7)\\delta_M L_Nr$, and induction yields\n\\[\n L_N^nh-h=\\frac17\\delta_M\\sum_{j=1}^{n}L_N^jr.\n\\]\nBecause $B^3=7I$,\n\\[\n L_N^{3n}f=\\sum_{k\\ne0}343^nc_{7^nk}e_k\n\\]\nfor a mean-zero $f$. For any $C^q$ norm choose $s>\\max(q,3)$. Smooth Fourier coefficients have finite weighted $\\ell^1$ norms of every order, and the displayed expression is bounded, up to a derivative convention constant, by\n\\[\n 7^{n(3-s)}\\sum_{k\\ne0}\\|k\\|^s|c_k|.\n\\]\nIt tends geometrically to zero and is summable. The other two residue classes are obtained by the fixed","\\begin{theorem}[Principal-norm rigidity]\\label{thm:principal-rigidity}\nLet $h,r$ be smooth mean-zero complex functions on $\\mathbb T^3$. If\n\\begin{equation}\\label{eq:norm-coboundary}\n S_Nh-h\\circ N=\\delta_Mr,\n\\end{equation}\nthen $h=\\delta_Mg$ for a smooth mean-zero $g$, with convergent formula\n\\[\n g=-\\frac17\\sum_{j\\ge1}L_N^jr.\n\\]\nIf $h,r$ extend holomorphically to all of $X$, so does $g$.\n\\end{theorem}\n\\begin{proof}\nApply $L_N/7$ to \\eqref{eq:norm-coboundary}. The trace relation gives $L_Nh-h=(1/7)\\delta_M L_Nr$, and induction yields\n\\[\n L_N^nh-h=\\frac17\\delta_M\\sum_{j=1}^{n}L_N^jr.\n\\]\nBecause $B^3=7I$,\n\\[\n L_N^{3n}f=\\sum_{k\\ne0}343^nc_{7^nk}e_k\n\\]\nfor a mean-zero $f$. For any $C^q$ norm choose $s>\\max(q,3)$. Smooth Fourier coefficients have finite weighted $\\ell^1$ norms of every order, and the displayed expression is bounded, up to a derivative convention constant, by\n\\[\n 7^{n(3-s)}\\sum_{k\\ne0}\\|k\\|^s|c_k|.\n\\]\nIt tends geometrically to zero and is summable. The other two residue classes are obtained by the fixed continuou","\\begin{theorem}[Principal-norm rigidity]\\label{thm:principal-rigidity}\nLet $h,r$ be smooth mean-zero complex functions on $\\mathbb T^3$. If\n\\begin{equation}\\label{eq:norm-coboundary}\n S_Nh-h\\circ N=\\delta_Mr,\n\\end{equation}\nthen $h=\\delta_Mg$ for a smooth mean-zero $g$, with convergent formula\n\\[\n g=-\\frac17\\sum_{j\\ge1}L_N^jr.\n\\]\nIf $h,r$ extend holomorphically to all of $X$, so does $g$.\n\\end{theorem}\n\\begin{proof}\nApply $L_N/7$ to \\eqref{eq:norm-coboundary}. The trace relation gives $L_Nh-h=(1/7)\\delta_M L_Nr$, and induction yields\n\\[\n L_N^nh-h=\\frac17\\delta_M\\sum_{j=1}^{n}L_N^jr.\n\\]\nBecause $B^3=7I$,\n\\[\n L_N^{3n}f=\\sum_{k\\ne0}343^nc_{7^nk}e_k\n\\]\nfor a mean-zero $f$. For any $C^q$ norm choose $s>\\max(q,3)$. Smooth Fourier coefficients have finite weighted $\\ell^1$ norms of every order, and the displayed expression is bounded, up to a derivative convention constant, by\n\\[\n 7^{n(3-s)}\\sum_{k\\ne0}\\|k\\|^s|c_k|.\n\\]\nIt tends geometrically to zero and is summable. The other two residue classes are obtained by the fixed continuous operators $L_N,L_N^2$. The series f","\\begin{theorem}[Principal-norm rigidity]\\label{thm:principal-rigidity}\nLet $h,r$ be smooth mean-zero complex functions on $\\mathbb T^3$. If\n\\begin{equation}\\label{eq:norm-coboundary}\n S_Nh-h\\circ N=\\delta_Mr,\n\\end{equation}\nthen $h=\\delta_Mg$ for a smooth mean-zero $g$, with convergent formula\n\\[\n g=-\\frac17\\sum_{j\\ge1}L_N^jr.\n\\]\nIf $h,r$ extend holomorphically to all of $X$, so does $g$.\n\\end{theorem}\n\\begin{proof}\nApply $L_N/7$ to \\eqref{eq:norm-coboundary}. The trace relation gives $L_Nh-h=(1/7)\\delta_M L_Nr$, and induction yields\n\\[\n L_N^nh-h=\\frac17\\delta_M\\sum_{j=1}^{n}L_N^jr.\n\\]\nBecause $B^3=7I$,\n\\[\n L_N^{3n}f=\\sum_{k\\ne0}343^nc_{7^nk}e_k\n\\]\nfor a mean-zero $f$. For any $C^q$ norm choose $s>\\max(q,3)$. Smooth Fourier coefficients have finite weighted $\\ell^1$ norms of every order, and the displayed expression is bounded, up to a derivative convention constant, by\n\\[\n 7^{n(3-s)}\\sum_{k\\ne0}\\|k\\|^s|c_k|.\n\\]\nIt tends geometrically to zero and is summable. The other two residue classes are obtained by the fixed continuous operators $L_N,L_N^2$. The series for $g$ conver","orm rigidity]\\label{thm:principal-rigidity}\nLet $h,r$ be smooth mean-zero complex functions on $\\mathbb T^3$. If\n\\begin{equation}\\label{eq:norm-coboundary}\n S_Nh-h\\circ N=\\delta_Mr,\n\\end{equation}\nthen $h=\\delta_Mg$ for a smooth mean-zero $g$, with convergent formula\n\\[\n g=-\\frac17\\sum_{j\\ge1}L_N^jr.\n\\]\nIf $h,r$ extend holomorphically to all of $X$, so does $g$.\n\\end{theorem}\n\\begin{proof}\nApply $L_N/7$ to \\eqref{eq:norm-coboundary}. The trace relation gives $L_Nh-h=(1/7)\\delta_M L_Nr$, and induction yields\n\\[\n L_N^nh-h=\\frac17\\delta_M\\sum_{j=1}^{n}L_N^jr.\n\\]\nBecause $B^3=7I$,\n\\[\n L_N^{3n}f=\\sum_{k\\ne0}343^nc_{7^nk}e_k\n\\]\nfor a mean-zero $f$. For any $C^q$ norm choose $s>\\max(q,3)$. Smooth Fourier coefficients have finite weighted $\\ell^1$ norms of every order, and the displayed expression is bounded, up to a derivative convention constant, by\n\\[\n 7^{n(3-s)}\\sum_{k\\ne0}\\|k\\|^s|c_k|.\n\\]\nIt tends geometrically to zero and is summable. The other two residue classes are obtained by the fixed continuous operators $L_N,L_N^2$. The series for $g$ converges in every $C^q$ norm; taking the li…","mean-zero complex functions on $\\mathbb T^3$. If\n\\begin{equation}\\label{eq:norm-coboundary}\n S_Nh-h\\circ N=\\delta_Mr,\n\\end{equation}\nthen $h=\\delta_Mg$ for a smooth mean-zero $g$, with convergent formula\n\\[\n g=-\\frac17\\sum_{j\\ge1}L_N^jr.\n\\]\nIf $h,r$ extend holomorphically to all of $X$, so does $g$.\n\\end{theorem}\n\\begin{proof}\nApply $L_N/7$ to \\eqref{eq:norm-coboundary}. The trace relation gives $L_Nh-h=(1/7)\\delta_M L_Nr$, and induction yields\n\\[\n L_N^nh-h=\\frac17\\delta_M\\sum_{j=1}^{n}L_N^jr.\n\\]\nBecause $B^3=7I$,\n\\[\n L_N^{3n}f=\\sum_{k\\ne0}343^nc_{7^nk}e_k\n\\]\nfor a mean-zero $f$. For any $C^q$ norm choose $s>\\max(q,3)$. Smooth Fourier coefficients have finite weighted $\\ell^1$ norms of every order, and the displayed expression is bounded, up to a derivative convention constant, by\n\\[\n 7^{n(3-s)}\\sum_{k\\ne0}\\|k\\|^s|c_k|.\n\\]\nIt tends geometrically to zero and is summable. The other two residue classes are obtained by the fixed continuous operators $L_N,L_N^2$. The series for $g$ converges in every $C^q$ norm; taking the limit gives $h=\\delta_Mg$. For a global holomorphic function on $X…","egin{equation}\\label{eq:norm-coboundary}\n S_Nh-h\\circ N=\\delta_Mr,\n\\end{equation}\nthen $h=\\delta_Mg$ for a smooth mean-zero $g$, with convergent formula\n\\[\n g=-\\frac17\\sum_{j\\ge1}L_N^jr.\n\\]\nIf $h,r$ extend holomorphically to all of $X$, so does $g$.\n\\end{theorem}\n\\begin{proof}\nApply $L_N/7$ to \\eqref{eq:norm-coboundary}. The trace relation gives $L_Nh-h=(1/7)\\delta_M L_Nr$, and induction yields\n\\[\n L_N^nh-h=\\frac17\\delta_M\\sum_{j=1}^{n}L_N^jr.\n\\]\nBecause $B^3=7I$,\n\\[\n L_N^{3n}f=\\sum_{k\\ne0}343^nc_{7^nk}e_k\n\\]\nfor a mean-zero $f$. For any $C^q$ norm choose $s>\\max(q,3)$. Smooth Fourier coefficients have finite weighted $\\ell^1$ norms of every order, and the displayed expression is bounded, up to a derivative convention constant, by\n\\[\n 7^{n(3-s)}\\sum_{k\\ne0}\\|k\\|^s|c_k|.\n\\]\nIt tends geometrically to zero and is summable. The other two residue classes are obtained by the fixed continuous operators $L_N,L_N^2$. The series for $g$ converges in every $C^q$ norm; taking the limit gives $h=\\delta_Mg$. For a global holomorphic function on $X$, its Laurent coefficients decay faster than every…","ta_Mr,\n\\end{equation}\nthen $h=\\delta_Mg$ for a smooth mean-zero $g$, with convergent formula\n\\[\n g=-\\frac17\\sum_{j\\ge1}L_N^jr.\n\\]\nIf $h,r$ extend holomorphically to all of $X$, so does $g$.\n\\end{theorem}\n\\begin{proof}\nApply $L_N/7$ to \\eqref{eq:norm-coboundary}. The trace relation gives $L_Nh-h=(1/7)\\delta_M L_Nr$, and induction yields\n\\[\n L_N^nh-h=\\frac17\\delta_M\\sum_{j=1}^{n}L_N^jr.\n\\]\nBecause $B^3=7I$,\n\\[\n L_N^{3n}f=\\sum_{k\\ne0}343^nc_{7^nk}e_k\n\\]\nfor a mean-zero $f$. For any $C^q$ norm choose $s>\\max(q,3)$. Smooth Fourier coefficients have finite weighted $\\ell^1$ norms of every order, and the displayed expression is bounded, up to a derivative convention constant, by\n\\[\n 7^{n(3-s)}\\sum_{k\\ne0}\\|k\\|^s|c_k|.\n\\]\nIt tends geometrically to zero and is summable. The other two residue classes are obtained by the fixed continuous operators $L_N,L_N^2$. The series for $g$ converges in every $C^q$ norm; taking the limit gives $h=\\delta_Mg$. For a global holomorphic function on $X$, its Laurent coefficients decay faster than every fixed exponential. On each compact product annulus the deca…","{equation}\nthen $h=\\delta_Mg$ for a smooth mean-zero $g$, with convergent formula\n\\[\n g=-\\frac17\\sum_{j\\ge1}L_N^jr.\n\\]\nIf $h,r$ extend holomorphically to all of $X$, so does $g$.\n\\end{theorem}\n\\begin{proof}\nApply $L_N/7$ to \\eqref{eq:norm-coboundary}. The trace relation gives $L_Nh-h=(1/7)\\delta_M L_Nr$, and induction yields\n\\[\n L_N^nh-h=\\frac17\\delta_M\\sum_{j=1}^{n}L_N^jr.\n\\]\nBecause $B^3=7I$,\n\\[\n L_N^{3n}f=\\sum_{k\\ne0}343^nc_{7^nk}e_k\n\\]\nfor a mean-zero $f$. For any $C^q$ norm choose $s>\\max(q,3)$. Smooth Fourier coefficients have finite weighted $\\ell^1$ norms of every order, and the displayed expression is bounded, up to a derivative convention constant, by\n\\[\n 7^{n(3-s)}\\sum_{k\\ne0}\\|k\\|^s|c_k|.\n\\]\nIt tends geometrically to zero and is summable. The other two residue classes are obtained by the fixed continuous operators $L_N,L_N^2$. The series for $g$ converges in every $C^q$ norm; taking the limit gives $h=\\delta_Mg$. For a global holomorphic function on $X$, its Laurent coefficients decay faster than every fixed exponential. On each compact product annulus the decay at $7^nk$…","h convergent formula\n\\[\n g=-\\frac17\\sum_{j\\ge1}L_N^jr.\n\\]\nIf $h,r$ extend holomorphically to all of $X$, so does $g$.\n\\end{theorem}\n\\begin{proof}\nApply $L_N/7$ to \\eqref{eq:norm-coboundary}. The trace relation gives $L_Nh-h=(1/7)\\delta_M L_Nr$, and induction yields\n\\[\n L_N^nh-h=\\frac17\\delta_M\\sum_{j=1}^{n}L_N^jr.\n\\]\nBecause $B^3=7I$,\n\\[\n L_N^{3n}f=\\sum_{k\\ne0}343^nc_{7^nk}e_k\n\\]\nfor a mean-zero $f$. For any $C^q$ norm choose $s>\\max(q,3)$. Smooth Fourier coefficients have finite weighted $\\ell^1$ norms of every order, and the displayed expression is bounded, up to a derivative convention constant, by\n\\[\n 7^{n(3-s)}\\sum_{k\\ne0}\\|k\\|^s|c_k|.\n\\]\nIt tends geometrically to zero and is summable. The other two residue classes are obtained by the fixed continuous operators $L_N,L_N^2$. The series for $g$ converges in every $C^q$ norm; taking the limit gives $h=\\delta_Mg$. For a global holomorphic function on $X$, its Laurent coefficients decay faster than every fixed exponential. On each compact product annulus the decay at $7^nk$ dominates both $343^n$ and the annular factor $e^{R\\|k\\|}$, …","formula\n\\[\n g=-\\frac17\\sum_{j\\ge1}L_N^jr.\n\\]\nIf $h,r$ extend holomorphically to all of $X$, so does $g$.\n\\end{theorem}\n\\begin{proof}\nApply $L_N/7$ to \\eqref{eq:norm-coboundary}. The trace relation gives $L_Nh-h=(1/7)\\delta_M L_Nr$, and induction yields\n\\[\n L_N^nh-h=\\frac17\\delta_M\\sum_{j=1}^{n}L_N^jr.\n\\]\nBecause $B^3=7I$,\n\\[\n L_N^{3n}f=\\sum_{k\\ne0}343^nc_{7^nk}e_k\n\\]\nfor a mean-zero $f$. For any $C^q$ norm choose $s>\\max(q,3)$. Smooth Fourier coefficients have finite weighted $\\ell^1$ norms of every order, and the displayed expression is bounded, up to a derivative convention constant, by\n\\[\n 7^{n(3-s)}\\sum_{k\\ne0}\\|k\\|^s|c_k|.\n\\]\nIt tends geometrically to zero and is summable. The other two residue classes are obtained by the fixed continuous operators $L_N,L_N^2$. The series for $g$ converges in every $C^q$ norm; taking the limit gives $h=\\delta_Mg$. For a global holomorphic function on $X$, its Laurent coefficients decay faster than every fixed exponential. On each compact product annulus the decay at $7^nk$ dominates both $343^n$ and the annular factor $e^{R\\|k\\|}$, proving norma…","ac17\\sum_{j\\ge1}L_N^jr.\n\\]\nIf $h,r$ extend holomorphically to all of $X$, so does $g$.\n\\end{theorem}\n\\begin{proof}\nApply $L_N/7$ to \\eqref{eq:norm-coboundary}. The trace relation gives $L_Nh-h=(1/7)\\delta_M L_Nr$, and induction yields\n\\[\n L_N^nh-h=\\frac17\\delta_M\\sum_{j=1}^{n}L_N^jr.\n\\]\nBecause $B^3=7I$,\n\\[\n L_N^{3n}f=\\sum_{k\\ne0}343^nc_{7^nk}e_k\n\\]\nfor a mean-zero $f$. For any $C^q$ norm choose $s>\\max(q,3)$. Smooth Fourier coefficients have finite weighted $\\ell^1$ norms of every order, and the displayed expression is bounded, up to a derivative convention constant, by\n\\[\n 7^{n(3-s)}\\sum_{k\\ne0}\\|k\\|^s|c_k|.\n\\]\nIt tends geometrically to zero and is summable. The other two residue classes are obtained by the fixed continuous operators $L_N,L_N^2$. The series for $g$ converges in every $C^q$ norm; taking the limit gives $h=\\delta_Mg$. For a global holomorphic function on $X$, its Laurent coefficients decay faster than every fixed exponential. On each compact product annulus the decay at $7^nk$ dominates both $343^n$ and the annular factor $e^{R\\|k\\|}$, proving normal convergence ther…","l of $X$, so does $g$.\n\\end{theorem}\n\\begin{proof}\nApply $L_N/7$ to \\eqref{eq:norm-coboundary}. The trace relation gives $L_Nh-h=(1/7)\\delta_M L_Nr$, and induction yields\n\\[\n L_N^nh-h=\\frac17\\delta_M\\sum_{j=1}^{n}L_N^jr.\n\\]\nBecause $B^3=7I$,\n\\[\n L_N^{3n}f=\\sum_{k\\ne0}343^nc_{7^nk}e_k\n\\]\nfor a mean-zero $f$. For any $C^q$ norm choose $s>\\max(q,3)$. Smooth Fourier coefficients have finite weighted $\\ell^1$ norms of every order, and the displayed expression is bounded, up to a derivative convention constant, by\n\\[\n 7^{n(3-s)}\\sum_{k\\ne0}\\|k\\|^s|c_k|.\n\\]\nIt tends geometrically to zero and is summable. The other two residue classes are obtained by the fixed continuous operators $L_N,L_N^2$. The series for $g$ converges in every $C^q$ norm; taking the limit gives $h=\\delta_Mg$. For a global holomorphic function on $X$, its Laurent coefficients decay faster than every fixed exponential. On each compact product annulus the decay at $7^nk$ dominates both $343^n$ and the annular factor $e^{R\\|k\\|}$, proving normal convergence there and a global holomorphic $g$.\n\\end{proof}\nFor a finite Laurent…","ives $L_Nh-h=(1/7)\\delta_M L_Nr$, and induction yields\n\\[\n L_N^nh-h=\\frac17\\delta_M\\sum_{j=1}^{n}L_N^jr.\n\\]\nBecause $B^3=7I$,\n\\[\n L_N^{3n}f=\\sum_{k\\ne0}343^nc_{7^nk}e_k\n\\]\nfor a mean-zero $f$. For any $C^q$ norm choose $s>\\max(q,3)$. Smooth Fourier coefficients have finite weighted $\\ell^1$ norms of every order, and the displayed expression is bounded, up to a derivative convention constant, by\n\\[\n 7^{n(3-s)}\\sum_{k\\ne0}\\|k\\|^s|c_k|.\n\\]\nIt tends geometrically to zero and is summable. The other two residue classes are obtained by the fixed continuous operators $L_N,L_N^2$. The series for $g$ converges in every $C^q$ norm; taking the limit gives $h=\\delta_Mg$. For a global holomorphic function on $X$, its Laurent coefficients decay faster than every fixed exponential. On each compact product annulus the decay at $7^nk$ dominates both $343^n$ and the annular factor $e^{R\\|k\\|}$, proving normal convergence there and a global holomorphic $g$.\n\\end{proof}\nFor a finite Laurent $r$ of coordinate height at most $H$, choose $n$ with $7^n>H$. Then $L_N^{3n}r=0$, so the series terminates by a de…","k\n\\]\nfor a mean-zero $f$. For any $C^q$ norm choose $s>\\max(q,3)$. Smooth Fourier coefficients have finite weighted $\\ell^1$ norms of every order, and the displayed expression is bounded, up to a derivative convention constant, by\n\\[\n 7^{n(3-s)}\\sum_{k\\ne0}\\|k\\|^s|c_k|.\n\\]\nIt tends geometrically to zero and is summable. The other two residue classes are obtained by the fixed continuous operators $L_N,L_N^2$. The series for $g$ converges in every $C^q$ norm; taking the limit gives $h=\\delta_Mg$. For a global holomorphic function on $X$, its Laurent coefficients decay faster than every fixed exponential. On each compact product annulus the decay at $7^nk$ dominates both $343^n$ and the annular factor $e^{R\\|k\\|}$, proving normal convergence there and a global holomorphic $g$.\n\\end{proof}\nFor a finite Laurent $r$ of coordinate height at most $H$, choose $n$ with $7^n>H$. Then $L_N^{3n}r=0$, so the series terminates by a derived bound. The finite coefficient recurrence has an integer Lean certificate; the smooth convergence and geometric interpretation above are ordinary analytic results…","r any $C^q$ norm choose $s>\\max(q,3)$. Smooth Fourier coefficients have finite weighted $\\ell^1$ norms of every order, and the displayed expression is bounded, up to a derivative convention constant, by\n\\[\n 7^{n(3-s)}\\sum_{k\\ne0}\\|k\\|^s|c_k|.\n\\]\nIt tends geometrically to zero and is summable. The other two residue classes are obtained by the fixed continuous operators $L_N,L_N^2$. The series for $g$ converges in every $C^q$ norm; taking the limit gives $h=\\delta_Mg$. For a global holomorphic function on $X$, its Laurent coefficients decay faster than every fixed exponential. On each compact product annulus the decay at $7^nk$ dominates both $343^n$ and the annular factor $e^{R\\|k\\|}$, proving normal convergence there and a global holomorphic $g$.\n\\end{proof}\nFor a finite Laurent $r$ of coordinate height at most $H$, choose $n$ with $7^n>H$. Then $L_N^{3n}r=0$, so the series terminates by a derived bound. The finite coefficient recurrence has an integer Lean certificate; the smooth convergence and geometric interpretation above are ordinary analytic results.","$s>\\max(q,3)$. Smooth Fourier coefficients have finite weighted $\\ell^1$ norms of every order, and the displayed expression is bounded, up to a derivative convention constant, by\n\\[\n 7^{n(3-s)}\\sum_{k\\ne0}\\|k\\|^s|c_k|.\n\\]\nIt tends geometrically to zero and is summable. The other two residue classes are obtained by the fixed continuous operators $L_N,L_N^2$. The series for $g$ converges in every $C^q$ norm; taking the limit gives $h=\\delta_Mg$. For a global holomorphic function on $X$, its Laurent coefficients decay faster than every fixed exponential. On each compact product annulus the decay at $7^nk$ dominates both $343^n$ and the annular factor $e^{R\\|k\\|}$, proving normal convergence there and a global holomorphic $g$.\n\\end{proof}\nFor a finite Laurent $r$ of coordinate height at most $H$, choose $n$ with $7^n>H$. Then $L_N^{3n}r=0$, so the series terminates by a derived bound. The finite coefficient recurrence has an integer Lean certificate; the smooth convergence and geometric interpretation above are ordinary analytic results.","ficients have finite weighted $\\ell^1$ norms of every order, and the displayed expression is bounded, up to a derivative convention constant, by\n\\[\n 7^{n(3-s)}\\sum_{k\\ne0}\\|k\\|^s|c_k|.\n\\]\nIt tends geometrically to zero and is summable. The other two residue classes are obtained by the fixed continuous operators $L_N,L_N^2$. The series for $g$ converges in every $C^q$ norm; taking the limit gives $h=\\delta_Mg$. For a global holomorphic function on $X$, its Laurent coefficients decay faster than every fixed exponential. On each compact product annulus the decay at $7^nk$ dominates both $343^n$ and the annular factor $e^{R\\|k\\|}$, proving normal convergence there and a global holomorphic $g$.\n\\end{proof}\nFor a finite Laurent $r$ of coordinate height at most $H$, choose $n$ with $7^n>H$. Then $L_N^{3n}r=0$, so the series terminates by a derived bound. The finite coefficient recurrence has an integer Lean certificate; the smooth convergence and geometric interpretation above are ordinary analytic results.","y order, and the displayed expression is bounded, up to a derivative convention constant, by\n\\[\n 7^{n(3-s)}\\sum_{k\\ne0}\\|k\\|^s|c_k|.\n\\]\nIt tends geometrically to zero and is summable. The other two residue classes are obtained by the fixed continuous operators $L_N,L_N^2$. The series for $g$ converges in every $C^q$ norm; taking the limit gives $h=\\delta_Mg$. For a global holomorphic function on $X$, its Laurent coefficients decay faster than every fixed exponential. On each compact product annulus the decay at $7^nk$ dominates both $343^n$ and the annular factor $e^{R\\|k\\|}$, proving normal convergence there and a global holomorphic $g$.\n\\end{proof}\nFor a finite Laurent $r$ of coordinate height at most $H$, choose $n$ with $7^n>H$. Then $L_N^{3n}r=0$, so the series terminates by a derived bound. The finite coefficient recurrence has an integer Lean certificate; the smooth convergence and geometric interpretation above are ordinary analytic results.","}\\|k\\|^s|c_k|.\n\\]\nIt tends geometrically to zero and is summable. The other two residue classes are obtained by the fixed continuous operators $L_N,L_N^2$. The series for $g$ converges in every $C^q$ norm; taking the limit gives $h=\\delta_Mg$. For a global holomorphic function on $X$, its Laurent coefficients decay faster than every fixed exponential. On each compact product annulus the decay at $7^nk$ dominates both $343^n$ and the annular factor $e^{R\\|k\\|}$, proving normal convergence there and a global holomorphic $g$.\n\\end{proof}\nFor a finite Laurent $r$ of coordinate height at most $H$, choose $n$ with $7^n>H$. Then $L_N^{3n}r=0$, so the series terminates by a derived bound. The finite coefficient recurrence has an integer Lean certificate; the smooth convergence and geometric interpretation above are ordinary analytic results.","ends geometrically to zero and is summable. The other two residue classes are obtained by the fixed continuous operators $L_N,L_N^2$. The series for $g$ converges in every $C^q$ norm; taking the limit gives $h=\\delta_Mg$. For a global holomorphic function on $X$, its Laurent coefficients decay faster than every fixed exponential. On each compact product annulus the decay at $7^nk$ dominates both $343^n$ and the annular factor $e^{R\\|k\\|}$, proving normal convergence there and a global holomorphic $g$.\n\\end{proof}\nFor a finite Laurent $r$ of coordinate height at most $H$, choose $n$ with $7^n>H$. Then $L_N^{3n}r=0$, so the series terminates by a derived bound. The finite coefficient recurrence has an integer Lean certificate; the smooth convergence and geometric interpretation above are ordinary analytic results.","is summable. The other two residue classes are obtained by the fixed continuous operators $L_N,L_N^2$. The series for $g$ converges in every $C^q$ norm; taking the limit gives $h=\\delta_Mg$. For a global holomorphic function on $X$, its Laurent coefficients decay faster than every fixed exponential. On each compact product annulus the decay at $7^nk$ dominates both $343^n$ and the annular factor $e^{R\\|k\\|}$, proving normal convergence there and a global holomorphic $g$.\n\\end{proof}\nFor a finite Laurent $r$ of coordinate height at most $H$, choose $n$ with $7^n>H$. Then $L_N^{3n}r=0$, so the series terminates by a derived bound. The finite coefficient recurrence has an integer Lean certificate; the smooth convergence and geometric interpretation above are ordinary analytic results.","tinuous operators $L_N,L_N^2$. The series for $g$ converges in every $C^q$ norm; taking the limit gives $h=\\delta_Mg$. For a global holomorphic function on $X$, its Laurent coefficients decay faster than every fixed exponential. On each compact product annulus the decay at $7^nk$ dominates both $343^n$ and the annular factor $e^{R\\|k\\|}$, proving normal convergence there and a global holomorphic $g$.\n\\end{proof}\nFor a finite Laurent $r$ of coordinate height at most $H$, choose $n$ with $7^n>H$. Then $L_N^{3n}r=0$, so the series terminates by a derived bound. The finite coefficient recurrence has an integer Lean certificate; the smooth convergence and geometric interpretation above are ordinary analytic results.","ies for $g$ converges in every $C^q$ norm; taking the limit gives $h=\\delta_Mg$. For a global holomorphic function on $X$, its Laurent coefficients decay faster than every fixed exponential. On each compact product annulus the decay at $7^nk$ dominates both $343^n$ and the annular factor $e^{R\\|k\\|}$, proving normal convergence there and a global holomorphic $g$.\n\\end{proof}\nFor a finite Laurent $r$ of coordinate height at most $H$, choose $n$ with $7^n>H$. Then $L_N^{3n}r=0$, so the series terminates by a derived bound. The finite coefficient recurrence has an integer Lean certificate; the smooth convergence and geometric interpretation above are ordinary analytic results.","^q$ norm; taking the limit gives $h=\\delta_Mg$. For a global holomorphic function on $X$, its Laurent coefficients decay faster than every fixed exponential. On each compact product annulus the decay at $7^nk$ dominates both $343^n$ and the annular factor $e^{R\\|k\\|}$, proving normal convergence there and a global holomorphic $g$.\n\\end{proof}\nFor a finite Laurent $r$ of coordinate height at most $H$, choose $n$ with $7^n>H$. Then $L_N^{3n}r=0$, so the series terminates by a derived bound. The finite coefficient recurrence has an integer Lean certificate; the smooth convergence and geometric interpretation above are ordinary analytic results.","king the limit gives $h=\\delta_Mg$. For a global holomorphic function on $X$, its Laurent coefficients decay faster than every fixed exponential. On each compact product annulus the decay at $7^nk$ dominates both $343^n$ and the annular factor $e^{R\\|k\\|}$, proving normal convergence there and a global holomorphic $g$.\n\\end{proof}\nFor a finite Laurent $r$ of coordinate height at most $H$, choose $n$ with $7^n>H$. Then $L_N^{3n}r=0$, so the series terminates by a derived bound. The finite coefficient recurrence has an integer Lean certificate; the smooth convergence and geometric interpretation above are ordinary analytic results.","limit gives $h=\\delta_Mg$. For a global holomorphic function on $X$, its Laurent coefficients decay faster than every fixed exponential. On each compact product annulus the decay at $7^nk$ dominates both $343^n$ and the annular factor $e^{R\\|k\\|}$, proving normal convergence there and a global holomorphic $g$.\n\\end{proof}\nFor a finite Laurent $r$ of coordinate height at most $H$, choose $n$ with $7^n>H$. Then $L_N^{3n}r=0$, so the series terminates by a derived bound. The finite coefficient recurrence has an integer Lean certificate; the smooth convergence and geometric interpretation above are ordinary analytic results.","=\\delta_Mg$. For a global holomorphic function on $X$, its Laurent coefficients decay faster than every fixed exponential. On each compact product annulus the decay at $7^nk$ dominates both $343^n$ and the annular factor $e^{R\\|k\\|}$, proving normal convergence there and a global holomorphic $g$.\n\\end{proof}\nFor a finite Laurent $r$ of coordinate height at most $H$, choose $n$ with $7^n>H$. Then $L_N^{3n}r=0$, so the series terminates by a derived bound. The finite coefficient recurrence has an integer Lean certificate; the smooth convergence and geometric interpretation above are ordinary analytic results.","\\subsection{Phase-only compatibility and the missing identification}\nSuppose a genuine nonzero relative multiplier has been constructed as\n\\[\n a_\\Delta(v)=\\alpha e^{2\\pi im^Tv}e^{h(v)},\\qquad\n m\\in\\mathbb Z^3,\\quad h\\text{ mean zero},\n\\]\nand its based ratio is actually identified with $U_\\Gamma/U_{\\rm alg}$. Write $h^\\#(v)=\\overline{h(\\overline v)}$. It suffices to establish \\eqref{eq:norm-coboundary} for $h-h^\\#$. The theorem gives $h-h^\\#=g-g\\circ M$. The gauge $e^{-g/2}$ replaces the periodic logarithm by $(h+h^\\#)/2$, real on the real torus. Since $Mp_3=p_3$ modulo the lattice, this periodic gauge has equal values at the two endpoints, leaving\n\\[\n \\frac{a_\\Delta(p_3)}{a_\\Delta(0)}=e^{2\\pi im_2/3}e^{\\text{real number}}.\n\\]\nAfter the based identification the exact modulus equality forces that real exponent to vanish. The ratio then has cube one, and an independent branch enclosure can select one. The constant $\\alpha$ c","\\subsection{Phase-only compatibility and the missing identification}\nSuppose a genuine nonzero relative multiplier has been constructed as\n\\[\n a_\\Delta(v)=\\alpha e^{2\\pi im^Tv}e^{h(v)},\\qquad\n m\\in\\mathbb Z^3,\\quad h\\text{ mean zero},\n\\]\nand its based ratio is actually identified with $U_\\Gamma/U_{\\rm alg}$. Write $h^\\#(v)=\\overline{h(\\overline v)}$. It suffices to establish \\eqref{eq:norm-coboundary} for $h-h^\\#$. The theorem gives $h-h^\\#=g-g\\circ M$. The gauge $e^{-g/2}$ replaces the periodic logarithm by $(h+h^\\#)/2$, real on the real torus. Since $Mp_3=p_3$ modulo the lattice, this periodic gauge has equal values at the two endpoints, leaving\n\\[\n \\frac{a_\\Delta(p_3)}{a_\\Delta(0)}=e^{2\\pi im_2/3}e^{\\text{real number}}.\n\\]\nAfter the based identification the exact modulus equality forces that real exponent to vanish. The ratio then has cube one, and an independent branch enclosure can select one. The constant $\\alpha$ cancels in this based ratio; it cannot simply be discarded in the unbased","\\subsection{Phase-only compatibility and the missing identification}\nSuppose a genuine nonzero relative multiplier has been constructed as\n\\[\n a_\\Delta(v)=\\alpha e^{2\\pi im^Tv}e^{h(v)},\\qquad\n m\\in\\mathbb Z^3,\\quad h\\text{ mean zero},\n\\]\nand its based ratio is actually identified with $U_\\Gamma/U_{\\rm alg}$. Write $h^\\#(v)=\\overline{h(\\overline v)}$. It suffices to establish \\eqref{eq:norm-coboundary} for $h-h^\\#$. The theorem gives $h-h^\\#=g-g\\circ M$. The gauge $e^{-g/2}$ replaces the periodic logarithm by $(h+h^\\#)/2$, real on the real torus. Since $Mp_3=p_3$ modulo the lattice, this periodic gauge has equal values at the two endpoints, leaving\n\\[\n \\frac{a_\\Delta(p_3)}{a_\\Delta(0)}=e^{2\\pi im_2/3}e^{\\text{real number}}.\n\\]\nAfter the based identification the exact modulus equality forces that real exponent to vanish. The ratio then has cube one, and an independent branch enclosure can select one. The constant $\\alpha$ cancels in this based ratio; it cannot simply be discarded in the unbased principal norm, where it contributes $\\alp","n{Phase-only compatibility and the missing identification}\nSuppose a genuine nonzero relative multiplier has been constructed as\n\\[\n a_\\Delta(v)=\\alpha e^{2\\pi im^Tv}e^{h(v)},\\qquad\n m\\in\\mathbb Z^3,\\quad h\\text{ mean zero},\n\\]\nand its based ratio is actually identified with $U_\\Gamma/U_{\\rm alg}$. Write $h^\\#(v)=\\overline{h(\\overline v)}$. It suffices to establish \\eqref{eq:norm-coboundary} for $h-h^\\#$. The theorem gives $h-h^\\#=g-g\\circ M$. The gauge $e^{-g/2}$ replaces the periodic logarithm by $(h+h^\\#)/2$, real on the real torus. Since $Mp_3=p_3$ modulo the lattice, this periodic gauge has equal values at the two endpoints, leaving\n\\[\n \\frac{a_\\Delta(p_3)}{a_\\Delta(0)}=e^{2\\pi im_2/3}e^{\\text{real number}}.\n\\]\nAfter the based identification the exact modulus equality forces that real exponent to vanish. The ratio then has cube one, and an independent branch enclosure can select one. The constant $\\alpha$ cancels in this based ratio; it cannot simply be discarded in the unbased principal norm, where it contributes $\\alpha^6$.","nd the missing identification}\nSuppose a genuine nonzero relative multiplier has been constructed as\n\\[\n a_\\Delta(v)=\\alpha e^{2\\pi im^Tv}e^{h(v)},\\qquad\n m\\in\\mathbb Z^3,\\quad h\\text{ mean zero},\n\\]\nand its based ratio is actually identified with $U_\\Gamma/U_{\\rm alg}$. Write $h^\\#(v)=\\overline{h(\\overline v)}$. It suffices to establish \\eqref{eq:norm-coboundary} for $h-h^\\#$. The theorem gives $h-h^\\#=g-g\\circ M$. The gauge $e^{-g/2}$ replaces the periodic logarithm by $(h+h^\\#)/2$, real on the real torus. Since $Mp_3=p_3$ modulo the lattice, this periodic gauge has equal values at the two endpoints, leaving\n\\[\n \\frac{a_\\Delta(p_3)}{a_\\Delta(0)}=e^{2\\pi im_2/3}e^{\\text{real number}}.\n\\]\nAfter the based identification the exact modulus equality forces that real exponent to vanish. The ratio then has cube one, and an independent branch enclosure can select one. The constant $\\alpha$ cancels in this based ratio; it cannot simply be discarded in the unbased principal norm, where it contributes $\\alpha^6$.","Suppose a genuine nonzero relative multiplier has been constructed as\n\\[\n a_\\Delta(v)=\\alpha e^{2\\pi im^Tv}e^{h(v)},\\qquad\n m\\in\\mathbb Z^3,\\quad h\\text{ mean zero},\n\\]\nand its based ratio is actually identified with $U_\\Gamma/U_{\\rm alg}$. Write $h^\\#(v)=\\overline{h(\\overline v)}$. It suffices to establish \\eqref{eq:norm-coboundary} for $h-h^\\#$. The theorem gives $h-h^\\#=g-g\\circ M$. The gauge $e^{-g/2}$ replaces the periodic logarithm by $(h+h^\\#)/2$, real on the real torus. Since $Mp_3=p_3$ modulo the lattice, this periodic gauge has equal values at the two endpoints, leaving\n\\[\n \\frac{a_\\Delta(p_3)}{a_\\Delta(0)}=e^{2\\pi im_2/3}e^{\\text{real number}}.\n\\]\nAfter the based identification the exact modulus equality forces that real exponent to vanish. The ratio then has cube one, and an independent branch enclosure can select one. The constant $\\alpha$ cancels in this based ratio; it cannot simply be discarded in the unbased principal norm, where it contributes $\\alpha^6$.","has been constructed as\n\\[\n a_\\Delta(v)=\\alpha e^{2\\pi im^Tv}e^{h(v)},\\qquad\n m\\in\\mathbb Z^3,\\quad h\\text{ mean zero},\n\\]\nand its based ratio is actually identified with $U_\\Gamma/U_{\\rm alg}$. Write $h^\\#(v)=\\overline{h(\\overline v)}$. It suffices to establish \\eqref{eq:norm-coboundary} for $h-h^\\#$. The theorem gives $h-h^\\#=g-g\\circ M$. The gauge $e^{-g/2}$ replaces the periodic logarithm by $(h+h^\\#)/2$, real on the real torus. Since $Mp_3=p_3$ modulo the lattice, this periodic gauge has equal values at the two endpoints, leaving\n\\[\n \\frac{a_\\Delta(p_3)}{a_\\Delta(0)}=e^{2\\pi im_2/3}e^{\\text{real number}}.\n\\]\nAfter the based identification the exact modulus equality forces that real exponent to vanish. The ratio then has cube one, and an independent branch enclosure can select one. The constant $\\alpha$ cancels in this based ratio; it cannot simply be discarded in the unbased principal norm, where it contributes $\\alpha^6$.","ha e^{2\\pi im^Tv}e^{h(v)},\\qquad\n m\\in\\mathbb Z^3,\\quad h\\text{ mean zero},\n\\]\nand its based ratio is actually identified with $U_\\Gamma/U_{\\rm alg}$. Write $h^\\#(v)=\\overline{h(\\overline v)}$. It suffices to establish \\eqref{eq:norm-coboundary} for $h-h^\\#$. The theorem gives $h-h^\\#=g-g\\circ M$. The gauge $e^{-g/2}$ replaces the periodic logarithm by $(h+h^\\#)/2$, real on the real torus. Since $Mp_3=p_3$ modulo the lattice, this periodic gauge has equal values at the two endpoints, leaving\n\\[\n \\frac{a_\\Delta(p_3)}{a_\\Delta(0)}=e^{2\\pi im_2/3}e^{\\text{real number}}.\n\\]\nAfter the based identification the exact modulus equality forces that real exponent to vanish. The ratio then has cube one, and an independent branch enclosure can select one. The constant $\\alpha$ cancels in this based ratio; it cannot simply be discarded in the unbased principal norm, where it contributes $\\alpha^6$.","is actually identified with $U_\\Gamma/U_{\\rm alg}$. Write $h^\\#(v)=\\overline{h(\\overline v)}$. It suffices to establish \\eqref{eq:norm-coboundary} for $h-h^\\#$. The theorem gives $h-h^\\#=g-g\\circ M$. The gauge $e^{-g/2}$ replaces the periodic logarithm by $(h+h^\\#)/2$, real on the real torus. Since $Mp_3=p_3$ modulo the lattice, this periodic gauge has equal values at the two endpoints, leaving\n\\[\n \\frac{a_\\Delta(p_3)}{a_\\Delta(0)}=e^{2\\pi im_2/3}e^{\\text{real number}}.\n\\]\nAfter the based identification the exact modulus equality forces that real exponent to vanish. The ratio then has cube one, and an independent branch enclosure can select one. The constant $\\alpha$ cancels in this based ratio; it cannot simply be discarded in the unbased principal norm, where it contributes $\\alpha^6$.",", real on the real torus. Since $Mp_3=p_3$ modulo the lattice, this periodic gauge has equal values at the two endpoints, leaving\n\\[\n \\frac{a_\\Delta(p_3)}{a_\\Delta(0)}=e^{2\\pi im_2/3}e^{\\text{real number}}.\n\\]\nAfter the based identification the exact modulus equality forces that real exponent to vanish. The ratio then has cube one, and an independent branch enclosure can select one. The constant $\\alpha$ cancels in this based ratio; it cannot simply be discarded in the unbased principal norm, where it contributes $\\alpha^6$.",", leaving\n\\[\n \\frac{a_\\Delta(p_3)}{a_\\Delta(0)}=e^{2\\pi im_2/3}e^{\\text{real number}}.\n\\]\nAfter the based identification the exact modulus equality forces that real exponent to vanish. The ratio then has cube one, and an independent branch enclosure can select one. The constant $\\alpha$ cancels in this based ratio; it cannot simply be discarded in the unbased principal norm, where it contributes $\\alpha^6$.","\\[\n\\omega=e^{2\\pi i/3},\\quad \\beta=7^{1/3}\\omega^2,\\quad\nJ=2+3\\omega,\\quad \\bar J=-1-3\\omega,\\quad J\\bar J=7.\n\\]","\\[\nF_0=\\mathbb Q(\\omega),\\qquad K=\\mathbb Q(\\beta),\\qquad\n\\epsilon=2-\\beta,\\qquad \\mathcal R=-\\log(2-7^{1/3})>0.\n\\]","Here \\(\\mathcal R\\) is the regulator of \\(K\\), and","\\[\n\\log|\\epsilon|=\\mathcal R/2. \\tag{A1.1}\n\\]","Indeed, \\(N_{K/\\mathbb Q}(\\epsilon)=1\\) and its real image is\n\\(2-7^{1/3}\\). The class number is three and \\(\\epsilon\\) is a\nfundamental unit. These arithmetic facts are recorded in BCG, section\n5.2; the earlier unramified-cubic note also proves the class-number and\nray-field identification used here.","\\[\n\\mathfrak f=(3,\\beta-1),\\qquad\nE=K(\\omega,\\theta),\\qquad \\theta=2\\cos(6\\pi/7).\n\\]","The inherited arithmetic proof identifies \\(E\\) as the narrow ray class\nfield of modulus \\(\\mathfrak f\\) and the real place. It has degree six\nover \\(K\\) and degree eighteen over \\(\\mathbb Q\\).","\\[\nX=\\frac1\\beta\\sum_{a=1}^{6}\\chi_7(a)e^{6\\pi ia/7},\n\\]","where the character values for \\(a=1,\\ldots,6\\) are\n\\(1,\\omega,\\omega^2,\\omega^2,\\omega,1\\). Then","\\[\nX^3=J,\\qquad E=\\mathbb Q(\\beta,X,\\omega). \\tag{A1.2}\n\\]","\\[\nX=\\frac{\\Omega^2}{\\wp_{\\Lambda_0}(1/J)},\\qquad\n\\Lambda_0=\\mathbb Z+\\mathbb Z\\omega,\\qquad\n\\Omega=e^{\\pi i/6}\\frac{\\Gamma_{\\rm Euler}(1/3)^3}{2\\pi}. \\tag{A1.3}\n\\]","The arithmetic Frobenius \\(\\sigma\\) at the norm-two prime acts by","\\[\n\\sigma(\\beta)=\\beta,\\quad \\sigma(\\omega)=\\omega^2,\\quad\n\\sigma(X)=\\omega\\beta/X. \\tag{A1.4}\n\\]","The norm-five prime has the same Artin class. In particular, \\(\\sigma\\)\ngenerates \\(\\operatorname{Gal}(E/K)\\).","\\textbf{Actual complex conjugation is a different automorphism.} Denote\nit by \\(\\kappa\\). In the chosen embedding,","\\[\n\\kappa(\\beta)=\\omega^2\\beta,\\quad \\kappa(\\omega)=\\omega^2,\n\\quad \\kappa(X)=\\omega\\beta/X. \\tag{A1.5}\n\\]","For the last formula, the conjugate Gauss resolvent is \\(7/G\\), and\n\\(\\bar\\beta=\\omega^2\\beta\\). Formula (A1.5) follows directly. It must\nnot be replaced by the involution \\(\\sigma^3\\), which fixes \\(K\\).","The splitting field \\(F_0(\\beta)/\\mathbb Q\\) has group \\(S_3\\), while\n\\(\\mathbb Q(\\theta)/\\mathbb Q\\) is cyclic cubic. Their intersection is\nGalois over \\(\\mathbb Q\\); it cannot have degree three because \\(S_3\\)\nhas no cyclic cubic quotient. Hence","\\[\n\\operatorname{Gal}(E/\\mathbb Q)=S_3\\times C_3. \\tag{A2.1}\n\\]","\\[\nr:\\beta\\mapsto\\omega\\beta,\\qquad\nc:\\omega\\mapsto\\omega^2,\\qquad\ns_0:\\theta\\mapsto\\theta^2-2,\n\\]","where each generator fixes the other specified primitive generators as\nappropriate. Thus \\(r,c\\) generate the \\(S_3\\) factor, \\(c\\) fixes\n\\(\\beta,\\theta\\), and \\(s_0\\) generates the second factor. We have","\\[\n\\operatorname{Gal}(E/K)=\\langle c,s_0\\rangle,\n\\qquad \\sigma=cs_0,\n\\qquad \\operatorname{Gal}(E/F_0)=\\langle r,s_0\\rangle.\n\\]","Let \\(\\varrho(r)=\\omega,\\varrho(s_0)=1\\), and let\n\\(\\eta(s_0)=\\omega,\\eta(r)=1\\). The three odd ray characters of \\(K\\)\nare","\\[\n\\chi_j(c)=-1,\\qquad \\chi_j(s_0)=\\omega^j,\\qquad j=0,1,2.\n\\]","\\[\n\\operatorname{Ind}_{C_2}^{S_3}(\\operatorname{sgn})\n=\\operatorname{sgn}\\oplus\\operatorname{Std},\n\\qquad\n\\operatorname{Std}=\\operatorname{Ind}_{C_3}^{S_3}(\\varrho)\n\\]","\\[\n\\boxed{L_K(s,\\chi_j)\n=L_{\\mathbb Q}(s,\\chi_{-3}\\chi_7^j)\\,\nL_{F_0}(s,\\varrho\\eta^j).} \\tag{A2.2}\n\\]","For \\(j=0\\), use the primitive Dirichlet character of conductor three,\nnot its imprimitive extension to modulus twenty-one. All three odd ray\ncharacters are ramified at \\(\\mathfrak f\\), so their Artin L-functions\nagree with the partial-zeta convention omitting that prime. Induction\nincludes the local factors at ramified primes through the\ninertia-invariant spaces. No unproved Artin holomorphy statement is\nused.","The even characters make no contribution to the derivative at zero. The\nnontrivial even characters have at least two archimedean zeros. The\ntrivial character has the zero of \\(\\zeta_K\\), together with the extra\nzero of \\(1-3^{-s}\\). This also agrees with BCG, section 3.1.","\\[\nD_0=L(0,\\chi_{-3})=1/3,\\qquad\nD_1=L(0,\\chi_{-3}\\chi_7)=-2\\omega,\\qquad\nD_2=-2\\omega^2. \\tag{A2.3}\n\\]","For example, the latter two use\n\\(-21^{-1}\\sum_{a=1}^{21}a\\chi_{-3}(a)\\chi_7(a)^j\\).","\\[\nL_{F_0}(s,\\varrho)=\\zeta_K(s)/\\zeta_{\\mathbb Q}(s).\n\\]","The analytic class number formula, \\(h_K=3\\), and\n\\(\\zeta_{\\mathbb Q}(0)=-1/2\\) give","\\[\nL_{F_0}'(0,\\varrho)=3\\mathcal R. \\tag{A2.4}\n\\]","Put \\(\\lambda=\\varrho\\eta\\). Since both \\(r\\) and \\(s_0\\) send \\(X\\) to\n\\(\\omega^2X\\), \\(\\lambda\\) is the inverse Kummer character for\n\\(F_0(X)/F_0\\). Its conductor is","\\[\n\\mathfrak m=3J\\mathscr O_{F_0}. \\tag{A3.1}\n\\]","Here is an explicit verification. The Eisenstein integers are principal,\nand the six units map bijectively to \\((\\mathscr O_{F_0}/3)^\\times\\).\nEvery ideal coprime to three therefore has a unique generator\n\\(\\alpha\\equiv1\\pmod3\\). Since \\(J\\equiv-1\\pmod3\\), cubic reciprocity\ngives","\\[\n\\lambda((\\alpha))=(J/(\\alpha))_3^2=(\\alpha/J)_3^2. \\tag{A3.2}\n\\]","This proves that the conductor divides \\(3J\\). It must contain \\(J\\),\nbecause the valuation of the Kummer radicand there is one. If its\nexponent at the prime above three were at most one, its conductor would\ndivide \\((1-\\omega)J\\). The ray class group of that modulus has order\ntwo, so it has no nontrivial cubic character. Hence the conductor is\nexactly (A3.1).","The ray group modulo \\(3J\\) has order","\\[\n\\frac{63(1-1/3)(1-1/7)}6=6.\n\\]","\\[\n1,4,10,13,16,19.\n\\]","Reduction modulo \\(J\\) sends \\(\\omega\\) to \\(4\\in\\mathbb F_7\\). Formula\n(A3.2) groups the representatives as follows:","\\begin{longtable}[]{@{}lll@{}}\n\\toprule\\noalign{}\nIndex \\(m\\) & Representatives & \\(\\lambda\\)-value","1 & 4, 10 & \\(\\omega\\)","2 & 16, 19 & \\(\\omega^2\\)","Let \\(Z_m(s)\\) be the sum of the two ray partial zeta functions in row\n\\(m\\), and write \\(\\ell_m=Z_m'(0)\\). Then","\\[\nL_{F_0}'(0,\\lambda)=\\ell_0+\\omega\\ell_1+\\omega^2\\ell_2,\n\\qquad\nL_{F_0}'(0,\\lambda^2)=\\ell_0+\\omega^2\\ell_1+\\omega\\ell_2. \\tag{A3.3}\n\\]","\\[\n\\zeta_{F_0}(s)(1-3^{-s})(1-7^{-s}),\n\\]","\\[\n\\ell_0+\\ell_1+\\ell_2=0. \\tag{A3.4}\n\\]","Conjugation of the base field by \\(c\\) sends \\(\\varrho\\eta^2\\) to\n\\(\\varrho^{-1}\\eta^2=\\lambda^2\\), so","\\[\nL_{F_0}(s,\\varrho\\eta^2)=L_{F_0}(s,\\lambda^2).\n\\]","Combining (A2.2)--(A3.4) and performing inverse Fourier transformation\non \\(C_2\\times C_3\\) yields","\\[\n\\boxed{\\zeta_{\\mathfrak f}'(c^e s_0^k,0)\n=(-1)^e\\left(\\frac{\\mathcal R}{6}-\\ell_{k-1}\\right).} \\tag{A3.5}\n\\]","All subscripts on \\(\\ell\\) are taken modulo three. To check the\nconstants directly, the two nontrivial Fourier contributions are","\\[\n-\\frac13\\left(\\omega^{1-k}L_{F_0}'(0,\\lambda)\n+\\omega^{2-2k}L_{F_0}'(0,\\lambda^2)\\right).\n\\]","In (A3.3), the coefficient of \\(\\ell_{k-1}\\) in the parentheses is two\nand the other two coefficients are minus one. Equation (A3.4) turns the\nparentheses into \\(3\\ell_{k-1}\\).","\\textbf{Euler-factor warning.} Replacing \\(3J\\) by \\(21=3J\\bar J\\) would\nremove the Euler factor at the other prime over seven. Its\n\\(\\lambda\\)-value is one, so this inserts \\(1-7^{-s}\\) and destroys the\ndesired first derivative. That imprimitive derivative is zero. The\nchecker verifies this as a separate normalization control, not as the\ndesired L-value.","For \\(z=a\\omega+b\\notin\\Lambda_0\\), define the positive quantity","\\[\nH(z)=e^{-\\pi\\operatorname{Im}(\\omega)B_2(a)}\n|1-e^{2\\pi iz}|\n\\prod_{n\\ge1}|(1-e^{2\\pi i(n\\omega+z)})(1-e^{2\\pi i(n\\omega-z)})|,\n\\tag{A4.1}\n\\]","where initially \\(0\\le a<1\\) and \\(B_2(a)=a^2-a+1/6\\). This is the\nabsolute value of the usual Siegel function. Its value is independent of\nthe representative of \\(z\\) modulo \\(\\Lambda_0\\).","\\[\n\\mathcal E_z(s)=\\sum_{m,n\\in\\mathbb Z}|m+n\\omega+z|^{-2s},\n\\]","continued from \\(\\Re s>1\\), the precise normalization is","\\[\n\\mathcal E_z(0)=0,\\qquad \\mathcal E_z'(0)=-2\\log H(z). \\tag{A4.2}\n\\]","For the ray class represented by \\(\\alpha\\), all its integral ideals are\ngenerated exactly once by the elements of \\(\\alpha+3J\\mathscr O_{F_0}\\).\nThe congruence condition eliminates all nontrivial units. Therefore its\npartial zeta function is","\\[\n63^{-s}\\mathcal E_{\\alpha/(3J)}(s).\n\\]","There is no scale contribution to its derivative, because\n\\(\\mathcal E_z(0)=0\\). Set","\\[\nV_0=H(1/(3J))H(13/(3J)),\n\\] \\[\nV_1=H(4/(3J))H(10/(3J)),\\qquad\nV_2=H(16/(3J))H(19/(3J)).\n\\]","\\[\n\\ell_m=-2\\log V_m,\n\\qquad V_0V_1V_2=1. \\tag{A4.3}\n\\]","Let \\(u=1/J\\) and \\(T=\\bar J/3\\equiv2/3\\pmod{\\Lambda_0}\\). The three\npairs of points above are, respectively,","\\[\n\\{T\\pm2u\\},\\quad\\{T\\pm u\\},\\quad\\{T\\pm3u\\}. \\tag{A5.1}\n\\]","Indeed, \\(T=7u/3\\), so their numerators over \\(3J\\) are \\(1,13\\),\n\\(4,10\\), and \\(-2,16\\). The last numerator \\(-2\\) may be replaced by\n\\(19\\) since \\(21/(3J)=\\bar J\\in\\Lambda_0\\).","On the normalized curve \\(E_0:y^2+y=x^3\\), set","\\[\nx(z)=\\wp_{\\Lambda_0}(z)/\\Omega^2.\n\\]","\\[\nx(u)=X^{-1},\\quad x(2u)=\\omega^2X^{-1},\\quad x(3u)=\\omega X^{-1}. \\tag{A5.2}\n\\]","\\[\nx(T)=-\\omega. \\tag{A5.3}\n\\]","For completeness, \\(T\\) is nonzero three-torsion and its normalized\nx-coordinate is a root of \\(3x(x^3+1)\\). The lattice is invariant under\nordinary conjugation, so \\(\\wp(1/3)\\) and its derivative are real. Since\n\\(g_3(\\Lambda_0)>0\\), the real differential equation\n\\(\\wp'^2=4\\wp^3-g_3\\) implies \\(\\wp(1/3)>0\\). The argument of\n\\(\\Omega^2\\) is \\(\\pi/3\\), selecting \\(-\\omega\\) among the three roots\nof \\(x^3=-1\\).","completeness, \\(T\\) is nonzero three-torsion and its normalized\nx-coordinate is a root of \\(3x(x^3+1)\\). The lattice is invariant under\nordinary conjugation, so \\(\\wp(1/3)\\) and its derivative are real. Since\n\\(g_3(\\Lambda_0)>0\\), the real differential equation\n\\(\\wp'^2=4\\wp^3-g_3\\) implies \\(\\wp(1/3)>0\\). The argument of\n\\(\\Omega^2\\) is \\(\\pi/3\\), selecting \\(-\\omega\\) among the three roots\nof \\(x^3=-1\\).","\\[\n\\frac{\\sigma_\\Lambda(T+z)\\sigma_\\Lambda(T-z)}{\n\\sigma_\\Lambda(T)^2\\sigma_\\Lambda(z)^2}\n=\\wp_\\Lambda(z)-\\wp_\\Lambda(T). \\tag{A5.4}\n\\]","The theta expression for (A4.1), or the sigma--theta relation, turns\n(A5.4) into the same identity for absolute Green functions up to a\nconstant independent of \\(z\\). The quadratic exponential corrections\ncancel by the parallelogram identity. Furthermore,\n\\(H(\\omega z)=H(z)=H(-z)\\), either from (A4.2) and the isometries of\n\\(\\Lambda_0\\), or from the theta transformations.","Since \\(2u\\equiv\\omega^2u\\) and \\(3u\\equiv-\\omega u\\), all three factors\n\\(H(ku)\\) are equal. Thus for one positive constant \\(C\\), the same for\nall three rows,","\\[\nV_0=C|\\omega^2/X+\\omega|,\\quad\nV_1=C|1/X+\\omega|,\\quad\nV_2=C|\\omega/X+\\omega|. \\tag{A5.5}\n\\]","Multiplication and (A4.3) give \\(C^3|X^{-3}+1|=1\\). Cubing each equation\nin (A5.5) now removes the unknown constant exactly:","\\[\n\\boxed{V_m^3=|a_m|,} \\tag{A5.6}\n\\]","\\[\n\\begin{aligned}\na_0&=\\frac{(\\omega^2+\\omega X)^3}{1+J}\n=(1+\\omega)X^2-X+1,\\\\\na_1&=\\frac{(1+\\omega X)^3}{1+J}\n=-X^2+(1+\\omega)X+1,\\\\\na_2&=\\frac{(\\omega+\\omega X)^3}{1+J}\n=-\\omega X^2-\\omega X+1.\n\\end{aligned} \\tag{A5.7}\n\\]","\\[\na_0+a_1+a_2=3,\\qquad\n\\sum_{i<j}a_i a_j=6\\omega,\\qquad a_0a_1a_2=1.\n\\]","In particular, the \\(a_m\\) are algebraic units, satisfying","\\[\na_m^3-3a_m^2+6\\omega a_m-1=0. \\tag{A5.8}\n\\]","\\[\n\\boxed{\\ell_m=-\\frac23\\log|a_m|.} \\tag{A5.9}\n\\]","\\[\n\\boxed{\\zeta_{\\mathfrak f}'(c^e s_0^k,0)\n=(-1)^e\\left(\\frac{\\mathcal R}{6}+\\frac23\\log|a_{k-1}|\\right).} \\tag{A5.10}\n\\]","\\[\n\\nu=(X+\\beta/X-\\beta^2)/3,\\quad v=-\\beta^2\\nu-2\\nu^2,\n\\qquad v^3-\\beta v^2+1=0.\n\\]","\\[\nW=(A_W+\\omega B_W)/3,\n\\]","\\[\n\\begin{aligned}\nA_W&=(21\\beta-42)v^2+(-15\\beta^2+30\\beta)v-\\beta^2-4\\beta+11,\\\\\nB_W&=3\\beta^2v^2+(18\\beta-42)v-11\\beta^2+13\\beta+7.\n\\end{aligned} \\tag{A6.1}\n\\]","Let \\(N_\\kappa(z)=z\\kappa(z)=|z|^2\\) in the fixed embedding. The six\nexact identities needed for the comparison are","\\[\n\\boxed{N_\\kappa(\\sigma^hW)\n=N_\\kappa(\\epsilon a_{h+1}^2)^{(-1)^h},\\quad 0\\le h<6.} \\tag{A6.2}\n\\]","\\[\n\\beta^3=7,\\quad X^3=2+3\\omega,\\quad \\omega^2+\\omega+1=0.\n\\]","A basis for the rational calculation is \\(\\beta^iX^j\\omega^k\\),\n\\(0\\le i,j<3\\), \\(0\\le k<2\\). For odd \\(h\\), multiply the two norms\ninstead of dividing: the residual is their product minus one. Both\nsupplied implementations obtain the exact zero polynomial in all six\ncases.","\\[\n\\log|\\sigma^h W|=(-1)^h\\left(\\frac{\\mathcal R}{2}+2\\log|a_{h+1}|\\right)\n=-3\\zeta_{\\mathfrak f}'(\\sigma^{h+5},0). \\tag{A6.3}\n\\]","Consequently \\(W_{\\rm St}=\\sigma(W)\\) satisfies","\\[\n\\zeta_{\\mathfrak f}'(\\sigma^h,0)\n=-\\frac13\\log|\\sigma^hW_{\\rm St}|\n=-\\frac16\\log|\\sigma^hW_{\\rm St}|^2. \\tag{A6.4}\n\\]","The one-step shift is essential. The unshifted \\(W\\) belongs to the\noriginal raw gamma value's ray-class indexing, not the identity class.","The earlier note proves that \\(W\\) is an algebraic unit,\n\\(\\mu(E)=\\mu_6\\), all its embeddings above the real place of \\(K\\) have\nmodulus one, and adjoining a sixth root produces an abelian extension of\n\\(K\\). These statements are preserved by \\(\\sigma\\). Together with\n(A6.4), this verifies the three conditions stated by BCG in section 3.3\nfor this specific field and modulus.","\\[\nW^5=\\sigma(W)U_{\\rm alg}^6,\n\\]","and, for \\(c_6=-\\omega^2\\),","\\[\nW^{21}=c_6U_{\\rm alg}^{25}\\sigma(U_{\\rm alg})^5\\sigma^2(U_{\\rm alg}).\n\\]","For \\(t^6=W\\), the lift \\(\\varrho(t)=t^5/U_{\\rm alg}\\) commutes with\n\\(t\\mapsto c_6t\\), and satisfies \\(\\varrho^6(t)=c_6^4t\\). Irreducibility\nis checked at the residue assignment \\((\\beta,v,\\omega)=(32,99,48)\\)\nover \\(\\mathbb F_{181}\\), where \\(W=23\\) and \\(T^6-23\\) is irreducible.\nThe earlier local-integrality argument validates this reduction. The\nresulting abelian group is \\(C_2\\times C_{18}\\).","These arithmetic statements concern \\(W\\), not the still unproved\nalgebraicity of the gamma value.","\\[\nU_\\Gamma=\\frac{\\Gamma_{\\rm ell}(1/3;\\tau,s_{\\rm per})^5}\n{\\Gamma_{\\rm ell}(5/3;5\\tau,5s_{\\rm per})},\n\\quad\n\\tau=(\\beta^2+2\\beta)/15,\\quad s_{\\rm per}=-(\\beta+2)/15.\n\\]","BCG section 5.2 identifies this raw expression with the geometric value\nat ray-class index five. The norm-five smoothing ideal has Artin class\n\\(\\sigma\\). Their Theorem 3.2 therefore gives, unconditionally,","\\[\n\\log|U_\\Gamma|^2\n=\\zeta_{\\mathfrak f}'(1,0)-5\\zeta_{\\mathfrak f}'(\\sigma^5,0). \\tag{A7.1}\n\\]","\\[\n\\zeta_{\\mathfrak f}'(1,0)-5\\zeta_{\\mathfrak f}'(\\sigma^5,0)\n=\\frac13\\left(5\\log|W|-\\log|\\sigma(W)|\\right).\n\\]","The exact algebraic smoothing identity \\(W^5/\\sigma(W)=U_{\\rm alg}^6\\)\nturns this into \\(2\\log|U_{\\rm alg}|\\). Thus","\\[\n\\boxed{|U_\\Gamma|=|U_{\\rm alg}|.} \\tag{A7.2}\n\\]","\\[\n\\log|U_\\Gamma|^2\n=\\mathcal R-\\ell_2-5\\ell_1\n=\\mathcal R+\\frac23\\log|a_2a_1^5|,\n\\]","\\[\n\\boxed{|U_\\Gamma|^3=|\\epsilon^3a_2a_1^5|.} \\tag{A7.3}\n\\]","In particular the positive real number \\(|U_\\Gamma|\\) is algebraic. This\nconclusion does not extend automatically to the complex number\n\\(U_\\Gamma\\).","\\[\n|\\mathbf\\Gamma_{\\mathfrak a,h}(\\mathfrak f\\mathfrak b^{-1})|^6\n=\\left|(\\sigma_{\\mathfrak b}W_{\\rm St})^{N(\\mathfrak a)-\\sigma_{\\mathfrak a}}\\right|.\n\\tag{A7.4}\n\\]","Let \\(\\tau=x+iy\\) with \\(y>0\\), and \\(z=a\\tau+b\\), initially \\(0<a<1\\).\nApply Poisson summation in the integer coordinate \\(m\\) of","\\[\n\\mathcal E_z(s)=\\sum_{n,m\\in\\mathbb Z}\n\\left((m+b+(n+a)x)^2+(n+a)^2y^2\\right)^{-s}.\n\\]","\\[\n\\sqrt\\pi\\,\\frac{\\Gamma(s-1/2)}{\\Gamma(s)}\\,\n y^{1-2s}\\bigl[\\zeta(2s-1,a)+\\zeta(2s-1,1-a)\\bigr].\n\\]","At zero its value vanishes. Its derivative equals \\(2\\pi y B_2(a)\\),\nusing \\(\\Gamma(-1/2)=-2\\sqrt\\pi\\), \\(1/\\Gamma(s)=s+O(s^2)\\), and\n\\(\\zeta(-1,a)=-B_2(a)/2\\).","The nonzero Fourier modes, obtained from the usual Gaussian integral,\nare exponentially convergent near \\(s=0\\) for \\(0<a<1\\). Their\nderivative sums to","\\[\n-2\\log|1-e^{2\\pi iz}|\n-2\\sum_{n\\ge1}\\log|(1-e^{2\\pi i(n\\tau+z)})(1-e^{2\\pi i(n\\tau-z)})|.\n\\]","Combining the two terms gives (A4.2). Extension to \\(a=0\\), away from\nlattice points, follows by continuation in \\(z\\). Equivalently,","\\[\nH_\\tau(z)=e^{-\\pi(\\Im z)^2/y}\n\\left|\\frac{\\theta_1(\\pi z;e^{\\pi i\\tau})}{\\eta_{\\rm Ded}(\\tau)}\\right|.\n\\]","Let \\(t\\) be any point of the upper half-plane and\n\\(E_t=\\mathbb C/(\\mathbb Z+\\mathbb Zt)\\). The classical product\nconvention is \\cite{BCG}","\\[\n \\theta_0(z;t)=\\prod_{n\\ge0}(1-e^{2\\pi i(z+nt)})(1-e^{2\\pi i((n+1)t-z)}).\n\\]","\\[\n \\operatorname{div}(q_t)=4[0]-[P]-[-P]-[2P]-[-2P],\\qquad P=1/5,\n\\]","and, for \\(n\\) prime to five,","\\[\n \\prod_{\\eta\\in E_t[n]}q_t(z+\\eta)=q_t(nz).                  \\tag{N1.1}\n\\]","\\[\n \\prod_{a=0}^{n-1}\\theta_0(w+a/n;t)=\\theta_0(nw;nt).\n\\]","\\[\n \\prod_{b=0}^{n-1}\\theta_0(nz+bt;nt)=\\theta_0(nz;t).\n\\]","For the smoothing denominator, multiplication by five permutes the real\nresidues modulo n; the same two steps apply with period 5t. The\nnumerator becomes \\(\\theta_0(nz;t)^5\\) and the denominator becomes\n\\(\\theta_0(5nz;5t)\\). This proves (N1.1) with no omitted multiplier and\nwithout using the unknown scalar \\(c_t\\).","\\[\n x(z)=\\wp_t(z),\\quad y(z)=\\wp_t'(z),\\quad\n f(X)=4X^3-g_2X-g_3.\n\\]","The classical equation is \\(y(z)^2=f(x(z))\\). The addition and\nduplication laws used below are those of this convention\n\\cite{DLMFWeierstrass}. In particular the y-coordinate here is the full\nWeierstrass derivative, not its half.","\\[\n a=x(P),\\quad b=x(2P),\\quad d=a-b,\\quad y_a=y(P),\\quad y_b=y(2P),\n\\]","\\[\n Q_5(X)=(X-a)(X-b).\n\\]","All of \\(d,y_a,y_b\\) are nonzero: a point of exact order five is neither\ntwo-torsion nor equal up to sign to its double. Comparing divisors\nyields a nonzero scalar \\(c_t\\) with","\\[\n q_t(z)=c_t/Q_5(x(z)).                                    \\tag{N1.2}\n\\]","Also \\(q_t(z)\\sim c_tz^4\\) at zero. The problem in this development is\nto determine this scalar exactly without a choice of cube root or eta\nmultiplier.","Suppose two nonzero elliptic functions with the same divisor differ by a\nscalar \\(r\\), and both satisfy the exactly normalized degree-two and\ndegree-three multiplication laws. On the degree-four and degree-nine\nfibers those laws imply","\\[\n r^4=r,\\qquad r^9=r.\n\\]","Thus \\(r^3=r^8=1\\), and","\\[\n r=(r^3)^3/r^8=1.                                       \\tag{N2.1}\n\\]","More generally, if \\(c^3=A\\) and \\(c^8=B\\) with \\(c\\ne0\\), then","\\[\n \\boxed{c=A^3/B.}                                       \\tag{N2.2}\n\\]","Since \\(2P\\) doubles to \\(4P=-P\\), let \\(\\ell,m\\) be the full-y tangent\nslopes at \\(P,2P\\). The addition law gives","\\[\n \\ell^2=4(2a+b),\\quad m^2=4(a+2b),\n\\]","\\[\n \\ell d=y_a+y_b,\\qquad md=y_a-y_b.                       \\tag{N3.1}\n\\]","\\[\n 4y_ay_b=(\\ell^2-m^2)d^2=4d^3.\n\\]","\\[\n \\boxed{y_ay_b=d^3,\\qquad f(a)f(b)=d^6.}                 \\tag{N3.2}\n\\]","\\[\n \\psi_3(X)=3X^4-\\tfrac32g_2X^2-3g_3X-\\tfrac1{16}g_2^2,\n\\]","\\[\n \\phi_2(X)=X^4+\\tfrac12g_2X^2+2g_3X+\\tfrac1{16}g_2^2.\n\\]","\\[\n Xf(X)-\\phi_2(X)=\\psi_3(X),\\qquad\n b=\\phi_2(a)/f(a),\\quad a=\\phi_2(b)/f(b)\n\\]","\\[\n \\psi_3(a)=f(a)d,\\qquad \\psi_3(b)=-f(b)d.\n\\]","\\[\n \\boxed{\\psi_3(a)\\psi_3(b)=-d^8.}                        \\tag{N3.3}\n\\]","Let \\(z\\to0\\) in (N1.1). The fourth-order zero gives","\\[\n \\boxed{\\prod_{\\eta\\in E_t[n]\\setminus\\{0\\}}q_t(\\eta)=n^4}\n \\quad(5\\nmid n).                                      \\tag{N4.1}\n\\]","The three x-coordinates of nonzero two-torsion are the roots of\n\\(f(X)/4\\). Hence","\\[\n \\prod_{\\eta\\in E_t[2]\\setminus\\{0\\}}Q_5(x(\\eta))\n =\\frac{f(a)f(b)}{16}.\n\\]","\\[\n c_t^3=16\\prod_{\\eta\\ne0}Q_5(x(\\eta))=f(a)f(b)=d^6.       \\tag{N4.2}\n\\]","Degree two alone determines \\(c_t/d^2\\) only up to a cube root of unity.","The eight nonzero three-torsion points have four x-coordinates, each\nrepeated twice; those coordinates are the roots of \\(\\psi_3(X)/3\\).\nTherefore","\\[\n \\prod_{\\eta\\in E_t[3]\\setminus\\{0\\}}Q_5(x(\\eta))\n =\\left(\\frac{\\psi_3(a)\\psi_3(b)}9\\right)^2.\n\\]","The \\(3^4=81\\) in (N4.1) cancels the denominator:","\\[\n c_t^8=(\\psi_3(a)\\psi_3(b))^2=d^{16}.                    \\tag{N4.3}\n\\]","\\[\n \\boxed{c_t=(c_t^3)^3/c_t^8=d^{18}/d^{16}=d^2.}           \\tag{N4.4}\n\\]","\\[\n \\boxed{\n q_t(z)=\\frac{(a-b)^2}{(x(z)-a)(x(z)-b)}\n       =\\frac{\\operatorname{disc}(Q_5)}{Q_5(x(z))}.\n }                                                       \\tag{N5.1}\n\\]","Writing \\(S_5=a+b\\) and \\(P_5=ab\\), the symmetric form is","\\[\n q_t(z)=\\frac{S_5^2-4P_5}{x(z)^2-S_5x(z)+P_5}.            \\tag{N5.2}\n\\]","Thus the formula depends on the unordered fifth-torsion kernel, not a\nchoice between its two x-coordinates. Under a change of generator\n\\(P\\mapsto2P\\), \\(a,b\\) swap and the formula is unchanged.","\\[\n \\mathsf X_t(z)=\\frac{x(z)-a}{a-b}.\n\\]","\\[\n \\boxed{q_t(z)=\\frac1{\\mathsf X_t(z)(\\mathsf X_t(z)+1)}.}   \\tag{N5.3}\n\\]","Swapping \\(a,b\\) changes \\(\\mathsf X\\) to \\(-1-\\mathsf X\\), leaving\n(N5.3) fixed. Scaling all Weierstrass x-coordinates by the same nonzero\nfactor also leaves the formula fixed. These field identities are\nformalized in \\texttt{H12CanonicalTheta}.","\\[\n u=-y_b/y_a,\\qquad \\lambda=y_a/d,\n\\]","and for an arbitrary point \\((x,y)\\), define","\\[\n X=\\frac{x-a}{\\lambda^2},\\qquad\n Y=\\frac{y-y_a-(1-u)\\lambda(x-a)}{2\\lambda^3}.\n\\]","\\[\n \\boxed{Y^2+(1-u)XY-uY=X^3-uX^2,}                        \\tag{N6.1}\n\\]","with \\(P=(0,0)\\), \\(2P=(u,u^2)\\), and","\\[\n \\boxed{q_t(z)=\\frac{u^2}{X(z)(X(z)-u)}.}                  \\tag{N6.2}\n\\]","\\[\n d=-u\\lambda^2,\\quad y_a=-u\\lambda^3,\\quad y_b=u^2\\lambda^3.\n\\]","The first tangent slope is \\((1-u)\\lambda\\). Equation (N3.1) therefore\ngives","\\[\n a=\\lambda^2(1-6u+u^2)/12.\n\\]","Writing \\(b_2=1-6u+u^2\\), \\(b_4=u^2-u\\), the derivative and curve\nequation imply","\\[\n g_2=\\lambda^4(b_2^2/12-2b_4),\\quad\n g_3=\\lambda^6(-b_2^3/216+b_2b_4/6-u^2).\n\\]","Substitution proves (N6.1). This coefficient calculation is\nindependently checked symbolically over \\(\\mathbb Q(u)\\) and numerically\nat the specified analytic lattices. The full geometric\nchange-of-variables theorem is not Lean-formalized in the corresponding\nsource checkpoint.","The parameter \\(u\\) remains analytically defined. Calling the curve a\nrational normal form does not assert that \\(u\\) is an algebraic number\nat the cubic period.","\\[\n \\beta^3=7,\\quad\\epsilon=2-\\beta,\\quad\\rho=\\epsilon^{-1},\n \\quad t=-(\\beta^2+2\\beta)/15,\\quad s=-(\\beta+2)/15.\n\\]","The retained modular relation gives \\(\\rho\\Lambda_s=\\Lambda_t\\). Since","\\[\n \\rho=4-15t,\\quad\\rho/5\\equiv-1/5\\pmod{\\Lambda_t},\n\\]","\\[\n a_s=\\rho^2a_t,\\quad b_s=\\rho^2b_t,\\quad c_s=\\rho^4c_t,\n\\]","\\[\n \\mathsf X_t(\\rho z)=\\mathsf X_s(z),\\qquad\n \\boxed{q_t(\\rho z)=q_s(z).}                              \\tag{N7.1}\n\\]","Thus the phase-free theta covariance has a proof through the canonical\nfifth-torsion normal form as well as its earlier eta-multiplier proof.\nThis statement is about two homothetic elliptic lattices, not an\nendomorphism of the single lattice \\(\\Lambda_t\\).","\\[\n w_v=\\tfrac13+(b\\tau+cs+a)/2,\\quad v=(b,c,a)\\in\\{0,1\\}^3,\n \\quad T(b,c,a)=(a,b,c),\\quad \\tau=-t.\n\\]","\\[\n F(\\rho w_v)=F(w_{Tv})B_v,\\qquad\n B_v=\\prod_{j=0}^{A_v-1}q_t(w_{T^2v}+js),\n \\quad A_v=5+4b-2c+7a.\n\\]","\\[\n \\boxed{\n B_v=\\prod_{j=0}^{A_v-1}\n \\frac1{\\mathsf X_t(w_{T^2v}+js)\n             (\\mathsf X_t(w_{T^2v}+js)+1)}.\n }                                                       \\tag{N8.1}\n\\]","\\[\n (76,-28,8,0,-16,0,0,0).\n\\]","When written with the unnormalized denominators \\(Q_5\\), the\ncorresponding known powers of \\(d=a-b\\) are","\\[\n (152,-56,16,0,-32,0,0,0).\n\\]","\\[\n c_t^{-36}=d^{-72}.                                      \\tag{N8.2}\n\\]","The same substitution works for every oriented three-step return product\n\\(J_{n,v}\\) from the predecessor. Consequently ALL seven nontrivial\nsigned-channel limits have finite rational elliptic expressions without\na separately supplied eta constant. The first-order limits still have\nonly their guaranteed first-order local cancellation; this normalization\nchange does not upgrade their convergence rate.","Coefficient values $\\{-4,-1,0,1,4\\}$ & Universal step-function and smoothing-weight bound & Identification with complex orders","\\subsection{Exact and numerical replay}\nThe current exact checker passes 20 named assertions with five negative controls. Its finite lift range contains 2,197 triples, and its family run checks 92,480 points for $1\\le n\\le16$. A separate finite extension passes nine named checks and four controls over 4,913 triples, 15,125 family points for $1\\le n\\le10$, 135 actual complex positions at $n=3$ and three exact polynomial samples.","At requested precisions 40 and 70 decimal digits (65 and 95 actual working digits, including 25 guards) the current numerical replay checks four reflection/anchor identities, one quadratic-jet diagnostic and three negative controls. The maximum identity errors are approximately $1.9204\\times10^{-62}$ and $1.2792\\times10^{-92}$. The finite-difference jet residual is approximately $3.9284\\times10^{-11}$ at both precisions: its fixed step causes a truncation residual, so it is a diagnostic rather than an exact identity proof. The independent extension checks 15 floating identities at 80 working digits, with maximum error approximately $2.1231\\times10^{-72}$.","mum identity errors are approximately $1.9204\\times10^{-62}$ and $1.2792\\times10^{-92}$. The finite-difference jet residual is approximately $3.9284\\times10^{-11}$ at both precisions: its fixed step causes a truncation residual, so it is a diagnostic rather than an exact identity proof. The independent extension checks 15 floating identities at 80 working digits, with maximum error approximately $2.1231\\times10^{-72}$.","During authoring, the first citation transformation attempted to replace the mathematical torsion notation $E_t[2]$ by a citation. The mathematical-token guard rejected this draft before it entered the manuscript. Protecting display and inline math before prose transformations corrected the conversion. This failure and fix are retained in the scoped proofread report. No mathematical assertion is inferred from a rejected authoring draft."]}