% Single-file LaTeX source; appendix tables and references included. % Source: innerlightr-wq/two-mode-oscillator-record-chronologies % Manuscript Git blob: d974bab605aad02283506abfab44a85cf6a54bd2 % Classification table Git blob: 67f2555c4ae8fc66f4a04ae4d67c1b04e8f8b49e % Exclusion table Git blob: 28742fa3a146443661c72f73f86f404f02c1e828 % Compile with pdflatex (two or more passes); no BibTeX or local input files needed. \documentclass[11pt]{article} \usepackage{amsmath,amsthm,amssymb} \usepackage[margin=1.1in]{geometry} \usepackage{booktabs,longtable,array} \usepackage[hidelinks]{hyperref} \usepackage{microtype} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\N}{\mathbb{N}} \newcommand{\pr}{\varrho} \newcommand{\gam}{\Gamma} \DeclareMathOperator{\Nm}{N} \DeclareMathOperator{\sgn}{sign} \newcommand{\mat}[4]{\left(\begin{smallmatrix}#1\\#3\end{smallmatrix}\right)} \title{Odd/Odd Records for a Relative Approximation Error:\\ Three Quadratic Classifications and an Exact Certificate} \author{Elias De Jes\'us\\ \small ORCID: \href{https://orcid.org/0009-0007-0190-9143}{0009-0007-0190-9143}} \date{7 November 2026} \begin{document} \maketitle \begin{abstract} For a quadratic irrational $\alpha>1$ and positive \emph{odd} integers $m,k$, set $H=m\alpha+k$ and $\varepsilon=|m\alpha-k|/(m\alpha+k)$. A state $(m,k)$ is a \emph{strict chronological record} if $\varepsilon$ is smaller than at every admissible state of smaller $H$. We classify the strict records completely for $\alpha=\varphi$, $\alpha=1+\sqrt3$ and $\alpha=\sqrt3$. In each case they are a finite explicit union of orbits of a single unimodular matrix, with signed defect cycles $(-1,+1,-5)$, $(-3,+1,-11)$ and $(-2)$ respectively. The proofs rest on two exact identities that make $\varepsilon$ a function of the defect $N$ and the conjugate factor $\eta$ alone, on a norm cutoff $|N|\le 18,204,27$ derived from heights, and on a finite exact certificate that decides every remaining comparison. Two phenomena require care and are isolated: at $\varphi$ the competition between the Fibonacci and Lucas ladders is a genuine tie at leading order, decided by an integer identity $N_B=-5N_A$; and at $\sqrt3$ a companion family has \emph{exactly} equal error at larger height, so it is excluded by equality rather than by loss. The results resolve two conjectures of the author's earlier note and correct one assertion there. Nothing here is formally verified or refereed, and the uniform generalisation is left open. \end{abstract} \section{Introduction}\label{sec:intro} Let $\alpha>1$ be irrational and consider pairs $(m,k)$ of positive odd integers. Write \begin{equation}\label{eq:Heps} H=H(m,k)=m\alpha+k,\qquad \varepsilon=\varepsilon(m,k)=\frac{|m\alpha-k|}{m\alpha+k}. \end{equation} Thus $\varepsilon$ is a \emph{relative} error, normalised by the same quantity $H$ that orders the states. Because $\alpha$ is irrational, distinct integer pairs have distinct $H$, so the states are linearly ordered by height with no ties. \begin{definition}[Records]\label{def:records} A state $(m,k)$ is a \emph{strict chronological record} if $\varepsilon(m,k)<\varepsilon(m',k')$ for every admissible state $(m',k')$ with $H(m',k')0,\qquad \xi=k-m\alpha,\qquad \eta=k-m\alpha',\qquad N=\xi\eta=k^2-tkm-sm^2 . \] $N$ is the integral defect. For the three slopes of this note: \[ \begin{aligned} \varphi:\quad &t=s=1, &&D_0=\sqrt5, &&N=k^2-km-m^2;\\ 1+\sqrt3:\quad &t=s=2, &&D_0=2\sqrt3, &&N=k^2-2km-2m^2;\\ \sqrt3:\quad &t=0,\ s=3, &&D_0=2\sqrt3, &&N=k^2-3m^2 . \end{aligned} \] In all three, $t\ge0$ and $\alpha'<0$, so $\eta>0$ on every positive state. Set \begin{equation}\label{eq:G} G=2\alpha\eta^2-tN . \end{equation} \begin{lemma}[Master identities]\label{lem:master} For every state with $\eta>0$, \begin{equation}\label{eq:masters} H=\frac{G}{D_0\,\eta},\qquad \varepsilon=\frac{|N|\,D_0}{G},\qquad\text{and in particular}\qquad G>0 . \end{equation} Thus $\varepsilon$ and $H$ are determined by the pair $(N,\eta)$ alone. \end{lemma} \begin{proof} From $\xi=k-m\alpha$ and $\eta=k-m\alpha'$ we get $\eta-\xi=mD_0$ and $\alpha\eta-\alpha'\xi=kD_0$, hence $m=(\eta-\xi)/D_0$ and $k=(\alpha\eta-\alpha'\xi)/D_0$, so \[ H=m\alpha+k=\frac{\alpha(\eta-\xi)+\alpha\eta-\alpha'\xi}{D_0} =\frac{2\alpha\eta-(\alpha+\alpha')\xi}{D_0}=\frac{2\alpha\eta-t\xi}{D_0}. \] Substituting $\xi=N/\eta$ gives $H=(2\alpha\eta^2-tN)/(D_0\eta)=G/(D_0\eta)$, and $H>0$ forces $G>0$. Then $\varepsilon=|\xi|/H=(|N|/\eta)\cdot D_0\eta/G=|N|D_0/G$. \end{proof} \begin{remark}[The error score, and why height is separate]\label{rem:score} Since $\varepsilon=|N|D_0/G$, records are the running minima of $|N|/G$, equivalently the running maxima of \[ S:=\frac{G}{|N|}=2\alpha W-t\,\sgn(N),\qquad W:=\frac{\eta^2}{|N|} . \] So the \emph{error score} is $2\alpha W$ corrected by a bounded term depending only on the \emph{sign} of $N$. The height order is a different question and is treated separately (Section~\ref{sec:compare}): by \eqref{eq:masters}, $H=G/(D_0\eta)$ depends on $N$ as well as $\eta$, so it is not a function of $S$, and in particular $\eta$-order is \emph{not} height-order across different defects. Example: at $\alpha=\varphi$ the primitive odd/odd states $(3,1)$ and $(1,3)$ have $\eta=3\varphi-2\approx2.8541<3.618\approx\varphi+2=\eta$ respectively, while $H=3\varphi+1\approx5.8541>4.618\approx\varphi+3$. Any argument that infers one order from the other must be checked. \end{remark} \begin{lemma}[Imprimitive states never record]\label{lem:imprim} If $g=\gcd(m,k)>1$ then $(m/g,k/g)$ is again an odd state, has the same $\varepsilon$, and has strictly smaller $H$. Consequently no imprimitive state is a strict record, and $N$ may be assumed primitively represented. \end{lemma} \begin{proof} $m,k$ odd forces $g$ odd, so $m/g$ and $k/g$ are odd and the reduced pair is admissible. Both $|m\alpha-k|$ and $m\alpha+k$ scale by $g$, so $\varepsilon$ is unchanged, while $H$ is divided by $g>1$. A strict record must beat every state of smaller height, in particular this one, which it cannot. \end{proof} The proof of Lemma~\ref{lem:imprim} uses that the gcd of two odd numbers is odd. This is special to the odd/odd class; see Section~\ref{sec:outlook}. \section{The admissible lattice and the admissible unit}\label{sec:lattice} Write $\eta=a+b\omega$ with $\omega=\varphi$ or $\sqrt{D}$ as appropriate, so that the state is recovered by \[ (m,k)=(b,\,a+b)\ \text{ for }\varphi\text{ and }1+\sqrt3,\qquad (m,k)=(b,\,a)\ \text{ for }\sqrt3 . \] Admissibility ($m,k$ both odd) becomes a congruence on $(a,b)$ modulo $2$: ``$a$ even, $b$ odd'' in the first two cases, ``$a$ odd, $b$ odd'' for $\sqrt3$. \begin{proposition}[Admissible unit step]\label{prop:unit} Let $u$ be the fundamental unit of the relevant order and let $U\in GL_2(\Z)$ be the matrix of multiplication by $u$ on the basis $(1,\omega)$. Reducing modulo $2$ places $U$ in the finite group $GL_2(\Z/2)$ of order $6$, so some power $U^j$ fixes the admissibility class; the least such $j$ gives the \emph{admissible unit} $\pr=u^j$. Multiplication by $\pr$ preserves admissibility and primitivity, and multiplies $\eta$ by $\pr$ while multiplying $N$ by $\delta:=\Nm(\pr)$. On the column $(m,k)$ it acts by a matrix with positive entries: \[ \varphi:\ \pr=\varphi^3=1+2\varphi,\ \ \begin{pmatrix}1&2\\2&3\end{pmatrix},\ \delta=-1;\qquad 1+\sqrt3:\ \pr=(2+\sqrt3)^2,\ \ \begin{pmatrix}3&4\\8&11\end{pmatrix},\ \delta=+1; \] \[ \sqrt3:\ \pr=2+\sqrt3,\ \ \begin{pmatrix}2&1\\3&2\end{pmatrix},\ \delta=+1 . \] \end{proposition} The minimal $j$ can be strictly smaller than the order of $U$ modulo $2$, because the class need only be \emph{fixed}, not acted on trivially: at $\sqrt3$ the class (odd,\,odd) is fixed by the swap $\mat{0}{1}{1}{0}$, so $j=1$ although $U\not\equiv I$. The values are $j=3,2,1$. \begin{remark}[$\varphi^3$ versus $\varphi^6$]\label{rem:phi3} The matrix $\mat{1}{2}{2}{3}$ sends $(1,1)\mapsto(3,5)$ and is the $\varphi^3$ step. The matrix $\mat{5}{8}{8}{13}$, which appears in \cite{DeJesus2026}, sends $(1,1)\mapsto(13,21)$ and is the $\varphi^6$ step, i.e. the step \emph{within} a fixed-defect orbit. Both occur below and must not be interchanged. \end{remark} \begin{lemma}[Signed transport]\label{lem:transport} Along an orbit, with $\pr$ applied $p$ times to a chosen representative, \begin{equation}\label{eq:transport} N_p=\delta^p\,N_{\mathrm{rep}},\qquad \delta=\Nm(\pr). \end{equation} At $\varphi$, $\delta=-1$, so the \emph{sign} of $N$ alternates along a $\pr$-orbit and the fixed-defect orbits are the $\pr^2=\varphi^6$ orbits. Taking $|N|$ therefore discards exactly the parity of $p$ relative to the representative. Every comparison below that needs an orientation transports the signed $N$ by \eqref{eq:transport}; magnitudes are used only where the quantity is $\delta$-invariant. \end{lemma} Also useful later: the ladders have positive entries, so once a state has $m,k>0$ all its successors do, and their heights increase without bound. Each orbit therefore has a finite initial segment of non-positive states followed by an infinite positive tail. \section{The height-only cutoff}\label{sec:cutoff} Fix a \emph{comparison ladder}: an admissible orbit of known defect $\pm L$, whose positive members we denote $A,C,\dots$ in increasing height. For $\varphi$ and $1+\sqrt3$ we take $L=1$; for $\sqrt3$ no admissible state has $|N|=1$, because odd $m,k$ force $k^2-3m^2\equiv1-3\equiv6\pmod 8$, and we take $L=2$ with the orbit of $(1,1)$. \begin{theorem}[Cutoff]\label{thm:cutoff} Let $B$ be a strict record whose height exceeds that of some positive ladder member, and let $A,C$ be consecutive positive ladder members with $H_A0$ then $0t$ (an exact inequality, checked in the certificate), and $Y$ is multiplied by $\pr^2>1$ at each step while $t,\alpha$ are fixed, so $u$ strictly decreases. The three closed forms with a $u$ in them are used only on $00$; and $\pr^2=\varphi^6=17.9442719\ldots$ \[ \varphi,\ n=+1:\quad F=\pr^2\frac{(1+u/\pr^2)^2}{(1-u)^2}, \] which is increasing in $u$, hence decreasing in $Y$; its maximum over the ladder is attained at the first member with $n=+1$, namely $Y=\varphi^8$, where $F=18.1960225\ldots<19$. \[ 1+\sqrt3:\quad F=\pr^2\Bigl(\frac{1-u/\pr^2}{1-u}\Bigr)^{\!2}, \] increasing in $u$, hence maximal at the first ladder member $Y=(2+\sqrt3)^2$, where $F=204.5520518\ldots<205$. \[ \sqrt3:\quad t=0\ \Longrightarrow\ F=L\pr^2=2(2+\sqrt3)^2=27.8564064\ldots<28 . \] Taking integer parts gives the three bounds. The endpoint inequalities are decided exactly in the certificate; the decimals are display only. \end{proof} \begin{remark}[Why the signs must be separated]\label{rem:signsep} At $\varphi$ the two branches move in \emph{opposite} directions along the ladder: the $n=-1$ values increase ($17.5116\ldots,17.9435\ldots,17.9443\ldots$) while the $n=+1$ values decrease ($18.1960\ldots,17.9450\ldots,17.9443\ldots$), both towards $\pr^2$. So $F$ is not monotone in $Y$, and no finite sample of ladder states bounds the infinite ladder. The separation above does: one branch is bounded uniformly by $\pr^2$, the other attains its maximum at an explicit first member. \end{remark} Combining Theorem~\ref{thm:cutoff} with Lemma~\ref{lem:imprim} and the congruence restrictions on $N$ leaves very little at $\varphi$. There $N=k^2-km-m^2$ is \emph{odd}, since $m,k$ are; $4N=(2k-m)^2-5m^2$ forces $4N$ to be a square modulo $5$, i.e.\ $N\equiv0,1,4\pmod5$, which among the odd $|N|\le18$ leaves $|N|\in\{1,5,9,11,15\}$; and $3$ is inert in $\Z[\varphi]$ (the discriminant $5$ is not a square modulo $3$), so $3\mid N$ forces $3\mid\gcd(m,k)$ and kills $|N|=9,15$ by Lemma~\ref{lem:imprim}. The admissible defects left at $\varphi$ are therefore $|N|\in\{1,5,11\}$, and the four orbits of Appendix~\ref{app:tables} realise exactly these. \section{Exhaustive enumeration of the surviving orbits}\label{sec:enum} \begin{proposition}[Bounded enumeration, no unique factorisation]\label{prop:enum} Let $|N|\le B$. Reduce $\eta$ by powers of $\pr$ until $1\le\eta<\pr$. Then $|\xi|=|N|/\eta\le B$ automatically, so $(\eta,\xi)$ lies in the rectangle $[1,\pr)\times[-B,B]$ in the two real embeddings, and since $m=(\eta-\xi)/D_0$ and $k=(\alpha\eta-\alpha'\xi)/D_0$ the integer coordinates of the representative are explicitly bounded. An exhaustive search over that box finds every orbit. \end{proposition} No class-group or principal-ideal hypothesis is used: this is the classical reduction theory of the generalised Pell equation, see \cite{Conrad}, Theorems 1.1 and 2.1. Reduced representatives may have $m\le0$ or $k\le0$; they must be retained, since the positive states appear later in the same orbit. The resulting counts are \[ \varphi:\ 4\ \text{orbits at }|N|\le18,\qquad 1+\sqrt3:\ 64\ \text{at }|N|\le204,\qquad \sqrt3:\ 6\ \text{at }|N|\le27 . \] \section{Exact comparisons: error and height}\label{sec:compare} Let $A,B$ be orbits with reduced representatives $r_A,r_B\in[1,\pr)$ and representative defects $n_A,n_B$. Consider $A$ at period $p-d$ and $B$ at period $p$, with offset $d\ge0$. \begin{proposition}[Error comparison]\label{prop:err} With $\gam:=|N_A|\eta_B^2-|N_B|\eta_A^2$, \begin{equation}\label{eq:crit} \varepsilon_B<\varepsilon_A \iff 2\alpha\gam>t\bigl(|N_A|N_B-|N_B|N_A\bigr), \end{equation} and $\gam=\pr^{2p-2d}\gam_d$ with $\gam_d=|N_A|r_B^2\pr^{2d}-|N_B|r_A^2$ independent of $p$. Hence $\sgn\gam=\sgn\gam_d$ is one exact sign per pair $(A,B)$ and offset $d$. The right-hand side of \eqref{eq:crit} is bounded in absolute value by $2t|N_A||N_B|$, so if $\gam_d\neq0$ then \eqref{eq:crit} is decided by $\sgn\gam_d$ as soon as \begin{equation}\label{eq:thr} 2\alpha\,\pr^{2p-2d}\,|\gam_d|\;>\;2t\,|N_A|\,|N_B| , \end{equation} a strict inequality which first holds at an explicitly computable period $P(A,B,d)$. The threshold starts at $p=d$, where $\pr^{2p-2d}=1$. \end{proposition} \begin{proposition}[Degenerate case $\gam_d=0$]\label{prop:gamma0} If $\gam_d=0$ then \eqref{eq:crit} reduces to the integer condition $0>t(|N_A|N_B-|N_B|N_A)$ with the defects transported by \eqref{eq:transport}: $N_A=\delta^{p-d}n_A$, $N_B=\delta^{p}n_B$. Hence \begin{equation}\label{eq:tiesign} \sgn(\varepsilon_B-\varepsilon_A)=\sgn\bigl(t(|N_A|N_B-|N_B|N_A)\bigr)\in\{-1,0,+1\}, \end{equation} a three-way outcome. If $\delta=-1$ this depends on the parity of $p$, so such a pair can give \emph{alternating} verdicts; if $t=0$ the right-hand side vanishes identically and the two states have \emph{exactly equal} error at every period. \end{proposition} \begin{proposition}[Height comparison]\label{prop:height} With $A$ at period $q$ and $B$ at period $q+d$, put \[ U=2\alpha(r_B\pr^{d}-r_A)\,r_A\,r_B\pr^{d},\qquad V=\delta^{q}\,t\bigl(\delta^{d}n_B r_A-n_A r_B\pr^{d}\bigr). \] Then $\sgn(H_B-H_A)=\sgn(\pr^{2q}U-V)$. Here $U$ is independent of $q$, and $V$ depends on $q$ only through $\delta^q$, hence only through the parity of $q$ when $\delta=-1$. If $U\neq0$ the sign is decided by $\sgn U$ once $\pr^{2q}|U|>|V|$, at an explicitly computable period. \end{proposition} \begin{proof} By \eqref{eq:masters}, $D_0H=2\alpha\eta-tN/\eta$, so $D_0(H_B-H_A)=2\alpha(\eta_B-\eta_A)-t(N_B/\eta_B-N_A/\eta_A)$. Substitute $\eta_A=r_A\pr^{q}$, $\eta_B=r_B\pr^{q+d}$ and the transported defects, then clear the positive factor $\pr^{-q}r_Ar_B\pr^{d}$. \end{proof} \begin{proposition}[$U=0$ is an index shift, not a tie]\label{prop:U0} $U=0$ if and only if $r_A=r_B\pr^{d}$. With distinct representatives in $[1,\pr)$ this is impossible: for $d\ge1$ it forces $r_A\ge\pr$, and for $d=0$ it forces $r_A=r_B$, i.e. the same orbit. Moreover $U=0$ would imply $V=0$: taking norms of $r_A=r_B\pr^d$ gives $n_A=\delta^dn_B$, and substituting both identities annihilates $V$. Hence $U=0$ denotes the same arithmetic state after an index shift, in contrast to $\gam_d=0$, which does occur between distinct states. \end{proposition} When the two thresholds are combined one must convert period coordinates: Proposition~\ref{prop:err} places $A$ at $p-d$ and $B$ at $p$, while Proposition~\ref{prop:height} places $A$ at $q$ and $B$ at $q+d$, so $q=p-d$ and the height threshold must be shifted by $+d$ before taking a maximum. \section{From a finite check to the infinite tail}\label{sec:tail} \begin{lemma}[Error monotonicity]\label{lem:mono} Along any orbit, $\varepsilon$ strictly decreases. Indeed $\varepsilon=|N|D_0/G$ and one unit step gives \begin{equation}\label{eq:Ginc} G_{\mathrm{next}}-G=2\alpha\eta^2(\pr^2-1)-tN(\delta-1). \end{equation} If $\delta=+1$ the second term vanishes and the first is positive, so $G$ increases at every step unconditionally. If $\delta=-1$ the second term is bounded by $2t|N|$ while the first grows like $\pr^{2p}$, so once $2\alpha\eta^2(\pr^2-1)>2t|N|$ it holds at every later step; the finitely many earlier steps are checked directly. \end{lemma} \begin{proposition}[Only offsets $d\in\{0,1\}$ matter, in the tail]\label{prop:window} Let $p_0$ be any period such that \emph{(a)} every orbit has a positive state at some period $\le p_0-1$, and \emph{(b)} every pairwise ordering threshold of Proposition~\ref{prop:height} is already decisive. Then for every orbit $B$ and every $p\ge p_0$, and for each orbit $A$, the latest $A$-state strictly below the period-$p$ state of $B$ in height occurs at period $p$ or $p-1$. If $B$ beats both of those, it beats every earlier state of $A$. \end{proposition} \begin{proof} Three ingredients. (i) Normalisation: all representatives lie in $[1,\pr)$, so for any orbits $A,A'$ one has $r_A\pr^{p-1}H_A$, the state $B$ meets the running minimum without improving it: it is a weak record and not a strict one. \end{proof} We emphasise that Theorem~\ref{thm:C}\,(ii) exhibits one infinite family of non-strict weak records; it is \emph{not} a classification of all weak records, and imprimitive weak records are not considered. \subsection{\texorpdfstring{The $\varphi$ tie, resolved by an integer}{The phi tie, resolved by an integer}}\label{sec:tie} The $|N|=1$ admissible states are exactly $(F_n,F_{n+1})$ with $n\equiv1\pmod3$: $\eta$ is then a unit $\pm\varphi^{\,n}$, the class condition modulo $2$ forces $n\equiv1\pmod3$, and $F_n$ and $F_{n+1}$ are both odd precisely when $3\nmid n(n+1)$. The $|N|=5$ admissible states are $\eta=\pm\sqrt5\,\varphi^{\,n}$ with the same congruence on $n$, since $5$ ramifies and $(\sqrt5)=(2\varphi-1)$ is the only ideal of norm $5$; their defect is $N=-5(-1)^n$. For such $\eta$ we abbreviate $\varepsilon(\eta)$ for the error of the corresponding state, by Lemma~\ref{lem:master} a function of $(N,\eta)$. \begin{theorem}[The tie and its resolution]\label{thm:tie} For $n\equiv1\pmod3$ put $A$ the unit state at $\eta_A=\varphi^{\,n}$ and $B$ the norm-$5$ state at $\eta_B=\sqrt5\,\varphi^{\,n}$. Then $\eta_B/\eta_A=\sqrt5$, so $(\eta_B/\eta_A)^2=5=|N_B|$: the leading terms cancel exactly and \begin{equation}\label{eq:tiepair} \varepsilon(\varphi^{\,n})=\frac{\sqrt5}{2\varphi^{\,2n+1}-(-1)^n},\qquad \varepsilon(\sqrt5\,\varphi^{\,n})=\frac{\sqrt5}{2\varphi^{\,2n+1}+(-1)^n}. \end{equation} Moreover $N_A=(-1)^n$ and $N_B=-5N_A$, so by \eqref{eq:tiesign} $B$ beats $A$ if and only if $N_A=+1$, i.e. $n$ even. Among $n\equiv1\pmod3$ this is $n\equiv4\pmod6$, and those states are exactly the Lucas pairs $(L_{6j+4},L_{6j+5})$. \end{theorem} So the competition is decided not by a multiplicative gap but by the bounded term $\pm1$ in \eqref{eq:tiepair}, equivalently by the integer identity $N_B=-5N_A$. The normalised representative of the norm-$5$ orbit has $N_{\mathrm{rep}}=+5$; the states that actually win are those with $N=-5$, reached at the other period parity. This is the ``asymptotically degenerate competition'' of \cite[Remark 5.10]{DeJesus2026}. \begin{remark}[Method note]\label{rem:signmethod} Two of the comparisons above are invisible to $|N|$: the alternating verdict of Proposition~\ref{prop:gamma0} when $\delta=-1$, and the $V$-term of Proposition~\ref{prop:height}. In the course of this work, asking of each computed quantity which distinctions it discards --- and whether a later step needs them --- located exactly these two places, and the certificate now transports signed defects by \eqref{eq:transport} and uses magnitudes only where the quantity is $\delta$-invariant. The mechanism itself is arithmetic: $N_B=-5N_A$, and the exact cancellation in \eqref{eq:tiepair}. \end{remark} \subsection{Classification summary}\label{sec:summary} \begin{center} \begin{tabular}{lrrcccc} \toprule $\alpha$ & cutoff & orbits & strict & alternating & equality & excluded \\ \midrule $\varphi$ & $18$ & $4$ & $1$ & $1$ & $0$ & $2$ \\ $1+\sqrt3$ & $204$ & $64$ & $3$ & $0$ & $0$ & $61$ \\ $\sqrt3$ & $27$ & $6$ & $1$ & $0$ & $1$ & $4$ \\ \bottomrule \end{tabular} \end{center} At $\varphi$ the alternating orbit is the norm-$5$ (Lucas) family of Theorem~\ref{thm:tie}; at $\sqrt3$ the equality orbit is the companion family of Theorem~\ref{thm:C}\,(ii). Appendix \ref{app:tables} lists every orbit with its representative, signed defect, verdict, and --- for each excluded orbit --- a competitor, an offset, the deciding sign and the period from which it is certified. \section{Relation to the earlier note}\label{sec:earlier} All references are to the pinned version of \cite{DeJesus2026} identified in Section~\ref{sec:intro}. \paragraph{Resolved.} Conjecture 5.11 ($\varphi$-completeness) is Theorem~\ref{thm:A}; Conjecture 5.13 ($(1+\sqrt3)$-completeness) is Theorem~\ref{thm:B}. The earlier note states that for these two slopes its pruned search range is labelled computational ``precisely because the analogous bound is'' those conjectures: that is, the norm cutoff was assumed. Theorem~\ref{thm:cutoff} derives it. Remark 5.10 of the baseline already identifies the mechanism of the degenerate competition correctly --- that $|N|=5$ is compensated to leading order, that the decision falls to the bounded term in $L_n^2-5F_n^2=4(-1)^n$, and that its sign selects the odd Lucas pairs with $n\equiv4\pmod6$ --- and says explicitly that this is ``an observed pattern with an identified finite mechanism, not a proved theorem''. Theorem~\ref{thm:tie} proves it, for every admissible $n$ at once, from the closed forms \eqref{eq:tiepair}; and it places the phenomenon in a general frame, as the case $\gam_d=0$ of Proposition~\ref{prop:gamma0}, where the same three-way integer rule also produces the permanent equality at $\sqrt3$. \paragraph{Corrected.} Observation 5.9 of the baseline describes the $\varphi$ chronology as having ``three records per multiplicative period $\varphi^6$ in height'', and Observation 5.12 says the same of $1+\sqrt3$ with period $(2+\sqrt3)^2$; both are labelled computationally verified. Exact multiplicativity holds for $\eta$, not for $H$: by \eqref{eq:masters}, \begin{equation}\label{eq:heightper} H_{\mathrm{next}}-\pr H=\frac{tN\,(\pr-\delta/\pr)}{D_0\,\eta}, \end{equation} which vanishes only when $tN=0$. Concretely, with $\mat{3}{4}{8}{11}(1,1)=(7,19)$ and $\varphi^6\cdot(1,1)=(13,21)$, \[ H(7,19)-\pr H(1,1)=-8\sqrt3,\qquad H(13,21)-\varphi^6H(1,1)=4-4\sqrt5, \] both nonzero. What is true is that $H_{\mathrm{next}}/H\to\pr$ along each orbit, by \eqref{eq:heightper}, since $\eta\to\infty$; the baseline's observed record lists are not in question, only the exactness of the height period. We do \emph{not} assert that every sufficiently large multiplicative height window contains exactly three records; that is a statement about the interleaving of three orbits with different multiplicative phases and would require a separate argument. At $\alpha=\sqrt3$, where $t=0$, one has $H=\eta$ and the height period $2+\sqrt3$ \emph{is} exact (Theorem~\ref{thm:C}). \paragraph{New.} The slope $\sqrt3$ does not appear in \cite{DeJesus2026}. It is the cleanest of the three: $t=0$ collapses the master identities to $H=\eta$ and $\varepsilon=|N|/H^2$, so $h^2\varepsilon$ is exactly $|N|/4$, and it is also the case in which no admissible state has $|N|=1$, which is why the cutoff is stated for a ladder of arbitrary known defect. \section{The certificate}\label{sec:cert} The finite content of the proofs is isolated in a small standalone bundle : the directory \texttt{certificate/} of the companion folder in the repository, documented by its own \texttt{README}. It is released under the tag \begin{center}\texttt{congruence-records-v1.0.0}\end{center} \noindent whose commit and per-file hashes are recorded in \texttt{docs/RELEASE\_MANIFEST.md}. Two programs are run: \texttt{certificate.py} prints the cutoff, the orbit enumeration, the classification and the threshold data; \texttt{verify\_records.py} runs 31 checks and reports \texttt{VERIFY PASSED}. The bundle uses the Python standard library only, reads no data files at runtime, and recomputes everything from the definitions. All ordering and comparison decisions are exact: arithmetic is carried out in $\Z[\sqrt D]$ with integer or rational coordinates, and signs are decided by integer comparisons. Floating point appears only as a presort key --- after which the sorted order is re-verified exactly --- and for display. The checks include the master identities, the congruence restrictions on $N$ for each slope, the imprimitivity reduction, the cutoff bound against every record found by unpruned search, an independent recount of the orbits by direct rectangle enumeration, the four-way classification against the claimed families, set equality of the record list with the recurrences of Theorems~\ref{thm:A}--\ref{thm:C}, the signed-transport ledger \eqref{eq:transport} and its invariance under change of representative, and the threshold data of Section~\ref{sec:tail}. \paragraph{Status of the verification.} The certificate is exact arithmetic, not a formal proof: it is not machine-checked in a proof assistant, and this note has not been refereed. Two independent audits of earlier versions of the bundle were carried out with assistance of an automated reviewer; they re-derived the classifications by separate orbit enumeration and tail logic, and they found and prompted the repair of several defects in the written proof and in the implementation, including the incorrect monotonicity justification repaired in Remark~\ref{rem:signsep} and a threshold initialisation error. Those audits re-used this bundle's arithmetic module, so they are independent in logic and in control flow but not in the underlying arithmetic implementation. The release history in \texttt{docs/CHANGELOG.md} records the versions and what changed. \section{Related work}\label{sec:related} The objective studied here --- a relative error $|m\alpha-k|/(m\alpha+k)$, with the states ordered by the same denominator $m\alpha+k$ --- is not one of the standard best-approximation notions, and we make only comparisons that the cited definitions support. Kim, Lee and Liao \cite{KLL2022} introduce the odd--odd continued fraction and prove that the best \emph{$1$-rational} approximations of an irrational $x$ --- their Definition 1.1: $p/q$ with $p,q$ odd and coprime such that $|qx-p|<|bx-a|$ for every $a/b$ of the same parity type with $00$ on positive states and $x\mapsto x(D_0+tx)$ is increasing. Other sign patterns need separate treatment. \item \textbf{Imprimitivity.} Lemma~\ref{lem:imprim} uses that the gcd of two odd numbers is odd, so dividing it preserves the class. This fails for a general class: at $\alpha=\sqrt6$ with the class $(m,k)\equiv(1,1)\pmod3$, the state $(4,10)$ is a strict record within the class although $\gcd=2$, and its reduction $(2,5)$ leaves the class. A general statement must quantify over primitive states or retain imprimitive bounded-defect orbits. \item \textbf{The comparison ladder.} Theorem~\ref{thm:cutoff} needs an admissible state of \emph{known} defect $L$. Existence of some primitive seed in an ordinary residue pair $(a,b)\bmod M$ with $\gcd(a,b,M)=1$ follows from a short construction: choose $m\equiv a$, let $P$ be the product of the primes dividing $m$ but not $M$, and use the Chinese remainder theorem to choose $k\equiv b\pmod M$ with $k\equiv1\pmod P$; common primes of $m$ and $M$ cannot divide $b$, so $\gcd(m,k)=1$, and a large enough $k$ makes $\eta>0$. Its defect is nonzero and computable. Obtaining a \emph{prescribed} or \emph{minimal} $|L|$ is a different and harder question; the values $L=1,1,2$ used here were found by inspection. \end{itemize} Assembling these into a theorem, with the quantifiers in order and the finite region described in general, is left as future work. Three worked slopes do not establish it. \section*{Acknowledgements and AI-assistance disclosure} \addcontentsline{toc}{section}{Acknowledgements} Anthropic's Claude was used throughout as an assistant: for exact-arithmetic computational design and execution, for drafting and revising this note and the certificate code, and as an adversarial reviewer of both. Several of the mathematical repairs recorded in \texttt{docs/CHANGELOG.md} originated in that review process, including the sign-separated cutoff argument of Theorem~\ref{thm:cutoff}, the pairwise height certificate of Proposition~\ref{prop:height}, and the correction of \eqref{eq:heightper}. All conceptual choices, and final responsibility for every claim here, belong to the author. No human referee has reviewed this note, and no part of it is formally verified. \begin{thebibliography}{9} \bibitem{ChaKim} B.~Cha and D.~H.~Kim. \emph{Intrinsic Diophantine approximation on the unit circle and its Lagrange spectrum.} Ann. Inst. Fourier (Grenoble) \textbf{73} (2023), no.~1, 101--161. \href{https://doi.org/10.5802/aif.3522}{doi:10.5802/aif.3522}. \bibitem{Conrad} \emph{Generalized Pell equation.} Math 154 course handout, Stanford University. Theorem 1.1: finiteness of the equivalence classes of solutions of $x^2-dy^2=n$ under the norm-one units. Theorem 2.1: an explicit bounded region of $\mathbb{R}^2$ containing a representative of every class. \url{https://virtualmath1.stanford.edu/~conrad/154Page/handouts/genpell.pdf}. The handout carries no printed author line; it is cited here by title, course and URL. \bibitem{DeJesus2026} E.~De Jes\'us. \emph{Boundary and Interior Record Chronologies for the Two-Mode Oscillator Spectrum.} Zenodo (2026). \href{https://doi.org/10.5281/zenodo.21796640}{doi:10.5281/zenodo.21796640}. Cited at repository commit \texttt{10ea1c62c6866990ded0a85a531883841d38d034}. \bibitem{Elsner} C.~Elsner. \emph{On Diophantine approximations with rationals restricted by arithmetical conditions.} Fibonacci Quart. \textbf{38} (2000), no.~1, 24--33. \url{https://www.fq.math.ca/Scanned/38-1/elsner.pdf}. \bibitem{KLL2022} D.~H.~Kim, S.~B.~Lee and L.~Liao. \emph{Odd-odd continued fraction algorithm.} Monatsh. Math. \textbf{198} (2022), 323--344. \href{https://doi.org/10.1007/s00605-022-01704-2}{doi:10.1007/s00605-022-01704-2}. Read in the author version arXiv:2002.06310v2. \bibitem{KLL2025} D.~H.~Kim, S.~B.~Lee and L.~Liao. \emph{Diophantine approximation by rational numbers of certain parity types.} Mathematika \textbf{71} (2025), e12285. \href{https://doi.org/10.1112/mtk.12285}{doi:10.1112/mtk.12285}. Read in the author version arXiv:2403.12341v1. \end{thebibliography} \appendix \section{Machine-generated orbit tables}\label{app:tables} The tables below are generated by \texttt{certificate/tables.py} and are reproduced verbatim; the same data is available as CSV in \texttt{tables/}. \textbf{Orbit identity.} Each orbit carries a stable identifier --- \texttt{P}, \texttt{T} or \texttt{S} for $\varphi$, $1+\sqrt3$ or $\sqrt3$, followed by its index in the canonical enumeration, which is sorted by the representative $\eta$. The identifier is necessary, because $N_{\mathrm{rep}}$ is \emph{not} unique: at $1+\sqrt3$ thirty values each label two distinct orbits and $N_{\mathrm{rep}}=-143$ labels four, and at $\sqrt3$ two values label two each. Here $N_{\mathrm{rep}}$ is the defect of the \emph{normalised representative} (reduced into $[1,\pr)$); by \eqref{eq:transport} the defect at period $p$ is $\delta^pN_{\mathrm{rep}}$, so at $\varphi$ the sign alternates. ``first positive state'' is the first $(m,k)$ with both coordinates positive. \textbf{What an exclusion row asserts.} A row ``$B$ excluded by $A$ at offset $d$ from period $P$'' asserts that for \emph{every} $p\ge P$, the state of $A$ at period $p-d$ lies below the state of $B$ at period $p$ in height and has strictly smaller error. Accordingly $P$ is the maximum of the error threshold \eqref{eq:thr} in $B$-period coordinates, the ordering threshold of Proposition~\ref{prop:height} converted by $+d$, and the first period at which both states are positive. The two states at $p=P$ are printed, so a reader can re-derive any row from Propositions~\ref{prop:err}--\ref{prop:height} without re-running a search. Every row is re-verified exactly at $p=P,P+1,P+2$ before it is emitted, and check C11 of \texttt{verify\_records.py} re-derives all $67$ rows independently from the orbit each one names and tests them at $p=P,\dots,P+11$. \bigskip \noindent\textbf{Classification of all admissible primitive orbits below the cutoff.} {\small % BEGIN embedded classification.tex % machine-generated by certificate/tables.py -- do not edit \begin{longtable}{llrrlll} \hline id & $\alpha$ & $|N|$ & $N_{\mathrm{rep}}$ & $\eta$ (rep) & first positive state & class \\ \hline \endfirsthead \hline id & $\alpha$ & $|N|$ & $N_{\mathrm{rep}}$ & $\eta$ (rep) & first positive state & class \\ \hline \endhead \hline \multicolumn{7}{r}{\footnotesize continued}\\ \endfoot \hline \endlastfoot P0 & $\varphi$ & 1 & -1 & 1.618034 & (1,1) & strict \\ P1 & $\varphi$ & 11 & +11 & 2.381966 & (5,7) & excluded \\ P2 & $\varphi$ & 11 & -11 & 2.854102 & (3,1) & excluded \\ P3 & $\varphi$ & 5 & +5 & 3.618034 & (1,3) & alternating \\ T0 & $1+\sqrt3$ & 143 & -143 & 1.014083 & (7,9) & excluded \\ T1 & $1+\sqrt3$ & 39 & -39 & 1.052559 & (5,11) & excluded \\ T2 & $1+\sqrt3$ & 73 & +73 & 1.091035 & (3,13) & excluded \\ T3 & $1+\sqrt3$ & 193 & +193 & 1.129510 & (1,15) & excluded \\ T4 & $1+\sqrt3$ & 131 & -131 & 1.157677 & (7,11) & excluded \\ T5 & $1+\sqrt3$ & 11 & -11 & 1.196152 & (5,13) & strict \\ T6 & $1+\sqrt3$ & 111 & -111 & 1.301270 & (7,13) & excluded \\ T7 & $1+\sqrt3$ & 169 & +169 & 1.378222 & (3,17) & excluded \\ T8 & $1+\sqrt3$ & 83 & -83 & 1.444864 & (7,15) & excluded \\ T9 & $1+\sqrt3$ & 69 & +69 & 1.483340 & (5,17) & excluded \\ T10 & $1+\sqrt3$ & 47 & -47 & 1.588457 & (7,17) & excluded \\ T11 & $1+\sqrt3$ & 121 & +121 & 1.626933 & (5,19) & excluded \\ T12 & $1+\sqrt3$ & 179 & -179 & 1.693575 & (9,17) & excluded \\ T13 & $1+\sqrt3$ & 3 & -3 & 1.732051 & (1,1) & strict \\ T14 & $1+\sqrt3$ & 181 & +181 & 1.770527 & (5,21) & excluded \\ T15 & $1+\sqrt3$ & 143 & -143 & 1.837169 & (9,19) & excluded \\ T16 & $1+\sqrt3$ & 109 & +109 & 2.019238 & (7,23) & excluded \\ T17 & $1+\sqrt3$ & 47 & -47 & 2.124356 & (9,23) & excluded \\ T18 & $1+\sqrt3$ & 177 & +177 & 2.162831 & (7,25) & excluded \\ T19 & $1+\sqrt3$ & 13 & +13 & 2.267949 & (9,25) & excluded \\ T20 & $1+\sqrt3$ & 167 & -167 & 2.373067 & (11,25) & excluded \\ T21 & $1+\sqrt3$ & 107 & -107 & 2.516660 & (11,27) & excluded \\ T22 & $1+\sqrt3$ & 157 & +157 & 2.555136 & (9,29) & excluded \\ T23 & $1+\sqrt3$ & 39 & -39 & 2.660254 & (11,29) & excluded \\ T24 & $1+\sqrt3$ & 37 & +37 & 2.803848 & (11,31) & excluded \\ T25 & $1+\sqrt3$ & 183 & -183 & 2.908965 & (13,31) & excluded \\ T26 & $1+\sqrt3$ & 107 & -107 & 3.052559 & (13,33) & excluded \\ T27 & $1+\sqrt3$ & 23 & -23 & 3.196152 & (3,1) & excluded \\ T28 & $1+\sqrt3$ & 69 & +69 & 3.339746 & (13,37) & excluded \\ T29 & $1+\sqrt3$ & 191 & -191 & 3.444864 & (15,37) & excluded \\ T30 & $1+\sqrt3$ & 1 & +1 & 3.732051 & (1,3) & strict \\ T31 & $1+\sqrt3$ & 109 & +109 & 3.875644 & (15,43) & excluded \\ T32 & $1+\sqrt3$ & 191 & -191 & 3.980762 & (17,43) & excluded \\ T33 & $1+\sqrt3$ & 83 & -83 & 4.124356 & (17,45) & excluded \\ T34 & $1+\sqrt3$ & 33 & +33 & 4.267949 & (17,47) & excluded \\ T35 & $1+\sqrt3$ & 157 & +157 & 4.411543 & (17,49) & excluded \\ T36 & $1+\sqrt3$ & 183 & -183 & 4.516660 & (19,49) & excluded \\ T37 & $1+\sqrt3$ & 59 & -59 & 4.660254 & (5,1) & excluded \\ T38 & $1+\sqrt3$ & 73 & +73 & 4.803848 & (19,53) & excluded \\ T39 & $1+\sqrt3$ & 167 & -167 & 5.052559 & (21,55) & excluded \\ T40 & $1+\sqrt3$ & 121 & +121 & 5.339746 & (21,59) & excluded \\ T41 & $1+\sqrt3$ & 143 & -143 & 5.588457 & (23,61) & excluded \\ T42 & $1+\sqrt3$ & 13 & +13 & 5.732051 & (1,5) & excluded \\ T43 & $1+\sqrt3$ & 177 & +177 & 5.875644 & (23,65) & excluded \\ T44 & $1+\sqrt3$ & 111 & -111 & 6.124356 & (7,1) & excluded \\ T45 & $1+\sqrt3$ & 61 & +61 & 6.267949 & (25,69) & excluded \\ T46 & $1+\sqrt3$ & 71 & -71 & 6.660254 & (5,3) & excluded \\ T47 & $1+\sqrt3$ & 23 & -23 & 7.196152 & (3,5) & excluded \\ T48 & $1+\sqrt3$ & 181 & +181 & 7.339746 & (29,81) & excluded \\ T49 & $1+\sqrt3$ & 179 & -179 & 7.588457 & (9,1) & excluded \\ T50 & $1+\sqrt3$ & 33 & +33 & 7.732051 & (1,7) & excluded \\ T51 & $1+\sqrt3$ & 131 & -131 & 8.124356 & (7,3) & excluded \\ T52 & $1+\sqrt3$ & 97 & +97 & 8.267949 & (33,91) & excluded \\ T53 & $1+\sqrt3$ & 169 & +169 & 8.803848 & (35,97) & excluded \\ T54 & $1+\sqrt3$ & 11 & -11 & 9.196152 & (3,7) & excluded \\ T55 & $1+\sqrt3$ & 61 & +61 & 9.732051 & (1,9) & excluded \\ T56 & $1+\sqrt3$ & 143 & -143 & 10.124356 & (7,5) & excluded \\ T57 & $1+\sqrt3$ & 141 & +141 & 10.267949 & (41,113) & excluded \\ T58 & $1+\sqrt3$ & 71 & -71 & 10.660254 & (5,7) & excluded \\ T59 & $1+\sqrt3$ & 97 & +97 & 11.732051 & (1,11) & excluded \\ T60 & $1+\sqrt3$ & 193 & +193 & 12.267949 & (49,135) & excluded \\ T61 & $1+\sqrt3$ & 59 & -59 & 12.660254 & (5,9) & excluded \\ T62 & $1+\sqrt3$ & 37 & +37 & 13.196152 & (3,11) & excluded \\ T63 & $1+\sqrt3$ & 141 & +141 & 13.732051 & (1,13) & excluded \\ S0 & $\sqrt3$ & 26 & -26 & 1.124356 & (5,7) & excluded \\ S1 & $\sqrt3$ & 6 & +6 & 1.267949 & (1,3) & equality \\ S2 & $\sqrt3$ & 26 & -26 & 1.660254 & (3,1) & excluded \\ S3 & $\sqrt3$ & 22 & +22 & 1.803848 & (1,5) & excluded \\ S4 & $\sqrt3$ & 2 & -2 & 2.732051 & (1,1) & strict \\ S5 & $\sqrt3$ & 22 & +22 & 3.267949 & (3,7) & excluded \\ \end{longtable} % END embedded classification.tex } \bigskip \noindent\textbf{Exclusion witnesses.} {\small % BEGIN embedded exclusions_summary.tex % machine-generated by certificate/tables.py -- do not edit \begin{longtable}{llrlrrrlll} \hline id & $\alpha$ & $N_{\mathrm{rep}}$ & by & $N_A$ & $d$ & $P$ & $A$ at $P-d$ & $B$ at $P$ & rule \\ \hline \endfirsthead \hline id & $\alpha$ & $N_{\mathrm{rep}}$ & by & $N_A$ & $d$ & $P$ & $A$ at $P-d$ & $B$ at $P$ & rule \\ \hline \endhead \hline \multicolumn{10}{r}{\footnotesize continued}\\ \endfoot \hline \endlastfoot P1 & $\varphi$ & +11 & P0 & -1 & 0 & 1 & (3,5) & (5,7) & $\Gamma_d<0$ \\ P2 & $\varphi$ & -11 & P0 & -1 & 0 & 0 & (1,1) & (3,1) & $\Gamma_d<0$ \\ T0 & $1+\sqrt3$ & -143 & T30 & +1 & 1 & 1 & (1,3) & (7,9) & $\Gamma_d<0$ \\ T1 & $1+\sqrt3$ & -39 & T30 & +1 & 1 & 1 & (1,3) & (5,11) & $\Gamma_d<0$ \\ T2 & $1+\sqrt3$ & +73 & T30 & +1 & 1 & 1 & (1,3) & (3,13) & $\Gamma_d<0$ \\ T3 & $1+\sqrt3$ & +193 & T30 & +1 & 1 & 1 & (1,3) & (1,15) & $\Gamma_d<0$ \\ T4 & $1+\sqrt3$ & -131 & T30 & +1 & 1 & 1 & (1,3) & (7,11) & $\Gamma_d<0$ \\ T6 & $1+\sqrt3$ & -111 & T1 & -39 & 0 & 1 & (5,11) & (7,13) & $\Gamma_d<0$ \\ T7 & $1+\sqrt3$ & +169 & T1 & -39 & 0 & 1 & (5,11) & (3,17) & $\Gamma_d<0$ \\ T8 & $1+\sqrt3$ & -83 & T5 & -11 & 0 & 1 & (5,13) & (7,15) & $\Gamma_d<0$ \\ T9 & $1+\sqrt3$ & +69 & T5 & -11 & 0 & 1 & (5,13) & (5,17) & $\Gamma_d<0$ \\ T10 & $1+\sqrt3$ & -47 & T5 & -11 & 0 & 1 & (5,13) & (7,17) & $\Gamma_d<0$ \\ T11 & $1+\sqrt3$ & +121 & T1 & -39 & 0 & 1 & (5,11) & (5,19) & $\Gamma_d<0$ \\ T12 & $1+\sqrt3$ & -179 & T1 & -39 & 0 & 1 & (5,11) & (9,17) & $\Gamma_d<0$ \\ T14 & $1+\sqrt3$ & +181 & T1 & -39 & 0 & 1 & (5,11) & (5,21) & $\Gamma_d<0$ \\ T15 & $1+\sqrt3$ & -143 & T1 & -39 & 0 & 1 & (5,11) & (9,19) & $\Gamma_d<0$ \\ T16 & $1+\sqrt3$ & +109 & T5 & -11 & 0 & 1 & (5,13) & (7,23) & $\Gamma_d<0$ \\ T17 & $1+\sqrt3$ & -47 & T5 & -11 & 0 & 1 & (5,13) & (9,23) & $\Gamma_d<0$ \\ T18 & $1+\sqrt3$ & +177 & T5 & -11 & 0 & 1 & (5,13) & (7,25) & $\Gamma_d<0$ \\ T19 & $1+\sqrt3$ & +13 & T13 & -3 & 0 & 1 & (7,19) & (9,25) & $\Gamma_d<0$ \\ T20 & $1+\sqrt3$ & -167 & T5 & -11 & 0 & 1 & (5,13) & (11,25) & $\Gamma_d<0$ \\ T21 & $1+\sqrt3$ & -107 & T5 & -11 & 0 & 1 & (5,13) & (11,27) & $\Gamma_d<0$ \\ T22 & $1+\sqrt3$ & +157 & T5 & -11 & 0 & 1 & (5,13) & (9,29) & $\Gamma_d<0$ \\ T23 & $1+\sqrt3$ & -39 & T13 & -3 & 0 & 1 & (7,19) & (11,29) & $\Gamma_d<0$ \\ T24 & $1+\sqrt3$ & +37 & T13 & -3 & 0 & 1 & (7,19) & (11,31) & $\Gamma_d<0$ \\ T25 & $1+\sqrt3$ & -183 & T5 & -11 & 0 & 1 & (5,13) & (13,31) & $\Gamma_d<0$ \\ T26 & $1+\sqrt3$ & -107 & T5 & -11 & 0 & 1 & (5,13) & (13,33) & $\Gamma_d<0$ \\ T27 & $1+\sqrt3$ & -23 & T13 & -3 & 0 & 1 & (7,19) & (13,35) & $\Gamma_d<0$ \\ T28 & $1+\sqrt3$ & +69 & T13 & -3 & 0 & 1 & (7,19) & (13,37) & $\Gamma_d<0$ \\ T29 & $1+\sqrt3$ & -191 & T5 & -11 & 0 & 1 & (5,13) & (15,37) & $\Gamma_d<0$ \\ T31 & $1+\sqrt3$ & +109 & T13 & -3 & 0 & 1 & (7,19) & (15,43) & $\Gamma_d<0$ \\ T32 & $1+\sqrt3$ & -191 & T5 & -11 & 0 & 1 & (5,13) & (17,43) & $\Gamma_d<0$ \\ T33 & $1+\sqrt3$ & -83 & T13 & -3 & 0 & 1 & (7,19) & (17,45) & $\Gamma_d<0$ \\ T34 & $1+\sqrt3$ & +33 & T13 & -3 & 0 & 1 & (7,19) & (17,47) & $\Gamma_d<0$ \\ T35 & $1+\sqrt3$ & +157 & T5 & -11 & 0 & 1 & (5,13) & (17,49) & $\Gamma_d<0$ \\ T36 & $1+\sqrt3$ & -183 & T5 & -11 & 0 & 1 & (5,13) & (19,49) & $\Gamma_d<0$ \\ T37 & $1+\sqrt3$ & -59 & T13 & -3 & 0 & 1 & (7,19) & (19,51) & $\Gamma_d<0$ \\ T38 & $1+\sqrt3$ & +73 & T13 & -3 & 0 & 1 & (7,19) & (19,53) & $\Gamma_d<0$ \\ T39 & $1+\sqrt3$ & -167 & T13 & -3 & 0 & 1 & (7,19) & (21,55) & $\Gamma_d<0$ \\ T40 & $1+\sqrt3$ & +121 & T13 & -3 & 0 & 1 & (7,19) & (21,59) & $\Gamma_d<0$ \\ T41 & $1+\sqrt3$ & -143 & T13 & -3 & 0 & 1 & (7,19) & (23,61) & $\Gamma_d<0$ \\ T42 & $1+\sqrt3$ & +13 & T30 & +1 & 0 & 0 & (1,3) & (1,5) & $\Gamma_d<0$ \\ T43 & $1+\sqrt3$ & +177 & T13 & -3 & 0 & 1 & (7,19) & (23,65) & $\Gamma_d<0$ \\ T44 & $1+\sqrt3$ & -111 & T13 & -3 & 0 & 1 & (7,19) & (25,67) & $\Gamma_d<0$ \\ T45 & $1+\sqrt3$ & +61 & T13 & -3 & 0 & 1 & (7,19) & (25,69) & $\Gamma_d<0$ \\ T46 & $1+\sqrt3$ & -71 & T13 & -3 & 0 & 1 & (7,19) & (27,73) & $\Gamma_d<0$ \\ T47 & $1+\sqrt3$ & -23 & T30 & +1 & 0 & 0 & (1,3) & (3,5) & $\Gamma_d<0$ \\ T48 & $1+\sqrt3$ & +181 & T13 & -3 & 0 & 1 & (7,19) & (29,81) & $\Gamma_d<0$ \\ T49 & $1+\sqrt3$ & -179 & T13 & -3 & 0 & 1 & (7,19) & (31,83) & $\Gamma_d<0$ \\ T50 & $1+\sqrt3$ & +33 & T30 & +1 & 0 & 0 & (1,3) & (1,7) & $\Gamma_d<0$ \\ T51 & $1+\sqrt3$ & -131 & T13 & -3 & 0 & 1 & (7,19) & (33,89) & $\Gamma_d<0$ \\ T52 & $1+\sqrt3$ & +97 & T13 & -3 & 0 & 1 & (7,19) & (33,91) & $\Gamma_d<0$ \\ T53 & $1+\sqrt3$ & +169 & T13 & -3 & 0 & 1 & (7,19) & (35,97) & $\Gamma_d<0$ \\ T54 & $1+\sqrt3$ & -11 & T30 & +1 & 0 & 0 & (1,3) & (3,7) & $\Gamma_d<0$ \\ T55 & $1+\sqrt3$ & +61 & T30 & +1 & 0 & 0 & (1,3) & (1,9) & $\Gamma_d<0$ \\ T56 & $1+\sqrt3$ & -143 & T30 & +1 & 0 & 0 & (1,3) & (7,5) & $\Gamma_d<0$ \\ T57 & $1+\sqrt3$ & +141 & T13 & -3 & 0 & 1 & (7,19) & (41,113) & $\Gamma_d<0$ \\ T58 & $1+\sqrt3$ & -71 & T30 & +1 & 0 & 0 & (1,3) & (5,7) & $\Gamma_d<0$ \\ T59 & $1+\sqrt3$ & +97 & T30 & +1 & 0 & 0 & (1,3) & (1,11) & $\Gamma_d<0$ \\ T60 & $1+\sqrt3$ & +193 & T13 & -3 & 0 & 1 & (7,19) & (49,135) & $\Gamma_d<0$ \\ T61 & $1+\sqrt3$ & -59 & T30 & +1 & 0 & 0 & (1,3) & (5,9) & $\Gamma_d<0$ \\ T62 & $1+\sqrt3$ & +37 & T30 & +1 & 0 & 0 & (1,3) & (3,11) & $\Gamma_d<0$ \\ T63 & $1+\sqrt3$ & +141 & T30 & +1 & 0 & 0 & (1,3) & (1,13) & $\Gamma_d<0$ \\ S0 & $\sqrt3$ & -26 & S4 & -2 & 1 & 2 & (3,5) & (5,7) & $\Gamma_d<0$ \\ S2 & $\sqrt3$ & -26 & S1 & +6 & 0 & 1 & (1,3) & (3,1) & $\Gamma_d<0$ \\ S3 & $\sqrt3$ & +22 & S1 & +6 & 0 & 1 & (1,3) & (1,5) & $\Gamma_d<0$ \\ S5 & $\sqrt3$ & +22 & S4 & -2 & 0 & 1 & (3,5) & (3,7) & $\Gamma_d<0$ \\ \end{longtable} % END embedded exclusions_summary.tex } \end{document}