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\title[An upper bound for the Zhang--Zagier essential minimum]{An explicit upper-bound certificate for the essential minimum of the Zhang--Zagier height}
\author{Ricardo Pinho}
\address{Independent Researcher}
\date{16 October 2026}
\keywords{Zhang--Zagier height, essential minimum, Mahler measure, rigorous computation}
\begin{document}
\begin{abstract}
We give an exact certificate proving $C_{82}<0.2534449$ for the essential
minimum of $h(\alpha)+h(1-\alpha)$, where $h$ is the absolute logarithmic
Weil height. We use Doche's limit-point method, building on Grinsztajn's
factor library and, in particular, \'{E}pi's unequal-multiplicity construction.
We retain \'{E}pi's denominator factors, add Doche's $P_7$, and add 25
numerator factors found by LP-dual-weighted evaluation lattices with LLL.
Joint linear-program re-optimisation is followed by exact rational rounding.
Two separately written ball-arithmetic checkers certify the bound; a further
adversarial computational audit accepts the certificate.
\end{abstract}
\maketitle

\section{Introduction}
For a primitive irreducible polynomial $f\in\ZZ[z]$ of degree $d$ and a root
$\alpha$, put
\[
 M(f)=|\lc(f)|\prod_{f(\beta)=0}\max(1,|\beta|),\qquad
 h(\alpha)=\frac{\log M(f)}d,\qquad
 \hzz(\alpha)=h(\alpha)+h(1-\alpha).
\]
All logarithms are natural. The essential minimum considered here is
\begin{equation}\label{eq:ess}
 C_{82}=\sup\{H\in\mathbb R:
       \{\alpha\in\Qbar:\hzz(\alpha)\le H\}\text{ is finite}\}.
\end{equation}
The notation follows constant 82a of Tao and contributors' optimization
constants table \cite{table}. Unlike the minimum of the nonzero values of
$\hzz$, this constant concerns infinitely many algebraic numbers of
unbounded degree.

The problem belongs to the arithmetic theory of small points and essential
minima developed by Zhang \cite{zhang1995}. Zagier \cite{zagier1993} proved
\[
 \hzz(\alpha)\ge \tfrac12\log\bigl((1+\sqrt5)/2\bigr)
\]
away from the roots of $z(z-1)(z^2-z+1)$. Doche developed the limit-point construction used here
\cite{doche2001a,doche2001b}. The table lists Grinsztajn's 2026 bound
$0.2536331090204145$ \cite{grinsztajn2026}. More recently, \'{E}pi (BakeLab,
September 2026) reported $0.2535744749068227$ \cite{epi2026}; an independent
re-enclosure of its unchanged construction gives approximately
$0.2535689302702101$. These earlier constructions are central to the present
one. The table lists the lower bound $0.24874$ due to Flammang
\cite{flammang2018}.
Burgos Gil, Menares, Qu and Sombra \cite{bmqs2026} prove strong duality for
upper- and lower-bound methods and theoretical computability of essential
minima. Their general results are not needed for the argument below.

\begin{theorem}\label{thm:main}
The essential minimum of the Zhang--Zagier height satisfies
\[
 C_{82}\le B<0.2534449.
\]
Here $B$ is the explicit functional in \eqref{eq:B} evaluated on the exact
certificate specified below.
\end{theorem}

A read-only check of the live table, its open pull requests, and the
\'{E}pi repository on 16 October 2026 found no smaller bound. This note
reports a verified upper bound, without a claim of optimality.

\section{Construction and method}
Write $T(z)=z(1-z)$ and
\[
 Q(X)=\prod_j R_j(X)^{m_j},\qquad
 \log|A(X)|=\sum_i q_i\log|S_i(X)|,\qquad D=\deg Q.
\]
The product $A$ is formal; no branches of rational powers are chosen.
The labels follow Doche, Grinsztajn and \'{E}pi; the exact integer
coefficient arrays (descending order) in \texttt{certificate.json}, rather
than the labels, define the polynomials. Table~\ref{tab:factors} gives all
active factors and exact weights.

\'{E}pi introduced unequal denominator multiplicities and the extra
factors $E_{d5}$ (degree 28) and $E_{n6}$ (degree 16). Our construction
keeps \'{E}pi's denominator factors, adds Doche's $P_7$, and adds 25
numerator factors found by LP-dual-weighted evaluation lattices with LLL
reduction. The factor $E_{n6}$ was included in the starting library but
is inactive in the final certificate. The perturbative factors $R_0,R_2$
come from Grinsztajn's public construction. Thus unequal multiplicities
and the \'{E}pi factors are inherited, rather than introduced here.

The sampled Doche functional is optimised by a linear program jointly in
denominator multiplicities and numerator exponents. Its dual weights
select evaluation points and external fields for integer evaluation
lattices. LLL reduction supplies candidate numerator polynomials;
exact factorisation and numerical screening select columns for further
LP re-optimisation. The retained weights are then rounded to positive
integer denominator multiplicities and nonnegative rational numerator
exponents. This is a heuristic search, not an exhaustive optimisation.
Only the rounded exact data enter the proof and the rigorous checkers.

The formal degree is $\deg A:=\sum_i q_i\deg S_i$.
Every denominator factor is monic, all numerator--denominator pairs are
coprime, and exact rational arithmetic verifies
\begin{equation}\label{eq:degrees}
 D=999999992,\qquad \deg A<944662259<D.
\end{equation}
Nonmonic numerator factors are permitted; their leading-coefficient
contributions are retained in every logarithmic evaluation.

\begin{table}[t]
\caption{Active factors of the exact certificate. Numerator labels
\texttt{select} and \texttt{merged} denote the 25 lattice-derived factors.}
\label{tab:factors}
\centering\small
\begin{tabular}{lrr}
\toprule Denominator & Degree & Multiplicity\\\midrule
Q1 & 28 & 9297152 \\
Q2 & 28 & 5575669 \\
R0 & 28 & 3376055 \\
R2 & 28 & 854210 \\
P7 & 12 & 4177785 \\
P9 & 16 & 14295157 \\
Ed5 & 28 & 6652059 \\
\bottomrule\end{tabular}
\medskip

\begin{minipage}[t]{0.49\textwidth}\centering\scriptsize
\begin{tabular}{lrl}
\toprule Numerator & Degree & Exponent $q_i$\\\midrule
P1 & 1 & $309690889482955/4194304$ \\
P2 & 1 & $630788667754879/8773059$ \\
P3 & 3 & $21260800937839/8055645$ \\
P4 & 4 & $52745169098993/7487927$ \\
P6 & 8 & $29007076082772/3169231$ \\
P8 & 12 & $3410042187423/8533421$ \\
select2\_1 & 34 & $1992723892935/9974618$ \\
select2\_11 & 24 & $702728210855/1985031$ \\
select2\_15 & 27 & $450900060781/9270228$ \\
select2\_22 & 28 & $6599318260539/9993640$ \\
select2\_25 & 32 & $562420442071/2264313$ \\
select2\_36 & 18 & $3362145432269/9532037$ \\
select3\_7 & 28 & $4352002110254/4887323$ \\
select4\_0 & 24 & $3578264620002/7819705$ \\
select4\_1 & 42 & $13531153728682/6884769$ \\
select4\_3 & 38 & $5381514591569/5240791$ \\
\bottomrule\end{tabular}\end{minipage}
\begin{minipage}[t]{0.49\textwidth}\centering\scriptsize
\begin{tabular}{lrl}
\toprule Numerator & Degree & Exponent $q_i$\\\midrule
select5\_0 & 39 & $4617249463684/4001225$ \\
select5\_7 & 34 & $4710839667389/9906458$ \\
select5\_9 & 32 & $1079755879201/2164874$ \\
select5\_10 & 35 & $6714480457493/8818907$ \\
select7\_0 & 42 & $9317325406677/3194929$ \\
select8\_0 & 22 & $10953669139763/9925694$ \\
select9\_1 & 24 & $3449087714362/9861087$ \\
select9\_2 & 37 & $4659890228461/8743052$ \\
select9\_3 & 42 & $6289381318901/9096035$ \\
select9\_5 & 42 & $651575276147/7044436$ \\
select9\_6 & 36 & $11838421919974/8950455$ \\
select9\_7 & 39 & $127103111402/1055161$ \\
select10\_0 & 41 & $4884194025059/4010746$ \\
select11\_0 & 42 & $6519687014701/5871539$ \\
merged\_0 & 30 & $3775427748764/6478995$ \\
\bottomrule\end{tabular}\end{minipage}
\end{table}

Set $u(t)=e^{2\pi it}$ and, away from zeros, define
\[
 g(t)=\sum_i q_i\log|S_i(T(u(t)))|-\log|Q(T(u(t)))|.
\]
The bound to be certified is
\begin{equation}\label{eq:B}
 B=\frac1D\left(\log M(Q\circ T)+\int_0^1\pos{g(t)}\,dt\right),
 \qquad x_+=\max(0,x).
\end{equation}
At numerator zeros the positive part is zero; isolated denominator zeros
are treated as integrable logarithmic singularities. Formula~\eqref{eq:B}
is the logarithmic form of Doche's formula \cite[equation (9)]{doche2001b}.
Notice that the Mahler term is for $Q\circ T$, not for $Q$.

\section{Why the certificate proves the theorem}
We give the infinite-sequence argument explicitly, including the step
that handles reducible polynomials.

\begin{proposition}\label{prop:limit}
Suppose $Q\in\ZZ[X]$ is monic up to sign with $D=\deg Q>0$,
$S_i\in\ZZ[X]\setminus\{0\}$, and $q_i\in\mathbb Q_{\ge0}$ satisfy
$\sum_i q_i\deg S_i<D$. Suppose also that each $S_i$ with $q_i>0$ is
coprime to $Q$. Then $C_{82}\le B$, with $B$ defined by \eqref{eq:B}.
In fact there are distinct algebraic integers $\alpha_k$ with
$\limsup_k\hzz(\alpha_k)\le B$.
\end{proposition}
\begin{proof}
Choose a positive integer $b$ clearing all the denominators and put
\[
 U(z)=\prod_i S_i(T(z))^{bq_i},\qquad W(z)=Q(T(z))^b,
 \qquad N=\deg W=2bD.
\]
These are coprime integer polynomials, since a common composed zero would
map under $T$ to a common original zero. We have $\deg U<N$, and $W$ is
monic up to sign. For $F_n=U^n-W^n$, factorization over roots of unity gives
\begin{equation}\label{eq:phase}
 \frac{\log M(F_n)}n
 =\frac1n\sum_{\zeta^n=1}\log M(U-\zeta W)
 \longrightarrow L:=\int_0^1\log\max(|U(u(t))|,|W(u(t))|)\,dt.
\end{equation}
Indeed, $U-\zeta W$ has fixed degree $N$ and leading coefficient of modulus
one. Continuity of its roots as a multiset implies continuity of its
logarithmic Mahler measure in $\zeta$. Thus the sums converge to its phase
mean, which Jensen's formula evaluates as $L$. The interchange of the
circle and phase integrals is justified as follows. The positive part of
$\log|U-\zeta W|$ is uniformly bounded on the two circles. Its phase mean
is $\log\max(|U|,|W|)$, which is bounded below because $U,W$ have no
common zero. Tonelli's theorem then makes the negative part integrable
as well. Splitting off $\log|W|$ and applying Jensen in $z$ gives
\begin{equation}\label{eq:L}
 L=b\log M(Q\circ T)+b\int_0^1\pos{g(t)}\,dt=bDB.
\end{equation}

For each prime $p$, consider
\[
 G_p(z)=\frac{U(z)^p-W(z)^p}{U(z)-W(z)}
       =\sum_{j=0}^{p-1}U(z)^{p-1-j}W(z)^j.
\]
This integer polynomial is monic up to sign, has degree $N(p-1)$, and is
invariant under $z\mapsto1-z$. A root cannot be a zero of $W$; its value of
$U/W$ is a nontrivial $p$th root of unity. At $U/W=1$ the sum is
$pW^{p-1}\ne0$. Consequently $G_p$ and $G_r$ are coprime for distinct
primes $p,r$.

Factor $G_p$ into primitive irreducible integer polynomials, retaining
multiplicities. Each factor is monic up to sign, and the height formula
shows that the average of $h(\alpha)$ over the roots is
$\log M(G_p)/\deg G_p$. Symmetry gives the same average for $h(1-\alpha)$.
Hence
\[
 \frac1{\deg G_p}\sum_{G_p(\alpha)=0}\hzz(\alpha)
 =\frac{2\log M(G_p)}{N(p-1)}\longrightarrow\frac{2L}{N}=B,
\]
by \eqref{eq:phase}, since
$\log M(G_p)=\log M(F_p)-\log M(U-W)$.
On each irreducible factor, $\hzz$ is constant on its Galois orbit.
Select one root of a factor with height sum at most this average.
Coprimality across primes makes the selected roots distinct, even if an
individual $G_p$ has repeated roots. Every threshold strictly greater
than $B$ therefore contains infinitely many such roots, which proves the
claim by \eqref{eq:ess}.
\end{proof}

The exact checks in \eqref{eq:degrees} verify the hypotheses. The certified
upper bounds in the next section then prove Theorem~\ref{thm:main}.
The factor two from $\deg T=2$ cancels the factor two in the root average;
this explains the divisor $D$ in \eqref{eq:B}.

\section{Computer verification}
The public repository supplies \texttt{verify.py} and
\texttt{independent\_check.py}, written separately at the application level.
Both compose the factors with $T$ exactly, certify complete root multisets
with FLINT/Arb, and compute the Mahler term by Jensen's identity. They use
exact rational weights and certify the integral over the full circle,
without assuming that a sample grid locates every positive arc.
They share FLINT/Arb and mathematical identities, so their agreement does
not constitute independent arithmetic-library verification.

The first checker integrates explicit Taylor upper majorants on exact
dyadic panels. Root-distance bounds exclude negative panels; separated
root balls bound the second derivative on remaining panels. The second
checker uses certified analytic quadrature on positive panels and full
interval enclosures on unresolved crossing panels. Both use real
coefficients to double the integral over $[0,1/2]$. The Taylor ball
bounds an upper majorant; it is not a two-sided enclosure of $B$.
The analytic checker encloses $B$ itself.

The following decimals are rounded upward from the certified endpoints:
\begin{center}
\begin{tabular}{lr}
\toprule Checker & Upper bound for $B$\\\midrule
\texttt{verify.py} & $0.253444687333627$\\
\texttt{independent\_check.py} & $0.253444687337420$\\
Additional adversarial evaluator & $0.253444687331849$\\
\bottomrule
\end{tabular}
\end{center}
The additional adversarial computational referee returned \emph{ACCEPT}.
It checked the same certificate hash, the strict degree inequality and all
217 cross-factor gcds, reconstructed the infinite-sequence proof, and
used a separate evaluator with Taylor sign isolation and Arb analytic
integration. Its \'{E}pi control reproduced the unchanged public construction;
a doubled-$Q_1$ numerical mutation failed the target, and malformed-input
controls were rejected. This is a computational audit, not journal peer
review. The certificate's stored floating search objective is not used
as evidence for the bound.

With Python and the repository's pinned \texttt{python-flint} dependency,
the principal checks are reproduced by
\begin{small}\begin{verbatim}
python verify.py certificate.json --tol 1e-10 --out taylor.json
python independent_check.py certificate.json analytic.json 1e-8
\end{verbatim}\end{small}
These tolerances control subdivision; the computed interval endpoints,
not a presumed tolerance error, certify the exact target $0.2534449$.
No search rerun or expanded product polynomial is required.

\section*{Data availability and limitations}
The certificate and both checkers are available at
\url{https://github.com/rpinho/c82-height-certificate}.
The SHA-256 digest of \texttt{certificate.json} is
\begin{center}\small\ttfamily
b33626f367b6f52922746c0fee7bb0084\\
49cd0bb18efcb983ab20ff073350fce.
\end{center}
The line break and final punctuation are not part of the digest.
The bound does not determine $C_{82}$ or establish optimality within
this factor family. Proposition~\ref{prop:limit} gives infinitely many
algebraic integers without expanding their defining polynomials.

\section*{AI assistance}
Analysis and proof drafting were done by AI agents (OpenAI Codex and Claude)
under the author's direction; the author reviewed the results at summary
level. AI agents also prepared the certificate, the checking software and
this note. This does not claim a line-by-line human verification of the
proof or the software; correctness rests on the exact certificate, checked
by two separately written ball-arithmetic checkers.

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