Constrained Padé ensembles for thermal <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi mathvariant="script">N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> SYM: Quantified uncerta

We quantify the transition between weak and strong coupling in thermal <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi mathvariant="script">N</a:mi> <a:mo>=</a:mo> <a:mn>4</a:mn> </a:math> supersymmetric Yang-Mills theory in four space-time dimensions by constructing an of log-aware Padé approximants that incorporate the weak- and strong-coupling expansions through <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"> <d:mrow> <d:mi mathvariant="script">O</d:mi> <d:mo stretchy="false">(</d:mo> <d:msup> <d:mrow> <d:mi>λ</d:mi> </d:mrow> <d:mrow> <d:mn>2</d:mn> </d:mrow> </d:msup> <d:mo stretchy="false">)</d:mo> </d:mrow> </d:math> and <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mrow> <i:mi mathvariant="script">O</i:mi> <i:mo stretchy="false">(</i:mo> <i:msup> <i:mrow> <i:mi>λ</i:mi> </i:mrow> <i:mrow> <i:mo>−</i:mo> <i:mn>3</i:mn> <i:mo>/</i:mo> <i:mn>2</i:mn> </i:mrow> </i:msup> <i:mo stretchy="false">)</i:mo> </i:mrow> </i:math> ( <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"> <n:mi>λ</n:mi> </n:math> is the ’t Hooft coupling), including the nonanalytic <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"> <p:msup> <p:mi>λ</p:mi> <p:mrow> <p:mn>3</p:mn> <p:mo>/</p:mo> <p:mn>2</p:mn> </p:mrow> </p:msup> </p:math> and <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"> <r:msup> <r:mi>λ</r:mi> <r:mn>2</r:mn> </r:msup> <r:mi>log</r:mi> <r:mi>λ</r:mi> </r:math> terms. This replaces single-curve estimates with a reproducible uncertainty band and a well-defined central curve across the intermediate regime. The framework is , setting testable benchmarks for forthcoming perturbative and holographic calculations.