<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mi>Be</mml:mi> <mml:mprescripts/> <mml:none/> <mml:mn>7</mml:mn> </mml:mmultiscripts> </mml:math> asymptotic normalization coefficients from <mml:math xmlns:mml="http:

We present the first direct determination of asymptotic normalization coefficients (ANCs) for the ground and first excited states of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mmultiscripts> <a:mi>Be</a:mi> <a:mprescripts/> <a:none/> <a:mn>7</a:mn> </a:mmultiscripts> </a:mrow> </a:math> from <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:msup> <b:mrow/> <b:mn>3</b:mn> </b:msup> <b:mi>He</b:mi> <b:mo>+</b:mo> <b:mmultiscripts> <b:mi>He</b:mi> <b:mprescripts/> <b:none/> <b:mn>4</b:mn> </b:mmultiscripts> </b:mrow> </b:math> elastic scattering phase shifts. The analysis draws upon the classic datasets of two older low-energy measurements, long considered difficult due to internal inconsistencies and the absence of systematic uncertainties. Employing single-level <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mi>R</c:mi> </c:math> -matrix fits, the mapping between reduced widths and ANCs, and Bayesian inference, we reconstruct posterior distributions and quantify uncertainties. A key innovation of this study is the application and validation of the extrapolation method developed by our group, specifically designed to extract ANC values in cases where direct low-energy information is lacking. Benchmarking against the latest measurements [Paneru , ], which reported only <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mi>R</d:mi> </d:math> -matrix fits with fixed ANCs, confirms the predictive power and robustness of our approach. Our findings show that ANCs deduced from older scattering measurements are systematically lower than those inferred from radiative capture, thereby confirming earlier indications of a discrepancy between the two methods. This work demonstrates both the originality and reliability of the extrapolation method as a powerful tool for ANC determination in nuclear astrophysics.