Investigating jet-induced identified hadron production from the relative transverse activity classifier in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>p</mml:mi> <mml:mi>p</mml:mi> </mml:math> collisions at the

In this work, we systematically investigate the jet associated identified hadron productions of pions, kaons, and protons based on the event topological separation method in proton-proton ( <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>p</a:mi> <a:mi>p</a:mi> </a:math> ) collisions at <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msqrt> <c:mi>s</c:mi> </c:msqrt> <c:mo>=</c:mo> <c:mn>13</c:mn> <c:mtext> </c:mtext> <c:mtext> </c:mtext> <c:mi>TeV</c:mi> </c:math> employing a multiphase transport model with ythia8 initial conditions. Enabling relative transverse event activity classifier <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msub> <e:mi>R</e:mi> <e:mi>T</e:mi> </e:msub> </e:math> , we analyze the transverse momentum ( <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:msub> <g:mi>p</g:mi> <g:mi>T</g:mi> </g:msub> </g:math> ) spectra and particle ratios in the jet aligned toward region and the underlying event contributions sensitive transverse region varying with <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:msub> <i:mi>R</i:mi> <i:mi>T</i:mi> </i:msub> </i:math> . The results indicate that a multi-phase transport model, incorporating both partonic and hadronic final-state interactions, provides a satisfactory description to the experimental data of the <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:msub> <k:mi>p</k:mi> <k:mi>T</k:mi> </k:msub> </k:math> differential yields, particle ratios, and average transverse momentum of identified particles in both the toward and transverse regions across different event activity classes. By subtracting the underlying event contribution from the toward region using the transverse region hadron yield, we introduce the in-jet hadron productions to analyze the modifications to the jet itself. We find that the in-jet baryon to meson ratios reveal a sensitive dependence on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:msub> <m:mi>R</m:mi> <m:mi>T</m:mi> </m:msub> </m:math> driven by final-state interactions, leading to a unique crossing pattern at intermediate <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:msub> <o:mi>p</o:mi> <o:mi>T</o:mi> </o:msub> </o:math> . This behavior can be regarded as a novel signature for probing jet medium interactions and energy loss effects in small collision systems.