<p>We study the one-phase Alt–Phillips free boundary problem, focusing on the case of negative exponents <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\gamma \in (-2,0)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mo>-</mo> <mn>2</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. The goal of this paper is twofold. On the one hand, we prove smoothness of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(C^{1,\alpha }\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>α</mi> </mrow> </msup> </math></EquationSource> </InlineEquation>-regular free boundaries by reducing the problem to a class of degenerate quasilinear PDEs, for which we establish Schauder estimates. Such a method provides a unified proof of the smoothness for general exponents. On the other hand, by exploiting the higher regularity of solutions, we derive a new stability condition for the Alt–Phillips problem in the negative exponent regime, ruling out the existence of nontrivial axially symmetric stable cones in low dimensions. Finally, we provide a variational criterion for the stability of cones in the Alt–Phillips problem, which recovers the one for minimal surfaces in the singular limit as <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\gamma \rightarrow -2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo stretchy="false">→</mo> <mo>-</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>