Principal components for model-agnostic modified gravity with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mn>3</mml:mn><mml:mo>×</mml:mo><mml:mn>2</mml:mn><mml:mtext> </mml:mtext><mml:mtext> </mml:mtext><m

To mitigate the severe information loss arising from widely adopted linear scale cuts in constraints on modified gravity parametrizations with weak lensing (WL) and large-scale structure (LSS) data, we introduce a novel alternative method for data reduction. This principal component analysis (PCA)-based framework extracts key features in the matter power spectrum arising from nonlinear effects in a set of representative gravity theories. By performing the analysis in the space of principal components, we can replace sweeping linear-only scale cuts with targeted cuts on the transformed data vector, ultimately reducing parameter bias and significantly tightening constraints. We forecast constraints on a minimal parametrized extension to <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi mathvariant="normal">Λ</a:mi><a:mi>CDM</a:mi></a:mrow></a:math> which includes modifications to the growth of structure and lensing of light (<d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:mi mathvariant="normal">Λ</d:mi><d:mi>CDM</d:mi><d:mo>+</d:mo><d:msub><d:mi>μ</d:mi><d:mn>0</d:mn></d:msub><d:mo>+</d:mo><d:msub><d:mi mathvariant="normal">Σ</d:mi><d:mn>0</d:mn></d:msub></d:math>) using mock stage-IV data for two simulated cosmologies: the <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:mi mathvariant="normal">Λ</h:mi><h:mi>CDM</h:mi></h:math> model and extended shift symmetric (ESS) gravity. Under the assumption of a Universe defined by <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi mathvariant="normal">Λ</k:mi><k:mi>CDM</k:mi></k:math> and general relativity, our method offers constraints on <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:msub><n:mi>μ</n:mi><n:mn>0</n:mn></n:msub></n:math> a factor of 1.65 tighter than traditional linear-only scale cuts. Crucially, our approach also provides the necessary constraining power to break key degeneracies in modified gravity without relying on <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mi>f</p:mi><p:msub><p:mi>σ</p:mi><p:mn>8</p:mn></p:msub></p:math> measurements, introducing a promising new tool for the analysis of present and future WL and LSS photometric surveys.