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Recent LIGO─Virgo─KAGRA analyses have revealed complex structure in the binary black hole population, including distinct features in the primary mass spectrum and nontrivial spin─mass correlations. However, phenomenological models used to capture these features often lack analytic transparency, making it difficult to isolate robust physical laws from modeling artifacts. To address this, symbolic regression is applied to posterior predictive samples from the GWTC-4 catalog, producing ensembles of closed-form surrogate expressions for four population relationships: (i) the merger-rate evolution with redshift, <inline-formula> <mml:math><mml:mi>R</mml:mi></mml:math> </inline-formula>(z); (ii) the mass-ratio dependence of the effective-spin distribution, χ<SUB>eff</SUB>(q); (iii) the redshift evolution of the effective-spin distribution, χ<SUB>eff</SUB>(z); and (iv) the conditional mass-ratio distributions associated with the 10 and 35 M<SUB>⊙</SUB> primary mass peaks. This framework compresses both rigid and highly flexible models into differentiable phenomenological laws, recovering a low-redshift merger rate slope of <inline-formula> <mml:math><mml:msub><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>3.1</mml:mn><mml:msubsup><mml:mrow><mml:mn>8</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>0.87</mml:mn></mml:mrow><mml:mrow><mml:mo>+</mml:mo><mml:mn>0.83</mml:mn></mml:mrow></mml:msubsup></mml:math> </inline-formula> without assuming an a priori power-law form. The exact analytic derivatives indicate that the q─χeff and z─χ<SUB>eff</SUB> correlations are driven more by broadening of the posterior widths than by shifts in the mean, with redshift broadening of <inline-formula> <mml:math><mml:msub><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mi>eff</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:math> </inline-formula> being the most securely recovered feature. Furthermore, distinct functional forms for the mass-ratio distributions conditioned on the 10 and 35 M<SUB>⊙</SUB> primary mass peaks are identified. These expressions enable exact analytic gradient diagnostics and compact surrogate summaries, particularly for flexible numerical posteriors not otherwise available in low-dimensional analytic form. They also facilitate downstream calculations for rate forecasting, formation channel comparison, and stochastic background estimation.