We further develop and systematically test a minimally assumption-dependent inversion framework for reconstructing the dark-energy density evolution from type Ia supernova distance-redshift measurements. Instead of adopting a parametric form for the equation of state or employing kernel-based smoothing, we derive an explicit inversion that expresses the normalized dark-energy density, $X(z)\equiv {\rho }_{\mathrm{DE}}(z)/{\rho }_{\mathrm{DE}}(0)$, directly in terms of the redshift derivative of the supernova distance modulus. We apply the method to the $\mathrm{Pantheon}+\text{compilation}$, restricting to $z>0.01$, and estimate the required derivative using a dynamic redshift-binning scheme with approximately equal occupancy per bin. To assess robustness, we compare a local central finite-difference estimator with covariance-based error propagation to an independent Savitzky-Golay local-polynomial reconstruction, finding qualitatively consistent trends, though with estimator-dependent differences in amplitude, for the binnings studied. The reconstructed $X(z)$ is broadly compatible with the $\mathrm{\Lambda }$ cold dark matter expectation $X\simeq 1$ over the low- and intermediate-redshift range, with a moderate enhancement near $z\approx 0.03–0.06$ and a clear, increasingly noisy rise above unity for $z\gtrsim 0.4$. We further show that increasing the number of bins systematically increases both the amplitude and the propagated uncertainty of the reconstructed values, an effect we trace to the quadratic amplification of derivative noise inherent to the inversion [$X(z)\propto D(z{)}^{-2}$]; among the binnings studied, coarser binning yields the more stable reconstruction. Our results demonstrate that direct inversion provides a transparent, data-driven consistency test for dark-energy evolution—including an explicit diagnostic of its own noise-amplification limitations—that can be straightforwardly extended to forthcoming high-precision distance measurements.

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• Dark energy