Local Partition Derivatives Near Equipartition Mean-Reverting Solar X-Ray Hardness During Deep Solar Minimum

This revised technical note re-analyzes GOES X-ray flux data from January 2020 during deep solar minimum using a local tangent-space version of the binary partition framework. Earlier RTLI-style interpretations of the same data emphasized global coherence constants, angular complementarity, four-sphere embeddings, and peak-ratio resonances. This revision substantially weakens those claims and reframes the dataset as a near-equipartition case. Using the GOES hardness ratio H = A/B, where A is the 0.5–4 Å channel and B is the 1–8 Å channel, we define a bounded partition coordinate χ = (H − H₀)/(H + H₀), with H₀ equal to the empirical median hardness. The resulting time series remains close to equipartition: σχ ≈ 0.0668, 95% of samples satisfy |χ| < 0.133, the median response is R ≈ 1.0018, and the maximum response is only R ≈ 1.205. Thus the data do not sample the nonlinear response corridor, Möbius self-duality regime, or boundary-response regime. Instead, the appropriate analysis is local. In this local regime, the Fisher–Thales Taylor identities R − 1 ≈ χ², ξ ≈ χ + χ³/3, and 1/2 − h ≈ χ²/4 hold to the expected leading orders. These are exact geometric identities and are not treated as independent solar evidence. The empirical content appears in the derivative structure. The local drift field satisfies μ(χ) = E[Δχ | χ] ≈ −kχ, with k ≈ 0.17–0.20 min⁻¹, corresponding to an empirical hardness perturbation half-life of approximately four minutes after correcting for a small measurement-noise component. The local diffusion is state-dependent, D(χ) ≈ D₀ + D₂χ², with D₂/D₀ ≈ 37, indicating that volatility grows away from the median hardness state. The dynamics are well reproduced by an AR(1)/Ornstein–Uhlenbeck surrogate, so the result is interpreted conservatively: quiet solar X-ray hardness behaves as a low-amplitude mean-reverting stochastic partition field near equipartition. The structural displacement channel L = |χ|/2 dominates local variation in every tested bin, as expected from its first-order sensitivity near χ = 0, while the coupling altitude h changes only at second order. No carrier sign-flips or irrationality-oscillation behavior are observed. The note therefore replaces the earlier global-coherence interpretation with a more limited and testable conclusion: the January 2020 GOES data provide a local equipartition baseline for partition-geometry diagnostics. Future work should apply the same drift–diffusion and effective-potential methods to flare-rich intervals, where larger excursions in χ and R may allow nonlinear response structure to be tested.