Reconstructed proton mechanical forces structure: Insights from Patterson function analysis

Using the gravitational form factor Di, the proton’s internal mechanical force distribution was reconstructed through Patterson function analysis combined with a scaling procedure. The reconstruction reveals a sharply localized central peak of approximately −1.5×105 fm−3, with a half‑maximum radius near 0.071 fm, highlighting the compactness of the distribution. Grid integration confirms a net mechanical force of −459 in normalized units, with strictly non‑positive values across the profile. This outcome suggests that the current form factor Di requires modification, as the absence of inward mechanical pressure would imply instability of the proton. Furthermore, the reconstructed mechanical radius exceeds the charge radius, providing additional evidence that the gravitational form factors demand refinement. Overall, Patterson analysis paired with scaling offers a reproducible, model‑independent framework for visualizing charge, magnetic, mass/energy, and mechanical structures of nucleons, light nuclei, and mesons.