σ = Em : Mass–Energy Genesis of Vacuum Information in Vacuum Folding Dynamics

The preceding paper in this series showed that the vacuum folding density σ_f is a reaction product of geometric capacity C_geo and vacuum fluctuation density F_vac. This paper postulates an identification of the two factors with mass and energy: C_geo = m (motivated by the Bekenstein bound S ≤ 2πRMc/ℏ at fixed radius) and F_vac = E (a normalisation of the Planck-scale zero-point energy). With these identifications the reactive equation σ_f = C_geo · F_vac takes the form σ = E · m in Planck-normalised units, which we call the Leonforte equation. It is a defining postulate of the VFD programme, not a derived result. In particular, it does not imply E = mc²: combined with E = mc² as an additional input it yields the identity c = √(σ/m²), which is a consistency check, not a derivation, because the Planck normalisation already contains c. The loop E · m → σ_f → {G, Λ, T, D} → E = mc² is therefore a consistency condition of the programme, not a derivation of mass–energy equivalence. The selection of D = 3+1 rests on the stability and propagation arguments of Paper II and does not follow from σ = Em alone. Version 3 (October 2026). Corrects and revises v1/v2 (v2 carries the text of v1 unchanged): the statement that every additional Planck mass adds one Planck area of horizon is withdrawn (the area grows as M²), and C_geo = m is stated as a postulate; F_vac = 1 is stated as a normalisation convention; the claims that σ = Em is the parent of E = mc², that the loop derives E = mc², and that σ = Em selects D = 4 are withdrawn; the statement that no Planck mass exists in D = 3 is corrected; predictions P1–P3 are withdrawn as tests of σ = Em; the epilog is marked as speculative; Table 1 now gives σ₀ = 1/l_P³; references carry titles and DOIs of the Zenodo records. Section 12 of the PDF lists all changes.