Manfred U. E. PohlIndependent Researcher, GermanyORCID: 0009-0009-0254-3133Diese E-Mail-Adresse ist vor Spambots geschützt! Zur Anzeige muss JavaScript eingeschaltet sein.July 2026
What the trilogy providesThe three papers form one derivation, read top to bottom. Together they take the PVT axioms (\( V_a=x V_b \), \( x\in\mathbb{Q} \); \( \pi\equiv T/L \); \( \delta V=0 \) on the six-dimensional volume) and close, in order, the static vacuum, the matter sector, and the general field equations — deriving what had previously only been cited in the corpus, and pinning down exactly how far the derivation reaches. The load-bearing posit throughout is the single one, \( \pi\equiv T/L \); everything else is either derived from \( \delta V=0 \) on the comparison chain or flagged as open.
View or Download (Zenodo) PART I Part I — the foundational vacuum closure. Supplies the missing foundation both later parts build on: the profile equation \( (rf)'=1 \), until now identified and used but not derived. Part I first makes the volume form exact — establishing the identity between the sine-product volume \( V_\Pi=L_1L_2L_3\prod\sin\theta_{jk} \) and the metric (Gram) volume \( V_G=\sqrt{\det h} \), which coincide in the \( \le 1 \)-active-angle sector where the whole vacuum lives — and then derives the static vacuum equations from \( \delta V=0 \) with no vacuum posit. The undeformed element \( f\,h=1 \) becomes a theorem, the constrained variation yields the multiplier form \( G^{\mu}{}_{\nu}=-\lambda(r)\,\delta^{\mu}{}_{\nu} \), the Bianchi identity collapses \( \lambda \) to a global constant \( \Lambda \), and the complete vacuum content is the single scalar equation \[ (rf)'=1-\Lambda r^2. \] The vacuum is thereby no longer defined as zero deficit; it is the derived stationary sector of the volume-constrained chain, with \( \Lambda \) the constant background deficit density, and the orthogonal reference boundary condition forces \( \Lambda=0 \) and Schwarzschild.
Part II — matter as an angular source. Within the metric class fixed by the axioms (\( f\,h=1 \)), the entire static vacuum content of general relativity collapses to the one scalar condition \( (rf)'=1 \), necessary and sufficient for \( R_{\mu\nu}=0 \) — the continuum counterpart of the single global condition \( H_{\mathrm{glob}}=0 \) of the discrete chain. A no-go theorem then shows that a single active angle forces \( p=-\rho \): genuine matter is impossible there. Matter requires the temporal and radial angles to split, \( \theta_t\neq\theta_r \), and the deformation \( D=\sin\theta_t/\sin\theta_r \) of the invariant volume element obeys the exact law \[ (\ln D)'=4\pi r(\rho+p)\,g_{rr}: \] matter is the local deformation of the volume element, and vacuum-energy-like sources (\( \rho+p=0 \)) leave it undeformed. The full PVT system — deficit equation, deformation law, chain conservation — is proved exactly equivalent to \( G_{\mu\nu}=8\pi T_{\mu\nu} \) for the static spherically symmetric perfect fluid, and the scale-free chain singles out the unique fixed-angle solution \( \sin^2\theta_\ast=(1+w)^2/(1+6w+w^2) \), giving \( \sin^2\theta_\ast=4/7 \) for the traceless source \( w=1/3 \).
Part III — the general field equations, and their exact reach. Closes the general case and states precisely what the general result is: it is not full general relativity, and it must not be. GRT admits an unconstrained local scale mode — the sector housing external time and an absolute volume scale — and with it a tenth field equation and a freely specifiable cosmological constant. The axiom \( \delta V=0 \) is an additional constraint GRT does not impose, and it deletes exactly that sector. Route B (variational): \( \delta V=0 \) is precisely a unimodular constraint; constrained variation gives the trace-free equations \[ R_{\mu\nu}-\tfrac14 R g_{\mu\nu}=8\pi\Bigl(T_{\mu\nu}-\tfrac14 T g_{\mu\nu}\Bigr), \] and Bianchi with chain conservation collapses the multiplier field to a single global constant \( \Lambda \) — an integration constant of the closed chain, not a source or coupling, so vacuum energy drops out of the bulk equations identically. In degree-of-freedom terms the twelve PVT volume operators are exactly a parametrization of the twelve dynamical ADM variables \( (h_{ij},\pi^{ij}) \), while GRT additionally carries the external-time apparatus and the scale excitation. Route A (uniqueness): under the structural premises P1–P4 extracted from the series, these equations are the only admissible closure (Lovelock). The static sector recovers Part II exactly; the FLRW sector yields the Friedmann equations with \( \Lambda \) as an integration constant.
The net result. The trilogy is a complete, axioms-to-field-equations derivation: the Einstein field equations — with matter, and with \( \Lambda \) — obtained as the volume-invariant core of general relativity, a strict restriction that reproduces GRT wherever GRT is tested and differs from it exactly where GRT is weakest, namely the status of the cosmological constant and the gravitation of vacuum energy. Every load-bearing step is verified symbolically and carries a provenance marker.