What question did this study set out to answer?

This research aims to establish a continuous field generalization of the previous tensional conservation framework for improved understanding of density interactions.

May 16, 2026Open Access

POLIS V13: The Unified Tensional Density Field — Continuous Field Generalisation of the Closed Tensional Conservation Framework

Key Points

This research aims to establish a continuous field generalization of the previous tensional conservation framework for improved understanding of density interactions.
Implemented the continuous tensional density field ρT(r) in place of the discrete closure condition from V12.
Presented three experimental proofs (hydrogen atom, K–Pg extinction, viral propagation) along with responses to four formal objections.
Validated 36 domains by showing the continuity of the density spectrum under the new framework.
The mother equation ε = ∫ ρT(2+ρT) dV = 0 replaced the previous discrete constraints, maintaining closure in all validated domains.
Demonstrated that all three V12 meshes (solid, liquid, gaseous) fit within a single continuous density spectrum.
Provided responses to the four formal objections, reaffirming the completeness of the continuous model.

Abstract

POLIS V13 presents the continuous field generalisation of the POLIS tensional conservation framework. V12 established the discrete closure condition ε = Σ Km(2+Km) = 0 across validated domains with zero free parameters. V13 replaces discrete polis Km with a continuous tensional density field ρT(r). The mother equation becomes ε = ∫ ρT(2+ρT) dV = 0. The three V12 meshes (solid, liquid, gaseous) and the tensional bubble are shown to be regions of a single continuous density spectrum. Everything known and unknown has a place in this metric by density alone. V12 is recovered as the limiting case ρT(r) = Σ Km δ(r−Tm). Three experimental proofs (hydrogen atom, K–Pg extinction, viral propagation) and four formal objections with responses are presented. All 36 domains validated in V12 remain closed under V13.

POLIS V13: The Unified Tensional Density Field — Continuous Field Generalisation of the Closed Tensional Conservation Framework

Key Points

Abstract

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