Apeiron: Foundations of a Universal Carrier Approach — An Effective Field Theory and Its Research Programme

Apeiron proposes a universal fundamental dynamical carrier as a common physical basis for differently organized manifestations of matter, dark-sector phenomena and effective causal geometry. This paper gives that research approach a concrete mathematical foundation through an explicitly specified local three-field scalar effective theory. The retained contribution comprises a fixed action and parameter benchmark, checked field equations and stress tensor, corrected phase kinetic normalization, a previously checked local spectrum, and a conditional characteristic relation. In the admissible timelike-gradient domain, the phase characteristic is positive and no faster than the metric cone; the three fields share the metric cone on the selected static background. The same action supplies a precise cosmological constraint: its null energy condition, together with minimal Einstein coupling and spatially flat homogeneous geometry, excludes a nonsingular bounce within that domain. This identifies assumptions that are insufficient for that cosmological transition, rather than testing every possible carrier realization or deciding the separate question of local black-hole reconversion. The original geometry, reconversion, entropy and conditional quantum branches define further physical connections to investigate. A staged research programme specifies the next outputs: carrier dynamics and effective matching, physical-sector and geometry bridges, controlled evolution, and independently calibrated observable tests. The effective theory is an explicit worked-out sector; deriving it and observed physical structures from the universal carrier remains the central foundational task. The exposition preserves the mathematical results, their assumptions and the existing scientific status without introducing new calculations or empirical confirmation.