This paper develops the rigorous mathematical results that follow from a single starting point: a Maximum-Entropy spectral density on an abstract frequency axis, taken as the fundamental object of a field theory for a complex substrate field. Rather than presenting a finished theory, it assembles the building blocks that can be established with full rigour, together with the negative results that delimit them, and marks every claim explicitly as derived, calibrated, or a negative result. The main results are: a uniqueness statement for the spectral density from the Maximum-Entropy principle of Jaynes, with a thermal reading in the internal mass-label; a closed-form gravitational kernel with a positive Källén–Lehmann representation and a verified Newtonian limit; a Derrick-theorem argument that localized matter cannot be a static scalar lump; a topological vortex matter sector from which charge quantisation, matter stability, and the existence of antiparticles follow without separate assumption, together with an explicit matter-existence threshold that couples the dark-energy and matter sectors through the single-field structure; a calibration-verified derivation of two emergent spatial axes from ground-state entanglement; a spin-2 no-go theorem showing that a scalar substrate cannot itself be the graviton; and a finite-range gravitation whose far field is a power law without a dynamical gap and Yukawa once a self-consistent gap is generated. For the matter sector, the paper states precisely what holds rigorously — sufficient degrees of freedom, exact Fermi statistics in one dimension via the Jordan–Wigner transformation with the actual spectrum, the avoidability of a fundamental Grassmann field, and a built-in ℤ₂ grading — and is explicit that the theory is at present bosonic, with a sharply delimited path toward fermions. Interpretive and not-yet-derived directions, including the propagating spin-2 mode, a third spatial axis, and fermionic matter in three dimensions, are stated as open problems and reserved for separate work. The work is independent research and is offered for scrutiny. Its aim is the explicit separation of what is proved from what is not. Keywordsemergent gravity; maximum entropy; spectral density; topological solitons; dark energy; modified gravity; entanglement; spin-2 no-go
Karol Frank (Thu,) studied this question.