Runaway solutions can be avoided in fourth order gravity by doubling the matter operator algebra with a symmetry constraint with respect to the exchange of observable and hidden degrees of freedom together with a change in sign of the ghost and the dilaton fields. The theory is classically equivalent to Einstein gravity, while its nonunitary Newtonian limit is shown to lead to a sharp transition, around 10 11 proton masses, from the wavelike properties of microscopic particles to the classical behavior of macroscopic bodies, as well as to a trans-Planckian regularization of collapse singularities. A unified reading of ordinary and black hole entropy emerges as entanglement entropy with hidden degrees of freedom. The emergent picture gives substantial agreement with Bekenstein-Hawking entropy and the Hawking temperature.

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The result of a numerical simulation of two interacting particles in the framework of Nonunitary Newtonian Gravity is presented here. Particles are held close together by a 3d harmonic trap and interact with each other via an 'electrical' delta-like potential and via the ordinary Newtonian term, together with a fluctuational nonunitary counterpart of the latter. Fundamental nonunitarity can be seen as arising from the interaction of the physical degrees of freedom with (gravitational) hidden copies of them. Starting from an energy eigenstate within the ordinary setting, it is shown that, while energy expectation remains constant, a slow net variation of the von Neumann entropy for the system as a whole takes place, with a small modulation induced on the relative entanglement entropy of the two particles. Besides, the simulation shows explicitly how fundamental gravity-induced entropy can be clearly distinguished from the subjective notion of coarse-grained entropy, i.e. from the entropy of one particle with respect to the 'environment' of the other.

We study a particular return map for a class of low-dimensional chaotic models called Kolmogorov-Lorenz systems, which received an elegant general Hamiltonian description and includes also the famous Lorenz-63 case, from the viewpoint of energy and Casimir balance. In particular it is considered in detail a subclass of these models, precisely those obtained from the Lorenz-63 by a small perturbation on the standard parameters, which includes for example the forced-Lorenz case in Ref. [6]. The paper is divided into two parts. In the first part the extremes of the mentioned state functions are considered, which define an invariant manifold, used to construct an appropriate Poincarè surface for our return map. From the 'experimental' observation of the simple orbital motion around the two unstable fixed points, together with the circumstance that these orbits are classified by their energy or Casimir maximum, we construct a conceptually simple skeletal dynamics valid within our sub-class, reproducing quite well the Lorenz map for Casimir. This energetic approach sheds some light on the 'physical' mechanism underlying regime transitions. The second part of the paper is devoted to the investigation of a new type of maximum energy-based long-term predictions, by which the knowledge of a particular maximum energy 'shell' amounts to the knowledge of the future (qualitative) behaviour of the system. It is shown that, in this respect, a local analysis of predictability is not appropriate for a complete characterization of this behaviour. A perspective on the possible extensions of this type of predictability analysis to more realistic cases in (geo)-fluid dynamics is discussed at the end of the paper.