A new context in which to apply geometry to: math, quantum physics, and the solar system, etc
Quantum physics assumes the global and descends to the local (ie random particle-spectral measures).
Is geometry a better vehicle to define the stability of quantum systems rather than function spaces?
Is the stable construct to be the very stable discrete hyperbolic shapes, in a many-dimensional context?
A geometrically stable and spectrally finite math construct, where, in adjacent dimensional levels, the bounding discrete hyperbolic and Euclidean shapes are defined, and then mixed as "metric-space states" in a Hermitian (or unitary) context, can provide a structure for stable properties.
Assume that math be consistent with (local) geometric-measures of stable shapes, which define finite spectral sets, contained in higher-dimensions.
The stable shapes in the different dimensional levels are con-formally similar, and resonate with a finite geometric-spectral set contained in a high-dimension space.
A new interaction type consists of a combination of hyperbolic and Euclidean components, but when in an "energy-size range" the system can resonate with the spectra of the containing space, and thus it can change to a new stable, discrete shape.
There are (moded-out) "cubical" simplexes in a many-dimensional context, whose structure is determined by hyperbolic metric-spaces, which can, themselves, be modeled as moded-out "cubical" simplexes. Transition between the different dimensions determine physical constants, and the value of these physical constants can imply that:
1. the different dimensional levels can be hidden from one another, ie the size of the interacting materials change from dimensional level to dimensional level and the geometry of the interaction can also change, ie material interactions are not usually spherically symmetric.
2. The over-all high-dimension containing space can be defined as having a finite spectral set.
3. The descriptions of both mathematical and the physical systems are (or can be) stable, because the "cubical" simplexes can define discrete hyperbolic shapes.
Metric-spaces have properties and subsequently an associated metric-space state, and this determines the dimensional distinction between Fermions and Bosons, as well as determining the unitary (invariant) mixing of (metric-space) states in subsets of complex-coordinates.
Unitary invariance implies continuity, or the conservation laws, eg the conservation of energy and material, etc.
Each dimensional level is a discrete hyperbolic shape, and this implies such a set can define a finite spectral set for the entire space.
A new interaction type consists of a combination of hyperbolic and Euclidean components which are one dimension less that the dimension of their containing metric-space. A 2-form construct emerges from this geometric context which is the same dimension as the adjacent (higher) dimension Euclidean base space of its fiber group which determines discrete spatial displacements. This interaction construct is either chaotic or it could begin to resonate during the interaction and, subsequently, to become a new stable spectral-orbital (discrete hyperbolic shape) structure, by means of its resonance with the spectra of the many-dimension containing space.