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A new way to apply geometry to both math and to quantum physics

Tradition and authority vs. (a very educated) Equal free-inquiry
This is a careful expression of ideas about a geometrically measurable set of systems associated to spectral-material and high-dimension math constructs, which are built from the sophisticated math patterns uncovered by this culture.
It is based on measurability, math stability, quantitative consistency, and finite spectral sets which have a unitary-invariant structure. That is, the coherent, or connected, or consistently resonant systems which are described conserve a many-dimensional, geometric-spectral math construct, ie the continuity of material existence is tied to a finite spectral, high-dimensional, geometrically based math structure.
It can describe the precise and stable properties of general quantum systems, and because it is based in geometry it is practically useful.
New ideas

Why are carefully expressed, new and fundamental, ideas which "more than any other ideas being expressed" address the core problems which face our society concerning "the failure of the society's intellectual and technical development," are so easily ignored and/or marginalized?


Where modern physics, which is, essentially, based on quantum physics, has only really contributed
1. The bomb (based on the rate or reaction being related to the probabilities of particle-collisions, ie based on 19th century theories of chemistry),
2. The laser (clearly invented based on quantum principles), and possibly
3. The transistor
(but really the transistor, like most technology associated to quantum properties of material, is mostly based on a classical thermal manipulation of material so as to allow for the existence of an observed quantum property in a semiconductor material)

That is, most technology related to quantum properties is about creating a classical system which can couple to a quantum property, ie the calculations and understandings associated to quantum physics is only very slightly associated to the technical development, and this is because the quantum description is mostly about systems composed of a relatively small numbers of components whose quantum properties can only be associated to random processes.
The quantum descriptions based on randomness, for systems built on relatively few components, do not provide a mechanism of system (or material) control, though the stable precise context of quantum properties suggests they existence in a controllable context.

Quantum physics has had the greatest impact on statistical physics, where the probabilities of a great number of system-components can be related to measurable and controllable properties as averages, but the quantum influence is essentially about partitioning a statistical system's energy structure and applying the statistical properties of Bosons and Fermions in a system's quantized energy structure.
That is, the impact of quantum physics, and its associated particle-physics, in regard to using knowledge to advance technical development has been very slight.


The marginalization of new and very important ideas is possible since we live in an unequal society, where the communication system (or propaganda system) only serves the interests (and the ideas) of society's "top few."
This control over institutional and public communication within society gives great credibility to the ideas, which the "top few" believe, and it is those same "top few" people, who control the resources, creativity (by investment), knowledge, and communication within all of society.
[Note: The education system is an arm of the propaganda system, and serves big-business interests. Education is not a ladder to high-culture and an authoritative truth, rather education is about challenging the authority of high-culture, especially if the high-culture is not developing the practical creative capacity of knowledge, but rather languishing in its relation to business interests. That is, education is not about wage-slavery, rather it is about free-inquiry, and the relation of knowledge to practical creativity]

The character traits which are desired and rewarded by the big business interests, the institution which most affects the nature of our society, are the traits of being: selfish, narrow, absolute, self-righteous, cruel, unfeeling, petty, and extremely violent (so as to be able to hold onto what one has which gives them social-value [as expressed by big business interests and the media]).

Thus, it is easy to marginalize new ideas which are not consistent with the ideas of the propaganda-education system.

This is exactly the same social structure by means of which the Catholics controlled thought in their (Holy-Roman-) Empire, but now it is not popes but rather the "lords of the monopolistic businesses" ... , eg oil, agri-oil-business, banking-oil, military, medicine-drugs... , who are in control the resources, creativity (by investment), knowledge, and communication within all of society. Thus, physics is about the interests of the big military businesses, etc.

However, if the discussion is about both new ideas and to show the substantial problems which permeate the traditional and authoritative ideas which today dominate thinking in the major US universities, and the discussion is at the level of assumption, interpretation, and (new) contexts, then the discussion can be very educational (in the strongest sense of what education means, namely, the relation that knowledge has to the practical creativity of individuals, as opposed to the creativity of large institutional businesses).
For example, the risk models used by the big banks failed because the math being used is based on ideas which, such as indefinable randomness (an arbitrary quantization math construct {or math scheme}) have fundamental flaws in their: assumptions, contexts, interpretations, the structure of set containment, and set size, ie its failure is based on elementary ideas, and thus it should be much more widely challenged.
But it is not widely challenged, thanks to the extreme fear, as well as extreme violence, by which the society is controlled, so that people are afraid to challenge authority since it leads to a one-sided contest in which the authorities get to destroy those who challenge their authority.
The social positions of those on top cannot be challenged in any context, especially in regard to how language is used.

Only a person outside the social institutions can express ideas about these topics and this is because such a person is so easy to marginalize, even the alternative media only give credence to the ideas of those people established within the big institutions, even if the new ideas being expressed are ideas derived from those same big (educational) institutions.
The only place where the public are allowed to express ideas, in the (so called) alternative media, is in regard to civil protests.
This is an idea of the legitimacy of public expression, which the authoritarian, traditionalist, and so called, progressive, N Chomsky, touts, ie only civil protest is a valid way to express ideas.
That is, one of the so called champions of the liberal cause, which is best defined as those who believe in equality, are, in fact, authoritarian traditionalists, and thus they are allowed onto the media, and in this position they are truly conservatives, who oppose change in regard to challenging the authority of traditional knowledge, and in fact, they support both tradition and authority.
Clearly the finer judgments of the likes of Chomsky lie with the W Lippmanns of the world, ie with those who support class warfare. Where early cultural sophistication, ie what is called intelligence in our society, should define one's social position, in an education system which filters for "cultural expression which will advance the authority of the culture."
There are
1. Socrates who supported equal free-inquiry, ie the inquirer is to be given the authority, and
2. Christ who expressed his belief in equality by the expression, love one's enemy, or "treat you neighbor as you want them to treat you," and then there are
3. those who populate the higher levels of the society, and who support the structures which determine and maintain their social positions, which are judged to be superior to the social positions of their neighbors.

The split is easy to determine, does one believe in equality, or does one believe in tradition and authority of the fixed way in which resources and knowledge is used, and thus such people are in opposition to new knowledge and in opposition to equal-creativity.
The evangelical religious people, the true protestant reformers, should believe that "each and everyone of us" is an equal creator, because this is the "best way" to interpret the golden rule [love one's enemy, ie we are all equal], and that equality is about the very "core nature of mankind" and our relation to creativity, where one cannot believe in the material world, wherein the value of human creativity is judged, there is a deeper relation that life has to creativity (than to material creations, or artistic creations marketed through material-based communication systems).
The new ideas place creativity into a "new way in which to consider" the "ideal" world, as opposed to the material world, and they provide a new context for human creativity, in the context of an ideal existence, in which the material context is a proper subset.

"What about ideas which are fundamental to the practical creativity of a culture (in regard to both its scientific and religious structures of belief)?" and not so much ideas which are concerned about "How power is organized within society (ie civil protest)?"

That is, even if people want knowledge to remain associated to practical creativity then the authority of traditional math ideas should be challenged in regard to the ideas of professional mathematicians (since practical creativity is still related to 19th century, materialistic (or scientific), view of existence) about:

1. Quantitative sets (set size, the "quantitative sets," eg the real numbers, are "too big" )
2. Quantitative consistency (non-linearity does not work in a quantitatively consistent manner, and indefinable randomness is not capable of forming a valid math structure [this is why the big banks models of risk failed])
3. Logical inconsistency (self-contradictory, eg when probability based constructs depend on geometry [particle-collisions])
4. We have an academic-industrial-vocational education system, eg physics departments are nuclear weapons engineering departments, which adheres to tradition and authority, and to the sources by which they get funded,
so as to develop data-fitting math-science constructs which support authority and tradition. These data-fitting math constructs are based on non-linearity and indefinable randomness (see below), and these are math constructs which are in opposition to the math properties of both stability and quantitative consistency.
5. Though the academic institutions developed the idea that:
(a ) precise language has clear limitations as to the patterns which it is capable of describing in a measurable context (Godel's incompleteness theorem), and
(b) the example of Copernicus vs. Ptolemy is often discussed (but the discussion is framed as "science is correct and religious authority is wrong in regard to the material world" ; rather than it being related to the idea that measurably verifiable constructs can be framed in the wrong context), and whereas
(c ) the usefulness of "classical physics descriptions" vs. "the technically un-useful quantum physics," where quantum physics is based on indefinable randomness and contributes very little to technical development; but on the other hand, most of the technical development today is still based on the classical physics of the 19th century, nonetheless, the stable preciseness of quantum systems suggest a very controllable context for their correct descriptive construct.

(this is a discussion which is fundamental to education, in regard to its relation to:
Creativity (the trait which best characterizes human life),
Knowledge, (knowledge is developed by means of equal free-inquiry),
Freedom (the freedom to inquire at an elementary level of language, and the freedom to use knowledge and create [gifts for the world]), and
Equality (we are all equal creators))



Abstract

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.


Introduction to the Abstract

If there are statements... , upon which this discussion of a new math context depend... ,
which people with expert status will be sure are wrong... ., then this belief is related to the context and interpretations of a description which the expert person in opposition is following, eg those who followed Ptolemy would surely find the ideas of Copernicus completely wrong... ., but according to the media we live in an age of science, where physical theories are measurably verified... , but in fact the theory of Ptolemy was also measurably verified... , but Ptolemy was in an incorrect context... ., but today's physics theories are also clearly in an incorrect context
... {since it is not possible to precisely describe the properties of general quantum systems using the laws of either quantum physics or particle physics... , and the candidates for the theories of everything describe much less that quantum physics (which can describe the H-atom) and particle physics (which can be used to describe the patterns of particle-collisions in particle-accelerators)... , can describe,... , but nonetheless... }
... , since fundamental physical systems, eg nuclei, general atoms, molecules, molecular shape, crystals and the properties of the stable solar system are stable and precisely discrete and observable [which means that these stable systems are continually emerging into their uniform stable structures, and thus they are emerging in an interaction context (ie a differential equation) which is linear, solvable, and controllable] yet the general types of the fundamental material systems, just listed, remain indescribable based on modern physical law (2012).
However, there is a new descriptive relation... related to the patterns of a "physical-form which has been observed" (or to the set of observed system properties, and whose properties {or their patterns} one is trying to describe)... . which is being identified in this discussion (in this article).

The new descriptive structure allows one to describe, within a new context, ie a new containment set, the stable properties of many-component systems, composed of charge and/or mass, whose geometry of interaction, in 3-spatial-dimensions is spherically symmetric... ,
[but {in the new descriptive context} the micro-structure of material systems is discrete hyperbolic shapes, where, in turn, these shapes and the angles between the system's components are influenced by the angular relations which exist between the Weyl chambers to maximal tori of the fiber group and the relation of these angular properties to angular changes (or angular shifts) in regard to the "toral sub-geometries" of the discrete hyperbolic shapes, ie causing an angular bending between the hole-structure of a discrete hyperbolic shape],
... , but the new descriptive context also describes how control can be affected over the subsystems of both living and the mental subsystems of living systems.

Whereas in the currently accepted math constructs of the material interactions between mass or charge these interactions are assumed to be (absolutely) spherically symmetric in 3-space, and in all other dimensions. This assumption, concerning an unmeasured property (of material interactions being spherically symmetric in higher-dimensions), is used as a proof that higher dimensions do not exist, where it is assumed that the higher-dimensional structure would form a continuum and that all material interactions would be spherically symmetric, since there are not observed higher powers than the inverse square powers, when materials interact, thus proving the non-existence of higher-dimensions. This is a context built on assumptions in a similar manner as Ptolemy's system was based on assumptions.

By describing both the fundamental building blocks which fit into the new context, as well as the properties of global systems, such as the solar system as well as the spectral structures of living systems, one is educated about both the details and assumptions of the descriptive construct as well as the large context and range of applicability in the new context of containment, ie both its details and its distant reach are identified in close juxtaposition.



Some principles which will likely be contested by the authorities in the (unnecessarily competitive) world of academic intellectualism:
-1. The physical systems which are being described are very stable and precise systems... , nuclei, general atoms, molecules and their shapes, crystals, and solar system stability, [as well as dark matter, and universe expansion], but also life and mind... , and this implies that the interaction structure, between the components which make-up these systems (ie the system's differential equations), are: linear, metric-invariant (for a fixed metric, with non-positive constant curvature), and separable, ie solvable and geometric, and thus these are controllable or determinable systems, and furthermore, their being modeled geometrically allows them to be both understandable (they can be pictured and measured {as opposed to "waiting for random events as a model of measuring" }) and practically useful.
That is, if the material interaction processes were non-linear and indefinably random then the physical properties of the resulting material systems would not be so uniform.
The uniform properties of so many fundamental systems implies that their interactions are in the realm of determinable mathematics, not in the realm of non-linear and indefinably random math structures.

How is this possible?
What context would allow for small material systems (quantum systems) to be described in a determinable manner?

This is possible within a context in which the very stable discrete hyperbolic shapes as well as the continuous (in shape) discrete Euclidean shapes are placed in a many-dimension context of containment [as well as the considerations of discrete shapes associated to other metric-spaces of various signatures] and these discrete shapes are used to model both material and metric-spaces (and these geometric shapes [as well as their determination of a large but fixed and finite spectral set] also contribute significantly to the "material" interaction process).
Furthermore the state-structure of both material (eg spin) and metric-spaces (eg metric-space states [eg positive and negative time directions associated to different subsets of complex coordinates]) requires that this description also be unitary.

One cannot simply build on the work of authorities, eg in quantum field theory this process (of honoring authority and its associated dogma and fixed interpretations) has led to a model of material interactions based on random point-particle-spectral (collision) interactions interpreted geometrically as particle-collisions which exist in a math structure where quantitative consistency and continuous geometry do not exist, yet these particle-collisions are modeled as if possessing a continuous geometry of object and path and identifiable collision, and this absurdity has led to other absurdities which show the plasticity of infinite-sets, indefinable randomness, and non-linearity, and a descriptive structure which logically proceeds from an assumed (but indefinable) global structure (of a function) to a local spectral measurement, ie a local particle-spectral (random) event in a 3-dimensional spatial subspace.
These assumptions have not worked, they are neither accurate for general systems, nor are they practically useful.

Mathematicians like to claim that they would consider any math pattern, ie in the math context of quantities and shapes, but this is apparently not true as they only consider peer reviewed literature, ie following authority and tradition within which a narrowly defined contest is constructed... , and apparently unknown to the professionals, their knowledge is channeled towards particular industrial (or big business) interests concerning engineering and production or about calculating risk (where it is clearly seen that top university PhD math and science people make calculations of risks which are clearly invalid and failed, the math of university PhD's are about math patterns which are not related to the world of our experience, rather the math experts are creating math patterns which describe a "world" which is an illusion, they are not math patterns which are practically useful, though they may be measurably verified, as Ptolemy's ideas were also measurably verified).

Some of the contested statements will be:

0. Stable physical systems must be based on a geometric descriptive construct (to be both measurable and usable).

1. Stable structures must be stable-geometric, ie based on the discrete hyperbolic shapes (and their associated discrete Euclidean shapes, which are related to material interactions as well as inertia).

2. Math structures must be quantitatively stable (metric-invariant, for metric-functions with constant coefficients), thus the only valid descriptions (quantitatively consistent descriptions) are diagonal, ie parallelizable and orthogonal at each point of the system's global geometry.

Note:
Only a linear model of either the derivative or of an equation which is defined in terms of derivatives acting on functions, ie such an equation must be linear, in order for these math constructs to be quantitatively consistent.
This means that "the derivative represented as a connection" identifies a non-linear context which is quantitatively inconsistent, since the local measures between the domain set and the set of function values do not relate the quantities in the domain set to the quantities which represent a function's values by a simple, diagonal, linear (matrix) relation.

3. Math structures must be quantitatively consistent, or linear and finite, and where the global structure is to be determined (from a local structure).

The new math context can give correct answers, and it can give consistent models of the organization of very complicated systems, eg living systems and their associated mental systems.

4. The fiber bundle becomes a part of the model (even more so than in quantum physics), not simply a "book-keeping set" used for the description of the properties of an external system, or an external math structure for a system, which is assumed to be non-linear, ie connections can be represented within "principle fiber bundles," and "function spaces" require that fundamental measurable properties be non-commutative, though quantum law wishes for commutativity. Thus the system [represented as a function space] {or represented as its set of physical operators, ie its measurable properties} is also non-linear.

Note: The commuting property for the "complete set of commuting Hermitan operators" which is supposed to define linear quantum systems has never been realizable for general quantum systems.

Thus, the linear superposition of spectral functions as a means by which to define a wave-function for a (general) quantum system "has been a wish" rather than a (true) principle of the laws of quantum physics.

Note:
The laws for a quantum system are (essentially):
(1) represented as a complex-number function space (but indefinably random), which is
(2) acted on by linear unitary operators (or in the context of local Lie algebras, function spaces are acted on as Hermitian operators) which represent physically measurable properties of the quantum system, (mostly the energy [or wave] operator and its decomposition into other Hermitian operators) and which is
(3) unitary invariant, ie conserves energy, and a
(4) linear quantum system is realized in regard to finding its "complete set of commuting Hermitan operators," etc.

Quantum physics is formally linear but in the context of
(1) a function space and its dual space being related to non-commutativity, ie the uncertainty principle, and
(2) in general quantum systems composed of 5 or more charged components, there does not exist simple symmetries, such as spherical symmetry, and
subsequently the non-commuting pairs of properties (operators and their associated function spaces) become a permanent, non-linear, fixture of the descriptive construct.
A non-commutative structure for a differential equation (for a set of operators) can (does) imply curvature so the system's differential equations are not diagonal, rather (for example) upper tri-angular, and then the matrix can have many solutions, the system is under-determined, and there can be bifurcations of value,
or
The differential equation can be made diagonal, but the coordinated in which it is diagonal are not quantitatively consistent with the metric-function, the local coordinate structure is different for the two cases of diagonalization, ie the metric-function and the differential equation, where again this causes quantitative inconsistency.
(3) the charged components identify a set of independent singular points, which seem to be able to arrange themselves so as to create a neutral system, ie no singularities,
but singularities lead to divergences.
Why do some systems composed of charged components have dipole and multi-pole (charge distribution) structure, and some are neutral?

The endless problems with math descriptions based on indefinable randomness (see below for definition of indefinable randomness)

A set of commuting (linear) Hermitian operators must be selected (found) in order to identify a quantum system's measurable properties, and so as to represent the system as a linear superposition of its spectral functions, but for general quantum systems, eg spherically symmetric quantum systems with five or more particle-components, this set of commuting operators cannot be found (has not yet been found).
They cannot be found for: nuclei, general atoms, molecules, nor for crystals.
Thus the set of a quantum system's measurable properties, eg measurable properties and their dual operators, together define a non-linear or curved space associated to the Lie algebra of operators, and their relation to a Lie group of operators which act on the infinite-dimension function space.
The description of a quantum system (in regard to its measurable properties, ie Hermitian operators) which has a linear structure (where the set of physical operators commute) is rare. Thus, there is an attempt to associate an approximate (linear) spectral structure to a non-linear context [non-commutative operator context] (for measuring [or modeling] quantum properties).

Problems with spherical symmetry

It should be noted that the singularities of the 1/r potential energy of a spherically symmetric structure for material interactions cause solutions to wave-equations to diverge.
Even for the H-atom the solution to the radial equation diverges, and thus it is truncated (the solution series cut-off) so as to fit data.
Furthermore, the singularity of the non-linear spherical geometry (associated to charged particles) is the focus for manipulating ways in which a spectral set, associated to the 1/r singularities of the system's various charges, can be manipulated to fit data.

Local symmetries of particle-physics

The (complex-number) function solutions of quantum wave-equations are energy invariant and thus they possess unitary invariance in regard to a global symmetry (ie global invariance).
On the other hand, the eigenvalues of the global functions are separated into (local) spin-eigenvalue-states, and thus in the model of "perturbation of wave-function spectral functions" by means of local eigenvalue-particle-state properties the global invariance (of the wave-function) is apparently assumed to be reduced to a local wave-function structure for the quantum system's eigenvalue structure.
These local particle-state invariances, which model material interactions as particle-collisions, which exist within a probabilistic construct (which should not allow the geometry of a particle-collision), which also models the random local spectral-particle event, ie the point about which the quantum description is trying to model, ie "how data is to be fit into a math construct."
Yet, how is this math construct of local small adjustments (due to particle-collisions in a context of indefinable randomness) to be related to the non-local properties of quantum systems, ie to the global wave-function properties of quantum systems?

The local gauge invariance ie how the wave-function is invariant to a local phase-transitions, is supposedly central to the construct of the math structure of physical law (based on the particle-collision model of material interactions). But a global wave-function needs a global relation to its phase.
These local symmetries, in regard to wave-phase, are related to the property of mass (and charge and field-particles etc) in the particle-collision models.

Thus, the question is about, "though the idea of gauge invariance" is supposed to be about the determination of the type of particle-collision interaction... ,
since local gauge-invariance requires that the local particle-state matrix be introduced (as a connection term) so as to assure gauge-invariance, and this allows the local phase term (due to differentiation of products of functions) to be a part of the differential equation of the material interaction-system, so as to define local gauge-invariance,
... , nonetheless, "how does a quantum system still have the property of non-localness?"
However, because local symmetry is about "how the type (or which type) of particle-collision interaction is a part of the differential equation of the material interaction-system," this results in a discussion (concerning these differential equations) about the multiplicative constants which are part of the description, eg q (charge), and photons, and m (mass) and g (gluons), and "D" dark-matter, etc (as elementary particle keep getting added to the elementary event space, ie indefinable randomness).
That is, the relation in regard to "How force-fields are related to charge and mass, and g, etc".

Symmetry breaking is about pulling the energy spectra, ie the mass, of particle-collisions away from zero. That is, the graphs of quantum properties (graphs of wave-functions) are given in the scalar values of energy. That is, the perturbation of a wave-function is described in the context of the wave-function's energy. That is, charged and strong-force interactions take place about the zero energy point, and mass is about spectral (energy) cut-offs associated to the manifestation of mass.
This type of a model is only of interest to bomb engineering and the manipulation of measurable parameters in regard to finding probabilities of collisions in particle-collisions eg mass and charge related parameters, the only measurable properties in the bubble chambers of the particle-accelerators (?).

Whereas, mass (and charge) are a part of the spectral-energy properties of particle-physics, and these are about local symmetries of wave-phase, but the spectral-energy properties of the wave-function must be about global invariance of wave-phase, so that non-local properties can manifest (as a property of the wave-function).
Are these un-resolvable properties of quantum systems, or are they defined away, by claiming that "away from the local particle-collision the average structure of the harmonic wave-function, the non-local wave-function re-manifests itself."
But how is this done?
Though spin can be placed into the global structure of a wave-equation, the local particle-state structure cannot be placed in an energy-invariant (non-local) wave-equation, because particle-physics, eg including renormalization, is all about the loss of a geometric continuum for space's small structure, yet the continuum of a particle-collision is maintained within the non-geometric context of particle-states.

A local particle spectral-value event, is mediated by a math construct of local, non-linear connection terms (in the system's differential equation, or in the system's operator space) which preserve local wave-phase invariance, where the spectra is about mass (since mass equals energy), but mass is now a construct which is only global in a different context, than in the local context in which the material-interaction (particle-collision) model is defining mass (and charge etc). Whereas a global wave-function has a global energy-invariant construct, which has a different relation to its operator space, than does the local wave-phase adjustments to a waves energy.

The problem with such a description is that, the global wave-function for general quantum systems cannot be found, thus the local adjustments are meaningless.
Local adjustments have no meaning in an indefinably random context, a context which has come to be about data-fitting, but a context which seems to have no validity.
The lack of validity, for the math structure of indefinable randomness, is also shown by the failures of the risk calculations for business-risk, where the calculation of business-risks are also defined in a context of indefinable randomness, and are used to try to fit data, and thus, to identify risk within the structure of function spaces, and their operator (or differential equation) fiber bundles.

The failure of indefinable randomness is clear since there are no valid calculations for the spectra of general quantum systems, eg nuclei or atoms with five or more components, and the fact that the descriptive context of quantum physics has not led to any (robust) form of technical development, yet the stable spectral structures of general quantum systems suggest a controllable context for the descriptions of general quantum systems.

The obsession (in regard to social interest, of the military industry) with particle-collisions is about the fact that university physics has been turned into nuclear weapons engineering (for the military industry), and the way by which Asperger's syndrome, ie autism, is sought-out (by those who guide social institutions) and exploited (obsessive and narrow and competitive, ie the measure of intelligence etc), so as to serve the interests of the military industry (essentially, the only industry in the US which still makes things).

The new context claims that, there are unitary symmetries, but it provides a stable, many-dimensional, geometric context, which is also a stable finite spectral context, within which to interpret these unitary symmetries, and it is a context which is consistent with classical physics, but based on stable geometry. That is, it is not a description based on manipulating, in a very artificial way, the quantitative structures of function spaces and an associated set of fanciful convergences, and thus it is a math structure which is stable, as the properties of quantum systems are stable, and it is quantitatively consistent, ie the measuring structure means something, and it is geometric so one can picture the descriptive context so as to facilitate "practical" creativity, where practical is newly defined in terms of higher-dimensional stable geometries and their stable spectral properties.

Apparently

The non-linear structure, it is believed, allows for a greater variety of ways in which data can be fit with cut-offs and bounds on measurable values. It is believed that this process, about "how to manipulate the difficult context of (spectral) analysis for a set of non-linear operators" is considered OK by the experts, because the containing math set is "very big" and the non-linear and indefinably random patterns are sensitive to small changes in a system's bounding conditions either spectral or spatial. But the description is now non-linear, and thus attempting a "linear superposition" of spectral functions results in "new spectral properties (different from the linear spectra)" for the (new) sum. That is, if one changes a spectral set of functions, in the context of non-linearity [or non-commutative operator context] then the spectra associated to that (same) set of functions can be radically changed.
It is a process of going from the frying pan into the fire.

The new descriptive context

In the new descriptive construct the fiber group (of a principle fiber bundle) has an active part of the interaction and the new description's base space is also related directly to a geometry composed of various dimensions, because the fiber group (of a classical Lie group) has a spectral storage structure, ie a set of maximal tori, and the discrete hyperbolic shapes, the geometry upon which the descriptive context is based, are stable spectral structures, thus resonances can exist between the base space and the maximal tori of the fiber group.

In the current description of quantum physics the system is the function space while its physical properties are operators in its fiber group (there is a division between the system and its measurable properties)

In quantum physics the space of operators is a (book-keeping) mechanism for manipulating the structure of a function space around singularities and other critical points of a system's operator structure (or differential equation structure), with a focus on fitting data.
However, if one considers the physical systems themselves, these stable spectral quantum systems would not have the same spectral properties so consistently as they do have (from the various interactions in which these stable systems emerge) if the physical structure of these interactions was the same as the descriptive structure, which is so non-linear.
It should be noted that a non-linear math structure is quantitatively inconsistent (and logically compromised), and thus from such an inconsistent set of measurable properties an "infinite spectral variety" would emerge as the measurable properties for such systems, ie from non-linear and indefinably random structures of the material interactions there would not be uniformly identifiable quantum systems (based on their stable and uniform spectra).

High dimension context of the new description within which the set structure could be finite

Since, in the new descriptive structure, stable existence depends on a well defined spectra and the bounded spectral-set contained within an 11-dimensional hyperbolic metric-space...
[resulting from the discrete hyperbolic shapes contained within this 11-dimensional hyperbolic metric-space (where the 11-dimensional hyperbolic metric-space models the containing metric-space)]
... , would be finite (or can be finite).
This is the basis for a (new) description based on a finite spectral set (though it is quite a large finite spectral set).

The nature of descriptive knowledge (for language which are precisely defined)

The point of Godel's incompleteness theorem is that the precise language of math (or of any specifically defined language) needs to be considered in terms of its (continual) re-organization at the level of assumption, context, interpretation, use, set-containment, and definition.
Tradition and authority are not valid determinants in regard to the validity of mathematics and science, rather these properties (of tradition and authority) define social class, and they define contests associated to using ideas in very narrow ways (where the contests are used to determine those who qualify for the upper social-intellectual class).

The current focus within the subject of math does not account for fundamental issues:

1. Quantitative consistency (eg it uses non-linearity as well as indefinable randomness as valid math constructs which are quantitatively inconsistent).

2. Set size (axiom of choice allows for "'too big' of a set," for quantities to be consistently defined). Sets which are "too big" cannot remain logically consistent.

3. Logical consistency which results both from sets which are "too big" as well as from quantitative inconsistency.

The data obtained from the local spectral-particle observations of general quantum systems

3b. There is another weird attribute, associated to the idea of logical consistency, which appears when contrasting the math of classical physics with the math techniques of quantum physics, where classical descriptions which are built upon local so as to determine a system's global properties, ie the laws of classical physics (Newton's laws), uses local model of linear measures (derivatives) to define differential equations, and the solutions of linear, separable differential equations (defined within Euclidean space) provide global information about the system.
Whereas "In the practice of quantum physics" the description assumes a global structure, ie it assumes a function space associated to a quantum system, which, in turn, is related to the local measuring of a system's spectral properties. The spectral functions identify locally-determined (or observed) properties.

The descriptions of micro-material properties seem to require that there be a fundamental change of assumptions wherein a stable global context is assumed.
When one is trying to find a measurable attribute which exists then it seems better to assume that such a description of measurable attributes be a result of a local linear measurements and their associated differential equations whose form is guided by an assumption of some stable bounding structure (or construct) [in classical physics this would be the material geometry surrounding a system's material component], but for micro-material properties "Is this stable structure a function-space along with a set of measurable properties (or set of Hermitian operators)?" or "Is it a stable geometry, eg a set of discrete hyperbolic shapes?"
It is a set of discrete hyperbolic shapes.

However, local, random particle-spectral events in space and time, ie the data one is trying to fit, and the set of global properties, ie the system's set of operators applied to the system's function space as well as the function-space itself, cannot be made (or used) to match the observed local spectra (in regard to some general quantum system).
The local spectral measures cannot be made to match-up with (or to fit) a general quantum system's calculated spectra which identify local particle-spectral measurements (or observations) [a general quantum system is assumed to be random, but the measurable nature of its randomness (its spectra and its spectral functions) is not determinable by physical law, ie it is an "indefinably random" system (see below, too).

4. Quantitative stability, the need for a descriptive and mathematical context of quantitative stability.

5. Accepting that creativity as well as observing a system's stable properties demand a stable geometric basis within which measuring is both reliable and useful for practical creativity.

The crux of the matter, in regard to basing descriptive language on tradition and authority

Simply because a math technique, or math process, or a math construct, can be made to fit (a set of cherry-picked) data does not mean that such a math construct is valid, the construct also needs to have a wide range of applicability, so as to (also) provide a framework within which practical creative development can take place.
Ptolemy's models fit data quite well.

The need for a descriptive structure to be widely applicable

The best example of wide applicability and great usefulness is classical physics, in regard to the linear, solvable physical systems (of a classical system's differential equations).
But similar context of linear, separable, and metric-invariant structure, within which the math techniques of quantum physics are applied, are not valid for systems with spherical symmetry and more than five-components composing the system (or geometric symmetries as properties of the quantum system's potential energy term), where quantum physics is framed in the context of function spaces and probabilities associated to spectral values, but the spectral values (that which one is trying to calculate) are local properties [not global properties]).

The failings of function spaces (as a basis for physical description)

Why is the function space context (for differential equations) not as reliable of a math construct for quantum systems as is the geometric context for the classical context?
The interpretation of materialism in classical physics means relatively local stable geometric structure while for quantum physics materialism means spectral discreteness, randomness, and mysterious spectral cut-offs which exist for quantum systems, cut-offs and spectral approximations which are needed to solve the differential equations.

These spectral cut-offs, which are needed to solve for the spectral structure of quantum wave-systems, exist for (all the) quantum systems (and are a characteristic of function space math-techniques), from the H-atom, ie the truncation of the radial equation's infinite series (diverging) solution,... .. to particle-physics, ie re-normalization due to (or based upon the idea of) "symmetry breaking," where the breaks in (mass-less) symmetry (the existence of mass) are supposed to identify an underlying spectral structure (spectral cut-offs, or spectral approximations) about which quantum systems are (supposed to eventually be) organized.
This might be true, except for the fact that symmetry breaking (and particle-physics) and the identification of elementary particles is only related to perturbations of spectral properties of quantum systems, where (it is assumed) the main spectral approximations have already been found from physical considerations (a complete set of commuting Hermitian operators)... ,
[which (without a sufficient amount of highly ordered geometric structure) are not sufficient constructs (ie sets of measurable properties [Hermitian operators] have not been found for quantum systems in general) by which to find a close approximation to a general quantum system's spectral properties]
... , where these (assumed to be) well defined approximations to the system's spectra can then be perturbed by using particle-physics,
[furthermore, the process of adjusting the spectra to be related to the singularity of a spherically symmetric energy {or interaction} structure (or set of such singularities) has not worked].
The lack of geometric structure within quantum description means that the function space techniques do not work for quantum systems which are (assumed to be) spherical symmetric but which have five or more components (particles), ie a nuclei and a set of electrons.
However, in such a system the spherical symmetry is lost, and instead the system has a set of 1/r singularities associated to each charged component.

The non-linear structure of function spaces

The reason for such a break-down of these math techniques should be clear (elementary) it is because function space methods are non-linear (and non-linear means quantitatively inconsistent), a property for differential equations which emerges (when solved by means of function spaces and their associated "spectral cut-off [or spectral approximation]" techniques) from the uncertainty principle related to a function space and its dual space [and a probability density-function's relation to standard deviations (or variances) {where the spectral functions of the function space are probability density-functions}].
The non-communitivity of, say, "momentum and position" [which identify dual sets of functions (or dual operators) under Fourier transformations] means that these properties define a non-linear structure in the Lie group of unitary operators (with its Hermitain Lie algebra) upon which (or within which) the wave-functions are defined to be invariant. However, these properties (eg momentum and position) are always physically present (it is assumed) [certainly for a system which does not possess spherical symmetry].


Furthermore, the spectral functions possess geometries which are not linearly compatible with one another, ie they are different shapes. Momentum functions and position functions have shapes which are incompatible with commutativity.

Solvability and commutativity

Linearity requires consistently parallel geometries, eg congruent or similar shapes, but similar shapes are not metric-invariant shapes (so similar shapes might exist in different dimensional levels so that the conformal structures defined by physical constants can intervene in the stable math relations which need to be defined for quantitative consistency).
That is, the physical properties, which are assumed to always be related to a measuring process, are quantitatively incompatible as "measured properties" and such physical properties would only be consistent in a commutative, linear description, but such descriptions cannot be found, (perhaps, mostly due to singularities associated to spherically symmetric of charged material interactions in systems with five or more charged components).
Thus, the quantum structures are indefinably random, since (1) the spectra of general quantum systems cannot be approximated in a valid manner and (2) the linear theory is non-linear in its formulation "based on sets of operators which identify physical properties," ie the math process is non-linear in regard to general operators acting on function spaces.

Indefinable randomness (defined)

Indefinable randomness is about elementary event spaces (the sets upon which probabilities are defined) are composed of: unstable events, un-define-able (or incalculable) events, the elementary event space is not fully identified, ie new events are added to the elementary event space at will... , eg dark-energy particles added to the set of elementary-particles etc... , where each (or any) of these properties for elementary-events makes these event spaces uncountable, ie one cannot rely on the counting process to determine probabilities. Thus, the probabilities cannot be determined for such elementary events spaces.
That is, even though it might be claimed that certain types of events are distinguishable, and thus countable (so as to be related to probability), however, if these events actually fit into a space which is "indefinably random" then one cannot rely on the probabilities which are determined from any counting process applied to such a probability space.

Where indefinable randomness is defined to be a set of elementary events of a random process which cannot be counted in a consistent manner, and thus it is not a quantitatively consistent structure, ie counting is not reliable within such a set of distinguishable (but unstable and undefined) elementary events, ie the structure is not a valid math structure.
Measuring (or defining measurements) on (or in) such a math structure is neither reliable nor conceptually useful at the level of practical creativity.

Geometry vs. sets of Hermitian operators (and their dual structure [which result form Fourier transforms])
eg The "position vs. motion" dual pair

The problem seems to be about material geometry being limited in its range, and material geometry deals with a finite range (or context) of locality (ie consistently relatable to local linear measures) (closely associated to the idea of closed and bounded (in a metric-space)), ie the system is (thought to be) a bounded object (when one considers a physical system in a geometric context).

The "too big" set structure of function spaces

The implicit infinity of function spaces, ie infinite dimensional, is not mathematically manageable in a quantitatively consistent manner (or in a consistent set containment structure) so that quantum descriptions cannot be relatable to a context of relatively stable local geometric properties, or local geometric stability, ie the quantum-foam of particle-physics.
Function space methods are non-linear due both to the uncertainty principle associated to dual spaces, and to the general context in which the operators are applied (or used), ie the geometric symmetries are not known for systems composed of five or more charged "particles."
Furthermore, the geometries of the spectral functions in function spaces which are discretely distinguished from one another, and thus the shapes of the different spectral functions cannot (in general) be continuously related to either shape or to topology ie holes in the space. Their local vector field geometries do not commute.

Classical physics assumes materialism and local linear measuring contained within a geometric and continuous context. However, classical inertial properties are non-local, while the properties of charged material possess "positive" and "negative" time-states associated to such physical systems.
Quantum physics assumes materialism, non-localness (ie action-at-a-distance), and a fundamental indefinable randomness which is contained in a non-linear context due to either the uncertainty principle (of a function space and its dual space) or the non-linear and indefinably random structure (of quantum description) is due to the indefinably random and non-linear model of particle-collision models of material interactions, as defined within quantum physics.

Preface

The new (alternative) model for mathematical description of existence is based on material and space both being composed of stable "cubical" simplexes (and their associated moded-out shapes) for both discrete Euclidean and (the very stable) discrete hyperbolic shapes, where these cubical simplexes are placed in a many-dimensional context, and where the properties of material and space are separated from one another by one dimensional level, ie n-dimensional space contains (n-1)-dimensional materials.
This simple math structure is needed so as to have mathematical stability, and to be able to describe over a wide range of general physical systems the observed stable set of physical properties associated to so many fundamental physical systems. Stable spectral-orbital properties are possessed by fundamental physical systems such as: nuclei, general atoms, molecules, crystals, as well as the mysterious stability of the solar system, which are all systems characterized by their stable spectral-orbital properties.
The stable basis for our experience, exists within a geometric context, where geometric descriptions can be used in practical ways when these descriptions are associated to measuring.
This math structure identifies a linear, metric-invariant (for fixed metric-functions with constant coefficients) math context, and it also provides a context for geometry which is based on parallelizable and orthogonal vector fields (which are associated to these simple geometries [of the cubical simplexes] which now model material systems as well as modeling the containment space for these material systems).
This metric-space structure has a clear relation to the classical Lie group isometries as well as the unitary fiber groups defined over the above mentioned metric-spaces and Hermitian-spaces, where complex coordinates are needed to describe the state-structure of both material (spin properties etc) and metric-spaces (which now possess "metric-space states" which are an important part of the dynamic interaction process of material interactions) where, in complex space, the real isometries are divided into the two subsets (real and pure imaginary) associated to positive and negative time directions, which represent opposite metric-space states of dynamical systems separated into subsets of the complex coordinates, thus leading to a unitary fiber group, and a subsequent mixing of these metric-space states.
The new context of containment is restricted to the spaces of non-positive constant curvature, ie Euclidean and hyperbolic spaces, as well as spaces of various divisions of spatial and temporal subspaces associated to a metric-function's signature (but not spherical space, which is a non-linear space).
However, the model for material interactions in 3-space (or in 3-dimensional spatial subspaces) is spherically symmetric, until the interaction approaches the size of the interacting material components, which have the geometry of discrete hyperbolic shapes.
The geometry of higher-dimensional interaction structures depend on the geometry of the fiber group, ie material interaction are not universally spherically symmetric. For example, SO(3) has the shape of a 3-sphere, but SO(4) has the shape of two 3-spheres, ie the inertia of the material interactions in hyperbolic 4-space is not spherically symmetric.

This math structure is both simple and very constrained, except for its higher-dimensional structure, which creates a context of great variety.

The higher-dimensional structure allows for descriptions of coherent higher-dimensional models of living systems which can control lower dimensions of their own set composition, and thus they can control chemistry, and such systems can also coordinate the functioning of the organs which compose their own living system.

Minds are also describable in such a context.

Rather than using infinity to ensure that the gaps of measuring structures are filled, might it be better "for the mathematical context to be stable" and quantitatively and logically and consistent, so that the descriptive structure (the stable geometric construct) is contained within higher-dimensions, and within which a finite spectral set can be defined (which is contained within a higher-dimensional [hyperbolic 11-dimensional metric-space] containment set), where the fixed finite spectral set is based on the stable, bounded, discrete hyperbolic shapes which compose the structure of the metric-spaces of the various dimensions.
That is, "gaps" (or incompleteness) in the quantitative structure can exist as long as the spectral set structure, and its associated stable geometric structure, is maintained within the over-all high-dimension containment set.

The new context is a stable geometric context so this will allow such a construct to be applicable to practical technical development, but it (also) seems to signal that human creativity is better focused outside of the realm of materialism, based on new higher-dimensional and "global" descriptions of life.

The new material interaction structure accounts for both
(1) the apparent indefinable randomness of small components in space, as well as
(2) the hidden properties of the higher-dimensions which compose existence, where physical constants also help hide the properties of higher-dimensional space and its material components.

How to build stable sets?

Inclusion in a set also provides the elements of the set all the properties which have already been attributed to the set.
But should the set be built from the ground upward to greater complication or should the set be based on known stable structures and set containment of elements be more about stability and consistency in regard to the models of physical measuring.
It could be noted that functions "model simultaneously" both measuring and set containment.
This is the main issue of quantum physics which assumes global structures and descends to the local (random particle-spectral measures), while the interpretation of the classical differential equation assumes a local measuring structure and finds a global relation.
Should the structure of math also be re-organized from the stable down to the measurable.
But is geometry a better vehicles to define stability rather than function spaces, and/or their operators?

That is, is the stable math construct to remain function spaces and/or their sets of operators (local Hermitian operators of physically measurable properties)?
or
Is the stable construct about which the math descriptions of physical existence are to "pivot (or revolve) around" the very stable discrete hyperbolic shapes, in a many-dimensional context, up to a dimension-11 hyperbolic metric-space (as identified by D Coxeter), (where it should be noted that hyperbolic 3-space is equivalent to space-time [one-to-one and onto]).

The Abstract:

Can a geometrically stable and spectrally finite math construct... ,
which is contained in a many-dimensional context, associated to an open-closed, but non-local (ie action-at-a-distance applies), set of bounding stable discrete hyperbolic shapes wherein adjacent dimensional levels are geometrically significant in regard to both containment and for a set of (partially) bounded dynamic-geometric math processes, which are defined within a bounding (containment set) structure of discrete hyperbolic, Euclidean, and unitary subgroups (or simply, linear discrete shapes),
... , provide a better math structure within which [or to be used] to frame the descriptions of observed physical properties?
Yes.

Rather than using infinity, might it be better "for the mathematical context to be stable" and quantitatively and logically and consistent, so that the descriptive structure is contained within higher-dimensions, wherein measurement is consistent with geometric measures of stable geometric shapes, and within which a finite spectral set can be defined (which is contained within a higher-dimensional [hyperbolic 11-dimensional metric-space] containment set), where the fixed finite spectral set is based on the stable, bounded, discrete hyperbolic shapes which compose the structure of the metric-spaces of the various dimensions. The geometries of the different dimensions have similar or consistent discrete structures, ie the geometries are compatible with one another, ie geometries can con-formally fit into one another, or as "free stable geometric structures" they can be resonant with a subset of a fixed, finite geometric-spectral containment-set structure.
When a metric-space modeled as a discrete hyperbolic shape is placed in (or contained within) an adjacent higher-dimensional metric-space then the original metric-space becomes a closed shape (which is easiest to think of as being a closed bounded shape in the higher dimensional metric-space within which it is contained). (This would be a model of a "free geometry" contained within a metric-space of some given dimension.)
That is, each dimensional level can contain within itself a variety of lower dimensional (lower than the dimension of the containing set itself) stable discrete hyperbolic shapes as well as a set of associated resonating Euclidean shapes, (which are resonating to a discrete hyperbolic geometries dominant {or occupied} energy structure [of the hyperbolic shapes various hole-spectral structures).
However, these stable lower dimensional shapes [of dimension (n-1) or less, when they are contained within a metric-space of some given dimension, n) can always be placed within (or can occupy) a higher-dimensional shape, up to shapes of dimension-(n-1). Thus, for an n-dimensional containment metric-space (which also has a shape) the main focus of interactions would be in regard to the (n-1)-dimensional shapes (it contains) which (naturally) interact with one another, by means of action-at-a-distance discrete Euclidean shapes, which fit into the geometric interaction construct in a geometrically consistent manner.

Note:

The interaction structure accounts for both random properties, as well as for the hidden properties of higher dimensions, (metric-spaces with spatial subspaces greater than three, seem to be hidden from our senses) where the higher dimensional spaces are hidden by both the interacting structure and by the interacting system's size which is (partly) determined by the physical constants (which are defined between the different dimensional levels, where physical constants determine the sizes of the interacting shapes (or systems)) within a metric-space.
It needs to be noted that both material and metric-spaces are discrete hyperbolic shapes, but with different dimensions, ie metric-spaces contain the discrete shapes of material systems.

The central focus of this new descriptive construct is spectra, as is also the case of function space descriptions

Since the new theory is a finite spectral theory, based on stable discrete (hyperbolic) geometries, it is a description which depends on spectral cut-offs (or spectral approximations) which are associated to physical constants.
In the new description Bosons are Euclidean structures attached to "infinite extent" discrete hyperbolic shapes, which are a part of the descriptive structure of a "material" system (the dimension of these Bosons is one less than the dimension of the metric-space which contains the given infinite-extent discrete hyperbolic shape), (or perhaps the same dimension as the dimension of the metric-space which contains the given infinite-extent discrete hyperbolic shape)...

... . However, there exist "interaction structures" which are higher-dimensional, and the spectral geometry of hyperbolic space (ie the discrete hyperbolic shapes) which are 6-dimensional and beyond... ,
(Note: up to 10-dimensional hyperbolic space, but (apparently, according to D Coxeter) 11-dimensional discrete hyperbolic shapes do not exist)
... , are determined by infinite-extent shapes, and the spectra of such infinite-extent discrete hyperbolic shapes depends either on intent or on the lower dimension compact space-material (discrete) geometric-spectral structures, of five-dimensions or less, which are a finite set (and contained within an over-all containing, hyperbolic, 11-dimensional metric-space), and any of the spectra (of the appropriate dimension) of this "finite spectral set" of the over-all containing space can be carried (or defined) upon an infinite extent shape.

Old and new

The new theory is not at odds with the observed properties of particle-physics it is unitary invariant and focuses of spectra (other than being at odds with the fact that particle-physics cannot be used describe any stable physical systems and its descriptive context is practically useless), but it is geometric and the construct of its interaction processes are simpler, and it provides a better defined context in which to identify "what the... 'mass resonances' or the 'broken symmetries' ... . actually are,"
in a geometric-spectral context.

Conjecture: A scalar boson would be a physical constant defined between dimensional levels (of higher dimensions than the spatial dimension defined by materialism) [and/or between different signature metric-spaces, of (say) the same spatial subspace dimension], ie it would define a different material size-scale in regard to significant (or easily observable) interaction forces of material within a certain dimension metric-space [and/or a metric-space which may have "varied" properties for the signature of its metric-function].
In the new description there is no need for scalar Bosons since mass is assumed to be an intrinsic part of the descriptive construct, namely the existence of discrete Euclidean shapes resonating to the sub-toral components of the stable discrete hyperbolic shapes




If physical description has a geometric context and global description can be based on local measures to find global structure then the description can be based on differential equations motivated by geometry where local linear measures allow the description to be quantitatively consistent but being contained within a metric-space the differential equation needs to be metric-invariant where the metric-functions have constant coefficients, but metric 2-forms are symmetric and thus in local coordinates can be found which diagonalize the metric 2-form, thus the linear structure will only be quantitatively consistent with the metric-function in these local coordinates but "are these local coordinates global?" but if the differential equations is separable, ie the geometry upon which it is defined is parallelizable and orthogonal, then the linear structure would be globally (on the "separable" shape) diagonal and thus both consistent with the geometric measures and invertible, and thus solvable.
In the new math context, if the description is determined by the a stable material system's shape, ie a discrete hyperbolic or Euclidean shape, then the metric-invariant, linear differential equation will be quantitatively consistent and solvable.
However, most interaction material systems are not defined on these stable discrete hyperbolic shapes, but rather they are defined between these shapes, and they are mostly non-linear interaction structures. However, if the interaction is defined within a "correct" range of energy and the material (or geometric) components of the interaction comes "close enough" to one another then the system can resonate with the spectra of the (over-all high-dimension) containing space (where the containing space is [also] composed of stable discrete hyperbolic shapes) and the interacting system can "deform" into a stable discrete hyperbolic shape (or adopt its new shape of its being) so as to form a stable system [of hyperbolic and Euclidean shapes together form a connected shape similar to some discrete hyperbolic shape, and thus a subset of this geometry could begin resonating, and, subsequently, instantaneously change to a discrete hyperbolic shape which does [exactly] resonate to the containing space's spectral properties] whose differential equation also changes into a linear, metric-invariant, separable differential equation associated to the newly emerged stable material system, (which emerged from the material interactions).
If these "correct" energy properties or the necessary property of "closeness" are not realized by the interacting system then the material interaction is non-linear and chaotic in its nature.

High-dimension discrete shapes and the relation of interactions to finite spectral sets

Inert material systems interact, and sometimes resonate and subsequently "jump" to a new stable, higher-dimensional material spectral-geometry.

Living systems can (now) be modeled to be some of the odd-dimensional, discrete hyperbolic shapes, which possess an odd-numbers of holes in their shapes, and all spectral flows are occupied within this shape, so as to form an unbalanced charged geometry within a stable hyperbolic shape. Thus, the shape possessing such an unbalanced charged distribution begins to oscillate, and to thus generate its own energy.
This is an elementary model of both life (and of a radioactive nucleus).

How do compact, living, oscillating, higher-dimensional "material" systems interact?

What is the spectral structure of an infinite-extent discrete hyperbolic shape?
Answer: It can take-on the spectral properties of the finite spectral set within which it is contained, ie it is contained in an over-all high-dimension containing space.

Can an infinite-extent living system "jump" between large finite spectral sets, where each spectral set is contained within an over-all high-dimension containing set (space), ie contained within a 10- or 11-dimensional hyperbolic space.

What is the relation between lower dimensional discrete shapes and the higher-dimensional metric-spaces within which they are contained?

How do lower-dimensional shapes relate geometrically to the higher dimensional discrete shapes, where these discrete shapes are based on "cubical" simplexes. Do they move about freely in the cube, or do they get confined to its faces?

Think of a given metric-space as an n-cube (or cubical simplex)... ,
{which can be moded-out to form an intrinsically n-dimensional shape whose extrinsic (n+1)-dimensional shape is contained in an (n+1)-dimensional metric-space (this is true for metric-spaces with non-positive constant curvature), eg the shape of the 2-torus is contained in Euclidean 3-space}
... , then the material system would be an intrinsic (n-1)-dimensional cube in the n-metric-space.
But the n-cube has (n-1)-faces, thus the (n-1)-cube, which defines a material system, can be one of the (n-1)-faces of the metric-space's n-cube structure, or it (the (n-1)-dimensional cube) can be contained in the n-cube so as to interact with another (n-1)-cube which is also contained in the n-cube (model of a metric-space). But the interaction of (n-1)-cubes (in a containing n-cube metric-space) in turn, also forms an (interaction) n-cube whose extrinsic (or moded-out) shape is contained in an (n+1)-dimensional cube. If the conformal constants defined between adjacent dimensional levels are such that only very large extrinsically (n+1)-dimensional shapes interact (by means of a detectable force) then the (n+1)-shape defined by (interacting) (n-1)-components can be stable, and in such a case would appear to move about as (n-1)-components in n-space, ie the n-cube model of the metric-space. Thus the (n-1)-dimensional shapes can either determine the (n-1)-flows of the moded-out n-cube within which it is contained, or they can move "freely" about the n-cube, ie the freely moving (n-1)-components are within the intrinsic (closed) geometry of the n-cube {whether the n-cube is moded-out or not}.
It can be noted that the n-cube (of the given n-dimensional metric-space) is part of the flow geometry of a moded-out (n+1)-cube, modeling an (n+1)-metric-space, when the (n+1)-metric-space is moded-out.

Now if infinite-extent discrete hyperbolic shapes are contained within a closed and bounded shape (ie a shape which is compact, in regard to metric-spaces, and also a moded-out shape) then such infinite-extent shapes would structurally prefer the spectra of the moded-out discrete hyperbolic shape.
Is the difference between light and a neutrino (both of which [in the new descriptive context] can have infinite-extent geometric properties) that of a difference of dimension, where the neutrino can be contained in a compact shape, while the "infinite extent geometric structure of light" is such that it is contained within the closed set structure of the next (or adjacent) dimensional level, and thus appears to be infinite-extent?

The finite spectral set which is built upon discrete hyperbolic shapes depends on the compact shapes of discrete hyperbolic shapes which are only defined up-to and including the 5-dimensional discrete hyperbolic shapes. (This is due to a Theorem by D Coxeter.)
Thus, if an over-all high-dimension containing space is 11-dimensional then the spectra of all the infinite-extent discrete hyperbolic shapes, 6-dimensions and above, would have to carry a given finite spectral set defined by the set's compact discrete hyperbolic shapes of dimension 1, 2, 3, 4, and 5.

Because the infinite extent discrete hyperbolic shape can carry on its geometric structure many different spectral values, since an infinite structure has no natural values for geometric measures, then what can hold such geometric structures to a particular finite spectral set?
or
How can such infinite extent geometries change to new spectral sets?

Can the infinite-extent discrete hyperbolic shapes of an (or contained within an) over-all 11-dimensional containing space, which contains a finite spectral set, "jump" to other, different, 11-dimensional containing spaces, defined by different (finite) spectral sets?
or
Is the structure to be considered the "jumping" of 10-dimensional infinite-extent shapes to other 10-dimensional infinite-extent shapes both of which are contained within an 11-dimensional hyperbolic metric-space?

Is this mostly a question of concern for the living infinite-extent systems in an 11-dimensional containing space, an 11-dimensional space defined by a finite spectral set. That is, for the infinite-extent 7-dimensional oscillating hyperbolic structures, and the infinite-extent 9-dimensional oscillating discrete hyperbolic shapes.
By the interactions of 9-dimensional discrete hyperbolic shapes the other 10-dimensional finite spectral structures can be contacted in the 11-dimensional hyperbolic metric-space, where the 11-dimensional hyperbolic metric-space has no spectral structure of its own. Thus in 11-dimensions all the different high-dimension, 10-dimensional spaces defined by finite spectral sets, can be accessed, and thus transitions between high-dimensional spectral sets is a possibility.
Thus, for high-dimension (infinite-extent) living systems, some of these other spectral sets could sustain the experience of the living system (survivability and being able to perceive), and others might not have the correct organization to sustain experience.
Are these issues of survivability about the "types" of infinite-extent 10-dimensional discrete hyperbolic shapes, which were identified by Coxeter?

What are the main issues concerning creativity for such a structure of existence?