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“Cubical” simplexes revisited
by m concoyle Ph D
Email: martinconcoyle (nospam) hotmail.com
10 May 2012
The failed dogmas of modern (2012) science and math,
Namely, the failed attempts to try to describe the following observed properties of the world, that:
1. Ordered, stable “physical systems” emerge from a context of randomness,
2. Life developed into greater complexity by evolving within a random context (which, supposedly, leads to greater complexity).
These are failures which result from the idea (belief) that math and science need to have a descriptive basis which depends on indefinable randomness and non-linearity.
The new ideas expressed in this paper, can claimed by anyone as their own, and used without fear of copyright infringement, except that they are original ideas, and thus as an originator of these ideas it must be stated that they cannot be used for military purposes, or for purposes of domination, or destruction, or they cannot be used within a society whose law is based on property rights and not based on equality, and in a society where the law requires that the power to govern is obtained from the people, who are all equal. That is, a dominant, unequal, group of people within a society (the owners of society) cannot derive value for their selfish purposes from these new ideas. These ideas are meant for practical creativity, motivated by selflessness.
The failed dogmas of science and math,
1. Ordered, stable “physical systems” emerge from a context of randomness,
2. Life developed into greater complexity by evolving within a random context (which, supposedly, leads to greater complexity)…,
[Note: This is a statement about ideas in general, it is not a statement in favor of religion, since religion has virtually nothing to contribute to understanding the observed property of order which is observed to exist within the world, rather “how order emerges from the world” is to be considered a question about the math-assumption (or math basis) for a valid mathematical description. In an equal (and free) society, people should be allowed to question authority in a meaningful way (this is how learning develops).] (This disclaimer is needed because of the distorting affects on people’s thinking (the belief that they belong to the learned class) brought on by the social condition of wage-slavery, in a society based on inequality.)
…, are ideas which result from a math and science which is based on indefinable randomness and non-linearity.
Such a mathematical basis (as indefinable randomness and non-linearity) can be used to describe many of the details which exist within an observed, measurable, descriptive context, but, these details are unstable and fleeting as are the math structures which are based on indefinable randomness and non-linearity. Apparently, indefinable randomness and non-linearity cannot be used to describe “from whence” the observed order of the world comes. Furthermore, indefinable randomness and non-linearity has not been able to do so during the 100 years within which indefinable randomness and non-linearity has been used to try to describe the observed order of the many fundamental material systems which are observed to possess the property of very stable order (nuclei, atoms, molecules, crystals, life, mind, planetary orbits, etc).
The failure of these assumptions (indefinable randomness and non-linearity) to be able to describe the observed order of matter (and life) is a result of the fact that indefinable randomness and non-linearity are math ideas which are quantitatively inconsistent, improperly associated with a quantitative system (by means of both a set structure, and the math structures which model measuring and/or counting), and dependent on very large sets (ie sets which are “too big”) wherein the meaning of words do not remain definitive (ie the type of thing to which the word refers changes, [the meaning of the word changes] while it [the thing] still appears to be a quantitative value, as it is used in a mathematical description), eg there is a plane-filling curve (a 1-dimensional pattern is equivalent to a 2-dimensional pattern).
Though the properties of indefinable randomness and non-linearity are observed in the “transitional structures” of both physical and living systems, the very stable, definitive, discrete spectral-orbital structures upon which our existence depends, as well as the math context through which order and complexity in living systems develop, are not being adequately described. This suggests a need to consider “how it is that very stable (yet still complex) math structures, eg ‘cubical’ simplexes, can be related to indefinable randomness and non-linearity?” rather than considering ideas which develop exclusively along the opposite logical path (of language’s logical structure), namely, that order emerges from indefinable randomness and non-linearity, ie the current assumption.
The assumption that “order emerges from indefinable randomness and non-linearity” is the basis of the current (2012) authoritative dogma, which is considered to be an “unchallengeable fundamental assumption,” ie a fundamental dogma, but this assumption needs to be challenged in fundamental ways.
There are a large number of mathematical contexts which are not considered. In math certain contexts are explored, where if certain contexts (apparently) possess many describable patterns, then these contexts become traditional topics of study within mathematics. Yet these traditional contexts may not be the patterns which are best used to describe the observed patterns of existence in a precise and practically useful way. There are very many mathematical contexts most of which are not considered.
The new assumption that very stable math structures, such as “cubical” simplexes, are primary, and that randomness is a derived property, is about re-organizing the language of math so as to develop a measurable description of existence which can be used in both a descriptive context (which can adequately describe the observed stable properties) and in a practically creative context, ie used within the context of existence itself [not simply existing within un-useful descriptive dogmas] Note: The new idea of existence extends beyond the idea of materialism into higher dimensions. The new context is that, there are layers of (adjacent) dimensions, so that each dimensional level is a stable shape, and that each dimensional level can contain many lower-dimensional stable shapes (which can be considered to be either closed metric-spaces or material systems).
This is essentially this is the constant curvature metric-space spaces of: SO(s,t) and SU(s,t) as well as Sp(2s, ?), where s + t = n, and s is the dimension of the spatial subspace, while t is the dimension of the temporal subspace, apparently up to SO(11,1) = H(11) (hyperbolic space).
This new context is not a traditional math context. Apparently, this is because the math structure with which this new context is composed are very simple, namely, closed, bounded, stable shapes associated to “cubical” simplexes. Spherical symmetry is non-linear, and thus unstable, and so it is not useful.
Whereas people will always enjoy creating in a context of art, ie a sense of balance, as well as developing the discipline for accurate replications (models) of the world’s beauty, and also in relation to an abstract sense of beauty.
But now, in a culture which “focuses on markets” which serve big financial interests, art is an expression which requires a notion of being caught-up in a narrow context, where careful judgments can be easily made so as to distinguish the so called superior art and superior artists, from the inferior [Note: Van Gough was placed in the inferior category.]
Thus, the artist became a producer of a “creative” commodity, in a context of distinguishable value.
This idea of distinguishing superior art, within a narrow context of a market, has been used to create the idea that people are not equal.
In professional math the same focus on superior and inferior is also required (by the owners of society) thus math traditions which are filled with a lot of complication are the desired set of contexts, and this leads to indefinable randomness and non-linear math patterns.
The ideas expressed in this article are very much pro-science, or equivalently pro-practical-creativity, where practical-creativity depends on “practically useful” knowledge. Note: The natural context for the creativity of a human being (an equal human being) may not be the context of materialism.
By science “it is meant” being able to use a set of described “measurable properties” to build things.
As opposed to an authoritative science whose descriptions depend on math structures which are unstable, inconsistent, and whose meanings cannot be pinned-down, so that in such an authoritative scientific context in order to identify a ‘precise’ description of patterns the measurable properties of such patterns only exist in a fictional world, and thus the quantitative descriptions are not practically useful. These visions of an authoritative science provide creative descriptions which only relate to a body of literature which represents the authoritative dogmas, which describe a fictional world, (ie science is no longer science, rather science is now science-fiction).
Traditionally, the idea of “measurable science” has meant descriptions of the material world. Note: Even the esoteric string-theory requires a special math structure (local space-time manifolds) to accommodate and maintain the idea of materialism.
Nonetheless, these traditional (professional) descriptions of a fictional world are claimed to be measurably verified, but this verification represents a process of data-fitting similar to the data-fitting associated to Ptolemy’s measurable verification of his fictional world, (where Ptolemy’s model of his fictional data-fitting math structure was also capable of being measurably verified).
A true useful measurable description, based on the stable math of the geometric structures of “cubical” simplexes, seems to be pointing towards human creativity which transcends materialism, and enters into dimensions which exist beyond the material world, but it is a description which contains the material world as a subset.
That is, the higher dimensions are not exclusively composed of abstract mental constructs, ie they are ideas which are not “literary creativity,” as the authoritative Physical Review is today, rather it is about a creativity which involves the intent of (knowledgeable) life interacting in a greater context of existence. It is a knowledge of the “true” structure of existence, which can be used in a context of measuring, in turn, used to form a new system, or so as to build something new.
But these new ideas about “how materialism (and a greater containing structure for existence) should be mathematically modeled” are outside of the current dogmas of science and math, ie outside their traditions of (absolute) authority, ie outside of what is considered to be the current belief, a belief which possesses an extremely high-cultural-value in regard to math and science (where science and math are represented as a difficult to understand descriptive structures which are used to define inequality amongst the people who compose a society). That is, science of authority is used in a similar manner as is commercial art, it is mainly used to identify superior people within society, so as to prove that people and not equal.
Simply because descriptions of non-linear curvature can be described as algebraic structures of local geometric measures (of curvature of a geometric shape), this does not mean that these algebraic patterns which are associated to curvature (which identify a local non-linear geometric measuring relation between a function and its domain space) are related to stable measurable system structures, or to stable descriptive math structures, eg the algebraic structures of systems of non-linear differential equations can be described, but they are inconsistent quantitative structures, which are also not practically useful.
On the other hand, “cubical” simplexes can be extended to higher dimensions in a stable manner, ie they remain stable geometries in higher dimensions. Furthermore, these high dimensional geometries are not only related to the structure of a material world, but they also are related to (or lead into) very interesting contexts, where both material and metric-spaces can have stable shapes, and many new math-physical properties can be defined, so as to be (or determine) a physical map which one can follow into higher dimensions. Both a map for the mind, and a map to practically, providing creatively useful patterns, including simple models of life, and its associated memory, where the intent of a living system can be modeled to interact with a further context of existence.
Perception can also be modeled.
Furthermore, as in the case of Copernicus, the new math-physical model has the capability to better predict (or fit) data. Remember, truth is about a measurable, precise description’s usefulness, while a descriptive language (based on new assumptions) organizes how the observed patterns (or the facts) fit together.
Truth is not about absolute truths.
**Particle-physics is a bad joke, a tribute the manipulation of obsessive people and to the social control of all aspects of the language which is used in society. The idea of the existence of a “scientific consensus” is due to the relation that science and math have to wage-slavery.
What does the observed property of precise quantum discreteness imply? Answer: (obviously) System discreteness, and (some might be surprised) metric-invariance of the descriptive context, and the need for stable math patterns to be used to describe the stable discreteness of quantum systems.
Precise quantum discreteness implies metric-invariance. Otherwise, if quantum interactions were truly non-linear then the local properties of indefinable randomness, associated to local non-linear curvature, would interfere with a quantum system’s stable discrete properties. [Note: The non-linearity of particle-physics is expressed in a math context where the fiber group is unitary, or Hermitian-form-invariant.] Thus, in order for the local properties of indefinable randomness to not interfere with a quantum system’s observed stable discrete properties, the non-linear property would have to be both local (as all curvature can be represented as local) and global, and thus the need to search for a non-linear global function, which contains “all information,” so that the local properties of indefinable randomness do not interfere with a quantum system’s observed stable discrete properties. That is, the global non-linear function would be needed to compensate for the condition of local randomness which is associated to non-linearity. Otherwise the discrete properties of quantum systems would not be stable, and a quantum system, such as an atom, could not be identified by their definitive spectra (because in the case of the existence of local non-linearity, a quantum system would not have a stable spectra associated to itself, because the local condition of the quantum system would be a condition of randomness, and thus no two quantum systems would have the same properties).
That is, the discrete spectral properties of quantum systems would change from point to point within its containing space because of the local randomness, which is associated to any point in space within which the quantum system exists.
Good luck at finding such a function, which contains how the local discrete spectra change from point to point in space due to local non-linearity, since what is observed is the same definitive stable discrete spectra of a quantum system which is independent of its position in space.
It is observed that quantum systems have the property of non-locality, or action-at-a-distance. Thus, the property of inertia is a property of Euclidean space, where it should be noted that this is not a part of a (gravitational) manifold which is locally space-time.
However, material interactions in 3-space are spherically symmetric in the new descriptive structures.
The physical sciences and
mathematics, along with
the foundations of biology, as well as (it also turns out)
“the basis for understanding the spirit,”
(all these subjects) need to be based on the math of:
1. Stability, the stable discrete shapes of spaces of non-positive constant curvature which are defined over (under):
(a) SO(s,t), so that s+t = n, and similar
(b) SU(s,t), and
(c ) Sp (2s), and possibly
(c2) Sp(2s,2t) (?),
principle fiber bundle spaces,
The exotic Lie groups may also be important, but the above are the simplest of the stable structures, in regard to both measuring and geometry (especially in regard to lower dimensions).
2. Sets which are defined, so as to be consistent with an idea of descriptive confinement, so that ideas actually have meaning, ie sets cannot be “too big,” and the “type of number” can remain well defined within the (simple) set structure, the limit process does not change the number-type, and the limit process (which is dependent on the order of numbers) is actually defined with a finite amount of information (so that it is not clear what the set structure “actually is” ),
[Note: When sets are “too big” then questions about “number order,” the “existence of a number,” and “number-type,” cannot be answered without an infinite amount of information (G Chaitin), ie the questions go unanswered. Yet answers are assumed to exist, but the meaning of the math patterns which are described, when one assumes that such answers exist, are quite questionable, ie such descriptive patterns are not to be trusted.
Either “finite sets” or some cut-off as to a limit to the capacity to measure within a quantitative structure, within which one’s descriptions of math patterns exist, are needed in order for one to truly believe in the consistent structure of one’s descriptions.]
3. Geometry, so that the description is both stable and use-able,
4. Each point of a coordinate space, upon which a stable system (both material as well as metric-spaces) is defined, is composed of a set of independent (and orthogonal) measurable sets (parallelizable and orthogonal, or geometric separability),
5. Local linear (models of local measures on systems), and the matrices, which are “local coordinate transformation matrices” of a stable system, are diagonal, though they are not so much (local) dynamic systems as they are stable geometric shapes,
6. The descriptive context transcends materialism (the description includes higher-dimensions) [this can lead to great complexity],
7. Physical constants are conformal factors which are defined between the different dimensional levels, which are (effectively) “geometrically independent” of one another, or constant factors defined between different subspaces of the same dimension,
8. Each metric-space can be identified with a (physical) property, a property which the stable shapes (of the metric-spaces) can preserve within a continuous context, where the continuous context is defined on the next dimensional level, ie defined on the metric-space within which the material property (or the discrete material shape) is contained, ie both metric-spaces and material are the same type of stable geometric construct (or shape) separated from one another by a dimension,
9. Along with a Real metric-space structure, R(n), there is also a Complex coordinate structure, C(n), so as to allow for the two independent “metric-space states” which can fit into a complex coordinate space for each dimensional level, ie the physical (material) property (associated to a stable geometric shape) has two states (see below), thus the stable geometric shapes, ie the stable material systems, are also a mixture (or built from “separated” ) opposite metric-space states, upon which spin-rotations of metric-space states are defined.
10. There is a spin-rotation between the pairs of metric-space states of each metric-space (or equivalently within each material system), and the time-period of the spin-rotation between the metric-space states can be used to identify a discrete process of change defined around the center-of-mass of (or between) two (or more) interacting material-shapes, which are relatively stable,
11. The (continuous) shapes involved in the force-fields of material interactions are tori, and for each new discrete time interval (defined by the period of the spin-rotation of metric-space states) these (interaction) tori re-fill the material-containing metric-space. For each such, re-invented, interaction associated to each such small time intervals, the relative positions of the interacting stable material shapes are changed in a relation to the local coordinate transformation fiber groups (of the principle fiber bundle). The tori of interaction can fill space because of an action-at-a-distance property which is a part of the material interaction process (or the tori form interaction shapes instantaneously) so as to define the context of a local measurement for each time interval defined by the spin-rotation of metric-space states (as well as providing values of geometric measurements and conformal factors [or physical constants]) which is (are) involved in describing a material interaction. These local and global geometric measures on a material containing metric-space are also related to the local coordinate transformations of the interacting material positions in that metric-space (in Euclidean space the spatial displacements [or inertial changes] are defined).
12. The stable shapes of material systems are discrete hyperbolic shapes (which are defined over a range of dimensions, but which seem to no longer exist beyond hyperbolic-dimension-10 [due to Coxeter Theorem]).
13. Finitely defined elementary-event spaces, so that the elementary-events are both defined (identifiable) and stable, are the types of event-spaces which are needed in regard to the descriptions of random-event phenomenon (so that the calculated probabilities can be trusted in a meaningful context of randomness),
14. That is, indefinable randomness and non-linearity are ruled-out of the descriptive structures of stable systems, where it is assumed that “one only needs to describe the stable structures of systems, or of existing (math) structures,” and therefore, one should deal with fleeting, unstable system structures in a qualitative manner, where only certain aspects of the description of an unstable system, (perhaps) defined as a set of differential equations (in a metric-invariant metric-space, in spaces of non-positive constant curvature), need to be considered in a descriptive structure, which is identified above.
Both the stability of a measurable description, and the stability of a measurable system, are both of great interest, in regard to the use of math patterns to provide an accurate and practically useful descriptions of a system.
The observed pattern is assumed to be stable (or has been seen to be stable) so that the described pattern needs to be stable and the descriptive words which identify properties and categories within the system’s description need to have a fixed narrow meaning (for the description to be precise).
In math, this seems to require that the description fit into a set which can be defined “to be finite,” but still within a continuous context of energy and material, for each dimensional level.
The above identified math structures which allow for stable math descriptions can be both:
(1) accurate for general systems within a sufficiently precise descriptive structure, and
(2) the description identifies a practically useful context, wherein technical development is widely related to the descriptive context (since it is a geometric descriptive context).
It should be noted that the idea of “peer reviewed” science and math is not valid, since it is exactly equivalent to requiring that the ideas of Copernicus be edited by the pope, but now it is the authoritative assumptions of “science.”
That is, absolute authoritative descriptive structures are too narrowly defined so as to not be able to assure anyone, that these absolute ideas contain the truth, where “truth” means, sufficiently precise descriptions which can also be applied to the observed properties of the world in a practical creative manner.
“Peer review” is about narrowly defined categories which have been identified by business interests, and then these narrow, absolutely based descriptive (measurably verifiable) languages define an institution (defined by its fixed traditional structure, which enshrines authority) wherein the expert authoritative leaders, of such absolute institutions, possess the same personality characteristics as do the owners of society. Namely, these authorities are: competitive, narrow, obsessive, manipulative, and they seek to be domineering. However, these authorities (many of whom are psychopaths, eg E Teller), are represented, to society, as being the supreme examples of human intellectual capability, so as to form a notion (or a belief) within the public, that within our society, “our culture really does understand everything.”
Scientific consensus is really the relation which scientific authority has to wage-slavery.
This is clear in the global warming debates, where the governing rule of the relation between carefully considered possibilities and public policy should be:
If problems are (truly) possible then society should correct itself, ie societies should be adaptable rather than being both rigid but (nonetheless) easily manipulated within its different levels of social authority.
(some of) The new mathematical descriptive structures
The two real, but opposite, metric-space states, naturally fit into the two subsets of R(n) and (iR(n)) which are contained in C(n) and are mixed by the spin-rotation of states in C(n), so as to be a part of each time interval of the “dynamic changes” of the material interaction, which involve material displacements whose dynamic paths are contained in the two opposite spaces, in the two subsets R(n) and (iR(n)).
At a point in C(n), which is along a dynamic path, one needs two (independent) global position functions of a dynamic interaction, which are only locally related as (opposite) independent directions of spatial displacements.
For stable systems the entire descriptive structure in C(n) needs to be “geometrically and temporally separable.”
For general dynamic paths the local set of opposite directions do not have a strictly causal, retrace-able path direction at each point (ie the path can be non-linear), though they are caused by both the locally measurable structure of toral shapes (which have properties of action-at-a-distance) and their relation to local coordinate transformations which identify spatial displacements due to the interaction (this is similar to the structure of a connection, ie a generalized derivative).
***During a material interaction a “total time of interaction” can be determined so as to define (with the help of the speed-of-light) a spherical ball which confines the two (opposite) dynamic paths, so that in the two separate balls, in the two subsets of R(n) and (iR(n)) respectively, there are two independent and opposite dynamic paths, whose relative times can be identified from the mid-point of the total time (interval) of the interaction, [where the two dynamic paths meet (or originate) at the mid-point time of the total time interval of the interaction]. Thus, the opposite states (of the dynamic path) can be made to coincide in regard to time measurements along this time interval. That is, from the time mid-point the two opposite time directions can both be given relative directions of time so that the two time values coincide at the mid-point and then are related to one another by the “relative opposite time directions” measured from the mid-point in time, for the two opposite time directions.
Consider two ways in which organize a complex structure for C(n):
(1) related to a spherical space, eg the Riemann n-sphere associated to R(n) [which can also be thought of as being in (iR(n))], and
(2) a disc (or loop) space of a disc associated to each C in C(n), (or a C-plane normal to each direction of R(n)).
The sphere has positive curvature and it has different curvature properties at different points (the metric-function of a sphere does not have constant coefficients), while the “disc-space” can maintain constant curvature, which would be a non-positive curvature (which could be zero curvature or a constant negative curvature).
Does one extend dimensional independence in C(n) by a pair of independent subsets R(n) and (iR(n)) [each containing an independent dynamic curve] or does one take R(n) and attach a C-disc to each independent direction?
The disc-space (at high enough dimension) can represent either the, high-dimension spectral context of a material interaction (which is contained in an over-all high-dimension containing space), or the spectral context of the (lower dimensional) material-toral interaction structure.
Can both structures be simultaneously true? Apparently, Yes.
The most significant example of identifying a mid-point in the time interval of interaction would be in relation to when a new system forms.
This could be modeled as a classical collision of relatively stable shapes. (or in the opposite sense when a stable system breaks apart, due to extra added energy). Where the new system depends on forming into a set of independent discs which are in relation to a fixed stable spectral structure of the over-all high-dimension containing space.
A large spectrum can be defined on a maximal torus and a (fairly large) set of multiplicative factors (or physical constants) associated to the set of sub-tori for each different dimensional level.
The spectra of a maximal torus can be related to either the entire spectra of the high-dimension over-all containing space,
To the slightly changing spectra of a material interaction.
This toral relation would be defined on a cube or rectangular shape, the n-cube could be represented as an n-torus in R(n+1).
The coordinate structure of the two opposite metric-space states associated to a [C(n),t] coordinate system are locally exactly opposite dynamic pathways, but globally, in [C(n),t], the spatial displacement paths diverge from one another as time changes, but can be made globally (locally) independent to one another by using two representations of the two “opposite position paths.” One a global spatial position function of the (spatial) displacements of material shapes contained in R(n), and the other time direction of the dynamic path identified as a “position path” in (iR(n)).
(the two directions of) time, t and -t, can be identified on the two subsets [R(n),t] and [(i(R(n)),-it] in space as containing oppositely directed paths.
In the complex coordinates, the (i(R(n))) subset is associated to (-t), and the (R(n)) subset is associated to (+t). Thus, there are the two sets which identify a pair of dynamic paths in C(n), during a material interaction, which can be identified as being [R(n),t] and [iR(n),-it], where for local measures of time, either dt or d(-t), (these local measures) identify oppositely directed paths, at the point where the displacements take place at a time interval so that the opposite path stays in both sets, [R(n),t] and [iR(n),-it].
The origin (or end-point) of these dynamic paths are either regions where the various material interactions are all about equal, or where, during a collision (or a break-up, or sudden change), the geometric-interaction-complex resonates with the containing (spectral) set, so as to form a stable system (contained in the next higher-dimension metric-space), where everything stabilizes and becomes “geometrically separable.”
From these origins there is defined a well identified dynamic (or spatial displacement) path. The midpoint, in time, for these two opposite paths can be identified.
The pairs of opposite paths are present at each point in the interaction process. That is, the dynamic paths exist at each time interval of the two opposite dynamic processes, yet each separate state seems to be individually determined (in relation to the geometry which exists during that time interval) for each short time-interval wherein certain changes are (causally) realized on (along) the (continuous) dynamic path so that these causal relations can be determined for either of the pair of opposite paths in an independent manner for each path, associated to its time direction.
However, locally the two path directions are opposite, but if they were determined in a causal manner and the path was a result of a non-linear relation, then the two directions at “the given point in time” would not necessarily be in opposite directions, this is because non-linear relations are indefinably random so if the paths at the same point in time are determined in a casual non-linear context for each separate time direction then they would not necessarily identify two opposite path directions.
At a point in C(n) there need to be two global position functions, which are only locally related (as opposites), in regard to dt and d(-t), while globally they are separate.
These two global functions (dynamic paths) form a pair of local independent metric-space states, as well as a pair of local independent coordinate structures which are the same path but spatially directed along the opposite direction.
From the time mid-point a real time relation between the two opposite times can be inter-related.
But if there is a property of the dynamic path being a pair of local linear and independent orthogonal coordinates as well as metric-space states “geometrically separable as well as being time-separable, eg (i(t)) vs. (t),” is only assured in the linear separable conditions of a stable system which exists in two independent metric-space states. A (periodic) position and its opposite position within a stable system relate the spatial time-path in one metric-space state with the opposite (spatial) time-path.
Thus, there could be a relation between opposite roots (of the fiber group’s Lie algebra root system) and the weights, or eigenvalues (or spectra), identified on the fiber group’s maximal tori, but which are related to of the systems which are contained in the base spaces of the principle fiber bundle. Thus one would “conjecture” that the pairs of opposite roots, in a fiber group’s root system, identify opposite dynamic paths and that these opposite paths naturally fit into orthogonal parallel geometric shapes so as to be geometrically consistent with the pairs of opposite roots in a fiber group’s root system, so that the local spatial translations are closely associated to the diagonal matrices of the fiber group’s maximal tori.
One-half of the roots in the root systems of fiber groups are opposites and this is consistent with a geometry where there also exists an equal balance of a set of opposite vectors which can identify the geometric shape, thus identifying a stable shape upon which can be defined dynamic material flows, so that there can exist a stable geometric spectral structure. That is, sets of opposite vectors identify stable geometric and stable mathematical (spectral) patterns.
Such a stable system depends on the system being resonant with the spectra of the over-all high-dimension containing space for the stable system, where that spectra can be represented on the maximal tori of the fiber groups.
A pair of globally opposite functions in [C(n),t], ie but time independent so as to be contained in R(n) and iR(n), requires a spectral memory in SU(n) related to a pair of “opposite” SO(n) fiber bundles.
symplectic group and the space of interactions
In symplectic space, ie x^p or Sp(2n), a material interaction is related to a geometric interaction structure of, T(n) x H(n-1)/#, which is contained in R(n+1) x H(n), or contained in an, n-cube x H(n), however, the dynamic path is contained in R(n) x H(n).
When n=2, then the interaction structure is a Euclidean discrete shape (or Euclidean space-form) is 2-dimensional, ie the dynamic interaction is contained in 3-space, that is the dynamic path of the interacting material system is contained in a symplectic space, and its spatial displacements and its changes in motion (or moving displacements) are contained in R(2) x H(2), ie the path of spatial displacement is in R(2), and thus the inertial spatial displacements are on a plane.
Does this also imply that the interaction will be based on a 2-body (material interaction) problem? No.
For a material interaction there is a Euclidean interaction discrete shape, T(n), and then there is the stable material discrete hyperbolic shapes, H(n-1)/#, where, #, is a discrete subgroup, which are the materials (which are interacting).
Thus, the interaction space, T(3) x H(2)/#, (where, #, is a discrete subgroup) is contained within, R(4) x H(3) so that the dynamic path is contained in R(3) x H(3).
This work is in the public domain