Corollary Cepts and Relationships
This manuscript is not written in indelible ink. It is a "living" extension of my thoughts and experiences. As such, I constantly re-write it, add to it and acknowledge other avenues and "potentials" I've noted but have not yet fully explored. "Glimpses of potential", you might call them. Therefore, with an eye toward focussing on future areas that need to be better enunciated I will mention some Corollary Topics, and, focus on important new paradigms and mathematical/behavioral relationships that come out of this new way of appreciating "Information" and "fluences". Some I have or will talk about more fully. Others will be cepts which occurred to me and seemed relevant and important enough to mention. The last one of this set (#23) should probably be ranked higher.
1. "Logic" and Set Theory, more fully evaluated as a Temporal öunction. Because considering it as spatial ffunction only, creates unavoidable anomalies. This is especially pertinent to the work by Russell, Gödel, Wittgenstein, Cantor.
2. Topological "information set" comparatives, indicate conditions and systems that need to be addressed:
Translations of information ...under at least two distinct environments..fluence to fluence of equal exponent definition ; fluence to fluence of unequal exponent definition;;;subsets of each are: a) equal scalarity - fixed or variable b) unequal scalarity - fixed or variable or fixed^variable or variable^fixed or variable^variable. Subset variability... a)smoothly open exponents <1,=1,>1 b)cyclic repetitive c)random "density" d) closed recursive.
3. Information storage and processing. Its presence in atomic and molecular structuring. Its expression in higher order forms and the transition to "behaviors". Form/ffunction limitations. Adaptability potentials. Adjustive elastic response behaviors.
4. Analysis and application to dynamic living systems: economics, metabolism, political systems, linguistics, et al.
5. In the midst of a recent discussion with Jim Feigenbaum on a network bulletin board, we talked about the nature and fundamental structure of subatomic particles and electron shells and ffunctions ... especially in regard to tensor and Quantum Mechanical formulations.
I was mentioning the Bohr model of the atom ... where the Schroedinger - Dirac quantum numbers specify the location and relationships of the electron shells. I jokingly described the paths of the electrons as "flying in their own "vapor trail" " ... and wondering how that would be possible, because obviously nothing would be left in that vacated region by the time the electron reached it again. Not even the magnetic field that the electron would ostensibly have generated. Ostensibly, the nucleus generates the magnetic field that the electron constantly balances. But, another stabilizing symmetry might be at work.
This perked a thought in me. Especially in regard to my reliance on the Apollonian relationship of "equal orthogonal information". Suppose that an electron displaces it own magnetic field at some point along its angular path of motion. By the "time" the electron arrives from the near side of the nucleus to the "opposite side"...the magnetic moment of the "near side" location would have already orthogonally traversed the interior of the atom and arrived at the far-side (!). The electron wave/particle would then have self-created, so to speak, its own environment in which to travel, interact and exist ( in addition to its relational configuration with the nucleus).
Now, keeping in mind the /Apollonian Circle, the location at which the wave establishes not only an equal "information" value but also a form equal in its orthogonal component, is when the wave travels in such a way as to establish an equilateral triangle ... a triple-phase symmetry. At that location the magnetic pulse propagates to the next node in synchronicity with the electric wave. There are probably other acceptable nodes at integral multiples. There would also be "impurities" and thus irregularities, caused by co-extant interactions with the nucleus. Every locational node constantly receives an indeterminant variable van der Waal type "kick" that electron motions adaptively alter to. The van der Waal type environment would effectively redirect the tangential angular momentum of orbital particles (electrons, photons, et al) so that the equilateral tri-phase symmetry occurs anywhere on the circumference of a circle "scribed" by a radial equilateral node.
Fundamentally, this equilaterality establishes a very specific "trisection" of the tensor/metric matrices ...or gauges... and the resultant combinants must necessarily be in factors of "thirds"! This implies that symmetry is not limited to spacial bi-directional balance, but can be involved with re-attainment of "original" (base) configuration. As stated in Physics: the symmetry of Action. For example, to return to a given EM phase state (renormalizable for "time") after having shifted forward 1/3, the particle/wave can either move +(2/3) or -(1/3). This is a loose but apropos analogy. Dimensional/fluencial compression can produce self-referential states and thereby create "internally" Integrity-stable configurations. The familiar "thirds" of quark architecture becomes an obvious product of these relationships.
The quantum chromodynamics quality called "color" can be thought of as one side of the equilateral triangle. Each of the 3 {!} is indistinguishable from the other yet are distinctly unique in that - all other factors being the same - the presence of any one of the 3 (representing uniquely different geometric configurations) - will establish a different resultant-particle. When physicists postulate the existence of "real extants" called quarks (none have been actually experimentally seen, and they are now considered to be "un-seeable"(!)), it could very well be that they are dealing with heterotelic dynamics: transfluencial relationships which are real, and do exist, but only as specific informational relationships that describe the gauge gradients enacted by matter/energy with actual physicality...or fluencial cross-dynamics. Quarks are heterotelic artifacts. Gravitons, similarly, may be heterotelic artifacts of multi-temporal fluences.
The multiplicity of triads can be envisioned as interrelated edges of a quadrilateral pyramid. An added level of Symmetry is present in bilateral splitability of the form into equal reflections of twin bi-isosceles groupings. Please remember, that these images I talk about are essentially geometric- analogues. Their forms still arise because of fluencial compression and expansion of information/ energy/gradients that those geometries represent.
It is here that my opening evaluation of the Liar's Paradox makes itself known in physical reality. Physicists talk about the "spontaneous breaking" of symmetry in the universe. What they mean is that if everything were perfectly balanced in the Universe then there would have been no differentials to move things off of a dead-centered pure balance of energy and events. Absolutely nothing would exist, because there would be nothing to allow for dynamic imbalanced interactions ... constantly adjusting to keep in accord with each other.
Imagine a disc perfectly balanced on a pin point. The disc is "symmetrical" in its environment. If you place even the tiniest smidgen of something somewhere on the disc other than the pivot the symmetry is "broken spontaneously". Therefore the disc invariable possesses - and has the potential to display - both possible characteristics or states of existence as part of its dynamic qualities. Its "set of behaviors" are both symmetric and asymmetric. Heterotelic existential potential. Self-consistent even as a "paradox". Holistically consistent when seem from "outside" the observed events ... from beyond the events-boundary. Seen from a different perspective. Even valid when observationally seen from "within" the bound called "existence" ... an observation done by parallax triangulation. Inherent qualities that become alternately expressed when there is interaction with environmentally external conditions.
The Aristotelian tautology "A=A" and the Formal Set Theory choice-state of "A or -A" are no longer valid. Those are enunciations of limited perceptions about existence. They don't sufficiently express the complete relational situation. They don't convey the essential Objectiveness of "the dynamics" through which individually Subjective events find expression.
Again, the corollary of this links the Fuzzy Logic expansion of Boolean Algebra with particle physics. It seems an extraordinary thing that a random group of 3-factor or 4-factor addition or subtraction combinants should be so explicit in specifying particle characteristics so exactly. (The specificity of spin, charge, strangeness, etc.). It becomes reasonable however, when we realize that those simple combinants of addition and subtraction are really the Boolean (plus) function. They are 1-fluencial transcription expressions of exponential dot and cross products. That is, they are the reductionist expanded form of tensor/metric information.
Alternatively, the factorial qualities that appear in the Calculus play a significant role here. But again I want to stress the relational exchange and expression of information that occurs when a closed- information dynamic is expressed as and "exponential whole" or natural log state versus its cumulative linear form.
6. <as of this writing {1994}I am exploring general formulations using the Law of Cosines, linking them with Euler exponents and topologies, with the cross product of complementary directional vectors I introduced to deal with opposite signed Nested Cantorian Infinities, and with imaginary numerical components>
Keeping in mind the desire to establish a new paradigm for "fluences" as determinant in the relationship of "exponent", and exemplified by Axo, I will first give some basic Euler equations as well as a basic equation for a derivative.
Euler's "2nd Formula": feigq where i = Ö-1
eig = Cosq + Isinq
ˆ for complex #s z = x + iy (similar form: ax+bi)
x = Rcosq
y = Rsinq
r,q represent z on an Argand Diagram; noting that q = p gives eir = -1
q = 2p gives e2ri = 1
Also: r(cosq + isinq) = x + iy
where x = rcosq
y = rsinq
r = eig.
Now then, ex is its own derivative (which is a situation related to sine and cosine derivitives retaining equal content) ; also, the derivative of a constant is 0 (zero).
Remembering that we want to establish a new definition for a special form of the exponent as being representative of a complete continuum in that active relational corresponding symbolic position...
...... we can then reconstruct it in this form :
Axo where: exponent "o" = the derivative of constant C
(d/dx)C = 0 (!)
and therefore X(d/dx)C = 1 by "definition".
Application of specific values f (aleph + C ) will construct domain continuums. And so, here, we apply Euler's formula which is an information relational dynamic: ÷ is the anti-derivative integral of {epi +I}, since it equals 0<zero> in the derivative.
This is Boolean Algebra applied to Calculus as a principly incorporated definition.
The important concept of Coherent Consistency needs to be recalled and applied here. Continuums related through the Calculus of differentials exists within the relationship of "exponent", besides the conventional situation where an exponent exists as an individual component-factor of a separate differentiation domain ffunction along another defined continuum.
Restating: Multiple mathematical continuums exist simultaneously. Not only do mathematical values ... particularly those expressed exponentially ...have a specific relationship and meaning for a base continuum they code relevant information for, they are also innately self-relevant and represent wholly distinct continuums.
Sets of fluences that have ffunctional relationships via which information can flow and translate correspondingly should be called "Confluences". Such is the case with a QuantaeDomain raised to an exponent. "Axn" is a conf luence. X as a distinct fluence informationally intersects with n as a distinct fluence. Compoundedly nested fluences (nominally called superpowers) are Nested Cantorian Infinities.
In the same way that de La'Hospitals Rule applies substitution of a derivative to find the limit of a function when the function's original form becomes net-sum-"zero" in the denominator of a larger function... I apply the same valid technique of substitution for "zero" that is found as net value for an exponent. This elucidates the total fluidity of our mathematical realizations. That "powers of powers" are fluidly natural nested relational continuums, and that it is proper and fitting to develop functional derivatives and integrals that only "work on" portions of a Confluence, as differentiated from its Base-fluence.
Modern re-definition of an older and previously "accepted" a priori function:
The Confluence of a QuantaeDomain, Axn, whose net exponent is the derivative of a constant (and thus such exponentially-local constant translates to a value that is a resultant Sum-Zero), for the configuration of all the bounded expressions shall have the net value Sum-One. (As comparative convergent series)
This is a primal Identity Principle definition.
It is the fundamental tool for designating and affirming information transcription through NESTED INFINITIES which are non-quantized fluencial continuums. That is, Nested Cantorian Infinities.
Additionally, another result comes into existence. That of Indefinite Integers. Since the new numerical matrix becomes:
"Ax(d/dy)C" (C = any complex ffunction)
Any "number" can in essence be a distinctive quantum value or a statistical range or a carrier of an "infinite continuum" of information...depending on the function form invoked, and the application involved. This is another re-enforcement of the wave/particle quality of extancy.
We can now coordinate this with the integration process proposed with the -Apollonian Circle. When we brought both Delta forms to the same side of the equation ( DA2 / DCa ) and then took the ratio to its infinitesimal limit to approach "zero", we came to the recognition that as Ca came to become "zero", the ratio took on the holistic quality of "one" (unity). This means we now have a new way of understanding division by zero. It no longer is problematically left as "undefined". We can now rewrite it as equivalent with it's limit form (per L'Hospital's Rule). Infinite-slope now has a place within coherent mathematical methodology.
This is also the way we can resolve the Primary Logical Dysjunction. In this relationship, there is a real and meaningful way in which 1 = 0. It depends on which fluences or confluences are considered. One equals zero as the environmental bounds are chosen.
0/0 = 1. 0/0 = 0.
In one instance it is a limit. In another instance it is a value. Considered holistically, it is a confluencial intersection.
7. <12/18/92> am currently looking at more general form for the equation of conic sections which incorporates the angle of the nappes. (represented in these discussions as the primary vertex of the isosceles triangle, aka, the asymptotes of the subsumed conic sections). That is, the equation for a circle, by way of example, must have a more general form which takes into account the specific conic it is being considered in relation to. Conics of different asymptotic vertex angles can generate seemingly identical circles, for example, except, that there are variances in the conic angle, and thereby, exactly how far along the conic axis that the circle is created by an intersecting plane. There is a crucial correspondence between these things and the relationships between masses...vis a vis Gravity and the relative motion of masses with each other! In some configurations, one or more of the component factors probably either cancels or becomes negligible or vanishes or balances. It is not unreasonable to look for correlations between previously unconsidered relationships. Just because all conic sections can be derived from conic nappes that have a constant vertex angle of 90E (recalling that this may be another reinforcement of orthogonal relationships), we may still find other valid correlations where the vertex angle and distance from the vertex may play a valid role in real physical systems.
8. The ratio of the mass of a proton to the mass of an electron <6p5> is a special quantity that affords us extremely valuable information. It is a window to the scenario of multiply integrated
fluences; to the real and mathematical fluenciality of dimensions not restricted to positive non-zero integer states; to the deeper connection between that ratio and denser confluences of time-space. 6p5, you will recall, now indicates (5+1) confluenced continuums. Additional co-references are indicated for special values of Bernoulli numbers and Fourier series.9. Gravity will be more clearly understood when seen as Tensor/Metric elastic Temporal Confluences mediated by entropic smoothing back to a non-stressed stable state for each component fluence. That is, mass creates a "displacement" in several temporal fluences, "away" from pure orthogonality. Time...in several relational fluences, is disturbed ... away from their (its) most stable entropic rest state of pure orthogonality. Gravity is the created felt-force of these several temporal fluences attempting to re-smooth back toward unperturbed orthogonality. The Entropic effort (force) is greater in the regions of greatest disturbance (closer to each center-of-mass) but none the less, for any examined region of space between any given masses, the temporal vectors will entropically attempt to re-smooth back to a base-rest state of net-zero. Gravitational "attraction" is the effort of temporal vectors to re-gain the lowest rest state of undisturbed highest temporal Entropy. Gravitons, per se, do not exist. Gravity is "temporal" information/energy in purest form. As discussed above, Relativity applies to everything. An extant can "be" information, as well as "carry" information. This is apropos for Time and Temporal fluences, as well as Space, or the energy constructs we find in Universe.
10. In evolutionary biology and environmental studies, Confluencial QuantaeDomains are habitually used to describe "Speciations over an observed Area of habitation" :
Cjz.
Data correlation is examined by separating and translating the component factors into the fluencial equalities inherently expressible in their Log functions.
log S = log(c) + (z)logJ.
This should be immediately recognized as the standard form of Complex numbers "ax + bi". As such, it points up again the corollary-normal situation where the derivative of "constant" in the exponent, while becoming "zero" in one state-of-expression ... and therefore "vanishing", can simultaneously take its confluence with a referential domain to the value of "one" in a translated equality ... and also "vanish" in the alternative form.
Similar relationships are abundant by plugging in the appropriate Trigonometric functions.
Lissajous figures are closed conic sections generated by coupled harmonic oscillators described by similarly phrased equations to the ones above:
x = Ax * sin (Wx*t)
y = Ay * sin (Wy*t+D)
A is amplitude of the sinusoid
W is angular frequency
t is time
D is the phase difference
Wx/Wy = 1 and D = 0 gives straight line @ 45 degrees
" D = 45 degrees gives a tilted ellipse
" D = 90 degrees gives a circle.
11. Additionally, tensor-metric concepts can be coordinated here. In a currently standard general equation such as
6x3 + 43y5x + 17 = k
the fluence that the Lorentzian translation adjustment value is a "ffunction of" goes unspecified (in this example it is the number 17). It could be
6x3 + 43y2x 1 + 17xo = k
or 6x3 + 43y2x 1 + 17yo = k.
Rationally, it could also be the expression of non(x,y) fluences. Or a net combinant. The "normal" current usage, is only by convention and habit.
In other words, there are "usual" situations where certain components are nominally "invariant", and others where it is crucial to specify nominal "variance" as making a difference in the behavior of the system or ffunction. Ie, the application and non-application of the Laws of Association and Commutation. In some formulations they are appropriate and useful. In others, they are not. To be explicit, in some circumstances "6 x 7" does not always equal "7 x 6" (using a sample ffunction); Symmetry is "conditional".
In everyday mathematical usage, all numerical terms that leave companion fluences unspecified become the specific location that all physicists use as their reference location for the extraordinary "symmetry" on the universe. Think about it this way:
1) All number values unfixed to or unassociated with specific fluence(s) can be and have been manipulated without restriction. E.g.: x + 4 = y - 3. This will map a topological graph in 2 coordinates. In this example, the unwritten (k) factor, which is comparable to the line parallel to the base hypotenuse string of the -Theorem, is the point (0,0). <also called "junction">
2) Those number values (4,-3) can be distributed in any ratio, combination, or fraction to the orthogonally pre-defined "dimensions" ... as long as the total "information/energy" of the number stays "constant". There is "conservation of energy", and there is no preferred reference. Physicists call this "symmetry". As the quantity of distinguishable factors increases (z,a,b,c,... or x3, x4, x5,...) the quantity of zero-dimensions <1st fluences> increases. The intertransmittal of information between fluences is open, unrestricted, and invariant through any and all of the multiple mutually-orthogonal strings n1. This is "Supersymmetry".
3) Specifying distinguishable conditions/states - such as affixing xo or yo or zo etc.; or requiring non-commutivity; or distinguishing alternating entropic vs. negentropic directions in alternating nested Cantorian domains (see#12 next) - allows us to designate different symmetries as "permanent", some as "conditional", with the net affect being that these designations qualify as types of "boundary conditions" for the ffunctions being evaluated. The differently designated symmetry relationships work the same way as limit functions f(x), or boundaries to Hilbert Spaces (after: Barankin <selected associate to A. Einstein>)
The essential requirement of Non-commutivity as a principle fundamental quality for mathematics, coincides very well with Grassman numbers and Super-Symmetry.
12. The net entropy of an n-level is the opposite of the (n-1) level and the (n+1) level. Notwithstanding the exact formulation, my premise infers that the " signs of the entropies will alternate in the nested levels...for a given matrix. (E.g., higher entropy of the electron cloud shell complex creates a lower entropy (higher negentropy) of the atoms-molecules matrix.) The "±" is relationally arbitrary and "relative". Any number is not positive or negative as a value, rather it is only a "comparator" vis a vis internal or environmental ffluencial Entropy domains.
The current assumption-by-convention is that exponential relationships are Hamiltonian in nature where values are normalized to be only "positive" rather than positive or negative. I believe that assumption to not be perfectly accurate. The fluencial multiplication of S Ä S carries opposite signings and generates correspondingly opposite (±) adjacent fluences.
13. Information transcription has a "normal-function" between fluences of a given f(xn) level.
that is, _ _ __ ___
dx /Ö2 Ä dy/Ö2 ³ (p/4) (h/ p2) qr (or: qxy )
<<I use the "overbar" as a temporary notation device to show which fluences and heterotelic relationships (like q ) are inclusively referenced>>
Ä is a function of information [multiplication] which gives a value
___
in the qxy matrix.
{note: must be defined better. The same for fluence functions Å, subtraction, division.}
"Law of Cosines" might be instrumental in transcription of fluences. I sense that Gamma Function will also be relevant.
14. The ceptualist paradigm of nested fluences in a holisticly integral universe, is one in which "information" is transcribed topologically.
Because certain partial derivative values, and, individual fluences can assume "null-sum" values in the exponential component within a general mathematical matrix...the result is a situation that unifies:
- electron tunneling
- Bell's theorem results for polarized light
- missing black-mass of the universe
- holography.
- sonoluminescence
All 4 affects occur when specific factor-components become "singularities" as a normal state of existence over a temporal duration. That is, it is normal and usual for a partial derivative of a wave form to collapse into a stable singularity (zero-th fluence) for any of the partial derivative components of its total structure. Those "compressed" states ostensibly "disappear" ... become non-interactive (or "dormant"..to use a biological vernacular) ... until conditionally regenerated by a field interaction.
eg: for {qn1,in2,xn3,yn4,zn5}
if in2,yn4,zn5 = 1 (ie: n2,n4,n5 = 0)
and the "information" of (i,y,z) compress to 3 singularities... (or more, or less, components)...
if the process is: then the result is:
Electron transient-state --> "Electron tunnelling"
Photon stable-state --> "Bell's polarization"
Particle stable-state --> "Black-body missing mass"
Temporal stable-state --> "Holographic imaging"
Another fascinating effect occurs when the bi-orthogonal -Apollonian triangle is applied in a 3 dimensional/4fluencial sphere. When a sonic pulse is directed at a bubble in a fluid, the bubble surface transmits the energy of the encountered wave and refocuses it inward toward the bubble's center point. The result is such a concentration of energy that photon emissions occur with extraordinarily high temperatures. Research work is now being done in this new field dubbed Sonoluminescence.
15. Integrity, though existing as a progression of higher and higher kinetic stability regions - and thus potentially diagrammable in that "upward" way as a "hill" or other plateau - is eventually more properly visualizable as higher inverted regions ... that is ... successively higher levels of "information processing Potential". A nesting of Weibull Distributions at successions of relationally stable plateaus.
16. A thought occurred to me after seeing Hal Puthoff's expressions of ZPE (Zero-Point Energy) in his articles in Physical Review(Series A #40 and #44; 1990.1992). One of the formulas was amended in the second article to include a Relativity factor. In that formula, Planck's constant was in the numerator and p2 was in the denominator. I realized at once that the format presented there was exactly the same as the format presented in Ivars Peterson's books when he discussed "sphere eversion" (turning a sphere inside out mathematically). In the eversion formulae, all components on the left side of the equations were 1-fluencial while those on the right were 2-fluencial. The Puthoff formulae went one step further. All factors on the left side of the equation were 1-fluencial while those on the left were 3-fluencial (!). Puthoff's approach for achieving ZPE required information condensation/compression/topological-translation in the format I indicate in my thesis ... that uses Heisenberg's Uncertainty Principle as the first-level example, in it's simplest form. If I take my formula of fluencial translation of Heisenberg's Uncertainty Principle x/Ö2 Ä y/Ö2 ³ (p/4) h/p2, and take (p/4) to the left side of the equation....then the concepts remain intact for both Heisenberg's U.P. and Topological Transcription of Information...as fundamentally expressible in the ratio of h/p2.
The presence of h/p2 in any other set of equations will completely infer every quality and relationship of the un-written left side portion of its principle equality/translation (the U.P.).
Therefore, by applying the principle of topological f luencial information transcription (as correlated with the Uncertainty Principle) ( 1992), Puthoff's defined expression of ZPE consolidates in a unified nomenclature both Quantum Mechanics and Relativity.
17. An interesting progression of ratios exists for geometric figures, as we go up through dimensions /fluences. If we utilize the standardly unitized radius of "one" (normalize the functions), then the ratio value for the "interior" relative to the "boundary" ... for each whole integer "dimension"(k) ... is "1:: k".
exponent # content bound Ratio {content:bound}
eg: point Ax * * 1:0 (?/!)
line Ax1 Length = r{1} Length = r{1} 1:1
plane Ax2 Area= pr2 Circumference=2pr 1:2
space Ax3 Volume= (4/3)pr3 Surface= 4pr2 1:3
hyper. Ax4 Hypervol= (p2r4)/2 Hypersurf= 2p2r3 1:4Ax5 " " 1:5
Ax6 " " 1:6
. .
. .
. .
Posit: A valid set of ratios might exist for all complex numbers, not just positive integers. That would allow for fractal ratios. Content v. Bound graphs out as a fractal surface of hyperbolic cross-sections. The number line applied to Content-Bound Ratios gives all fractal fluencial possibilities including rational, irrational, transcendental, and complex fluences.
This exponential expansion of fluences can also link Fermat's Conjecture - his Last Theorem - with a polydimensional Pascal Triangle to give us a geometric image of where the conjecture is within the mathematical hierarchy
This is relevant to all those possible geometric states where an initially chosen information fluence set remains not just homologous, but de facto, unaltered, through any kind of Lorenztian translation, rotation, and exponential increase to (n+1)...fluences; and beyond...through (n+m).
X0
X1 = X1
(X)2 = (-X)2 ; (X1,2)2 = (X1)2 + (X2)2 ; (X123)2 = (X1)2 + (X2)2 + (Z3)2 ; . . . . .
(X)3 =
.
.
.
18. The imagery I used of 3 temporal vectors depicted as 3 joined near light-speed engines might also be useful to re-interpret cosmological Doppler shifts that exceed the speed of light. A shift in excess of 'c' is probably information transcription residue indicative of the several temporal fluences. In one frame, it could be the net-vector of several Gravitational vectors, or even of just the simplest photonic motion. Distances and time-scales would necessarily need to be re-thought and adjusted.
19. There are subtle connections between Brouwer, Pascal's Triangle, and new multi-matrix functions
eg: (a,b,c,d) x (e,f,g,h).
The resultants generate triangular Pascal configurations. {ref: "quad-math" work sent to me Spring, 1993 for evaluation by Jay Hall of San Diego}. Self-relevant boundary condition arise as well as internal patterns. These are related to quaternions devised by Hamilton.
20. Gravity - as a blend of temporal asymmetries - calls into question how gravity is expressed when mass is converted to energy and vice versa. As mass is "vaporized" into energy does the temporal symmetry of that mass/energy fall to zero or to a value conserved in equal but opposite portion. Is this a "violation" of symmetry laws, or an affirmation that gravitational and temporal asymmetry are indeed identical? What are all the possible forms that time or temporal stress can take?
21. Nested Cantorian Infinities probably hold for each x,tx,y,ty,z,tz ... with each having its own nested Derivative-of-a-Constant progression in exponential relation. A tensor-metric grouping.
22. Confluencial exponential-Calculus might be able to handle difficult 3-body problems and more. The process will allow simultaneous flows of a plurality of force gradients, moving toward different entropy states. This goes beyond the limitations of tensor-metric formulations which are restricted to identical entropy gradients (symmetry restrictions).
23. Boundary Statement.
The process of defining this relationship creates two distinct types of sets, even though the process is monotonously singular. It is a Turing sequence qualified by quantization. Not of identical quantum "values", but of closed elements regardless of size.
Tj is a "thing" or extant. Ej is its immediate conditionally required "environment"... what ever is minimally required to completely enclose Tj.
T1 + E1 = T2
T2 + E2 = T3
T3 = E2 + E1 + T1
advancing indefinitely Tå j = Eå j + T1
until T¥ = E¥ + (DT1 -> i) {as 1:T¥ = -¥}.
There is no "formal" closure or boundedness. Formality is conditional on the Heisenberg observation event (which I refer to as the "Heisenberg Intersection). Potential requires a variable Identity-bound, and Informality. An extant can simultaneously "be" environment AND be "in" environment. Concurrently, this establishes an alternative representation for "non-existence": Traditionally the Null-set has been written as various forms of "zero", "empty set" and so forth (using a variety of notations). It may also be represented by "-¥". "Absence" from observational existence now becomes a matter of several conditional factors, rather than a fixed-formality. Nested Cantorian Infinities are functionally enhanced by diminution-to-zero in neighboring exponential continuums.
24. Exploring several new classes of functions which have a bearing on quantum-physics and wave function descriptions. In particular, there seems to be a family of ffunctions which can vary to infinity yet the difference between any two of these bound-ffunctions will always be a fixed constant ... which is correlatable with quantum units. The simple relationship is f(A) - f(B) = f(C) where f(C) is a constant, M, based on a single parameter shared by A,B.
25. Complexity is an Aleph-function. The intersection of two inverse gradients is a new region where their entropies vary directly. Essentially, a new gaussian-bubble region is stable in this intersection and is tantamount to an emergent-state ... a new "integrity".
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