Sunday, May 11, 2014

Part 2 of 2. 'Musean' Hypernumbers and ‘‘‘Dark Matter/Energy’’’.




Full Title -- 

Part 2 of 2. 

Musèan Convolute Hypernumber Qualifiers and ‘‘‘Dark Matter’’’ / ‘‘‘Dark Energy’’’.

Dear Readers,

Preface to Part 2.

Background.  This blog-entry is the second part of a series of blog-entries that began last year --

The predicted next phases of the dialectic of Nature -- of the ‘self-meta-evolution’ of the cosmos in general, and of its human[oid]-social species in particular, as predicted by the F.E.D. ‘Dialectical Theory of Everything Meta-Equation’ itself, and by the F.E.D. ‘Psychohistorical-Dialectical Meta-Equations’, which zoom-in on the human-social and ‘meta-human’ components of that master ‘meta-equation’ -- all seem to presuppose the development of superluminal interstellar drive capability in the coming years, among many other conditions and developments that their fruition requires, if the possibilities that these predictions identify are to be actualized.

In late September of 2011, this blog, and various internet forum threads, featured a series of posts about an old F.E.D. hypothesis, addressing a potential pathway to a superluminal interstellar drive --


The purpose of the present post is not to recapitulate the specifics of that old hypothesis as to a possible physical mechanism that might be harnessed to engineer an interstellar drive, since that hypothesis has already been addressed in those earlier blog-entries.

Rather, it is to share with you a related new hypothesis that has emerged within the current work of the Foundation research collective, regarding a potential solution to the two greatest mysteries of contemporary cosmology, and of contemporary cosmological physics.

Many contemporary mainstream theoretical physicists are, in our observation, experiencing a deep sense of bankruptcy, of late, as a result of recent superb and precision work by the community of experimental and observational physicists.

In particular, recent observational work by astronomers, cosmologists, and other ‘‘‘observational physicists’’’ has revealed that only ~ 4.9% of the mass-energy of the known cosmos is encompassed by the “Visible Matter” upon which standard model physical theorizing, to date, has been focused almost exclusively.

The majority of that known cosmological mass-energy -- ~ 95% -- is in the form of “InvisibleMatter-Energy:  an estimated ~ 26.8% taking the form of “Dark Matter”, and another ~ 68.3% taking the form of “Dark Energy”. 

Much of the mainstream theoretical physics community admits that it presently “hasn’t a clue” as to the nature of these majority constituents of Nature.  


Others have proliferated a veritable “Tower of Babel” of new, mostly “ad hoc” theories, or, really, of theory-fragments, or theoretessimals, isolated or scrambled in relation to the rest of physics, in attempting to explain these still-mysterious ‘majority phenomena’.

Lacking scientific explanations for the vast majority of the contents of the cosmos constitutes a kind of “scientific insolvency” for modern theoretical physics.

This text is a summary record of a “thought-experiment” currently being explored by the Foundation research collective. 

The question that this “thought-experiment” addresses is this --


¿What explanation of the nature of this majority of Nature might emerge if the alternative, “hypernumber”, ‘qualifier, [meta-]finitary solution to the v = c singularity of the Einstein Special-Relativistic Momentum Equation is “physical”, i.e., if it has empirical counterparts, rather than it being merely an ‘aphysical’, ‘anempirical’, “extraneous” solution?

This question turns out to be related to another question:  ¿What would result if we required that the values of the square-roots involved in the evaluation of the Einstein Special-Relativistic Momentum Equation, alike, for both subluminal and ‘‘‘luminal’’’ velocities, be ‘‘‘proper square roots’’’ [which are hypernumber-valued], as opposed to ‘‘‘improper square-roots’’’ [which are “Real-valued”], but required if and only if these hypernumber, ‘‘‘proper square root’’’ values turn out to be “physical?

Part 2. of this inquiry is set forth in the HYPOTHESIS section below.



HYPOTHESIS.  We have already noted that the dialectical theory of everything embodied and encoded in the F.E.D. Dialectic of Nature Meta-Models, falls short of covering and addressing everything as presently known to Terran humanity -- all known ontological categories -- in that it does not explicitly address the still largely unknown entities presently referenced as “Dark Energy” and “Dark Matter”.

Foundation researchers are also at work on a still highly speculative hypothesis, one that, if not falsified by empirical data, could ground a new standard version of our Dialectic of Nature Meta-Model.

This hypothesis in effect de-lumps’ the ALLn as «arché» category, by seeking a category for a

deeper, earlier-irrupted physical ontology, explicitly addressing Dark Energy, denoted ALLx

as the new «arché», plus a new, next-irrupted physical ontology, described algebraically as   

ALLqxx, basing an also pre-ALLn, first meta-«physis» category, explicitly addressing  

non-baryonic Dark Matter,  ALLc, i.e., the vast majority, by mass, of all Dark Matter.

This hypothesis thus amends our Dialectic of Nature Meta-Model meta-equationto the following for its epoch 2 equation [with |-.= serving as the solution-assertion equality sign] --

ALL>-|-<2  =  < ALLx >22  =   

< ALLx >4  = 

< ALLx2 >2  =   

< ALLx2 >  x < ALLx2 >   =  

< ALLx  +ALLqxx >  x < ALLx  +ALLqxx >  =

< ALLx  +ALLc >  x < ALLx  +ALLc >  =

ALLx  +ALLc   +ALLqcx  +ALLqcc |-.=

ALLx  +ALLc   +ALLqcx +ALLn

-- with ALLqcx representing the category of the interactions between non-baryonic 

Dark Matter” and “Dark Energy”, and such that ALLn, the present first, «arché» category

of our present standard Dialectic  of Nature Meta-Model, now arises, instead, in this new,

candidate ‘meta-model’, as the fourth, or second meta-«physis»’ category, namely, as --

ALLqxx[xx] |-.= ALLqcc |-.= ALLn.

This means that, if the generic primary interpretation of these solutions for the meanings of the

terms/category-descriptions ALLqxx, and ALLqcc, as expressed in the  ‘solution

sub-equations’ ALLqxx |-.= ALLc, and ALLqcc |-.= ALLn, holds in the

specific case of the ‘taxonomy level 1’, everything universe[-of-discourse], for the  

ALLc «monads» and the ALLn «monads» of this new candidate universal

meta-model’ -- as it does, they have found, for so many other domains, which are

sub-domains of this ‘ALL’ domain, and as it also holds for all of the other,

later-arising ‘self-hybrid’ categories of, e.g., the


< ALLr >2tALL 

meta-model’ version -- then we have two entailed sub-hypotheses regarding the composition

of the ALLc non-baryonic Dark Matter” «monads», and of the ALLn -- no-longer-

elementarystandard elementary particles” «monads». 

They are that the «monads» of the non-baryonic Dark MatterALLc  

«arithmos»/category aremeta-«monads»’ of the «monads» of the

Dark EnergyALLx «arithmos»/category. 

And that «monads» of the “Ordinary MatterALLn «arithmos»/category are  

meta-«monads»’of the «monads» of the non-baryonic Dark MatterALLc

«arithmos»/category, and are therefore also describable as meta-meta-«monads»’

of the «monads» of the “Dark EnergyALLx «arithmos»/category. 

That is, the two entailed sub-hypotheses hold that (1)  the 0 < v < c capable ‘contracton

«monads» of the non-baryonic Dark MatterALLc «arithmos»/category are in some sense  

made of the ‘‘‘tachyonic’’’, v > c capable ‘expandon’ «monads» of the “Dark Energy 

ALLx «arithmos»/category, and (2) the “tardyons” of the “Ordinary Matter 

ALLn «arithmos»/category  are in some sense made of the light-speed-capable

contractons’ of the non-baryonicDark MatterALLc «arithmos»/category.  

This also means that, in this ‘deeper «arché»’ version of the Dialectic of Nature Meta-Model,  
if it works out empirically, we will still, as in the present standard ‘meta-model’, have

composite sub-atomicparticles, such as protons, ALLs, arising as ALLqnn, and

atoms, ALLa, arising as ALLqss, and molecules, ALLm, arising as ALLqaa, and  

pre-eukaryotic living cells, ALLp, arising as ALLqmm, and so on, as before.

But, in addition, our ‘meta-model’ will be enriched with additional terms/category-descriptions,

relative to our current ‘standard meta-model’, covering the possible interactions among “Dark 

Energy”, ALLx, non-baryonic Dark Matter”, ALLc, and “Ordinary, Visible Matter

 categories, starting from ALLn, denoting the category of the “standard elementary particles”,

and onward from there. 

For example, for epoch 3, the new candidate ‘meta-equation’ yields the equation -- 

ALL>-|-<3  =  < ALLx >23  =   

< ALLx >8  =

ALLx  +ALLc   +ALLqcx  +ALLn  +

ALLqnx  +ALLqnc  +ALLqncx  + ALLs

-- including 3 additional categories/terms --  

ALLqnx,  ALLqnc, and ALLqncx --

inserted, with respect to our present standard ‘meta-model’, in-between ALLn and ALLs, and

addressing, respectively, the possible interaction-products of “standard elementary particle

«monads» with (1)Dark Energy” «monads», with (2) non-baryonic Dark Matter

«monads», and with (3) possible hybrid «monads», ALLqcx, the hybrid products

of the interaction of non-baryonic Dark Matter” «monads» with “Dark Energy” «monads».

This new research hypothesis seeks to anchor the entire Dialectic of Nature Meta-Model,
in what, per this hypothesis, represents the primary intra-duality, or self-duality, of our
cosmos -- in the older language of dialectics, the primary internal, immanent, or dialectical self-contradiction [ a relationship that we signify via the relation-sign ‘#’] of this cosmos
as totality, as expressed this meta-model’s first pair of terms, its first “antithesis sum” --

ALL>-|-<1  =  < ALLx >21  =   

< ALLx >2  =

< ALLx1 >  x < ALLx1 >   =


ALLx   + ALLqxx  |-.=

ALLx   + ALLc.

The “first antithesis” expressed by the ‘oppositional sum’ of the first two terms of this tentative new version of the Dialectic of Nature Meta-Model might be formulated as --  

Dark Energy   #  non-baryonic Dark Matter.

However, the Foundation researchers prefer to formulate that first antithesis in terms of the mutually opposing cosmological tendencies of -- 

Dark Anti-Gravity  #  non-baryonic Dark Gravity

-- or --

Unknown-Source Anti-Gravity  #  Unknown-Source Dark Gravity

-- or, better still, in terms of the mutually counter-acting tendencies of --  

space-time pro-expansivity  vsspace-time anti-expansivity.  

-- or --

space-time expansivity  vsspace-time contractivity.  

They see the space-time expansivity side, or ‘‘‘dual’’’, of this dialectical internal duality,

or self-duality, of the whole cosmos, as manifesting first in the history of our cosmos, making

it appropriate to choose ALLx as the new, candidate «arché» category of our sought new,

Dark Energy”/non-baryonic Dark Matter” comprehending, deeper-«arché»  

Dialectic of Nature Meta-Model. 

They also see the cosmological-expansive ALLx  Dark Energy” ‘anti-gravitic’ ‘‘‘force’’’ as manifesting in three major manners, so far, in the history of our cosmos, to present human
knowledge -- (1) as the initial “Big Bang” expansion; (2) as the theorized early post-“Big Bang” episode of faster-than-light-speed cosmological inflation expansion of the ‘space-time-mass-energy’ of the total cosmos, and; (3) as the currently-observed, slower-than-light-speed acceleration in the rate of expansion of the “space-time” of our cosmos.

They see the cosmological-contractive ALLc  non-baryonic Dark Matter  ‘‘‘gravitic’’’

force as manifesting, so far in the history of our cosmos, to present human knowledge, in the

cosmological formation of -- galactic ‘superw-clusters’:  super2-clusters or

super-super-clusters”, super1-clusters, “mere” [super0-]clusters, and galaxies

themselves -- also enabling the formation of stars and of stellar-systems, including of

stellar/planetary systems, etc., within galaxies.

They see the faster-than-light-speed expansion-capability of the ALLx  Dark Energy” ‘‘‘force’’’

as a possible key clue to the classification and identification of that ” ‘anti-gravitic’ ‘‘‘force’’’, as

well as to the classification and identification of cosmologically-contractive ALLc  non-baryonic

Dark Matter”.

That super-luminal capability suggests to us that a form of ‘‘‘materiality’’’ that is capable of faster-than-light-speed celerity will also be an extremely “ghostly” -- or “subtle” and “intangible” -- form of ‘‘‘materiality’’’, one that will lack almost all of the basic interaction capabilities and behaviors associated with “Ordinary Matter”, ALLn and beyond [ ‘ALLn+’ ]. 

That is, such a ‘‘‘materiality’’’ will lack electromagnetic interaction capability, making it “Dark”, as well as lacking capability for the “strong” and the “weak” interactions, retaining only the ‘‘‘gravitic’’’ ‘interactivity’, in the form of an anti-graviticity’.

These observations further suggests to them that the ‘‘‘materiality’’’ of cosmologically-contractive  

ALLc  non-baryonic Dark Matter” will be less “ghostly”, less “subtle” and “intangible”,

 relative to the ‘‘‘materiality’’’ of ALLx  Dark Energy”, but still very “ghostly”,

 or “subtle” / “intangible” relative to the tangible ‘‘‘materiality’’’ of ‘ALLn+

Ordinary Matter”.

In particular, they suspect that the «monads» of ALLc  non-baryonic Dark Matter” are --

like the photonic ‘‘‘materiality’’’* -- capable of traveling at light speed, c, but no greater

[thus distinguishing it from ALLx  Dark Energy”], and, unlike 0 rest mass

photon «monads», have > 0 rest mass, and are thus capable of traveling at speeds

less than c as well, unlike the photonic “luxons”, which are observed traveling only

at light speed, and never at speeds either greater than or less than the speed of light.   

*[For this reason, they group photons -- those ‘luxonic bosons’ -- together with
contractons’, as two speciesarithmoi» of a common «genos»-«arithmos», which
they sometimes call ‘alacritons’, signifying the «genos»-«arithmos» of all  
light-speed-capable «monads»].

As a consequence of these celerity characteristics, they further understand that the > 0 rest

mass «monads» of ALLc  non-baryonic Dark Matter” will lack electromagnetic interaction

capability, and hold that those «monads» will also lack capability for the “strong” interaction,

retaining only the “weak” interaction, and the ‘‘‘gravitic’’’ interaction, similar to the 0 rest

mass photon «monads», which are the ‘‘‘bosonic’’’ carriers of the electromagnetic

interaction, but which lack capability for the “strongand the “weak” interactions,

but which also retain ‘‘‘gravitic’’’ ‘interactivity’.

Conjecture -- 

Epsilonicity’ ofOrdinary Matter”; Hybridicity of “Dark Matter”; Ioticity of “Dark Energy”.

These hypotheses enable us to use the Einstein special-relativistic momentum equation

as a possible classification/identification mechanism for ALLx  Dark Energy”,  

ALLc  non-baryonic Dark Matter”, and  ‘ALLn+ Ordinary Matter

relative to one-another.

Newtonian momentum [p] of a body of mass-energy is the product of the velocity [v] and the inertial mass [m] of that body:  p = mv.

Einstein’s Special Theory of Relativity applies a correction factor, 1/( 1 - (v2/c2) )1/2,
to mv, in order to track the way in which the mass of such a body increases at increased
velocities --

p = mv/( 1 - (v2/c2) )1/2.

This latter equation exhibits a “singularity” -- a division by the square-root [or half-power] 
of zero -- should v = c --

p = mc/( 1 - (c2/c2) )1/2  =  mc/(1 - 1)1/2  =  

mc/(0)1/2  =  mc/0  =  oo”.

Division by zero is often evaluated as the ‘‘‘unphysical’’’ value “infinity”, or as signifying an indeterminate” state, or an undefined” value. 

The value x/0 for any number/value x in the ‘“Real”’ number set, R, is not a ‘“Real”’ number value; it is a value that breaks out of the “closure” of the ‘“Real”’ numbers.

The Einstein momentum equation’s escalation, in finite time, “to[ward]” an unboundedly-high magnitude as v ---> c, is empirically fit, as a tendency, given the measured pattern of escalation of the mass of a body of “ordinary” mass-energy as its velocity, v, approaches c.

However, in the actual universe, the velocity of any “ordinary” body accelerated toward v = c would stop well short of v = c, and, hence, well short of its momentum reaching mc/(0)1/2,
at some sub-luminal v, some v < c, and therefore also at some finite value of p.

The energy available from the rest of the universe to further accelerate the eventually
ultra-massive “ordinary mass-energy” body -- ultra-massive due to its ultra-high but still
sub-light velocity -- would run out, and the escalating mass of the ultra-high-velocity
“ordinary mass-energy” body would anyway make it ever-more-difficult to accelerate
further, and would also begin to pull-in the rest of the “ordinary matter” universe, at least,
due to this “ordinary” body’s ultra-high gravitational mass.

It would become a [¿universal?] “black hole” well short of reaching a velocity of v = c.

As a result, physicists often say that it is PHYSICALLY impossible for anything “material” or “substantial” -- i.e., that is more “substantial” than the somewhat “ghostly” photons -- to
reach light speed, let alone to travel at a speed faster-than-light-speed.

Indeed, this statement may be correct for “Ordinary Matter, for which the Einstein equation --   

p = mv/( 1 - (v2/c2) )1/2

-- describes a “speed-of-light barrier, one that, perhaps, “Ordinary”, “Visible”, “Bright”,  Electromagnetically-Reactive Matter” can never breach, at least not by ‘velocitarian’ means.

However, it is not MATHEMATICALLY impossible for v to equal, or even to exceed, c, in the Einstein special-relativistic momentum equation, and yet to still yield finite, determinate, defined values, values just as real as “Real” number values, but values which also transcend the space of the “Real” numbers.

To see this involves recalling a mathematical notion of “propriety”. 

The result of a mathematical operation upon a mathematical value is not a “proper” result if the result is the same value, again, as the value that was so operated upon, unchanged.

For example, the “subsets” function, operating upon a given set, includes that set itself in its result.  The given set is among its own subsets, is a subset of itself.  But that set is not a “proper” subset of itself.  A given set is only an improper” subset of itself.

Similarly we hold, with Musès, that 1 is not the “proper” square root of 1; 1 is, instead, the “improper” square root of 1 itself, and that 0 is not a “proper” square root of 0; 0 is, instead, the “improper” square root of 0 itself.

Let us, then, re-calculate Einstein special-relativistic momentum for the v < c, for the v = c, and for the v > c ranges, using only “proper” values for the square roots involved therein.

Ever since the European Renaissance, humanity has been working with square roots of
“negative” numbers, which involve the so-called “imaginary” number, the unit i, such that,
i = +(-1)1/2, or i2 = -1 -- although this unit is no more “imaginary” than the “Real”
unit 1, and is also, equally, no less “real” -- and is the positive “proper” square root of
negative unity, -1. 

Recall that i = iiii, so that also 1/i = -i, i.e., the “multiplicative inverse” of i equals its 
 “additive inverse”, that is, +i and -i both solve the algebraic equation  x-1  =  -x
thus exhibiting a characteristic which is alien to the realm of the so-called Reals” --

1/i = iiii/i = iiii/i  = iii/1 = (ii)i/1 = (i2)i/1  =  (-1)i/1 = -i.

The unit i is also ‘contra-Boolean’ -- i2 does not = i -- and is qualitatively, not quantitatively, unequal to the “Real” unit, in that i is not < 1, and i is not = 1, and i is not > 1.

We will therefore, henceforth, in this blog-entry, notate +(-1)1/2 as i, applying our standard underscoring, for ‘contra-Boolean operators’, to i too, even though this underscore-notation is non-standard” for standard” C arithmetic, algebra, and analysis.

Thus, the i unit is a mathematical qualifier’, not a ‘‘‘mathematical quantifier’’’, although i is also
a quantifiable qualifier’, e.g., i  +  i   =  2i.  

The i unit, in the form of 1/i, or -i, potentially qualifies’, or  ‘‘‘classifies’’’ and  ‘‘‘identifies’’’,

the kinds of “materialities” that can move [e.g., “expand”] at speeds

faster-than-the-speed-of-light, e.g., per our hypothesis, the ‘‘‘materiality’’’ of ALLx  

Dark Energy”, as the driver of primordial,  faster-than-light-speed,

cosmological inflation. 

It does so per the following chain of implications [ ‘==>’ ] --

v > c  ==>  v2/c2 > 1  ==>  1 - (v2/c2) < 0   ==> 

( 1 - (v2/c2) )1/2  = ( (-1)( (v2/c2) - 1) )1/2  = 

(-1)1/2 x ( (v2/c2) - 1 )1/2  =  (i)( (v2/c2) - 1 )1/2  =

i( (v2/c2) - 1 )1/2  ==> 

mv/( 1 - (v2/c2) )1/2  =  mv/(i)( (v2/c2) - 1 )1/2  =

mv/i( (v2/c2) - 1 )1/2  =  -imv/( (v2/c2) - 1 )1/2

-- so that 1/i, or -i, is the ‘arithmetical qualifier’ for the ALLx  Dark Energy” “materiality

when it expands ‘space-time-mass-energy’ at a faster-than-the-speed-of-light rate, as it can,

per the cosmological inflation theory, although the ALLx  Dark Energy” “materiality

must also be capable of accelerating “space-time” expansion at an accelerating but  

slower-than-the-speed-of-light rate, as observed in the present cosmos.

They do not associate the “minus sign” in this ‘-iqualifier as signifying that the superluminal velocities that it designates must involve travel backward in time, as does the “tachyon theory” -- a hypothesis that they regard as contra-empirical’ [unscientific], and rather fanciful.

They do consider the “minus sign” in this ‘-iqualifier as possibly related to the negative pressureassociated with anti-gravityforces.

There is also another “hypernumber unit[y]”, besides i, or “iota”, which can also be coherently
constructed, that was perhaps first identified as such by Dr. Charles Musès, and which
he named “Epsilon”, or E.

This value, E, or +E, is the positive “proper” square root of positive unity, of +1 -- the unit[y],

E = +(+1)1/2, or E2 = +1, and 1/E  =  E, [E is the [“improper”] reciprocal of itself],

since -- 1/E = EE/E  = EE/E  = E/1  = E, that is, +E and -E solve the algebraic equation

x-1  =  +x, thus exhibiting a characteristic that is alien to the realm of the so-called “Real” 

numbers, +1 and -1 excepted.

The unit E is also ‘contra-Boolean’ -- E2 does not = E -- and is qualitatively, not quantitatively, unequal to the “Real” unit, 1, in that E is not < 1, and E is not = 1, and E is not > 1.

We will, therefore, henceforth, in this blog-entry, notate +(+1)1/2 as E, applying our standard underscoring, for ‘contra-Boolean operators’, to E too, even though this underscore-notation was not part of Dr. Musès’s notational conventions.

Thus, the E unit is a mathematical qualifier’, not a ‘‘‘mathematical quantifier’’’, although E is also a quantifiable qualifier’, e.g., E  +  E   =  2E.   

The E unit potentially qualifies’, or  ‘‘‘classifies’’’ and  ‘‘‘identifies’’’ -- and, in a sense,  

‘‘‘characterizes’’’ -- the kinds of “materialities” that move only at less than the speed of light,

that is, the ‘‘‘materiality’’’ of ‘ALLn+  Ordinary”, “Visible”, “Bright”,   

Electromagnetically-Reactive Matter

The E unit can do so per the following chain of implications [ ‘==>’ ] --

v < c  ==>  v2/c2 < 1  ==>  1 - (v2/c2) > 0   ==> 

( 1 - (v2/c2) )1/2  = ( (+1)( 1 - (v2/c2)) )1/2  = 

(+1)1/2 x ( 1 - (v2/c2) )1/2  =  ( E )( 1 - (v2/c2) )1/2  =

E( (1 - (v2/c2) )1/2  ==> 

mv/( 1 - (v2/c2) )1/2  =  mv/( E )( 1 - (v2/c2) )1/2  =

mv/E(1 - (v2/c2) )1/2  =  +Emv/( 1 - (v2/c2) )1/2

-- so that 1/E, or +E, is, potentially, an ‘arithmetical qualifier’ for the kinds of “materialities

that can move only at less than the speed of light, that is, the ‘‘‘materialities’’’ of ‘ALLn+’ 

‘“Tardyonic, Ordinary”, “Visible”, “Bright”,  Electromagnetically-Reactive Matter”,

photons-reactive Matter’.

They therefore also call ‘ALLn+’  ‘“Tardyonic, Ordinary”, “Visible”, “Bright”,  Electromagnetically-Reactive Matter”, photons-reactive Matter’, by the name -- 

epsilonic matter.

They therefore also call their hypothesized “materialities” that can move [e.g., “expand”] at speeds faster-than-the-speed-of-light -- e.g., per their hypothesis, the ‘‘‘materiality’’’ of ALLx  Dark Energy” --

iotic matter.

Now, given what we have just seen -- that E, in the [equivalent] form of 1/E, can qualify and classify epsilonic, “ordinary”, SUBluminal ‘“bright matter”’, and that i, in the [non-equivalent]
form of 1/i can potentially qualify and classify their hypothesized, SUPERluminal,  iotic
“exotic” ‘“dark force matter”’, one might, “naively”, suspect that, in-between these two, the intervening category  of ‘‘‘LUMINAL’’’-velocity-capable, non-baryonic “Dark Matter”,
might be qualified and classified by a “hypernumber” value that is a mixture, or combination,
or hybrid of the “hypernumber”values 1/E and 1/i.

And, indeed, they find that their hypothetical ‘‘‘LUMINAL’’’-velocity-capable non-baryonic
Dark Matter”, potentially can be qualified, and classified, by a classic Musèan “hypernumber” value, a value that is, precisely, a ‘reciprocal-additive’ mixture, combination, or hybrid of the “hypernumber” values  1/E and 1/i -- namely, the reciprocalnilpotent hypernumber value --

1/( E + i ).

¿But how did they arrive at this determination?

The answer is that, once again, this determination involves the concept of a “proper” square root.

In particular this determination involves the concept of a “proper” square root of 0, already mentioned, in passing, above.

As we have seen, if v = c as, per their hypothesis, it can for Luminal’’’-velocity-capable,
non-baryonicDark Matter”, as well as for ‘‘‘luxonic matter’’’ [photons], then the Einstein special-relativistic kinetic momentum equation involves a division by the square root of 0, i.e., a division by (0)1/2  --

p = mc/( 1 - (c2/c2) )1/2  =  mc/(1 - 1)1/2  =  mc/(0)1/2

Typically, only the improper square root of 0 -- (0)1/2  =  0 -- is considered when evaluating this equation for its v = c possibility -- 

p = mc/( 1 - (c2/c2) )1/2  =  mc/(1 - 1)1/2  =  

mc/(0)1/2  =  mc/0

-- which results in an undefined value for the momentum for v = c, but a value which can readily be interpreted, given the finite escalation in the kinetic momentum as v increases toward c, that precedes the undefined value, for p, per that equation, that arises if v actually achieves c.

But let us consider an alternative evaluation of the relativistic kinetic momentum equation at velocity v = c, one based upon a proper square root of 0

Let us consider a nilpotent”, that is, a [hyper]number, call it n, which is not nil, not equal to 0 --

n  =  n1   ~=   0

-- but whose “potent” -- whose “power”; specifically whose second power, or square -- is nil, is equal to 0 --

n2   =   0

-- a value which can therefore serve as a proper  square root of 0 --

0   =   n2, so (0)1/2   =   (n2)1/2   =   n2 x (1/2)   =   ±n1   =   ±n.

By means of such a proper square root of 0 -- (0)1/2  ±n -- we would no longer be confined to the undefined” value for the v = c calculation of the special-relativistic kinetic momentum -- 

p = P(v) = P(c)  =  mc/( 1 - (c2/c2) )1/2  =  mc/(1 - 1)1/2  =  

mc/(0)1/2  =  mc/0

-- but could consider, as well, the possible physical meaningfulness of the calculation --

p = P(v) = P(c)  =  p = mc/( 1 - (c2/c2) )1/2  =  mc/(1 - 1)1/2  =  

mc/(0)1/2  =  mc/±n

-- which might be a quite finite, defined, determinate, meaningful value, albeit a hypernumber value, for p = P(c), a value which might identify, or be somehow associated with, other kinds, other qualities, of “matter-energy” than the familiar, “normal” -- actually RARE -- kind:  the epsilonic’,
ALLn+’,  ‘“Tardyonic, Ordinary”, “Visible”, “Bright”,  Electromagnetically-Reactive”,  
photons-reactive’ kind of “matter-energy”.

Now, such nilpotents -- of which many species are now known to be constructible -- are rather exotic constructions of the human mind, unfamiliar from ordinary arithmetic.

However, one of the most accessible nilpotents takes the form of the sum of the vary pair of hypernumbers that we have already invoked in this blog-entry:  ( E + i ).

Now, clearly,  n   =   n1  =   ( E + i )1   ~=   0.

But look what happens when we square ( E + i ) --

n2   =   ( E + i )2   =   ( E + i )x( E + i )   =  

( EE + Ei + iE + ii )   =   ( E2 + Ei + iE + i2 )   =

( 1 + Ei + iE + -1)   =   ( 1  + -1 + Ei + iE )   =

( 1  - 1 + Ei + iE )   =    ( 0 + Ei + iE )   =

( Ei + iE ).   

Now, it turns out that E and i exhibit a mathematical phenomenon called anti-commutativity”, which means that Ei  = -iE, or, equivalently, iE  = -Ei -- take this on my testimony for the moment, because we will show why it is so further below -- so that --  

( Ei + iE )  =  ( -iE + iE )  =  ( +iE - iE )  =  0

-- and therefore --

n2   =   ( E + i )2   =    0.

Thus, an alternative “solution” for the v = c value of the Einstein relativistic kinetic momentum, using “proper” square-roots, is --

p = mv/( 1 - (v2/c2) )1/2, which, for v = c, becomes --

p = mc/( 1 - (c2/c2) )1/2  =  mc/(1 - 1)1/2  =  

mc/(0)1/2  =  mc/( E + i )

-- which is a finite, defined, determinate, meaningful value, albeit a hypernumber value, for p.

To further illustrate why ( E + i )2   =    0, we have recourse to the realm of 2-by-2 ‘‘‘arrays of numbers’’’, or “matrices”, in which ordinary positive unity, +1, takes the following form, ‘1s on the diagonal, 0s off-diagonal’ --

| 1    0 |
| 0    1 |         

-- in which the standard product rule, for the multiplication of matrices, is defined as follows --

._      _............._......_............._..........................._
| a    b |.....x.....| e......f |....=....| ae + + bh |        
| c    d |...........| g.....h |..........| ce + + dh  |         
._      _............._......_............._..........................._

-- and in which perhaps the simplest species of the «genos» that Musès named E, or En, was designated by him as the unit E3, and is mimicked by the counter-diagonal” 2-by-2 matrix --

 |0    1 |
 |1    0 |         

-- via which we can easily see that E32 = +1 --

._....... _........... _........_............_........_
| 0    1  |.....x....| 0.....1 |...=.... | 1.....0 |        
| 1    0  |...........| 1.....0 |..........| 0.....1 |         

-- and via which we can also see, given that i is represented by the 2-by-2 matrix --

 | 0    +1 |
 | -1     0 |         

-- noting that this 2-by-2 matrix, squared, models i x i  =  -1 --

| 0....+1 |.....x.....| 0.....+1 |....=.....| -1......0 |.....=..-1..x..| +1......0 |       
| -1....0  |............| -1......0 |............| 0.....-1 |....................| 0.....+1  |         

-- that E3i  = -iE3 --

.._           _..............._..........._............._..........._
|  0      +1 |.....x......| 0......+1 |....=....| -1......0 |        
| +1     0  |.............| -1.......0 |...........| 0.....+1 |         
.._           _..............._..........._............._..........._

.._           _..............._..........._............._.........._
|  0      +1 |.....x......| 0......+1 |....=....|+1......0 |        
| -1       0  |.............| +1.....0 |...........| 0......-1 |         
.._           _..............._..........._............_..........._

 _            _..............._..........._
| -1........0 |.....=......| +1......0 |....x..-1        
|  0.....+1 |.............| 0........-1 |         
 _           _................_..........._

-- so that E3i  + iE3   =  0  --
.._            _............._..........._.............._...................._..............._........_
| -1........0 |....+.....| +1......0 |.....=.....| -1+1.....0+0 |....=......| 0......0 |.        
|  0.....+1 |............| 0.......-1 |............|   0+0...+1-1 |............| 0......0 |         
.._           ._.............._..........._.............._...................._..............._........._

-- and, given that ( E3 + i ) is matrix-modeled as follows --

|  0      +1 |.....+......| 0......+1 |....=....| 0......+2 |        
| -1       0  |.............| +1......0 |...........| 0.........0 |         

-- we can also see directly that ( E3 + i )x( E3 + i )   =  0 --

| 0......+2 |.....x......| 0......+2 |....=.....| 0......0 |.        
| 0.........0 |.............| 0.........0 |...........| 0......0 |         

Thus, in particular, 1/( E3 + i ) is, potentially, a ‘«genos»-eric ‘arithmetical qualifier’ for the ALLc  non-baryonic “Dark Matter” “materialityspecies, as well as for the “materiality” of photons species, which, per this hypothesis, are grouped together under a single «genos» «arithmos» whose «monads» are, collectively, named ‘alacritons’.

The category of ‘alacritons’ thus combines the photonic luxon species together with the non-baryonic “Dark Matter” species of ‘contractons’, all of the «monads» of that «genos» being light-speed-capable «monads» that are all not capable of speeds greater than the speed of light.

The photons, or luxons, are only capable of traveling at exactly the speed of light.

Per this hypothesis, the ALLc  non-baryonic “Dark Matter” ‘contractons’ are capable of traveling at speeds less than or equal to the speed of light, but not capable of traveling at speeds greater than the speed of light.  

Therefore, in overall summary, we have --
(1)ALLn+’  ‘“Tardyonic, Ordinary”, “Visible”, “Bright”, sub-light-velocity Matter‘‘‘classified’’’ and ‘‘‘qualified’’’ by the hypernumber quantifiable qualifier’ value 1/E3, 

(2) ALLc’ , light-velocity-capable, non-baryonicDark Matter”, as well as ‘‘‘luxonic matter’’’ [photons], ‘‘‘classified’’’ and ‘‘‘qualified’’’ by the hypernumber quantifiable qualifier’ -- the reciprocal “mixed” or “hybrid” hypernumber value -- 1/( E3 + i ), and;

(3) ALLx  super-light-velocity-capable, non-baryonicDark Energy”, ‘‘‘classified’’’ and ‘‘‘qualified’’’ by the hypernumber quantifiable qualifier’ value 1/i.    

If we assume that each of these three values, analytically-geometrically, ‘uni[ti]fies’ its own special axis in hypernumber space, perpendicular to the axis ‘uni[ti]fied’ by each of the other two, then each of these values is qualitatively unequal, not quantitatively unequal, to each of the other two values, so that the three kinds of “matter” are tagged with three qualitatively distinct ‘hypernumber qualifier’ “labels”.


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