In The Case Of A D-Field

A d-Field, is a substringular field -- that is here to exhibit a minimum of six spatial dimensions, plus time, -- over the course of the process of any given arbitrary directly affiliated Fourier Transformation, in which such an inferred Ward-Cauchy-related phenomenon, is here to act as being of a "d-field."  Aside from the three directly pertinent spatial dimensions that we are most familiar with, the fourth spatial dimension, is here to act as an elongated spiraling spatial dimension.  The next two spatial dimensions that are less obvious to us as people, are curled-up -- from within the Ward-Neumman confines of the four already inferred spatial dimensions, that I have here just eluded-to.  The "fifth" spatial dimension is curled-up, in such a manner -- to where it is here to be orthogonal in a Nijenhuis manner, to the fourth said spatial dimension, -- as may here be imagined in accordance to the right-hand-rule.  The "sixth" spatial dimension is curled-up, in such a manner -- to where it is here to be orthogonal in a Nijenhuis manner, to the fifth said spatial dimension, -- as may here be imagined in accordance to the right-hand-rule.  When it comes to any given arbitrary field, that is here to work to bear even more spatial dimensions than the minimum number of spatial dimensions, that may be exhibited by a d-field, -- this is what the general pattern, as to the presence of any added proximal local spatial dimensions -- that may potentially be attributed to the spatial parameters of the  Gliosis-related topological surface, of any superstring of discrete energy permittivity, -- that is here to incorporate the innate condition, of working to bear any more than six spatial dimensions plus time. Sam Roach.

Conditioned Substringular Orientation

Whenever an orbifold eigenset is conditioned to be of an unorientable nature, both its correlative composite superstrings of discrete energy permittivity as well as its correlative composite gauge-bosons -- will be put into such a Ward-Cauchy-related set of conditions, to have thus been made unorientable.  Yet, whenever the superstrings of discrete energy permittivity, that are here to be of an orbifold eigenset, are instead, to have been conditioned to be made orientable -- then its correlative composite superstrings as well as its correlative composite gauge-bosons, will then be of such a nature -- to where these will ensue to have been made orientable.  Gauge-Bosons will always tend to display the spatial dimensionality, that is of the correlative field of discrete energy permittivity that these indirectly help at working to form, via the nature of their kinematic Hamiltonian function -- yet, in such a manner to where such a general genus of heterotic superstrings -- such as the just mentioned gauge-bosons -- will always be of a symplectic homology, when this is to here to be taken simply in and of itself.  You see -- gauge-bosons are a certain form of heterotic stringular phenomenology, that are here to exist from within the multiplicit Ward-Neumman bounds of the core-field-density of any one individually taken discrete quantum of energy, which act in my particular string theory model -- in so as to pluck the correlative second-order light-cone-gauge eigenstates, that directly correspond to the wave-functionality of the discrete energy impedance of any one discrete quantum of energy, -- in so as to multiplicitly work to form that general genus of Schwinger-Indices, that are necessary, in so as to be able to help in the process of then to be working to form the general physical forces, that are to be exhibited by the said multiplicit discrete energy (such as either the strong force, the electrostatic force, the gravitational force, and/or the weak force).  Metaphorically think of this biological example.  Animals have mitochondria in their cells. Mitochondria are certain organisms that exist in all animal cells -- that act, in so as to help-out in the process of working to form most of the chemical energy of any one said animal.  Likewise -- gauge-bosons exist from within the confines of discrete energy quanta, -- in so as to help-out in the perpetuity of the existence of those said discrete energy quanta.
I will continue with the suspense later!  To Be Continued!  Sincerely, Samuel David Roach.

The Fourth Test Solution Of The Last Test Of Course 19 -- The Klein Bottle And Orbifold Differentiation

4)  The core-field-density of the general phenomenology of superstrings of discrete energy permittivity, is the topological surface that is Gliosis to the Ward-Caucy-based holonomic substrate of an eigenstate of any given arbitrary respective superstring of discrete energy permittivity -- as this is taken at the Poincare level to the Laplacian-based Ward-Neumman bounds of the respective incoming given arbitrary mini-stringular segmentation.  As the so-eluded-to mini-stringular segmentation works here, in so as to come together or to integrate as a Reimman scattering, into such a said so-eluded-to node of compiled norm-state-projection-like phenomenology, in such a manner in so as to work to form that relative chiral-based organization of particles -- that bear a level of relatively half full compactification per each individual "mer" -- as to those so-stated first-ordered point particles, that have here to come together into a proximity of their homotopic proximal localization, in so as to form that relative order of the multiplicit disturbance of space, to where this works to form a discrete increment of the pointal-based quanta of energy permittivity.  The directly corresponding counterstrings, of what are typically phenomenology that are positioned relatively parallel to the correlative superstring of any respective case during BRST, are positioned pretty much, just to the relative holomorphic side to the said individually taken superstring, to where such said counterstrings, bear a much greater webbing of mini-stringular segmentation Just Outside Of these first-ordered point particles that work to form their topological-based holonomic substrate.  This is why the counterparts of superstrings of discrete energy permittivity -- act as more of a multiplicit metrical-gauge-based Hamiltonian operator, that is of the wave-based function of the quanta of the said discrete energy permittivity.  The homotopic inter-binding of pointal/wavelike Ward-Caucy Fourier-related covariant differentiation, that is of those eigenindices that work to form the so-eluded-to discrete energy permititivty -- that is at the Poincare level to such an organized disturbance in space, works to form the topological stratum of the core-field-density of superstrings.

As To Physical Re-Delineation Of Masses

Let us consider a set of discrete masses.  Let us say that the masses that are to be considered here, are in a relationship that is to then involve the initial Laplacian-based condition of three basic hydrogen atoms.  In this given arbitrary case scenario, one is to here be considering the fundamental sub-atomic particles that are to then work to comprise the three so-stated basic respective hydrogen atoms -- that I have eluded-to in this given arbitrary case.  What I am more specifically referring to as the physical substrate of each of such individually taken discrete masses -- of which are to be considered here as the eigenmembers of this respective given arbitrary case scenario, are the two fundamental basic cases of what always exist in any stable atom, by which I am simply referring to as the initial Laplacian-based condition of the existence of one relatively stable proton, that is being surrounded by the kinematic-based orbiting of one relatively stable electron, -- that are of the nature of as to being of either the holonomic substrate of a proton, or of the nature of being of the holonomic substrate of an electron.  Let us say that all three of such mentioned fundamental hydrogen atoms (each of which are a stable atom that is comprised of one proton that is being orbited by one electron), are to then each become positively charged all of the sudden -- by losing their respective multiplicit electron -- due to a proximal localized group-attractor.  The then resulting three loose protons would then tend to move towards whatever loose negatively charged adjacent source, is most aptly to be able to bring this so-eluded-to loose set of three positive charges into a state of optimum rest.  In the course of such a basic situation -- one is to then have a re-adjustment of the local delineation of the distribution of the proximal gluonic force of this specific case scenario -- since the three directly corresponding protons of such a basic case that I have described here, are made unstable, in order to move into a changed delineation, in so as to move into the respective individually taken spots where these are then to be at more of a state of optimum rest.  Individual protons each consist of two quarks that bind to one lepton.  Any given quark has a gluon that is situated within the Ward-Neumman bounds of its topological-based holonomic substrate.  Gluons -- of which are an example of what works to act as being a metrical-gauge-based Hamiltonian operator of the strong force -- are an example of a general condition as to the cite as to where there is a centralized knotting of the Rarita Structure.  (Where there is a gluon, there is an eigenstate of the centralized knotting of the Rarita Structure.)  So, whenever a proton is re-distributed in so as to form a re-delineation of its component gluons that are directly associated with the here so-eluded-to  Ward-Caucy bounds of the so-stated respective proton -- there is then to be a re-delineation of an eigenstate of the strong force.  So, whenever there is a re-delineation of the strong force -- there is thereby a directly corresponding re-delineation of an eigenstate of the centralized knotting of the respective multiplicit Rarita Structure eigenstate.  I will continue with the suspense later!  To Be Continued!
Sincerely, Samuel David Roach.

A Different Tense Of Relative Covariance

Let us say that one were to have an orbifold eigenset, that was comprised of a set of smaller orbifolds, of which functioned as Hamiltonian operators from within the Ward-Neumman confines of the initially said orbifold eigenset.  Let us next say that the initially stated orbifold eigenset, was just acted upon by a group-attractor eigenbase -- in so as to go from moving in a divergent tense, via its respective given arbitrary Fourier-based translation, into moving as a Hamiltonian operator that is then put into a state of static equilibrium -- to where the overall orbifold eigenset of such a case, is put into a tense of conformal invariance over the course of a Wess-Zumino interaction -- that would exist here between the group-attractor and the said overall eigenset, over a relatively transient duration of time.  Let us then say that the orbifolds that are here to exist from within the Ward-Neumman bounds of the holonomic substrate of the orbifold eigenset, had here just been perturbated in a harmonic manner -- to where these are differentiating in a Fourier-based manner, over time, in such a manner that is cyclical, in a divergent manner, that is of less of a conformally invariant tense of kinematic motion.  This is to where, ironically enough, the interior-based orbifolds that are of a Ward-Caucy relationship with the overall orbifold eigenset that is of an external Ward-Neumman bounds, are in less of what may here be considered as a tense of static equilibrium, than the initially so-stated orbifold eigenset -- that had just been brought into a higher state or optimum relaxation -- over a sequential series of instantons.  Even though the orbifolds that are of a Ward-Caucy relationship with the externally constructed orbifold eigenset -- that work to contain the Ward-Neumman boundaries of the internally proximal localized orbifolds -- are of a relatively less stable covariant tense of both codetermination and codifferentiation, over time -- the internally proximal localized eigenindices, that are of the internal genre of a multiplicit framework, are of more of an entropic state of Fourier Transformation than the relative state of rest,that is associated instead with the Fourier Transformation of the external framework of codetermination and codifferentiation -- as the initially so-stated orbifold eigenset is oscillating in a tense of what may be here considered as a state of relative static equilibrium.  So -- this means that what would here be an externalized physical framework of Ward-Caucy Hamiltonian operation, is acting as a state of holonomic substrate that is in a  Majorana-Weyl-Invariant-Mode -- even though the action of the so-stated group-attractor that had brought the initially said orbifold eigenset into a Wess-Zumino interaction, in so as to bring the said eigenset into a harmonic perturbation that had then brought it into a tense of conformal invariance, did not simultaneously, through the vantage-point of a central conipoint, work to bear a Wess-Zumino interaction upon the Yukawa-based topology of the holonomic substrate of the internally-based orbifold-related fields that are Gliosis to the Hamiltonian operation of the overall said orbifold eigenset.  To Be Continued!  Sincerely, Sam Roach.

Different Tenses Of Covariance

Let us first initially consider an orbifold eigenset -- that is here to be differentiating in a Fourier-based manner, over a sequential series of group-related instantons.  Let us next say that one were to consider the index of the covariance of such a so-stated orbifold eigenset, as of a divergent genus of kinematic display -- since such a set  of superstrings that would operate here in so as to perform a specific function, would be effected here by a Cevita interaction among the relatively proximal neighboring substringular members -- that are of the general proximal superstringular region.  Let us then consider that the so-stated divergence -- that would here act as an annharmonic perturbation of the discrete quanta of energy of the said general substringular region -- is caused here by a set of ghost-inhibitors, that would then be acting upon the cohomology-based topological entity, that is directly appertaining to the physical memory of the initially stated orbifold eigenset of holonomic substrate, to where the projection of the trajectory of the superstringular indices that would work here in so as to form the cohomological mappable tracing of the said orbifold eigenset, is then pulled, in such a manner in so as to cause the initially stated orbifold eigenset to be moved out of its given arbitrary beginning tense of Majorana-Weyl-Invariant-Mode -- into a relative propagation of its Hamiltonian operation, in so as to move out of the general Yukawa field of the so-stated ghost-inhibitor in such a manner in so as to gradually but possibly either spuriously or not spuriously, move into a region of more optimum rest.  Let us next say that the said orbifold eigenset that is in question here, is comprised of tinier orbifolds -- that would then be within both the Ward-Neumman and the Ward-Derichlet physical boundaries of the substringular confines of the initially stated orbifold eigenset.  Let us then consider the individually taken orbifolds -- that are of a smaller scalar magnitude of Hodge-Index of discrete energy quanta -- are acting in a Fourier-based manner, as a framework of Hamiltonian-based operation that is of a smaller and of a more proximal-based localized set superstringular region of kinematic-based operation, over time.  Let us then say that this so-stated smaller framework of Fourier-based differentiation, is in a tense of conformal invariance -- to where these orbifolds, that would here acts as a smaller Ward-Caucy-based boundary cite -- would then be existing in a Majorana-Weyl-Invariant-Mode, that, when taken at a fractal of its externalized field -- that is then of the larger and also initially said orbifold eigenset that I have mentioned here -- is of both a covariant, codeterminable, and of a codifferentiable field index, that is within a divergent field, that is not conformally invariant, at the Poincare level to the Gliosis-based topological surface of the core-field-density that is of the initially stated orbifold eigenset.  This is of a case -- to where -- one is to have a Fourier-based translation of delineatory inidces, that are of an orbifold eigenset -- that would here work to bear in composition the existence of internal eigenindices, that are in a tense of a Majorana-Weyl-Invariant-Mode, over a sequential series of the correlative iterations of group-related instantons.  I will continue with the suspense later!  To Be Continued!  Sam Roach.

More As To The Indistinguishable Differences In Charge

When there is an indistinguishable difference in the Real Reimmanian-based charges of any given arbitrary orbifold eigenset, that works to exhibit such a said condition -- there is, as well, an indistinguishable difference in the directly associated Imaginary or Njenhuis-based charges, that are also associated with the so-stated orbifold eigenset.  For instance, let us say that one is to extrapolate as to what is going on with a given arbitrary orbifold eigenset -- to where this said eigenset, of which here is detected -- in so as to be stable in its charge.  This means -- over the time in which such an eigenset is charge-wise stable -- that for every charge that works to enter the said orbifold eigenset, that there is simulteneously (through the vantage-point of a central conipoint) the equivalence of a like charge (in terms of the general genus of parity) that works to leave the so-mentioned orbifold eigenset.  As such charges are leaving and entering the Ward-Caucy bounds of the said eigenset (mainly, as such said charges are leaving and entering the Ward-Neumman bounds of the said eigenset), as taken per each of such individually taken Real Reimmanian charges, that there is a Njenhuis or an Imaginary-based charge, that is of the same general genus of polarity -- that is then stemming from the so-eluded-to Real Reimmanian-based charge -- in a manner that is orphoganal to both the wave-tug/wave-pull of that holonomic substrate that is here correlative to the motion of the so-stated Real-based charge's tendency of propagation in the relative holomorphic direction, as well as simultaneously (through the vantage-point of a central conipoint) there also being a condition of an orphoganal stemnming of the so-eluded-to Njenhuis-based charge, that is edge-wise tangent at the locus of a relative endpoint-based singularity, to the Real Reimmanian-based plane of such a respective given arbitrary case scenario.  Such a multiplicitly-taken endpoint of a covariant-based tangency, will tend to be of a Lagrangian-based Chern-Simmons singularity, over time.  To Be Continued! Sam Roach.

Part Four of Session 10 of Course 19 -- The Klein Bottle and Orbifold Differentiation

If a superstringular manifold works to help bear the condition of describing just the Real Reimmanian-based dimensional object of holonomic substrate, that is implied by the Hamiltonian operation of any respective given arbitrary orbifold eigenset in question -- the so-eluded-to superstringular manifold, that would then thus be described of as such, would be what may be termed of as a substringular membrane that would here involve only the consideration of three spatial dimensions, since such a limited consideration would then not be including a monitoring of those Njenhuis spatial parameters of dimensionalilty -- that are as well inherent in the extrapolation of both the Ward-Neumman existence and the overall motion of the so-mentioned and eluded-to orbifold eigenset, that is of this respective given arbitrary consideration.  This would then work to add more of a lack of an appropriate tense of expectation values, to the effort of then working to bear the otherwise applicable probabilities as to both the Laplacian-based differentiation and the Fourier-based differentiation of that orbifold that is then not being fully studied in such a genus of a limited case.  So, the more that is understood as to both the Laplacian-based conditions and the Fourier-based conditions of an orbifold eigenset -- the better that both the so-stated Ward-Neumman existence and the so-stated kinematic-based activity of such a said orbifold eigenset may be predicted, as to the what, where, how, and why, that such a so-stated group of superstrings that operate in so as to perform a specific function is to behave, over time and space.  That is why it is necessary to understand all of the directly applicable respective Njenhuis spatial parameters of dimensionality, that work to bear a Yakawa-based wave-tug/wave-pull upon the Gliossi-based topological surface of any respective given arbitrary single functioning holonomic substrate of a respective tense of Ward-Caucy bounds, of any respective given arbitrary orbifold eigenset that is to be extrapolated -- in order to maximize those expectation values, that are needed in order to better predict any discrete quanta of Hamiltonian operation, over time.
I will continue with the suspense later!  To Be Continued!  Sincerely, Sam Roach.

The First Part of an Aside to Course 16

When a superstring reiterates through a trajectory, the given superstring that was just mentioned works to form a ghost anomaly -- that acts as the physical memory of the just eluded to world-sheet.  (A world-sheet is a discrete trajectory of a superstring.)  When either a one-dimensional superstring of discrete energy permittivity, or, when a two-dimensional superstring of discrete energy permittivity, forms a world-sheet -- there is often an annulus formed from within the Ward-Neumman bounds of the ghost anomaly that acts as the physical memory of the just eluded to world-sheet.  At the substringular level, the just mentioned genus of annulus is here an oriface or "hole" where there are no first-ordered point particles present -- at any given arbitrary iteration of group instanton, that works to directly correspond to the said condition of there being an annulus -- from within the Ward-Neumman bounds of the Hamiltonian operand of the projection of the trajectory of any given arbitrary superstring of discrete energy permittivity -- that would work to form such a said annulus over the course of the kinematic activity of such a given arbitrary superstring.  An annulus here is an empty projection that works to bear no first-ordered point particles that would otherwise contain such eluded to first-ordered point particles.  World-Sheet projection that bears both a Lagrangian-based smoothness and a metrical-based smoothness over time is said to bear a Yau-Exact genus of Real cohomology.  A Yau-Exact genus of Real cohomology does not in and of itself bear any directly corresponding Chern-Simmons singularities, when considering this from within the premises of its Ward-Neumman bounds, from within either the Lagrangian-based and/or from within the metrical-based kinematic orientation that works to directly affiliate with any given arbitrary ghost anomaly, of which corresponds to the trajectory of the motion of the corresponding superstrings of discrete energy permittivity that may be considered here as forming the eluded to projection over time.  A Yau-Exact genus of a Real cohomology tends to kinematically differentiate within the Ward-Caucy bounds, over time, in such a manner that bears a harmonic oscillation during the directly corresponding sequential series of iterations of group instanton -- in which the correlative superstrings of discrete energy permittivity that I have here eluded to are differentially associating as having the said Yau-Exact condition of a Real-based cohomology.  The directly associated Fadeev-Popov-Trace eigenstates that would here thus correlate to the said superstrings of discrete energy permittivity -- of which act as the corresponding discrete units of energy impedance -- would thus act accordingly as physical phenomena that bear a Yau-Exact genus of Real cohomology.  The difference in operational function that exists between a discrete unit of energy permittivity and a discrete unit of energy impedance is that the respective superstrings work to pull the discrete energy in the general manner of the relative holomorphic directoral topological-based sway, while the respective Fadeev-Popov-Trace eigenstates work to hold the discrete energy, in a homotopic manner,  back enough to keep the said energy from resonating.   Superstrings act as discrete units ofenergy permittivity, while, Fadeev-Popov-Trace eigenstates act as discrete units of energy impedance.  I will continue with the suspense later! Sam Roach.

The Fifth Part of the Fifteenth Session of Course 14

When two-dimensional superstrings map out a trajectory, with only the existence of relatively transient ghosts, one may still call this format of mapping-out of the so-stated two-dimensional string -- whose mapping works to form a twisted shaft in the form of three-dimensional world-sheets, the said detection of the ghost anomalies -- as  formed by what is here the forward-holomorphic norm-states -- is Imaginary in this given scenario.  The two-dimensional world-sheets that work to form from the kinematic differentiation of one-dimensional superstrings often forms a twisted cylinder, because space not only displays jointal conditions, yet also, space also displays smooth curvedness.  The three-dimensional world-sheets that form from the kinematic differentiation of two-dimensional strings forms a twisted shaft, because space not only displays jointal conditions, yet also, space also displays smooth curvedness.  World-Sheets that are relatively Real Reimmanian are relatively limited in trajectory, because ghost anomalies are annhilated periodically in so as to balance gravitational force.  This causes the condition that both Real and Imaginary (Njenhuis here) cohomologies must always simultaneously exist.  Loci that display Imaginary cohomology are often shifted to display Real cohomology in other neighboring loci that involve multiple superstrings.  Homology that is between two and three-dimensional world-sheets is Imaginary, even when the directly corresponding Ward-Neumman conditions that exist here  in-between these given world-sheets is even.  The reason for this is because the Ward-Derichlet conditions of these world-sheets is centrally variant.  Central variance does not tend to allow for smooth interconnection.  As world-sheets codifferentiate, when involving Real-based ghost anomalies, the loci of the directly corresponding Real and Imaginary (Njenhuis) cohomology shifts in so as to allow the corelative superstrings and orbifolds to display differential tangencies in the multiple spatial dimensions that these said superstrings kinematically differentiate in.  I will continue with the suspense later, in the form of the test questions for this corresponding course!  Sincerely, Sam Roach.

Solution to Ninth Part of Test One of Course 5

9)  During ths sub-metric in which the Bases of Light are manifested, there are countless mini-string or substringular field eigenstates that interconnect partial loci of individual Bases of Light to the topological holonomic surface of superstrings which are to subsequently be retied relatively proximal to their respective Planck Phenomenon related phenomena.  As mini-string ebbs back-and-forth during the untying of the Bases of Light, the corrersponding superstrings, which act as discrete units of energy permittivity condenses the anharmonically translated to harmonic flow of substringular field holonomy into the countless discrete Ward Caucy neighborhoods which define the sundry eigenstates of interbound energy permittivity that is balanced by discrete energy impedance via the wave residue in-between these Laplacian physical Ward Neumman states that acts as the light-cone-gauge.  The decompactification of the Bases of Light serves to spacially delineate and redistribute discrete energy phenomena via the sum of the partial integrands of the said Bases' holonomic structures as this basis as a whole is unwoven via the sub-metrics of the sequential series of the multiplicit dot product operation of the enfoldment of each respective Hodge-based perturbation.  The bases of the respective Hodge indices is the supplemental norm conditions that are mapped Yau-Exact if Noether and Chern-Simmons if tachyonic via the group action of the gauge-metrical activity of stringular encoders upon the sum holonomic manifestation that transpires during the sub-metric that is known as the instanton-quaternionic-field-impulse-mode.  I hope that the solutions that I have posted today are helpful!  Sincerely, Sam.