Torsion Applied Upon A Homotopic Divergence/Convergence

 A Euclidean-Related torsion, that is to be applied upon a homotopic divergence/ homotopic convergence, is to work to bear a directly associated Yukawa Coupling, that will often tend to be eminently associated, with a Poisson-Related recursion. Whereas; A Clifford-Related torsion, that is to be applied upon a homotopic divergence/homotopic convergence, is to work to bear a directly associated Yukawa Coupling, that will often tend to be eminently associated, with a Slater-Related recursion. SINCERELY, SAM ROACH. 

Alteration In Type Of Expansion -- Lagrangian-Based Chern-Simons Singularities

 When a given arbitrary Noether-Based mass-bearing cohesive set of discrete energy quanta, is to spontaneously alter, from exhibiting a euclidean expansion, into ensuing to exhibit a Clifford expansion, the stated cohesive set of discrete energy quanta, at the locus in which such a change in the type of expansion, that is here to be exhibited in its consequentially resultant kinematic display, will often tend to work to bear a certain given arbitrary respective tense, of a Lagrangian-Based Chern-Simons singularity. Likewise; When a given arbitrary Noether-Based mass-bearing cohesive set of discrete energy quanta, is to spontaneously alter, from exhibiting a Clifford expansion, into ensuing to exhibit a euclidean expansion, the stated cohesive set of discrete energy quanta, at the locus in which such a change into the type of expansion, that is here to be exhibited in its consequentially resultant kinematic display, will often tend to work to bear a certain given arbitrary respective tense, of a Lagrangian-Based Chern-Simons singularity. TO BE CONTINUED! SAM.

Hyperbolic Euclidean/Clifford Expansion

 A given arbitrary hyperbolic euclidean expansion, is often to have a relatively heightened general potential, of working to asymptotically approach its general locus of attempted convergence, in a heuristically tangential manner. Whereas; A given arbitrary hyperbolic Clifford expansion, is often to have a relatively heightened general potential, of working to asymptotically approach its general locus of attempted convergence, in an inversely tangential manner. TO BE CONTINUED! SINCERELY, SAM.

Time — Delineation

 Discrete energy quanta, that directly work to involve forward moving time momenta, generally have a heightened tendency,  of working to bear second-order light-cone-gauge eigenstates, that are often to tend to bear the eminent incursion, of a Euclidean delineation of positional distribution, as taken over the course of a given arbitrary iteration of group-related instanton; Whereas, —  Discrete energy quanta, that directly work to involve backward moving time momenta, generally have a heightened tendency, of working to bear the eminent incursion, of working to bear second-order light-cone-gauge eigenstates, that are often to tend to bear the eminent incursion, of a Clifford delineation of positional distribution, as taken over the course of a given arbitrary iteration of group-related instanton. To Be Continued Later! Sincerely, Sam Roach. 

Initial Propagation

 The initial set of Higgs Boson eigenstates, that came into existence right after the Big-Bang happened, worked to create discrete energy, which was based upon the interdependent interaction of a correlative set of gauge-metric-related eigenstates, in what managed to be directly associated with two main physical attributes of expansion. These two expansion-related characteristics, were: 1) Discrete energy that was based upon the reductional flow of a Euclidean expansion; And — 2) Discrete energy that was based upon the reductional flow of a Clifford expansion. The inferred Euclidean flow, primarily worked to form superstrings of discrete energy permittivity; Whereas —The inferred Clifford flow, primarily worked to form gauge-bosons. Superstrings of discrete energy permittivity, work to form actual energy; Whereas, propagated vibrations that are “indirectly” formed by the interactions of gauge-bosons, eminently work to transfer the reductional forces of physical nature.  TO BE CONTINUED! Sincerely, Samuel David Roach. (1989).

Escalating Lagrangian — Chern-Simons Singularities, — My View

 Those Lagrangian-based Chern-Simons singularities, that are thence to tend to be formed, by the gauged-action of the heteromorphic escalation of a correlative Lagrangian, that is here to be of a Clifford Expansion, tend to bear an anharmonically permutative tree-based nature; whereas, — those Lagrangian-based Chern-Simons singularities, that are thence to tend to be formed, by the gauged-action of the heteromorphic escalation of a correlative Lagrangian, that is here to be of a Euclidean Expansion, tend to bear an anharmonically permutative commutative-trace-based nature. To Be Continued! Sincerely, SAMUEL DAVID ROACH. (1989).

Compounded Homotopic Residue

 In general — a binary homeomorphic escalating Lagrangian, that is here to be undergoing a Euclidean/Clifford Expansion, tends to work to exhibit a greater scalar attribute, of a proximal local tense of compounded homotopic residue, than a unitary homeomorphic escalating respective Lagrangian, that is here to be undergoing a respective Euclidean/Clifford Lagrangian does; a tertiary homeomorphic escalating Lagrangian, that is here to be undergoing a Euclidean/Clifford Expansion, tends to work to exhibit a greater scalar attribute, of a proximal local tense of compounded homotopic residue, than a binary homeomorpjhic escalating respective Lagrangian, that is here to be exhibiting a Euclidean/Clifford Expansion does; etc... . I will continue with the suspense later! Sincerely, SAMUEL DAVID ROACH.

Escalating Lagrangian — Isotropic Spur

 When one is to have a given arbitrary escalating Lagrangian of Euclidean/Clifford Expansion, that is here to work to bear a mappable isotropic “spur,” at a semi-set locus that is Virtually of a pure Laplacian-related nature, as this is here to be mapped-out, along the path of the inferred determinable projected trajectory, of the Euclidean/Clifford Lagrangian-based mappable-tracing of the said expansion, — the Flow of the inferred general genus, of such an escalating Lagrangian of Euclidean/Clifford Expansion, as taken through the coni-point, of the proximal locus of the inferred mapped-out semi-set locus, of such an isotropic “spur,” will consequently tend to work to bear the manner, of being of a hermitian nature. However; when one is to have a given arbitrary escalating Lagrangian of Euclidean/Clifford Expansion, that is here to work to bear a mappable isotropically Unstable “spur,” at a semi-set virtually Laplacian-related locus, as this is here to be mapped-out, along the path of the inferred determinable projected trajectory, of the Euclidean/Clifford Lagrangian-based mappable-tracing of the said expansion, — the Flow of the inferred general genus, of such an escalating Lagrangian of Euclidean/Clifford Expansion, as taken through the coni-point, of the proximal locus of the inferred mapped-out semi-set virtually Laplacian-based non-isotropic “spur,” will then consequently tend to work to bear the nature, of working to bear the general condition, of exhibiting a Lagrangian-based Chern-Simons singularity. SAM.

From Accelerating Via A Euclidean-Related Path Into Accelerating Via A Clifford-Related Path

When any one given arbitrary Noether-based orbifold eigenset, that is here to work to maintain its general tense of spatial dimensionality, is to initially go from being spatially transferred into a tense of a smooth transversally accelerating pattern of motion, via a mappable cohomology that may be traced in a Laplacian-based euclidean-related path, Into then subsequently altering in the general genus of its Lagrangian-based path, -- by being smoothly delineated via a mappable cohomology-related path, that is instead here to be of a Clifford-related manner, in spite of working to bear what would otherwise be the same consistent general pattern of being spatially transferred, via a tense of what would otherwise be the same general type of a smooth transversally accelerating manner of motion, over an evenly-gauged Hamiltonian eigenmetric, that is here to be of the same duration as the initially inferred general tense of motion; then, at the point in time and space at which such a said orbifold eigenset is here to be altering from being spatially transferred as a Hamiltonian Operator, that is here to initially be working to form a euclidean mappable cohomology-related path, via the operational translation of its Lagrangian, over a discrete quantum of time, Into subsequently being spatially transferred as a Hamiltonian operator, that is here to work to form a hyperbolic mappable cohomology-related path, via the operational translation of its Lagrangian, over a discrete quantum of time, -- this general type of an occurrence will consequently tend to work to cause, at the proscribed point of such an alteration in the general genus of a mappable-tracing, a set of one or more Lagrangian-based Chern-Simons singularities; -- although to where such a respective general genus of a scenario, will Often not tend to work to form metric-related Chern-Simons singularities, on account of the stated condition here proscribed of such an orbifold eigenset of such a respective given arbitrary case, to be a case in which one is here to be considering a Hamiltonian Operator, that is to be maintaining its condition of a smoothly-accelerating transversal-based pattern of motion, through time and space. Samuel Roach.

Change In Tense Of Slater-Related Propagation

Let us consider an orbifold eigenset, that is here to be moving via a Lagrangian-based path, that is of a euclidean nature, -- to where the here inferred propagation of this said orbifold eigenset, is here to be of a Slater-related nature.  This implies, that the propagation of the said eigenset -- is here to be in the process of being translated through space, via a mean Lagrangian-related Hamiltonian operand, -- to where such a said eigenset, may consequently be depicted of as a set of a discrete quanta of energy, -- that are here to be in the process of being transferred in a relatively "straight" manner, through some sort of a tense of an identity function, (F(y) = F(x)).  Furthermore; let's now say, that the ensuing nature of the motion of such a said orbifold eigenset, via its correlative Lagrangian-based path, is here to alter from its initial euclidean-related nature, -- to where even though the said orbifold eigenset is still to be propagated in a Slater-related manner, it is to instead to be of a Clifford-related nature.  Consequently; the said orbifold eigenset, is now to work to bear a hyperbolic translational flow of its spatial eigenindices, in the process of its spatial transference over time, -- to where, instead of the said respective eigenset to thence to be moving in a manner that may be depicted of as being translated in a relatively "straight" way, via some sort of an identity function, it is instead to be moving in a manner that may be depicted of as being translated in a relatively concave-up way, such as with (F(y) = F(x)^(a constant)).  Sam Roach.

Part Two Of Session 13 Of Course 20

By the way, -- the general condition of the torsion that alters the order of any one said light-cone-gauge eigenstate by 90i degrees, works to cause the correlative composite second-order light-cone-gauge eigenstates to either go from working to bear a sinusoidal delineation -- to working to bear a relatively supplemental delineation, when there is a light-cone-gauge perturbation that is to go from working to exhibit a Yang-Mills or non abelian light-cone-gauge topology, to working to exhibit a Kaluza-Klein or abelian light-cone-gauge topology, or, in the ulterior case, to cause the composite second-order light-cone-gauge eigenstates to go from working to bear a relatively supplemental delineation, to working to bear a sinusoidal delineation, when there is a light-cone-gauge perturbation that is to go from working to exhibit a Kaluza-Klein or abelian light-cone-gauge topology to a Yang-Mills or non abelian light-cone-gauge topology.  Such a said torsioning is necessary, in order to work to allow for electromagnetic energy to initially to be able to re-arrange as entropic photons, when it is first disorganized by a correlative scattering -- while to then to be able to re-group and re-quantize as non entropic photons, once such electromagnetic energy has torsioned back into the general genus of the proximal local electromagnetic radiation.  Once the multiplicit photonic energy is to be re-quantized back into the general proximal local electromagnetic radiation, in so as to here to be having a Yang-Mills light-cone-gauge topology -- its Clifford algebra will then have more of a tendency of going from working to bear a primarily euler and a Dirac geometry, into working, instead, to bear a primarily euclidean and a hermitian geometry, over time.
 I will continue with the suspense later!  To Be Continued!  Sincerely, Samuel David Roach.

Affiliated With Light-Cone-Gauge Expansion

Let us initially consider a given arbitrary discrete quantum of energy, during the course of one specific given arbitrary iteration of BRST.  Over the course of such a so-eluded-to duration -- both the directly corresponding superstring of discrete energy permittivity, and, its immediate superstringular counterpart -- are decompactified to the inverse of the extent that these are Lorentz-Four-Contracted.  This so-stated process of decompactification, may be called the Polyakov Action.  Due to the condition, that the correlative Fadeev-Popov-Trace eigenstate, that is to directly correspond to both the said superstring and its said counterpart -- does not work to decompactify with the so-eluded-to discrete quantum of energy permittivity, -- as both the so-stated superstring and its superstringular counterpart are in the process of being decompactified, there is a hyperbollic-based Clifford Expansion that is to here occur, in the Laplacian-based Hamiltonian-based region, that is to exist in-between the correlative superstring and its directly corresponding Fadeev-Popov-Trace eigenstate, over the course of the said iteration of BRST.  Any given arbitrary superstring, is to bear an oscillation during BRST -- that is to be both codeterminable and codifferentiable in a pseudo-based Fourier-related tense, as being of a of a euclidean tense of orphoganation -- with any adjacent respective given arbitrary superstrings, that are of the same universal-based setting in the substringular.  Adjacent superstrings that are of the same universe, may potentially vary completely in the extent by which their Lorentz-Four-Contractions are to bear as a Laplacian-based scalar amplitude by, and thus, by which their Polyakov Action eigenstates are to then bear as their individually taken scalar amplitude -- in the manner in which these do.  This will then tend to cause the correlative tense of the wobbling of the directly affiliated second-ordered light-cone-gauge eigenstates, as well as the correlative tense of the vibrational oscillations of both the respective superstring, its counterpart, and the correlative Fadeev-Popov-Trace eigenstate, -- to tend to bear such a tense of conformal invariance -- that will work to help suit both the directly associated Laplacian conditions and the directly corresponding Fourier-based conditions, in so as to bear the necessary tense of wobble, while still working to bear the necessary tense of the inverse activity of the so-stated Polyakov Action eigenstate, that may here be attributed to the correlative and the respective scalar amplitude of the here pertinent tense of its Lorentz-Four-Contraction, over the course of the said iteration of BRST of this respective given arbitrary case scenario. 

I will continue with the suspense later!  To Be Continued!  Sincerely, Samuel David Roach.

About Antiholomorphic Divergences

Let us here consider two different orbifold eigensets, that are initially moving in a kinematic manner -- with what may be here viewed of as a supplemental line, that is not of a Wilson-based linearity, that works to inter-bind the topological index of the physical entity of their mutual condition of holonomic substrate.  Let us now say that the two so-stated orbifold eigensets have been differentiating in a Fourier-based manner, for a relatively preminent duration of time -- to where these may both be considered here to be moving in the same general directoral-based genus of holomorphicity -- over the so-eluded-to eminent sequential series of group-related instantons.  Let us now say that there is a group-inhibitor index of one general given arbitrary nature -- that acts upon the two individually taken substringular entities of holonomic substrate of Hamiltonian operation -- that acts in so as to work to prohibit the continued covariant Fourier-based activity, that is of the motion of the two so-stated orbifold eigensets that are here initially flushly moving in a common general direction, with a Ward-supplemental subtension that is to be here drawn-out between the two said eigensets of superstrings, in a Laplacian-based manner.  Let us now say that the Fourier-based operation of the Hamiltonian-based delineation of the functional index of the so-stated group-inhibitor, would here work to cause a relatively sudden Njenhuis-based Ward-supplemental motion -- that would here be enacted upon both of the said orbifold eigensets of this respective given arbitrary case -- to where the two so-eluded-to sets of superstrings will then be moving with the opposite general index of delineatory operation, over the directly ensuing sequential series of group-related instantons.  This immenent and swift perturbation in the immediate field density -- that would then exist between the two said orbifold eigensets would then work to form a hyperbolic tangential-based covariant field of divergence -- that would be exhibited here by a Cevita interaction that would be operational between the two Clifford-based Hamiltonian field operators (the two orbifolds of which would here be the two said operators).  The so-eluded-to perturbation of divergence would then be an annharmonic perturbation -- that would form both Lagrangian-based Chern-Simons singularities and metrical-based Chern-Simons singularities -- of which would then work to form an antiholomorphc Kahler condition, of which would then work to set-on an ensuing Kahler-Metric.
I will continue with the suspense later!  To Be Continued!  Samuel David Roach.

An Explaination Of Topology

Superstrings are comprised of first-ordered point particles that integrate into the general shape-like formats of either vibrating strands or vibrating hoops that form the discrete energy that makes-up the Continuum. Superstrings are generally vibrating hoops of such phenomena. First-Ordered point particles are "yarned together" mini-string that forms the general compositional genus of tiny spheres that organize to form the prior mentioned superstrings. The mentioned vibrating hoops are bosonic strings -- of which are primordially considered as two-dimensional. The mentioned vibrating strands are fermionc strings -- of which are primordially considered as one-dimensional. This does not directly consider the condition of heterotic superstrings, even though heterotic strings are completely real. This goes beyond the scope of what I am discussing here. The prior said mini-string is the fundamental basis of the topology that works to form the basis of substringular fields. Superstrings are primarily interconnected by the said mini-string. Topology is the general term for the "substance" that forms both the Poincaire-based and the Clifford-based interconnectivity that works to form the respective composition and the networking of superstrings -- in so that superstrings may not only exist in such a manner in so that these may be able to operate as the discrete foundation for the existence of energy, yet, also, so that superstrings may be able to bind into a covariant relationship that allows these said superstrings to be able to interact in both a direct and in an indirect manner. The said topology interacts in both field-based segments and in field-based curvatures that cause both the Laplacian-based mapping of the "topography" of superstrings, as well as to be able to cause the Laplacian-based mapping of the "topography" of the field indices that work to interbind the said superstrings, to act as the operational basis as to what this said topology is. The interconnectivity of all unfrayed topology to all other unfrayed topology per instanton is homotopy. The tendency of the said homotopy is the foundation of Cassimer Invariance. Sam Roach.

Non-Euclidean-based Traces

Substringular phenomena tends to move in such a manner in so that it leaves physical traces as to where it just was before it reached a different spot. Such physical traces are known of as ghost anomalies. Such ghosts may move in either form a euclidean-based mapping or a euler-based tracing. The tracing may be either proportional to its immediate surroundings, or, the expansion of the mapping as may be derived by a Laplacian-based tracing of the ghost-like delineations that here may be Clifford in nature. These just mentioned ghosts form a Laplacian-basis as to how one may work to determine where certain substringular phenomena were, and, from such knowledge, such a mapping may work to help determine -- through non-euclidean math -- where such substringular phenomena will be in the future. So, the activity that forms such traces happens over Fourier-based Transformations, yet, the actual mapping of such traces often involves a timeless determination that involves relative "snapshots" of what would here exist in such a manner in so as to work to determine the physical evidence as to where, what, and how certain substringular phenomena had moved over the directly prior period in which it had moved in so as to form such relatively stagnant traces. Such ghosts may be Rham or Doubolt in terms of their covariant cohomology. Often, though, the topological-based framework of certain ghost anomalies may be kinematic -- causing the basis of such a framework to instead bear the need to be mapped over a tracing that here involves a Fourier-based Transformation through an onset of redistributional extrapolation. Sincerely, Sam Roach.