Part Four of the Second Session of Course 16

Often, a Doubolt cohomology approximates a Rham cohomology.  This is due to a condition that a cohomology that has only one or so Chern-Simmons singularities, but is otherwise of a Rham nature -- when in terms of its cohomology -- is basically a Rham-based cohomology that has only a limited spike in the nature of the harmonics of either the Lagrangian-based and/or the metrical-based flow of the Hamiltonian wave-tug/wave-pull of that physical memory of the substringular trajectory that works to comprise the said given arbitrary cohomological traceable mapping.  This is if the interconnection of world-sheets that works to form the so-stated extrapolation-based depiction of the directly corresponding cohomology is mapped over either a Lagrangian and/or a group metric in which the said given arbitrary cohomology goes from first bearing a purely hermitian harmonic-based flow, while then converting into having the bearing of a Chern-Simmons spike in the harmonics of its flow (whether such a spike is a spur in the path of the Lagrangian that the given superstring is projecting through over time, and/or, whether such a spike is a spur in the metrical delneation of the said superstring over the time in which such a directly corresponding superstring is kinematically differentiating in as it is reiterated from a sequential series of redistributions), while then re-converting back into bearing a purely hermitian-based flow in both the Lagrangian-based path of its motion and the metrical-based pulse of its harmonics over time.  Especially if a Chern-Simmons-based spike is low in its Hodge-Index over that sequential series of iterations in which the directly related cohomology is delineaeted, then, what may otherwise be termed of as a Rham-based cohomology that is instead Doubolt may appear to have a relatively insignificant spur in either its permittivity of path-based projection and/or its harmonic-based pulsation over time.  Likewise, a cohomology that is very perturbative over either a path-wise and/or a metrical-wise traceable mapping of its projection, may appear as basically a Doubolt cohomology, yet, if one were to extrapolate the basis of the directlty corresponding cohomology in-between the spikes in its cohomological path, the perception of the directly corresponding singularities that would otherwise work to determine a Doubolt cohomology may appear as a Rham-based cohomology.  Yet, this so-stated latter mentioned tendency may often break down into so few iterations of group instanton, that it may be too impractical to conceive of certain cohomologies to be considered as Rham at all -- over certain delineations of certain superstrings that are extremely perturbative in both Lagrangian-based and/or metrical-based anharmonic-based flow over time.  So, although a Doubolt cohomology may be basically Rham in terms of a relative lack of Chern-Simmons singularities over time -- when given a relatively more vast range of mappable tracing, no Doubolt cohomology is basically Rham, since any Chern-Simmons singularities that may work to define an extrapolation of the motion and the metrical-basis of a superstring in the course of its path would here cause the so-stated mapping to be of a purely Doubolt nature instead of Rham in nature.  The latter tendency is especially true if both the mappable trace of a superstring is perturbative and/or if the metrical flow of the said given arbitrary superstring is very perturbative -- due to many anharmonic spikes in either the path of its flow and/or the pulsation of its metical delineations over time.  To Be Continued.  I will continue with the suspense later!  Sincerely, Samuel David Roach.