The Rayleigh MeasureAn Oscillator ModuleBob Price, Chief ScientistApplied Devices(a division of Midas Energy Group)September the 12th, AD 2016

Abstract. A bewildering maze of modern modules is given uniformity.The treatment, which is developed relative to a protoring module,

emphasizes the inductive discovery of the weakest sense under whicha module may properly be said to exist. In other words, our “experiment”

is used to show (in a physical setting), the avuncular nature of ourgeneral module concept: It is shown that the Rayleigh System is in

every sense representative of the deviceable module.

0. Introduction0.0 An Overview

This introduction opens with 0.) an overview of twelve subsections (from 0 to 11): 1.) the general systemdescribes the handling of an arbitrary system, given that no specific data is available (such a system mayor may not be generic, it’s just that we have no data strong enough to specify the system); 2.) a specialsystem describes the handling of an arbitrary system, given that specific data is available (the Rayleighsystem is used to illustrate a tractable system with highly nontrivial behavior); 3.) a pathological systemdescribes my personal involvement with a family of systems which were introduced to me during my stayas a student at Berkeley; 4.) an error system is proposed to handle the inevitable accumulation of discrep-ancy between a system as it is, and one or more faces of that system as our tools and instrumentsenable us to perceive it; 5.) an intelligence system is inferred (under self-organization), using as itsexample a family of invertebrate deep-sea crawlers; 6.) a mechanical system is contrived to the handlingof an otherwise intractable intelligence, and an emphasis on modularity is introduced; 7.) a system offusion describes the innate capacity of a system to absorb (and to reconcile) multiple types of systemwhile persisting itself in a condition of organic resolution (into many diverse modules); 8.) a resolvingsystem describes the methods and modalities of resolution via a tension, the anxieties, and someephemeral bits of discordant requite; 9.) the general resolution is defined, and its ephemeridians areintroduced as a system of modules; 10.) some consequences include a mass-aggregation algorithm,suggesting (but not necessarily endorsing) the Higgs mechanism; and, 11.) the logical notion of compre-hension carries the foregoing into the modular consideration of twenty-five subsequent sections on somevery specific types of module.

0.1 The General SystemThis document is a survey of modern techniques used to resolve an arbitrarily given system into aderived system of modules, where the concept of a module is found to vary broadly across multipledisciplines. And because the term module is preserved (in one special form or another) across eachconstituent discipline, we may think of the ambiguous term module as indicating a unifying conceptacross a great many systems of expression (provided that some attaching range of validity, and ofapplication, might be indicated relative to each respective domain of use). Thus, to speak of a modularresolution is to accept the limiting features of a modular hypothesis as enunciated below [section 23]. Itis to accept (as plausible, if not as vital) the organic assumption, maintained against any “garden varietysystem”, that its corresponding modular resolution will converge (under reasonably good approximation)to the given system -- a cohesive whole or, in the broader sense, an ordinary generic slippy mix.

0.2 A Special SystemOwing to the seminal matter propounded by the course of my own body of research, I do feel with increas-ingly palpable immediacy a sense of misgiving concerning the neglect of a broad interdisciplinaryapproach to the many departments of human knowledge (and to the many divisions of human experi-ence). What is more, in this review of effort expended to embrace (in simple and whole fashion) somerudimentary artifact held jointly by most (if not all) of the concerned disciplines, I have stumbled (in theusual way) upon an altogether fortuitous distinction: I seem to have found that all physical phenomenamay be forced into an artificial (but useful) classification of oscillatory (and non-oscillatory) patterns ofbehavior.

In fact, much of what follows below may be itself resolved into a study of the many diverseshades of gray emerging between the two spectral extremes of purely oscillatory, and of strictly non-oscillatory, regimes of device behavior. What is more, in the search for a practical technology whichcomprises (in itself) the broad, broad range of deviceable phenomena, we have seen the merit of aparametric model of dynamical systems in the second order differential equation studied by LordRayleigh (Baron John Strutt) in his monumental treatise titled, “Theory of Sound”. The so-calledRayleigh equation is studied below as a special system. It is a system whose spectacular range ofdynamical phenomena cuts across the sciences as a reminder of the ubiquitous nature of a reaction-diffusion system whose simplicity conceals (against all expectation) the stark and unreasoning complex-ity which lies at the very heart of every species of animism entertained by the playful soul of our ownhumanity striving (with or without harmony) after the truths which govern our changing environment.

0.3 A Pathological SystemFollowing on the work of Carson Jeffries, a Berkeley Physics professor of my personal acquaintance, Idevised a concept of chaos whose expression is illustrated (with ubiquity) across many practical devicesused in today’s world of rapidly expanding technological encroach. And though the assumption of adriving force can be accommodated with little effort, we find nonetheless broad general utility in theisolation of the regime of chaos which lies quietly untapped in ordinary (soft- and hard-wired) devices.And, consonant with this reality, it is further useful (and imminently advisable) that we take some cog-nizance of the range of influence afforded by the identification of an abstract device which I have seen fitto name the chaos engine. This engine of change, and of mind-boggling complexity, was studied (assuch) and reported in a brief note published by Ulam and von Neumann (probably in the journal, PhysicalReview Letters). It was studied by S. Smale and R.F. Williams (1976) as a biological system controlled

2 The Rayleigh Measure (An Oscillator Module).nb

Review Letters). It was studied by S. Smale and R.F. Williams (1976) as a biological system controlledby a certain “spirit of life” parameter, and I myself took up its study in the late 1980’s under Steve Smale,and again under Carson Jeffries.

After a hiatus from the chaos engine (lasting more than a decade), I returned to the pathologicalsystem (which we know as the dynamical system governed by a logistic equation of growth and evolu-tion) as embedded within the studies of FitzHugh and Nagumo, of Bonnhoffer and Van der Pol, and ofLord Rayleigh himself. In recent days, I have shown (yet again) that this canonical instance of the gen-eral chaos engine is embedded very nicely within every living cell of a nervous system shared by everyliving member of the animal kingdom. And, if I should add that the human body affords no exception tothis finding of dynamic animation, then too should I observe that not all is well in our handling of this, ourlittle nervous system. For, as we ourselves do venture into an uncharted territory of biological experimen-tation, so too do we invite that old bugaboo -- that haunting spirit lurking behind (and within) much innova-tion: It is the spirit known to ancient Greece as techne (pronounced tech-knee).

0.4 An Error SystemIn respect of the data which current technology has enabled us to collect concerning the behavior ofsome highly sophisticated systems (both physical and non-physical), we do begin to realize that even thesmallest constituents of matter (and of the subtle noumenal matter inhabiting all systems, both definedand undefined) regale themselves with a structural complexity with which it is profoundly difficult tocontend. And though a great body of reductive method has grown up around a perceived need to masterevery data set, we do nonetheless encounter the most beguiling species of confusion in respect of theultimate source of big data (inhabiting little experiments) with the physical and metaphysical worlds ofshared acquaintance.

Nevertheless, we do see into the matter perturbing all good and tractable theory a certain envi-able prospect. And, to give this matter its proper situation relative to our own limited capacity to digest(and to apply, in device-oriented fashion), we may frame the problem (together with its typical handling)under the following play of analytic-reductive protocol: An experiment is conducted; data is collected;and findings are digested consistent with a statement asserted (to relatively low precision) as follows.The result is determined to fit quantitative specification at a confidence level measured to a threshold ofeighty percent, give or take five or ten (percentage) points, more or less.. we think. Here, the degree oferror is indicated by a nested measure of data goodness-of-fit whose center is eighty, whose range ofvariation is bounded (loosely) by five or ten, and whose overriding degree of uncertainty is estimated‘more or less’.

The upshot, here, is that to any system of measures through which an experimental apparatusmight be subjected, there is an accompanying system of error which it is our duty to estimate, todescribe, and (if at all possible) to interpret according to the dictates of reason and method. Moreover,given some intelligible degree of success in this very important effort, we may further seek to accommo-date the extraordinary diversity of consequence, implication, and entailment respecting the too real bodyof error which crops up against all scientific endeavor of experimental nature. Specifically, we realizethat to each error signal (within a given error system), there corresponds a control manifold (which mayor may not conceal singularity, and genericity). And, given the existence of a non-unique control mani-fold, there also corresponds a command structure and an operating system. The problem, therefore, inany responsible system of experiment, is to identify the overarching system of interaction among seem-ingly isolated systems of complex data. Thus, we inevitably ascribe some mode of transaction betweena system of interest and its great unknown complement, which we have (for reasons of propriety) desig-nated “the environment”.

The Rayleigh Measure (An Oscillator Module).nb 3

0.5 An Intelligence SystemI think that within the regime of systems analysis which concerns itself with the design and implementa-tion of a self-organizing intelligence system, we are (or should be) interested first and foremost with theinvention and maintenance of a cognate system of metrics dedicated to gauging all aspects of perfor-mance (and of potential) in the matter of organic intelligence limited (say) to a semi-autonomous applica-tion of known scientific principles (and of the more ‘human’ aspects) of our shared evolution in thisenvironment of natural (and artificial) systems of interaction. This is not to say that every intelligencesystem is a system of intelligence, nor is it to brow-beat those of us who have dedicated our professionalcareers to the study and formation of such systems. Rather, it is given as a friendly reminder to any whomight consider joining this effort, that the integrity of our shared endeavor is itself a matter of primeimportance. In fact, the level of scholastic transparency is in every respect absolutely paramount to thesuccess of this very, very competitive calling.

My own exposure to this particular line of investigation began with a perfectly innocent (andunassuming) bit of research into the nervous systems of diverse species populating our shared animalkingdom. And, though I did pride myself on collecting no data (no surgically invasive bit of signal sam-pling machinery), I did nonetheless refer in the most glancing manner to a bit of data harvested byHartline & Kuffler in their studies of Horseshoe Crab. To be fair in the matter of nomenclature, theyacknowledged that limulus polyphemus (the horseshoe crab) is not really a crab at all. Rather, thisparticular species of limulus is a member of the arachnid family. That is, their so-called crab was in allscientific reality a spider! And though this finding does not really discount the nature of their investiga-tion, it does shed a bit of light on our capacity (or, rather, incapacity) to interpret (or to contextualize) thedata set accumulated on crab, spider, and other species of invertebrate deep-sea crawler.

Here, in fairness to those of us who work (out of subtlety) without the benefit of a truly rigorousdata set, it must then be acknowledged that much of what our virtual environment takes for granted mayexhibit (within the range of predictive feedback technique) a measurably valid set of findings: We areourselves, then, concerned with those very certain aspects of a shared environment which do conform tothe broader settings of purely metaphysical inquiry drawn systematically from the stream of humanexperience without explicit control mechanisms at work upon our personal anatomy. And if this considera-tion should strike the reader as unduly disquisitive of life in the modern world, we should be reminded(early, hard and often) how easily our good society might descend along the trajectory of a slippery slopeinto the dark, seductive realm of a man-machine future driven more by fantasy and technology than bythe abiding sense of humanity which (in times of desperation, and of retributive conquest) has beenknown to fade from view.

0.6 A Mechanical SystemIn the world of science regaling nature with findings, methods, and techniques of unforeseen prescienceand capacity for the prediction and control of virtually all aspects of our own existence within a oncehostile environment, we do find a great many investigative minds pursuing the benefits promised byevery ilk of prospecting admen throwing light and huzzah upon a seemingly endless race for dominance.It is, to be sure, a veritable gold rush: It is fever!

My own pet attraction in this little mechanizing nightmare run wild of purpose is known by myreaders to involve not merely the pursuit of intelligence, but also the mastery of space and time in respectof their deep, deep foundations now opening through each progressive installment of inquiry into the trueand unfathomable nature of a God-given reality rendered approachable through a broad investigationinto the analytic-synthetic dichotomy or, as I have found it, the analytic-synthetic integrity. Yet, to be

4 The Rayleigh Measure (An Oscillator Module).nb

honest, we must see from the outset that this work reduces (in the final analysis) to a sheer andunabashed study of mechanics. In fact, the species of mechanics to which I have been by degreesdrawn is still known in the world of astrophysical speculation as “Celestial Mechanics”. And, as the bulkof my present effort remains concentrated in that area of kinematic investigation (even as my methodcarries me further into a general study of dynamical procession), I find that the entire corpus has openedvery nicely to the touch of a noninvasive line of inquiry.

Accordingly, in the work which follows in the present article, I have concentrated my effort to athorough-going study of modular systems. That is, I feel as though the modular approach enables me tosee with abundant clarity much of the structure which big data might (at great expense) corroborate.And, because I feel this degree of access without reference to the usual expenditures of fiduciary (andkarmic) burden, I have chosen to set down in painstakingly explicit form, the most exacting study ofcomparative literature on the module concept. My hope, then, is that my own approach to modularsystems will enjoy its due season of use (and remuneration).

0.7 A System of FusionAmong the attributes of a system (qua cohesive whole) which we find most enticing is its character as asystem (qua slippy mix). That is, an ill-defined system may host any number of a great variety of noume-nal systems, merely through the affirmation that a noumenality (the monic embedding of a contact sys-tem within a host system) has been successfully asserted. And, from the perspective of a randomelement machine [section 21, below], this scenario is entirely typical of device operation!

In other words, a host system S (which we may call an environment prepared for our use) iscapable of rendering multiple contact systems without the least adumbration of fidelity, or of functionaldiscontinuity. That is, a given host (the system S) may render into simultaneity (verging at times into averitable overlap of multiple universes) many diverse instantiations of the noumenal construct. And, wethink, this capacity is very interesting! Not merely for the ease with which it enables us to study thewormhole mechanics of a celestial system, but also for its implications in the realm of human psychology.In fact, we feel that this singularly fascinating attribute lies at the very heart of a problem which has longplagued the great scientific minds of our planetary civilization: It is the problem studied in all times andplaces (of deep contemplation, and introspective scrutiny) as the problem of fusion.

Because of this ubiquity of the presumptively human feature of fusion (within diverse reaches ofspace and time), we may suppose (or blatantly guess) the matter to be shared across more species ofsentient existence than mere hubris would suggest. In fact, we may with reasonable confidence surmisethe existence of sentient forms (be they possessed of life and limb, or not) across countless dimensionsof vitalizing existence, both here on the planet, and elsewhere in the galaxy and beyond. Accordingly,we assert with no small trepidation the social imperative that we do with good intention, faithful ability,and cogent means, explore the broader reaches of our own spacetime continuum. And, should thisimperative carry mind, or body (or both) into worlds unguessed and unseen, then too might we surmiseat this early outset the capacity to be resolvable in a manner considered across so many fields of studyas yet another resolution of an arbitrary system into its constituent clientele -- its scheme of many, manymodules.

0.8 A Resolving SystemA fluxion is a processive extension: It passes, for instance, from a point to a horizon (and back again); itruns from a center to its periphery -- from an event singularity to an event entirety (and back again). Atension is an impeded fluxion: It is an arrow of the form Φ: α → Y, with contact α and scheme Y. Here,the scheme Y resolves the tension Φ into many overlapping systems of anxiety. That is, each bit of

Φ Φ α

The Rayleigh Measure (An Oscillator Module).nb 5

Φ α → αΦimpeded fluxion conspires to the elaboration of a multiplicity of tensive subcomponents, where eachtensive subcomponent bears its image (within the broader image Im(Φ) = Φ(α)) as a species of countlessindividual anxiety modules: They are alike only in their approximate effect, which differs in every respectof permutivity attaching (It is a permutivity attaching an unspecified command ξ within an unresolvedoperating system Ξ at work in consequence of a single random element machine (rem) of which it is theisomorphism.) Thus, in the theoretical vernacular of academic psychology, our tension Φ gives rise to aresolving system of countless anxieties populating the given scheme Y.

In order to see the precise relationship subtending an impedance and the many modules which itengenders, we may delve into the composition of an arbitrary impedance δM = δ∘M = δ(M), where thenoumenality M: α → S (monic) specifies an embedding of type α, and where the dither δ: S → Γ(Q,X)suggests a range of contingency under the indirect control of our contact system α (which, of course, isprecisely the desired effect here -- the effect now in demonstrative exhibit). And, we may peer into thiscomposition, asking after the way that its resultant tends toward the expression of its great, great manyconstituent modules. Moreover, should we succeed in the proposition that each overlapping subsystemof anxiety modules draws into scheme Y precisely those parts of an impedance required in aspect ofspectrum Q (say, a subsystem specified in Q by a partial phase activation of its parent grid Γ(Q,X)), thenwe do perceive and affirm the ephemeral stasis of a coding within Y (by virtue of a mitigated subsystemof Q). That is, our scheme Y meets the purpose intended in the noumenal grid Γ(Q,X) in virtue of theimpedance δM intending (via the specific noumenality M, as adapted by the attending dither δ). And, inconsequence of our ephemerides (our specific balance of noumenal type, portune sort, and accommodat-ing venue), we do apprehend the causal matter of projective requite -- that “spooky action at a distance”.

0.9 The General ResolutionWe have earned access to the following bit of “what Steenrod called abstract nonsense”. And, thoughthe reference [Lang, Algebra, pp. 761-763] produces the matter to considerable generality, we shall herelimit consideration to a level of abstraction including, but not exceeding, the specific abelian categoryknown (to general homology theory) “as complexes of modules over a ring”. And, to be formidablyconcrete in our use of this specific abelian category, we may concentrate the entirety of its “well-oiledmachinery” to the elucidation of the preceding matter -- the geometric matter of the celestial ephemeridi-ans (and of their projective horology). For, in the matter aligned of aggregate terms dwelling fine into ourJungian psychosphere (our shared realm of transpersonal entelechy), we do remotely perceive (anddirectly infer) the combined effect of diverse kinematic action with respect to each ongoing system offluxion (irregardless of the presenting tensions -- the tensions resplending impeded procession). And,should their joint and several computations confound ability, we may further take refuge in the availabilityof a certain notational convention associated with “the standard complex” [Lang, p.764]. Finally, inrespect of the man who did avail us of this fruitful body of technique, we shall continue this subsectionprimarily in the notation used by Lang. And, if I might interpret this man’s effort, then I should like to saythat each module considered is (in our sense) an ephemeridian E i, that each module homomorphism iswritten d i: E i → E i+1, that the index i ranges over all integers, and that the entire machinery is defined asa system of modules over a ring A. That is, we may say (here) that a modulation is a system of modulesover a ring, and we may (accordingly) say that a complex is a differential modulation. Thus, an exactcomplex is sometimes called an exact differential modulation.

Now to the matter of resolution, we may with daring spoil brave the following hypothetical defini-tion with as much vaunting optimism as surreptitious jest: A resolution is a compact modulation whoseindividual morphisms pair as exact differentials. And, while this particular definition runs considerably farinto the wilds of unexplored algebra, it also bears the merit of generality. And, because it is so general

6 The Rayleigh Measure (An Oscillator Module).nb

(given the sequential standard invoked by Lang, and most others), we may feel ourselves justified inasserting the following provisional definition: A standard resolution is an exact complex indexed by asequence of integers. And, in consequence of this acknowledged contribution, we have the followingdesideratum: A general resolution is an exact system of modules over a ring: It is a resolution that isexact relative to the successor arrow over its indicial (i.e., its index system).

0.10 Some ConsequencesIn respect of the coding depth at which the notion of exactness is programmed into our general resolution(just given), we may with profit ascertain the sense of sequence (and its attaching matter of conse-quence) furtively imputed of our desired generality (above). We may suggest (by way of building momen-tum) that much (if not all) of the mass-aggregation algorithm associated with the Higgs mechanism iselaborated in direct entailment of fusion [in subsection 0.7, above], and communicated through a restric-tive filament of pseudo-modus ponens as the fullest import of material implication [due, I think, toBertrand Russell]. And, though countless other bits of restrictive comprisal might further elaborate therange of subject-matter established here, I feel that a certain degree of clarity might be had on an insis-tence that we focus (with extreme coherence, possibly singular) the development of this experiment to acompletely novel explication of the Higgs consequence as a corollary of the general resolution.

Accordingly, I should like to invest this opportunity with the advertisement of a seemingly innocu-ous experiment in the science of thin-film polymers. Specifically, I should like to revisit my little device ofblue mylar sandwiched between a pair of conducting plates -- one silver, the other gold. To this end, Irefer the interested reader to my second book, the one titled, “@ Progress Report”: Its third part eluci-dates my hoped for experiment on the mylar propagator -- the dielectric capacity in mylar (a field-theo-retic interaction). Here, the point of contact with the Higgs mechanism relates directly my work on thegeneral field transform [op.cit. Appendix A].Remark. If we pause to consider the actual occurrence of a wormhole event within a pre-existing α-cosmos (of known α), then we might easily arrive at a shared consensus in the view that we have beforeour consideration a demonstrative proof that “kinematic parenthesis is episodic!”

0.11 The Logical Notion of ComprehensionWithin the range of human understanding, where the natural and the supernatural do occasionally blendinto a sense of life expecting life, we find that our too, too human nature does spoil and belie a crucialdistinction maintained across the many worlds of personal (and transpersonal) experience: We find thatin the balance of diverse opinion asserted, there proceeds a randomizing feature through which our mostpersistent efforts to mechanize, to automate, to classify, and to otherwise categorically repudiate the firstthings of our shared nature is systematically thwarted. We find that within our underlying need todescribe, to predict, and to control an environment not of our own making, there runs a process ofrenewal (and of regeneration) through which a population of psychological automata does strive to create(and, again, to transcend) every useful form of human expression given (by God above, or by the Dick-ens below) as an innate means of becoming more fully human.

Thus, in the prosecution of the following program of study, we do admit with all necessary regretthe more mechanistic features of analysis as comprising (even in the broadest sense), a curt and perfunc-tory treatise of applied systems analysis. And, if this regret is to bear fruit of a sort deemed superior tothe loss incurred through mere systematization of an unapologetically anthropomorphic vitalism, thenalso should some measure of cost and benefit be weighed in light of the pervasive themes introduced(bit-by-bit) as the guiding theme for our own peculiar journey through a world of constant incongruity,

The Rayleigh Measure (An Oscillator Module).nb 7

through a world of effervescent inconsistency, and through a world of brazen and outright contradiction.For, in the world of our experience, the capacity to hold (within the realm of comprehension) everydiverse reality must receive its portion of credit for the genius of a species whose nature we admittedlyconcede does defy all “logic and proportion” -- all vain and perspicacious scrutiny.

This matter notwithstanding, we do with apology of forethought assert the following program ofanalysis:

1. The Hungerford Module1.0 Setup

Modules are central to the resolution of an arbitrary system. This is seen through the following paradigm:“Modules over a ring are a generalization of abelian groups (which are modules over ).” [Thomas W.Hungerford, Algebra, p.168]

1.1 DefinitionThe following definitions are due to Thomas W. Hungerford [op. cit., p.168]:

Let R be a ring. A (left) R-module is an additive abelian group A together with a function R×A →A (the image of (r,a) being denoted by ra) such that for all r,s ∈ R and a,b ∈ A:

(i) r(a+b) = ra + rb.(ii) (r+s)a = ra + sa.(iii) r(sa) = (rs)a.

If R has an identity 1R and(iv) 1Ra = a for all a∈A,

then A is said to be a unitary R-module. If R is a division ring, then a unitary R-module is called a (left)vector space.

2. The Lang Module2.0 Setup

A vector space over a field has a basis. But, a module over a ring may or may not have a basis. Specifi-cally, a free module is a module which admits a basis, or the zero module. [S. Lang, Algebra (RevisedThird Edition), p.135]

2.1 DefinitionLet A be a ring. A left module over A, or a left-module M is an abelian group, usually written additively,together with an operation of A on M (viewing A as a multiplicative monoid by [the axiom] RI2), such that,for all a,b ∈ A and x,y ∈ M we have

(a + b)x = ax + bx and a(x + y) = ax + ay.

8 The Rayleigh Measure (An Oscillator Module).nb

3. The Jacobson Module3.0 Setup

“The concept of a left module is the ring analogue of a group acting on a set. As in the group case, thisarises in considering a homomorphism of a given ring R into the ring of endomorphisms, End M, of anabelian group M.” [N. Jacobson, Basic Algebra I (Second Edition), p.163]

3.1 DefinitionIf R is a ring, a left R-module is an abelian group M together with a map (a,x) ↦ ax of R×M into M satisfy-ing the following properties:

1. a(x+y) = ax + ay2. (a+b)x = ax + bx3. (ab)x = a(bx)4. 1x = x

for x, y ∈M, a,b,1 ∈R.

4. The MacLane - Birkhoff Module4.0 Setup

Let R be any ring. We call it the “ring of scalars” and write its elements as lowercase Greek letters κ, λ,μ, … . [MacLane & Birkhoff, Algebra, p.160]

4.1 DefinitionAn R-module A is an additive abelian group together with a function R×A → A, written (κ,a) ↦ κa, andsubject to the following axioms, for all elements κ, λ ∈ R and a,b ∈ A:κ(a+b) = κa + κb,

(κ+λ)a = κa + λa,(κλ)a = κ(λa),1a = a.

More explicitly, such a module A is a left module, because in forming κa the scalar κ is written on the leftof the module element a.

5. The Macaulay ModuleA modular system is an infinite aggregate of polynomials, or whole (rational) functions of n variables x1,..., xn defined by the property that if F, F1, F2 belong to the system F1+F2 and AF also belong to thesystem, where A is any polynomial in x1, ..., xn. [F.S. Macaulay, The Algebraic Theory of Modular Sys-tems (1916)]

The Rayleigh Measure (An Oscillator Module).nb 9

6. The Standard Module6.0 Setup

A contact engine is an arrow of the form d: d → d. A contact is a subsystem of a contact engine; it iswritten α in d, and we say that α is in d (as a subsumptive inclusion, which is a subsystem). A contactlemma is the power system of a contact engine; it is written ℘ = {α in d, d: d → d}. A contact ring is aprincipal ring in a contact lemma ℘. A noumenal is an ideal in a contact ring.

6.1 Basic DefinitionA module is a workable ring-action on an abelian group A: It is an action whose type (α,β) is specified byα (the type of A) as much as β (the co-type of ring R = ∇β in the protoring ℘-1); it is written R×A → A; (b,x)↦ ∇b x, with b a metric in the gauge β, with x an element of the abelian group A, with ∇b a metric homo-geneity (the b-derivative) in ℘-1, and with ∇b x the b-derivative of x within A.Note. The presumption here is that our reciprocity between α and β will give a workable instance of theaction -- the instance which forms a module. The details are given in section 6.3 below.

6.2 Support DefinitionsA contact engine is an arrow of the form d: d → d. A contact lemma is the system of subsystems for acontact engine d; it is written ℘ = {α a subsystem of d, with d: d → d}. A contact restriction monoid is asemigroup with identity τ of the form (℘,|,d), where ℘ is the contact lemma for a contact engine d, wherethe restriction | is an arrow of the form |: ℘×℘ → ℘; (a,a’) ↦ a’|a, and where d is the identity element; themonoid is written τ = (℘,|,d).

A contact system is a subsystem α of a contact engine d; this α is sometimes called a contact. Acontact group is the group π defined as canonical reduct of a contact restriction monoid τ relative to thekernel Ker(τ→π); it is written (delightfully) as π = τ / Ker(τ→π). The contact system α = Ker(τ→π) iscalled a symmetry-inversion kernel.Remark. The symmetry-inversion kernel α well-defines its contact group π = τ/α.

6.3 Main DefinitionsA standard module is an arrow of the form

A : R×A → A,with abelian group A, with scalar ring R = End A, and with working conditions:

(s+t)x = sx + tx,t(x+y) = tx + ty,(st)x = s(tx),

where s,t∈R, and x,y∈A. A standard contact module is a standard module whose group A is the abelian-ization of a contact group π, so that A = Ab(π) and R = End[Ab(π)].Remark. To any abelian group A there corresponds an α-measure group π (given by the contact groupπ), such that A is isomorphic (as an abelian group) to the abelianization Ab(π). Here, the abelianizationis constructed by writing Ab(π) = π/[π,π], where the commutator [π,π] is given by the condition [π,π] =

π

10 The Rayleigh Measure (An Oscillator Module).nb

π ππ π π π π π π π{xyx-1y-1, x and y within π}.

6.4 WorkThe work associated with a module is expressed by the action of a scalar, say t within R. The effect ofthis action is conditioned by three axioms: 1.) an axiom of right distribution, which is the distribution of agroup element x over the scalar terms s and t; 2.) an axiom of left distribution, which is the distribution ofthe scalar t over the group terms x and y; and, 3.) an order axiom for the evaluation of scalars s and tover the group term x.

7. A Reciprocity Module7.1 Basic Definitions

A dither is an arrow of the form δ: S → Γ(Q,X), with host system S, and with grid Γ(Q,X). A quiver is anarrow of the form ψ: Γ(Q,X) → Y, with noumenal grid Γ(Q,X), with scheme Y, and with fluxion (processiveextension) ψ. A translocation is an arrow of the form j: Y → Z, with scheme Y, and with zero condition Z.A zero-noise is an arrow of the form W: S → Z, with host system S, and with zero condition Z; the effectof W is given under the composition j∘ψ∘δ, with dither δ, with quiver ψ, and with translocation j.

7.2 Support DefinitionsA noumenality is an arrow of the form M: α → S (monic), with contact α, and with host system S; it issometimes called a contact embedding (of type α). A zero-gauge is an arrow of the form E: Z → β(epic), with zero condition Z, and with gauge β. A reciprocity is an arrow †: α → β whose effect is givenby the composition E∘W∘M, with noumenality M, with zero noise W, and with zero-gauge E.

7.3 Main DefinitionA nonstandard module is a module R×A → A, whose ring of scalars R is not the ring of endomorphismsEnd A over the group A. A nonstandard contact module is a nonstandard module M: R×Ab(π) → Ab(π),and where Ab(π) is the abelianization of a contact group π. A reciprocity module is a nonstandardcontact module of the form R × Ab(π) → Ab(π), where R is a ring whose co-type β is the reciprocity of thetype α used to well-define π; it is written co-type β = α†.

8. The Rayleigh ModuleThe Rayleigh module is a module of the form (End R)×R → R, where R is the abelian group derived outof the integral curves of the Rayleigh oscillator, and where End R is the ring of endomorphisms on R.

9. A Wolfram ModuleA Wolfram module is a module in the sense that its variables (and their transformations) are localized to(or wrapped within) the confines of the module: It is merely a formality which restricts the scope of

The Rayleigh Measure (An Oscillator Module).nb 11

internal variables.

10. A Simplex ModuleA simplex module is a module of the form R×A → A, where A is an abelianized simplex group, and whereR = End A.

11. A Noumenal ModuleA noumenal module is a module of the form R×Ab(π) → Ab(π), where π is the contact group well-definedby its symmetry-inversion kernel (the contact noumenal α), and where R is a ring.

12. An Anxiety ModuleAn anxiety module is a noumenal module R×Ab(π) → Ab(π), where R is a local ring in the scheme (oftension) Y.

13. A Cluster of ModulesA condition is a system U that describes the action of a binding principle between two systems (say, Sand T): It is a system U that binds a system S to a system T. Thus, each condition is (in its activeaspect) a binding condition.Remark. A condition is sometimes characterized as the determinative factor within the fusion of twosystems.An echelon is a triad (S,T;U) comprised of a pair of systems (S,T) together with a condition U; it is some-times written (S,T;U) = [(S,T),U]. An articulation is a pairing between a channel S::T and a condition U: Itis a pairing through which U (the articulant) is said to describe S::T (the articulent); it is written S::T;U. Afusion is an arrow V from an echelon (S,T;U) to an articulation S::T;U and, its effect is written V(S,T;U) =S::T;U. A collation is a conditioned filament VU of a fusion V to be evaluated at the given condition U; itseffect is written VU(S, T) =V(S, T ; U), and this effect is sometimes called the formative principle (ofcollation). The arrow of work is an arrow e that assigns to each echelon (S,T;U) the valuee(S, T ; U) =V(S, T ; U) /VU: This arrow e: Echelons → Energies; (S,T;U) ↦ V(S, T ; U) /VU is sometimescalled the arrow of collate fusion, and its value at the echelon (S,T;U) is called the energy of formation.

Accordingly, it is natural that we should look for an isomorphism between work and collate fusion.To this end, the collate fusion (or the work) may be evaluated for a cluster of modules as follows. Givena condition U, the binding energy between two systems S and T is expressed under the filament eU (ofthe energy of formation e), and it is written eU(S, T) = e(S, T ; U). Here, the system U is sometimes calleda binding condition for the energy eU. Also, a U-cluster is a system of entities (which may or may not besystems) that satisfy a condition U. A cluster of modules is a U-cluster (of modules).

14. A Scheme of ModulesLet β = α† be a gauge (a modular system of metrics). Let b be a metric (a measure of α-proximity, whichis an α-measure of proximity) within β. Let b be a b-ring (a local ring of metric scalars, with metric b).Φ α → α Φ

12 The Rayleigh Measure (An Oscillator Module).nb

αLet Φ: α → Y be a tension (with noumenal foot given by the contact α), with scheme of tension Y = Cod Φ≠ Im Φ. Then the local ring b in the scheme Y (a locally ringed system) satisfies b = (Ej)(b), given that β= E(j(Y)) = (Ej)(Y). Here, the arrow j: Y → Z is called the translocation, and the arrow E: Z → β (epic) iscalled the zero gauge (or a fuzzy wiggle).

The contact noumenal α well-defines a contact group π, and this relation of definedness is givenexplicitly by writing α = Ker(τ → π). The corresponding abelian group is written Ab(π), and the correspond-ing local module (over metric b) is written b×Ab(π) → Ab(π) (with working conditions as written in section6.3). In sum, we get that a β-module is a scheme of b-rings (with b a metric within the gauge β = α†); thisβ-module is sometimes called a gauge-module.Remark. Notice that the tension Φ is an open variable: It is an impeded fluxion with impedance givenas a dithered noumenality δ∘M: α → Γ(Q,X), and with fluxion given as a quiver ψ: Γ(Q,X) → Y. Thus,each choice of tension Φ = ψ∘δ∘M gives its own characteristic flavor to the reciprocity †: α → β. In otherwords, each variation on the tension Φ leaves its distinct mark on the reciprocity † = (Ej)(Φ). This meansthat instead of getting a single monolithic gauge-module within the scheme Y, we get a scheme-module(with β fluctuating and, perhaps, adapting to some cooked up application of the gauge formalism).A scheme of modules is a cluster of modules, whose condition U is a scheme Y, so that our binding U isthe pullback base Y of a noumenal grid Γ(Q,X).

15. A Hyper Module15.1 Basic Correspondence Theory

A correspondence is a pairing between two bits of whatnot (say a and b), such that the entailed moduleis deviceable; it is written a::b. A hyper correspondence is a pairing between two correspondences, suchthat the entailed module is deviceable; it is written (a::b)::(c::d); its its entailed module (which is device-able) is sometimes called a hyper module.

A symbolic correspondence is a correspondence between two symbols (say μ and ν); it is writtenμ::ν. A rational correspondence is a correspondence a::b whose pairing is interpreted to a simple ratioa:b. A birational correspondence is a correspondence between two rational correspondences (say a:band c:d); it is written a:b :: c:d, and it is sometimes called an analogy.Remark. The hyper correspondence (a::b) :: (c::d) is read, ‘the correspondence a::b corresponds to thecorrespondence c::d’. Also, the analogy a:b :: c:d is read, ‘a corresponds to b in the way that c corre-sponds to d’ or, more simply, “a is to b as c is to d”.Remark. The entailed module for an analogy is a module over a field; and, it is sometimes called avector space. The field is a division ring with unit, where the commutativity is evident upon writing a:b ::c:d iff b:a :: d:c.

15.2 Semiotic DevicesA device is syntagmatic if it gives instance to a symbolic correspondence. A formal analogy is the anal-ogy derived out of a syntagmatic device (it is a matter of syntax, grammar-one). A device is paradigmaticif it gives instance to a birational correspondence. A rational analogy is the analogy derived out of aparadigmatic device (it is a matter of semantics, grammar-two). A primitive correspondence is a correspondence free of analogy. Alternatively, an articulation is a corre-spondence (say a::b) that holds on the strength of some explanans (say c), and it is written a::b; c. Forα η ψδ α α ηψδ

The Rayleigh Measure (An Oscillator Module).nb 13

example, the articulation α::η;ψδ is used to express a protoring of type α, where the correspondence α::ηis articulated by the term ψδ.

15.3 ExampleThe rational analogy 1:2 :: 2:4 asserts that the ratio 1:2 corresponds to the ratio 2:4, but it does not asserta condition of equality. Here, our example supports the articulation (1::2) :: (2::4); 0.5.Remark. A constructive remark on notation affirms that the viation a;b asserts the reverse entailment,“a holds on the strength of b”. Here, the same notation is used to assert that b is evidence of a, or that bis a witness of a.

15.4 ConditionalsA conditional semantics is said to hold under each instantiation of the formula, “If A means B, then Cmeans D.”A conditional entailment is said to hold under each instantiation of the formula, “If A entails B, then Centails D.”

15.5 Constructive Semanticsinviat Ωo (impassable entity)inchoacy Ω (impredicable entity)ad liberatum Ω/Ωo (system)cipitat Ωα (semantic filament of inchoacy)viatem Ωα/Ωo (semantic system, type α)

15.6 Special Semanticsfilament fil(x,f) = filx(f) = fxevaluation @(x,f) = x@f = evx(f) = ev(x,f) = f(x)restriction |(f,x) = f|x

16. The Tableau ModuleA tableau module is a module whose underlying group is generated by the template for a simplex tableau.

17. The Protoring ModuleRemark. Modules over a homogeneity are a generalization of modules over a ring (which areephemeridians). ∴ A module over a gauge-homogeneity (say, a control factor k(β) = ∇β = ∂/∂β) aregauge-ephemeridians.

18. The Deviceable ModuleAn echelon is a pair of systems S and T together with a binding condition U; it is written (S,T;U). A niceexample of the echelon concept is given by the semidirect product S ×UT.

× →

14 The Rayleigh Measure (An Oscillator Module).nb

A device is a fused echelon; it is the fusion V(S,T;U) of an echelon (S,T;U). An R-module is aworkable action of a ring R on an abelian group A, and it is written R×A → A; here, the working conditionU = (U1, U2, U3) is given by the following axioms:

U1 :: s(x+y) = sx + syU2 :: (s+t)x = sx + txU3 :: s(tx) = (st)x

A module (R,A;U) is deviceable if the working condition U binds the pair (R,A) as an echelon. Becausethere are several interesting ways by which a module can be fused to a device [they are listed in thesubsections below], the following construction of an error process may be studied to advantage:

1. (S,T) ; pair of systems2. U ; binding condition3. (S,T;U) ; echelon 1,24. U : (S, T) → (S, T ; U) ; binding process 1,35. V(S,T;U) ; device 36. V : (S, T ; U) → V(S, T ; U) ; fusion process 3,57. θ ; error signal8. W : V(S, T ; U) → θ ; zero noise 5,89. ϵ : (S, T) → U ; error process 4,6,8

The above systems (1-9) satisfy the tope condition ϵ =W ∘V ∘U , as depicted in the following diagram:

The noumenal case is given if we put θ in Z; relabel (S,T) to (α,S); put V(α,S;U) in S; and, put (α,S;U)into the noumenality M : α → S (monic).

Also, the protoring example is given by an α-protoring, which is an articulated channel α :: η; ψδ.Here, the corresponding diagram is drawn from the third tope of a random element machine (rem), giving

I think that the most interesting feature of the above tope is expressed in the equation W = j ∘η, whichasserts that our zero noise W : S → Z can be written as a composition of the translocation j : Y → Zover the α-protoring noise η : S → Y . In other words, the zero noise W is the translocated α-noise η.

Finally, we say that a module is deviceable if any one of the following twenty-five conditions holds:

The Rayleigh Measure (An Oscillator Module).nb 15

18.1 The Module is Standard18.2 The Module is Iso-Standard18.3 The Module Resolves into A System of Standards18.4 The Module Resolves into A System of Iso-Standards18.5 The Module is Echelon-Specific18.6 The Module is Gauge-Definable18.7 The Module is Flip-Compatible18.8 The Module is Pre-Algebraic18.9 The Module is Translocatable18.10 The Module is Null-Ready18.11 The Module is Error-Supported18.12 The Module is Homogenizable18.13 The Module is Quasi-Heterogeneous18.14 The Module is Viatable18.15 The Module is Not Intrinsic18.16 The Module is Gauge-Compact18.17 The Module is Attachable18.18 The Module Shows Non-Trivial Monodromy18.19 The Module Shows Measurable Evidence18.20 The Module Reduces (via the noumenal) to a Standard Witness18.21 The Module Co-Restricts to an Evaluation18.22 The Module Embeds within an Inchoacy18.23 The Module Defines a Reverse-Entailment18.24 The Module is Scheme-Complete18.25 The Module is Noise-Exigent (mode-discernible)

16 The Rayleigh Measure (An Oscillator Module).nb

19. The Non-Deviceable ModuleAn ultra-module is a transcardinal module without the inviat Ωo; it is constructed within the subsystems ofthe star-engine d*: d* → d*, with Ωo star-excluded.Remark. Here, the abelian group A is given by adding exponents under subsystem composition [say,dm ·dn = dm+n], the multiplicative system T is given by multiplying exponents under subsystem exponentia-tion [say, (dm)n = dmn], and the ring R = (A,T) is given by the usual rules of arithmetic on the exponent.Notice, however, that division is included only if we can embed the radical (in its own domain, which isthe whole engine). Note also, that if d is a subsystem of d*, then R is a ring with identity, and U is aunitary R-module.A non-deviceable module is a module that is not deviceable. It happens to be an ultra-module, or simplya co-module.Remark. A module can be written as the split (direct) sum of a deviceable module and a co-module.This is odd (I think), but the co-module appears to be a subsystem of its own module. That is, to eachmodule M (with deviceable submodule D), there is an ultra module U = M/D, and this U is precisely theco-module of M. Put differently, the ultra module U satisfies a condition of equality with the co-moduleM/D. In other words, M = D⊕U gives the direct representation of the sum (it is ultra); and, M = D⊕(M/D)gives the split representation of the sum (it is co-module). Note further that if D is not deviceable, thenthe condition of equality between U and M/D may fail on any one of its three axioms: 1.) indiscernibility;2.) identicality; or, 3.) translocated equilibrium is zero sum.

20. An Operating SystemAn operating system is a system Ξ of commands ξ within Ξ, where each ξ is an attachable locus ofmonodromy Perm(ξ,S), which is a locus of permittivity perm(S). Thus, the monodromy group for a hostsystem S over a locus ξ is given by

Perm(ξ,S) = loc[ξ, perm(S)] = ξ-1perm(S) = ξ-1[Aut(S)/S]

Remark. We tune a module Mμν = Dμ⊕Uν by messing with its device parameter μ (brain surgery), or byadjusting its ultra co-parameter ν (rocket science), or both (wormhole mechanics).

21. The Random Element MachineA channel is a four-tuple of systems (say A0,A1,A2,A3). A hyperchannel is a pair of channels (sayA0,A1,A2,A3; and B0,B1,B2,B3). An articulated hyperchannel is a hyperchannel together with the arrows

f1 : A0 → A1, f2 : A1 → A2, f3 : A2 → A3;g1 : B0 → B1, g2 : B1 → B2, g3 : B2 →B3;h0 : A0 → B0, h1 : A1 → B1, h2 : A2 →B2, h3 : A3 → B3.

The following diagram (of an articulated hyperchannel) gives the right idea:

The Rayleigh Measure (An Oscillator Module).nb 17

A random element machine is an articulated hyperchannel with the valuesA0 = ℘, A1 = α, A2 =S, A3 = Γ(Q, X );B0 = ℘-1, B1 = β, B2 = Z, B3 = Y ;f1 = σ :℘→α, f2 =M : α →S, f3 = δ : S→Γ(Q, X );g1 = k :℘-1 →β, g2 = E : Z →β, g3 = j : Y → Z;h0 = ζ :℘→℘-1, h1 = † : α →β, h2 =W : S→ Z, h3 = ψ : Γ(Q, X ) →Y .

The following diagram (of a random element machine) gives the right idea:

22. The Chaos EngineA chaos engine is a spanning module generated by a contact engine d: d → d. And, since our contactengine d is a transcardinal continuum, its spanning modules are transcardinal. Thus, a chaos enginemay be characterized as a transcardinal module.

23. The Modular HypothesisThere is a bold piece of conjecture about the behavior of modular systems, and it is used as a workinghypothesis on the proper setting for an arbitrary module. It asserts that each module is isomorphic to asubmodule of the transcardinal continuum. This conjecture is sometimes called the modular hypothesis.

24. Systematic Modularity As a broad corollary of the modular hypothesis, we encounter the notion that each system S may beresolved into a generalized sequence of chaos modules (by drawing each module as an isomorphism ofa corresponding submodule in the chaos engine). This approach to systems analysis is sometimescalled systematic modularity.

25. A Unified Theory of Modules

18 The Rayleigh Measure (An Oscillator Module).nb

If we accept the dogma of systematic modularity, then we run into a system of foundations whose fullexplication remains to be given as a unified theory of modules: It is homologous to the foundations ofgrammar, which are eight in number. There are four natural grammars (syntax, semantics, semiotics,and epinetics); there are three supernatural grammars (these are transactional); and, there is one praeter-natural grammar (it is vitalistic animism, and it is studied as comprehensive possession).

The Rayleigh Measure (An Oscillator Module).nb 19