*Note: You might be wondering, what in the h#$ is this? Well, its weird. As early as I can remember I’ve been fascinated with how nature works. Over the years I came to learn that this was in fact a huge, gigantic riddle that some of the smartest people in the world have been working on for a long, long time – and no dice. So, over the years as I thought through this problem I developed a picture of what I thought was a solution. Being a narcissist and all you’d think I’d be strangely comfortable with this. But no, this riddle is way too daunting even for me to think I could have possibly solved this all on my lonesome. So what to do? I mean, this is serious crackpot territory and I’m not even that well versed in math and physics. Well, I don’ tknow but for what its worth there it is. Its not completed and I know there have to be errors, but I was convicted of the belief that it was overall correct … until I come back to reality and appreciate once again the scale of this riddle.*

**Space**

*C. Komrik *[i]

*ABSTRACT. A systematic application of the principles of necessity and sufficiency regarding the properties of space and events thererin leads to a comprehensive algorithm for measuring and predicting all natural behaviors. *

**1.****Introduction**

We consider a set of defined conditions under which we will pretend observation:

(a) All observers discussed possess a total energy sufficiently small.

(b) The space in which these observations are made is sufficiently large.

(c) The space considered is sufficiently vacuous

(d) All observers discussed possess a history sufficiently brief.

(e) All events observed occur in a time interval sufficiently small.

(f) No extraneous event with a causal relationship to the events observed presents.

We next begin by considering a space, denoted ** S**. We then introduce an observer,

**ᶄ**

_{1, }into that space. All observations made are to be carefully qualified as observations made solely from the reference frame of the class of observers,

**ᶄ**

*, and not from any other reference frames. Observable events at*

_{n}**ᶄ**

_{1}, in order of causality, are denominated:

**2.****Historical Acts (***Gedanken*1.0)

In the beginning of the history of **ᶄ**_{1}there is an event, **u**_{1}, in which energy is exchanged from some undefined source[1] to **ᶄ**_{1}. In this energy exchange the event **u**_{1} is incident upon **ᶄ**_{1}at a presumptive time |**ᵯ**|_{0} . Let **ᶄ**_{1} record this event and thus record a history, call it ** n**, belonging to a set of histories,

**. With only event**

*S*_{n}**u**

_{1}an interval, and hence a magnitude and direction called

**, is undefined. But**

*n***ᶄ**

_{1 }has observed the event

**and has thus observed the history of**

*n***without**

*n**presuming*an endpoint for it. Thus, by definition, the appropriate representation of this history is:

|** n**| = √ a

^{2}

*+ ɨ*b

^{2}where

** n** = a

*+ ɨ*b and

**is linearly independent.**

*n*We will later show that it is in fact the component “b” that is presently under description. The key experiment herein will be for observer **ᶄ**_{1} to attempt to experimentally show what information is necessary and sufficient to define some event **u**_{1}, qualified only by the constraint that it is the first event observed by **ᶄ**_{1}, but otherwise arbitrary.[ii] And we shall see that,

**u**_{1}⊧ ** S_{n }**(⊧ means entails, if A then B goes with it. ⊢ means is derived from; infers).

**ᶄ**_{1} measures its own properties to derive the shortest sequence of events ∃ necessity and sufficiency avail complete definition of **u**_{1}:

The event **u**_{1 }provides for the measurement of an open-ended interval (**u**_{1}, **φ**_{1}) where **φ**_{1} is undefined. Though **ᶄ**_{1 }cannot assume the existence of this interval it can be observed by itself. Upon some second event **u**_{2} the history, ** n**, is an open interval:

(**u**_{1}, **u**_{2}),

Thus, by definition, the appropriate representation of this history is:

|** n**| = √ a

^{2}

*+ ɨ*b

^{2}where

** n** = a

*+ ɨ*b and

**is linearly independent.**

*n*With events **u**_{1} and **u**_{2}, and requiring that all effects have a corresponding set of causes,the following four intervals are observable:

- An interval [
**u**_{1},**u**_{2}] which includes events**u**_{1}and**u**_{2}, call it basis*i* - A parameter of that same interval, [
**u**_{1},**u**_{2}], measured as an independent quantity. Call it basis**ᵲ**. - An undefined interval (
**u**_{1},**u**_{2}) contained within [**u**_{1},**u**_{2}], call it basis.*n* - An undefined interval (
**u**_{1},**u**_{2}) contained within [**u**_{1},**u**_{2}], call it basis**ᵯ**.

A parameter of that same interval measured as an independent quantity presents. As before, another interval, (**u**, **φ**), exists with events **u**_{1} and **u**_{2}; namely, (**u**_{2}, **φ**_{2}) and similar observations avail. But necessity and sufficiency require only two observables. Since the parameter **ᵯ **exists on the interval (**u**_{1}, **u**_{2}) and is thus observable from within [**u**_{1}, **u**_{2}], the variable is observed to parameterize ** i**. But change in

**can only be parameterized by**

*i***ᵯ**if the relationship between them is itself defined, which cannot occur without an event

**u**

_{3}.

A parametric quantity less than [**u**_{1}, **u**_{2}] is undefined with respect to that same interval, and any parameter of [**u**_{1}, **u**_{2}] must be greater than | (**u**_{1}, **u**_{2}) |. This cannot be without some event **u**_{3}, the point at which the parameter **ᵯ** may be meaningfully rounded up to a unit value of 2 (without **u**_{3}, 1 > |(**u**_{1}, **u**_{2})|) .

- This, then, necessesarily makes measurement of some
’th basis,*j* - along with a requisite imaginary couplet,
, observable. And we can see that*o***ᶄ**_{1 }is effectively observing parametric behavior without an explicit parameter; that is, a change in order or the observation of linearly independent bases in lieu of the observability of independent parameterization.

Only upon event **u**_{3} does such parameterization present, making **u**_{3} the cause of the effect of parameterization.

- Due to that dependency, two additional bases,
**k**in the reals - and its imaginary couple
**p**, are measureable at**u**_{3}that do not depend on**ᵯ**or**ᵲ**.

Upon observation of the three events **u**_{1}, **u**_{2} and **u** _{3}, **ᶄ**_{1} finds that another observable presents that cannot be sufficiently well defined in order 8. It may be loosely described as a “sense of direction” and each basis is in fact a vector linearly independent in itself.

Thus, upon some event **u**_{4} there is observed an event history basis, again linearly independent in itself, whose key property is that, from 0 up to a Natural Limit, it lies parallel to all bases in order 8, then alternates periodically between parallel and perpendicular orientations with respect to the order 8 system.

- and here designated
in the reals*ℓ* - and its counterpart the imaginary
.*q*

The list above will continue in this pattern without such time as necessity and sufficiency preclude it, that is, when *S** _{n}* becomes linearly dependent. The reader may wish to verify by a continuation of this Gedanken that all subsequent events observed at

**ᶄ**

_{1}are dependent solely on

**with no further event histories necessary to define them; that**

*S*_{n }**is linearly dependent in order 10. We will leave one condition on this statement but for now, we can show by mathematical induction that if:**

*S*_{n}**0 = **a_{0}|** n|** + a

_{1}|

**| + a**

*i*_{2}|

**| + a**

*j*_{3}|

**ᵯ**| + a

_{4}|

**| + a**

*o*_{5}|

**| +**

*k*a_{6}|** ℓ**| + a

_{7}|

**| + a**

*p*_{8}|

**| + a**

*q*_{9}|

**ᵲ**| ∀ a ∈ ℝ and ∀

**∈ ℝ**

*v*^{10}

where

**0** ≠ _{i = 0}Σ^{i = 9} |a_{i}| ∀ a ∈ *S** _{10}*.

Then if by *Gedanken* One,

**0** ≠ _{i = 0}Σ^{i = 10} |a_{i}| ∀ a ∈ *S** _{11}* and

**0** ≠ _{i = 0}Σ^{i = 11} |a_{i}| ∀ a ∈ *S** _{12}* and

**…**

**0** ≠ _{i = 0}Σ^{i = k} |a_{i}| ∀ a ∈ *S** _{k + 1}* where

*k*= 19

then

**0** ≠ _{i = 0}Σ^{i = k} |a_{i}| ∀ a ∈ *S** _{k + 1}* and ∀

*k*∈ ℝ. Q.E.D. [§ 0.0201]

So, sufficiency makes with event **u**_{3}, save for two special cases of events. So, a real valued event history ** ℓ** and an imaginary valued event history

**are special cases whose causal linkage to the event history**

*q***ℋ**described is more subtle. Assume events

**u**

_{1},

**u**

_{2},

**u**

_{3},

**u**

_{4},

**u**

_{4 }≔

**u**

_{x}observed by

**ᶄ**

_{1 }is linearly dependent in

**iff the event**

*S*_{x}**u**

_{x}=

**u**

_{4}and

**ℋ**_{ℓ}**≔**** ℓ** and

**ℋ**_{q}**≔**** q**.

where the two resultants above are histories of the real and imaginary spaces (from hereon referred to as **r**Space and **m**Space respectively) linearly independent in ** S_{n}** ∀ events

**u**

*and ∀ histories*

_{c}

*v**; c, d ∈ ℝ. The necessity of this independent, ‘global’ event history presents with event*

_{d}**u**

_{4}. Thus far we have not examined in detail the creation of the quantization of event histories, the Natural Limits (which will be more rigorously developed later), but for now we will note that for any measurement of

**ᶄ**

_{1}of each basis

**s**∈

**there exists a Natural Limit denoted**

*S*_{n}**₵**. With appropriate equipment

_{s}**ᶄ**

_{1 }can measure the real end points of the Natural Limit interval (since the Natural Limit itself lies within that interval). Thus, any one dimensional event,

**u**

_{4}, perpendicular to the necessarily orthogonal basis,

**s**, that

**ᶄ**

_{1}employs for measuring it, will itself present the width,

**₵**. So,

_{s}**ᶄ**

_{1 }may observe that should

**u**

_{5}obtain, a linearly independent basis event history that necessarily captures all event histories in one event may take place.

**ᶄ**

_{1}can observe this subtlety only because the width on

**s**of

**u**

_{4}need not align with

**₵**, that fact being the special case we mentioned. This ‘offset’, which is more accurately referred to as a general moment, is possible because we have no guarantee that all Natural Limits of any observer

_{s}**ᶄ**

_{0}will have the same real magnitude

**₵**so that, in general:

**₵**_{ij} ≠ **₵**_{ji} ∀ i ∈ ℝ

which compels us to derive this ratio between the Natural Limit on ** ℓ** and

**and the average Natural Limit of the remaining bases. This fundamental quantity will help us understand the moment magnitude, at least for local observers. This ratio will then be used to derive a relation to define periodicity.**

*q*Let all bases of a spatial system ** S_{n}** excepting bases

**and**

*ℓ***be the directional components of a defined moment of rate change given by the rate change in bases**

*q***and**

*ℓ***upon an event**

*q***u**

_{4}∃

**ᶄ**

_{1}measures a quantity, call it

**τ**, whose direction is perpendicular to all bases and whose behavior is characterized thusly:

**τ_{uℓ} = ** (d

**u**

_{5}/ d

**ᵯ)**⨂ (d

**d**

*s /***ᵯ)**and

**τ_{uq} = ** (d

**u**

_{5}/ d

**ᵯ)**⨂ (d

**d**

*s /***ᵲ)**

|** ℓ**|

**|**

*=***⨀**

*ℓ***τ**| and

_{uℓ}**with Natural Limit**

*ℓ***₵**

_{ℓ}|** q**| = |

**⨀**

*q***τ**| and

_{uq}**with Natural Limit**

*q***₵**

_{q}But the Natural Limits **₵_{ℓ}** and

**₵**are atypical since the real, closed interval within which they are undefined is

_{q }|**₵_{ℓ}**|= |

**₵**

*| ≤ |*

_{i}**₵**

*| ≤ |*

_{i}**₵**

*| ≤ |*

_{i}**₵**

*| ≤ |*

_{i}**₵**

*| ≤ |*

_{i}**₵**

*| ≤ |*

_{i}**₵**

*| ≤ |*

_{i}**₵**

*| ≤ |*

_{i}**₵**

*|*

_{i}|**₵_{q}**|= |

**₵**

*| ≤ |*

_{i}**₵**

*| ≤ |*

_{j}**₵**

*| ≤ |*

_{j}**₵**

*| ≤ |*

_{j}**₵**

_{j}| ≤ |

**₵**

*| ≤ |*

_{j}**₵**

*| ≤ |*

_{j}**₵**

*| ≤ |*

_{j}**₵**

_{j}|

where each**₵ **denotes a **₵*** _{i}* ∈

**which is a unique basis of**

*S*_{n}**ᶄ**

_{1 }and

**u**

_{4 }presenting to each basis, relative to

**ᶄ**

_{1}, a magnitude of width parallel to each basis measuring |

**₵**| for

_{ℓ}**and |**

*ℓ***₵**| for

_{q}**. So**

*q***ᶄ**

_{1}measures and sums the product of real magnitudes and some scalar of each basis Natural Limit (except for the basis being evaluated), then divides the result by 9.

**ᶄ**

_{1 }advances the hypothesis

that the ratio of this average to the width presented by

**u**

_{4}is a constant of significant local import in that it is a factor of the rate change in each basis and the First Act, to be discussed later. Until we can look more closely at its meaning,

**ᶄ**

_{1}’s measured quantity is denoted

**∃**

*h**h* = *a* / |**₵_{ℓ}**|

where

*a* = _{i=0}Σ^{i=9} |**₵_{i}**| ∀ bases

**s**, ∈

**relative to**

*S*_{n}**ᶄ**

_{1}

and we shall confirm later that |**₵_{ℓ}**| = |

**₵**|.

_{q}Thus, in the Second Act we will expound on the relation that describes the geometry of an event in which two linearly independent Natural Limits of unequal real magntitude present to each other:

(c cos ϕ)^{2} + (*h* – |**₵_{ℓ}**|)

^{2}= c

^{2}

(*h* – |**₵_{ℓ}**|)

^{2}= c

^{2}– 𝔵

^{2}, 𝔵 ≔ abscissa

ð c = (*h* – |**₵_{ℓ}**|) / sin(ϕ)

where ϕ = (π / 2) – θ

and θ ≔ maximum presentation interval on ** ℓ** or

**. And we note the derivation of the important quantity to which we’ve alluded:**

*q*c = (*h* – |**₵_{ℓ}**|) / sin(ϕ)

ð π ≔ c (*h* – |**₵_{ℓ}**|) / (

*h*– |

**₵**|)

_{ℓ}ð π ≔ c which is independent of *h* or |**₵_{ℓ}**|. [§ 0.0202]

So, **ᶄ**_{1} observes that in the Natural Limits any given basis, **s**, will experience one full but arbitrary cycle of event history in which the rate change in that event history is not constant. Note that the value derived is in vector component form.

As a result of this oblique interaction should **ᶄ**_{1 }measure the rate change in event history on any given basis, **s**, then **ᶄ**_{1} observes two salient facts:

- Any coordinate transformation between
**ᶄ**_{1 }and any other observer,**ᶄ**_{0}, is guaranteed covariant by this fact. - One cycle of turns, that is, one turn for each basis, occurs within the smallest Natural Limit.

This is because the general moment aforementioned allows for the persistence of event histories without requiring a parameter **t**, defined in some spatial system ** S_{z}** of order > 10. This is possible because the effect of the moment is contained within the smallest Natural Limit

**₵**∈

**. The second observation will be proven by walking through**

*S*_{n}**ᶄ**

_{1}’s observation of the cycles to derive the ratio between the largest and smallest Natural Limits.We shall also see that another basis,

**, exceeds the bounds of necessity and sufficiency and the same observations avail by a of**

*t***ᵯ**and

**ᵲ.**and self-parameterization which we will describe shortly.

An intrinsic rate of change in each event history of **ᶄ**_{1}is observed to exist inasmuch as there is logical order in causality. But it is parameterized by an undefined parameter.

So, generally, **u**_{4} is observed by **ᶄ**_{1} as incident upon itself as to produce a linearly independent basis vector, 𝕋, tangent to the vector weighted rates of change in all other basis axes observed. The vector function 𝕋 takes dependent variables which are rates of change. Its range is the n + 1 derivative of that rate of change. **ᶄ**_{1, }having recorded its history, observes that 𝕋 is unique and linearly independent in that all other tangent vectors 𝕋 (f^{ q}(x_{0}), …, g^{r}(x* _{n}*)) are linearly dependent in 𝕋 (x→λ); which is both necessary and sufficient to account for

**ᶄ**

_{1}’s measurements. The details of this observation will be developed in what follows.

Conditions necessary and sufficient exist for linearly dependent events to be observed from a linearly independent measurement angle. Ergo, a Spatial System *S** _{n}* is linearly dependent and contains 8 subsets of linearly independent basis vectors and two parameterizations,

**ᵯ**,

**ᵲ**, that are both necessary and sufficient to fully define its history and all natural events therein. The preceding we shall reference from here on as the Historical Acts.

We now describe the behaviors observed in *Gedanken* One with a preliminary algorithm that, at least as a first order approximation appears reasonable. Let **c** = constant, **v** be an imparted rate change along any event history, and **a** be the order 10 tensor resultant consequent to applying **v**:

**a**^{2} = |**c**||**v**| [(∂|** i**| / ∂)

^{2}+ (∂|

**| / ∂)**

*j*^{2}+ (∂|

**| / ∂)**

*k*^{2}+(∂|

**| / ∂)**

*ℓ*^{2}+ (∂|

**ᵯ**| / ∂)

^{2}+

–*ɨ(*∂|** n**| / ∂ )

^{2}–

*ɨ(*∂|

**| / ∂ )**

*o*^{2}–

*ɨ(*∂|

**| / ∂ )**

*p*^{2}–

*ɨ(*∂|

**| / ∂ )**

*q*^{2}–

*ɨ(*∂|

**ᵲ**| / ∂ )

^{2}]

⨀

[(∂ / ∂ |** i**| + ∂ / ∂ |

**| +∂ / ∂ |**

*j***| + ∂ / ∂ |**

*k***| + ∂ / ∂ |**

*ℓ***ᵯ**|) + (

**r**Space)

(∂ / ∂ |** n**| + ∂ / ∂ |

**| + ∂ / ∂ |**

*o***| + ∂ / ∂|**

*p***| + ∂ / ∂|**

*q***ᵲ**|] (

**m**Space)

In which we take the area of the Argand planes considered as a dot product scaling factor along with the factor following it; which is a vector of the generators used in tensor form. Note that **c **is distributed into both parts and we shall find shortly that **e*** ^{x}* is

*the*definition of this function. We motivate that finding with the following observation made at

**ᶄ**

_{1}. For each unit |

**| observed**

*n***ᶄ**

_{1}also measures one unit in |

**|. But a possible scenario, and one that we shall see is common, is for a unit advance in |**

*i***| to occur whilst unit advances > 1 occur in |**

*n***|; or**

*i**m*|

**|, where**

*i**n*is the magnitude of the |

**|’th basis vector. Following this observation, we can characterize it logically as a ratio of |**

*i***| to**

*n**m*|

**|, which will provide for us the length along |**

*i***| within which any |**

*i***| is contained ϵ |**

*n***|. The exact quantities of position within |**

*i***| is unimportant for our purposes (and cannot be defined). Our condition for this example is that no further events occur once these two basis vectors are defined. ∴**

*i*When *m* = 2 the region in |** i**| in which |

**| is contained is**

*n*|** n**| /

*m*|

**| = 1/2 and when**

*i**m*= 3 we have,

|** n**| /

*m*|

**| = 1/3,**

*i*but this relation is incomplete. The region within which |** n**| is contained must be 1/3 of ½, that is, 1/3 of the previous value; i.e

p = |** n**| /

*m*|

**| = 1/2 * 1/3 for**

*i**m*= 3 and p is a placeholder ratio variable. And as we progress a factorial relation emerges:

p = |** n**| /

*m*|

**| = 1/2 * 1/3 * 1/4 for**

*i**m*= 4 and

p = |** n**| /

*m*|

**| = 1/2 * 1/3 * 1/4 * 1/5 for**

*i**m*= 5 or, more generally,

p = |** n**| /

*m*|

**| = 1/**

*i**m*!

The region of containment is, of course, the sum of all these ratios since we seek the *total possible* region of confinement. That is;

p = |** n**| /

*m*|

**| = Σ**

*i**m*=0,

*m*=q 1/

*m*!

We note a few interesting features of this relation. By *Gendanken* One, any integer multiple of |** n**| generates the same integer multiple in |

**|; that is, the event that initiated the**

*i***’th history depends on the event that initiated the**

*i***’th event history. And that relation is 1 to 1. This means that the magnitude of this ratio, the rate at which that magnitude changes with respect to the reals, and the rate at which the rate changes are all equal (for q up to a Natural Limit – to be discussed later). So, the function that we are deriving will be unique since it’s first and second derivatives will equal the functions range. Any proof of this case for any n number of derivatives we will leave as an exercise, but one can clearly see from**

*n**Gendanken*One that this must hold for any n number of derivatives. And, we can see more generally that

p = *o*|** n**| /

*m*|

**| = Σ**

*i**m*=0,

*m*=q

*o*/

^{m}*m*! where

*o*is any real integer.

If not obvious, we can frame the question as, “within what region of ** n** is any |

**| confined”? The answer is trivial as it is simply**

*n**o*. That is, for any increment in

^{m}*o*= 2 in |

**| the real interval over which**

*n**o*|

**| is confined is the product of the greatest increment and the least increment since the greater depends on the lesser**

*n**and*depends on

**i**

*n**.*

and for q sufficiently large;

q ≈ **e ^{1}**. This is a fundamental

*causality ratio*of nature and does not depend on any system of formal logic. We can also state it as the ratio of a Natural Limit to the real length of the region in any unit 1 real space within which the Natural Limit ₵

**(see following section for definition) is confined:**

_{ᶄ}ℓ ≈ 1 / q, where ℓ is the length of the real region of space within which its Natural Limit is confined with both the real and imaginary axes set to 1.

We will develop this and the derivation more fully later.

**3.****Principle of a Natural Limit**

*An event is sufficiently well defined in any spatial system S_{n}, iff the event information necessary and sufficient to observe the event exists in S_{n}, regardless of its form. And any s *

*⊂*

**S**_{n}, in which any such event information cannot exist is a natural limitation to that event and the event is undefined relative to any observer**ᶄ**ϵ**S**_{n}.[2]Any quantized Natural Limit relative to some observer ** ᶄ**we shall, from here on, denote as ₵

**, provided a function, f, exists to define its spatial extent in**

_{ᶄ}**r**Space. It is read as pipe ‘C’ relative to an observer

**:**

*ᶄ*Let ** ξ = **(d|

**ℓ**| * ₵

**)/ d**

_{ℓ}**t**

_{re}denote a rate of change in the axis

**ℓ**and

**ℓ ≥ **₵** _{ℓ}** = 1

Let ** χ** = |

**ℓ**| * ₵

_{ℓ}Should d|₵** _{ℓ}**| / d

**ᵲ**retard

**/**

*ξ***approaches 1 and by the Natural Limit, ₵**

*χ***, cannot proceed to anything < ₵**

_{ℓ}**as it is undefined. Therefore, we will proceed first with the much delayed operational definition of what we mean by a Natural Limit:**

_{ℓ}**Definition:** Let ℱ be a function defined on some real, open interval that contains the number *s*. Then we say that the Natural Limit of ℱ(*r*) when *r* equals (*s* – 2γ) is ₵, and we denote it

*nlm _{r}*

_{ →s}[ℱ(

*s*) – ℱ(

*s*– γ)] = ₵ [§ 0.0301]

if for every number ϵ > 0 there is a corresponding number ρ > 0 ∃

|ℱ(*s* – γ)| < | ℱ(*s*) – 2₵| < ϵ whenever |(*s* – γ)| < |(* s* – 2γ)| < ρ [§ 0.0302]

Which we contrast with the traditional definition of a limit which might be:

“Let ℱ be a function defined on some open interval that contains the number *s*, except possibly at *s* itself. Then we declare that the limit of ℱ(*r*) as *r* approache *s* is ₵, and we denote it

lim_{r}_{→s} ℱ(*r*) = ₵

if for every number ϵ > 0 there is a corresponding number ρ > 0 ∃

| ℱ(*r*) – ₵| < ϵ whenever 0 < |*r* – *s*| < ρ

And we refer to this as a General Limit, or *glm*. This definition does not depend on properties of nature.[3]

In accordance with the findings from above, **ᶄ**_{1}initiates an experiment to establish an operational definition of “spatial definition” and “lack of spatial definition”. Measuring from the **r**Space surface (|** i**| = 0) of the Natural Limit

**ᶄ**

_{1}records that not only is a measurement of any one linearly dependent basis not possible, but any unobservable space defined at the surface of a Natural Limit is linearly independent.

**ᶄ**

_{1}proceeds with an experiment to measure the ratio of the rates of change in the definition between the two parameters

**ᵯ**and

**ᵲ**.

**ᶄ**

_{1}proceeds in the following manner:

First, it arbitrarily sets **ᵲ** to a unit 1 value. It then proceeds to measure the Natural Limit of the **ᵲ** axis by measuring from the only two defined points defined and about which ₵ is confined; to-wit, 0 and ₵ both of which are found in the interval [**u**_{1}**, u**_{2}] along the **ᵲ**’th basis axis. **ᶄ**_{1} is careful to measure a unit value sufficiently large ∃ 1 ≫ ₵. **ᶄ**_{1} observes that for each new dilation on **ᵲ** = **ᵲ** + ₵,the current rate of dilation, **v**_{1ᵲ}, in the Natural Limit is the immediately smaller dilation v_{0} * |**v**|_{1ᵲ} . That is, **ᶄ**_{1} parameterizes the new increment in **ᵲ** on the immediately smaller value |**ᵲ**| – ₵. Thus |**v**|** _{ᵲ}** are sufficiently well defined only in quanta of

**r**Space. Stated generally, this means:

1 / **₵ **= |**ᵯ**|

ð the ratio of the two rates of change is r^{m} = **v _{ᵲ}** /

**v**;

_{ᵯ}And **ᶄ**_{1} observes that for

**v**_{finalᵲ} = (|**v**_{mᵲ}| / |**v**_{mᵯ}|) > 1; **r**Space is expanding on **ᵯ**

**v**_{finalᵲ} = (|**v**_{mᵲ}| / |**v**_{mᵯ}|) < 1; **m**Space is expanding on **t**_{re} and if

**v**_{finalᵲ} = (|**v**_{mᵲ}| / |**v**_{mᵯ}|) = 1; no expansion or contraction is occurring.

**ᶄ**_{1}therefore defines “definition of a basis axis” thusly:

**Definition: **A basis axis, *ℓ*, is defined to the extent that the *ℓ*’th event history of the Historical Acts is defined in magnitude and direction up to the Natural Limit of *ℓ*. The quantifiable measure of definition is the ratio of the real numbered magnitude and direction of the Natural Limit, ₵_{ℓ} (pipe C), of *ℓ* to the magnitude and direction of the *ℓ*’th event history.

**Corollary: **Let q_{α} be defined as a magnitude of the same type and kind as ℓ (i.e. a “length”). A basis axis, *ℓ*, is sufficiently well defined to the extent that the *ℓ*’th event history of the Historical Acts is sufficiently well defined in magnitude and direction up to the Natural Limit of *ℓ* iff:

Upon a cause, α, of an event’s effect, ω, on ℓ q_{α} < ₵_{ℓ} a dependency ω = f(q_{α}) does not present.

In summary, the Historical Acts are canonical ∵ it is the history only as given by the Historical Acts. **r**Space creates histories and **m**Space destroys (retraces) them.[4] Though it is possible to reduce, or back track an event history, if it is given that the canonical historical acts exist for some observer **ᶄ**_{0}, there can be created no event history **v** with any combination of scalars a* _{n}* ∃

a_{0}|**v|** + a_{1}|** i**| + a

_{2}|

**| + , … , a**

*j**|*

_{n}**| =**

*z***0**(not all a’s = 0)

regardless of whether

a_{1}|** i**| + a

_{2}|

**| + , … , a**

*j**|*

_{n}**| =**

*z***0**(not all a’s = 0)

holds.

**4.****Parameterizations**

We next engage the same phenomenon with parameterizations of the basis vectors observed.[5]

Let an observer *ᶄ*_{0}place two instruments in its frame of reference, one with spatial extension only in the ** i**’th direction and the other only in the

**’th direction. The instrument**

*m*

*ᶄ*_{re}, for the purpose of observing “change in the event history”[6] with respect to the

**r**Space basis vector |

**| and |**

*i***ᵯ**|, that is, from

**r**Space, and holds all other basis vectors constant (though it appears that we are measuring a ‘speed’ the choice of a spatial axis

**as the parameter for**

*i***ᵯ**is a gentle introduction to self-parameterization. This will allow us to calculate the rate of the event history of time). An instrument

*ᶄ*_{im}on

*ᶄ*_{0}performs a similar observation with respect to the basis axes

**and**

*n***ᵲ**that is, from

**m**Space.

**ᵲ**is time and

**and**

*i***are corresponding spatial axes sharing the same Natural Limit as**

*n***ᵲ,**and derived from event interval (

**u**

_{1}

**, u**

_{2}).

*ᶄ*_{re}proceeds to calculate the rate of the event history – the

**’th basis axis.**

*i*

*ᶄ*_{re}establishes a system of units by setting the rate of change in the event history of

**ᵯ**= 1 (from the

**re**reference frame this is necessary due to the Natural Limit at

**ᵯ**of 1)[7].

*ᶄ*_{re}first notes that:

When *ᶄ*_{re} measures its own parameter, |**ᵯ**|, it measures the age of *ᶄ*_{re}’s current Spatial System, *S _{ᶄ}*

_{re}. It’s value will be directly dependent on the largest magnitude of the

**’th basis axis and has meaning only relative to that reference frame.**

*z*

*ᶄ*_{im}measures a different value for |

**ᵯ**|, either dilated or constricted and the ages of

*S*_{ᶄ}_{re}and

*S*_{ᶄ}_{im}do not agree. |

**ᵯ**| presents as a function of the ratio of the scalar values on each

**’th basis axis ϵ**

*z***m**Space to the scalar values for each

**’th basis axis ϵ**

*z***r**Space where:

|**ᵯ**|_{ᶄ}_{im} ≠ |**ᵯ**|_{ᶄ}_{re}; for any |**ᵯ**| ϵ ℝ.

So, when we impute a scalar, |** n**|, defined for

*ᶄ*_{im}onto the corresponding basis in

*ᶄ*_{re}, relative to

*ᶄ*_{re}, |

**| is undefined, that is, it is rather the Natural Limit on**

*n***, ₵**

*i*

_{ᶄ}_{re}|

**|. Notice that the antecedent sets the scalar using its own Natural Limits, not those of the precedent.[8]**

*i***5.****Principle of Repose**

*For any parameterizations, t_{i}, and any spatial geometry *

*ℝ*

^{j}chosen, all observers defined by that space are immutably reposed. That is, motions, speeds, and velocities have no possibility of existence.**6.****The Three Acts**

Here we will classify “Acts” which are the three Canonical Acts of Nature principally observable. They are:

- The First Act: This relation reveals a natural curvature of the spatial system that maintains symmetry in spatial curvature with equal values of volume, rate change and change in the rate change of the event history at any given point in Space.
- The Second Act: The combined effect of the First Act and Natural Limits.
- The Third Act: The combined effect of Acts one and two and relative rotation of spatial systems.

At bottom, all ‘forces’ of nature are in fact various modes, or Acts, of spatial change. The First Act can be seen below in the graph in which the ordinate reveals the volumetric growth of e/1. In a full rendering the ordinate would be the spatial axis that is expanding and the surface would look like a sombrero turned over.

**7.****The First Act**

Witness an invariant metric that gives us a generator to apply to any change in any basis axis:

ð**ᵯ**_{f}^{2} = (|**ᵯ*** _{i}*| / |

**ᵯ**

*|)*

_{f}^{2}

_{re}–

*ɨ*(|

**ᵲ**

*| / |*

_{f}**ᵯ**

*|)*

_{f}^{2}

_{im};

which should provide us with an invariant parameter (time) for measuring all other axes.[9] So, the experimental observations of *ᶄ*_{re} and *ᶄ*_{im} are given by the relations:

1 = [∂|**ᵯ*** _{i}*| / ∂]

_{re}; in the

**re**reference frame; operated on by

**m**Space

⨀ = [-*i* (∂|**ᵲ*** _{f}*| / ∂)]

_{im}; in the

**im**reference frame

Which shows the two components on an Argand plane, one for **r**Space and the other for **m**Space. Let **v**_{ξ} denote a vector of the event history rate of change on some axis, ξ. Here this quantity is always negative since we are evaluating the rate at which the event history is being “back tracked”. We take the absolute value of the components to evaluate it.

We next derive a relation to characterize the effect this will have generally on the spatial system, ** S_{n}**. Since we need to characterize behaviors in both

**r**Space and

**m**Space we’ll need to reference the Argand plane for some of the evaluations we perform. We begin by outlining in more detail the general form of what we shall show is the appropriate geometric representation of space as a rank 2, order 10 tensor:

_{2}**|**_{10} = |**v**_{ξ}||**e**|^{𝔵} [(∂|** i**| / ∂) + (∂|

**| / ∂) + (∂|**

*j***| / ∂) +(∂|**

*k***| / ∂) + (∂|**

*ℓ***ᵯ**| / ∂) +

–*ɨ(*∂|** n**| / ∂ ) –

*ɨ(*∂|

**| / ∂ ) –**

*o**ɨ(*∂|

**| / ∂ ) –**

*p**ɨ(*∂|

**| / ∂ ) –**

*q**ɨ(*∂|

**ᵲ**| / ∂ )]

⨀

|**e**| [(∂ / ∂ |** i**| + ∂ / ∂ |

**| +∂ / ∂ |**

*j***| + ∂ / ∂ |**

*k***| + ∂ / ∂ |**

*ℓ***ᵯ**|) + (

**r**Space)

(∂ / ∂ |** n**| + ∂ / ∂ |

**| + ∂ / ∂ |**

*o***| + ∂ / ∂|**

*p***| + ∂ / ∂|**

*q***ᵲ**|] (

**m**Space)

Where ** is referred to as “check”, and ****v**_{ξ} refers to the vector being transformed by the tensor.

ð|**v**_{ξ}|[(∂|** m**| / ∂) + –

*ɨ(*∂|

**r**| / ∂ )] *

passing in [ ∂ / ∂|** m**|] for metric of real time to imaginary time

** = |****e**|^{𝔵} |**v**_{ξ}|[∂|** m**| / ∂|

**| + –**

*m**ɨ(*∂|

**r**| / ∂|

**|)];**

*m*** = |****e**|^{𝔵} |**v**_{ξ}|[ 1 + –*ɨ(*∂|** r**| / ∂|

**|)];**

*m*|**v**_{ξ}| ^{2} = |**v**_{ξ}||**e**|^{𝔵} [ 1^{2} + –*ɨ(*∂|** r**| / ∂|

**|)**

*m*^{2}];

|**v**_{ξ}|** = |****v**_{ξ}| |**e**|^{𝔵} √ [ 1^{2} + –*ɨ(*∂|** r**| / ∂|

**|)**

*m*^{2}];

β = ^{2} = 1^{2} + –*ɨ(*∂|** r**| / ∂|

**|)**

*m*^{2}; [§ 0.0801]

where |** | is the generator to apply to any unweighted space variable. This value represents the relative ambient rate change between any two bases and is generated solely by the First Act. As the reader may suspect, “checking” (invoking the **** tensor) some axis **** z** locating two observers neglecting all other Acts produces in full fidelity the

*fundamental metric tensor of General Relativity*. The key difference is that no observation under this tensor will result in infinite quantities due to the limitations imposed by natural limits.

From event **u**→**v** all **m**Space dims ** m**,

**,**

*n***,**

*o***have intervals**

*p*(**u**, **v**), (**v**, **w**), (**w**, **x**), (**x**, **y**). This is defined as *anything but ***r**Space.

**r**Space dims ** i**,

**,**

*j***,**

*k***have intervals**

*ℓ*[**u**, **v**]*n*, [**v**, **w**]*n*, [**w**, **x**]*n*, [**x**, **y**]*n*.

where *n* denotes the historical magnitude of the axis. Note that this includes a potential surface of matter, a plane (or manifold) of tangent vectors ** p** and

**.**

*ℓ*Let us now propose the hypothesis that the event histories of *S** _{n}* are an example of canonical wave behavior with period T. Let

*ᶄ**measure the rates of definition to determine, first, if the rate of event histories observes any kind of conservation principle.*

_{1}

*ᶄ**reports that we have already observed the fact that for any unit length |*

_{1}**|**

*i*

^{u}_{re}and for any unit length measured over the Natural Limit, ₵

^{u}

**, there is a corresponding Natural Limit, ₵**

_{i}**, ∃**

_{i}₵_{i}* / *₵^{u}**_{i }** = [|

**|**

*i*_{re}/ |

**|**

*i*

^{u}_{re}]

_{,}

the point being that, for each unit value of |** i**|

^{u}_{re}defined there is exactly one unit of the Natural Limit, ₵

^{u}

**, that is undefined. So**

_{i}

*ᶄ**measures under covariant parameterization (for both*

_{1}**m**Space and

**r**Space) the magnitude of both and finds that, provided no other perturbations are present, the proportions of magnitude, rate change and rate of rate change repose equal, as the value e

^{1}requires. That is, should we reflect the previous experiment used to derive the volumetric definition per Natural Limit from

**r**Space to

**m**Space,

*ᶄ**gets the same answer. The key observation is whether or not this behavior is periodic. For each invariant unit value of time passed, t, in*

_{1}**m**Space and

**r**Space,

*ᶄ**notes that, for any arbitrarily selected length in both Spaces the value is equal. However, when measured under covariant time in*

_{1}**r**Space the periodicity appears to disappear (for a macroscopic observer, the periodicity is reflected within the Natural Limit and is not readily observable). The same is true for any observation strictly from

**m**Space. ∴

*ᶄ*_{1}concludes that rates of event histories are a conserved quantity and that, summed over its full breadth, the spatial system

*S**is the*

_{n}**0**matrix. And the period of this canonical wave, the First Act, is |

*v**|e / 4, where*

_{i}

*v**is the current velocity of event history at a point*

_{i}*a*on |

**|**

*i*_{re, im}at which the period is measured. e ≈ 2.718 is the necessary consequence of the manner in which Space is defined and undefined, indeed, the spatial system,

*S**is*

_{n}**definition of e.[10]**

*the*To specify the behaviors of the First Act **ᶄ**_{1} calculates the total defined area created as observed by an invariant observer (in this case, **ᶄ**_{1}). Calculating **r**Space and **m**Space area **ᶄ**_{1} determines the surface area created for each unit 1 of **r**Space created and each ₵ of **m**Space created. Thus, we’d expect

∫ ∫ (1/₵_{im} * 1/|**ᵵ**|) d₵_{im} d|**ᵵ**| [§ 0.0802]

= ((₵_{im}) ln(₵_{im}) – ₵_{im} + c) / |**ᵵ**| and let c = 0.

(1 / |**ᵵ**|) ∫ ∫ d₵_{im} d|**ᵵ**| (1/₵_{im} * 1/|**ᵵ**|)

= ((₵_{im}) ln(₵_{im}) – ₵_{im} + c) and let c = 0

A = ∫ |**ᵵ**| ln(|**ᵵ**|) – |**ᵵ**|

*nlm* as |**ᵵ**|→A = |**ᵵ**|e^{(|ᵵ|)}

and A = 1;

But recall that for each |**ᵵ**|_{ℓ} found there exists a real space |** ℓ**| whose exposure to measurement depends on |

**ᵵ**|

_{ℓ}. Consider the representation of an event history. If the event history |ℓ| should increase with some parameterization

*t*_{ℓ},then it must do so with its dependency; that is, for each unit increase in |ℓ| there is a corresponding change in time with respect to |

**ᵵ**|

_{ℓ}occurring at the rate of |

*t*_{ℓ}| / |

**ᵵ**|

_{ℓ}∃

|**ℓ**|_{1} * |**ℓ**|_{2} = 1 / |**ᵵ**| (|*t*_{ℓ}| = 1; a unit value)

if we assume 1 real increment of |**ℓ**| per unit time. So that generally, and working only with the defined endpoints of the interval (|**ᵵ**|_{ℓ0}*,*|**ᵵ**|_{ℓ1})

|**ᵵ**|_{ℓn} = |**ᵵ**|_{ℓ1 }* |**ᵵ**|_{ℓ2} * … |**ᵵ**|_{ℓn} and (as in *n*! = |**ᵵ**|_{ℓn})

|**ℓ**|* ^{n}* = |

**ℓ**|

_{1 }* |

**ℓ**|

_{2}* … |

**ℓ**|

*∀ increments*

_{n }*n*∈ |

**ℓ**| where

*n* = 1 / |**ᵵ**|_{ℓ} and

|**ℓ**|* ^{n}* = 1/ |

**ᵵ**|

_{ℓn}, [§ 0.0803]

And adding unity to represent the unit values of |** ℓ**|, we have:

Δ|**ℓ**| = (|** ℓ**|

*+*

_{unit}*n*)

^{(1/n)}

^{ð }Δ|**ℓ**| = *glm _{n}*

_{→0}(1 +

*n*)

^{(1/n)}

^{ð }Δ|**ℓ**| = |**e**^{1}|. [§ 0.0804]

for each increment *n* in |**ℓ**| per unit time |*t*_{ℓ}|. If there are Δ|**ℓ**| increments then |**ℓ**| = |**e**^{Δ|ℓ|}|,

ð|**ℓ**| = |**e*** ^{n}*|,

ð|**ℓ**|` = |**e*** ^{n}*|,

ð|**ℓ**|“ = |**e*** ^{n}*|.

Thus, the rate change |**ℓ**|` is given by Euler’s formula:

|**e*** ^{in}*| = |

**ℓ**| = |

**ℓ**|`= |

**ℓ**|“= cos (

*n*)

*+ ɨ*sin (

*n*);

Recalling the complex plane representation of **m**Space, we can directly deduce the physical meaning of this equation as the rate and manner at which some basis axis |**ℓ**| is being defined and undefined. Here we’ve set the increment count to the time, both in unit values. ∴

*n* = 𝓎 = |*t*_{ℓ}|, 𝓎 a periodic angle, ∃

for 𝓎_{i }= 𝓎_{j},𝓎_{i}cos (*n*) *+ ɨ* 𝓎* _{j}*sin (

*n*)

ð|**e*** ^{in}*| = |

**e**

^{i}^{𝓎}| = |

**e**

^{i}^{|t|}|

showing that the periodic change in event histories occurs as a pattern of cosine and sine and all observers *ᶄ*_{m }∈ *S** _{n}*are surfing different points on the same canonical wave,

**ℳ**.

We now examine its corollary measured in **m**Space. *ᶄ** _{1}*, having measured the properties denominated in the Historical Acts, is presented with the following proposition: if the Natural Limit of the

**r**Space imaginary parameter |

**ᵯ**| implies ponderable observables ∈ |

**₵**|

_{ᶄ}*then we are now advised to set up another experiment to investigate. Since we know that |*

_{1m}**ᵯ**| < |

**₵**|

_{ᶄ}*and that |*

_{1m}**ᵯ**| lies within the open interval (0, |

**ᵯ**|) contained ∈ |

**₵**|

_{ᶄ}*, observables within |*

_{1m}**₵**|

_{ᶄ}*must exist; namely, the value for |*

_{1m }**ᵯ**| and its rate changes. In addition, we can determine whether or not its change is like the change we identified as above, i.e. |

**e**|

*. ∴*

^{it}

*ᶄ**locates measuring instruments on*

_{1}**ᵯ**and

**a**designed to measure the rate of the time event history relative to

**ᵯ**and some arbitrary real axis

**a**. Using clocks of sufficiently high precision

*ᶄ**measures the rate passage of time on*

_{1 }**ᵯ**and

**a**and indeed finds a very small difference between them. This difference, however, is larger than any of the Natural Limits in

**thus confirming the physical significance of the measurement.**

*S*_{n}We can interpret this data by starting with the supposition to be demonstrated that |**ᵯ**| ≪ |**₵**|_{ᶄ}* _{1m}* in general for any observer

*ᶄ**∈*

_{1}**. If the supposition holds, we have:**

*S*_{n}|**₵**|_{ᶄ}* _{1m}* / |

**ᵯ**| ≫ 1

and for any measurement of **ᵯ**, call it *p,* in |**₵**|_{ᶄ}* _{1m}* (which we now know to be observable) the Principle of a Natural Limit suggests that:

*nlm _{t}*

_{ →p}[ℱ(

*p*) – ℱ(

*p*– |

**ᵯ**|)] = |

**₵**|

_{ᶄ}

_{1m}and we have obtained the condition that if for every number ϵ > 0 there is a corresponding number ρ > 0 ∃

|ℱ(*p* – |**ᵯ**|)| < | ℱ(*p*) – 2₵| < ϵ whenever |(*p* – |**ᵯ**|)| < |(*p* – 2|**ᵵ**|)| < ρ

it follows immediately that the undefined region swept out in a unit change of **ᵯ** of |**₵**|_{ᶄ}* _{1m}* is given by:

A = (|**₵**|_{ᶄ}* _{1m}* / |

**ᵯ**|)

^{2}

where A is the area “swept out” and that generally we have:

V = (|**₵**|_{ᶄ}* _{1m}* / |

**ᵯ**|)

^{(|₵|ᶄ1m / |ᵵ|)}

where V is a volumetric change between the imaginary time (**r**Space time) and an **r**Space spatial axis which changes at the rate:

d/d** t** V = ₭ = d/d

**[(|**

*t***₵**|

_{ᶄ}*/ |*

_{1m}**ᵵ**|)

^{(|₵|ᶄ1m / |ᵵ|)}]

ð₭ = [|**ᵵ**| / (|**₵**|_{ᶄ}* _{1m}* / |

**ᵵ**|)

^{(|₵|ᶄ1m / |ᵵ|)}]

ð₭ = |**ᵵ**| |**₵**|_{ᶄ}_{1m}^{-(|ᵵ| |₵|ᶄ1m)}

let **₵**|_{ᶄ}* _{1m}* = 1 unit value.

ð₭ = |**e**|^{–it}. [§ 0.0805]

And this is the same result we got for **r**Space except for a change in the sign of the exponent. *ᶄ** _{1}* compares the experimental results to this formula and finds agreement up to the limits allowed by the precision of its clocks. And the supposition holds to the precision afforded by the clocks.

**Definition: **Let **S**_{n} = Β where the Principle of a Natural Limit holds. Suppose i_{n} ϵ δ and j_{m} ϵ Β exist such that each value j is the natural effect of a corresponding natural cause i_{n} ϵ δ. Now let β ⊂ Β be a set of effected natural events i with natural causes {i_{0},… i_{k}} ϵ δ. Then there can be found ∈ Β no function ℱ ∃

ℱ(i) = j_{m} ϵ Β where i_{n} ϵ δ. [§ 0.0806]

And the existence of β ∈ Β is observable ∈ Β but principally inscrutable ∈ Β, which we *define* to be the Inscrutable Subset. That is, the scientific method when defined as a process for applying necessary and sufficient conditions for the hypothesized value of a dependent variable, upon reaching some value of an independent variable, to exist fails. And,

Any system of formal logic ∈ Β is necessarily incomplete and possibly internally inconsistent.[11]

**Definition:** We *define* The First Act as a spatial wave, **ℳ**, whose behavior |**e**^{i}^{t}| is the effect of an inscrutable cause.

**Definition:** We*define*the unique event history scalar |**e*** ^{n}*| ∃ it is equal to the natural volume of any sufficiently well defined space measured over a defined set of parametric increments n, and which always equals the rate of change in volume and the subsequent curvature of that space, all with respect to some real parameter, |

**ᵯ**

_{ ℓ}|, assuming it exists.

**Corollary:** The First Act is the initiation point of causality in the spatial system, ** S_{n}**, and a definable precedent event ∈

**has no possibility of existence. That is, no system of formal logic alone, ζ, can suffice to define e sufficiently well.**

*S*_{n}**Corollary:** Corollary to the Principle of Repose: For any order p spatial system **S**_{n}, all representations of natural logic defined therein are a special case of nature and are linear. Thus, the methods of the The Calculus represent the general case. Spatial ‘curvature’ is a combination of linear geometries on order p constrained to some integer value 0 < q < p.

**Definition:** Any arbitrarily chosen observer, **ᶄ**_{0}, is guaranteed to have what we shall here denote to be a local spatial buoyancy, 𝒷, of zero ∃:

_{2}**|**_{10}*=* **e*** ^{it }*+ 𝒷; (see endnotes[12])

** **_{2}**|**_{10}*=* **e*** ^{it }*[§ 0.0807]

where |**e**^{i|t|}| is a universal quantity. Checking **ᶄ**_{0} we get the resultant of the local variation in buoyancy, 𝒷, relative to **ᶄ**_{0}, and the First Act, |**e**^{i|t|}|, which tensor we shall derive directly.

_{2}**|**** _{10 }** ∝ (d / dt) [(∂ f / ∂ u) * (∂ u / ∂ t)] – (∂ f / ∂ u),

and something wonderful happens[13]. The First Act confirms the conservation of energy generally.

On the Manner of Causality

Having discussed the First Act’s behavior, we next observe it’s implications as they relate to the scope of this work. An event α occurs on an axis *q** _{im}* in

*t**∃ an event ω occurs a distance of |*

_{re}**₵**|

_{re}over the Natural Limit, ₵

_{re}, in a time

*t**that locates orders of magnitude in the future of α which is in fact just ω (but this can only be observed by*

_{im}**ᶄ**

_{1}, not assumed). The linear separation is infinitesimal but the time separation is on the order of 10

^{9}years or more. We pose the following hypothesis to observer

**ᶄ**

_{1}and seek a means to measure it’s veracity. The hypothesis is as follows:

Iff the event history of ω renders one definition relative to **m**Space and another definition relative to **r**Space separate and distinct even though we can observe that it is not, causality is preserved. The reasoning is that, ∵ the definition of an event history is relative to each observer, event α when viewed from **m**Space may spontaneously convert to event ω when viewed from **r**Space. The alternative would be that an event α located distantly in the past, due to the conservation implied by e, could have an effect, ω, in the present and vice versa. Causality would then be bidirectional on ** t**. For our purposes, this would be equivalent to destroying, or “back tracking” the event history toward –

**.**

*t***ᶄ**_{1} begins by measuring the time |** t**|

_{imω}of event ω and records the result. With that result

**ᶄ**

_{1 }measures the time |

**|**

*t*_{reα }of event α and records the result. But

**ᶄ**

_{1}notes the value recorded for |

**|**

*t*_{reα}is undefined and that it is not an observable quantity. Upon investigation

**ᶄ**

_{1}observes that during the Historical Acts the event history |

**|**

*t*_{im}was

*adapted*from

**m**Space for use in

**r**Space for the reasons already shown. They were swapped. Let the period be 2T, T =

*n*π where

*n*is an integer multiple. So, in the last semi-period of π; i.e. T – 1, it swaps again. That is, |

**|**

*t*_{im}is imaginary at the value |

**|**

*t*_{imω}(no transformation specific to

**m**Space

_{T–1}exists for it and |

**|**

*t*_{imω}was inside a Natural Limit of

**m**Space

_{T-1}) in

**r**Space

_{T}and in

**m**Space

_{T}the point |

**|**

*t*_{reα}that parameterizes the event α also falls in its Natural Limit. This is because the event history |

**|**

*t*_{re }referenced is the t ∈ T, the period containing ω, not the |

**|**

*t*_{re }∈ T – 1 which parameterizes α.[14]

Likewise, **ᶄ**_{1} can observe that any time ** t** is not reversible. The rate of change in the event history

**can be increased or decreased, but it cannot suffer passage through its Natural Limit[15].**

*t***ᶄ**

_{1}observes this by increasing the destructive flux on

**given an energy E sufficient to measure a change in the rate of change, parameterized by |**

*t*

*t*_{0}

**, in all 10 event histories. Confirming this means that any**

*|***drawn to full reversal violates the conservation implied by a symmetric periodic change, |**

*t***e**

*|. If we let a total event history be the total positive energy of a system*

^{it}**ᶄ**

_{1},over some time |

**, then**

*t|*E = ∫ |**e*** ^{it}*| d|

**|**

*t*ðE = [ |**e*** ^{it}*|

_{t = nπ/2 }– |

**e**

*|*

^{it}_{t=(n-1)π/2 }] = 0 for any

*n*∈ {1, 2, …,

*n*– 1,

*n*}.

Consider |**v**_{re}| / |**v**_{im}| and let:

**v**_{ijkℓ} = 1 / ₵|_{re}

(contraction/expansion between **m**Space and **r**Space).

We will now attempt to calculate the rate ratio of **v**_{re} to **v**_{im} to determine if:

**v**_{finalre} = (|**v**_{mre}| / |**v**_{mim}|) > 1; **r**Space is expanding on **t**_{im}

**v**_{finalre} = (|**v**_{mre}| / |**v**_{mim}|) < 1; **m**Space is expanding on **t**_{re} and if

**v**_{finalre} = (|**v**_{mre}| / |**v**_{mim}|) = 1; no expansion or contraction is occurring.[16]

So, the action of the First Act alone is given as:

{*n* = 1 / |**₵**|_{ℓ}, |**ℓ**|* ^{n}* = 1/ |

**₵**|

_{ℓn}}

To now we have seen the relation (1 + *n*)^{(1/n)} present as a natural result of the Gedanken. We will now perform the same thing more rigorously by applying the Natural Tensor and getting the same result.

[(∂|**ᵯ**| / ∂ ) + –*ɨ(*∂|**ᵲ**| / ∂)]; the real and imaginary comparison operator and

_{2}**|**_{10} = |**e**|^{𝔵} |**v**_{ξ}| [(∂|** i**| / ∂) + (∂|

**| / ∂) + (∂|**

*j***| / ∂) +(∂|**

*k***| / ∂) + (∂|**

*ℓ***ᵯ**| / ∂) +

–*ɨ(*∂|** n**| / ∂ ) –

*ɨ(*∂|

**| / ∂ ) –**

*o**ɨ(*∂|

**| / ∂ ) –**

*p**ɨ(*∂|

**| / ∂ ) –**

*q**ɨ*(∂|

**ᵲ**| / ∂ )]

⨀

|**e**| [(∂ / ∂ |** i**| + ∂ / ∂ |

**| +∂ / ∂ |**

*j***| + ∂ / ∂ |**

*k***| + ∂ / ∂ |**

*ℓ***ᵯ**|) + (

**r**Space)

(∂ / ∂ |** n**| + ∂ / ∂ |

**| + ∂ / ∂ |**

*o***| + ∂ / ∂|**

*p***| + ∂ / ∂|**

*q***ᵲ**|] (

**m**Space)

_{2}**|**_{10} = |**v**_{re}| / |**v**_{im}| = |**v**_{ξ}| [(∂|**ᵯ**| / ∂|**ᵲ**|) + –*ɨ*(∂|**ᵲ**| / ∂|**ᵯ**|)];

We seek the magnitude of change in real event histories with respect to the imaginary event histories. We can take a one dimension (1 real, 1 imaginary) case for demonstration purposes. We seek this value at equilibrium, so we set |**v**_{ξ}| = |** i**| / |

**| or |**

*n***v**

_{ξ}| = |

*i*_{re}| / |

*i*_{im}| for consistency in how we set up various relationships in the tensor. At equilibrium the difference in these magnitudes should reveal the relative rates of event history change. We also set the sign of the second term of the operator based on whether or not that term is real or imaginary. We keep or remove the imaginary number of the second operator term based on whether or not the component is, in the context being calculated, real or imaginary.

So,

(**p**_{ijkℓ})^{1/₵} = (|**v**_{re}|₵ / |**v**_{im}|)^{1/₵ }=

*nlm *|**ᵯ**|→₵ of (|**v**_{re}|₵ / |**v**_{im}|)^{1/₵} = (|**v**_{im}|₵ / |**v**_{im}|)^{1/₵}

note that we do NOT need to continue using *nlm* notation because we have established that the quantities that follow are physically meaningful up to the Natural Limit, which is the limit of definition.

ð**p**_{ijkℓ} = (|**v**_{im}|₵ / |**v**_{im}|) =

(|**ᵯ**|₵ / |**ᵯ**|) * [(∂|**ᵲ**| / ∂|**ᵯ**|) – *ɨ(*∂|**ᵲ**|₵ / ∂|**ᵯ**|)],

ð**p**_{ijkℓ} = [∂|**ᵲ**| / ∂|**ᵯ**|)₵ – *ɨ*₵(∂|**ᵯ**| / ∂|**ᵯ**|)^{2}],

(real ratio both l and r)

ð**p**_{ijkℓ} = ₵ * [(∂|**ᵲ**| / ∂|**ᵯ**|)₵ – *ɨ*(₵)],

ð₵^{-1/₵}(**p**_{ijkℓ})^{1/₵} = [1 – *ɨ*(₵)]^{-1},

ð₵^{-1/₵}(**p**_{ijkℓ})^{1/₵} = [1 – *ɨ*(₵)]^{1/₵},

ð(**p**_{ijkℓ})^{1/₵} = [1 – *ɨ*(₵)]^{1/₵} and for **p**_{ijkℓ}

Finally, we take the General Limit (taking the complex conjugate):

*glm* |₵| →0 of [1 + (|₵|_{ijkℓ})]^{1/|₵|}_{ijkℓ }= |**e|**^{1} [§ 0.0808]

for each ** i, j, k, ℓ**. This is, for the first unit value of time, the First Act’s addition to event histories, the rate of change of those histories and the acceleration of those histories for an observer,

*ᶄ**. For all subsequent changes we have:*

_{1}|**e|**^{|m|} = |*x*_{re}| = d|*x*_{re}| / d|*m*_{re}| = d^{2} |*x*_{re}| / d |*m*_{re}|^{2} ∀ basis axes; |** i**|, |

**|, …, |**

*j***| of observer**

*r*

*ᶄ**.*

_{1}free body, E = 0;

**[17]***ᶄ** _{1}*observes that for ₵|

_{re}set to unity (

**v**

_{ijkℓ}= 1 / ₵|

_{re})

**v**

_{ijkℓ}slows to 0 and for ₵|

_{re}> 1 the rate of the event history reverses (destroys). At the surface of a Natural Limit any basis axis

**whose |**

*i***| = ₵**

*i**, where*

^{n}*n*is the number of base axes measured – in this case 1 – does not alter the event history. And finally, we can combine the

**m**Space and

**r**Space results to get the full representation of the First Act which is:

_{2}**|**_{10} *=* **e**^{ɨ(}^{ᵯ1 – ᵯ2)} [§ 0.0809]

And if **κ** denotes curvature at the surface,

**κ** = |**p`**_{i}(|₵|_{i}) ⨂ **p“**_{i}(|₵|_{i}) | / |** p`**_{i} (|₵|_{i})|^{3},

ð**κ** = |** v**_{i} ⨂ **v**_{i`} | / |** v**_{i }|,

ð**κ** = **e**; as expected and the result is independent of |**v**_{i}|.

*ᶄ** _{1}* finally addresses the question regarding the meaning of unity and the choice of units. The question posed is, at what ratio of |₵| to unit values does the Natural Limit obtain ∃

[1 – *ɨ*(|₵|_{ijkℓ})]^{1/|₵|}_{ijkℓ }= e,

holds with perfect precision? *ᶄ** _{1}* may evaluate this question as follows:

Let e_{1} denote a value for e set to arbitrary precision based on a unit value 1 for **t**_{re} and **t**_{im}. Next let e_{2} denote another like value for e derived of the same time unit 1 + μ ∃

ϕ = e_{2 }– e_{1}; ϕ > |₵|.

Now, repeat the process setting e_{1 }= e_{2} and e_{2 }= e_{2 }+ μ *n* times without

e* _{n}* – e

*> |₵| > e*

_{n – 1 }*– e*

_{n + 1}*and*

_{n}₵ ϵ [(√ hG / c^{3}) – e* _{n}* – e

*, (√ hG / c*

_{n – 1}^{3}) + e

_{n }_{+ 1}– e

*];*

_{n}and the Natural Limit of **v**_{i} obtains as an explicit calculation shows. [18]

**8.****The Second Act**

[19]We next consider an energy exchange between observer **ᶄ**_{1} and a defined observer **ᶄ _{2}**. For simplicities sake we will evaluate only one basis component and denote the exchange

**x**

*. We shall consider the exchange relative to the Spatial System*

_{ℓ}

*S**of*

_{n}**ᶄ**

_{1}. The event history value on

*ℓ*_{1}of

**ᶄ**

_{2}relative to

**ᶄ**

_{1}is observable (with conditions) up to the Natural Limit, that is,

**ᶄ**

_{1}locates

**ᶄ**

_{2}at some distance on

*ℓ*_{1}, φ. Let

*ℓ*_{0 }be the origin of

**. The exact length along**

*ℓ***of φ is undefined since, for**

*ℓ***ᶄ**

_{1}, there is a corresponding ₵

_{ℓᶄ}*. We can parameterize the axis with a variable, |*

_{1}**ᵯ**|. Now let

*s*be any point on that axis. So, the distance is observed to be on the interval:

[(φ (*s*), φ (*s* – γ))] and γ = *s*_{2} – *s*_{1} ϵ **ᵯ** and γ > ₵_{mᶄ1} ()[20]

then [φ (*s – γ*), φ (*s*) – φ (2γ)) is the most accurate defined quantity in |** ℓ**| observable to

**ᶄ**

_{1}. Notice that it does not equal φ (

*s*). And the length measured therefore, lies in the interval:

(|** ℓ**|

_{φ}) =

**(**

*p***) * [|**

*ℓ***|**

*ℓ*_{φ (s – γ)}, – |

**|**

*ℓ*_{φ (s) – 2γ}].

where ** p** (

**) is a “probability” function which is the object of this digression. And we can show a similar interval for the parameter as well:**

*ℓ*|**ᵯ**|_{β} = ** p** (|

**ᵯ**|) * [|

**ᵯ**|

_{β (t – ξ )}, – |

**ᵯ**|

_{β (t) – 2ξ}].

So, the ratio of the Natural Limits of these two axes squared indicates the Argand plane area change per unit time |** m**| in the Natural Limit. Let each interval be denoted

|**ᵯ**|* _{nlm }* = [|

**ᵯ**|

_{β (t – ξ )}, – |

**ᵯ**|

_{β (t) – 2 ξ}] and

|** ℓ**|

*= [|*

_{nlm}**|**

*ℓ*_{φ (s – γ)}, – |

**|**

*ℓ*_{φ (s) – 2γ}] and the rate of change in

**is:**

*ℓ**v*_{ℓ} = |**ᵯ**|* _{nlm}* / |

**|**

*ℓ*

_{nlm}ð*v*^{2}_{ℓ} = (|**ᵯ**|* _{nlm}* / |

**|**

*ℓ**)*

_{nlm}^{2}is the rate of change in the plane.

Since |**ᵯ**|* _{nlm}* is the parameter we can set it to unit values:

*v*^{2}_{ℓ} = *ɨ(*1 / |** ℓ**|

*)*

_{nlm}^{2 }and thus restrict ourselves once again to the complex component.

So,* ɨ(*1 / |** ℓ**|

*)*

_{nlm}^{2}denotes the area of Natural Limit swept out in ℓ per unit in time. And we know that the ratio of magnitude, increase and acceleration of

**r**Space to

**m**Space is |

**e**

^{1}| at unit time 1. More generally, we can say that for some time |

**ᵯ**|

*the indicated changes are in the ratio |*

_{nlm}**e**|

^{|m|nlm}. This is a mathematical description of the wave pattern exhibited by the First Act; e.g.

e^{ɨ(}^{ω|m|nlm)} = cos(ω|** m**|

*)*

_{nlm}

*i**+ ɨ*sin(ω|

**|**

*m**)*

_{nlm}**; [§ 0.0901]**

*j*where ω is a periodic constant.

**ᶄ**_{1} initiates another experiment to provide further details and precision regarding the aforementioned properties of Natural Limits. **ᶄ**_{1 }assembles two precision clocks, one for its observations and another for those of **ᶄ**_{2}. First **ᶄ**_{1}, measures a ‘time’delay on **ᵯ** to determine internal consistency. That is, **ᶄ**_{1} seeks to experimentally validate that **ᵯ** observes the constraints of the First Act by maintaining a magnitude, rate change and rate of rate change. After measuring an interval on **ᵯ** sufficiently large to expose any natural time loss or gain that might exist, **ᶄ**_{1}, observes that **ᵯ** does not appear to behave as the First Act requires. Namely, **ᵯ** appears to be losing time (within the tolerances of the clocks in use) by either running fast or slow. Before the experiment **ᶄ**_{1} had no formal means of predicting these changes in advance. **ᶄ**_{1} then calibrates its clock against the clock in use by **ᶄ**_{2 }and requests that **ᶄ**_{2} repeat the experiments. **ᶄ**_{2 }reports the same variability but cannot predict the outcome in advance. The behavior appears to both observers to be truly random inasmuch as there is no experiment they can devise, and no relation they are aware of, that can reliably predict this variability.

**ᶄ**_{1} next measures the Natural Limit of **ᵯ** and records the result as |**ᵵ**|. Comparing that value with the time increments of the clock, **ᶄ**_{1} observes that the unit value of *t* on its clock is 100 times greater than |**ᵵ**|. **ᶄ**_{1 }then restarts its clock measuring a precise interval of exactly 100 units in *t* and records the result. Next, we advance the hypothesis that the Natural Limit **ᵵ** is affecting the progression of time on **ᵯ** by magnifying the impact of **ᵵ** at scales significantly larger than **ᵵ**. **ᶄ**_{1} records each unit time advance noting that it falls on 100|**ᵵ**| for each cycle. However, for each advance in unit *t* **ᶄ**_{1} observes that the exact value of *t* on **ᵯ** cannot be observed, but that **ᶄ**_{1} can observe only the interval:

[100|**ᵵ**| – |**ᵵ**|, 100|**ᵵ**| + |**ᵵ**|] [§ 0.0902]

because it cannot observe a value within a Natural Limit. And that upon each increment *t* on **ᵯ** there is a left minimum and a right maximum (if we read the axis as left negative, right positive) ∃ for any interval advance:

[100|**ᵵ**| – |**ᵵ**|, 100|**ᵵ**| + |**ᵵ**|] there exists the possible values of outcome,

[99|**ᵵ**|, 101|**ᵵ**|]. [§ 0.0903]

But the clock must then *assume* (to the extent its tolerances allow) that value – which is actually an interval – for the next increment *t* = 2 on **ᵯ**. So, the value of *t*, like any other natural quantity, is not a quantity but an interval of insufficient definition within which the value sought lies. That is, left 99 or right 101…or any value in between. Thus **ᶄ**_{1} notes that there is an inherent definitional absence on the interval [99|**ᵵ**|, 101|**ᵵ**|] as to what exact value the time *t* on **ᵯ** makes at any given moment relative to **ᶄ**_{1}. Then **ᶄ**_{1} observes that this definitional absence grows with each increment. If left then the next increment occurs at 199 with a definitional absence in time a numerically defined value of

[198|**ᵵ**|, 200|**ᵵ**|] or if right [§ 0.0904]

[201|**ᵵ**|, 202|**ᵵ**|] ∃ [§ 0.0905]

the difference between the two (in this example, [198|**ᵵ**|, 202|**ᵵ**|]) creates a definitional absence in *t* on **ᵯ** that is an inherent property of a Natural Limit. **ᶄ**_{1} next proceeds to follow up on these observations with several more to confirm that the Natural Limits behave according to the relation:

1/ (r – l), r ≠ l => definitional loss.

where l is the left loss, given by the simple relations *t*(*n* – 1) and r, right loss, by *t*(n + 1), where *n* is the number of |**ᵵ**| in each increment *t*. *t* denotes the number of time increments evaluated. Of course, the definitional loss is simply a ratio of the difference in the absolute value of right and left losses to the number of increments *t*,

*t*(*n* + 1) – *t*(*n*-1) / *t* which quickly collapses to,

1/2*t* which we will write in the more general form,

1/2u = ½ * u^{-1}. [§ 0.0906]

But this is just the probability function we already described in disguise. Recalling the relation 1/₵ and adjusting we get:[21]

|ψ|^{2} ∝ 1 – u^{-1}/2.

To complete the analysis, **ᶄ**_{1 }runs experiments on a repeated trial basis to exploit the central limit theorem and discovers that the majority of outcomes are closer to the mean than not; especially when the interval *t* is taken very large. So, for any interval evaluated [*t*(*n*-1), *t*(*n* + 1)], there exists a normal distribution over that interval for any measurement

x ∈ [*t*(*n*-1), *t*(*n* + 1)], x ϵ ℝ. [§ 0.0907]

**ᶄ**_{1 }further notes that this figure represents the probability of a given discrete value |** ℓ**| occurring at a discrete time |

**ᵯ**| (for the sake of parameterization we’ll use this value. We could, in principle, use any defined axis ∈

**) and warns that the probability defined as such is not physically meaningful but is referenced here for the sake of illustration and comparison to traditional concepts of nature. It would be useful, then, to rigorously define what we are quantifying in this example:**

*S*_{n}**Definition:** Let an event history **ᵯ**change monotonically from left, ℓ, to right, *r*, on |**ᵯ**| ∈ ℝ with one to one changes in **ᵯ** by a magnitude |ῡ_{ᵯ}| relative to the rate change on **ᵯ** attributable to the First Act, *a*|**e**^{1}|. Then the magnitude of this change in **ᵯ**, |ῡ_{ᵯ}|, is certainly *defined* as the rank 1, order 2 *tensor*:

|ῡ_{ᵯ}|^{2} = |**ῡ_{r}**| ([|

**ᵯ**|

_{0 + |ᵯ0|(n – 1)}] / [|

**e**|

_{0 + |e0|(n – 1)}])

^{2}+

*i*|**ῡ_{i}**| ([|

**ᵯ**|

_{0 + |ᵯ0|(n + 1)}– |

**ᵯ**|

_{0 + |ᵯ0|(n – 1)}] / [

*a*|

**e**|

_{0 + |e0|(n + 1)}–

*a*|

**e**|

_{0 + |e0|(n – 1)}])

^{2},

ð|ῡ_{ᵯ}|^{2} = |**ῡ_{r}**| (f(|

**ῡ**|))

_{left}^{2}

*+ ɨ*|

**ῡ**| ((g(|

_{i}**ῡ**|) – f(|

_{right}**ῡ**|))

_{left}^{2}. [§ 0.0908]

The definition above is not a *choice* of formal logic but an *observable* of nature. The above relation is a relation between the state of equilibrium provided by the First Act and changes occurring in **ᵯ**. If no other acts applied, we would expect the ratio to be 1. But it is not. So, we can see now that neglecting the Natural Limit of **ᵯ** leads directly to the necessity of “uncertainty” accomodation. But nature contains no such uncertainty and “God does not play dice with the universe”. We will now prove it.

Setting this up for use in the Natural Tensor we can think of this quantity as a rate change ratio between *a*|**e**| and |**ᵯ**|, that is, |**e**| / |**ᵯ**|, and we can see right away that this is not an unreasonable assumption since we are attempting to measure the degree of definition in **ᵯ**. We note that this ratio will also have an effect on the other spatial axes as well, in addition to the fact that each spatial axis will exhibit the same behavior we are seeing in **ᵯ**. (we will see in the one dimensional, time dependent example the hints of what will happen when we run the tensor over order 10). Note as well that the ratio of definition to undefinition appears here as an angle ϕ, where

ϕ = π / 2 iff [*a*|**e**|_{0 + |ℓ0|(n – 1)}] = [*a*|**e**|_{0 + |ℓ0|(n + 1)} – *a*|**e**|_{0 + |ℓ0|(n – 1)}].

[22]^{, [23]}We now look at this function not as a vector but a scalar, with a more familiar symbol, |ψ**|** (*a***e**) = (1 / |**ᵵ**|) (which we will shortly prove) and looking only at the real components now. For comparison, we will take the dot product of ῡ_{ᵯ} at the zero’th angle against an observer, **ᶄ**_{1}‘s event history rate change on **e** as a function of momentum; i.e. ῡ**_{ℓ}** •

**q**/m. This is the analog to the “uncertainty of momentum” of the observer,

**ᶄ**

_{1}.

First, we will derive a “probability” function from Natural Limits. In the literature the equations below are called “probability density functions”. We will then apply that to other calculations. Recalling that the effect of a Natural Limit on time can be characterized by [§ 0.0903] and [§ 0.0908]. Any arbitrary location on **ᵯ**, call it **ᵯ*** _{m}*, can be located as an interval outside of which

**ᵯ**is fully defined; i.e.

[|**ᵯ**| (*n* – 1), |**ᵯ**| (*n* + 1)] and *n* = 1 / |**ᵵ**| where 1 denotes unit time.

ð**ᵯ*** _{m}* = [|

**ᵯ**|

*(1 / |*

_{m}**ᵵ**| – 1), |

**ᵯ**|

*(1 / |*

_{m}**ᵵ**| + 1)],

which implies a vector rather than a scalar, as required. That is,

**|ᵯ|**_{ℓ }**/ **|**ᵵ**| |**ᵯ**|* _{m}*(1 / |

**ᵵ**| – 1)

**|ᵯ|**

_{r }**/**|

**ᵵ**|

**|ᵯ|**^{2}* _{m}* =

**ᵯ**

_{ℓ}[-|

**ᵯ**|

*(1 / |*

_{m}**ᵵ**| – 1)]

^{2}/ |

**ᵯ**|

_{m}+ **ᵯ**_{r} [|**ᵯ**|* _{m}* (1 / |

**ᵵ**| + 1)]

^{2 }/ |

**ᵯ**|

*[§ 0.0909]*

_{m }Examining this relation as a Riemann sum over successive values of |**ᵵ**| (hence the dot product with the resultant), and setting *n*_{1} equal to the number of units of time elapsed since the last measurement of time, we see the left and right pattern emerge as:

γ_{1}^{2}|**ᵯ**|^{n}^{12} ([γ_{1} … ([γ_{1} ([γ_{1} ([γ_{1} / |**ᵵ**| – |**ᵯ**|_{1} / |**ᵯ**|^{2 }_{0}]^{2} /|**ᵵ**|)

– |**ᵯ**|_{2} / |**ᵯ**|^{4}_{0}]^{2}) / |**ᵵ**| – |**ᵯ**|_{3} / |**ᵯ**|^{8}_{0}]^{2})

/ |**ᵵ**|– |**ᵯ**|_{4} / |**ᵯ**|^{16}_{0}]^{2})… / |**ᵵ**|– |**ᵯ**|* _{n}* / |

**ᵯ**|

^{n}^{12}

_{0}]

^{2})/ |

**ᵵ**|

γ_{2}^{2}|**ᵯ**|^{n}^{12} ([γ … ([γ ([γ ([γ / |**ᵵ**| + |**ᵯ**|_{1} / |**ᵯ**|^{2 }_{0}]^{2} /|**ᵵ**|)

+ |**ᵯ**|_{2} / |**ᵯ**|^{4}_{0}]^{2}) / |**ᵵ**| + |**ᵯ**|_{3} / |**ᵯ**|^{8}_{0}]^{2})

/ |**ᵵ**|+ |**ᵯ**|_{4} / |**ᵯ**|^{16}_{0}]^{2})… / |**ᵵ**|+ |**ᵯ**|* _{n}* / |

**ᵯ**|

^{n}^{12}

_{0}]

^{2})/ |

**ᵵ**|

where γ_{1} = (1 / |**ᵵ**| – 1) for **ᵯ** left and γ_{2} = (1 / |**ᵵ**| + 1) for **ᵯ** right. And, under equilibrium where the only rate change in **ᵯ** is attributable to the First Act, we have the identity, first for the left and then for the right:

|**ᵯ**|_{ℓ} = γ^{2}_{1}|**e**|^{n}^{2} ([γ_{1} … ([γ_{1} ([γ_{1} ([γ_{1} / |**ᵵ**| – |**e**|_{1} / |**e**|^{2 }_{0}]^{2} /|**ᵵ**|)

– |**e**|_{2} / |**e**|^{4}_{0}]^{2}) / |**ᵵ**| – |**e**|_{3} / |**e**|^{8}_{0}]^{2})

/ |**ᵵ**|– |**e**|_{4} / |**e**|^{16}_{0}]^{2})… / |**ᵵ**|– |**e**|* _{n}* / |

**e**|

^{n}^{2}

_{0}]

^{2})/ |

**ᵵ**| [§ 0.0910]

|**ᵯ**|_{r} = γ^{2}_{2}|**e**|^{n}^{2} ([γ_{2} … ([γ_{2} ([γ_{2} ([γ_{2} / |**ᵵ**| + |**e**|_{1} / |**e**|^{2 }_{0}]^{2} /|**ᵵ**|)

+ |**e**|_{2} / |**e**|^{4}_{0}]^{2}) / |**ᵵ**| + |**e**|_{3} / |**e**|^{8}_{0}]^{2})

/ |**ᵵ**|+ |**e**|_{4} / |**e**|^{16}_{0}]^{2})… / |**ᵵ**|+ |**e**|* _{n}* / |

**e**|

^{n}^{2}

_{0}]

^{2})/ |

**ᵵ**| [§ 0.0911]

Letting

β_{1} = γ^{2}_{1} = (1 / |**ᵵ**| – 1)^{2} = (1/ |**ᵵ**|^{2} – 2 / |**ᵵ**| + 1)

for the left and

β_{2} = γ^{2}_{2} = (1 / |**ᵵ**| + 1)^{2} = (1/ |**ᵵ**|^{2} + 2 / |**ᵵ**| + 1)

for the right

We now let *n*_{2} = *n*_{1 }* 1 / |**ᵵ**|. Finally, let the total real probability of two clocks in two reference frames **ᶄ**_{1} and **ᶄ**_{2} reporting times 100% congruent as a function of the First Act, |**e**|, be denoted as a function, ψ. Then, under the sole influence of the First Act,

ψ (*n*) ≈ γ^{2}_{2}∫_{0}^{|ᵯ|m} |**e**|^{(|ᵯ||ᵵ|)2/2} d**e**^{(|ᵯ||ᵵ|)} + ∫_{0}^{|ᵯ|m} |**ᵯ**|* _{m}* * |

**e**|

^{-(|ᵯ||ᵵ|)2/2}d

**e**

^{(|ᵯ||ᵵ|)}

– [γ^{2}_{1}∫_{0}^{|ᵯ|m} |**e**|^{(|ᵯ||ᵵ|)2/2} d**e**^{(|ᵯ||ᵵ|)} – ∫_{0}^{|ᵯ|m} |**ᵯ**|* _{m}* * |

**e**|

^{-(|ᵯ||ᵵ|)2/2}d

**e**

^{(|ᵯ||ᵵ|)}]

ψ (*n*) ≈ ∫_{0}^{|ᵯ|m} γ^{2}_{2}|**e**|^{(|ᵯ||ᵵ|)2/2} – γ^{2}_{1}|**e**|^{(|ᵯ||ᵵ|)2/2} d**e**^{(|ᵯ||ᵵ|)} + 2*∫_{0}^{|ᵯ|m} *n _{f}* |

**e**|

^{–n2/2}d

**e**

^{(|ᵯ||ᵵ|)}

ψ (*n*) ≈ ∫_{0}^{|ᵯ|m} γ^{2}_{2}|**e**|^{(|ᵯ||ᵵ|)2/2} – γ^{2}_{1}|**e**|^{(|ᵯ||ᵵ|)2/2} d**e**^{(|ᵯ||ᵵ|)} + 2∫_{0}^{|ᵯ|m} *n _{f}* |

**e**|

^{–n2/2}d

**e**

^{(|ᵯ||ᵵ|)}

and letting *n _{f }* = |

**ᵯ**|

_{m}ð**ᵯ*** _{nlm}*(|

**ᵯ**|) =

*n*2∫

_{f}_{0}

^{n}*|*

^{f}**e**|

^{-(|ᵯ||ᵵ|)2/2}d

**e**

^{(|ᵯ||ᵵ|)}(preliminary)

and, noting that when we began in [§ 0.0908] we saw that the proper treatment of the integration above was as a vector resultant rate change in time. Thus the ratio of the initial area, a* _{i}*, to the enlarged area generated by the First Act, a

_{1}, in which the First Act serves as the parameter for the observed time,

**ᵯ**, is a

*/ a*

_{i}_{1}. Thus, we have:

*|***ῡ |**

^{2}

*=***[**

*r**n*2∫

_{f}_{0}

^{n}*|*

^{f}**e**|

^{-(|ᵯ||ᵵ|)2/2}d

**e**

^{(|ᵯ||ᵵ|)}] ⨀

**ῡ**

_{r}*+*[

**i***n*2∫

_{f}_{0}

^{n}*|*

^{f}**e**|

^{-(|ᵯ||ᵵ|)2/2}d

**e**

^{(|ᵯ||ᵵ|)}] ⨀

*i*|

**ῡ**|

_{i}and since

|** r**[

*n*∫

_{f}_{0}

^{n}*|*

^{f}**e**|

^{-(|ᵯ||ᵵ|)2/2}d

**e**

^{(|ᵯ||ᵵ|)}]|

*=*|

**[**

*i**n*∫

_{f}_{0}

^{n}*|*

^{f}**e**|

^{-(|ᵯ||ᵵ|)2/2}d

**e**

^{(|ᵯ||ᵵ|)}]|

*|***ῡ |**

^{2}

*=***(1/√ 2) [**

*r**n*2∫

_{f}_{0}

^{n}*|*

^{f}**e**|

^{-(|ᵯ||ᵵ|)2/2}d

**e**

^{(|ᵯ||ᵵ|)}] ⨀

**ῡ**

_{r}*+ i *(1/√ 2) [

*n*2∫

_{f}_{0}

^{n}*|*

^{f}**e**|

^{-(|ᵯ||ᵵ|)2/2}d

**e**

^{(|ᵯ||ᵵ|)}] ⨀

*i*

**ῡ**

_{i}and, appropriately, setting our Natural Limit as the unit value and taking its ratio with unity we get:

*|ᵶ|*^{2}** =** |

**|**

*r**+*|

**|**

*i*ð** |ᵶ| =** √ |

**|**

*r**+ √*|

**|**

*i*then the rate of change in ** |ᵶ|** at unit value is:

1 / |**ᵵ**| = ** |¯ ᵶ |** / |

**ᵵ**|

**√ (|**

*=***| / |**

*r***ᵵ**|)

*+ √*(|

**| / |**

*i***ᵵ**|). (basis vector resultant).

and each basis alone is:

√ (1 / |**ᵵ**| * |**ᵵ**|_{φ}) = 1 / √|**ᵵ**| * |**ᵵ**|`

where |**ᵵ**|` is the expanded total magnitude of |**ᵵ**| for each unit time elapsed, |**ᵯ**|. So that,

*|***ῡ |**

^{2}

*=***[1/√ (2 |**

*r***ᵵ**| * |

**ᵵ**|

_{φ})

*n*2∫

_{f}_{0}

^{n}*|*

^{f}**e**|

^{-(|ᵯ||ᵵ|)2/2}d

**e**

^{(|ᵯ||ᵵ|)}] ⨀

**ῡ**

_{r}*+ i*[1/√ (2 |

**ᵵ**| * |

**ᵵ**|

_{φ})

*n*2∫

_{f}_{0}

^{n}*|*

^{f}**e**|

^{-(|ᵯ||ᵵ|)2/2 }d

**e**

^{(|ᵯ||ᵵ|)}] ⨀

*i*|

**ῡ**|

_{i}And

|**ᵵ**|_{φ} = [[2 *n _{f}* ∫

_{0}

^{n}*|*

^{f}**e**|

^{-(|ᵯ||ᵵ|)2/2}d

**e**

^{(|ᵯ||ᵵ|)}] *

*|ᵶ|*^{-2}]

^{2}

**(2|**

*/***ᵵ**|)

ð√ |**ᵵ**|_{φ} = √2|**ᵵ**| |**ᵯ**|* _{m }*/

*|ᵶ|*^{-2}) ∫

_{0}

^{n}*|*

^{f}**e**|

^{-(|ᵯ||ᵵ|)2/2}d

**e**

^{(|ᵯ||ᵵ|)}

ð√ |**ᵵ**|_{φ} = 1 / (** |ᵶ| **√2|

**ᵵ**|) * 2∫

_{0}

^{n}*|*

^{f}**e**|

^{-(|ᵯ||ᵵ|)2/2}d

**e**

^{(|ᵯ||ᵵ|)};

|**ᵵ**|_{φ} = 3.14… = π [vector component form]. So,

we can *again* define the quantity π thusly:

**Definition:** The value π is the ratio of a Natural Limit, |**𝔑**|_{φ,} at some parameterization, |**ᵯ**| + 1 to the Natural Limit, |**𝔑**|_{,} at parameterization |**ᵯ**|, expressed as a component of an order 2 vector. And π^{2} = ** i** [|

**𝔑**|

_{φ}/ |

**𝔑**|].

And the above is not a derivation in formal logic but an observable of nature. And note that the value π is not a constant and varies depending on the ‘location’ of an observer, **ᶄ**_{1}. So, any internally consistent system of formal logic, 𝔏, changes with the ‘location’ of the evaluator performing it. The same is true for e, except that we have not yet shown that e varies, as a practical matter, with any other physical quantity. We also note that this derivation generates the same result as achieved in the Historical Acts, [§ 0.0202]. So, we will expound on that relationship.

And we can clearly delineate now the difference between the probability function and the velocity. Taking either component of **ῡ**, we have:

*|***ῡ**_{a}*|*^{2} = ** a**[1/√ (2 |

**ᵵ**| * |

**ᵵ**|

_{φ})

*n*2∫

_{f}_{0}

^{n}*|*

^{f}**e**|

^{-(|ᵯ||ᵵ|)2/2 }d

**e**

^{(|ᵯ||ᵵ|)}]

which contains the velocity which we’ll set to unity and factor out singly:

*|***ῡ**_{a}*|*^{2} = ** a**[1/√ (2 |

**ᵵ**| * |

**ᵵ**|

_{φ})

*n*2∫

_{f}_{0}

^{n}*|*

^{f}**e**|

^{-(|ᵯ||ᵵ|)2/2 }d

**e**

^{(|ᵯ||ᵵ|)}]

ð*|***ῡ**_{a}** | ** = 1/2 |

**ᵵ**|*π |

**ᵯ**| 2∫

_{0}

^{n}*|*

^{f}**e**|

^{-(|ᵯ||ᵵ|)2/2 }d

**e**

^{(|ᵯ||ᵵ|)}

ðπ *|*ῡ* _{a}|*|

**ᵵ**||

**e**|

^{(|ᵯ||ᵵ|)2/2}= ∫

_{0}

^{|ᵯ|}1 / |

**ᵯ**|d

**e**

^{(|ᵯ||ᵵ|)}

ð*|***ῡ**_{a}** |** = (-1 / π |

**ᵵ**||

**e**|

^{(|ᵯ||ᵵ|)2/2 }|

**ᵯ**|

^{2})

ð*|***ῡ**_{a}** |** = – (π |

**ᵵ**|)|

**ᵯ**|

^{-2}|

**e**|

^{(-|ᵯ||ᵵ|)2/2}

ð|**ᵯ**|^{2}|**e**|^{(|ᵯ||ᵵ|)2/2}*|*ῡ* _{a}|* = – (π |

**ᵵ**|)

One will note that the relation between volume, the normal distribution and π can be shown for any normalized integration and √ π is used as a normalizing factor for just such a situation. What is different in this derivation is the physical meaning behind the quantities and that is the theme of what we are going to expound upon here. In the derivation above we set the ‘new volume’ of an undefined region that is generated by the First Act to a unit value of 1. Working backward allowed us to generate the value for π, that is, for any unit parameter of a Natural Limit there is a √ π increase in the volume of that Natural Limit upon each subsequent increment in that parameter. And that, in turn, is due to the compounding effect of the First Act upon each unit increase in the Natural Limit. So, the question remains, how does this tie into [§ 0.0202]?

Let an observer **ᶄ**_{1} present upon event **u**_{5} of the Historical Acts its Natural Limits from a linearly independent direction. And let the Spatial System, ** S_{n}**, of

**ᶄ**

_{1}, present all 9 bases for the event. Let the average real magnitude of the intervals that contain the Natural Limits be as before:

*h* = *a* / |**₵_{ℓ}**|

where

*a* = _{i=0}Σ^{i=9} |**₵_{i}**| ∀ bases

**s**, ∈

**relative to**

*S*_{n}**ᶄ**

_{1}.

Then we can simplify the illustration by taking the mean to be the corresponding value of one basis. Of course, upon event **u**_{5} it is a given that all 9 bases are party to the event, but we shall see that taking their mean as one basis Natural Limit is a valid approach for what we are illustrating. So, the average length, presented as one basis Natural Limit, we’ll here denote as |**₵_{s}**| and the Natural Limit width for event

**u**

_{5}we’ll denote |

**₵**|. So, the moment of the event is |

_{ℓ}**₵**| – |

_{s}**₵**|, that is,

_{ℓ}**τ_{ℓ} = ** (d

**u**

_{5}/ d

**ᵯ)**⨂ (d

**d**

*s /***ᵯ)**

|**₵_{s}**| – |

**₵**|

_{ℓ}|** C`**|

**|(|**

*=***₵**`| – |

_{s}**₵**`|) ⨀

_{ℓ}**τ**|.

_{ℓ}So, the length traversed over one Natural Limit of time must satisfy:

|**ᵵ**| * |** C**`|

**||**

*=***ᵵ**|[|

**₵**`| – |

_{s}**₵**`|] ⨀

_{ℓ}**τ**|

_{ℓ}ð|** C**|

**||**

*=***ᵵ**|[|

**₵**| – |

_{s}**₵**|] ⨀

_{ℓ}**τ**| for time units of 1 Natural Limit.

_{ℓ}The satisfaction criteria is that the real endpoints that define ** C** must themselves be defined. So, without the full traversal, or change in value, of the length of |

**₵**| along

_{s}**, combined with the same change in |**

*s***₵**| along

_{ℓ}**(the new basis), even if the proportionate length on |**

*ℓ***₵**| or even |

_{ℓ}**₵**| – |

_{s}**₵**| makes, the endpoints that contain

_{ℓ}**are undefined. And, |**

*C***₵**| > |

_{s}**₵**| by the definition given in the Historical Acts. So, unless this traversal occurs congruently between

_{ℓ}**and**

*s***, the endpoint on**

*ℓ***is not reached and it**

*C*remains undefined. That is, the Natural Limit |

**₵**| = |

_{ℓ}**| at one end of**

*C***u**

_{5}and a|

**₵**| = b|

_{ℓ}**| at the other.**

*C*The interpretation of this behavior macroscopically is interpreted as a formal logic called “trigonometry” and the value of π is limited to the allowed definitional range given by the final value of |** C**|. This leads us to another definition:

**Definition: **A system of formal logic, ξ, is sufficiently well defined only in the reference frame in which it is testable. That is, any constant ∈ ξ invariant to changes in the magnitude by which two or more observers locate, has no possibility of existence.

Let two observers **ᶄ**_{1 }and **ᶄ**_{2} locate on the axis that parts them and the magnitude of that axis, call it 𝔵, be sufficiently large to exceed certain limitations placed by Natural Limits. Then the value of π measured by **ᶄ**_{1 }differs from the same value measured by **ᶄ**_{2}. And the value of π as derived by any system of formal logic, ξ, applied in the reference frame of **ᶄ**_{1 }will not equal the same purported value derived with ξ in the reference frame of **ᶄ**_{2}. In a nutshell, as we saw from the above, π is not a constant and any formal logic that references it, to be complete and consistent, must change under the appropriate change in the reference system in which it is ostensibly testable. To be careful, we must point out that a system of formal logic that is dynamic *is not precluded from existence*, and if it is, it can accommodate these variations. But π in that case is no longer a constant under the system of logic applied. We suspect that such general systems may exist in the body of the current literature.

And the final form becomes:

*|***ῡ |**

^{2}

*=***(1/√ 2π|**

*r***ᵵ**|) [

**|ᵯ**|

*∫*

_{m}_{0}

^{|ᵯ}^{|}

*|*

^{m}**e**|

^{-(|ᵯ||ᵵ|)2/2}d

**e**

^{(|ᵯ||ᵵ|)}] ⨀

**ῡ**

_{r}*+ i *(1/√ 2π|

**ᵵ**|) [

**|ᵯ**|

*∫*

_{m}_{0}

^{|ᵯ}^{|}

*|*

^{m}**e**|

^{-(|ᵯ||ᵵ|)2/2}d

**e**

^{(|ᵯ||ᵵ|)}] ⨀

*i*

**ῡ**[§ 0.0912]

_{i }and solving for the common component we can see that it is a

solution to the equation:

*|***ῡ_{r}|**

^{2}

**1/√ 2π|**

*=***ᵵ**|) 2

**|ᵯ**|

*∫*

_{m}_{0}

^{|ᵯ}^{|}

*|*

^{m}**e**|

^{-(|ᵯ||ᵵ|)2/2}d

**e**

^{(|ᵯ||ᵵ|)}

ð2**|ᵯ**| / √ (2π|**ᵵ**|) * e^{-(|ᵵ|)2(|ᵯ|)2/2} = – |**ᵵ**|^{2}**|ᵯ**| / (2π|**ᵵ**|) * e^{-(|ᵵ|)2(|ᵯ|)2/2}

And we can also readily see that each component is the most general solution to the well known Schrodinger equation. Taking dot products with state variables (such as momentum) that generate the appropriate constants will generate exactly the form seen in the Schrodinger relations. But the physical meaning behind the approaches is what is so starkly different; the Schrodinger equation is meaningless while the Natural Tensor reveals that what we are in fact measuring is a value interval (whose discrete value is undefined) of a natural state variable ** relative to** the quantity of Natural Limits present in that same interval.

Let *n* = |**ᵯ**|. Precision can be increased, in principle, to full fidelity by dropping General Limits in favor of Natural Limits in a McLaurin Series using |**e**|_{0}^{–n2}* n _{f}* ∫

_{0}

^{n}*|*

^{f}**e**|

^{-(|ᵯ||ᵵ|)2/2}d

**e**

^{(|ᵯ||ᵵ|)}

|**e**|_{0}^{-(|ᵯ||ᵵ|)2/2} = [|**ᵯ**||**ᵵ**| = 0 → |**ᵯ**||**ᵵ**| = k] Σ(|**ᵯ**||**ᵵ**|)* ^{x}* /

*x*

**!**

**= **[|**ᵯ**||**ᵵ**| = 0 → |**ᵯ**||**ᵵ**| = k] Σ (-1)^{|ᵯ||ᵵ|}(|**ᵯ**||**ᵵ**|)^{2x} / (|**ᵯ**||**ᵵ**|)**! **

**=** 1 – (|**ᵯ**||**ᵵ**|)^{2} / 1**! +**(|**ᵯ**||**ᵵ**|)^{4} / 2**! – **(|**ᵯ**||**ᵵ**|)^{6} / 3**! + …**

ð* a ***∫**|**e**|_{0}^{-(|ᵯ||ᵵ|)2/2}d**e**^{(|ᵯ||ᵵ|)}

*=** ∫* (1 – (|**ᵯ**||**ᵵ**|)^{2} / 1**! +**(|**ᵯ**||**ᵵ**|)^{4} / 2**! – **(|**ᵯ**||**ᵵ**|)^{6} / 3**! + … **

**+** (-1)^{x} (|**ᵯ**||**ᵵ**|)^{2x} / x**! + …**) d**e**^{(|ᵯ||ᵵ|)}

where k = |**ᵵ**| < (-(|**ᵯ**||**ᵵ**|)^{2})* ^{x}* /

*x*

**! <**2 * |

**ᵵ**|. This quantity is an exact measurement and admits of no “uncertainty”. Which is what we sought to show, Q.E.D.[24] [25]

We recall the (for “charged, zero spin”, rest energy μ_{0}) Klein-Gordon equations:

( ^{2} – μ_{0}^{2} ) ϕ = 0

ðϕ ^{2} = ϕ μ_{0}^{2} and

( ^{2} – μ_{0}^{2} ) ϕ^{*} = 0

ðϕ^{* } ^{2} = μ_{0}^{2} ϕ^{*}; where ϕ and ϕ^{* }are the complex and conjugate forms of the Langrangian density.

of which both can be derived from the Langrange-Euler equation (for a complex scalar field). L is a function derived of Newtonian equations of ‘motion’ in three variables, F, F` and t. F represents the “correct path” for a Lagrangian. That is, we seek a function F that satisfies:

0 = (∂ L / ∂ F) – d/dt [∂ L / ∂ F`];

But this is just the result of applying the Natural Tensor condition:

_{2}**|**_{10}*=* **e*** ^{it }*.

to the total energy of a system **ᶄ**_{0}, which we can prove is:

_{2}**|**_{10} = |**e**|^{𝔵} |**v**_{ξ}| [(∂|** i**| / ∂) + (∂|

**| / ∂) + (∂|**

*j***| / ∂) +(∂|**

*k***| / ∂) + (∂|**

*ℓ***ᵯ**| / ∂) +

–*ɨ(*∂|** n**| / ∂ ) –

*ɨ(*∂|

**| / ∂ ) –**

*o**ɨ(*∂|

**| / ∂ ) –**

*p**ɨ(*∂|

**| / ∂ ) –**

*q**ɨ(*∂|

**ᵲ**| / ∂ )]

⨀

|**e**| [(∂ / ∂ |** i**| + ∂ / ∂ |

**| +∂ / ∂ |**

*j***| + ∂ / ∂ |**

*k***| + ∂ / ∂ |**

*ℓ***ᵯ**|) + (

**r**Space)

(∂ / ∂ |** n**| + ∂ / ∂ |

**| + ∂ / ∂ |**

*o***| + ∂ / ∂|**

*p***| + ∂ / ∂|**

*q***ᵲ**|] (

**m**Space)

Letting

L (|** n_{i}**|) = |

**|; ∵**

*i***= [**

*i***v**

_{1},

**u**

_{2}] * |

**| and**

*r***= [**

*n*_{i}**u**

_{1},

**v**

_{1}] * |

**|**

*t*_{re}

F (|** n_{m}**|) = |

**|; ∵**

*m***= [**

*m***u**

_{1},

**v**

_{1}] * |

**|**

*i*_{re }and

**= [**

*n*_{m}**u**

_{1},

**v**

_{1}] * |

**|**

*t*_{re}

G (|** n_{r}**|) = |

**|; ∵ |**

*r***| = |**

*r***| / [**

*i***u**

_{1},

**v**

_{2}] and

**= [**

*n*_{r}**u**

_{1},

**v**

_{1}] * |

**|**

*t*_{re}

(** n** may obtain without

**and**

*m***depends on**

*m***).**

*n**r*_{c}^{2} = [1^{2} + –*ɨ(*∂|** r**| / ∂|

**|)**

*m*^{2}]; We want a ‘relativistic’ result.

ð√*r*_{r} = –*ɨ(*∂|** r**| / ∂|

**|)**

*m*^{2}.

[(∂|** m**| / ∂ ) + –

*ɨ(*∂|

**r**| / ∂)]; the real and imaginary comparison operator

ð|** p**| = [ (∂|

**| / ∂|**

*m***|) + –**

*i**ɨ(*∂ |

*r*_{r}| / ∂|

**|)]; operand: <∂ / ∂ |**

*m***| + ∂ |**

*i*

*r*_{r}| / ∂>.

ð|** p**| = [ (∂|

**| / ∂|**

*m***|) + –**

*i**i*[∂ (|-

*ɨ(*∂|

**| / ∂|**

*r***|)**

*m*^{2}| / ∂|

**|)]; operand:**

*m*

*r*_{r}

ð|** p**| = [ (∂|

**| / ∂|**

*m***|) + –**

*i**i*[∂ | –

*ɨ(*∂|

**| / ∂|**

*r***|)*-**

*m**ɨ(*∂|

**| / ∂|**

*r***|)| / ∂|**

*m***|)];**

*m*ð|** p**| = |

**v**

_{ξ}|[ (∂|

**| / ∂|**

*m***|) + –**

*i**i*[∂ | –

*ɨ(*∂|

**| / ∂|**

*r***|)*-**

*m**ɨ(*∂|

**| / ∂|**

*r***|)| / ∂|**

*m***|)]; operand: (∂|**

*m***| / ∂|**

*i***|)**

*m*^{2}, the event history plane of interest, one dimensional case[26]. Recall that we are trying to solve for an

**r**Space path along a real axis.

ð|** p**| = [(∂|

**| / ∂|**

*i***|) + -ɨ(∂ ((∂ |**

*m***| / ∂|**

*r***|)*-**

*m**i*(∂|

**| / ∂|**

*r***|)) / ∂|**

*m***|)];**

*m*ð|** p**| = [(∂|

**| / ∂|**

*i***|) + ((∂ / ∂ |**

*m***|) (∂ |**

*m***| ∂ |**

*r***| / ∂ |**

*r***| ∂ |**

*m***|)];**

*m*ð|** p**| = [(∂|

**| / ∂|**

*i***|) + ((∂ |**

*m***| / ∂ |**

*r***|) (∂ |**

*m***| / |**

*r***|`)];**

*m*And substituting our event functions back in to the equation yields:

ð |** p**| = [(L / F) + ((d / d|

**|) (∂ |**

*m***||**

*r***| / F`)];**

*r*And using the identity that |** r**| = |

**|**

*t*^{2}

_{xy}we substitute |

**| for |**

*i***|**

*r*^{2}:

|** p**| = [(L / F) + ((d / d

**t**

_{im}) (∂ |

**| / F`)];**

*i*Since this tensor was run only on net energy gain or loss, this is the state of equilibrium, which is just the buoyancy 𝒷 (to be discussed shortly). That is

|** p**| = 𝒷 = [(L / F) + ((d / d

**t**

_{im}) (∂ |

**| / F`)];**

*i*which is riding the wave surface of the First Act, that is, 𝒷 = 0.

** **_{2}**|**_{10}*=* 0 = 𝒷 = [(L / F) + ((d / d**t**_{im}) (∂ |** i**| / F`)] ∀ |

**| ∈**

*m***. which will always hold for a closed system,**

*m***ᶄ**

_{0. }and

0 = [(L / F) + ((d / d**t**_{im}) (∂ |** i**| / F`)],

is just the Euler-Lagrange equation. Q.E.D.

**Definition:** Thus, we *define* the Second Act to be a perturbation in the event history of the First Act itself at some local value *ℓ* defined up to a Natural Limit. And each such perturbation, to the extent that the perturbation is of sufficient energy, is an “observer” **ᶄ** with a spatial system *S** _{n}* unique to it.

**9.****The Third Act**

We now put together a hypothesis that comports with what we’ve seen so far but remains unproven. Let us hypothesize that mass is proportional to the surface area of the Argand plane tangential to the surface of a Natural Limit.

[27]The possible perturbations of the plane originating tangentially from a Natural Limit:

order 10

parameters 2 (**r**Space and **m**Space)

tangential axes 2 (**r**Space and **m**Space)

[28]axes of rotation 2 (**r**Space and **m**Space)

Before beginning with the perturbations themselves we need to discern whether or not any properties change if we switch the ordering of the cross products (a change in chirality) used in the perturbations. **ᶄ**_{1} observes that for each combination **ᶄ**_{1}→ **ᶄ**_{r} that is mixed handed, that is, where the two observers are oscillating in a plane with opposite cross product ordering conventions, the following change will occur:

Let **ᶄ**_{r } lay a **𝒵 _{im}**‘ th event history rate change by rotation in the cross product of

**⊗**

*j*

*i =***𝒵**. Then, relative to

_{im}**ᶄ**

_{1}, with counterpart,

**𝒵**

_{imk}_{1},

**ᶄ**

_{1}is in a state of slow but steady constriction in accordance with the First Act, with magnitude = rate change = acceleration of event history. That is, its value is “ambient”. Were

**ᶄ**

_{r}in the same state, it’s

**𝒵**would also be constricting at the already known rate of the First Act. With an event of sufficient energy

_{imkr}**ᶄ**

_{1}now ‘spins up’ on

**𝒵**

_{imk}_{1}to provide an increased magnitude of constriction to run this experiment.

**ᶄ**

_{1}measures the value and direction of the cross product and observes that rather than attracting, that is, rather than the axis that relatively locates them constricting,

**ᶄ**

_{r}appears to be pulling away from

**ᶄ**

_{1 }with a “magnitude, rate and acceleration” that makes well outside the constraints of the First Act and may contain any plausible physical value, |

**𝒵**| – |

_{imkr}**𝒵**

_{imk}_{1}|.

**ᶄ**_{1 }consults its history for an answer and finds the answer in Section 6 on multiple observers. First, **ᶄ**_{1} has confirmed that, with respect to its coordinate system, the wave perturbations being created act to constrict the |**𝒵 _{imk}**

_{1}| event history but, when

**ᶄ**

_{1}removes and positions itself at the |

**𝒵**| event history the axis |

_{imkr}**𝒵**|, relative to

_{imkr}**ᶄ**

_{r}, is not equal to the previous measure it made of |

**𝒵**

_{imk}_{1}|. In fact, upon taking derivatives of both it finds that the axis |

**𝒵**

_{imk}_{1}| is collapsing (destroying) at a slower rate than |

**𝒵**|’s event history is growing. That is, as we’ve stated previously, a spatial system

_{imkr}

*S**is relative to each observer,*

_{n}**ᶄ**

*ϵ*

_{n}

*S**and a ‘common’ space*

_{n}*does not exist*. So,

**ᶄ**

_{1}, when at the

**ᶄ**

_{r}position, observes a wave flux propogating in the opposite direction of its own (chirality changes). This is for no other reason than that

**ᶄ**

_{r}’s oscillation happens to be in the direction at which the left hand rule applies, not the right.

**ᶄ**

_{r}‘s canonical reference – and it’s reality – is its own spatial system,

**, not that of**

*S*_{r}**ᶄ**

_{1},

*S**.*

_{1}**lemma:** The geometry and magnitude of any observable in the event history of any arbitrarily chosen observer (**ᶄ**_{1}, **ᶄ**_{2}, … , **ᶄ**_{n}) ϵ **S*** _{n}* is the tensor-valued function,

**𝒵**,whose domain is restricted to only the chosen observer’s state variables. That is, any observer state variable shared by two or more observers supposes an undefined relationship.

_{i }We next let **ᶄ**_{1}oscillate an observer **ᶄ**_{r }such that **ᶄ**_{r }is oscillated through all possible orientations. This will require corresponding changes in the spatial system *S** _{n}* relative to

**ᶄ**

_{1}.

**ᶄ**

_{1}begins with a table showing all of the measured wave behaviors observable with the Historical Acts have attained; the total count being 288. That is, there are a total of 288 possible ways in which two observers may interact during oscillation. In general, the number of ‘particles’ (our choice of this word will be clear momentarily) would be expected to grow very large if summed over all observers, however,

**ᶄ**

_{1}notes that 2 observers are both necessary and sufficient for relative behaviors between them to be characterized. In other words, multiple observers simply suggests iterative comparisons. Each observer has two axes about which to rotate, one in each space but shared between them when interacting. And each observer may be a wave form as a plane wave operating in order 3. The total number of possible configurations per observer is 24 (see tables). So, the total count of possible combinations, the total number of possible ‘particles’, is

12 *24 = 288. {as of 2006 237 ‘particles’ have been discovered}

We begin with the **ᶄ**_{1 }permutation and, for each row in the first table, we’ll need to cycle through each remaining table once:

ᶄ_{1 }Count |
Notation |
Chirality |
Flux |
Axes |

1 | ⟨ │i│ɠ ⟩j |
Right | 𝒵, 1½_{re} |
½ , i ½j |

2 | ⟨ │j│ɠ ⟩i |
Left | 𝒵, -½_{im} |
½ , j ½i |

3 | ⟨ │i│ɠ⟩m |
Right | 𝒵, ½_{re} |
½ , i ½m |

4 | ⟨│j│ɠ⟩m |
Right | 𝒵, ½_{re} |
½ , j ½m |

5 | ⟨ │i│ɠ⟩n |
Left | 𝒵, ½_{im} |
½ , i ½n |

6 | ⟨ │j│ɠ⟩n |
Right | 𝒵, ½_{re} |
½ , j ½n |

7 | ⟨│n│ɠ⟩ i |
Right | 𝒵, ½_{re} |
½ , n ½i |

8 | ⟨│ m│ɠ⟩i |
Left | 𝒵, ½_{re} |
½ , m ½i |

9 | ⟨│m│ɠ⟩j |
Left | 𝒵, ½_{re} |
½, m½j |

10 | ⟨│n│ɠ⟩j |
Left | 𝒵, ½_{re} |
½, n½j |

11 | ⟨ │m│ɠ⟩n |
Right | 𝒵,-½_{re} |
½ , m ½n |

12 | ⟨│n│ɠ⟩m |
Left | 𝒵,-½_{re} |
½, n½m |

13 | ⟨ │i│ɠ ⟩j |
Right | 𝒵,-½_{im} |
½ , i ½j |

14 | ⟨ │j│ɠ ⟩i |
Left | 𝒵, 1½_{re} |
½ , j ½i |

15 | ⟨ │i│ɠ⟩m |
Right | 𝒵, ½_{im} |
½ , i ½m |

16 | ⟨│j│ɠ⟩m |
Right | 𝒵, ½_{im} |
½ , j ½m |

17 | ⟨ │i│ɠ⟩n |
Left | 𝒵, ½_{re} |
½ , i ½n |

18 | ⟨ │j│ɠ⟩n |
Right | 𝒵, ½_{im} |
½ , j ½n |

19 | ⟨│n│ɠ⟩i |
Right | 𝒵, ½_{im} |
½ , n ½i |

20 | ⟨│ m│ɠ⟩i |
Left | 𝒵, ½_{im} |
½ , m ½i |

21 | ⟨│m│ɠ⟩j |
Left | 𝒵, ½_{im} |
½, m½j |

22 | ⟨│n│ɠ⟩j |
Left | 𝒵, ½_{im} |
½, n½j |

23 | ⟨ │m│ɠ⟩n |
Right | 𝒵, 1½_{im} |
½ , m ½n |

24 | ⟨│n│ɠ⟩m |
Left | 𝒵, 1½_{im} |
½, n½m |

[n=ij,c=r,c=re1½] shortened to []_{x,y}

The resultant tables can be found in the appendix. All 288 ‘particles’ are listed there.

Writing out the products for each vector we get (orthonormal system) all possible components of revolution:

|*i*_{r}| => **W**_{im}, **W**_{im},**W**_{re}

|*j*_{r}| => **W**_{im},**W**_{im},**W**_{re}

|*m*_{r}| => **W**_{im},**W**_{re},**W**_{re}

|*n*_{r}| => **W**_{im},**W**_{re},**W**_{re}

Which are the components of revolution for one, the other or both rotational axes.[29]

For example:

|*k*_{re}|^{2} = |*i*_{r}|^{2} + |*j*_{r}|^{2};

And/or

|*o*_{im}|^{2} = |*m*_{r}|^{2}** + **|*n*_{r}|^{2}

And we see at once that, relative to the observer for whom these values are taken, both |*i*_{r}| and |*j*_{r}| are removing definition from |*k*_{re}|. And conversely, both |*m*_{r}| and |*n*_{r}| are adding definition to |*k*_{re}|. That is, the First Act is experiencing perturbation from what we shall from here on refer to as the Third Act; tertiary behavior within the spatial system *S** _{n}* caused by relative rotation of spatial systems between 2 or more observers. And note the rotation (cross product) relations:

|*k*_{re}| = **⟨ **|*i*_{k1}| ⨂ |*j*_{k1}| **│** ** ɠ^{nlm}**|

_{10}

^{2}

**│**|

*j*_{k2}| ⨂ |

*n*_{k2}|

**⟩**, and |

*k*_{re}| is contracting

|*k*_{re}| = **⟨** |*m*_{k1}| ⨂ |*n*_{k1}| **│** ** ɠ^{nlm}**|

_{10}

^{2}

**│**|

*n*_{k2}| ⨂ |

*i*_{k2}|

**⟩**, and |

*k*_{re}| is expanding

[30]Following the permutations we also see other phenomenon, ‘forces’, that repel and attract respectively, namely

when |*k*_{re}| = |*i*_{re}|; (weak nuclear)

and when |*k*_{re}| = |*n*_{im}|; (strong nuclear).

We can make the identifications with the ‘forces’ mentioned by elimination; however, we will now show how we can be certain of these. The proportions between [To be continued … ].

In sum, the Third Act is tertiary behavior made possible by the First and Second Acts in which the interactions between defined observers occur.

**Definition: **we define a perturbation of the First Act as any event history, **ᶄ**, where

|**ᶄ**| = |**ᶄ**_{E}| – |**e**|^{i}^{|t|} |** t**| any valid parameterization of

**ᶄ**and E, the total energy of the sysem

**ᶄ**.

**Definition:** we *define* the Third Act to be the set of all possible events ζrelative to any arbitrary perturbation of the First Act as defined by the Second Act if each 𝒶 ∈ ζ is sufficiently well defined.[31]

**10.****The Principle of Delusion**

*Any arbitrarily chosen point on the surface of the wave of the Third Act retains ponderable quantities of degrees of freedom, f, equal to the order of S_{n}; and, when measured relative to any other point’s degrees of freedom, is a delusion created by the measurement of the point itself. That is, particles of any kind, point or extendend, have no possibility of existence. Rather, they are a relative measure of two numeric points on the wave of the Third Act whose degrees of freedom, f, may present any allowable wave values.*

And the particle ‘zoo’ is merely a collection of the various ways in which any given point on the wave of the Third Act can be observed. And, as previously noted, there are 288 possible permutations thereof. So, the delusion cannot be downgraded to a simple illusion: treating a point on the wave as a particle is a choice made upon measurement. By choice, we allude to the manner of observing ‘chosen’ by the phylogenetics of the observer, a topic best left to Biology. And we note that the search for the most fundamental particles, as we can now see, is rather less informative than one might have previously (though it can serve as valuable confirmation of predictions herein) thought considering the physical meaning behind the effort. There are no ‘particles’ to observe, just a rich set of different ways of observing the same thing.

The final implication of the Third Act can now be enunciated, if not already evident from the previous discussion. The ‘particle mass’, ‘particle energy’, and quantities for the degrees of freedom of some point on the wave of the Third Act corresponding to that ‘particle’ are one and the same. The degrees of freedom represent the magnitude, rate change and rate of rate change of the wave of the First Act, for each order 10 component, as an immutably reposed point.

Having explored these facts and not at all satisfied with the vagueness of the terms ‘illusion’ and ‘delusion’, we will now attempt to more rigorously describe the source of the illusions and delusion aforementioned in order to establish an operational definition. Suppose some observer, **ᶄ**_{1}, is tasked, or otherwise engaged, with making some observation of an event **u**_{k}, in which the bases of observation for **ᶄ**_{1 }are restricted to the ** i**,

**,**

*j***, and**

*k***ᵯ**bases, not by natural limitations but by some circumstantial limitation, that is, a chosen limitation. Then all events,

**u**

_{k}, occurring in

**,**

*i***,**

*j***, and**

*k***ᵯ**appear strictly local to

**ᶄ**

_{1}. But all other degrees of freedom of

**u**

_{k }with respect to

**ᶄ**

_{1 }need not be local, thus potentially obscuring most of the degrees of freedom of

**u**

_{k}from immediate observation by

**ᶄ**

_{1}. That is, it is not possible, provided the initial ‘choice’ is enforced, for

**ᶄ**

_{1}to observe all degrees of freedom of an event,

**u**

_{k}. So,

**ᶄ**

_{1}records a history ‘deceptive’ to interpretation by dint of incompleteness. Because this requires a ‘choice’ of observation that is not principally controlling, we refer to this as a delusion. But illusions, such as motion, are produced as something inherent to nature and have no ‘choice’ to which we can ascribe cause. But, for both illusions and delusions, the effect is the same; the axiomatic property of “uncertainty” ubiquitous in quantum field theory is merely a result of trying to measure values that themselves don’t exist, but are artifacts of the delusions of the observer who in fact doesn’t understand what is physically occurring.

**11.****A Natural Tensor**

We will advance a conservation theorem to show that the resultant of summing over all observers is the **0** matrix.[32]

So, we make three assumptions upon which to predicate a Theorem on the Conservation of Space:

- The universe began by an energy exchange at the origin of dim = 10.
- That in dim 10 this exchange was equal in magnitude in all directions.
- That any sum of components of each observers m
**v**over the surface of the exchange would, at any arbitrary time, result in

∳∳∳∳∳∳∳∳∳∳ d**v** f(m**v**)** = **0.

Thus we conclude that the result of summation of the Natural Tensor (*ɠ*_{0}) over all observers does in fact generate the **0** matrix. Any observations without full accounting for all observers will be, to some possibly insignificant degree, inaccurate. In most cases that fact would not be locally apparent. Indeed, the only reason we don’t observe the **0** matrix at localities sufficiently large is because of Natural Limits.

Let **ᶄ**_{1 }observe the aforementioned rate change given by:

**ᶄ**_{1 }begins by taking a measurement of the historical magnitude of one of its spatial axes, |** i**|, and notes that should it change |

**| such that it locates itself at |**

*i***|/2, |**

*i***| and -|**

*i***| cancel and the rate of change in the event history of |**

*i***|, and the rate of change in that rate of change, shall all be identically ₵**

*i**and |*

_{i}**| will maintain the ratio e/1.[33]**

*i*We next derive this relation more generally:

_{a}**|**_{b} = |**e**|^{𝔵}|**v**_{ξ}| [(∂|** i**| / ∂) + (∂|

**| / ∂) + (∂|**

*j***| / ∂) +(∂|**

*k***| / ∂) + (∂|**

*ℓ***ᵯ**| / ∂) +

–*ɨ(*∂|** n**| / ∂ ) –

*ɨ(*∂|

**| / ∂ ) –**

*o**ɨ(*∂|

**| / ∂ ) –**

*p**ɨ(*∂|

**| / ∂ ) –**

*q**ɨ(*∂|

**ᵲ**| / ∂ )]

⨀

|**e**| [(∂ / ∂ |** i**| + ∂ / ∂ |

**| +∂ / ∂ |**

*j***| + ∂ / ∂ |**

*k***| + ∂ / ∂ |**

*ℓ***ᵯ**|) + (

**r**Space)

(∂ / ∂ |** n**| + ∂ / ∂ |

**| + ∂ / ∂ |**

*o***| + ∂ / ∂|**

*p***| + ∂ / ∂|**

*q***ᵲ**|] (

**m**Space)

Or more compactly we can write

_{i}**|*** _{n}* = |

**e**|

^{𝔵}<

**+**

*i***,… ,**

*j +***> ∃,**

*ᵲ*|**v*** _{f}*| =

_{a}

**|**

_{b}|

**v**

*|*

_{i}ð**p** = |**v**_{ξ}|[∂|** m**| / ∂|

**| + –**

*m**i*[(∂ (∂|

**r**| / ∂|

**|)) / ∂|**

*m***|]];**

*r*ð**p** = |**v**_{ξ}|[ 1 + –*i*[(∂ (∂|** r**| / ∂|

**|)) / ∂|**

*m***|) ]];**

*r*ð**p** = |**v**_{ξ}|[ 1 + –*i*[- 3|** m**| / |

**|**

*r*^{5}]; using

*n*= |

**| / |**

*m***|, |**

*r***| = ₵**

*r**;*

_{m}ð**p** = β / |**v**_{ξ}| = |**v**_{ξ}|[ 1 *+ ɨ*[3|** m**| / |

**|**

*r*^{5}];

ð**p**^{2} = |**v**_{ξ}|[1^{2} + 9(|** m**|

^{2}/ |

**|**

*r*^{10})];

_{i}**|*** _{n}* =

_{ n}**|**

_{i}|**v**|∇** K**= |

**v**| [(∂|

**m**| / ∂) + –

*ɨ(*∂|

**r**| / ∂)] *

[(∂ / ∂c* _{i}*|

**| + ∂ / ∂ c**

*i**|*

_{j}**| +∂ / ∂c**

*j**|*

_{k}**| + ∂ / ∂c**

*k**|*

_{ℓ}**|) + (**

*ℓ***r**Space)

(∂ / ∂c* _{n}*|

**| + ∂ / ∂c**

*n**|*

_{o}**| + ∂ / ∂c**

*o**|*

_{p}**| + ∂ / ∂c**

*p**|*

_{q}**|]; (**

*q***m**Space)

Multiplying out the terms gives:

det|**V**|∇** K**=

det|**V**| <∂|**m**| / ∂c* _{i}*|

**| + ∂|**

*i***m**| / ∂c

*|*

_{j}**| +∂|**

*j***m**| / ∂c

*|*

_{k}**| + ∂|**

*k***m**| / ∂c

*|*

_{ℓ}**| +**

*ℓ*∂|**m**| / ∂c* _{n}*|

**| + ∂|**

*n***m**| / ∂c

*|*

_{o}**| + ∂|**

*o***m**| / ∂c

*|*

_{p}**| + ∂|**

*p***m**| / ∂c

*|*

_{q}**| +**

*q*–*ɨ(*∂|**r**| / ∂|**m**|) (∂ / ∂c* _{i}*|

**|) + –**

*i**ɨ(*∂|

**r**| / ∂|

**m**|) (∂ / ∂ c

*|*

_{j}**|) +**

*j*–*ɨ(*∂|**r**| / ∂|**m**|) (∂ / ∂c* _{k}*|

**|) + –**

*k**ɨ(*∂|

**r**| / ∂|

**m**|) (∂ / ∂c

*|*

_{ℓ}**|) +**

*ℓ*–*ɨ(*∂|**r**| / ∂|**m**|) (∂ / ∂c* _{n}*|

**|) + –**

*n**ɨ(*∂|

**r**| / ∂|

**m**|) (∂ / ∂c

*|*

_{o}**|) +**

*o*–*ɨ(*∂|**r**| / ∂|**m**|) (∂ / ∂c* _{p}*|

**|) + –**

*p**ɨ(*∂|

**r**| / ∂|

**m**|) (∂ / ∂c

*|*

_{q}**|) >**

*q*where det|**V**| represents the rate change for which we are seeking a solution (a rank 2 or rank 1 tensor run over the natural tensor). It is always a rate change in a quantity of an axis. ** K** is the unit vector showing the direction of most rapid expansion. det|

**V**| is another derivative applied using the chain rule on each term therein.

A Natural Metric Tensor.

_{2}|_{10} |
i |
j |
k |
ℓ |
ᵯ |
n |
o |
p |
q |
ᵲ |

i |
∂||/∂|i|i |
∂||/∂|j|i |
∂||∂|k|i |
∂||∂|ℓ|i |
∂|ᵯ|/∂||i |
∂||/∂|n|i |
∂||/∂|o|i |
∂||/∂|p|i |
∂||/∂|q|i |
∂||/∂|ᵲ|ᵲ |

j |
∂||/∂|i|j |
∂||/∂|j|j |
∂||∂|k|j |
∂||∂|ℓ|j |
∂|ᵯ|/∂||j |
∂||/∂|n|j |
∂||/∂|o|j |
∂||/∂|p|j |
∂||/∂|q|j |
∂||/∂|ᵲ|j |

k |
∂||/∂|i|k |
∂||/∂|j|k |
∂||∂|k|k |
∂||∂|ℓ|k |
∂|ᵯ|/∂||k |
∂||/∂|n|k |
∂||/∂|o|k |
∂||/∂|p|k |
∂||/∂|q|k |
∂||/∂|ᵲ|k |

ℓ |
∂||/∂|i|ℓ |
∂||/∂|j|ℓ |
∂||∂|k|ℓ |
∂||∂|ℓ|ℓ |
∂|ᵯ|/∂||ℓ |
∂||/∂|n|ℓ |
∂||/∂|o|ℓ |
∂||/∂|p|ℓ |
∂||/∂|q|ℓ |
∂||/∂|ᵲ|ℓ |

ᵯ |
∂||/∂|iᵯ| |
∂||/∂jᵯ |
∂||∂kᵯ |
∂||∂ℓᵯ |
∂|ᵯ|/∂ᵯ |
∂||/∂n ᵯ |
∂||/∂oᵯ |
∂||/∂p ᵯ |
∂||/∂ qᵯ |
∂||/∂ᵲ ᵯ |

n |
∂|/∂|i|n |
∂|/∂|j|n |
∂||∂|k|n |
∂||∂|ℓ|n |
∂|ᵯ|/∂||n |
∂||/∂|n|n |
∂||/∂|o|n |
∂|/∂|p|n |
∂||/∂|q|n |
∂||/∂|ᵲ|n |

o |
∂||/∂|i|o |
∂||/∂|j|o |
∂||∂|k|o |
∂||∂|ℓ|o |
∂||/∂|mo| |
∂||/∂|n|o |
∂||/∂|o|o |
∂||/∂|p|o |
∂||/∂|q|o |
∂||/∂|ᵲ|o |

p |
∂||/∂|i|p |
∂||/∂j|p |
∂||∂|k|p |
∂||∂|ℓ|p |
∂|ᵯ|/∂||p |
∂||/∂|n|p |
∂||/∂|o|p |
∂||/∂|p|p |
∂||/∂|q|p |
∂||/∂|ᵲ|p |

q |
∂|/∂|i|q |
∂||/∂|j|q |
∂||∂|k|q |
∂||∂|ℓ|q |
∂|ᵯ|/∂|q| |
∂||/∂|n|q |
∂||/∂|o|q |
∂||/∂|p|q |
∂||/∂|q|q |
∂||/∂|ᵲ|q |

ᵲ |
∂||/∂|i|ᵲ |
∂||/∂|j|ᵲ |
∂||∂|k|ᵲ |
∂||∂|ℓ|ᵲ |
∂|ᵯ|/∂||ᵲ |
∂||/∂|n|ᵲ |
∂||/∂|o|ᵲ |
∂||/∂|p|ᵲ |
∂||/∂|q|ᵲ |
∂||/∂|ᵲ|ᵲ |

**12.****The Principle of the Fundamental Axiom**

Given the First Act as axiomatic to formal logic and Nature, an event, ξ, sufficiently well defined in a spatial system ∈ *S** _{n}* is necessary and sufficient for ξ to be sufficiently well defined in a formal logic, ℒ, ∈

*S**. Otherwise, ℒ is, at best, incomplete or possibly internally inconsistent. That is, formal logic seeks the definition, and nature the fulfillment, of necessities and sufficiencies ∈*

_{n}

*S**.[34]*

_{n}So, we now begin putting together the Natural Tensor aforementioned. It is the most general solution limited only by the Natural Limits inherent in the system observed. But, by definition, that is complete. Therefore, a general solution for local conditions is needed such that we sum over only the local observers whose properties dominate above the noise of Natural Limits; to-wit, **ᶄ**_{1}**, ᶄ _{2}, . . ., ᶄ_{m}**. Beginning with the general form above and with

**sufficiently large we note again:**

*m*_{2}**|**^{10}_{i}_{ → r}*=* **e**^{it}

That is, the local deviation from the **0** matrix in **r**Space is about 2.718 meters per second faster in **r**Space than in **m**Space. This fact accounts for everything we’ve examined herein. This curvature is intrinsic to nature and represents spatial tension created by Natural Limits. We can also now state a conservation principle on the basis of the above:

*The rate of change in the observable event histories of a Spatial System S_{ᶄ}_{n}, is constant. *

Let ** G** =

*e*

^{1};

**=**

*S**e*

^{iπ}^{ / 2};

**=**

*W**e*

^{π}^{ / 2}and

**= e**

*E*^{+}^{π / 2 }* e

^{π / 2}, e

^{i}^{π / 2 }* e

^{i}^{π / 2};

**= e**

*E*^{–}^{π / 2 }* e

^{i}^{π / 2}.

Checking the curvature of any arbitrary point ‘a’ linearly independent in a Spatial System, ** S_{n}**:

and if { r | if i cross i or r cross r; i | if i cross r or r cross i} this tensor simply reduces to:

_{8}**|**_{10} =

ᶄ_{deluded} |
ᶄ_{1} |
^{L}z^{i | r} |
^{L}z^{i | r} |
^{L}z^{i | r} |
^{L}z^{i | r} |
^{L}z^{i | r} |
^{R}z^{i | r} |
^{R}z^{i | r} |
^{R}z^{i | r} |
^{R}z^{i | r} |
^{R}z^{i | r} |

ᶄ_{2} |
ᶄ_{abs} |
p_{1}_{ | }p_{1} |
o_{1 }_{| }o_{1} |
n_{1 }_{| }n_{1} |
m_{1 }_{| }m_{1} |
ℓ_{1 }_{| }ℓ_{1} |
p_{1 }_{| }p_{1} |
o_{1 }_{| }o_{1} |
n_{1 }_{| }n_{1} |
m_{1 }_{| }m_{1} |
ℓ_{1 }_{| }ℓ_{1} |

^{R}z^{ r | i} |
ℓ_{2 }|ℓ_{2} |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |
1^{2},( -2, 1) |
1^{2},( -2, 1) |
1^{2},( -2, 1) |
1^{2},( -2, 1) |
1^{2},( -2, 1) |

^{R}z^{ r | i} |
m_{2 }| m_{2} |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |
1^{2},( -2, 1) |
1^{2},( -2, 1) |
1^{2},( -2, 1) |
1^{2},( -2, 1) |
1^{2},( -2, 1) |

^{R}z^{ r | i} |
n_{2 }| n_{2} |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |
1^{2},( -2, 1) |
1^{2},( -2, 1) |
1^{2},( -2, 1) |
1^{2},( -2, 1) |
1^{2},( -2, 1) |

^{R}z^{ r | i} |
o_{2 }| o_{2} |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |
1^{2},( -2, 1) |
1^{2},( -2, 1) |
1^{2},( -2, 1) |
1^{2},( -2, 1) |
1^{2},( -2, 1) |

^{R}z^{ r | i} |
p_{2 }| p_{2} |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |
1^{2},( -2, 1) |
1^{2},( -2, 1) |
1^{2},( -2, 1) |
1^{2},( -2, 1) |
1^{2},( -2, 1) |

^{L}z^{ r | i} |
ℓ_{2 }| ℓ_{2} |
1^{2},( -2, 1) |
1^{2},( -2, 1) |
1^{2},( -2, 1) |
1^{2},( -2, 1) |
1^{2},( -2, 1) |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |

^{L}z^{ r | i} |
m_{2 }| m_{2} |
1^{2},( -2, 1) |
1^{2},( -2, 1) |
1^{2},( -2, 1) |
1^{2},( -2, 1) |
1^{2},( -2, 1) |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |

^{L}z^{ r | i} |
n_{2 }| n_{2} |
1^{2},( -2, 1) |
1^{2},( -2, 1) |
1^{2},( -2, 1) |
1^{2},( -2, 1) |
1^{2},( -2, 1) |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |

^{L}z^{ r | i} |
o_{2 }| o_{2} |
1^{2},( -2, 1) |
1^{2},( -2, 1) |
1^{2},( -2, 1) |
1^{2},( -2, 1) |
1^{2},( -2, 1) |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |

^{L}z^{ r | i} |
p_{2 }| p_{2} |
1^{2},( -2, 1) |
1^{2},( -2, 1) |
1^{2},( -2, 1) |
1^{2},( -2, 1) |
1^{2},( -2, 1) |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |
3^{2}, (-4, 3) |

From Gedanken One, we begin by considering the two periodic bases, one imaginary and one real, and the operands that generate them via a cross product. Thus, we begin by considering four basis axes by which the imaginary and real periodic bases are generated. From these, we can generate, via a suitable transform, six basic forms to complete the tensor:

*1.) **For like chirality and bases crossed both imaginary, we proportion the magnitude of dilation in the reals as 1;*

*2.) **For like chirality and bases crossed both real, we proportion the magnitude of dilation in the reals as 1;*

*3.) **For like chirality and bases crossed one imaginary and one real, we proportion the magnitude of dilation in the imaginaries as 1 and constriction in the reals as 2;*

*4.) **For unlike chirality and both bases crossed imaginary, we proportion the magnitude of dilation in the reals as 3;*

*5.) **For unlike chirality and bases crossed both real, we proportion the magnitude of dilation in the reals as 3;*

*6.) **For unlike chirality and bases crossed one imaginary and one real, we proportion the magnitude of dilation in the imaginaries as 3 and constriction in the reals as 4;*

And these constitute the six forms aforementioned, which generates 4 different proportions of dilation/constriction in a statistically normalized universe of component values.

Which is the high-level view of the Natural Tensor we sought to derive. Note that the lower left and upper right quadrants contain entries that can be created in six ways for the ** E,S,G** scalar types, each possibly

**i**Space or

**r**Space, that is, a total of 6 different types of generators all based on periodic behavior as given by

**. In the absence of a cross product, the entry is simply the component curvature caused by the First Natural Act alone; that is,**

*e*

*e*^{1}; denoted here as

**. The ligher shades indicate curvature of an imaginary axis while darker shades indicate curvature of a real axis. And the vertical bar,**

*G***|**, means “or”. Therefore, the remaining four available component values will be guaranteed to always be a vector sum of one or more of the basic six generator component results.[35]

Note that in each ** E** entry we see a phenomenon in which, wherever observer bases have like chirality in their cross product, and wherever the sign conventions of imaginary and real products are applied, we see a “charge” or “pole” behavior in which two observers appear to “repel” from each other when the bases of the observers crossed are like and to “attract” the two observers when they differ. Whether we observe this behavior or not in any given measurement depends on whether or not the curvature involved has components in these respective quadrants; the same reasoning applying equally to any other phenomenon (or “force”) we’ll encounter. The “force” of “unification” is

*e*^{1}itself, observed as “forces” consequent to the peridocities presenting and the inscrutable cause of the geometric magnitude ratio

*e*^{1}. Note further that, all combinations unlike cross products in observable behavior will generate an attractive “force”, which

**. Thus,**

*G***is universal to all observations. This is because wherever we take a cross product of r**

*G*

_{ᶄ}_{1}and r

_{ᶄ}_{2}or i

_{ᶄ}_{1}and i

_{ᶄ}_{2}, it necessarily follows that an unlike cross product is principally observable as well (in the simple case of periodicity only along one axis in both observers means there is a many-to-many relation of 4 cross products between two observers).

Since the generators alluded to in the tensor above are denoted in an approximate fashion there, we will now explicitly derive each generator algorithm and describe it’s role in the tensor. We’ll start by establishing a coordinate designation system, based on Gedanken One and of order 10, for three observers **ᶄ**_{1},** ᶄ _{2}**, and

**ᶄ**:

_{3}** i,j,k**; real-valued spatial basis axes

**ℓ**; a real-valued spin basis axis

** m**; a real-valued parameter basis axis

** n, o, p**; imaginary-valued spatial basis axes

** q**; a imaginary-valued spin basis axis and

** r**; an imaginary-valued parameter basis axis

We next locate **ᶄ**_{1} and **ᶄ**_{2} in their respective spatial systems ** S_{ᶄ1}** and

**such that all scalar values are equal. We propose to measure events in the**

*S*_{ᶄ2}**coordinate system. The plan will be to incrementally increase the magnitude of an arbitrary vector**

*S*_{ᶄ3 }**∈**

*z***which we will use to locate observers**

*S*_{ᶄ3 }**ᶄ**

_{1}and

**ᶄ**

_{2 }in

**. The initial value of**

*S*_{ᶄ3}**,**

*z*

*z*_{0}, we shall set to

**0**. For all observers their respective spatial systems are expanding in all directions, dim 10, as according to Gedanken One. Our focus here will be on observing the effect of the spin basis axes,

**and**

*ℓ***, and how their orientation and spin affects the measurements made by**

*q***ᶄ**

_{3}. The reader may care to note from Gedanken One that the chirality of spin in

**i**Space is necessarily the opposite of chirality in

**r**Space. Under initial conditions

**ᶄ**

_{3}reports that spin angles are undefined.

**is then increased by one natural limit,**

*z**nlm*

**ᶄ**

_{3n}, and

**ᶄ**

_{3}observes the following:

**ᶄ**_{1} reports a spin angle ω_{1} with magnitude |** q**| ∈

**in**

*S*_{ᶄ1}**i**Space only.

**ᶄ _{2 }**reports a spin angle ω

_{2}with magnitude |

**| ∈**

*q***in**

*S*_{ᶄ2}**i**Space only.

Since, at this point, **ᶄ**_{3} measures only one natural limit separating observers **ᶄ**_{1} and **ᶄ**_{2}, no real-valued separation is present. Consequently, the real component of ** z** is measured only as endpoints of a natural limit, all other points in-between undefined.

Both **ᶄ**_{1} and **ᶄ**_{2} report that they can observe any combination of spins:

through any two imaginary bases

through any three imaginary bases

through any three imaginary bases and one imaginary parameter basis.

with any angle ω, with respect to each basis, possible on the range [0, π/2].

However, they note, any rotation through reals does not begin without a |** z**| of greater real value. Thus, |

**| is increased just beyond the natural limit and both**

*z***ᶄ**

_{1}and

**ᶄ**

_{2}report that spin through real basis axes is observable but values of ω and |

**ℓ|**are undefined. Both observers report that these values cannot be observed until |

**| > |**

*z***1**| > |

**2**| > |

**3**|, where 1,2 and 3 are the imaginary basis axes magnitude of

**,**

*n***and**

*o***, not necessarily in corresponding order. Both**

*p***ᶄ**

_{1}and

**ᶄ**

_{2}further report that for any spin through only one real basis and spin through more than one imaginary basis, just beyond the natural limit,

**ᶄ**

_{3 }observes a positive acceleration along |

**| between observers**

*z***ᶄ**

_{1}and

**ᶄ**

_{2}which, in

**, appears to be increasing the real distance between them. Since the condition for observation of values depends on the condition that |**

*S*_{ᶄ3}**| > |**

*z***1**| > |

**2**| > |

**3**|, neither observer can report values for this phenomenon. A concomitant phenomenon presents independent of spin configuration whose values are also not observable until the condition |

**| > |**

*z***1**| > |

**2**| > |

**3**| is met, which introduces what is observed in

**as a proportionally much larger (as an absolute value) negative acceleration on |**

*S*_{ᶄ3}**| occurring for any observer on the interval (|**

*z**nlm*

**ᶄ**

_{3n}|, |

**| = |**

*z***1**| > |

**2**| > |

**3**|). However, the former depends on the values in

**i**Space but does not depend on any given position on (|

*nlm*

**ᶄ**

_{3n}|, |

**| = |**

*z***1**| > |

**2**| > |

**3**|), while the latter does. Thus, the negative acceleration on (|

*nlm*

**ᶄ**

_{3n}|, |

**| = |**

*z***1**| > |

**2**| > |

**3**|) observed by

**ᶄ**

_{3 }is observed as a magnitude whose value is constant, independent of location on (|

*nlm*

**ᶄ**

_{3n}|, |

**| = |**

*z***1**| > |

**2**| > |

**3**|) and does not exist outside it. Observers

**ᶄ**

_{1}and

**ᶄ**

_{2}are next placed on |

**| at the point (|**

*z**nlm*

**ᶄ**

_{3n}| + (|

**| = |**

*z***1**| > |

**2**| > |

**3**|) + |

*nlm*

**ᶄ**

_{3n}|. Observer

**ᶄ**

_{3 }reports the instantaneous loss of the negative accelerations supra at point (|

**| = |**

*z***1**| > |

**2**| > |

**3**|) + |

*nlm*

**ᶄ**

_{3n}|.

However, at point (|*nlm* **ᶄ**_{3n} | + (|** z**| = |

**1**| > |

**2**| > |

**3**|) + |

*nlm*

**ᶄ**

_{3n}|

**ᶄ**

_{3 }reports the instantaneous appearance of another negative acceleration consequent to the fact that real values of ω at this point are maximum and decrease as the real magnitude of |

**| increases. Real values for ω > 0 result in smaller expansion on |**

*z***| for each unit value of the parameter |**

*z***| than that given by the First Natural Act acting alone. Given the large difference between 2 or 3 integer multiples of |**

*m**nlm*

**ᶄ**

_{3n}| (representing the magnitude of length through which imaginary spin occurs) and the value of |

**| at (|**

*z***| = |**

*z***1**| > |

**2**| > |

**3**|) + |

*nlm*

**ᶄ**

_{3n}| one can correctly suspect a very large proportionate difference in the magnitudes of the first and second negative accelerations observed by

**ᶄ**

_{3}. And we note that because

*z*_{im}and

*z*_{re}constitute a plane swept out by spin, the negative acceleration in the second case will decrease with the square of the real distance locating

**ᶄ**

_{1}and

**ᶄ**

_{2}on |

**. It is of peculiar note that this spin through the**

*z|***plane, if restricted thereto, is not observable as spin when observed solely from either**

*z***r**Space or

**i**Space.

**ᶄ**_{3} also notes another phenomenon that requires us to return to the condition |** z**| =

**0**in order to elaborate. Up to this point, we have only generally considered the various combinations of rotations permissible under order 10.

**ᶄ**

_{3}can observe that there are four other spin configurations that can also result in accelerations as perturbations of the First Natural Act. These configurations are as follows:

**ᶄ**_{1} reports spin through 2 imaginary and 3 real basis axes.

**ᶄ**_{2} reports spin through 3 imaginary and 2 real basis axes.

or vice versa or

Both **ᶄ**_{1 }and**ᶄ**_{2 }report a spin configuration identical to one of the above.

We take each case as an example. Should **ᶄ**_{3 }observe that both **ᶄ**_{1} and**ᶄ**_{2} report spin through 3 real basis axes but only 2 imaginary, then the spin on *z*_{re} is less than that of the First Natural Act and a negative acceleration presents on *z*_{re}. Conversely, if both observers report spin through 3 imaginary and only 2 real basis axes, then the spin on *z*_{re} is greater than that of the First Natural Act and a positive acceleration presents on *z*_{re}. On the other hand, if the two observers report two different configurations, then the resultant of the two yields a negative acceleration on ** z** as observed from

**. This exhausts all physical possibilities for spin through a spatial system,**

*S*_{ᶄ3}**.**

*S*The letter notation is used to denote what each component result corresponds to in the traditional language of physics for any curvature composed solely of that notation;

** G** => “a gravitational force”

** S** => “a strong nuclear force”

** W** => “a weak nuclear force”

** E** => “an electromagnetic force”

The ratio of the total curvatures of each quadrants, as can be explicitly calculated from the Natural Tensor, is:

** S** = 1

ð** E** /

**= 1/137**

*S*ð** W** /

**= 10**

*S*^{-6}

ð** G** /

**= 10**

*S*^{-38}

So, the following relation is the general relation, by the theory here advanced, by which all nature must behave :

_{8 }**|**_{10} = **<**(*e*^{1}| *e*^{ πi}^{ / 2}|* e*^{π}^{ / 2} | *e*^{π / 2 }* *e*^{π / 2 }| *e*^{i}^{π / 2 }* *e*^{i}^{π / 2} | *e*^{π / 2 }* e^{i}^{π / 2})*i + *

(*e*^{1}| *e*^{ πi}^{ / 2}|* e*^{π}^{ / 2} | *e*^{π / 2 }* *e*^{π / 2 }| *e*^{i}^{π / 2 }* *e*^{i}^{π / 2} | *e*^{π / 2 }* e^{i}^{π / 2})* j + *

(*e*^{1}| *e*^{ πi}^{ / 2}|* e*^{π}^{ / 2} | *e*^{π / 2 }* *e*^{π / 2 }| *e*^{i}^{π / 2 }* *e*^{i}^{π / 2} | *e*^{π / 2 }* e^{i}^{π / 2})* k + *

(*e*^{1}| *e*^{ πi}^{ / 2}|* e*^{π}^{ / 2} | *e*^{π / 2 }* *e*^{π / 2 }| *e*^{i}^{π / 2 }* *e*^{i}^{π / 2} | *e*^{π / 2 }* e^{i}^{π / 2})* l + *

(*e*^{1}| *e*^{ πi}^{ / 2}|* e*^{π}^{ / 2} | *e*^{π / 2 }* *e*^{π / 2 }| *e*^{i}^{π / 2 }* *e*^{i}^{π / 2} | *e*^{π / 2 }* e^{i}^{π / 2})* m + *

(*e*^{1}| *e*^{ πi}^{ / 2}|* e*^{π}^{ / 2} | *e*^{π / 2 }* *e*^{π / 2 }| *e*^{i}^{π / 2 }* *e*^{i}^{π / 2} | *e*^{π / 2 }* e^{i}^{π / 2})*n + *

(*e*^{1}| *e*^{ πi}^{ / 2}|* e*^{π}^{ / 2} | *e*^{π / 2 }* *e*^{π / 2 }| *e*^{i}^{π / 2 }* *e*^{i}^{π / 2} | *e*^{π / 2 }* e^{i}^{π / 2})*o + *

(*e*^{1}| *e*^{ πi}^{ / 2}|* e*^{π}^{ / 2} | *e*^{π / 2 }* *e*^{π / 2 }| *e*^{i}^{π / 2 }* *e*^{i}^{π / 2} | *e*^{π / 2 }* e^{i}^{π / 2})*p + *

(*e*^{1}| *e*^{ πi}^{ / 2}|* e*^{π}^{ / 2} | *e*^{π / 2 }* *e*^{π / 2 }| *e*^{i}^{π / 2 }* *e*^{i}^{π / 2} | *e*^{π / 2 }* e^{i}^{π / 2})*q + *

(*e*^{1}| *e*^{ πi}^{ / 2}|* e*^{π}^{ / 2} | *e*^{π / 2 }* *e*^{π / 2 }| *e*^{i}^{π / 2 }* *e*^{i}^{π / 2} | *e*^{π / 2 }* e^{i}^{π / 2})*r *>

[§ 1.00].

In closing I would like to dedicate this work to Eugene Wesley Roddenberry, whose passion to fly higher, faster and farther than any before us inspired me to continue work on this grandest of mysteries.

Anonymous 2009-04-18

Appendix

Introduction

Lorentz

Heisenberg relations

QM wave mechanics (probability measured as a wave)

Combined EM:

- Strong nuclear
- Weak nuclear
- Electromagnetic

When a new theoretical edifice is proposed, it is very desirable to identify distinctive testable experimental predictions. In the case of superstring theory there have been no detailed computations of the properties of elementary particles or the structure of the universe that are convincing, though many valiant attempts have been made

I’ll borrow this line when my theory is proven wrong:

“It is believed that the reason that these particles have not yet been observed is because supersymmetry is a broken symmetry, and as a result the superpartners are heavier than the known elementary particles. Experiments carried out so far have not had particle beams of sufficient energy and intensity to produce them in observable numbers.”

Derivations

Basic Lorentz generators

We begin with basic form of the Natural Tensor Generator:

_{2}**|**_{10} = |**e**|^{𝔵} |**v**_{ξ}| [(∂|** i**| / ∂) + (∂|

**| / ∂) + (∂|**

*j***| / ∂) +(∂|**

*k***| / ∂) + (∂|**

*ℓ***ᵯ**| / ∂) +

–*ɨ(*∂|** n**| / ∂ ) –

*ɨ(*∂|

**| / ∂ ) –**

*o**ɨ(*∂|

**| / ∂ ) –**

*p**ɨ(*∂|

**| / ∂ ) –**

*q**ɨ(*∂|

**ᵲ**| / ∂ )]

⨀

|**e**| [(∂ / ∂ |** i**| + ∂ / ∂ |

**| +∂ / ∂ |**

*j***| + ∂ / ∂ |**

*k***| + ∂ / ∂ |**

*ℓ***ᵯ**|) + (

**r**Space)

(∂ / ∂ |** n**| + ∂ / ∂ |

**| + ∂ / ∂ |**

*o***| + ∂ / ∂|**

*p***| + ∂ / ∂|**

*q***ᵲ**|] (

**m**Space)

Program: the Lorentz generator is the ratio of the imaginary parts of both parameters whilst their real components are held constant and equal. This requires that the generator receives the |**ᵲ**|’th value as a factor. Reducing the basic form to just that focus we get:

|**e**|^{𝔵} |**v**_{ξ}|[(∂|**ᵲ**| / ∂) + –*ɨ(*∂|**ᵯ**| / ∂ )] *

passing in [ ∂ / ∂|**ᵲ**|] for metric of imaginary time to real time

ð = |**e**|^{𝔵} |**v**_{ξ}|[∂|**ᵲ**| / ∂|**ᵲ**| + –*ɨ(*∂|**ᵯ**| / ∂|**ᵲ**|)];

ð = |**e**|^{𝔵} |**v**_{ξ}|[ 1 + –*ɨ(*∂|**ᵯ**| / ∂|**ᵲ**|)];

ð|**v**_{ξ}| ^{2} = |**v**_{ξ}||**e**|^{𝔵} [ 1^{2} + –*ɨ(*∂|**ᵯ**| / ∂|**ᵲ**|)^{2}];

ð|**v**_{ξ}| = |**v**_{ξ}| |**e**|^{𝔵} √ [ 1^{2} + –*ɨ(*∂|**ᵯ**| / ∂|**ᵲ**|)^{2}];

β = ^{2} = 1^{2} + –*ɨ(*∂|**ᵯ**| / ∂|**ᵲ**|)^{2};

which is just the Lorentz transform. Alternately, we can perform the same calculation for any couplet of spatial bases as well, and calculate any combination of them by following the general form above.

(**p**_{ijkℓ})^{1/₵} = (|**v**_{re}|₵ / |**v**_{im}|)^{1/₵ }=

*nlm *|**ᵯ**|→₵ of (|**v**_{re}|₵ / |**v**_{im}|)^{1/₵} = (|**v**_{im}|₵ / |**v**_{im}|)^{1/₵}

note that we do NOT need to continue using *nlm* notation because we have established that the quantities that follow are physically meaningful up to the Natural Limit, which is the limit of definition.

Definition of e

The basic program of the First Act is characterizing natural behaviors consequent to the variable ratio of the rate change in a basis Natural Limit to the rate change of that same basis in the reals. This can be simply represented as:

₵ (|**v**_{im}| / |**v**_{re}|) and invoking the general form

_{2}**|**_{10} = |**e**|^{𝔵} |**v**_{ξ}| [(∂|** i**| / ∂) + (∂|

**| / ∂) + (∂|**

*j***| / ∂) +(∂|**

*k***| / ∂) + (∂|**

*ℓ***ᵯ**| / ∂) +

–*ɨ(*∂|** n**| / ∂ ) –

*ɨ(*∂|

**| / ∂ ) –**

*o**ɨ(*∂|

**| / ∂ ) –**

*p**ɨ(*∂|

**| / ∂ ) –**

*q**ɨ(*∂|

**ᵲ**| / ∂ )]

⨀

|**e**| [(∂ / ∂ |** i**| + ∂ / ∂ |

**| +∂ / ∂ |**

*j***| + ∂ / ∂ |**

*k***| + ∂ / ∂ |**

*ℓ***ᵯ**|) + (

**r**Space)

(∂ / ∂ |** n**| + ∂ / ∂ |

**| + ∂ / ∂ |**

*o***| + ∂ / ∂|**

*p***| + ∂ / ∂|**

*q***ᵲ**|] (

**m**Space)

We can at once remove the |**e**|^{𝔵} factor as that is what we are deriving and which has already been derived by other means. So, for the one-dimensional case we have:

_{2}**|**_{1} = |**v**_{ξ}| [(∂|** i**| / ∂) –

*ɨ*(∂|

**| / ∂ )] ⨀ [(∂ / ∂ |**

*n***|]**

*i*ð _{2}**|**_{1} = |**v**_{ξ}| [(∂|** i**| / ∂|

**|) –**

*i**ɨ*(∂|

**| / ∂|**

*n***| )]**

*i*ð _{2}**|**_{1} = ₵ (|**v**_{im}| / |**v**_{re}|) [(∂|** i**| / ∂|

**|) –**

*i**ɨ*(∂|

**| / ∂|**

*n***| )]**

*i*ð _{2}**|**_{1} = ₵ (∂|** n**| / ∂|

**ᵯ**|) * (∂|

**ᵯ**| / ∂|

**|) [(∂|**

*i***| / ∂|**

*i***|) –**

*i**ɨ*(∂|

**| / ∂|**

*n***| )]**

*i*ð _{2}**|**_{1} = ₵ (∂|** n**| / ∂|

**|) [(∂|**

*i***| / ∂|**

*i***|) + –**

*i**ɨ*(∂|

**| / ∂|**

*n***| )]**

*i*ð _{2}**|**_{1} = [₵ (∂|** n**| / ∂|

**|) + –**

*i**ɨ*₵(∂|

**| / ∂|**

*n***| )**

*i*^{2}]

ð₵ (∂|** n**| / ∂|

**|) ***

*i*_{2}

**|**

_{1}= [1 + –

*ɨ*(∂|

**| / ∂|**

*n***| )]**

*i*^{-1}and setting

*i*= 1.

ð₵^{2} * _{2}**|**_{1} = [1 + –*ɨ*₵]^{-1}

ð₵^{2} * _{2}**|**_{1} = [1 + –*ɨ*₵]^{-1}

ð₵^{2} * ( _{2}**|**_{1} )^{-1/₵} = [1 + –*ɨ*₵]^{1/₵}

ð₵^{-2/₵} ( _{2}**|**_{1} )^{-1/₵} = [1 + –*ɨ*₵]^{1/₵}

ð*glm* |₵| →0 of [1 + –*ɨ*₵]^{1/₵}= |**e|**^{1} and

ð│ _{2}**|**_{1} │ = 1 / ₵^{2}|**e|**^{₵}

A derivation of π and the Schrodinger equation

β_{1} = γ^{2}_{1} = (1 / |**ᵵ**| – 1)^{2} = (1/ |**ᵵ**|^{2} – 2 / |**ᵵ**| + 1)

for the left and

β_{2} = γ^{2}_{2} = (1 / |**ᵵ**| + 1)^{2} = (1/ |**ᵵ**|^{2} + 2 / |**ᵵ**| + 1)

for the right

We now let *n*_{2} = *n*_{1 }* 1 / |**ᵵ**|. Finally, let the total real probability of two clocks in two reference frames **ᶄ**_{1} and **ᶄ**_{2} reporting times 100% congruent as a function of the First Act, |**e**|, be denoted as a function, ψ. Then, under the sole influence of the First Act,

ψ (*n*) ≈ γ^{2}_{2}∫_{0}^{|ᵯ|m} |**e**|^{(|ᵯ||ᵵ|)2/2} d**e**^{(|ᵯ||ᵵ|)} + ∫_{0}^{|ᵯ|m} |**ᵯ**|* _{m}* * |

**e**|

^{-(|ᵯ||ᵵ|)2/2}d

**e**

^{(|ᵯ||ᵵ|)}

– [γ^{2}_{1}∫_{0}^{|ᵯ|m} |**e**|^{(|ᵯ||ᵵ|)2/2} d**e**^{(|ᵯ||ᵵ|)} – ∫_{0}^{|ᵯ|m} |**ᵯ**|* _{m}* * |

**e**|

^{-(|ᵯ||ᵵ|)2/2}d

**e**

^{(|ᵯ||ᵵ|)}]

ψ (*n*) ≈ ∫_{0}^{|ᵯ|m} γ^{2}_{2}|**e**|^{(|ᵯ||ᵵ|)2/2} – γ^{2}_{1}|**e**|^{(|ᵯ||ᵵ|)2/2} d**e**^{(|ᵯ||ᵵ|)} + 2*∫_{0}^{|ᵯ|m} *n _{f}* |

**e**|

^{–n2/2}d

**e**

^{(|ᵯ||ᵵ|)}

ψ (*n*) ≈ ∫_{0}^{|ᵯ|m} γ^{2}_{2}|**e**|^{(|ᵯ||ᵵ|)2/2} – γ^{2}_{1}|**e**|^{(|ᵯ||ᵵ|)2/2} d**e**^{(|ᵯ||ᵵ|)} + 2∫_{0}^{|ᵯ|m} *n _{f}* |

**e**|

^{–n2/2}d

**e**

^{(|ᵯ||ᵵ|)}

and letting *n _{f }* = |

**ᵯ**|

_{m+}ð**ᵯ*** _{nlm}*(|

**ᵯ**|) =

*n*2∫

_{f}_{0}

^{n}*|*

^{f}**e**|

^{-(|ᵯ||ᵵ|)2/2}d

**e**

^{(|ᵯ||ᵵ|)}(preliminary)

and, noting that when we began in [§ 0.0908] we saw that the proper treatment of the integration above was as a vector resultant rate change in time. Thus the ratio of the initial area, a* _{i}*, to the enlarged area generated by the First Act, a

_{1}, in which the First Act serves as the parameter for the observed time,

**ᵯ**, is a

*/ a*

_{i}_{1}. Thus, we have:

*|***ῡ |**

^{2}

*=***[**

*r**n*2∫

_{f}_{0}

^{n}*|*

^{f}**e**|

^{-(|ᵯ||ᵵ|)2/2}d

**e**

^{(|ᵯ||ᵵ|)}] ⨀

**ῡ**

_{r}*+*[

**i***n*2∫

_{f}_{0}

^{n}*|*

^{f}**e**|

^{-(|ᵯ||ᵵ|)2/2}d

**e**

^{(|ᵯ||ᵵ|)}] ⨀

*i*|

**ῡ**|

_{i}and since

|** r**[

*n*∫

_{f}_{0}

^{n}*|*

^{f}**e**|

^{-(|ᵯ||ᵵ|)2/2}d

**e**

^{(|ᵯ||ᵵ|)}]|

*=*|

**[**

*i**n*∫

_{f}_{0}

^{n}*|*

^{f}**e**|

^{-(|ᵯ||ᵵ|)2/2}d

**e**

^{(|ᵯ||ᵵ|)}]|

*|***ῡ |**

^{2}

*=***(1/√ 2) [**

*r**n*2∫

_{f}_{0}

^{n}*|*

^{f}**e**|

^{-(|ᵯ||ᵵ|)2/2}d

**e**

^{(|ᵯ||ᵵ|)}] ⨀

**ῡ**

_{r}*+ i *(1/√ 2) [

*n*2∫

_{f}_{0}

^{n}*|*

^{f}**e**|

^{-(|ᵯ||ᵵ|)2/2}d

**e**

^{(|ᵯ||ᵵ|)}] ⨀

*i*

**ῡ**

_{i}and, appropriately, setting our Natural Limit as the unit value and taking its ratio with unity we get:

*|ᵶ|*^{2}** =** |

**|**

*r**+*|

**|**

*i*ð** |ᵶ| =** √ |

**|**

*r**+ √*|

**|**

*i*then the rate of change in ** |ᵶ|** at unit value is:

1 / |**ᵵ**| = ** |¯ ᵶ |** / |

**ᵵ**|

**√ (|**

*=***| / |**

*r***ᵵ**|)

*+ √*(|

**| / |**

*i***ᵵ**|). (basis vector resultant).

and each basis alone is:

√ (1 / |**ᵵ**| * |**ᵵ**|_{φ}) = 1 / √|**ᵵ**| * |**ᵵ**|`

where |**ᵵ**|` is the expanded total magnitude of |**ᵵ**| for each unit time elapsed, |**ᵯ**|. So that,

*|***ῡ |**

^{2}

*=***[1/√ (2 |**

*r***ᵵ**| * |

**ᵵ**|

_{φ})

*n*2∫

_{f}_{0}

^{n}*|*

^{f}**e**|

^{-(|ᵯ||ᵵ|)2/2}d

**e**

^{(|ᵯ||ᵵ|)}] ⨀

**ῡ**

_{r}*+ i*[1/√ (2 |

**ᵵ**| * |

**ᵵ**|

_{φ})

*n*2∫

_{f}_{0}

^{n}*|*

^{f}**e**|

^{-(|ᵯ||ᵵ|)2/2 }d

**e**

^{(|ᵯ||ᵵ|)}] ⨀

*i*|

**ῡ**|

_{i}And

|**ᵵ**|_{φ} = [[2 *n _{f}* ∫

_{0}

^{n}*|*

^{f}**e**|

^{-(|ᵯ||ᵵ|)2/2}d

**e**

^{(|ᵯ||ᵵ|)}] *

*|ᵶ|*^{-2}]

^{2}

**(2|**

*/***ᵵ**|)

ð√ |**ᵵ**|_{φ} = √2|**ᵵ**| |**ᵯ**|* _{m }*/

*|ᵶ|*^{-2}) ∫

_{0}

^{n}*|*

^{f}**e**|

^{-(|ᵯ||ᵵ|)2/2}d

**e**

^{(|ᵯ||ᵵ|)}

ð√ |**ᵵ**|_{φ} = 1 / (** |ᵶ| **√2|

**ᵵ**|) * 2∫

_{0}

^{n}*|*

^{f}**e**|

^{-(|ᵯ||ᵵ|)2/2}d

**e**

^{(|ᵯ||ᵵ|)};

|**ᵵ**|_{φ} = 3.14… = π [vector component form]. So,

we can *again* define the quantity π thusly:

**Definition:** The value π is the ratio of a Natural Limit, |**𝔑**|_{φ,} at some parameterization |**ᵯ**| to the Natural Limit, |**𝔑**|_{,} at parameterization |**ᵯ**| – 1, expressed as a component of an order 2 vector. And π^{2} = ** i** [|

**𝔑**|

_{φ}/ |

**𝔑**|].

*|***ῡ |**

^{2}

*=***(1/√ 2π|**

*r***ᵵ**|) [

**|ᵯ**|

*∫*

_{m}_{0}

^{|ᵯ}^{|}

*|*

^{m}**e**|

^{-(|ᵯ||ᵵ|)2/2}d

**e**

^{(|ᵯ||ᵵ|)}] ⨀

**ῡ**

_{r}*+ i *(1/√ 2π|

**ᵵ**|) [

**|ᵯ**|

*∫*

_{m}_{0}

^{|ᵯ}^{|}

*|*

^{m}**e**|

^{-(|ᵯ||ᵵ|)2/2}d

**e**

^{(|ᵯ||ᵵ|)}] ⨀

*i*

**ῡ**[§ 0.0912]

_{i }and solving for the common component we can see that it is a solution to the equation:

*|***ῡ_{r}|**

^{2}

**1/√ 2π|**

*=***ᵵ**|) 2

**|ᵯ**|

*∫*

_{m}_{0}

^{|ᵯ}^{|}

*|*

^{m}**e**|

^{-(|ᵯ||ᵵ|)2/2}d

**e**

^{(|ᵯ||ᵵ|)}

ð2**|ᵯ**| / √ (2π|**ᵵ**|) * e^{-(|ᵵ|)2(|ᵯ|)2/2} = – |**ᵵ**|^{2}**|ᵯ**| / (2π|**ᵵ**|) * e^{-(|ᵵ|)2(|ᵯ|)2/2}

And we can also readily see that each component is the most general solution to the well known Schrodinger equation.

A derivation of the relativistic Schrodinger equation

L is a function derived of Newtonian equations of ‘motion’ in three variables, F, F` and t. F represents the “correct path” for a Lagrangian. That is, we seek a function F that satisfies:

0 = (∂ L / ∂ F) – d/dt [∂ L / ∂ F`];

But this is just the result of applying the Natural Tensor condition:

_{2}**|**_{10}*=* **e*** ^{it }*.

to the total energy of a system **ᶄ**_{0}, which we can prove is:

_{2}**|**_{10} = |**e**|^{𝔵} |**v**_{ξ}| [(∂|** i**| / ∂) + (∂|

**| / ∂) + (∂|**

*j***| / ∂) +(∂|**

*k***| / ∂) + (∂|**

*ℓ***ᵯ**| / ∂) +

–*ɨ(*∂|** n**| / ∂ ) –

*ɨ(*∂|

**| / ∂ ) –**

*o**ɨ(*∂|

**| / ∂ ) –**

*p**ɨ(*∂|

**| / ∂ ) –**

*q**ɨ(*∂|

**ᵲ**| / ∂ )]

⨀

|**e**| [(∂ / ∂ |** i**| + ∂ / ∂ |

**| +∂ / ∂ |**

*j***| + ∂ / ∂ |**

*k***| + ∂ / ∂ |**

*ℓ***ᵯ**|) + (

**r**Space)

(∂ / ∂ |** n**| + ∂ / ∂ |

**| + ∂ / ∂ |**

*o***| + ∂ / ∂|**

*p***| + ∂ / ∂|**

*q***ᵲ**|] (

**m**Space)

Letting

L (|** n_{i}**|) = |

**|; ∵**

*i***= [**

*i***u**

_{1},

**u**

_{2}] * |

**ᵲ**| and

**= |**

*n*_{i}**ᵯ**| * |

*v*_{i}|

F (|** n_{m}**|) = |

**ᵯ**|; ∵ |

**ᵯ**| = (

**u**

_{1},

**u**

_{2}) * |

**|and**

*i***= |**

*n*_{m}**ᵯ**| * |

*v*_{m}|

G (|** n_{r}**|) = |

**ᵲ**|; ∵ |

**ᵲ**| = [

**u**

_{1},

**u**

_{2}] and

**= |**

*n*_{r}**ᵯ**| * |

*v*_{r}|

(** n** may obtain without

**and**

*m***depends on**

*m***).**

*n*|**ᵲ**|_{c}^{2} = [1^{2} + –*ɨ*(∂|**ᵲ**| / ∂|**ᵯ**|)^{2}]; We want a ‘relativistic’ result.

ð√|**ᵲ**|_{r} = –*ɨ*(∂|**ᵲ**| / ∂|**ᵯ**|)^{2}.

[(∂|**ᵯ**| / ∂ ) + –*ɨ*(∂|**ᵲ**| / ∂)]; the real and imaginary comparison operator

ð|** p**| = [ (∂|

**ᵯ**| / ∂|

**|) + –**

*i**ɨ*(∂ |

**ᵲ**|

_{r}/ ∂|

**ᵯ**|)]; operand: <∂ / ∂ |

**| + ∂ |**

*i***ᵲ**|

_{r}/ ∂>.

ð|** p**| = [ (∂|

**ᵯ**| / ∂|

**|) + –**

*i**i*[∂ (|-

*ɨ*(∂|

**ᵲ**| / ∂|

**ᵯ**|)

^{2}| / ∂|

**ᵯ**|)]; operand: |

**ᵲ**|

_{r}

ð|** p**| = [ (∂|

**ᵯ**| / ∂|

**|) + –**

*i**i*[∂ | –

*ɨ*(∂|

**ᵲ**| / ∂|

**ᵯ**|)*-

*ɨ*(∂|

**ᵲ**| / ∂|

**ᵯ**|)| / ∂|

**ᵯ**|)];

ð|** p**| = |

**v**

_{ξ}|[ (∂|

**ᵯ**| / ∂|

**|) + –**

*i**i*[∂ | –

*ɨ*(∂|

**ᵲ**| / ∂|

**ᵯ**|)*-

*ɨ*(∂|

**ᵲ**| / ∂|

**ᵯ**|)| / ∂|

**ᵯ**|)]; operand: (∂|

**| / ∂|**

*i***ᵯ**|)

^{2}, the event history plane of interest, one dimensional case[36]. Recall that we are trying to solve for an

**r**Space path along a real axis.

ð|** p**| = [(∂|

**| / ∂|**

*i***ᵯ**|) + -ɨ(∂ ((∂|

**ᵲ**| / ∂|

**ᵯ**|)*-

*i*(∂|

**ᵲ**| / ∂|

**ᵯ**|)) / ∂|

**ᵯ**|)];

ð|** p**| = [(∂|

**| / ∂|**

*i***ᵯ**|) + ((∂ / ∂ |

**ᵯ**|) (∂|

**ᵲ**| ∂ |

**ᵲ**| / ∂ |

**ᵯ**| ∂ |

**ᵯ**|)];

ð|** p**| = [(∂|

**| / ∂|**

*i***ᵯ**|) + ((∂|

**ᵲ**| / ∂ |

**ᵯ**|) (∂|

**ᵲ**| / |

**ᵯ**|`)];

And substituting our event functions back in to the equation yields:

ð |** p**| = [(L / F) + ((d / d|

**ᵯ**|) (∂|

**ᵲ**||

**ᵲ**| / F`)];

And using the identity that |**ᵲ**| = |** t**|

^{2}

_{xy}we substitute |

**| for |**

*i***ᵲ**|

^{2}:

|** p**| = [(L / F) + ((d / d

**t**

_{im}) (∂ |

**| / F`)];**

*i*Since this tensor was run only on net energy gain or loss, this is the state of equilibrium, which is just the buoyancy 𝒷 (to be discussed shortly). That is

|** p**| = 𝒷 = [(L / F) + ((d / d

**t**

_{im}) (∂ |

**| / F`)];**

*i*which is riding the wave surface of the First Act, that is, 𝒷 = 0.

ð _{2}**|**_{10}*=* 0 = 𝒷 = [(L / F) + ((d / d**t**_{im}) (∂ |** i**| / F`)] ∀ |

**ᵯ**| ∈

**ᵯ**. which will always hold for a closed system,

**ᶄ**

_{0. }and

0 = [(L / F) + ((d / d**t**_{im}) (∂ |** i**| / F`)],

is just the Euler-Lagrange equation. Q.E.D.

Derivation of the Action of conservative forces

Having already derived an action principle for a relativistic Schrodinger derviation we are now prepared to generalize that result and introduce the fundamental action principle.

_{2}**|**_{10} = |**e**|^{𝔵} |**v**_{ξ}| [(∂|** i**| / ∂) + (∂|

**| / ∂) + (∂|**

*j***| / ∂) +(∂|**

*k***| / ∂) + (∂|**

*ℓ***ᵯ**| / ∂) +

–*ɨ(*∂|** n**| / ∂ ) –

*ɨ(*∂|

**| / ∂ ) –**

*o**ɨ(*∂|

**| / ∂ ) –**

*p**ɨ(*∂|

**| / ∂ ) –**

*q**ɨ(*∂|

**ᵲ**| / ∂ )]

⨀

|**e**| [(∂ / ∂ |** i**| + ∂ / ∂ |

**| +∂ / ∂ |**

*j***| + ∂ / ∂ |**

*k***| + ∂ / ∂ |**

*ℓ***ᵯ**|) + (

**r**Space)

(∂ / ∂ |** n**| + ∂ / ∂ |

**| + ∂ / ∂ |**

*o***| + ∂ / ∂|**

*p***| + ∂ / ∂|**

*q***ᵲ**|] (

**m**Space)

Letting

L (|** n_{i}**|) = |

**|; ∵**

*i***= [**

*i***u**

_{1},

**u**

_{2}] * |

**ᵲ**| and

**= |**

*n*_{i}**ᵯ**| * |

*v*_{i}|

F (|** n_{m}**|) = |

**ᵯ**|; ∵ |

**ᵯ**| = (

**u**

_{1},

**u**

_{2}) * |

**|and**

*i***= |**

*n*_{m}**ᵯ**| * |

*v*_{m}|

G (|** n_{r}**|) = |

**ᵲ**|; ∵ |

**ᵲ**| = [

**u**

_{1},

**u**

_{2}] and

**= |**

*n*_{r}**ᵯ**| * |

*v*_{r}|

(** n** may obtain without

**and**

*m***depends on**

*m***).**

*n*|**ᵲ**|_{c}^{2} = [1^{2} + –*ɨ*(∂|**ᵲ**| / ∂|**ᵯ**|)^{2}]; We account for ‘relativistic’ action.

ð√|**ᵲ**|_{r} = –*ɨ*(∂|**ᵲ**| / ∂|**ᵯ**|)^{2}.

[(∂|**ᵯ**| / ∂ ) + –*ɨ*(∂|**ᵲ**| / ∂)]; the real and imaginary comparison operator

ð|** p**| = [ (∂|

**ᵯ**| / ∂|

**|) + –**

*i**ɨ*(∂|

**ᵲ**|

_{r}/ ∂|

**|)]; operand: <∂ / ∂ |**

*i***| + ∂ |**

*i***ᵲ**|

_{r}/ ∂>.

ð|** p**| = [ (∂|

**ᵯ**| / ∂|

**|) + –**

*i**i*[∂ (|-

*ɨ*(∂|

**ᵲ**| / ∂|

**ᵯ**|)

^{2}| / ∂|

**|)]; operand: |**

*i***ᵲ**|

_{r}

ð|** p**| = [ (∂|

**ᵯ**| / ∂|

**|) + –**

*i**i*[∂ |-

*ɨ*(∂|

**ᵲ**| / ∂|

**ᵯ**|)*-

*ɨ*(∂|

**ᵲ**| / ∂|

**ᵯ**|)| / ∂|

**|)];**

*i*ð|** p**| = |

**v**

_{ξ}|[ (∂|

**ᵯ**| / ∂|

**|) + –**

*i**i*[∂ |-

*ɨ*(∂|

**ᵲ**| / ∂|

**ᵯ**|)*-

*ɨ*(∂|

**ᵲ**| / ∂|

**ᵯ**|)| / ∂|

**ᵯ**|)]; operand: (∂|

**| / ∂|**

*i***ᵯ**|)

^{2}, the event history plane of interest, one dimensional case[37]. Recall that we are trying to solve for an

**r**Space path along a real axis.

ð|** p**| = [(∂|

**| / ∂|**

*i***ᵯ**|) + -ɨ(∂ ((∂|

**ᵲ**| / ∂|

**ᵯ**|)*-

*i*(∂|

**ᵲ**| / ∂|

**ᵯ**|)) / ∂|

**ᵯ**|)];

ð|** p**| = [(∂|

**| / ∂|**

*i***ᵯ**|) + ((∂ / ∂ |

**ᵯ**|) (∂|

**ᵲ**| ∂ |

**ᵲ**| / ∂ |

**ᵯ**| ∂ |

**ᵯ**|)];

ð|** p**| = [(∂|

**| / ∂|**

*i***ᵯ**|) + ((∂|

**ᵲ**| / ∂ |

**ᵯ**|) (∂|

**ᵲ**| / |

**ᵯ**|`)];

Gedanken One with magnitudes very large

c * H_{0 }= “magnitude of Pioneer effect”

(8.74 ± 1.33) × 10^{−10} m/s² acceleration toward sun

|**ᵯ**|_{max} = 1 / H_{0} = √ (3c^{2} / 8π G ρ)

ðc * H_{0} = 1 / √ (3 / 8π G ρ)

relative expansion rate = H_{0} * distance

** |v** ⨀

**e**| = |

**ᵯ**|

_{max}/ √ (3c

^{2}/ 8π G ρ)

** |v** ⨀

**e**| = |

**ᵯ**|

_{max}√ 8πGρ / c √ (3)

Let ** |v**|

_{r}denote the absolute rate of expansion of some point x at some distance d.

*|v*_{r}| = |**ᵯ**|_{max} √ 8πGρ / c √ (3)

ð*|v*_{r}| / |**ᵯ**|_{max} = √ 8πGρ / c √ (3)

ðd = |**ᵯ**|^{2}_{max} √ 8πGρ / c √ (3)

Hubble constant read in km of distance per second per 1 million LY. Converting to standard SI we get:

H_{0} = 1000m / s / (3.1536 * 10^{13 }s * 299,792,458 = |** i**|)

ðH_{0} = (*v*_{r} * 1000) / (3.1536 * 10^{13 }s * 299,792,458 = |** i**|)

ðH_{0} = (*v*_{r} * 1000) / (9.454254955488 * 10^{21})

ðH_{0} = (*v*_{r} * 1000) / (9.454254955488 * 10^{21})

ðH_{0} = *v*_{r} * 1.05772480719860492635583686731011704652645078591442

* 10^{-19}

*v*_{ᶄ}_{1} / *v*_{ᶄ}_{2 }= *v*_{r}

And is thus now read as meters per second per meter. Let

h_{c} = 1.05772480719860492635583686731011704652645078591442

* 10^{19} ∃

H_{0} = *v*_{r} (1 / h_{c}).

ðh_{c} H_{0} = *v*_{r}

ðh_{c} H_{0} / |**ᵯ**| = d*v*_{r} / d|**ᵯ**|

ð*v*_{T} H_{0} = *a*_{r}.

Where *v*_{T} is the locally measured rate change in ** z** relative to both observers. But

*v*_{T}is just the speed of light, c,

cH_{0} = *a*_{r}.

Which indicates that observers **ᶄ**_{1}and** ᶄ**_{2} appear to be accelerating toward each other, in a manner additive to the apparent accelerations given by the First Act, by an amount *a*_{r}.

Obsrvr right/left symmetrically opposing chirality – positive growth in reals **k**_{1} all left chiral, **k**_{2} all right chiral; resultant eigenvector is **0**

=> force **F** = **0**. Off-angle values represent forces exerted under the *shared* spatial systems of **k**_{1} and **k**_{2} when both *combine* to interact with any *other* observer on that axis.

ᶄ |
ᶄ_{1} |
^{L}z^{i} |
^{L}z^{i} |
^{L}z^{i} |
^{L}z^{i} |
^{L}z^{i} |
^{R}z^{i} |
^{R}z^{i} |
^{R}z^{i} |
^{R}z^{i} |
^{R}z^{i} |
^{R}z^{r} |
^{R}z^{r} |
^{R}z^{r} |
^{R}z^{r} |
^{R}z^{r} |

ᶄ_{2} |
ᶄ_{abs} |
p_{1}_{ | }p_{1} |
o_{1 }_{| }o_{1} |
n_{1 }_{| }n_{1} |
m_{1 }_{| }m_{1} |
ℓ_{1 }_{| }ℓ_{1} |
p_{1 }_{| }p_{1} |
o_{1 }_{| }o_{1} |
n_{1 }_{| }n_{1} |
m_{1 }_{| }m_{1} |
ℓ_{1 }_{| }ℓ_{1} |
p_{1 }_{| }p_{1} |
o_{1 }_{| }o_{1} |
n_{1 }_{| }n_{1} |
m_{1 }_{| }m_{1} |
ℓ_{1 }_{| }ℓ_{1} |

^{R}z^{ r} |
ℓ_{2 }|ℓ_{2} |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |
-2, 1 |
-2, 1 |
-2, 1 |
-2, 1 |
-2, 1 |
1 |
1 |
1 |
1 |
1 |

^{R}z^{ r} |
m_{2 }| m_{2} |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |
-2, 1 |
-2, 1 |
-2, 1 |
-2, 1 |
-2, 1 |
1 |
1 |
1 |
1 |
1 |

^{R}z^{ r} |
n_{2 }| n_{2} |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |
-2, 1 |
-2, 1 |
-2, 1 |
-2, 1 |
-2, 1 |
1 |
1 |
1 |
1 |
1 |

^{R}z^{ r} |
o_{2 }| o_{2} |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |
-2, 1 |
-2, 1 |
-2, 1 |
-2, 1 |
-2, 1 |
1 |
1 |
1 |
1 |
1 |

^{R}z^{ r} |
p_{2 }| p_{2} |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |
-2, 1 |
-2, 1 |
-2, 1 |
-2, 1 |
-2, 1 |
1 |
1 |
1 |
1 |
1 |

^{L}z^{ r} |
ℓ_{2 }| ℓ_{2} |
-2, 1 |
-2, 1 |
-2, 1 |
-2, 1 |
-2, 1 |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |
3 |
3 |
3 |
3 |
3 |

^{L}z^{ r} |
m_{2 }| m_{2} |
-2, 1 |
-2, 1 |
-2, 1 |
-2, 1 |
-2, 1 |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |
3 |
3 |
3 |
3 |
3 |

^{L}z^{ r} |
n_{2 }| n_{2} |
-2, 1 |
-2, 1 |
-2, 1 |
-2, 1 |
-2, 1 |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |
3 |
3 |
3 |
3 |
3 |

^{L}z^{ r} |
o_{2 }| o_{2} |
-2, 1 |
-2, 1 |
-2, 1 |
-2, 1 |
-2, 1 |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |
3 |
3 |
3 |
3 |
3 |

^{L}z^{ r} |
p_{2 }| p_{2} |
-2, 1 |
-2, 1 |
-2, 1 |
-2, 1 |
-2, 1 |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |
3 |
3 |
3 |
3 |
3 |

^{L}z^{ i} |
ℓ_{2 }| ℓ_{2} |
1 |
1 |
1 |
1 |
1 |
3 |
3 |
3 |
3 |
3 |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |

^{L}z^{ i} |
m_{2 }| m_{2} |
1 |
1 |
1 |
1 |
1 |
3 |
3 |
3 |
3 |
3 |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |

^{L}z^{ i} |
n_{2 }| n_{2} |
1 |
1 |
1 |
1 |
1 |
3 |
3 |
3 |
3 |
3 |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |

^{L}z^{ i} |
o_{2 }| o_{2} |
1 |
1 |
1 |
1 |
1 |
3 |
3 |
3 |
3 |
3 |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |

^{L}z^{ i} |
p_{2 }| p_{2} |
1 |
1 |
1 |
1 |
1 |
3 |
3 |
3 |
3 |
3 |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |

*1.) **For like chirality and bases crossed both imaginary, we proportion the magnitude of dilation in the reals as 1;*

*2.) **For like chirality and bases crossed both real, we proportion the magnitude of dilation in the reals as 1;*

*3.) **For like chirality and bases crossed one imaginary and one real, we proportion the magnitude of dilation in the imaginaries as 1 and constriction in the reals as 2;*

*4.) **For unlike chirality and both bases crossed imaginary, we proportion the magnitude of dilation in the reals as 3;*

*5.) **For unlike chirality and bases crossed both real, we proportion the magnitude of dilation in the reals as 3;*

*6.) **For unlike chirality and bases crossed one imaginary and one real, we proportion the magnitude of dilation in the imaginaries as 3 and constriction in the reals as 4;*

ᶄ |
ᶄ_{1} |
^{Lk}z^{i} |
^{Lk}z^{i} |
^{Lk}z^{i} |
^{Lk}z^{i} |
^{Lk}z^{i} |
^{Lk}z^{r} |
^{Lk}z^{r} |
^{Lk}z^{r} |
^{Lk}z^{r} |
^{Lk}z^{r} |

ᶄ_{2} |
ᶄ_{abs} |
p_{1 }_{| }p_{1} |
o_{1 }_{| }o_{1} |
n_{1 }_{| }n_{1} |
m_{1 }_{| }m_{1} |
ℓ_{1 }_{| }ℓ_{1} |
p_{1 }_{| }p_{1} |
o_{1 }_{| }o_{1} |
n_{1 }_{| }n_{1} |
m_{1 }_{| }m_{1} |
ℓ_{1 }_{| }ℓ_{1} |

^{Lk}z^{ r} |
ℓ_{2 }|ℓ_{2} |
(ℓ_{2 / }p_{1})∙ ℓ_{2} |
o_{1}∙ ℓ_{2} |
n_{1}∙ℓ_{2} |
m_{1}∙ℓ_{2} |
ℓ_{1}∙ℓ_{2} |
<ℓ_{2 + }p_{1}> |
<ℓ_{2 + }o_{1}> |
<ℓ_{2 + }n_{1}> |
<ℓ_{2 + }m_{1}> |
<ℓ_{2 + }ℓ_{1}> |

^{Lk}z^{ r} |
m_{2 }| m_{2} |
p_{1 }∙m_{2} |
o_{1}∙ m_{2} |
n_{1}∙m_{2} |
m_{1}∙m_{2} |
ℓ_{1}∙m_{2} |
< m_{2 + }p_{1}> |
< m_{2 + }o_{1}> |
< m_{2 + }n_{1}> |
< m_{2 + }m_{1}> |
< m_{2 + }ℓ_{1}> |

^{Lk}z^{ r} |
n_{2 }| n_{2} |
p_{1 }∙n_{2} |
o_{1}∙ n_{2} |
n_{1}∙n_{2} |
m_{1}∙n_{2} |
ℓ_{1}∙n_{2} |
< n_{2 + }p_{1}> |
< n_{2 + }o_{1}> |
< n_{2 + }n_{1}> |
< n_{2 + }m_{1}> |
< n_{2 + }ℓ_{1}> |

^{Lk}z^{ r} |
o_{2 }| o_{2} |
p_{1 }∙o_{2} |
o_{1}∙ o_{2} |
n_{1}∙o_{2} |
m_{1}∙o_{2} |
ℓ_{1}∙o_{2} |
< o_{2 + }p_{1}> |
< o_{2 + }o_{1}> |
< o_{2 + }n_{1}> |
< o_{2 + }m_{1}> |
< o_{2 + }ℓ_{1}> |

^{Lk}z^{ r} |
p_{2 }| p_{2} |
p_{1 }∙p_{2} |
o_{1 }∙p_{2} |
n_{1}∙p_{2} |
m_{1}∙p_{2} |
ℓ_{1}∙p_{2} |
< p_{2 + }p_{1}> |
< p_{2 + }o_{1}> |
< p_{2 + }n_{1}> |
< p_{2 + }m_{1}> |
< p_{2 + }ℓ_{1}> |

ᶄ |
ᶄ_{1} |
^{Un}z^{i} |
^{Un}z^{i} |
^{Un}z^{i} |
^{Un}z^{i} |
^{Un}z^{i} |
^{Un}z^{r} |
^{Un}z^{r} |
^{Un}z^{r} |
^{Un}z^{r} |
^{Un}z^{r} |

ᶄ_{2} |
ᶄ_{abs} |
p_{1 }_{| }p_{1} |
o_{1 }_{| }o_{1} |
n_{1 }_{| }n_{1} |
m_{1 }_{| }m_{1} |
ℓ_{1 }_{| }ℓ_{1} |
p_{1 }_{| }p_{1} |
o_{1 }_{| }o_{1} |
n_{1 }_{| }n_{1} |
m_{1 }_{| }m_{1} |
ℓ_{1 }_{| }ℓ_{1} |

^{Un}z^{ r} |
ℓ_{2 }| ℓ_{2} |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |
3 |
3 |
3 |
3 |
3 |

^{Un}z^{ r} |
m_{2 }| m_{2} |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |
3 |
3 |
3 |
3 |
3 |

^{Un}z^{ r} |
n_{2 }| n_{2} |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |
3 |
3 |
3 |
3 |
3 |

^{Un}z^{ r} |
o_{2 }| o_{2} |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |
3 |
3 |
3 |
3 |
3 |

^{Un}z^{ r} |
p_{2 }| p_{2} |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |
-4, 3 |
3 |
3 |
3 |
3 |
3 |

**(|****ℓ _{2}**

**|**

**/**

**|**

*p*

_{1}**|)**

*∙*

**ℓ**

_{2}[1] For our purposes the identification of such a source is not important, only that the event occurs.

[2] If not already obvious each respective basis scalar – the property itself – and events occurring in that interval are undefined. That is, **t _{2 }**–

**t**

_{1},

**t**–

_{3 }**t**,

_{2}**t**–

_{3 }**t**and

_{4}

*t*_{λ}

*–*are quantized scalars fundamental to the Spatial System

**t**_{4 }**S**

_{n}itself. It can be observed as a pseudo-surface, geometric property.

[3] Note that, technically speaking, all the mathematical relations of nature which characterize events via the traditional limit laws are physically meaningless. The full implications for the mathematical methods of nature are involved and outside the scope of this work. However, we shall require use of Natural Limits herein.

[4] These definitions are clear from the observations already made; namely, that without a necessary and sufficient paired event the axis locating the event is undefined. Conversely, with a paired event the axis obtains definition.

Relative to **ᶄ**_{1} there is a suitable parameterization that presents for any event **u**,** v**,** w, x**, **λ** whereby:

- Two changes have occurred (energy exchanges received by
**ᶄ**_{1}). - An associated basis vector
+*i|*u*|*+*j|*v*|*+*k|*w*|*|λ| is observable for each*ℓ***u**,**v**,**w**,**x**and**λ**respectively.

∃, upon observation of the initial basis vector ** i|u|**, there may be observed an appropriate parameterization,

**t**, both necessary and sufficient for any rate of change in any linearly independent axes measured in

**r**Space.[5] We note first that from

**u**→

**v**, relative to

**ᶄ**

_{1}, there is observed an interval of undefined Space ∃ for each

**’th component |**

*n***|, |**

*i***|, |**

*j***|, |**

*k***| that need not satisfy an equality constraint. ∵ of this potential inequality we have:**

*ℓ*₵_{iᶄ}* _{1}i*, ₵

_{j ᶄ}*, ₵*

_{1}**j**

_{kᶄ}*, ₵*

_{1}**k**

_{ℓᶄ}*₵*

_{1}**ℓ,**

_{iᶄ}*, ₵*

_{1}**m**

_{j ᶄ}*, ₵*

_{1}**n**

_{kᶄ}*, ₵*

_{1}**o**

_{ℓᶄ}

_{1}**p**as a source from which to provide a parameter for **r**Space. But any two parameters, one parameter such that **t**₵* _{1ᶄ1}1 *= ₵

_{1ᶄ1}**1***i*₵_{nᶄ1}**n ≤ ***i*₵* _{nᶄ1}n* ≤

*i*₵

*≤*

_{nᶄ1}**n***i*₵

_{nᶄ1}**n**and

*i***t**₵* _{1ᶄ}*1

*≤**i*₵

*≤*

_{nᶄ1}**n***i*₵

*≤*

_{nᶄ1}**n***i*₵

_{nᶄ1}**n**is necessary and sufficient for parameterization of ℝ^{4}.

Similarly, the other parameter is defined by

a parameter such that **t**₵_{4ᶄ}* _{1}4 *= ₵

_{4ᶄ}

_{1}**4**₵_{mᶄ}* _{1}m ≤ * ₵

*≤ ₵*

_{nᶄ1}**n**

_{oᶄ}*≤ ₵*

_{1}**o**

_{pᶄ}

_{1}**p**and

₵_{mᶄ}* _{1}m ≤ * ₵

*≤ ₵*

_{nᶄ1}**n**

_{oᶄ}*≤*

_{1}**o****t**₵

_{4ᶄ}

_{1}**4**is necessary and sufficient for parameterization of *i*ℝ^{4}.[5]

Thus, for **m**Space we have:

*i i|*

**u**+

*|**i*

**j|****v**+

*|**i*

**k|****w**

*| + il|*t_{im}

**;**

*|***t**=

**f (**

*|***)**

*u, v, w, λ*

*|*And for each of these basis axes in ℝ^{4} **r**Space:

** |u|** +

**+**

*j|*v*|*

*k|*w*| +**i*

**l|****t**

_{re}

**;**

*|**i*

**t**=

**g (**

*|***)**

*u, v, w, λ*

*|*Motivating our discussion for ₵**_{ᶄT} **< (

**∆**mc

^{2})

*****

_{ᶄT}**∆t**we may now characterize rates of change in the Spatial System,

_{ᶄT}

*S*_{ᶄ}*using the following relation:*

_{1}|** n**| /

*i*|

**| =**

*t*

*v*which **ᶄ**_{1} reads as the rate at which the basis axis ** n** is dilating with respect to the parameter

*i*

**t**. But we note that

*i|*

**t|**is also changing at the same rate as |

**t**₵|

*= |₵|*

_{1ᶄ1}**1***. That is, the relative rates of change for each*

_{1ᶄ1}**1****are observable by**

*n***ᶄ**

_{1}as ratios of velocities between basis vector change rates. This can be mirrored for the

**m**Space as well.[5]

[6] if this should concern the reader we will take this phraseology up later and give it a concrete definition.

[7] This is the first hint that time reversal cannot occur and that causality is preserved. This is also internally consistent since the number and allowed values of an observer’s degrees of freedom are predicated on its own kinematic history.

Therefore, from the reference frame of **im**, |*m*_{im}| imputes a corresponding basis axis, that is, it is the Natural Limit on *m*_{im}, ₵_{ᶄ}_{im}|*ℓ*_{re}|** ^{im}** and is designated

*i*_{re}. What we need is a parameter transformation so that one parameter can be used for all metrics in

*S*_{ᶄ}_{0}.[8] We will do that under the section entitled the First Act.

We now have cause to apply what we shall refer to as *The Mirror Test*: Testing our postulate that an **m**Space exists that is characterized by the mirror image of **re**, **ᶄ**_{1}, by measuring a parameter for **r**Space which is defined only in **m**Space, to the extent that our results agree with experiment, has proven the postulate.[8]

A rotation of -ϕ toward π / 2 results in:

- An increase in
*i|*| / |**t**| = |-*n*; ⇓ |*v|*|, ⇑*n**i|*|.**t** - A corresponding change in the proportion of undefined Space to defined Space

the latter due to the virtual “co-location” of a large number of observers versus one observer (see the previous discussion of momentum).

Rather than be overly pedantic we shall, from here on, imply the application of appropriate Natural Limits to all equations without specifying them explicitly.

∵ the relation above affects the parameterization of all real and complex ‘Space-like’ basis vectors it should exist wherever **m**Space is present. **ᶄ**_{0} observes that the rate of the event history of a **r**Space basis vector ** n** is given by:

** n**/

*i*|

**t**| =

*v*

_{d}and that **v*** _{d}* is sufficiently well defined for any value

*i*|

**t**| ≥ ₵

_{ᶄ}*, undefined for any other value and where*

_{0}**is the current maximum.[8]**

*n*[9] Note that we are comparing time in one frame of reference to time in another; **m**Spacewith respect to** r**Space and** r**Space with respect to **m**Space. We hope to input a rate of change in definition of a spatial axis and return a scaling factor that can be used to adjust our parameters. The program is straightforward: **ᶄ**_{1} measures rate change from two reference frames, each treated as a vector component. And we’d like it to be as general as possible since we suspect that these rates are key to understanding the overall system.

[10] The First Act is currently known as the “Gravitational Force” and is well approximated by the fundamental metric tensor of General Relativity. We shall see shortly how it acts like a “force”.

[11] So, relative to the spatial system *S** _{n}*, there exists at least one hypothetical spatial system implied by

*S**. We here denote it*

_{n}

*Q**and its effect is the wave of the First Act. But any real*

_{n}

*Q**is inscrutable and is therefore undefined relative to*

_{n}

*S**. It is implied but not demonstrable relative to*

_{n}

*S**as a cause necessary and sufficient, for two reasons:*

_{n}- The cause of the effect that is the expanding and/or contracting of
*S*remains unidentified._{n} - The properties of the spatial system
*S*do not appear to be organized randomly._{n}

[12] The unfamiliar symbol here is applied and proposed for convention because the indicated operation is unique, of a special nature and should stand out in any notation of formal logic wherever used. It should be understood to be a tensor using the Einstein Summation convention abbreviated ∃ i → r indicates a summation over order 5 or greater –depending on the lexicographical position – from index *i* to index *r*; i.e. *i **⇀** r* is defined as the Einstein Summation over *i, j, …, q, r*. To indicate contravariance and covariance distinctions, the author should use explicit Einstein Summation notation, otherwise all indexes are assumed covariant. Strictly speaking, this example would be called a definite check as the indexing is explicitly stated as “*i **⇀** r”*. But if the ASCII representation is immaterial we don’t see a need to always provide that if the order is given and the tensor type known. So, should the indexing not appear at all the tensor should be considered in a general context – as an indefinite check – and the indexing is implied by the given matrix order (which, by using a unique symbol, ensures symbolic consistency).

[13] If we also look at the integral form of the Langrangian (as a derivation from Hamilton’s Principle) we see immediately the First Act; that is, we see that we are integrating the sum of the rate of change in energy and the rate of change in that rate. This amounts to an equality between volume and rate change. The above equation re-frames the same argument as the rate of change in the Langrangian equals the rate of change in the rate change of u with respect to its parameterization, t. The Langrangian is indirectly referencing the very thing we’ve discussed regarding the definition of a spatial system *S** _{n}*; to-wit, the common proportion maintained between magnitude, rate and rate of rate in the event history of any observer

**ᶄ**

_{0}. The primary difference is that the Langrangian is a representation ignorant of higher dimensions at work, of the significance of the meaning of e and the intrinsic capacity of the check tensor to calculate energy change for

*any*force of nature. The Langrangian elegantly ignores the deeper details and simply ‘gives us the right answer’ in a conserved environment. Now we know why. We feel that this is one of the most significant connections between Acts and what we’ve known up to this point; one of the most elegant traditional equations is surprisingly similar to a complete equation.

[14] It amounts to a circularity: attempting to parameterize an event from **r**Space to **m**Space, then trying to parameterize a related event in that mSpace back to rSpace; merely returns us to the **r**Space in which we started and we are never able to measure a defined parameterization in **m**Space for the event α. It can be observed that the two events are identical in every regard, but it cannot be defined as such. So, again, there is no transformation definable between these variables and causality is preserved.

[15] With the exception that the First Act will do so twice per period.

[16] The result will be a scaling factor between the two parameters, a necessary calculation to proceed before much else can be done:

[17] And **r**Space, relative to that reference frame, is expanding/contracting faster than **m**Space, the rate being approximately 2.718 times faster. But the reverse is true when observed from the **m**Space reference frame.

[18] The variable names should be self-explanatory. Upon measuring ₵, *ᶄ** _{1}* interprets the findings as a unit square of side length 1 each side representing unit 1 values for corresponding parameters

**t**

_{ie}and

**t**

_{re}, sweeping out e

^{1}units of the ‘volume’ of the |

**|’th axis in one invariant unit time.**

*i*

*ᶄ**reports this as the natural velocity of event histories of a spatial system of order 3 (we will examine the order 10 case shortly).*

_{1}[19] We will begin this portion with a derivation of the Schrodinger equation. (probability densities, uncertainty).

strong and weak are bonded like e and m so that they both take one axis.

[20] If this condition is not satisfied, setting the Natural Limit on m will be required instead of ℓ. So, we must take the largest Natural Limit as the controlling limit.

[22] For electron orbits we know that:

mvr = nh / 2π

ðv = nh / m2πr;

ðh/t = h / m2πr;

ðm2πr = t; m = t / 2πr; Substituting,

ð₵_{tᶄ}_{0}**t **= m2πr; m = ₵_{tᶄ}_{0}**t _{R}** / 2π

*i*₵

_{tᶄ}

_{0}**t**

_{A}[23] Since we are using unit values our function does not have the form most are used to seeing. In order to make it clear what exactly **ᶄ**_{1}is observing in this context we’ll switch lenses briefly

[24] A heuristic might be in order at this point. The probability wave function above is a unit value based probability that a time measurement will be precisely *defined* to be some value,|**ᵯ**|_{m} , upon a measurement which may reduce the lack of definition generated over the interval |**ᵯ**|_{m} by forcing the observer’s own value of |**ᵯ**|_{m} as a reference frame for the time measured (presumably more precisely in the macroscopic) at **ᶄ**_{1} relative to another observer, **ᶄ**_{2}. For those interested in the ramifications for formal logic, this means that while uncertainty is physically meaningless the possibility of unpredictability, and hence true randomness, remains in a natural system, but not necessarily so in a system of formal logic.

The apparent back and forth nature of epistemological priority between formal logic and nature is no accident since the two are in fact the same thing. Neither is prior, both are identity. This is a very important point.

[25] The purpose of this exercise was to show the connection between two models. The method of using the Natural Tensor as above wouldn’t be a preferred method. It is also worthy to note that what we are encountering in this exercise is not probability, but a Natural Limit. If a region of Space is not sufficiently well defined then the ‘location’ of an observer in that Space is undefined. It is outside the scope of this work but we point out that a Natural Limit, as a cause for an effect, is a truly non-algorithmic process.

[26] We have two choices here. Depending on the context it may be correct to apply the change vector to one or both components of the comparison operator. If we know that the change vector being transformed has only components in **r**Space, then we do not apply it to the imaginary component and vice versa. If we know that it has components in both real and imaginary space then we apply the vector to both spaces. So, this is no different than what the tensor will do for us automatically, but we make this distinction here so the reader can follow the discussion.

[27] To explain the ‘forces’ of the Third Act, we begin with the construction of a basic matrix in which we attempt to identify all possible combinations of what we will call “spin”.

[28] The axes of rotation were established when the tangential basis axes were ‘created’ since rotation about some axis separately in rSpace and mSpace was required for that ‘motion’. This can be seen by having k1 observe what happens between the initiating and concluding events for this rotation scheme. Note that during this interlude a real axis about which the rotation is occurring is, by definition, orthogonal to its counterpart in mSpace. It must be orthogonal to its counterpart since the rotation has presumably already begun before the concluding event.

[29] We now include the effect of *which* axis of rotation is involved. Since each cross product will be 2 vectors producing one resultant, we can see from above that each basis has 3 possible candidates for matching up in a cross product. Since we also know that each space is matched with its own, we have

|**W**_{re}|*i*_{re} ⨂ |**W**_{re}|*j*_{re}

(Weak nuclear and like charges/poles must rotate here to repel)

|**W**_{im}|*m*_{im} ⨂ |**W**_{im}|*n*_{im}

(EM and Strong must rotate here to attract)

Note that 2πr = c

ð2π = c / r

and that for small orbits for which r ≈ |**₵**|_{i} for some ‘r’ on ** i** and for high arc speeds (energetic) the event history

**begins to collapse. We will now derive a representation for this behavior. Let observer**

*i*

*ᶄ*_{1 }measure the following quantities:

orbit radius, r ≈ |**₵**|_{i}

For both **z**_{re} and **z**_{im}. These will be the two axes about which *ᶄ*_{1} will rotate

Measure both while *ᶄ*_{1} is ‘moving’ linearly and while stationary.

Notes: motion affects particle states like spin does.

[30] And note that observers *ᶄ** _{1}* and

**experience a ‘force’ that, in the first equation, is ‘pulling’ them toward each other and in the second equation is ‘pushing’ them apart. If likes repel and opposites attract, we can see how with the appropriate permutation we can have either the first condition or the second, depending on the rotational axes involved in the rotation for each of the respective observers. Additional permutations will also reveal the different cases of magnetism and charge. We would expect a more or less random process involved and thus a relatively equal number of both types if summed over the spatial system. Of course, there is nothing like a ‘force’ here, but that is the illusion of direct observation. The foregoing is presently known as the electromagnetic force, which we shall add to our Natural Tensor shortly.**

*ᶄ*_{2}[31] Though purely heuristic, it would not be inaccurate to refer to the universe as a record of all that was, is and will be.

[32] Inquiring into the events **u**, **v**, **w** we note the following: prior to the observation of the aforementioned events we could say that all Space was undefined (not just illusory – but not even mathematically representable). If we again stipulate the necessary but sufficient view of definition we can say that **m**Space is not **r**Space. Though that statement is remarkably general it does offer some insight. But let’s start with some terminology clarification. Relative to **ᶄ**_{1}, we confirmed that Space was not observable until event **v. **But this means that that which **ᶄ**_{1 }already traversed, or that property already observed, is something but is not Space. Hence, we referred to it as undefined Space, **m**Space, imputed an imaginary coordinate system into it, and denoted it with ₵_{ᶄ}* _{1}*. This imputation is only valid ∵ the region of

**m**Space is a mirror reflection or image of

**r**Space. This is evident when we consider the observations of

**ᶄ**

_{1}by running the

*Gedanken*in reflection. The result is the same except that angular rotations are forward (not backward) and

**r**Space and

**r**Space are reversed.

**r**Space becomes particulate and

**m**Space becomes ‘spatial’. This rearward rotation makes

**r**Space, by definition, complex. So, we know we have the correct mathematical relation in complex numbers. If we were to trace each observer backward in time through all events we would presumably reach a point at which perceived Space collapses to the origin; that is, that it constitutes a

**0**matrix. We should be careful to note that an actual trace isn’t necessary, only that it could be calculated with a Natural Tensor by working backward from the initial conditions of ‘now’.

It should now be evident that all degrees of freedom in any natural phenomena depend solely on the relative histories of their participants, ** ᶄ_{p}**. From a practical standpoint, we characterize nature geometrically to the extent that all properties and quantities have a geometric meaning presented on a pseudo-surface that depicts the logic of nature. But we know, in reality, that this is a reflection of deeper properties inscrutable to an observer

*ᶄ*

_{n}[33] We point this out explicitly ∵ the ‘forces’ we are about to examine are central, so-called attractive ‘forces’ (we shall see that these ‘forces’ are all different manifestations of the same behavior). This is ∵ for |** i**| sufficiently large, we can take a general limit as an approximation and consider the local conditions to be endpoints on |

**|. Therefore, the spatial change appears as a shrinkage on |**

*i***| between two observers, not an expansion. The axis of shrinkage |**

*i***| is often denoted as a power of r, that is, as the power of the radius of a body. As**

*i***ᶄ**

_{1}continues its observation we will better understand the significance of the powers involved.

[34] Upon a complete understanding of the arbitrary divisions of both nature and formal logic the division therein vanishes. If established by experiment to the satisfaction of the stake holders, this paper and/or its followup work renders Physics complete. We await the same for Mathematics. A step forward would be to create a discipline, call it “Nature”, that sets the First Act as axiomatic and see what happens. It could take generations of study but is well worth it if the ideas we’ve put forth are tenable. Currently mathematical physics, usually explored in mathematics departments, is the closest exponent we have. A secondary proposal would be to just make it a separate discipline by just substituting in naming convention.

[35] This now leads us to a slight digression which better explains, in more concrete terms, what is so “special” about the 3 spatial axes and their parameter with which human beings appear to have a biological capacity to comprehend so readily. Though outside the scope of this work, we note briefly that, by definition, a delusion requires the substitution of a thing revealed for a thing hidden. If the least number of bases sufficient to define a thing hidden is 6, then whatever remaining bases exist, in no case less than 1 and in our case 4, are sufficient to sustain the delusion in a thing revealed, that is, the remaining 4 bases. Biological cognition may be so constructed as to follow the simplest path to a practical solution. Though we establish no hypothesis along these lines, we encourage biological research into this area

[36] We have two choices here. Depending on the context it may be correct to apply the change vector to one or both components of the comparison operator. If we know that the change vector being transformed has only components in **r**Space, then we do not apply it to the imaginary component and vice versa. If we know that it has components in both real and imaginary space then we apply the vector to both spaces. So, this is no different than what the tensor will do for us automatically, but we make this distinction here so the reader can follow the discussion.

[37] We have two choices here. Depending on the context it may be correct to apply the change vector to one or both components of the comparison operator. If we know that the change vector being transformed has only components in **r**Space, then we do not apply it to the imaginary component and vice versa. If we know that it has components in both real and imaginary space then we apply the vector to both spaces. So, this is no different than what the tensor will do for us automatically, but we make this distinction here so the reader can follow the discussion.

[i] Notice to the editor and readers

This is a preprint version. This work has been date stamped multiple times, is copyrighted by this author, it’s author retains irrefutable means of identification and all principles, lemmas and equations are copyrighted broadly as notional constructs as well. Any attempt to claim credit for them, regardless of how mathematically constructed or represented, will result in aggressive prosecution in all available jurisdictions. This work is a plagiary trap.

[ii] We will accept for now the possible ambiguity in definition of the “first event” since it will become clear as we proceed that this is in fact a reasonable assumption that we can better define then.