So here it is, the pinnacle of my research effort thus far. I'll start by the definitions:

**Definition 1.** Let and be surfaces so that . The star operator takes two surfaces and creates another in the following way:

with the central transformation being defined by

and the last transformation that takes

we will call

. Thus

**Definition 2.** Define a continuous, bounded surface , with , and let be true regardless of the value of . In other words, integrating such surface with respect to yields the uniform probability distribution , . We will call this a *strict Pasquali patch*, and such is intimately related to probability notions. With , we have a more general definition for a *Pasquali patch*.

**Construction 1.** Let , a function which consists of a finite sum of pairs of functions of and . In the spirit of conciseness, we omit the transpose symbology, thusly understanding the first vector listed in the dot product as a row vector and the second vector as a column vector. Then is a *Pasquali patch* provided

and

. Thus, we may choose

arbitrary functions of

,

arbitrary functions of

, an

th function of

so that

, and

We may write the normalized version as:

and again observe that the unit contribution to the integral of the

*Pasquali patch* is provided by

, so that

.

**Claim 1.** *Pasquali patches* constructed as by **Construction 1** are closed.

**Proof.** To make the proof clear, let us relabel the normalized version of **Construction 1** as

with

so as to manipulate the equation more simply, and

with

and

. Then

with

,

, and

. We're not too concerned of the

*form* of the resultant star product as much as its

*structure*. Observe

can be folded back into function vectors

and

. Thus the structure of

**Construction 1** functions is preserved when we multiply one by another, showing closure. Of course the property of

**Construction 1** being

*Pasquali patches* means

is closed under

*that* property, and so is a

*Pasquali patch* also, as can be seen when we integrate across

:

and the unit contribution is given by

. \qed

**Claim 2.** *Pasquali patches* constructed as by **Construction 1** have powers:

with

and

**Proof by Induction. **First, using the formula observe

and the second power

which is exactly what we expect from the definition of

**Construction 1** and

**Claim 1**. Next, let us assume that the formula works and

Let us examine

term by term upon dotting. The first dot the first term is:

The first dot the last term is:

The last dot the first term is:

The last dot the last term is:

The middle dot the first term is:

Finally, the middle dot the last term vanishes:

Putting all this information together we get:

\qed

**Claim 3.** It follows that

and

,

bounded, both conditions necessary and sufficient to establish that such a limiting surface indeed exists (convergence criterion). Furthermore, we check that this is indeed a

*Pasquali patch*.

**Proof.** To reach a steady state limit,

Next, the steady state limit must be solely functions of

, so the functions of

must vanish at the limit. Thus, it follows that

. We have now established bounds on

, which happen to be exactly the radius of convergence of the geometric series:

and

gives the desired result:

As a check, we integrate across

to corroborate the definition of

*Pasquali patch*:

**Claim 4.**

is the eigenfunction corresponding to eigenvalue

of all

**Construction 1** functions, through each power independently.

**Proof.** An eigenfunction has the property

where the eigenfunction's corresponding eigenvalue is

. The claim is more ambitious, and we will show that

for any

. The left-hand side is

Observe the first term dotted with the middle and last term produce

which annihilates the results, so that the only relevant term is the first dot the first:

The second term dot the first produces:

The second term and the second:

and the second by the last term gives

Factoring gives

The parenthetical part of this last formulation is equivalent to

within the bounds already established for

, and the result of the star product is

as we wanted to show. \qed

**Claim 5.**

is a constant, is the eigenfunction corresponding to eigenvalue

of

**Proof.** The eigenfunction equation is suggestive of what we must do to prove the claim: . We must show that, starring the eigenfunction with , we obtain times the eigenfunction.

Thus:

\qed

Part I v27