PROOF HIGHLIGHTS

Nowadays, some math experts are saying that we can not solve the famous twin prime conjecture without some new tools.  Well, the Kruse Super Sieve is likely a new tool for this job because it naturally organizes things about number factorization and number patterns to explain how/why any/all "qualified" random potentiality prime number patterns (including twins), which can be seen within a certain area on this sieve, become an infinite prime number pattern.  Here are some of the things that the Kruse Super Sieve has provided help with:

✓  The use of endless unique contiguous least common multiple segments to manage the realm of the infinite allows us to see that any/all "qualified" random potentiality prime number patterns are already infinite in count at their very first appearance along the X-axis of the Kruse Super Sieve.  Being "qualified" simply means that the count/magnitudes of the individual potentiality prime uni-factored numbers contained within the number pattern that you can see within a certain area on the Kruse Super Sieve must completely fit within the boundaries of some established least common multiple segment and it's associated factorization level in the ways that these two things are defined on this sieve.  Note that if you don't like any of the qualified number patterns that you can see inside of a particular least common multiple segment, then look at a larger/smaller segment (within it's associated factorization level) to find a qualified number pattern that you do like.  In any case, your chosen qualified number pattern will become an infinite prime number pattern for you.

✓  We can accurately count potentially prime uni-factored numbers (or their patterns), in their various stages of factorization, within the least common multiple segments and their associated factorization levels on the Kruse Super Sieve.  This information allows us to conclude that the qualified potentially prime uni-factored number pattern which you first saw in that certain area on this sieve will always remain infinite in count (but at a lesser non-zero rate of occurrence) in spite of the endless strike outs of that same uni-factored number pattern by an infinite number of prime number factors along the Y-axis of the Kruse Super Sieve.

✓  As increasing factorization levels and expanding least common multiple segments on the Kruse Super Sieve give birth to more potentially prime uni-factored numbers or more potentially prime uni-factored number patterns, this sieve allows us to realize that any/all of the infinite unique uni-factored numbers (or their patterns like twins/sextuplets), which are spread out along the X-axis of this sieve, are all in their permanent fixed locations forever and that they just exist there with their unchanging values all the while that factorization levels progress towards infinity along the Y-axis of this sieve.  The strike outs of potentially prime uni-factored numbers (or their patterns like twins/sextuplets) along the X-axis, which occur during the endless progressive factorization levels along the Y-axis, do not change this situation regarding the location and the values of these infinite unique uni-factored numbers (or their patterns like twins/sextuplets).  So in order for prime numbers (or their patterns like twins/sextuplets) to be finite in count, ALL potentially prime uni-factored numbers (or their patterns like twins/sextuplets) on the Kruse Super Sieve, greater than some proposed largest prime number (or prime number pattern), must be immediately and completely struck out forever along the X-axis of this sieve towards infinity.  This is the basis for the TRUE conditional statement "(E)" (found in Exhibit "E") which declares that: "IF confirmed prime sextuplets are finite in number, THEN uni-factored sextuplets are finite in number".  The logical contrapositive of this same TRUE statement worded as: "IF uni-factored sextuplets are infinite in number, THEN confirmed prime sextuplets are infinite in number" must have the same truth value of "TRUE".  Since we know that uni-factored numbers (or their patterns like twins/sextuplets) ALWAYS remain infinite in count on the Kruse Super Sieve, then we also know that prime numbers (or their patterns like twins/sextuplets) are also infinite in count on this sieve.

✓  Realizing that the lowest valued uni-factored sextuplet contained within a least common multiple base segment will always eventually be confirmed as a prime sextuplet because there are no lower valued uni-factored sextuplets available within any template segments to act as multiplicands with any prime number factors along the Y-axis to cause any strike outs of this same lowest valued uni-factored sextuplet.  When this same lowest valued uni-factored sextuplet begins confirmation within some future expanded unique base segment, then the next larger unconfirmed uni-factored sextuplet present within this same future base segment will then become the lowest valued uni-factored sextuplet and this process goes on forever with ever expanding unique base segments.  Since there are an infinite number of expanding unique base segments on the Kruse Super Sieve which contain unique uni-factored sextuplets within their boundaries, then there are an infinite number of lowest valued uni-factored sextuplets available for eventual confirmation as an infinite number of prime sextuplets.  Each prime sextuplet shown in a typical list of traditional sextuplet patterns held the status of being the lowest valued uni-factored sextuplet contained within some collection of expanding unique base segments on the Kruse Super Sieve during the strike out activity of one or more prime number factorization levels.

✓  Because of the inherent checks and balances naturally incorporated into the design of the Kruse Super Sieve, we can be 100% confident that the infinite nature of qualified number patterns will never be compromised during the normal factorization process on this sieve.  Challenging certain aspects of this sieve does, in fact, lead to contradictions such as when you try to assume that you can limit prime sextuplets as being finite in number while, at the same time, you allow uni-factored sextuplets to be infinite in number.  Working through this challenge results in discovering that prime sextuplets are infinite in number which contradicts the original assumption.

✓  Many other proof details have been developed using some other strategic ideas along with the productive features of the Kruse Super Sieve.  Things such as: (1) the active/inactive factorization areas and the square root function curve overlays; (2) the template segment protocol for multiplication assistance with factorization events; (3) the "closed system" properties of "(6n +/- 1)" type numbers; (4) that each prime number factor can only make one strike out within the linear span of a uni-factored sextuplet thus treating sextuplets like a single object for counting purposes; (5) the unique target segment strike out remainders modulo (the size of the template segment) are equal in count/magnitude to the unique template segment uni-factored numbers; (6) the substitute pseudo-multiplications showing that we can use the simple post factorization congruent relationships between the unique target/template segment uni-factored numbers to avoid the actual complex nature of the multiplication strike outs; (7) the rule that a lower valued unique uni-factored sextuplet within a template segment will participate in the strike outs of six other larger valued unique uni-factored sextuplets from within a target segment; (8) the related but separate number confirmation process on the 45° line (y = x); (9) developing conditional statements and logical sequences showing that prime sextuplets are infinite in number.