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“Finite or Infinite is not the question; it’s not which side of the room you’re on, it’s what’s in it”.

 

 

Before we get started  on our adventure,  if anyone decides they  like what they see on here and want to make a contribution, I would very much appreciate any amount that you choose. You can do so by clicking the link below.

 

 

Thank you very much

 

 

Enter N (1–200):
Choose transform: Root 1 (identity) Root 2 (square root) Root 3 (cube root) Root 4 Root 5 Root 6 Root 7 Root 8 Root 9 Root 10 ln(N) ln(ln(N))
Generate → PAIR

 

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Here is a peak at something else I’ve been working on below.

 

Whole Theory

 

 

 

Now to something else:

One morning as a small mental math exercise I was curious about both, non integer factorial values and finding the factorial for a relatively large number, since it’s more probable that this “factorial root” would   be a non integer for most numbers, especially the larger they are, I set out to make one. Despite ankle pains, (hey math is a great passion of mine so a great relaxation as well), I decided to make a basic formula. Now based just on first principals I came up with a simple linear scaler, but decided a log linear one would be more accurate.

 

Here is that formula:

 

r = n+(ln N – ln n!) / ln(n+1)

n!<N<(n+1)!

Formula Created by Russell B

 

where r=the factorial root, N=the number we are looking for the factorial of,  and n is the largest expanded factorial<N

 

That got me curious,  because this formula was so easy to create,  what else already existed, and I learned of the inverse gamma function, which was designed to give exact results, but is frequently looked as reasonably math heavy. My little expression gives quite accurate results for unimportant day to day activities,  especially considering its comparative lack of great complexity.

 

Let’s compare it to inv gamma:

Enter N: Run → Roots

N	Log Scaler Root	Inverse Gamma Root	Δr

Some people may have heard of a google plex or Graham’s number,  which a far larger than “ordinary” numbers. I was thinking about a possible growth rule myself, and created one that I found interesting.

 

I call it PIE CHEF, it’s a pretty quick growing function.

Acronym: PIE = Pi + Irrational + Euler’s number; CHEF = Cyclic Hierarchical Exponential Factorial

 

Abstract

 

PIE CHEF is a giant‑class operator law defined by a factorial‑cadence exponential coil and an irrational digit alignment test. It runs on two clocks: a dynamic loop end value (W) that drives the coil forward, and a static streak requirement (w) frozen at the first match (but reset to (W) on failure). Each cycle promotes itself until the frozen streak is satisfied, at which point the process halts finitely.

Step Structure (Coil Backbone)

The coil is built step by step, nesting each result into a descending exponential tower. The initial run uses 104 steps (a-z, a-Z, A-z, and A-Z…):

(a = 2^1)

(b = a^{2^1})

(c = b^{3^{2^1}})

(d = c^{4^{3^{2^1}}})

(e = d^{5^{4^{3^{2^1}}}})

… continuing through the alphabet …

up to the 104th step (Z).

The value at the end of the coil is the dynamic loop end value (W).

 

The process is

Output (W): 

Evaluate the coil to produce a decimal string (W).

This is the “establishment” cycle.

 

Compare the first (|W|) digits of (W) against the first (|W|) digits of an irrational constant (π by default, but any irrational is valid).

If no match occurs, then same process:

New cycle,  step count W from precious cycle and so on.

Static Freeze Law:

On the first successful match, freeze that (W) as static (w), the required number of consecutive matches.

If a later cycle fails before the streak is satisfied, reset (w := W) (the most recent output) until the next successful match.

 

Promotion Law: 

After each cycle, set the step count for the next coil to the current (W). Thus the coil length is always dynamic.

 

Halting: 

Continue until exactly (w) consecutive matches occur. The final (W) is the output of PIE CHEF.

 

Growth Rules

Rule 1 — Dynamic promotion: Each cycle’s output (W) becomes the step count for the next coil.

Rule 2 — Static freeze: The first successful match freezes (W) as (w), the streak requirement.

Rule 3 — Reset on failure: If a cycle fails before the streak is satisfied, reset (w := W) until the next successful match.

Rule 4 — Closure: The process halts exactly when the streak requirement (w) is met.

Rule 5 — Universality: Any irrational constant may be used for comparison; the law is invariant under substitution.

 

Example Trace (π)

Cycle 1: (W = 3) → matches “3” → freeze (w = 3), step count = 3.

Cycle 2: (W = 314) → matches “314” → streak = 2, step count = 314.

Cycle 3: (W = 31415936) → matches “31415936” → streak = 3, step count = 31415936.

Halt after 3 consecutive matches (static (w = 3)).

 

Features

Irrational universality: Works with π, e, φ, ζ(3), √2, or any irrational expansion.

 

Dual clocks:

Dynamic (W) = loop end value, promoted each cycle as the new step count.

Static (w) = streak requirement, frozen at first match but reset to (W) on failure.

 

Finite closure: Always halts once the streak requirement is satisfied.

 

Factorial cadence: The “F” in CHEF of course refers to the integer‑stepped rhythm of the coil, not a literal factorial at each stage — the factorial grammar is structural, I thought it had a nice “feel” to it.

 

*I have since developed a different version of PIE CHEF. It was  soon after the one presented above, making the first one effectively zero size in comparison.

 

 

Here it is:

 

☆PIE CHEF! — Summary (STAR reapplication count corrected)

Coil (Factorial‑Exponential Tower)

· Initial step count: 104!
· Steps (first pass only, no matching):
a_1 = (2!)^{1!}!
a_2 = (a_1!)^{3!^{2!}}!
a_3 = (a_2!)^{4!^{3!^{2!}}}!
… for 104! steps → yields first W.
(All W are implicitly factorialised at every occurrence: W to W!.)
· This first cycle is ignored for matching. It is purely generative to establish the initial W and reveal internal structure.
· Next cycle: step count becomes W!; the coil is rebuilt, and now the real matching process begins.

Even‑Number Rarity
Every W is even — a structural consequence of the initial 2^1 base.

Parity‑Induced Rarity
Because all W are even, their decimal strings always end in an even digit. When matching against an irrational’s digits, the chance that a randomly selected target string begins with an even digit is roughly 1/2 (assuming uniform digit distribution). This compounding 1/2 factor at each matching stage significantly increases the rarity of a successful chain — the process must repeatedly align not just a specific digit sequence, but one that is weighted towards even starts. (This is not the only rarity factor, but it amplifies the difficulty.)

STAR Amplification (applied to the decimal string of W)

1. Substrings: All contiguous substrings, forwards and reversed; palindromes counted twice.
2. Digit transform: 0 \to 9, 1 \to 8 (others unchanged).
3. Factorialize: Each transformed substring → integer → factorial.
4. Ascending tower: Sort values ascending; build nested exponential tower → T.
5. S = T^{T^{^{^{T}}}} (a tower of T copies of T).
6. Factorial stack: S! reapplied S times (factorial tower of height S).
7. Reapplication count = first STAR output of the current W:
· Within each coil step, STAR is reapplied a number of times equal to the first STAR output obtained from the current W.
· Between coil cycles, the full STAR treatment is also reapplied that many times (value = first STAR output of the current W).

Multi‑Irrational Gauntlet (begins after first matching cycle with pi)

· After pi matches establish static w, the same digit string is reused for the next irrational.
· Number of irrationals required = current W.
· Each new irrational must match its first |W| digits on the very first attempt with the current W.
— Success → move to next irrational.
— Any single mismatch → irrational counter resets to 0 (start again from pi), but W itself never decreases; it only grows.
· The static streak requirement w (factorialised to w!) stays locked from the first successful match; only the sequence of irrationals restarts.

Final Closure
After all irrationals (exactly W of them) have been passed without a single failure, the final successful W must be repeated as a full successful cycle W times with zero failures. Then ☆PIE CHEF! halts.

Guarantee of Finiteness
Under the assumption that the irrationals used are normal, every finite digit string exists somewhere. Hence a perfect chain — though extraordinarily rare — must eventually occur. The process always halts, producing a finite, even, unimaginably large number.

 

**This is a custom scheduler to generate the irrationals for ☆PIE CHEF! that I created.

 

Decimal points are ignored and the first digits starting from the first one are used  (including the one(s) before the decimal point).

*First pi, then e, 2^1/2, phi, or whatever chosen, then:

ln x, ln(x^2), ln(ln(x!)), ln(ln(x!)^2), ln(x!)^(1/2), ln(ln((x!)^(1/2)), ln(ln(x^(1/4)), ln(x^(1/4)).

 

x=1,2,3,4,…….. 

All should be irrationals if any were not to be, then it can be skipped to the next one.

 

 

Explore the finite horizon with an infinitive walk.

 

 

Just a notable curiosity,  it may not have a deeper meaning other that structural numerical alignment, but it is interesting nonetheless:

 

Pi^(sq root 2) is very close to the value of e^phi.

Another number that I created in about a second that felt right, (when I was considering near-intergers related to irratuonals) is Phi^(7(4^1/3)).

A couple close to pi:

((2^1/2)^phi)/(ln(2^1/2)(ln5))

And 306^1/5

 

 

Off topic, but here is an acronym:

Appearing Condensed Revealing Obvious Names You Memorize 

 

This is the first entry ive had in a while,  here is something interesting.

 

This was similar to that time when I wanted to know the inverse of any factorial,  integer or otherwise. 

Another morning, I was mentally exploring relationships of the differences of consecutive ratios of towers with operands and bases of the same value.

I did the computatuon mentally as well, and I noticed that the value quickly converged from above to barely under 2.72 and stabilized slowing significantly.

This , I immediately recognized at what the rest of the world calls “e”, and I had found it independently , not even having set out to find any certain value, especially a well known constant.

 

((x+1)^(x+1))/(x^x) – (x^x)/((x-1)^(x-1))

 

Example:

4^4/3^3-3^3/2^2

Quick converging

Also:

(x^(x+1))/((x+1)^x)-((x-1)^x)/(x^(x-1))

= 1/e

Also quickly converging

Example:

(4^5/5^4)-(3^4/4^3)

 

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