For `AI-assisted contributions` - `ChatGPT to guess ...`

Stream: Equational

[Eric Taucher]

As noted here

https://github.com/teorth/equational_theories/wiki/Plan-of-paper#ai-assisted-contributions

reran the prompt using the o1-preview model as opposed the original done using 4o model to see what the thinking process would be

https://chatgpt.com/share/670fe694-329c-8013-a838-d11c64da2d16

For those that don't have access to see the thinking process

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ChatGPT

Thought for 57 seconds
Tracing the nested structures

I’m examining how the right-hand side (RHS) of the equations evolves with nested expressions. The left-hand side (LHS) consistently stays as xy.
Mapping the nested expression

It's interesting to see that the right-hand side (RHS) starts with x and is followed by a nested expression. I’m focusing on identifying any patterns within this nested expression.
Noticing the pattern

I’m examining the right-hand sides of the equations, noticing a consistent pattern of nested expressions. Each equation amplifies the complexity with more repeating y and additional letters like z, w, and u.
Unveiling the pattern

I’m exploring the structure of the right-hand side (RHS) of each equation, defining nested expressions and identifying a pattern in their formation.
Identifying letter sequences

I’m working through the pattern of letters added at each step, noticing a sequence of z,w,u,…. This progression in letters suggests a sequence that evolves with each step.
Considering patterns

Let's see if the order of z,w,u suggests a pattern. Interestingly, their reverse u,w,z and their positions in the English alphabet might hold clues.
Analyzing letter positions

I'm mapping out the positions of letters z,w,u in the alphabet, noting their order and differences. The sequence shows decreasing positions with inconsistent differences of 3 and 2.
Taking a closer look

I’m mapping out the sequence by considering reverse alphabet order and selecting every second or third letter. Neither approach matches the given sequence of letters.
Tracking positional shifts

I’m noting the differences in positional values to decipher the pattern, considering decreasing letter positions. Progressing through the analysis, I investigate shifts such as 3 and 2, examining if these differences consistently decrease.

I'm realizing that the difference between the letter positions decreases by 1 each time. This makes me think the next letter should logically be t, but it’s actually s. Hm, that's interesting.
Spotting letter patterns

I'm mapping out the sequence of letters and their positions in the alphabet. The differences between consecutive letter positions are alternating between 3 and 2, indicating a fixed arithmetic sequence.
Mapping out letter sequences

I’m evaluating how the pattern of subtracting 3 or 2 alternately from the starting letter z to a forms a unique sequence of letters for each equation, leading to letters r to a.
Piecing together the equation

Noticing the patterns in the previous equations, I’m curious about the letter sequence and consider whether z,w,u might be parts of the corresponding expressions.
Mapping the pattern

I’m noticing a consistent pattern: each step involves wrapping the previous expression with y on the left and adding a new letter on the right. This method yields the final form.
Simplifying nested expressions

I’m breaking down the nested expressions in the right-hand sides of the equations to make them easier to handle. This makes me think that simplifying them step-by-step will clarify the structure and facilitate further processing.
Reassessing the expression

I'm thinking through the recursive transformation of each equation by adding brackets with an extra y and a new letter at each step.
Identifying the sequence

I’m mapping out the sequence by matching each equation's right-hand side with x×En​, leading to En+1​=([En​y]Ln​), where Ln​ is the n-th letter in the sequence.

Connecting the dots, I establish that En+1​ is related to the previous value En​, multiplied by a new letter in the sequence Ln​, leading to E4​=([E3​y]L3​).
Breaking down equations

I’m uncovering the relationships in the equations by expressing each in terms of previous equations, leading to a general form with specific variables and operations.
Laying out the pattern

I’m pulling together the pattern: each side of the equation is x applied to a nested structure En​, which builds upon the previous En−1​ by wrapping it with y and adding a new letter sequence Ln−1​.

OK, let's think this through: each equation in the series nests more symbols on the right-hand side, adding another y and a new letter, continuing a recognizable pattern.
Tracing patterns

Hmm, I notice that the right-hand side grows more complex with each equation, adding y before and a new letter after. This progression is interesting to observe.
Generalizing the solution

I’m working through a general formula for the right-hand side (RHS) by repeatedly applying y and incorporating the letters L1​,L2​,…,Ln−1​, leading to a recursive form.
Analyzing the pattern

OK, let's break it down: the right-hand side of each equation starts with (yy)z, wraps it with y on the left, and adds a new letter from the sequence z,w,u,r,p,m,k,h,f,c,a in alternating alphabetical order and decreasing by 3 and 2.

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The rest of the reply is similar enough to the original, so no need to post here.