The Fundamental Theorem of Cadoggle – as stated above – can be explained as follows with the backstory included:

• Cadoggle is a nickname for a classic game of Word Ladder attributed to Lewis Carroll where two words are bridged letter by letter using only valid words from a given dictionary. Put another way:
• Cadoggle is a word game where you aim to bridge the difference between two given words of the same length in the minimum number of steps by changing the letters one by one to create a chain of valid words forming each step
• The name started as The Cat-Dog Puzzle – which was CS Weaver’s nickname for the Word Ladder Game
• The term Cadoggle was coined by CD Weaver in Portland, Oregon in 2022 as a dual reference and portmanteau of The Cat-Dog Puzzle and Wordle – which took the world by storm in 2021 – 2022 amidst the Covid-19 pandemic that was happening at the time. 
• Other names for the game are Doublets, Word-links, Change-the-Word Puzzles, Paragrams, Laddergrams,and Word Golf
• The Word Ladder game is described in wikipedia as “A word ladder puzzle begins with two words, and to solve the puzzle one must find a chain of other words to link the two, in which two adjacent words (that is, words in successive steps) differ by one letter”
• Given that alphabets are numerical by nature Cadoggle – Word Ladder -etc – creates a mathematical branch which is subject to its own set of mathematical inquiries, theorums, proofs, etc.
• Cadoggle math is a subranch of the mathematical branch of Combinatorics 
• The Fundamental Theorem of Cadoggle is a logical theorum about the game conceived by CS Weaver of Ann Arbor Michigan and posted in 2022
• The Fundamental Theorem of Cadoggle (FTC) states “If you have two words that differ in n letters, it takes at least n steps to change one to the other, and if you can solve it in exactly n steps, youvcan do this by replacing each beginning letter by its corresponding end letter in some order.”
• For example – if you have two words such as cat and dog – which differ in all three letters – then to bridge them you have to change each letter at least once – maintaining the rule that each word used in the bridge must be in the given dictionary or list – of words
• In other words – there is no way to bridge the gap between cat and dog in any less than 3 changes – given that there are 3 letters different between cat and dog
• FTC provides the minimum number of letters that need to be changed  -and also states that each letter must change at least once. So this provides guidance to the player on how to get from word 1 to word 2
•  A corollary to this is that for two words that can be connected in n steps and differ in n-1 letters any solution in n steps requires changing the original n-1 positions that differ and leaving alone the letter that is the same in both words.