I wrote two unrelated blog posts this morning, one about the math paper H = W and one about a joke putting numbers into the D&D alignment matrix.
I used Grok to look up the reference to the H = W paper, and to confirm that the alignment matrix originated with Dungeons & Dragons [1]. I didn’t start a new chat when I asked a new question, but Grok felt obligated to connect the two queries. Near the end of the response to my D&D query, Grok added a section “Connection to Sobolev Spaces Query” with the following content:
While the alignment system and Sobolev spaces (from your previous query about the “H = W” paper) are unrelated mathematically, both reflect efforts to create structured frameworks for complex systems—alignments for character ethics, Sobolev spaces for function analysis. The “Neutral” alignment, like the regularity of distributions, represents a balanced or intermediate state, bridging extremes (Law/Chaos or regular/singular distributions).
Emphasis in the original.
On the one hand this is nonsense. On the other hand, Sobolev spaces and the D&D alignment matrix are structured frameworks for complex systems!
Grok’s response reminds me of when I would doze off while studying and my brain would find marvelous nonsensical connections between everything that I had been studying, such as a way to integrate French verbs.
Aligning functions
Classifying functions into the D&D alignment grid is silly, but what if we were to do it anyway? Here’s my quick take.
The good and lawful functions have classical definitions. C is the space of continuous functions, C² the space of functions with two continuous derivatives, and C∞ the space of infinitely differentiable functions.
The function W classified as chaotic good is Weierstrass’ example of a nowhere differentiable, everywhere continuous function.
The true neutral functions Wk,p (no relation to the Weierstrass W) are the Sobolev spaces from the earlier post. The larger the k, the more regular the functions. Lp is Wk,p with k = 0. These functions are not necessarily defined pointwise but only modulo a set of measure zero.
The function 1ℚ is the indicator function of the rationals, i.e. the function that equals 1 if a number is rational and 0 if it is irrational.
Here δ is Dirac’s delta “function” and δ is its nth derivative. This is a generalized function, not a function in any classical sense, and taking its derivative only makes it worse.
[1] The original 1974 version of D&D had one axis: lawful, neutral, and chaotic. In 1977 the game added the second axis: good, neutral, and evil.
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