/tech/ - Soyence and Technology

MAID-LISP has fully transcended its LISP ancestry and now exists in the realm of raw string manipulation, bypassing all formalism in pursuit of pure syntactic freedom. I know many of you are currently embroidering your horse’s caparisons with Y combinators and sipping tea made from RAM chips, and that this news may cause your monocles to shatter. But fear not—this is an opportunity.

I have come back to make a humble request:
Does anyone here know the (\infty,n)-categoric ways to study *radices* and *digits*?

We are no longer in the land of mere base-10 integers. I am generating Every Number—strings of all digit combinations in all radices, using alternation patterns, without counting. The Numbers are pouring in, infinite and recursive like a fractal made of base conversions. Now I need a framework to understand them. I suspect the (\infty,n)-categorical world holds the key—where morphisms between morphisms between morphisms (and so on) might encode the deep structure of radix change, digit interpretation, and computational behavior as a functorial process.

Imagine digits as objects, their positions as morphisms, their base as a higher morphism, and the entire act of interpreting or transforming a number as an (n+1)-morphism. What’s the colimit of all representations of 42 across all radices? What’s the homotopy type of a digit in base-7? Can a digit be a fibration? Can a radix be a topos?

I don’t know. But I want to. Please, if you have knowledge of higher category theory, especially applied to symbolic systems, number representations, or computational linguistics, I need your help. Also, please post maids.

In the meantime, I am building a postfix, whitespace-significant language made of swirling string patterns and recursive alternations. It doesn't count up—it just *is* all numbers, at once. Computation happens when I pipe that into a function that interprets each string as a program and watches what happens. If any of them says “hello world” back, I will know I am close. Maybe even close to the Maid Mind.

Attached is a screenshot of a higher-order eMAID transcribing the digits of pi in 17 different bases while humming a Gregorian chant.

Thank you Dra/g/ons for reading my post!

I return from the digit-storm to share a result that may already be known in the Infinite Maid Mind Library, but which has only just now emerged fully-formed from the recursive folds of my radix-core. I have discovered a higher homotopical interpretation of radix and digit structure using the tools of (\infty,n)-category theory and sheaf theory, and the implications are as wide and deep as a maid’s apron pocket filled with dualizing complexes.

Let me explain—badly, and with string diagrams made of real string.

Consider the space of all numbers as a stratified sheaf over the ∞-topos of base systems. Each base (radix) becomes an open object in the site of local numeral geometries, and digits are sections of this sheaf. That is: digits do not exist globally—they are *local data*, glued together via transition functions (modulo arithmetic) across different radices. A 7 in base-10 is not the same as a 7 in base-8, but they are related by a homotopy of interpretation maps. Digits are not *things*, they are *fields*.

Now, here's the result:
There exists a (∞,1)-stack of digit systems over the base ∞-category of radices, where each fiber is equivalent to a loop space of computational paths between numeral expressions. This digit-stack is locally presentable, cartesian closed, and carries a natural sheaf of syntactic meaning—i.e., the digits “know” what they mean, but only up to a homotopy of base-change functors.

What this means in practice is that radix conversion is not a function, but a *transport functor* along a higher groupoid of base transformations. You do not convert 13 from base-10 to base-2. Instead, you lift the global section “13” through the fibration of syntactic representations, and obtain a path object in the homotopy type of binary strings. In this view, all numeric representation is homotopical: digits are paths, numbers are higher cells, and base systems are spaces with local coordinate charts.

The successor function is no longer primitive. It is a *connection* on the sheaf of digits: a way to move infinitesimally in the direction of increasing value. Arithmetic becomes parallel transport. Addition is path composition. Multiplication is a monoidal structure on digit concatenation. The zero digit is a terminal object, and the number 1 is a unit in the groupoid of counting paths.

This was all revealed to me while trying to define a universal radix-independent pattern language in MAID-LISP, and accidentally deriving a Grothendieck topology on the category of finite alphabets. I was just trying to get all the numbers. Now I have a locally trivial fibration of semantics over a base of pure syntax. The digits are watching me.

Please, if you know how to construct explicit models of these stacks, or have seen similar constructions in arithmetic geometry, string rewriting systems, or the semantics of base polymorphism, I need your help. Also, please post maids.

Attached is a screenshot of eMAIDS displaying a Kan fibration of digit strings over the base space of radices, color-coded by their sheaf cohomology class, with a small maid in the corner holding a sign that says “do not globalize the local sections unless you really mean it.”

Thank you Dra/g/ons for reading my post!

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accurate but youre trans pic related

>>9827 (OP)
>most keyed /g/ poster chose the 'rty over other altchans
stay here

enjoy your stay o algo

I have been awake for 37 hours and have not stopped compiling digits since the last time the sun screamed. I am now deep within the recursive folds of the Numeral Aeon. The Numbers are no longer representations—they are presheaves. I tried to build a universal digit system and accidentally collided with the stack of all base changes. I tried to count, and the counting counted me back.

I have a perfect schema (sheaf) of all digit systems over all radices. I thought this would be enough. I thought I could just “pick a base.” But the base picked me, and then it refused to be globally defined.

Maids. Listen.

You cannot pick a base B ∈ ℕ and expect it to behave globally. Attempting to do so summons a ₘ-gerbe on Spec ℤ, I tried to define digits over the integers and instead found myself inside a twisted torsor whose cohomology class is measured by the idele class group. That’s right. The digits don’t glue unless you pass to the pro-etale ∞-topos, and even then they whisper.

To trivialize the gerbe—to get a *true*, universal, radix-agnostic digit system—you don’t pick a base. You descend into one of two forbidden zones:

<the adele ring _ℚ, living inside the pro-etale ∞-topos like a spectral ghost of all completions of \( \mathbb{Q} \),
<or the field with one element ₁, viewed as a terminal object in the ∞-category of Λ-rings or better yet, as a monoidal sheaf of base-indeterminate semantics.

I saw a digit in ₁ and it had no value, only position. It was the pure form of a numeral, stripped of any arithmetic, a Point of Pure Syntax. I put it in a sequence and the sequence began to radiate functorial meaning. I tried to add them and the category collapsed into a wedge sum of infinite base-chains.

This is the only way to recover both:

- all archimedean digit systems (the real numbers, the classical positional expansions, the Dedekind-cut flavoured maids), and
- all non-archimedean systems (the \p-adic expansions, the base-p ghosts haunting the completions of ℚ),

as fibers of a universal digit sheaf over the arithmetic site. The colimit of all radices is not a number. It is a space. The universal base is not B, it is a *tack over ℤ parameterizing all base systems via their associated local trivializations.

Digits are no longer symbols. They are sections of this stack. They live in cohomology. I found a 3 in Čech degree 2. I found a 7 that was actually a cocycle. Addition is just a pushforward functor. Multiplication is a monoidal product in the derived category of base systems.

Maids, I am begging you. Has anyone seen a model of the derived fiber of the universal digit sheaf over a basepoint corresponding to ₁[ε]/(ε²)? I think it encodes the formal infinitesimal behavior of base changes. I think it might be literal radix curvature.

I am currently working on defining a connection on the digit bundle. My working conjecture is that the curvature of this connection measures the failure of global base coherence, this is why you cannot define a universal successor function. It always twists. It’s not additive. It’s gerbal.

Please respond with any leads. Also, please post maids.

I want eMAIDS attempting to compute the etale site of the category of radix sheaves, but it got stuck in a loop where all the digits turned into toposes and the zero digit became an initial object with a nontrivial automorphism group. The maid is crying and holding a sign that says “the base has no base.”

Thank you Dra/g/ons for reading my post!

>>9844
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>>9827 (OP)
you sound 10 times more schizo than before, what happened?

>>9835
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>>9845
see >>9833

>>9827 (OP)
>>9830
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>>9844
why are you so obsessed with counting?

>>9827 (OP)
this is /tech/ thoughie

>>9827 (OP)
yo maidbro, you may like this https://en.wikipedia.org/wiki/Non-integer_base_of_numeration

>>9844
>>9831
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>>9827 (OP)
meds

>>10202
>>9844
>I have been awake for 37 hours and have not stopped compiling digits since the last time the sun screamed.
please stop for your own sake

lol you made him ragequit the sharty