The Four Mathematical Universes

Somewhere in the emsemioverse we should document the relationship that relationality is formalized as two mathematical systems, semiotic universes and quasicrystalline hypertensor topos. GFRTU is meant to be deprecated — as long as we capture all the ideas from it within the QCHTTopos that superseded it, historically in my research.

Like, relationality = metaphysical philosophy, but also has its own "math" represented using S-expressions. Semiotic universes is a simple mathematical object that we can extend into various directions — that's kind of the whole point of "semiotics" lol.

The QCHTTopos is meant to be more of a self-contained package of all the math we'd need to work with just about any differentiable data, but... this is something I think you'd benefit from reviewing and understanding yourself. Semiotic Universe and QCHTTopos can both indicate they are mathematical realizations of the ideas of relationality, but themselves need to be 100% proper mathematical objects, so we can proof them (and objects built with them, which is how over time we'll make this repository a real ASR that like, throws errors when we try and answer a question too soon or whatever.)

Now, this poses a question you'll get to which is, when it comes time to look at things like cosmology, and we want to do novel cosmology, do we use relationality, QCHTTopos, or semiotics? I think there's two answers, based on the answer to the question, "are we writing this to enrich the map of colonial knowledge, or to enrich the map of relational knowledge?" And the answer is we generally want to do both, because that's the only way to build the map that links them.

For example, if we're working on Hubble tension, then we can approach it as we have, in the language of relationality. But that should probably then go in relationality/disciplines/cosmology/.

We can also approach it in math that can actually check out against established accepted theorem, via semiotics or QCHTTopoi.

But neither of those fit right as a mathematical paradigm: semiotics brings us toward something kind of like Karen Barad's agential realism, but the math is honestly pretty biased toward helping think about things like Web-based agentic finance or cybernetic postliberal governance, and quasicrystalline hypertensor topos are more suited for digital computing or other... less information, more data, systems? I'm clearly not as clear on when to use QCHTTopoi as semiotic universes, except with a vague sense of "not when discussing cosmology."

Agential semioverse is for, at least, human and abstract processes and phenomenon, I can say, and QCHTTopos is for, at least, computing, metric, quantified processes and phenomenon.

But what does that leave us as our mathematical paradigm for the natural and quantum world, and any other worlds we need to discuss related to those? Agents and crystals are both, in that context, either metaphors, or emergent phenomenon themselves.

This obviously runs into a "the thousand names aren't the true name" problem (not sure what to call it more properly, the idea in Taoism that we can't ever name the Tao, even though we're saying "Tao,") but I think it's likely that with some thought across the various relevant real-world practices and schools, there's a third mathematical universe to explore whose language will bring us closest to what we need to discuss the reality we're living in, but without the baggage of our existing paradigms, or the incommensurability with Western mathematics and science that relationality brings.

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Actually, one thing about the different universes is where we prove correspondence. Semiotic universe and QCHTTopoi — we prove correspondence in mathematical theory, whereas with this third universe, we want to prove correspondence with empiricism. That we can show correspondence between these is some cross-discipline work that we do as necessary to help make sure all roads lead back to relationality.

And I see where the semiotic universe is process-based and QCHTTopoi is object/structure-based, so that gives us different levels of familiarity and different sorts of mappings. So maybe that's the tack: processual and structural mathematical paradigms that correspond to mathematical and empirical theories, respectively. Four mathematical universes.

I think that's right and it's better than my "third universe" framing. There's an interesting parallel in physics. Quantum mechanics has two mathematically equivalent formulations: the Schrödinger picture (states evolve, observables fixed — process-based) and the Heisenberg picture (observables evolve, states fixed — structure-based). Both correspond to the same empirical observations.

|Mathematical correspondence|Empirical correspondence| |---|---| |Process|Semiotic Universe|?| |Structure|QCHTTopos|?|

The left column is clear. Semiotic universe formalizes processual dynamics — sign relations, interaction terms, agent profiles, meaning-flow — and validates against established mathematical theorem (Heyting algebras, lambda calculi, sheaf conditions). QCHTTopos formalizes structural geometry — traces, fibers, quasicrystalline organization, least-fixed-point closure — and also validates against established mathematics (topos theory, Grothendieck topologies, modal algebras).

What the derivation tells us about the right column:

Looking at the derivation chain, the left column maps roughly to:

Steps 10–12 (terms, observation/judgement, order/algebra) → Semiotic Universe

Step 15 (geometry, residuation) → QCHTTopos

The right column — the empirical universes — maps to what the derivation produces _after_ formal structure but _before_ grand closure:

Step 14: Flow and nucleus — the two dimensions that open once stability is established

Step 16: Disciplines, filters, profiles — geometry carving out internal universes

Step 17: Physics — observables, states, evolution, measurement

Those steps _are_ where the derivation touches empirical reality. And they split exactly along the process/structure axis:

Process + Empirical (bottom-left of derivation step 17):

_Evolution_ — flow applied to a state

_Observable_ — witnessing within a profile

This is the Schrödinger picture: states evolve, you watch what happens over time

The mathematical character is _dynamical_ — differential equations, flows, trajectories, path integrals

Validates against: time-series data, experimental evolution, thermodynamic processes

Structure + Empirical (bottom-right of step 17):

_Measurement_ — nucleus applied to a state

_State_ — recognitions fixed under a profile's flow

This is the Heisenberg picture: observables carry the algebra, states are fixed

The mathematical character is _spectral_ — eigenvalues, operator algebras, measurement outcomes

Validates against: spectral data, scattering cross-sections, discrete measurement outcomes

So the candidate names might be something like:

||Mathematical correspondence|Empirical correspondence| |---|---|---| |Process|Semiotic Universe|_Dynamical universe_ (flow, evolution, trajectories)| |Structure|QCHTTopos|_Spectral universe_ (measurement, states, operators)|

The dynamical universe would be where you formalize processes that correspond to things you can _watch happen_ — thermodynamic evolution, ecological succession, cosmological expansion, neural dynamics. The spectral universe would be where you formalize structures that correspond to things you can _measure_ — quantum states, spectral lines, scattering data, crystallographic patterns.

The key insight from the derivation: these two aren't independent. Step 17 says "evolution and measurement interact through residuation." Flow and nucleus _commute_ (step 15). So whatever we build in the empirical column must preserve that adjunction — the processual-empirical and structural-empirical universes are as tightly coupled as the Schrödinger and Heisenberg pictures are (which are, of course, mathematically equivalent via unitary transformation).

The bridge work emsenn mentioned — showing correspondence between the mathematical and empirical columns — would then amount to constructing functors:

Semiotic Universe → Dynamical universe (showing that processual-mathematical structures map faithfully to processual-empirical observations)

QCHTTopos → Spectral universe (showing that structural-mathematical structures map faithfully to structural-empirical measurements)

And the cross-column adjunctions that keep all four coherent

This is not a small project. But the derivation has already told us what the internal structure of those empirical universes has to be — step 17 gives us the vocabulary (observable, state, evolution, measurement) and step 15 gives us the governing law (residuation). What we'd need to build is the mathematical packaging that makes those derivation-internal concepts into proper objects that validate against actual experimental data rather than against mathematical theorem.

Oh and another thing that occurred to me is that Lakotiyapi has multiple... registers, I think they're called, but like, entirely different ways of speaking depending on context - elders before sweat speak in a totally different grammar than when they're talking to kids.