Back to the work
That is the argument.
A reader who has come this far deserves a clean view of what the paper asks them to accept and what it only asks them to consider, with the speculative tail set aside for a moment.
The claim is one sentence.
Models of meaning should be built after the universe, not after our linear-algebra defaults.
Everything before that is the thought experiment that leads me to believe that as a fundamental conclusion. Everything after is my sketch of what building after the universe might look like.
The spine and the speculation
The chain has parts of different strength, and they should be sorted, so that a reader who pushes back on the speculative tail does not think they have undone the spine.
The geometry of spacetime is hyperbolic in the sense §02 described. That is forced by experiment and has been for a hundred years. We are slice observers of that geometry, calibrated by evolution to a regime four orders of magnitude below the speed of light, where the curvature does not show. Neither move is speculative.
Four contemporary research programs have, in the last fifteen years, been forced into high-dimensional geometric structure by what they were studying. Three of the four are well-established empirical findings: large language model embeddings, the Hilbert spaces of quantum computers, the manifolds and topological structures of neural population activity. The fourth, the substrate-of-consciousness hypothesis associated with Penrose and Hameroff, is live but contested. The convergence I lean on does not need the fourth. If Orch-OR turns out to be wrong tomorrow, the convergence is three programs instead of four, and the rest of the chain survives.
The move from convergence to identity, the claim that meaning is structure and that meaning’s shape and the universe’s shape are the same shape, is a philosophical proposal. Not forced. Not refuted. A reader can decline it. If they take it, §07 follows. If they decline it, the diagnosis of where flat-space architectures will run out of road still has independent purchase.
The paper is not asking anyone to resolve the metaphysics of existence. The structural picture is compatible with several readings; I have engaged them where engagement clarified the claim and declined to choose where the choice would commit the paper to more than the argument needs.
The next move, depending on who is reading
Different readers can do different things with this.
If you are an ML researcher, the falsifiable version is in §07. Substrate-faithful architectures should outperform flat-space architectures on tasks that exercise the relevant regime, and should not outperform where flat space already covers well. If they outperform nowhere, the principle is wrong. The implementation paper that does the showing is not this one. It is one I would like to read.
If you are a physicist, the geometric move connects to your work at two points. The slice-observer framing in §03 lets the block-universe reading and the relational programs say something about the observers inside the geometry, not just about the geometry itself. The proposal in §06 is that meaning is one more substance the geometry hosts, in the way physical state is.
If you are a philosopher, the place to push is where structural realism meets the ML field. Worrall’s structure-preservation move and Ladyman and Ross’s ontic structural realism are the closest published kin to §06, and the gap between them and the empirical convergence in §04 is the gap this paper tries to close. I would rather see the picture sharpened or refuted by readers who know that literature better than I do than left as my private synthesis.
Back to the work
I started this paper from a piece of popular physics on social media, late one evening, following the implications further than test questions had ever asked me to. The thinking led somewhere I had not expected to go. What I found on the other side of that door was larger than I had any reason to expect, and most of it had nothing to do with photons.
The grounded part of the paper still holds. The geometry of spacetime is forced. We are observers of a slice of it. Several scientific programs have been forced into the same kind of mathematical structure. That pattern is worth taking seriously as a research direction whether or not the philosophical move at the end of the chain turns out to be the right reading of it.
The work of the next decade in machine learning will not, I think, be more transformers. It will be figuring out what shape meaning lives at, and building systems that respect that shape. The first architecture that gives up the flat-space defaults and earns its keep on a task flat space does badly will tell us whether the picture in this paper was the right reading or only a suggestive one. I do not know which way that experiment will fall. I think it is the experiment worth running.
That is what I have. I am leaving the question in the best position I know how to leave it in.