Fractal Embeddings

Complete

Complexity ScienceRepresentation GeometryPre-Registered

Hypothesis

A large body of knowledge, represented as machine-readable meaning, might have a fractal fingerprint: a low effective dimension that stays the same as you zoom in and out, a link structure distinguishable from a random one with the same connectivity, and a rough boundary between inputs a language model settles on and inputs it does not. We froze the pass and fail thresholds before touching any data, and most of what came back was a no.

Overview

This is the experiment we point at when someone asks whether this program actually does what it says. Every threshold was frozen in writing before any data was touched, the tests were run, and most of them came back negative. They are reported here exactly as they came back.

The question

Complex systems that grow by accumulation often turn out to be fractal — the same structure repeating as you zoom in and out. Coastlines do it, blood vessels do it, river networks do it. A body of knowledge that has been built up over years, note by note and link by link, is that kind of system. So: does it carry the same fingerprint?

This matters beyond curiosity. If meaning-spaces are fractal, that constrains what kind of thing they are and how measurements on them should behave — including the integration measurements the rest of this program depends on.

What came back

Confirmed — the meaning-space is far smaller than it looks. Each document is represented by 1,024 numbers, but the whole collection actually occupies a space of about 11 effective dimensions. Two completely independent ways of measuring that agreed to within 1%. The knowledge lives on roughly 1% of the room it appears to take up.

Did not hold — it is not self-similar. The obvious follow-up was whether that number stays the same as you zoom in on smaller and smaller subsets. It does not. It shifts systematically with scale, and the measurement is clear enough to rule out coincidence. So: low-dimensional, yes. Fractal in that sense, no.

Did not hold — and this one is the most instructive. The link network does show a clean fractal power law, which on its own looks like a finding. But compared against 100 randomly rewired versions that preserve only how many links each note has, the real network sits inside the random range. Meaning: the fractal appearance is fully explained by how many links things have. The actual wiring — which note connects to which — adds nothing measurable. Without that null comparison we would have reported a discovery. This is exactly what the pre-registered control was for.

Confirmed — a language model's internal trajectories separate into regimes. Tracking how far a model's internal representation moves at each layer, across 200 inputs, the movement profiles cluster cleanly into distinguishable groups. Inputs the model settles on and inputs it does not travel differently through the network.

Did not hold — the boundary is smooth. The highest-risk test: slide an input gradually from one the model resolves toward one it does not, and see whether the frontier between those behaviours is rough and fractal or a single clean switch. It is a single clean switch, at every resolution tested. We flagged this in advance as the result most likely to come back negative, and it did.

One test could not run at all. It required the same documents represented by a second, lower-quality model, and only one model's representations existed. Rather than quietly regenerate the data and risk tuning the test to the answer, it is recorded as blocked, with no claim made in either direction.

Why the negative results are the point

Three of six tests came back negative, one was blocked, and two confirmed. That ratio is what a pre-registration is for. Every one of those nulls is a result we could have avoided reporting by deciding the threshold after seeing the number — and the link-network result would have been a false discovery had the random control not been frozen into the design in advance.

There is one loose thread we are explicitly not claiming. A second measure of the same network did land outside the random range, in the opposite direction from the first. That was not something we pre-registered, so it is recorded as an observation for follow-up rather than a finding. Noticing something interesting after the fact is not the same as having predicted it, and the distinction is the whole discipline.

Methodology

Freeze every success threshold, every no-effect threshold, and every method-failure threshold in writing before looking at a single number. Nothing below was decided after seeing results.
Thread A — measure the effective dimension of a 5,749-document meaning-space using two independent estimators, then re-measure it at nested zoom levels to see whether the number stays put.
Thread B — measure the link network's fractal dimension, then compare it against 100 randomly rewired versions of the same network that preserve exactly how many links each note has. If the real structure is no different from the shuffle, the wiring adds nothing.
Thread C — take a language model, feed it 200 inputs, and track how far its internal representation moves at each layer. Cluster those movement profiles and test whether distinguishable regimes exist.
Thread C part two, the riskiest — slide an input by 256 steps from one that the model settles on toward one it does not, and measure whether the boundary between the two behaviours is smooth or rough.
Report every result against its frozen threshold, including and especially the ones that failed.

Status