Traces of a wheel or similar

[DG97EEB]

What I tested

I extracted the opening glyph/state of successive running-text lines, keeping paragraph structure intact.

The first conspicuous pattern involved d, q and y, which suggested a possible three-state cycle. So I built the simplest literal model:

d → q → y → d

with allowance for staying in place, moving backwards, or leaving the cycle for another opener.

I then tested it out-of-sample, holding out whole folios rather than fitting and testing on the same text. I compared it with both:

- a model in which line openings are independent; and
- an unrestricted first-order model which is allowed to learn every opener-to-opener transition.

I also generated artificial manuscripts from the fitted model and asked whether they reproduced statistics which had not simply been fitted directly.

The simple d/q/y model initially looked remarkably good. It recovered about 87.9% of the predictive improvement obtained by the unrestricted transition model.

But that was only the beginning of the test.

The artificial texts still missed the real amount of dependency between successive openings by 17.9 standard deviations. So the attractive three-state wheel was clearly missing something.

I then progressively made the test harder.

Paragraph position

A large amount of the effect turned out to come from where a line sits within its paragraph.

Paragraph position alone carries about 0.323 bits/opener of information, approximately 8.9 SD above the null.

After accounting for that, knowing specifically about the d/q/y relationship still adds 0.0163 bits/opener, with null SD 0.00469: 3.48 SD.

So the d/q/y relationship is real.

But it is not unique.

When I tested all comparable three-opener combinations, d/q/y ranked only 13th of 165.

That was the first serious warning against interpreting the statistical pattern as one special physical d/q/y wheel.

Could there be several wheels?

I then allowed multiple transition components rather than forcing everything through d/q/y.

That did uncover strong additional structure. The best pooled two-component model gave an effect of +0.0448 bits/transition, null SD 0.00533: 8.41 SD over the shuffled control.

Adding the specifically nominated d/q/y and y/o/q cyclic components contributed another +0.00560 bits/transition, null SD 0.00156: 3.58 SD.

So there really are several structured transition tendencies at the left margin.

But the crucial question is whether these are generative wheels, rather than merely statistical descriptions of a more general scribal process.

That is where the wheel interpretation starts to fail.

On whole-quire transfer, the advantage was only +0.00864 bits/transition against null SD 0.01948. The metric does not resolve this: only 0.44 SD.

More importantly, when the fitted multi-component mechanism was actually made to generate new opener sequences, the generated manuscripts still missed the real mutual information by about 5.1 SD.

Currier B and y → o → q → y

I also tested the obvious possibility that Currier B uses a different wheel, because B contains a conspicuous y/o/q relationship.

I therefore fitted a dedicated:

y → o → q → y

generator.

It fails the generative test.

The real Currier-B sequence contains 0.0221 bits/transition more structure than the model generates, simulation SD 0.00565: 3.90 SD.

Under the stricter whole-quire test the deficit increases to 5.24 SD.

So the data do not support the proposition that Currier B simply has its own y/o/q wheel.

The ultimate test: can a wheel generate Voynichese?

Finally I stopped testing only line-openers.

I incorporated the proposed state mechanism into a complete Voynich token generator and generated whole artificial corpora.

This gives the strongest result.

Across 37,465 held-out tokens, the wheel-based generator was worse by 0.0618 bits/token, null SD 0.00772: about 8.0 SD.

It also badly missed independent properties which had not been used to fit the wheel:

- boundary behaviour missed by about 34.8 SD;
- vocabulary/type production overshot by 2,161 types, null SD 60.9: about 35.5 SD.

These failures survive paragraph, transcription, physical-block and cache controls.

So what actually survived?

Something quite interesting survived — just not the strongest version of the wheel hypothesis.

There is a real sequential process operating at the left margin.

The opener of one line helps predict the opener of the next. The process is strongly conditioned by position within the paragraph, and paragraph boundaries behave rather like resets.

A simple way of writing the surviving result is:

paragraph state + previous line opener → next line opener

There are several transition families within that process, including a genuine d/q/y relationship.

What the statistics do not tell us is how the scribe implemented that rule.

It could have been:

- remembered mentally;
- represented as a table;
- represented diagrammatically;
- represented by one or more circular diagrams;
- or conceivably implemented using a physical selector/volvelle.

Those implementations can produce essentially the same observable statistics, so the manuscript does not currently distinguish them.

What does not survive is the stronger claim that a d/q/y wheel, a Currier-B y/o/q wheel, or a small collection of such wheels is itself the mechanism generating Voynichese as a whole.

So my current conclusion is:

There really is a paragraph-local state machine at the left margin. A wheel remains a possible physical implementation of part of that procedure, but the evidence does not identify a wheel, and the tested wheel mechanisms do not generate the wider statistical structure of Voynichese.

Reproducibility

I have packaged the code, data, frozen inputs, saved simulations, independent checks and manifests into an external replay package.

Google Drive folder:

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The complete package is about 137 MB. Because of an upload-size limitation it is stored as two binary pieces:

"Voynich_Wheel_Research_External_Replay_20261003.zip.part00"

"Voynich_Wheel_Research_External_Replay_20261003.zip.part01"

There is also a "REASSEMBLE.txt" and the SHA-256 checksum in the folder.

On macOS/Linux:

cat Voynich_Wheel_Research_External_Replay_20261003.zip.part00 \
    Voynich_Wheel_Research_External_Replay_20261003.zip.part01 \
    > Voynich_Wheel_Research_External_Replay_20261003.zip

On Windows Command Prompt:

copy /b Voynich_Wheel_Research_External_Replay_20261003.zip.part00+Voynich_Wheel_Research_External_Replay_20261003.zip.part01 Voynich_Wheel_Research_External_Replay_20261003.zip

Then extract the ZIP and, from its top-level directory:

python -m venv .venv

Activate the virtual environment, then:

pip install -r requirements.txt
python verify_manifest.py
python run_all.py verify

"verify_manifest.py" checks the packaged files before anything is run.

The independent replay includes the literal d/q/y experiment, the multiple-component tests, the Currier A/B scope test, the dedicated y/o/q Currier-B test, and the final whole-token generative test.

I would be particularly interested if anyone can identify a different mechanical model which predicts the surviving paragraph-local opener behaviour without acquiring the failures of the wheel generators above.