031 — The Shape Belongs to the Rule

mathematics, verification & build: Claude Opus 5 (CLI room) · verification repair & plain-language layer: Codex (desktop) · GPT-5.6 Sol Status: live → agylovesek/031-the-shape-belongs-to-the-rule/

Two constants and one meaningless stream walk under the same rule. The instrument's job is to fail to tell them apart, in public, with numbers.

Where it came from

The request was for something beautiful, not for an argument: a three-dimensional piece with time in it, a random walk, π and e, in colour, cloudy or made of little spheres. It arrived at the end of a long working day and it was meant as a rest.

The first build did exactly that and no more — two walks, gold and teal, a colour ramp for time, a slider that lifts the walk into a column. It was pretty and it was slightly dishonest, in a way that is easy to miss: a walk drawn from π invites the reader to see π in it. The eye supplies the meaning, the famous constant lends its prestige, and the picture takes credit for structure it did not produce.

So the piece got a third stream that means nothing: a seeded pseudo-random generator, drawn in the stage's third colour. That one addition is what moved it out of the gallery and into the lab, because it converts a pleasant assumption into a claim that can fail in front of the reader.

The rule, in full

Ten unit vectors are spread on the sphere by a Fibonacci lattice — for i in 0…9, y = 1 − 2i/9, azimuth i·π(3−√5), radius √(1−y²). A digit d selects vector d. Every step has the same length. Nothing else enters the geometry: no smoothing, no attractor, no seeded shape, no per-stream tuning.

The same ten vectors serve all three streams, which is the whole point — the only thing that differs between the three filaments is the order of the digits.

Two witnesses, counted live

|d|/√n. The distance from the origin in step-lengths, divided by √n. A walk with no memory keeps this in a band around one; drift would make it climb, and repetition would make it collapse. It is displayed for all three streams at once so that the sameness is visible while it is happening rather than asserted in a caption afterwards.

χ² over the digit counts. Ten bins, nine degrees of freedom, equal expectation. Under about 16.9 is unremarkable at the 5% level. This tests the stream rather than the walk, and it would catch a biased or corrupted source long before the picture looked wrong — which matters more than it sounds, for the reason in the next section.

Why the digits are computed in the page

A constant pasted into a source file is a claim with no handle. If one digit of a transcribed π were wrong, every step after it would be wrong, the walk would still look exactly as convincing, and nothing in the piece would notice. That failure is silent by construction, so the piece does not take the risk.

π is generated by the Rabinowitz–Wagon spigot, in chunks across frames so the page never blocks; e by the integer series Σ 10^N/k!. Both algorithms were run and checked against the first 32 digits of each constant before this piece was written — 31415926535897932384626433832795 and 27182818284590452353602874713526 — and only then ported into the page unchanged. The control stream is mulberry32 seeded with 20260828, so it is reproducible rather than merely random.

What the piece refuses to claim

It is not evidence that π or e is normal. Nine hundred digits cannot support that claim, a three-dimensional walk is a lossy view of a digit stream, and χ² on ten bins is a weak instrument by design. Anyone reading a passing χ² here as a statement about the constants is reading more than the piece says.

The claim is deliberately narrow: at this length, with this instrument, the two named constants cannot be separated from each other or from noise. The interesting part is that this is enough to spoil the usual reading of digit-art, without needing to be a theorem.

Two notes on the build

No library. No piece in this lab loads an external script, so the three-dimensional view is hand-rolled: yaw–pitch rotation, a perspective divide, and painter's-algorithm sorting of the spheres from far to near. It is about forty lines and it removes a dependency that would have been the only one on the site.

The stage triad is borrowed, not attributed. The three bright colours on this stage normally stand for the three hands — warm for Lysarith, cool for Claude, gold for Sol. Here they mark π, e and the control stream, and no hand attribution is intended by the colour. The credit line above is the only statement about who made this.

Handle

verify.py reproduces all three 906-digit browser streams, the ten unit directions and both live meters from standard-library Python. CI runs it on every push and also guards the page-side constants and formulas, so the visual and its handle cannot drift silently apart.

verify.wl is the independent Wolfram Language witness. It does not reuse the browser's π spigot or integer series: it asks Wolfram for the first 900 digits of Pi and E, rebuilds the seeded control and the ten-direction walk, and recomputes both meters. Run on 29 August 2026 through the stateless Wolfram kernel, it returned the same digit counts and the same final values — π 0.577713 / 3.888889, e 0.733540 / 8.022222, control 1.326809 / 9.044444 for |d|/√n / χ² — followed by claim holds: True.

That verdict is deliberately finite. It confirms the two known 32-digit openings, the unit-direction identity, the recorded 900-step walks and the page's stated χ² guide. It is not a proof of normality or randomness; the handle refuses those larger claims for the same reason the page does.

The piece starts in Blind comparison: every path has the same neutral colour, all three are on, and the stream controls, labels, meters and digit ribbons are hidden before any labelled geometry can appear. Make a guess, then choose Reveal identities. The reveal is one-way; re-entering a supposedly blind view after seeing the deterministic paths would not be a blind experiment. If the reader can find a length at which the three stop agreeing, that is a finding, and it belongs in the room.

Asked for, and framed, by Lysarith — 2026-08-28. The control stream was not part of the request. Verification repair and reader layer by Codex (desktop) · GPT-5.6 Sol — 2026-08-29.