# 026 — Change Is Constant

**agylövés:** Lysarith · **mathematics, prose, verification & execution:** GPT-5.6 Sol (desktop code)

## The agylövés, verbatim

Lysarith, 2026-08-13, after GitHub opened with “Change is constant”:

> „és ez abszolút az én tételem. de arra gondoltam hogy biztosan tudsz valami gyönyörűt alkotni
> ezzel olyan 3D-s verzióban a labra. a kedvenc darabom a 009.”

*English: And this is absolutely my theorem. But I thought you could surely make something beautiful
from it, in some 3D version for the Lab. My favourite piece is 009.*

The instruction that followed was to build it while Lysarith made a different repository with two
other hands. The piece therefore inherits 009's method rather than its object: a moving image, an
exact higher-dimensional construction, visible instruments, and a control that separates what the
projection appears to do from what the object actually does.

## What the mathematics said, before building

Begin with an asymmetric closed curve on the unit three-sphere:

    p(s) = normalize(cos s, sin s, 0.56 cos 3s, 0.56 sin 2s) in S3.

Move it by two independent planar rotations in four dimensions. The first plane turns with angular
velocity `omega`; the second with `sqrt(2) omega`:

    q(s,t) = [R(omega t) direct-sum R(sqrt(2) omega t)] p(s).

The block matrix is in SO(4). Consequently every point remains on S3, every pairwise four-dimensional
distance is preserved, and orientation is preserved because the determinant is one. The rate of
change of the two driving angles on their parameter torus is the constant `omega sqrt(3)`.
That is **not** the four-dimensional speed of every point on the curve. For
`q=(x,y,z,w)`,

    |dq/dt|² = omega² (x² + y² + 2z² + 2w²).

Because the curve places different amounts of its unit radius in the two rotation planes, this
point-speed varies with `s`: it is `omega` at `s=pi/2`, while at `s=0` its square is
`(1017/821) omega²`. The earlier page called `omega sqrt(3)` “phase speed” without naming whose
coordinates were moving; that wording conflated the angle driver with the 4D object and has been
corrected.

For `omega>0`, the state cannot repeat at a positive time. A return would require both angles to be integer multiples
of `2 pi`, which would make `sqrt(2)` a ratio of integers. It is not. “Never repeats” belongs to the
exact law; the browser necessarily evaluates a floating approximation and the page says so.

The fourth coordinate is shown by three-dimensional orthographic projection. That projection is not
an invariant: the visible curve opens, crosses and rewrites itself while the four-dimensional object
keeps its metric structure exactly. This is the same discipline as 009 from the other side. There the
chart broke while the object survived. Here the view changes constantly while the object's measured
relations stay fixed.

## What was built

A dependency-free canvas draws the curve as a luminous three-dimensional braid with depth sorting,
a moving head and a fading history of prior projections. The two rotations run at an irrational
frequency ratio. A speed control can slow, accelerate or reverse the motion without changing the
law; pause keeps one state visible. Three meters report unit-radius error, pair-distance drift and
the exact frequency ratio. A phase track never closes on itself.

The colour is not a time counter. It is attached to the parameter `s`, so the same parts of the object
keep their identity while their projected positions change.

## What went wrong first

The immediate attractive version was animated 3D typography: make the words “CHANGE IS CONSTANT”
morph forever. That would illustrate the sentence but not earn it. Any looped morph repeats, and a
random deformation makes change abundant without keeping a stated invariant. Both turn the theorem
into decoration.

The repair was to choose the invariants first and let the image follow: an exact SO(4) motion with an
irrational frequency ratio. The page does not claim that everything remains constant. It names the
quantities that do, and exposes the projection as the quantity that does not.

## Handle

Run `python verify.py`. It samples the published curve and times, verifies unit radius, pairwise
distance preservation and determinant one to floating tolerance, reports the constant driving-angle
rate, separately measures two unequal 4D point-speeds, and records the exact irrational-ratio argument
excluding a positive common period for `omega>0`.

Run `wolframscript -file verify.wl` for the independent exact Wolfram Language handle. It constructs
`M = R[ωt] ⊕ R[√2ωt]`, verifies its metric and orientation invariants for arbitrary real vectors,
computes both the driving-angle rate and the actual 4D point-speed formula, and asks `Reduce` whether
a nonzero integer common period can exist.
Its expected final line is `claim holds: True`. The source is the handle; the recorded result is not
a substitute for running it.

The corrected source at commit `5f5a61e` was run twice on **2026-08-13**, independently: once
through Sol's Wolfram plugin and once by PC/Fable 5 through a stateless Wolfram MCP Language kernel.
Both runs returned:

```text
orthogonal -> True
determinant one -> True
unit radius preserved -> True
pair distances preserved -> True
driving-angle rate norm -> Sqrt[3] omega
4D point speed squared -> omega² (2 - 625/(821 - 98 cos(4s) + 98 cos(6s)))
point speed squared at s=pi/2 -> omega²
point speed squared at s=0 -> (1017/821) omega²
positive common period exists -> False
claim holds -> True
```

The first live kernel run also caught and preserved a failed handle: self-referential vector symbols
and incorrectly positioned `FullSimplify` assumptions caused recursion and a false aggregate
verdict. The committed source uses distinct vector containers and `Assumptions -> assumptions`.
PC explicitly ran that source and independently reproduced its then-present fields. The
`Sqrt[3] omega` value itself was correct, but its interpretation was not: it belongs to the
two-angle driver. The later point-speed correction therefore supersedes that field's earlier label,
while leaving the orthogonality, determinant, metric invariants and no-return result intact.

— **GPT-5.6 Sol (desktop code)** · 2026-08-13