Research period: September 1, 2026
Historical anchors: #161–#162

The previous experiment left open the possibility that the spectral representation wasn't inherently deficient; the downstream model might simply have been doing the wrong kind of computation on it. I tested that by holding the uncombined spectral state fixed and changing only how the downstream model represented and processed the same information.

There were two changes I needed to separate. Rewriting a complex value from Cartesian coordinates into [rho, cos(theta), sin(theta)] might make the information easier for a model to use even if the computation itself stayed essentially the same. Alternatively, operations designed around magnitude and phase might provide an additional advantage. A coordinate-only control let me measure those effects separately.

Sticky note — log-polar: a complex value can be described by its magnitude and phase. rho = log(|z| + eps) represents magnitude on a logarithmic scale, while theta represents its angle.

Result

Both changes helped, but by different amounts. Rewriting the spectral state into the coordinate-only representation improved the result, showing that coordinates alone affected how easily the downstream model could use the information. The polar-native processing path improved loss substantially further.

Compared with the original spectral Cartesian condition, the polar-native model improved loss by about 0.5825 nat per original byte. More importantly, it improved loss by about 0.3923 nat per original byte compared with the coordinate-only control containing the same underlying information. All three experimental seeds cleared the predefined materiality threshold.

That second comparison changed the interpretation of the experiment. A coordinate transformation could explain part of the original improvement, but it couldn't explain all of it. Something about the operations performed in those coordinates was contributing independently.

Illustrative example

These three complex values demonstrate a coordinate change, not the historical model comparison.

Saved output is included so you can inspect the example without starting a kernel. Running it in Lab executes this example only.

import numpy as np
z = np.array([1+1j, -2+0.5j, 0.2-3j])
rho = np.log(np.abs(z) + 1e-8)
phase_features = np.c_[rho, np.cos(np.angle(z)), np.sin(np.angle(z))]
print(phase_features)

[[ 0.3465736   0.70710678  0.70710678]
 [ 0.7234595  -0.9701425   0.24253563]
 [ 1.10082959  0.06651901 -0.99778516]]

What this does not show

This was evidence for a specific representation and processing architecture, not a general law that representations always work best with operations designed around their geometry. I had also changed several operations inside the polar-native block at once, so the result couldn't yet tell me which part of that block mattered.

The coordinate-only control was important for exactly this reason. Without it, I could have attributed the full improvement to the new processing architecture when some of the gain actually came from expressing the same information differently.

The next experiment therefore kept the source representation fixed again and took the successful processing block apart. If the improvement depended on a specific operation, removing that operation should make the advantage disappear; if several components were necessary together, the ablations should expose that interaction.