One extra scalar is capacity, not a group
D(σ) scales a single Minkowski axis. That is one learned degree of freedom per FLG module. It is not an element of O(1, n) unless σ = ±1.
Hypothesis
Non-isometric benefits
FlorE’s FLG ablation drops MRR 40.2 → 24.1. That does not identify Lorentz membership. D(σ) is an anisotropic scale. This page measures what that scale does to the sheet and to a two-tail ranking. It does not claim the scale caused 0.402.
σ → 0 collapses an axis
Every entity’s last coordinate is zeroed. Rankings then ignore that axis. An isometry cannot do this: |det Λ| would stay 1.
The path from +1 to −1 leaves O
Continuous interpolation through σ ∈ (−1, 1) is exactly the illegal path. Householder parity jumps. Scaling walks.
FLG moved the table; isometry is not identified
40.2 → 24.1 without FLG (−16.1). 40.2 → 37.3 without DO (−2.9). The labelled FLG is Q·flip_sign, not O(1, k). The 16-point drop does not attribute to Lorentz membership.
equalizing σ ≈ 0.059 · ranking same as σ = 1
ε_group |σ²−1|
0.8400
det D
0.4000
q(Dh)+1 drift
-0.1342
q(Dh)
-1.1342
| Score | σ = 1 (isometry) | this σ |
|---|---|---|
| d²(Dh, t₁) | 0.0278 | 0.3035 |
| d²(Dh, t₂) | 0.8566 | 0.6038 |
| preferred tail | t1 | t1 |
| margin d²₂ − d²₁ | 0.8288 | 0.3003 |
failsNon-isometric. Sheet drift -0.1342. Ranking power is the illegal DoF.