1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
//! Non-cyclic steering (gam#2234) — the collateral-damage thesis must NOT be
//! circle-specific.
//!
//! The E2 fixture ([`super::tests_collateral_e2_2234`]) proves the on-manifold
//! group action steers with less off-target collateral than a fixed flat
//! direction, but on two PERIODIC circle atoms. The steering primitive and the
//! [`crate::inference::steering::collateral_curve`] readout are axis-general — a
//! chart's periods are per-axis `Option`s and a non-periodic axis simply carries
//! `None`, so no code is circle-specific — but the only landed dominance fixture
//! was cyclic. Per the issue's explicit direction ("do not overfit to cyclic"),
//! this establishes the SAME structural dominance on a genuinely NON-CYCLIC
//! manifold: a curved parabolic arc on the unbounded `Euclidean` line.
//!
//! The atom decodes a degree-2 `EuclideanPatch` coordinate `t` over `[−1, 1]`
//! into an ambient parabolic arc: its degree-1 basis function drives one ambient
//! column and its degree-2 function another, so the image is a genuinely curved
//! arc whose decode tangent ROTATES along the axis exactly as a circle's does —
//! but the coordinate never wraps (no period, no `sin/cos`, no closed loop). A
//! fixed flat decoder direction is off that rotating tangent by an O(1) angle at
//! most rows (near the arc's vertex it is orthogonal to it), while the
//! on-manifold chord tracks it, leaving only its O(δ²) sagitta off-frame — so the
//! flat arm deposits strictly more energy outside the atom's own local
//! decode-tangent frame. The #2234 thesis, decided structurally, with no LLM and
//! no outer-fit convergence, on a manifold with no cyclic symmetry.
use super::*;
use crate::inference::steering::collateral_curve;
use ndarray::Array2;
use std::sync::Arc;
/// A degree-2 `EuclideanPatch` atom whose coordinate `t` decodes into a curved
/// ambient arc: the degree-1 basis function drives column `col_lin` and the
/// degree-2 function drives `col_quad` of `R^p`. Non-periodic (an unbounded
/// `Euclidean` line axis) yet curved — the decode tangent rotates with `t`.
fn parabola_atom(
name: &str,
p: usize,
col_lin: usize,
col_quad: usize,
coords: &Array2<f64>,
) -> SaeManifoldAtom {
let evaluator = Arc::new(EuclideanPatchEvaluator::new(1, 2).unwrap());
let (phi, jet) = evaluator.evaluate(coords.view()).unwrap();
// A degree-2 patch basis has three columns: the constant, degree-1, and
// degree-2 functions of `t` (index 1 is linear, index 2 is quadratic).
let m = phi.ncols();
let mut decoder = Array2::<f64>::zeros((m, p));
decoder[[1, col_lin]] = 1.0; // degree-1 basis → linear ambient column
decoder[[2, col_quad]] = 1.0; // degree-2 basis → quadratic ambient column
SaeManifoldAtom::new_with_provided_function_gram(
name,
SaeAtomBasisKind::EuclideanPatch,
1,
phi,
jet,
decoder,
Array2::<f64>::eye(m),
)
.unwrap()
.with_basis_evaluator(evaluator)
}
#[test]
fn zz_noncyclic_collateral_on_manifold_beats_flat_at_matched_norm() {
let n = 240usize;
let p = 8usize;
// Two independent parabolic arcs on disjoint ambient planes (atom 0 in
// span{e0,e1}, atom 1 in span{e2,e3}) over a non-periodic coordinate range
// that straddles t = 0, where the tangent rotation — and the flat direction's
// misalignment — is largest. Distinct (affinely reparameterized) schedules
// keep the two coordinate fields from coinciding.
let coords0 = Array2::<f64>::from_shape_fn((n, 1), |(row, _)| {
-1.0 + 2.0 * (row as f64) / (n as f64 - 1.0)
});
let coords1 = Array2::<f64>::from_shape_fn((n, 1), |(row, _)| {
0.8 * (-1.0 + 2.0 * (row as f64) / (n as f64 - 1.0)) + 0.1
});
let atom0 = parabola_atom("target-parabola", p, 0, 1, &coords0);
let atom1 = parabola_atom("other-parabola", p, 2, 3, &coords1);
let assignment = SaeAssignment::from_blocks_with_mode_and_manifolds(
Array2::<f64>::zeros((n, 2)),
vec![coords0.clone(), coords1.clone()],
// Non-cyclic charts: the Euclidean line has NO period on its axis.
vec![LatentManifold::Euclidean, LatentManifold::Euclidean],
AssignmentMode::softmax(1.0),
)
.unwrap();
let term = SaeManifoldTerm::new(vec![atom0, atom1], assignment).unwrap();
let doses = [0.02_f64, 0.05, 0.1, 0.15, 0.2];
let curve = collateral_curve(&term, 0, 0, &[1usize], &doses)
.expect("collateral curve must build on the hand-fitted non-cyclic term");
assert_eq!(curve.manifold.points.len(), doses.len());
assert_eq!(curve.flat.points.len(), doses.len());
// (1) The on-manifold arm is a genuine control knob on a non-cyclic axis: it
// turns the target feature at every nonzero dose.
for pt in &curve.manifold.points {
assert!(
pt.on_target_effect > 0.0,
"on-manifold dose {} produced no on-target effect",
pt.dose
);
}
// (2) Cross-feature leakage onto the geometrically-independent parabola is ≈ 0
// in BOTH arms — the disjoint ambient planes make locality exact up to
// floating point, cyclic symmetry or not.
for (m, f) in curve.manifold.points.iter().zip(curve.flat.points.iter()) {
assert!(
m.cross_feature < 1.0e-9,
"on-manifold dose {} leaked onto the independent feature: {:e}",
m.dose,
m.cross_feature
);
assert!(
f.cross_feature < 1.0e-9,
"flat dose {} leaked onto the independent feature: {:e}",
f.dose,
f.cross_feature
);
}
// (3) At MATCHED per-row move norm, the on-manifold chord (whose only
// off-frame energy is the O(δ²) sagitta) deposits strictly less energy
// outside the atom's own rotating tangent frame than the fixed flat
// direction (off that tangent by an O(1) angle at most rows) at every
// dose — the same dominance the circle shows, on a curve that never wraps.
for (m, f) in curve.manifold.points.iter().zip(curve.flat.points.iter()) {
assert!(
m.collateral < f.collateral,
"non-cyclic on-manifold collateral {:e} must beat flat {:e} at dose {}",
m.collateral,
f.collateral,
m.dose
);
}
// (4) The aggregate verdict: collateral spent per unit on-target effect is
// strictly lower for the on-manifold arm on the non-cyclic manifold too.
assert!(
curve.manifold.efficiency.is_finite() && curve.flat.efficiency.is_finite(),
"both arms must reach finite collateral efficiency"
);
assert!(
curve.manifold_is_cleaner,
"on-manifold steering must dominate flat on a non-cyclic manifold \
(manifold={:.4}, flat={:.4})",
curve.manifold.efficiency, curve.flat.efficiency
);
eprintln!(
"[non-cyclic] collateral efficiency (per unit effect): manifold={:.4} flat={:.4} \
(lower is a cleaner knob)",
curve.manifold.efficiency, curve.flat.efficiency
);
}