Candidate Invariants Probe for qdim=3 Levels

This probe tests the cheapest next-step invariants suggested after the first canonical family comparison.

level	status	h	T2|Q minpoly	nil-index	w_N|Q minpoly	w_N coords in {I,E,N}	proj group	ord(2)	ord(3)	ord(11)	dlog ratio
163	canonical	1	x^2	2	x^2 + 2	[1, 1, 0]	C6	162	162	162	55
577	degenerate	8	x^2	2	x + 1	None	C6	144	48	96	None
773	degenerate	26	x	1	x^2 + 2	None	C2	772	772	386	686
811	misaligned	7	x^2	2	x^2 + 2	[0, 2, 1]	C6	270	810	405	118
829	canonical	22	x^2	2	x^2 + 2	[1, 1, 0]	C6	828	207	23	None
883	degenerate	3	x	1	x^2 + 2	None	C8	882	126	882	None
1009	degenerate	20	x^2	2	x^2 + 2	None	C6	504	168	1008	None
1093	misaligned	10	x	1	x + 2	[1, 0, 0]	C13	364	7	13	None

Immediate reads

• T_2 does not give a new separator here. Since Q = ker(T_2^2), the only real distinction is whether T_2|Q = 0 or has nilpotency index 2, and that is already the old x versus x^2 minimal-polynomial data.
• The raw Atkin-Lehner eigenvalue pattern is not enough: 163, 829, 773, 811, 883, and 1009 all have w_N|Q with minimal polynomial x^2 + 2, so the spectrum {-1,+1,+1} alone does not isolate the canonical class.
• The projective Heegner group is also too coarse: both canonical levels have projective group C6, but so do 577, 811, and 1009.
• The Atkin-Lehner coordinates in the Hecke basis are sharper. The current levels with {I,E,N} independent split as: 163: status canonical, w_N = 1 I + 1 E + 0 N. 811: status misaligned, w_N = 0 I + 2 E + 1 N. 829: status canonical, w_N = 1 I + 1 E + 0 N. 1093: status misaligned, w_N = 1 I + 0 E + 0 N.
• On the current sample, both canonical levels have exactly the same Atkin-Lehner coordinate law:

w_N|Q = I + E

in the {I,E,N} basis, while the misaligned levels do not. - The dlog-ratio idea is not uniformly well-posed on the current sample. It requires log(3) to be a unit mod p-1, and that already fails at 829, one of the two canonical levels.

Best surviving cheap invariant

Among the cheap tests run here, the best current candidate is the Atkin-Lehner coordinate identity

w_N|Q = I + T_3|Q

equivalently w_N|Q has coordinates [1,1,0] in the basis {I, T_3|Q, T_11|Q}. This matches both 163 and 829, and fails on the present misaligned levels 811 and 1093.

It is not a complete theorem yet, because the degenerate levels do not admit an {I,E,N} basis at all. But it is the first cheap invariant in this session that actually tracks the canonical pair rather than just one of its shadows.