Residuals: strange (GI-style)

Model: with GI momentum-sandwiched smeared contact hyperfine (every L), fixed-sector L·S (vector+Thomas) and OGE-tensor diagonalization with GI momentum-factor sandwiches and smeared-G/S derivative kernels; GI Table II b, c, masses, ε factors, and Fig. 2 α_s(r); closed-form GI G̃(r), S̃(r), plus central Coulomb momentum sandwich A(p)G̃A(p) in the solver's native p² representation, see the GIModel package (src/ at the repository root) for the computation and GIPaper (GIPaper/src/) for the comparison layer.

Numerics: OscillatorSolver — nbasis = 24:8:80 adaptive, ΔE ≤ 0.1 MeV, beta in [0.25, 2.35] GeV (22 bracket points, refined to 0.002 GeV), 6 levels/channel.

state	reference GeV	baseline GeV	residual MeV	confidence
1^1S_0	0.470	0.461	-8.7	high
2^1S_0	1.450	1.453	+3.4	high
3^1S_0	2.020	2.014	-6.4	high
1^3S_1	0.900	0.903	+2.8	high
2^3S_1	1.580	1.579	-1.2	high
1^3D_1	1.780	1.777	-3.0	high
3^3S_1	2.110	2.106	-4.0	high
2^3D_1	2.250	2.251	+0.9	high
1^3P_0	1.240	1.234	-6.4	high
2^3P_0	1.890	1.890	+0.1	high
1^1P_1	1.340	1.352	+11.6	high
1^3P_1	1.380	1.366	-14.4	high
2^1P_1	1.900	1.895	-4.7	high
2^3P_1	1.930	1.930	+0.2	high
1^3P_2	1.430	1.428	-1.9	high
2^3P_2	1.940	1.938	-1.6	high
1^3F_2	2.150	2.151	+1.4	high
1^1D_2	1.780	1.790	+9.6	high
1^3D_2	1.810	1.804	-5.6	high
2^1D_2	2.230	2.236	+5.9	high
2^3D_2	2.260	2.257	-2.9	high
1^3D_3	1.790	1.794	+4.1	high
2^3D_3	2.240	2.237	-3.0	high
1^3G_3	2.460	2.458	-1.8	medium
1^1F_3	2.120	2.130	+10.0	high
1^3F_3	2.150	2.143	-6.6	high
1^3F_4	2.110	2.108	-1.8	high
1^1G_4	2.410	2.422	+12.0	medium
1^3G_4	2.440	2.433	-6.7	medium
1^3G_5	2.390	2.388	-1.8	medium

Contribution Breakdown (diagnostic)

The central, contact, spin-orbit, and tensor columns are operator contributions in the same fully diagonalized fixed-sector eigenstate. They are not differences from a separate central-only solve.

state	central contribution GeV	contact MeV	L·S(vec) MeV	L·S(Thomas) MeV	L·S total MeV	tensor MeV	annihilation MeV	spin + annihilation MeV	predicted GeV
1^1S_0	0.876	-415.1	+0.0	+0.0	+0.0	+0.0	+0.0	-415.1	0.461
2^1S_0	1.536	-82.6	+0.0	+0.0	+0.0	+0.0	+0.0	-82.6	1.453
3^1S_0	2.073	-58.9	+0.0	+0.0	+0.0	+0.0	+0.0	-58.9	2.014
1^3S_1	0.821	+81.4	+0.0	+0.0	+0.0	+0.0	+0.0	+81.4	0.903
2^3S_1	1.545	+34.0	+0.0	+0.0	+0.0	+0.0	+0.0	+34.0	1.579
1^3D_1	1.806	+2.5	-113.3	+89.3	-24.0	-8.7	+0.0	-30.2	1.777
3^3S_1	2.080	+26.1	+0.0	+0.0	+0.0	+0.0	+0.0	+26.1	2.106
2^3D_1	2.246	+2.3	-67.4	+74.4	+7.0	-4.8	+0.0	+4.5	2.251
1^3P_0	1.405	+14.5	-232.2	+93.9	-138.4	-48.1	+0.0	-171.9	1.234
2^3P_0	1.920	+8.3	-95.8	+75.8	-20.0	-18.4	+0.0	-30.1	1.890
1^1P_1	1.386	-34.6	+0.0	+0.0	+0.0	+0.0	+0.0	-34.6	1.352
1^3P_1	1.387	+11.4	-104.6	+49.8	-54.8	+22.5	+0.0	-20.9	1.366
2^1P_1	1.923	-26.1	+0.0	+0.0	+0.0	+0.0	+0.0	-26.1	1.895
2^3P_1	1.921	+7.8	-47.7	+37.3	-10.4	+9.2	+0.0	+6.6	1.930
1^3P_2	1.391	+7.4	+87.3	-54.1	+33.2	-4.0	+0.0	+36.7	1.428
2^3P_2	1.922	+6.9	+47.6	-36.5	+11.1	-1.9	+0.0	+16.1	1.938
1^3F_2	2.137	+0.6	-67.3	+84.1	+16.7	-3.3	+0.0	+14.1	2.151
1^1D_2	1.796	-5.7	+0.0	+0.0	+0.0	+0.0	+0.0	-5.7	1.790
1^3D_2	1.797	+2.0	-34.4	+31.3	-3.1	+8.2	+0.0	+7.0	1.804
2^1D_2	2.244	-6.2	+0.0	+0.0	+0.0	+0.0	+0.0	-6.2	2.236
2^3D_2	2.244	+2.0	-21.7	+25.0	+3.3	+4.7	+0.0	+10.0	2.257
1^3D_3	1.800	+1.4	+60.9	-66.4	-5.5	-2.1	+0.0	-6.2	1.794
2^3D_3	2.245	+1.8	+41.6	-50.4	-8.8	-1.3	+0.0	-8.3	2.237
1^3G_3	2.426	+0.2	-45.5	+79.1	+33.6	-1.6	+0.0	+32.2	2.458
1^1F_3	2.132	-1.4	+0.0	+0.0	+0.0	+0.0	+0.0	-1.4	2.130
1^3F_3	2.132	+0.5	-15.7	+21.9	+6.2	+3.9	+0.0	+10.6	2.143
1^3F_4	2.135	+0.4	+43.2	-68.9	-25.7	-1.2	+0.0	-26.5	2.108
1^1G_4	2.423	-0.4	+0.0	+0.0	+0.0	+0.0	+0.0	-0.4	2.422
1^3G_4	2.423	+0.1	-8.6	+16.4	+7.8	+2.2	+0.0	+10.1	2.433
1^3G_5	2.425	+0.1	+32.3	-68.1	-35.9	-0.7	+0.0	-36.5	2.388

Fine-Structure Mass Convention (audit note)

Unequal-mass diagonal fine structure still uses the symmetric L·S contraction. Same-J ^1L_J/^3L_J rows are then corrected by the antisymmetric spin-orbit block when both partner rows are present; other unequal-mass rows remain under the symmetric convention.

state	m1 GeV	m2 GeV	convention
1^1S_0	0.220000	0.419000	disabled
2^1S_0	0.220000	0.419000	disabled
3^1S_0	0.220000	0.419000	disabled
1^3S_1	0.220000	0.419000	disabled
2^3S_1	0.220000	0.419000	disabled
1^3D_1	0.220000	0.419000	unequal_mass_equal_share_LdotS
3^3S_1	0.220000	0.419000	disabled
2^3D_1	0.220000	0.419000	unequal_mass_equal_share_LdotS
1^3P_0	0.220000	0.419000	unequal_mass_equal_share_LdotS
2^3P_0	0.220000	0.419000	unequal_mass_equal_share_LdotS
1^1P_1	0.220000	0.419000	unequal_mass_same_j_mixed
1^3P_1	0.220000	0.419000	unequal_mass_same_j_mixed
2^1P_1	0.220000	0.419000	unequal_mass_same_j_mixed
2^3P_1	0.220000	0.419000	unequal_mass_same_j_mixed
1^3P_2	0.220000	0.419000	unequal_mass_equal_share_LdotS
2^3P_2	0.220000	0.419000	unequal_mass_equal_share_LdotS
1^3F_2	0.220000	0.419000	unequal_mass_equal_share_LdotS
1^1D_2	0.220000	0.419000	unequal_mass_same_j_mixed
1^3D_2	0.220000	0.419000	unequal_mass_same_j_mixed
2^1D_2	0.220000	0.419000	unequal_mass_same_j_mixed
2^3D_2	0.220000	0.419000	unequal_mass_same_j_mixed
1^3D_3	0.220000	0.419000	unequal_mass_equal_share_LdotS
2^3D_3	0.220000	0.419000	unequal_mass_equal_share_LdotS
1^3G_3	0.220000	0.419000	unequal_mass_equal_share_LdotS
1^1F_3	0.220000	0.419000	unequal_mass_same_j_mixed
1^3F_3	0.220000	0.419000	unequal_mass_same_j_mixed
1^3F_4	0.220000	0.419000	unequal_mass_equal_share_LdotS
1^1G_4	0.220000	0.419000	unequal_mass_same_j_mixed
1^3G_4	0.220000	0.419000	unequal_mass_same_j_mixed
1^3G_5	0.220000	0.419000	unequal_mass_equal_share_LdotS

Same-J Antisymmetric Spin-Orbit Mixing

Rows below use the mixed eigenvalues from the (^1L_J, ^3L_J) mass block. Components are ordered as singlet/triplet in the unmixed basis.

state	unmixed GeV	mixed GeV	offdiag MeV	theta deg	singlet component	triplet component
1^1P_1	1.352	1.352	-1.1	+4.32	+0.997	+0.075
1^3P_1	1.366	1.366	-1.1	+4.32	-0.075	+0.997
2^1P_1	1.897	1.895	-8.3	+14.21	+0.969	+0.245
2^3P_1	1.928	1.930	-8.3	+14.21	-0.245	+0.969
1^1D_2	1.791	1.790	-3.2	+12.85	+0.975	+0.222
1^3D_2	1.804	1.804	-3.2	+12.85	-0.222	+0.975
2^1D_2	2.238	2.236	-6.9	+20.17	+0.939	+0.345
2^3D_2	2.254	2.257	-6.9	+20.17	-0.345	+0.939
1^1F_3	2.131	2.130	-2.9	+12.60	+0.976	+0.218
1^3F_3	2.143	2.143	-2.9	+12.60	-0.218	+0.976
1^1G_4	2.422	2.422	-2.1	+10.82	+0.982	+0.188
1^3G_4	2.433	2.433	-2.1	+10.82	-0.188	+0.982

Same-J Tensor Mixing

Rows below use the mixed eigenvalues from triplet L=J-1 / L=J+1 tensor blocks. Components are ordered as lower-L/higher-L in the unmixed basis.

state	unmixed GeV	mixed GeV	offdiag MeV	lower-L component	higher-L component
2^3S_1	1.579	1.579	+7.8	+0.999	-0.039
1^3D_1	1.776	1.777	+7.8	+0.039	+0.999
3^3S_1	2.106	2.106	+2.3	+1.000	-0.016
2^3D_1	2.251	2.251	+2.3	+0.016	+1.000
2^3P_2	1.938	1.938	+1.2	+1.000	-0.006
1^3F_2	2.151	2.151	+1.2	+0.006	+1.000
2^3D_3	2.237	2.237	+0.2	+1.000	-0.001
1^3G_3	2.458	2.458	+0.2	+0.001	+1.000

Mean absolute residual: 4.8 MeV. Max absolute residual: 14.4 MeV.

Spin-Averaged Diagnostics

Weighted by 2J+1 within each available (n, L) group.

multiplet	states	reference GeV	baseline GeV	residual MeV
1D	4	1.791	1.793	+2.0
2D	4	2.244	2.244	-0.2
1F	4	2.130	2.130	+0.5
1G	4	2.421	2.422	+0.4
1P	4	1.379	1.377	-2.0
2P	4	1.923	1.922	-1.8
1S	2	0.792	0.792	-0.1
2S	2	1.548	1.547	-0.1
3S	2	2.087	2.083	-4.6

The spin-independent central path is the current GI reproduction candidate. Remaining heavy-quarkonium residuals should be read mainly as spin-dependent/operator-ordering and extraction-audit targets.