BGS VLIM log M_* ≥ 10.75, z < 0.31 — detailed systematic analysis

Intermediate-mass BGS VLIM sample (2,802,710 galaxies, z < 0.31). The LRT rejects the additive null at NSIDE 64 (\(\lambda_{\rm LR} = 89.1\)).

Sample statistics 

Parameter	NSIDE 32	NSIDE 64	NSIDE 128	NSIDE 256
Stellar-mass threshold	log M* ≥ 10.75	log M* ≥ 10.75	log M* ≥ 10.75	log M* ≥ 10.75
Redshift limit	z < 0.31	z < 0.31	z < 0.31	z < 0.31
N_gal	2,802,710	2,802,710	2,802,710	2,802,710
N_pix (good footprint)	5,618	21,662	84,824	332,812
N_templates	11	11	11	11
MCMC walkers	210	210	210	210
MCMC steps after burn-in	1500	1500	1500	1500

Goodness-of-fit: \(\hat{\sigma}\) by method and resolution 

The noise parameter \(\hat{\sigma}\) measures residual scatter after systematic subtraction — lower is better. Results are shown for NSIDE 32, 64, 128, and 256. ISD-3 is unavailable at NSIDE 128 and 256 (no partial files generated at those resolutions).

Method	\(\hat{\sigma}\) (N32)	\(\hat{\sigma}\) (N64)	\(\hat{\sigma}\) (N128)	\(\hat{\sigma}\) (N256)	Notes
OLS	0.3057	0.2831	0.4190	0.6422	closed-form least-squares
ElasticNet	0.3069	0.2831	0.4194	0.6423	L1+L2 regularised; 3-fold CV
ISD-1	0.3057	0.2831	0.4190	0.6422	iterative self-calibration; poly order 1
ISD-3 †	0.6315	0.4658	0.9265	0.6894	† degree-3 polynomial; unavailable at NSIDE 128/256
MCMC-add	0.3061	0.2831	0.4191	0.6422	MCMC additive; acc N32=0.388 N64=0.392
MCMC-comb	0.3982	0.3009	0.4300	0.6305	MCMC combined; acc N32=0.295 N64=0.281

† ISD-3 uses a degree-3 polynomial expansion and is ill-conditioned with correlated templates.

Likelihood Ratio Test (additive vs combined model)

Resolution	\(\lambda_{\rm LR}\)	dof	p-value	Reject H₀
NSIDE 32	1169.2	11	< 10^-244	Yes
NSIDE 64	89.1	11	< 10^-14	Yes
NSIDE 128	287.4	11	< 10^-55	Yes
NSIDE 256	580.6	11	< 10^-117	Yes

MCMC acceptance fractions: NSIDE 32: add 0.388, comb 0.295 NSIDE 64: add 0.392, comb 0.281 NSIDE 128: add 0.388, comb 0.293 NSIDE 256: add 0.388, comb 0.297. Healthy range: 0.15–0.50.

Template amplitude ranking — additive model (MCMC-add, NSIDE 64)

All 11 templates sorted by absolute MCMC-add additive amplitude \(|\hat{a}_i|\). OLS shown for comparison.

Rank	Template	NSIDE	\(\hat{a}_i\) (MCMC-add)	\(\hat{a}_i\) (OLS)	Physical meaning
1	GAIA_nstar_faint	64	+0.3222	+0.3268	GAIA faint stellar density (photometric mis-classification of faint stars as galaxies)
2	GAIA_nstar_medium	64	-0.2216	-0.2251	GAIA medium stellar density (crowding and deblending near bright stars)
3	GAIA_phot_rp_mean_flux	64	-0.0415	-0.0418	GAIA mean stellar flux in RP band (red scattered light)
4	GAIA_phot_g_mean_flux	64	+0.0290	+0.0292	GAIA mean stellar flux in G band (scattered-light / sky-background variations)
5	GAIA_phot_bp_mean_flux	64	-0.0149	-0.0146	GAIA mean stellar flux in BP band (blue scattered light)
6	LS10_EBV_NSIDE_0064	?	-0.0122	-0.0122	LS10_EBV_NSIDE_0064
7	LS10_GALDEPTH_R_NSIDE_0064	?	+0.0076	+0.0075	LS10_GALDEPTH_R_NSIDE_0064
8	LS10_PSFSIZE_R_NSIDE_0064	?	-0.0047	-0.0048	LS10_PSFSIZE_R_NSIDE_0064
9	LS10_GALDEPTH_Z_NSIDE_0064	?	+0.0045	+0.0045	LS10_GALDEPTH_Z_NSIDE_0064
10	LS10_NOBS_R_NSIDE_0064	?	-0.0023	-0.0022	LS10_NOBS_R_NSIDE_0064
11	LS10_GALDEPTH_G_NSIDE_0064	?	+0.0017	+0.0016	LS10_GALDEPTH_G_NSIDE_0064

Template amplitude ranking — multiplicative model (MCMC-comb, NSIDE 64)

All 11 templates sorted by absolute MCMC-comb multiplicative amplitude \(|\hat{b}_i|\).

Rank	Template	NSIDE	\(\hat{b}_i\) (MCMC-comb)	Physical meaning
1	GAIA_nstar_faint	64	+0.6170	GAIA faint stellar density (photometric mis-classification of faint stars as galaxies)
2	GAIA_nstar_medium	64	-0.3227	GAIA medium stellar density (crowding and deblending near bright stars)
3	GAIA_phot_rp_mean_flux	64	-0.0498	GAIA mean stellar flux in RP band (red scattered light)
4	GAIA_phot_g_mean_flux	64	+0.0250	GAIA mean stellar flux in G band (scattered-light / sky-background variations)
5	LS10_NOBS_R_NSIDE_0064	?	+0.0166	LS10_NOBS_R_NSIDE_0064
6	LS10_PSFSIZE_R_NSIDE_0064	?	+0.0129	LS10_PSFSIZE_R_NSIDE_0064
7	LS10_EBV_NSIDE_0064	?	+0.0117	LS10_EBV_NSIDE_0064
8	LS10_GALDEPTH_R_NSIDE_0064	?	+0.0109	LS10_GALDEPTH_R_NSIDE_0064
9	GAIA_phot_bp_mean_flux	64	-0.0108	GAIA mean stellar flux in BP band (blue scattered light)
10	LS10_GALDEPTH_Z_NSIDE_0064	?	-0.0076	LS10_GALDEPTH_Z_NSIDE_0064
11	LS10_GALDEPTH_G_NSIDE_0064	?	-0.0050	LS10_GALDEPTH_G_NSIDE_0064

Per-galaxy weight statistics (NSIDE 64)

From the *_NSIDE0064_WEIGHTS.fits file. Mean ≈ 1 and small std indicate a well-behaved weight distribution.

Method	N	mean	std	p1	p5	p50	p95	p99
OLS	2,802,710	0.9976	0.0325	0.9371	0.9566	0.9932	1.0548	1.0866
ElasticNet	2,802,710	0.9970	0.0311	0.9376	0.9573	0.9931	1.0509	1.0817
ISD-1	2,802,710	0.9975	0.0326	0.9368	0.9565	0.9932	1.0548	1.0869
ISD-3 †	2,802,710	1.4256	6.3201	0.6910	0.8561	0.9801	1.2456	1.7699
MCMC-add	2,802,710	0.9975	0.0324	0.9368	0.9567	0.9932	1.0544	1.0861
MCMC-comb	2,802,710	1.0658	0.0391	0.9604	1.0028	1.0661	1.1213	1.1747

Systematic weight maps 

Mollweide weight maps — all six methods

weight maps NSIDE 32 weight maps — NSIDE 32 weight maps

weight maps NSIDE 64 weight maps — NSIDE 64 weight maps

weight maps NSIDE 128 weight maps — NSIDE 128 weight maps

weight maps NSIDE 256 weight maps — NSIDE 256 weight maps

Systematic weight distributions 

Narrow peaks near 1 indicate stable weight estimates. ElasticNet weights may be exactly 1 when cross-validation selects zero amplitudes.

Per-galaxy weight distributions — all six methods

weight distributions NSIDE 32 weight distributions — NSIDE 32 weight distributions

weight distributions NSIDE 64 weight distributions — NSIDE 64 weight distributions

weight distributions NSIDE 128 weight distributions — NSIDE 128 weight distributions

weight distributions NSIDE 256 weight distributions — NSIDE 256 weight distributions

Angular clustering w(θ) before and after correction 

Each panel shows the observed angular two-point correlation function (solid black) and the corrected \(w(\theta)\) for all six methods. A well-corrected sample shows suppressed excess clustering at all scales. Each panel corresponds to one map resolution.

w(θ): observed vs corrected — all six methods

Cosmological analysis verdict 

Sub-degree scales: unknown (no empirical two-point measurement available).

Without correction: uncertain.
After correction: suitable after applying WEIGHT_COMB.

LRT (NSIDE 64): \(\lambda_{\rm LR} = 89.1\) (dof = 11), p = 2.5e-14 → Reject H₀ — multiplicative contamination is statistically detected.

Recommendation: use WEIGHT_COMB (WEIGHT_SYS) for all science-grade analyses.

BGS VLIM log M* ≥ 10.75, z < 0.31 — detailed systematic analysis

BGS VLIM log M_* ≥ 10.75, z < 0.31 — detailed systematic analysis