Fitting PS1 weights to flux and dipole — 2026-10-19

Conclusion Four newly fitted objectives were compared on the same spatially held-out F1 stars. Flux-only fitting leaves a strong dipole; dipole-only suppresses it while retaining a red flux deficit. Joint objectives trade between the two. No tested weight mixture closes both flux and spatial residuals. These are conditional tests of the historical October 9 images with frozen band kernels, not evidence for changing production weights or for the current paper products.

Inputs and execution Parent record: ../README.md. Same F1 FFI, band images, background and catalogue as the initial three-way comparison. Exact paths in compare.py; read-only producing code worktree HEAD 157dc95d9048bf630569094bfb1c600ca20b6cef. Script and products live in this directory. No ePSF retraining, scene-fit evaluation or manuscript edit.

source /astro/armin/koji/syndiff/dev_runs/faint_cog_20261009/scripts/env.sh
python /astro/armin/koji/syndiff/dev_runs/weight_residual_stacks_20261019/objective_comparison/compare.py

Comparison contract Only T14–16 and BP−RP 0.3–3.0, same purity selection as parent. Split by parity of 128-pixel image tiles, with counts recorded below; all new weights use only training stars. Evaluation stars, coordinates, sky subtraction, bilinear stamp interpolation, pixel support and catalogue-flux denominators are identical between arms. Previously saved weight candidates may have seen these stars in their original fitting: their columns are reference comparisons, not fully independent validation. The ePSF and background are frozen historical products, not retrained on this split. Neighbour light and model dependence remain possible; checkerboard splitting is not independence from all shared scene systematics.

One exact global normalization constraint per arm matches the mean radius-3 aperture flux of mid-colour (BP−RP 0.8–1.2) TRAINING stars. This prevents dipole-only fitting from arbitrarily changing overall brightness. It differs from the previous diagnostic's all-sample median normalization, so absolute numbers should not be spliced between the two experiments.

Method The residual stamp is $$R_s(\mathbf{x})=D_s(\mathbf{x})-\sum_b\alpha_b X_{bs}(\mathbf{x}).$$ Here $s$ indexes a selected star, $\mathbf{x}$ is relative position in native TESS pixels, $b$ indexes r/i/z/y, $D_s$ is the measured sky-subtracted stamp divided by fixed catalogue flux, $X_{bs}$ is the similarly normalized per-band stamp INCLUDING its original adopted weight, and $\alpha_b\ge0$ is a fitted dimensionless band multiplier. Stamp values are dimensionless fractions of catalogue flux per native pixel. Reported sum-normalized synthesis weights are $w_b=\alpha_b w_b^0/\sum_c\alpha_c w_c^0$, where $w_b^0$ is the adopted weight and $c$ also indexes bands. The denominator is the fitted overall scale, recorded in results.json.

A fixed linear projection inside radius 3 pixels decomposes each residual into the adopted model profile, its x/y gradients, and a constant. All four basis columns are solved simultaneously, separating gradient coefficients from profile amplitude and constant sky. The adopted model includes the frozen local scene; consequently these are gradient-mode coefficients, not independent astrometric centroid measurements. The x/y gradient coefficients $d_{xs},d_{ys}$ have equivalent-displacement units of pixels. Their signs follow the gradient basis, not an asserted physical displacement direction. An asymmetric profile mismatch can contribute to them.

The joint objective is $$J=\sum_s\frac{1}{n_{g(s)}}\left[\left(\frac{f_s}{0.01}\right)^2+\lambda\frac{d_{xs}^2+d_{ys}^2}{(0.01\,{\rm pixel})^2}\right].$$ Here $f_s$ is the data-minus-model radius-3 aperture flux divided by fixed catalogue flux (dimensionless); $g(s)$ assigns a star to one of six colour × magnitude bins; $n_{g(s)}$ is the number of training stars in that bin, giving each bin equal total influence. The denominators are chosen trade-off scales (1% flux and 10 millipixels), NOT estimated measurement errors. $\lambda$ is dimensionless: 0 for flux-only, 1 and 10 for the joint trials. Dipole-only omits the flux term but retains the common mid-colour normalization constraint. Fits use non-negative coefficients and SLSQP on the convex quadratic objective with an exact linear constraint.

Validation and limits All solvers converged from two different starts. A synthetic 0.02-pixel gradient injection is recovered to numerical precision by the fixed dipole projector; exact maximum error and unconstrained design-matrix condition numbers are in results.json. The dipole-only unconstrained matrix is rank deficient by construction: its projector removes the baseline profile, so adding equal band multipliers leaves dipole coefficients unchanged. The independent flux-anchor constraint fixes that amplitude direction; the raw condition number must not be interpreted as a failure of the constrained solve. This tests the implemented projection, not physical PSF correctness or recovery of a genuinely displaced independent PSF. No nonlinear centroid injection has been claimed. Selected stars and train/test membership are saved in sample.csv.

Bin errors in results.json are star-bootstrap scatter with weights and normalization fixed; they omit fitted-weight uncertainty and correlated spatial/calibration errors. No formal significance is attached to small differences between joint methods. The red T14–15 held-out bin has only 18 stars; T15–16 has 37. This is one spatial split on one historical FFI, so no universal optimum is established. Colour/magnitude grids and pooled 3×3 spatial stacks are provided; the sample is too small for precise joint colour × magnitude × cell claims.

Held-out scores Flux score is the RMS of six bin-median aperture residuals, percent of catalogue flux. Dipole score is the RMS length of the six bin-median gradient vectors, in millipixels. Pixel score is the RMS of the median residual-stack pixels inside radius 3, pooled equally across the six bins, percent of catalogue flux per pixel. These measure systematic residual stacks, not per-star noise or reduced chi-square. Lower is better; none is a complete image-quality criterion.

Train: 310; test: 303.

Images (all visually inspected)

The saved and flux-only mixtures leave a coherent diagonal positive–negative core in both magnitude bins. Joint and dipole-only arms attenuate it, but the red negative core returns. The blue and middle-colour positive cores also remain. All residual panels share the same ±1.5% scale; annotations are per-star median aperture/gradient measurements, not sums of median stack pixels.

The data and adopted model use the same 0–20% display scale. Each other row is the median paired residual change from adopted, using the same stars and ±1.5% scale. These show how altering weights moves light spatially as well as changing total subtraction.

Nine detector cells, pooled colours/magnitudes, same ±1.5% scale; counts shown. Spatial heterogeneity persists. Pooled cells are not matched for their colour/magnitude composition.

Interpretation and next test Joint 1 reduces both measured systematic scores relative to adopted on this held-out sample, while the saved weights and new flux-only fit achieve the best aperture-flux scores but worsen the dipole. Joint 10 and dipole-only prioritize the spatial gradient at a cost in red flux. The free bright-star z=0 mixture is competitive in stack pixel RMS, but has remaining flux and dipole structure; no reason to force z=0 follows.

Conclusion: fitting a dipole penalty works, but the fixed four-band mixture has a trade-off on these images. The next discriminating experiment is to allow a separately constrained spatial displacement alongside the spectral weights, keeping held-out evaluation and flux/shape metrics separate, then replicate on current paper products and other fields. A broad conclusion about true bandpass weights is not warranted by this single-field test.