Paper 1 · W4 · S23 camera 4 CCD 2 · 2026-10-08

Why C4 keeps a blue/red colour residual

The paper1_final reference fit on C4 (tess2020087042920) leaves a colour residual that the other four Paper 1 fields do not show. This page covers Phase A and Phase B: what the residual is, what causes it, and how large it is in photometry. Phase B repeated the same measurements on the final e2e run3 C4 products and found no change.

Conclusion

The residual is a property of camera 4 CCD 2's PSF. Crowding, blended Gaia colours and the template are ruled out as causes. At 2–4 px from a red star (BP−RP 1.5–2.2) the data hold about 5–7 ppt of the star's flux less than the A3 model predicts, relative to mid-colour stars. Blue stars show the mirror excess. The other four fields show about 1 ppt. The raw FFI of the same CCD in a different sector shows the same deficit (−7.4 ± 1.0 ppt, against about −1.7 on raw FFIs of other CCDs).

Photometry: on the exact 15×15 fit stamps this is a redistribution of light (2–4 px: −5.1 ± 0.5 ppt; 4–7 px: +5.0 ppt). It adds no net flux bias beyond what every field already has. Separately, all five fields under-model red stars' far wing (4–7 px, +2 to +6 ppt), so red stars' fitted flux is 0.4–0.8% low relative to mid stars. That generic term belongs to the red-core work (W5).

Confidence: high that it is the camera-4 CCD-2 PSF and not crowding, colour or template. The raw-FFI replication is about 4.4σ against the other CCDs, and the C4 products alone are more than 10σ from the other fields. The physical mechanism inside the camera is not established. Next: Phase B (automatic) checks the final per-band products. Phase C, which would add a model term able to express the halo (for example the existing chroma_halo leaf), runs only on request.

The residual

C4 residual grid: colour by Tmag stacks in a 3 by 3 CCD grid — **Standard residual grid, C4 product fit (in-sample).** Rows are Tmag 8–10, 10–12 and 12–13; columns are BP−RP blue, mid and red; the outer 3×3 is the CCD. Mid-colour stamps are clean. Blue stars show a broad positive lobe toward the upper left and red stars a broad negative one, in every cell. On this CCD the optical axis lies just outside the upper-left corner, at about (−60, 2090), so "toward the axis" points up and left from every cell.

Stacked residuals blue minus mid and red minus mid for four data sets — **Stacked residual/flux images behind the numbers** (data − model, T10–13, 5-MAD-clipped mean, one colour scale ±6×10⁻⁴ per px²). Row 1: C4 held-out stars. Row 2: F1 held-out stars (also a CCD 2, same axis geometry). Row 3: raw FFI of camera 4 CCD 2, sector 20, no template. Row 4: raw FFI of camera 3 CCD 3. Rows 1 and 3 match: red − mid has a negative crescent at 2–3.5 px on the axis side, and blue − mid the mirror image. Rows 2 and 4 are weak and close to round. Right column: azimuthal profiles of the same stacks.

Method

Each star's residual stamp $R(\mathbf{u})$ is the data minus the model, divided by the star's fitted flux. It is sampled on a 33×33 raster with 0.25 px cells out to ±4 px. Every star is fitted with the same basis, built from the field's median model image $P$:

$$R(\mathbf{u}) \approx a\,P + b_x\,\partial_x P + b_y\,\partial_y P + w_x\,\partial_x^2 P + w_y\,\partial_y^2 P + c,\qquad |\mathbf{u}|<3.6\ \text{px}$$

$\mathbf{u}$: position relative to the star centre [TESS px].
$P$: median model/flux image of T10–13 stars in the field, the same for every star.
$a$: flux error, as a fraction of the star's flux.
$(d_x,d_y) = -(b_x,b_y)$: centroid displacement of the data relative to the model [px], rotated into $d_{ax}$ (toward the optical axis) and $d_{tan}$ (perpendicular to it).
$w_x, w_y$: width terms [px²]; negative means the data are sharper than the model.
$c$: flat pedestal [fraction of flux per px²].

$$I_{\rm ring} = A_{\rm ring}\,\langle R\rangle_{\rm ring}$$

$I_{\rm ring}$: residual light in a ring (r<1, 1–2, 2–3.6 px), as a fraction of the star's flux.
$A_{\rm ring}$: ring area [px²].
$\langle R\rangle_{\rm ring}$: mean of the filled raster cells in the ring. The rasters fill only about 25% of cells per star; see Bugs below.

Red − mid is the median over red stars (BP−RP 1.5–2.2) minus the median over mid stars (BP−RP 0.85–1.15). Errors are 1σ bootstrap over stars. Stars are binned by Gaia BP−RP throughout; the fit itself uses the Gaia-XP colour $u$, which on C4 is typically 0.02 mag from BP−RP. The exact-pixel ring test skips the raster: for each star it divides Σ residual over the valid pixels of the 15×15 fit stamp in a ring by the star's fitted flux. Sign and scale were checked on synthetic shifts (0.02 and −0.03 px recovered to 3%) and on a sparse 1% flux error.

Tests

Cause tested	Test	Result	Verdict
H1 blended Gaia colours pull the colour terms toward zero	A2 colour coefficients; A3 BP/RP contamination within 4″, u − BP−RP	C4 terms match the other fields and are stable across folds; 390 of 409 red stars have no contaminant; deficit flat across flags	ruled out
H2 environment: neighbour or background light that correlates with colour	A1b scaling with Tmag; A3 neighbour light <2 px, 2–4 px, density	Deficit is a fixed fraction of the star's flux (e⁻/s falls about 20×); flat in every environment flag	ruled out
H3 camera-4 optics or detector	A4 the same metric on 11 chroma_v2 A3 lanes, including raw FFIs	Every camera-4 CCD-2 fit shows it (raw S20 −7.4, old C4 lane −5.2, C4 product −5.45); 13 other fits −0.6 to −2.7	supported
H4 template colour mismatch	A4 raw FFI (no template); Phase B per-band final image	Present without any template; unchanged with the new per-band templates	ruled out (Phase B)

Per-star residual features versus colour for five fields — **A1. Per-star features against colour** (held-out stars, T10–13, median per bin). The 2–3.6 px integral (bottom left) and the pedestal split C4 (thick red) from the other four fields at both ends of the colour range. C4's slope is −9.0 ± 0.3 ppt per mag of BP−RP, against −2.4 to −2.9. The flux term $a$ mirrors it. The core integral, the 1–2 px integral, the shifts and the widths look like the other fields: C4's shift toward the axis is +0.9 ± 0.3 mpx per mag, no larger than elsewhere.

Red minus mid halo deficit versus Tmag and per CCD cell — **A1b. Light from the star, nearly uniform on the CCD.** Left: as a fraction of the star's flux, C4's red − mid halo deficit is −4.1 to −6.7 ppt from T8.75 to T12.6 (other fields −0.5 to −2). Centre: in e⁻/s it falls about 20× over the same range, which background or neighbour light would not do. Right: the deficit is −5.4 to −7.9 ppt in 8 of 9 cells and −3.0 ± 1.0 in the cell nearest the optical axis.

Red minus mid halo deficit in tertiles of blending flags — **A3. Blending and environment, fixed colour.** Tertiles of red stars by neighbour light inside 2 px, neighbour light in 2–4 px, Gaia G contamination within 4″, u − BP−RP, and local density. C4 (red) stays at −5.5 to −6.8 ppt across every flag; F1 is shown for reference. Only u − BP−RP shows a weak 2.5σ trend.

Red minus mid halo deficit across sixteen fits — **A4. Specific to camera 4 CCD 2.** The same metric on the chroma_v2 A3 lanes (in-sample) and on the five paper1_final products. All three fits on camera 4 CCD 2 (red) stand apart: the raw S20 FFI with no template, the older C4 difference-image lane, and the C4 product. The other 13 fits on cameras 1–3 sit between −0.6 and −2.7 ppt.

Exact stamp-pixel ring test, red minus mid, five fields — **Photometric size, exact stamp pixels** (product fits; ring residual divided by the star's fitted flux). r<2 px: no colour difference in any field. 2–4 px: C4 −5.1 ± 0.5 ppt, others −0.2 to −0.8. 4–7 px: every field +2.4 to +6.0 (C4 +5.0). Whole stamp: C4 +2.2 ± 1.3, others +4.4 to +8.3. C4's feature moves light outward; the outer excess common to all fields is a separate, generic red far-wing term.

A3 colour-term coefficients per field — **A2. Colour coefficients.** Product values, with error bars showing the spread over the five folds. C4's shift polynomial, kurtosis, blur, dilation and colour planes are in family with the other fields. The CCD colour translation is frozen at (0, 0) in every field.

Ring average or one-sided? (follow-up)

To check whether this is a halo, each star's residual is split per 0.25 px ring in the axis frame, with $\theta = 0$ pointing from the star to the optical axis:

$$R(r,\theta) \approx m_0(r) + c_1(r)\cos\theta + s_1(r)\sin\theta + c_2(r)\cos 2\theta$$

$m_0$: ring average.
$c_1$: one-sided part along the axis; positive means more light toward the axis.
$s_1$: one-sided part across the axis.
$c_2$: radial-minus-tangential elongation.
$I_0 = 2\pi r\,\Delta r\,m_0$: ring light [fraction of flux].
$D_1 = \pi r^2\,\Delta r\,c_1$: the ring's contribution to the residual light's centroid along the axis [px].

Ring-average, one-sided and elongation profiles versus radius — **Profiles, T10–13.** Thick solid: camera 4 CCD 2 (C4 held-out, raw S20 FFI, older C4 lane). Dashed: other CCDs. Ring average (left): every CCD has the same red − mid shape, positive at about 1.1 px and negative at 1.8–3.5 px. Camera 4's outer trough is about 5× deeper and wider. One-sided part (centre): red shoulder and wing light sits away from the axis relative to the model on every CCD, about 2× more on camera 4. Elongation (right): small beyond 2 px.

Rotated red minus mid stacks split into ring average and remainder — **Rotated red − mid stacks** (axis to the right, one colour scale). Left: the stack. Middle: its ring average. Right: the remainder, the one-sided and higher parts. For C4 and the raw camera-4 FFI, the ring average is a strong positive ring at about 1 px inside a negative band at 2–3.5 px, and the remainder is positive on the side away from the axis and negative toward it. Raw camera 3 and F1 show weaker versions.

red − mid	other paper fields	raw cam2 / cam3	C4	raw cam4 CCD2	old C4 lane
$I_0$, 2–3.75 px [ppt]	−1.1…−1.3	−1.0, −1.9	−6.5 ± 0.2	−8.1 ± 0.5	−5.8 ± 0.4
$D_1$, 2–3.75 px [mpx]	−2.3…+1.7	−5.0, −5.0	−10.4 ± 0.3	−9.3 ± 0.8	−5.6 ± 0.5
$D_1$, 1–2 px [mpx]	−1.8…−2.9	−1.1, −3.4	−4.8	−5.8	−3.7

Reading: this is not a halo. On every CCD, the shoulder and inner wing (1–3.5 px) change with colour more than the A3 terms allow, both symmetrically and as coma. Camera 4 CCD 2 shows the same pattern 2–5× stronger. The A3 shift, blur, dilation and kurtosis are fitted per CCD but constrained mostly by the bright core, so the wing cannot follow colour on its own. One model change for all CCDs (a colour × wing-profile term and a colour × coma term) would let each CCD's fit set its own amplitude. This is untested (Phase C, on request).

Key numbers

Fit	red − mid, 2–3.6 px [ppt]	2–4 px, exact pixels [ppt]	whole stamp [ppt]	halo slope [ppt per mag BP−RP]
F1 (S24 cam2 CCD2)	−1.36 ± 0.36	−0.71 ± 0.28	+4.38 ± 1.03	−2.40 ± 0.24
F2 (S20 cam3 CCD3)	−1.00 ± 0.32	−0.79 ± 0.36	+6.40 ± 0.76	−2.87 ± 0.25
S22 (S22 cam3 CCD3)	−1.08 ± 0.29	−0.17 ± 0.35	+6.16 ± 0.77	−2.73 ± 0.25
C1 (S24 cam1 CCD2)	−1.17 ± 0.26	−0.36 ± 0.41	+8.26 ± 1.06	−2.80 ± 0.24
C4 (S23 cam4 CCD2)	−5.45 ± 0.32	−5.08 ± 0.46	+2.24 ± 1.33	−9.03 ± 0.27
Raw FFI S20 cam4 CCD2 (chroma_v2 A3)	−7.41 ± 0.95	–	–	–

Column 2: product rasters, in-sample, T10–13. Columns 3–4: exact-pixel ring test, product fits, T10–13. Column 5: held-out stars, T10–13, OLS slope over BP−RP 0.5–2.2.

Bugs found and fixed

Both were in this investigation's own analysis code and were fixed before any number was reported.

Sparse-raster integrals 4× too small. Each star's raster fills only about 25% of its cells, so the first nansum × H² ring integrals undercounted by 4×. They are now computed as area × mean of filled cells; a test is in code/test_sign.py. The fitted coefficients and all field-to-field ratios were unaffected.
Wrong normalisation in the first ring test. It divided by Σ model in the ring, but the stamp model includes neighbours' light. It now divides by the star's fitted flux.

Phase B: the final run3 C4 products

A read-only watcher reran these measurements once run3 C4 finished (10-08 19:56), using the run's own code pin. Run3 uses the verified ledger neighbours, new per-band templates and a refit on the final image. The residual is unchanged.

Held-out blue minus mid and red minus mid stacks and azimuthal profiles, reference against run3 final — **Held-out stacks, reference (top) vs run3 final (bottom).** Blue − mid and red − mid residual divided by flux, T10–13, on one colour scale (±6×10⁻⁴ per px²), with azimuthal means on the right. The two rows look the same: the red deficit and blue excess sit at 2–3.6 px with the same upper-left lobe. Red − mid halo: −6.34 ± 0.37 ppt in the reference, −5.81 ± 0.42 in run3 (0.9σ apart).

red − mid, C4, T10–13	reference	run3 boot	run3 final
held-out halo 2–3.6 px [ppt]	−6.34 ± 0.37	−6.16 ± 0.31	−5.81 ± 0.42
exact stamp ring 2–4 px [ppt]	−5.08 ± 0.46	−5.13 ± 0.43	−4.91 ± 0.37
exact stamp ring 4–7 px [ppt]	+5.01 ± 0.62	+4.88 ± 0.67	+4.50 ± 0.54
whole 15×15 stamp [ppt]	+2.2 ± 1.3	+2.2 ± 1.3	+1.9 ± 1.0
radial colour shift d_ax [mpx]	+1.2 ± 0.5	+1.4 ± 0.4	+1.3 ± 0.4

All 17 A3 colour terms in run3's fit and refit lie within about one fold-to-fold σ of the reference, so they are not pulled toward zero. This rules out the template and the neighbour list as causes and confirms the camera-4 PSF explanation. The run3 per-band kernels are distilled from this ePSF, which fits C4's larger red-core residual and its failed red-star G4 criterion (see the C4 G4 canvas). A model term that can express it (Phase C) runs only on request.

Records

Run record: /astro/armin/koji/syndiff/dev_runs/c4_colour_dipole_20261008/README.md (code in code/, figures in figs/, numbers in out/)
Inputs: dev_runs/paper1_final_fits_20261007/eval/ and dev_runs/chroma_v2_20260928/eval/*_A3_u_quad_elong/
Shared with other sessions: A2 table to the W3 lead; blending flags out/a3_blending_flags_{C4,F1}.csv and the generic red-wing finding to the red-core session
Canvas source: /home/kshukawa/syndiff_pipeline/doc/c4_colour_dipole_canvas/