# Paper 1 outline — PS1-assisted calibration of an individual TESS FFI

**Adopted field/exposure (2026-09-22): Sector 24, camera 2, CCD 2; `tess2020120182919-s0024-2-2`, the minimum-background `kernel_fit` reference.** See [data selection and provenance](DATA_SELECTION.md). Earlier fit figures use a different exposure and must be regenerated for this choice.

2026-09-16; structure updated 2026-09-22. Updated to the user's clarified narrative. **Outline only: section purposes, planned content, and required evidence; no manuscript prose or asserted results.** This narrative supersedes the earlier suggestion to make native-FFI fitting the primary method.

**Model freeze (2026-09-29): see [MODEL_FREEZE.md](MODEL_FREEZE.md).** Adopted model = #14 + quadratic colour shift + Gaia-XP colour $u$ + CCD-frame elongation planes (chroma v2 arm A3), moment-blind fine-neighbour prior, all metrics out-of-fold. Brought into Paper 1 on 2026-09-29, each behind a gate: a brightness-dependent width term (§4.6) and a third pass with per-PS1-band templates convolved with ePSF-distilled kernels instead of Hotpants (§5 since 2026-09-30). Where this outline and MODEL_FREEZE.md disagree, MODEL_FREEZE.md governs.

**Pipeline order changed 2026-09-29 ~18:45 (MODEL_FREEZE D15; its F=1 preliminary image is superseded by the 2026-09-30 structure note below):** the ePSF/WCS calibration is fitted on the **F=1** difference image; the F=4 per-band ePSF-kernel template (§5) produces the **final de-crowded image**; a refit on that image (D14) confirms the calibration does not change. The F=4 *Hotpants* image is no longer a fitting input, only a comparison baseline. §3.2 and the narrative below are superseded where they say the final fit uses the F=4 Hotpants image.

**Structure revised 2026-09-30 (user; figure-agent session b53efe8b; second revision the same day). This governs the section layout below.** The "pass 1/2/3" framing is retired. Sections follow the data flow, and each takes the previous one's product:
- §2 builds the PS1 scene as four band templates, mapped onto the TESS pixel grid at F=1 and F=4.
- §3 **Preliminary de-crowded image: PS1 template matched with an empirical kernel.**
- §4 **Scene modelling: simultaneous fit of stellar fluxes, WCS and chromatic ePSF.**
- §5 **Final de-crowded image: per-band templates convolved with the calibrated chromatic ePSF**, followed by forced photometry and targeted de-crowding.
- §6 is discussion and conclusions.
- Appendices: A (F=1 vs F=4), B (colour-model selection), C (band-synthesis verification).

**Terminology (user 2026-09-30):** "preliminary de-crowded image" and "final de-crowded image", not "difference image", because the bright ($T<13$) calibration stars are deliberately still in them. The two differ only in the kernel: one fitted Hotpants kernel $K$ versus calibrated per-band kernels $E_b$. Avoid "bootstrap": it means resampling in statistics.

Paper 1 obtains the ePSF/WCS from F=4 (user 09-30). F=1 is kept for the multi-frame temporal work in Paper 2, where one template per frame at F=4 is too costly; an appendix compares the two. **Open:** every calibration fit so far used the F=1 image, so the delivered ePSF either comes from a refit on the F=4 preliminary de-crowded image or is the D14 refit on the final image (see "Outline decisions").

Working title: **PS1-assisted astrometric and chromatic ePSF calibration of TESS full-frame images**.

## Narrative and terminology

**[Current chain, 2026-09-30:** PS1 scene as per-band templates (F=1 and F=4) → band-combined F=4 template + oversampling-aware Hotpants → preliminary de-crowded image → Gaia-constrained A3 WCS/chromatic ePSF calibration → per-band F=4 template convolved with ePSF kernels → final de-crowded image, and star extraction for photometry → refit confirms the calibration is unchanged.**]** Earlier D15 chain (F=1 preliminary image) superseded. Previous chain: **Understand the instrument → PS1 reference scene → native-scale template matching and de-crowding → improved initial ePSF/WCS → rebuild template at 4× finer sampling → oversampling-aware Hotpants → Gaia-constrained WCS/chromatic ePSF refinement of the second-pass image → pass 3: per-band templates matched with kernels distilled from the calibrated chromatic ePSF (gated, 2026-09-29) → demonstrate the gains.**

The two external data sources have complementary roles: PS1 supplies resolved scene information for removing contaminating light; Gaia supplies astrometric and color constraints for calibration and can also guide bright-source removal during template preparation. Neither is exact TESS-image truth.

Use **synthetic TESS-like reference image**, **high-resolution reference scene**, and **template-assisted crowding reduction** in the outline. “De-crowded image” is useful shorthand after defining which components are removed and which calibration-star signals are retained. Avoid “ground truth” for observed templates; reserve it for known simulated inputs.

**Forward modeling** is appropriate for the refinement: predict pixel values from a source scene, WCS, ePSF, and stellar flux parameters, then compare predictions with background-subtracted data. There is no fitted local background in the final optimizer. **Joint astrometric and chromatic ePSF fitting** is a more specific description. Template construction/crowding reduction and image-level background subtraction are preparation steps; rough centroid/ePSF estimation is initialization.

## 1. Introduction: instrument calibration as a route to more science

**Purpose:** motivate better use of the information in individual TESS images.

Planned points:

- Introduce the untapped scientific potential of TESS: improved instrument calibration and crowding treatment can help extract more of the information already present in its wide-field, repeated imaging. Motivate the opportunity without asserting unmeasured light-curve gains.
- Explain why position, image response, and contaminating light matter for extracting reliable measurements.
- Describe the coupled difficulty of estimating an undersampled ePSF and WCS in a crowded scene.
- Motivate spatial and color dependence of the effective response; acknowledge existing TESS/Kepler/Gaia work from the shared literature review.
- Introduce the complementary roles of a PS1-derived scene and Gaia constraints.
- State the paper's single-FFI question and the measured quantities it will evaluate. Improvement is to be demonstrated, not assumed from an assertion that standard products are inadequate.

**Boundary:** no extended survey of detrending, long-period recovery, orbital light-curve systematics, or promised science discoveries. Science motivation does not require a light-curve demonstration in this paper.

## 2. PS1 reference scene: per-band templates on the TESS grid

**Purpose:** describe the shared scene preparation once, then explain how the two passes render it at different sampling scales.

### 2.1 Inputs, footprint, and units

- Identify one TESS exposure/region, the PS1 images/bands, and catalog inputs.
- Explain selection of overlapping PS1 skycells and their image, mask, and uncertainty products; show the sector footprint and its nested relationship to the PS1 survey and skycells.
- State epochs and registration conventions, skycell overlaps, incomplete band/sky coverage, and quality rules. Catalog epoch propagation and PS1/TESS epoch mismatch are separate issues.
- Specify conversion from stored PS1 values to common linear flux units before band combination, and the later scaling to the TESS science units. Distinguish flux per pixel from surface brightness when changing grids.

**Figure 1 — two panels, left and right:** left: full-sky equatorial Mollweide view made with `healpy.mollview`, showing the nominal PS1 3π footprint and all sixteen CCDs of the selected sector (four cameras, four CCDs each); distinguish the adopted camera 2 / CCD 2. Right: only that CCD's science crop, with native TESS $x,y$ axes and no FFI image underneath. Read **every skycell** from the mapping CSV, construct its own PS1 WCS, transform its image corners through sky coordinates into the adopted FFI's TESS coordinates, and draw the resulting polygons clipped to the science crop. Record the raw-FFI-to-science-crop coordinate offset. Do not subsample skycells or infer their boundaries from the TESS mapping-image WCS. Distinguish nominal PS1 survey extent from measured usable coverage.

### 2.2 PS1 bands and the TESS band (main text kept short; verification in Appendix C, user 2026-09-30)

- The template is a weighted sum of PS1 $r,i,z,y$, which acts as a filter approximating the TESS response.
- **Adopted weights** $w_{r,i,z,y}=0.254/0.437/0.165/0.144$ (D13) were fitted on Gaia-XP spectra through the TESS POC v2.0 response (D17).
- **Accuracy:** the synthesis error is ≤0.7% held-out for stars bluer than late M. Late M has a ~1–2% floor for any four-band mix.
- **Why it can't be exact:** the PS1 curves carry atmospheric absorption, $r$ responds at 550–580 nm where TESS does not, and the match holds only when integrated over smooth stellar spectra. Narrow emission lines are mis-weighted by ~35% (a limit, §6.2).
- **Figure 2** (bandpasses).
- Appendix C gives the fit, the weights table by spectral type, and the residual-based checks (S-bandpass, S-response, S-weights).
- Keep band synthesis separate from the photometric scale, which Hotpants sets in §3.

**Band templates are kept separate (2026-09-30).** §2's product is the four band templates $T_b$ ($b\in\{r,i,z,y\}$, flux per template pixel), not one combined image. The combined template $\sum_b w_bT_b$ ($w_b$: adopted weights) is what the preliminary match uses (§3). The band-resolved set is convolved band by band in §5, because the TESS response is colour dependent (§4).

### 2.3 Removing PS1-saturated stars while preserving resolved neighbors

- Describe source segmentation and suppression of PS1 background structure before aggregation onto the coarser TESS grid; current code uses SEP. Record detection, mask, and dilation settings from the selected recipe.
- Emphasize the advantage of removal in high-resolution PS1 skycell space: where the saturated star occupies its own segment, that star can be omitted while neighboring stars in separate segments remain in the template, even when blended together at TESS sampling. Their subtraction then exposes the retained star in the TESS image.
- Explain catalog-based bright-source selection and PS1 quality flags separately; a brightness cut is not proof of saturation. Show that the representative segment contains the intended star and measure neighbor preservation. Merged segments, saturated wings, and unseparated companions are exceptions to document, rather than assuming every segment contains exactly one star.
- Make the source accounting explicit: a source omitted from the subtraction template can remain in the TESS residual; a source included in the template is intended to be subtracted. Masking a science pixel is a different operation.
- Trace whether any bright-source model is reinserted, and into which product. The draft's TIC/Gaussian reinjection is historical, not an adopted step for the calibration image.
- **Forward reference (2026-09-30):** the same segmentation later lets any fainter target be removed from the final template and measured (targeted de-crowding, §5.5). Bright-star omission here and star extraction there are the same operation.
- Develop the depth argument: the expectation is that PS1 resolves and detects sources whose remaining unmodeled flux is small relative to the relevant single-FFI TESS sensitivity. Test it by mapping the excluded component through the same smoothing, sampling, flux scaling, and measurement aperture, then comparing its contribution to the TESS noise budget. Survey depth alone does not establish that summed unresolved light or structured background is negligible. Separate residual astrophysical light from PS1 sky artifacts.

**Figure 3 — full-skycell four-panel source decomposition:** original band-combined PS1 skycell; removed saturated/bright-source component; excluded PS1 background component; scene retained for template creation. Show the **entire stored skycell**, approximately $6000\times6000$ pixels (use and record its actual dimensions), rather than the previous small cutout or a narrow strip. This full extent allows examination of possible chip-gap-associated background structure; identify its origin before attributing a visible feature to a chip gap. Use logarithmic normalization (`LogNorm`), recording positive display limits and how zero, negative, invalid, and excluded pixels are shown; retain signed values in the data. The current full-skycell render uses $8\times8$ native-pixel area means for display only, preserving the full footprint and native arrays. Use a common logarithmic scale where practical, with any deeper background scale explicitly labeled. Define disjoint masks whose components reconstruct the original on valid pixels. Label catalog-selected bright-source removal accurately; not every removed star is necessarily saturated. Put any segment-boundary/neighbor-preservation close-up in supporting material so it does not replace the full-skycell view. Quantify the excluded component's TESS-equivalent contribution separately from its display contrast.

### 2.4 Smoothing and assembling a continuous scene

- Explain the staged computational design: Gaussian preprocessing brings the PS1 response closer to the broad TESS response while remaining independent of the particular TESS camera, CCD, and observing orientation. The same prepared sky scene can therefore be reused when TESS observes it on a different detector or orientation.
- Specify a common smoothing recipe in PS1/angular units and establish the width and support used. It is an intermediate response, not a detector-specific TESS PSF. Leave the remaining spatial and exposure-specific response differences to later kernel matching and ePSF inference; verify that the chosen smoothing still permits those matches.
- Distinguish the PS1 image response, preprocessing Gaussian, fitted matching kernel, and final empirical TESS ePSF. Their roles and normalizations must not be conflated.
- Explain overlap handling, padding from adjacent skycells/projections, and coverage bookkeeping so the reader understands how seams and edge losses are controlled.
- State the spatial remapping/aggregation operator used for the selected image, including flux conservation and pixel-area treatment. Do not reuse the draft's blanket claim that center-assignment errors are negligible.

### 2.5 Mapping the scene onto the TESS pixel grid (F=1 and F=4)

- **Mapping** (the pipeline's mapping stage; it uses only the TESS WCS, never pixel values):
  - Find the PS1 skycells overlapping the buffered TESS footprint.
  - Assign every TESS pixel, or sub-pixel at $F=4$, to one skycell; where skycells overlap, the one whose centre projects closest in TESS pixel space wins.
  - Record which TESS (sub-)pixel each PS1 pixel falls in.
- **Aggregation:** each band template is the flux-conserving sum of PS1 pixels into those TESS (sub-)pixels. State the pixel-area treatment and seam/padding bookkeeping (§2.4).
- **Drift:** small WCS changes are applied as integer PS1-pixel rolls of the frozen mapping, not by re-running it. 1 PS1 px ≈ 0.012 TESS px. Record which geometry mode built the adopted templates.
- **The mapping depends on the WCS:** §5.1 re-runs it on the calibrated WCS, so the final template shares the ePSF's WCS.

- The same band templates are rendered at two samplings: the native TESS grid ($F=1$) and a grid four times finer per axis ($F=4$, 4×4 sub-pixels per native pixel). Both are remapped from the PS1 scene, not resampled from each other.
- **Paper 1 uses $F=4$** for the preliminary de-crowded image (§3) and the calibrated template (§5), because the template must be convolved with a sub-pixel kernel before it is summed onto native pixels. **$F=1$ is kept** for the multi-frame temporal work (Paper 2), where one template per frame is needed and $F=4$ costs too much. Appendix A compares their performance on the same frame.
- The $F=4$ template for §5 is remapped on the calibrated WCS; the template WCS must equal the ePSF WCS.
- Separate template sampling F from the independent oversampling used to represent the fitted ePSF.

**Table 1:** exposure/region, source products, sample coverage, and grid conventions. Include the adopted $F=1$ and $F=4$ values without presenting old documentation defaults as the actual experiment configuration.

**Figure 4 — aligned sampling comparison:** Gaussian-convolved PS1 skycell, template sampled at $F=1$, and template sampled at $F=4$, showing the same sky footprint and orientation. Overlay native TESS pixel boundaries and align source markers. For this sampling demonstration use a common WCS so the comparison isolates sampling; label the actual initial-WCS and improved-WCS realizations separately if also shown. Use common surface-brightness units or explicitly account for differing pixel areas: raw flux per pixel changes with sampling. $F=4$ is four times finer per axis, not finer TESS observations.

**Detail budget:** this is now a substantive methods section. Keep the scientific choices, source accounting, and error pathways in the main text; defer scheduling, storage, caches, and full remapping derivations to supporting material. Explain the shared synthesis once, followed by its two renderings.

**Sources and status:** useful concepts come from the [archived older draft](../archive/2026-09-16_legacy_manuscript/README.md). Current implementation checks use [band processing and source removal](../../../syndiff_pipeline/template_creation/processing/band_utils.py) and [PS1 processing](../../../syndiff_pipeline/template_creation/processing/ps1_process.py); the [technical guide](../../../docs/markdown/stages/ps1_process_technical.md) includes historical implementation details. Code supports available operations, not provenance of an unexamined run. Draft figure assets and bibliography are not included in the attachment.

## 3. Preliminary de-crowded image: PS1 template matched with an empirical kernel (rewritten 2026-09-30)

**Purpose:** break the circularity. An exact template needs the TESS PSF and WCS, and measuring those needs a de-crowded image. A first match with a kernel fitted to the image itself removes enough neighbouring light to calibrate on; §5 replaces it with a template built from the calibration.

### 3.1 The preliminary match

$$D_{\rm prel}=(I-B)-K\ast\sum_b w_bT_b$$

Symbols:
- $D_{\rm prel}$: the preliminary de-crowded image (native pixels, e⁻ s⁻¹).
- $I$: the FFI.
- $B$: the image-level background (§3.2).
- $T_b$: the F=4 band templates of §2 on the initial WCS.
- $w_b$: adopted weights.
- $K$: a spatially varying kernel fitted by oversampling-aware Hotpants. It maps the finer template onto native science pixels and sets the flux scale.

The matched template has to reproduce the PSF (spatially varying and colour dependent), the sub-pixel WCS and the flux scale. None of these is known yet, so an empirical kernel is used, accepting that one kernel cannot follow a colour-dependent PSF. Record the kernel basis, spatial order, stamps and masks.

### 3.2 Background removal: alternating kernel fits and background estimates (new, user 2026-09-30)

- **Why:** the TESS sky background can't be estimated reliably on the raw crowded FFI, where stars cover most pixels. It is estimated on the residual after the modelled stars are subtracted.
- **Procedure** (pipeline name `kernel_fit`; not used in the paper). On the one adopted frame:
  1. A Hotpants match that includes a polynomial background term.
  2. Estimate the background robustly on $I-K_1\ast T$, with the bright omitted stars and strong residuals masked, and gaps filled by an exact harmonic (Laplace) solve.
  3. A second Hotpants match on $I-B_1$, without a background term, followed by a new background estimate $B_2$ on $I-K_2\ast T$.
  4. A third, final match on $I-B_2$ gives the kernel $K$.
  5. The last background estimate on $I-K\ast T$ gives $B$, and the final Hotpants run on $I-B$ gives $D_{\rm prel}$.
- **Record:** the masked fraction per round, the robust estimator settings, and whether $B$ is a total map or a residual correction, so the background is not subtracted twice.
- **Confirm** that the F=4 run used this recipe; the F=1 lane's frozen config did.
- **Figure 6 (re-assigned 2026-09-30):** the background loop on one sky area in common units. Panels: FFI; round-1 residual with its mask; $B_1$; final $B$; $B-B_1$. Annotate masked fractions.

### 3.3 What the preliminary image enables, and why it is not enough

- **Enables:** bright ($T<13$) stars omitted in PS1 skycell space keep their TESS signal while modelled neighbours are removed (Figure 18). This image is the input to §4. The production-style initial ePSF and centroids are initialization only and get no figure (user 2026-09-30).
- **Not enough:** calibration at this level leaves percent-level errors that depend on detector position and colour:
  - production ePSF photometry against Gaia-XP has a ±25 mmag gradient across the CCD (map rms 16.6 mmag) and a colour slope;
  - the single kernel leaves a red−blue residual dipole of +117 mpx and a red-core deficit.
- **Why that matters without light curves:** any time variation of such errors (pointing jitter, focus, scattered light) appears as trends that must be detrended out of light curves. This paper does not make light curves (Paper 2). It removes the static, position- and colour-dependent part in one exposure, which is the prerequisite.
- **Figure 7 (re-assigned 2026-09-30):** "Why the preliminary products are not enough". Panels: production-ePSF $\Delta m$ map and $\Delta m$ against colour (from the Fig 17 production arm), and the red−blue residual stack in $D_{\rm prel}$ (from the Fig 16 baseline column). Same scales as the calibrated counterparts in §4.6 and §5.3.

**Figure 5** (the calibration loop) opens §3 as the roadmap for §3–§5.

## 4. Scene modelling: simultaneous fit of stellar fluxes, WCS and chromatic ePSF (renamed 2026-09-30)

**Purpose:** explain the adopted single-image calibration and show the evidence for its performance beside the relevant technique. Separate the model definition from the fitting procedure, then present spatial calibration and chromatic results. This is not a chronological account of every model tried.

**Starting point:** stars saturated in PS1 were omitted from the subtraction template, so their observed TESS profiles are intended to remain virtually untouched while modeled fainter objects and neighboring light are removed. Select stars usable in TESS and verify retained-signal preservation. The input is the preliminary de-crowded image of §3 (F=4 template, native pixels; 2026-09-30). **No local background is fitted.**

**Scope and evidence status:** the adopted paper exposure is `tess2020120182919-s0024-2-2`. The existing development notes describe a different, superseded FFI (`tess2020127055918-s0024-2-2`), 8560 groups, and a reported 8511-star color-stack sample. These counts do not describe the adopted exposure and must not be transferred to it. Derive fresh sample counts and fitting results for the adopted FFI. Existing comparisons are historical in-sample development evidence, not completed independent validation. Keep numerical provenance and alternative-model experiments in the [archived model notes](../archive/2026-09-16_chroma_model_notes/README.md) and [model-form notes](../../forward_epsf_wcs/docs/CHROMA_MODEL_FORM_ABLATION_20260916.md). Present the adopted model and a matched color-independent comparison in the main paper; a full model ladder is not required.

### 4.1 The scene model and stellar fluxes

- The scene model predicts every native pixel of $D_{\rm prel}$ as a sum of stars, each rendered at its Gaia position (§4.2) with the spatial, colour-dependent ePSF (§4.3–4.4).
- **Each star's flux is a parameter.** Fluxes are solved conditionally (linear least squares at fixed WCS and ePSF) rather than as optimizer leaves.
  - Bright ePSF contributors ($8\le T<11$) and fainter WCS anchors ($T<13$) constrain different parameters.
  - Nuisance neighbours get fluxes only.
- The fitted fluxes are the delivered bright-star photometry; their accuracy is quoted held-out (§4.6).
- No local background is fitted; the background comes from §3.2.


**Question:** given a star's sky position, color, flux, and detector location, what native TESS pixel values does the model predict?

**Inputs and fitted quantities:**

- Enumerate the background-subtracted image, noise plane, masks, source groups, epoch-propagated Gaia coordinates and colors, and initial WCS/ePSF. Distinguish fixed catalog information, initialization, shared fitted fields, and source fluxes.
- Use “Gaia constraints” unless catalog covariance actually enters as a probabilistic prior. State the treatment of missing colors, unreliable astrometry, companions, and proper-motion outliers.
- Define the reference color $c_{\mathrm{ref}}$, detector coordinates, exposure epoch, and units. Reserve the following schematic model, with displacement sign checked against the renderer:

$$
\begin{aligned}
m_p &= \sum_i f_i\,E\!\left(\boldsymbol{p}-\boldsymbol{q}_i;\boldsymbol{r}_i,\delta_i,\boldsymbol{\theta}\right), \\
\boldsymbol{q}_i &= W\!\left(\boldsymbol{s}_i(t_{\mathrm{exp}});\boldsymbol{a}\right)
  + \delta_i\,\boldsymbol{d}(\boldsymbol{r}_i), \\
\delta_i &= (\mathrm{BP}-\mathrm{RP})_i-c_{\mathrm{ref}}.
\end{aligned}
$$

- Here $\boldsymbol{s}_i(t_{\mathrm{exp}})$ is the source sky position propagated to the exposure epoch, $\boldsymbol{r}_i$ is detector field position, $W$ the reference-color mapping, $\boldsymbol{d}$ a displacement field, and $E$ the resulting native-pixel response. Distinguish this rendered response from the stored oversampled flux-fraction grid $P$: the current renderer interpolates $P$ and sums subpixel contributions as specified below. Verify renderer-to-displacement sign with a known-shift test.
- Stellar fluxes are profiled/solved at fixed shared parameters rather than independent optimizer leaves. Explain the actual single-star or grouped companion solve and weights. Residual background is a preparation uncertainty, not a local parameter.


### 4.2 Astrometry: Gaia positions and the WCS

- Gaia DR3 positions are propagated to the exposure epoch with proper motion and parallax, then projected with a reference-colour WCS: a tangent plane plus a degree-5 Chebyshev distortion.
- The WCS is defined for a star of the reference colour $c_{\rm ref}$. A star of another colour sits at the WCS position plus the chromatic displacement of §4.4, so a FITS WCS header alone cannot encode source positions.
- **Gauge:** the ePSF's core-centroid origin fixes where "the position" is, so an ePSF translation cannot trade against the WCS.
- One static WCS per exposure; its time dependence is Paper 2.
- Accuracy against held-out Gaia stars is in §4.6.

**Reference-color geometry and profile representation:**

- Describe the frozen tangent-plane projection and static detector correction, including coordinate normalization, crop origin, degree, and coefficient count. A degree-5 Chebyshev correction and 6×6 ePSF node grid are current example settings, not universal requirements.
### 4.3 The spatial ePSF and rendering onto native pixels

- A 6×6 grid of 4×-oversampled node images, blended bilinearly across the detector, with the moment-blind fine-neighbour prior (D6).

- Describe the spatial node images, support, normalization, positivity if enforced, and detector-position interpolation. The initial 5×5 display in Section 3 is not the fitted node grid.
- Define the finite-support flux convention and core-centroid origin. The core origin is not the whole-profile center of mass; differences between these conventions must not be mistaken for astrometric improvement.
- Explain the position/shape ambiguity: arbitrary ePSF translations cannot float independently of the WCS. Keep chromatic displacement outside profile recentering and constrain shape directions against reproducing flux or core displacement.
- Introduce $E$ as the native-pixel response and $P$ as the stored oversampled profile. Keep the conceptual rendering explanation in the main text; the full interpolation derivation below can move to an appendix if it interrupts the narrative. The adopted chromatic dependence is developed alongside its evidence in Section 4.4.

**Rendering the 4× representation onto native pixels — planned explanation:**

1. **Interpolate across detector position:** blend the four surrounding spatial node images to obtain the local profile, applying the model's chromatic shape and core-origin conventions. These detector-field weights are distinct from the within-profile interpolation weights below.
2. **Determine pixel phase:** Gaia propagated through the current WCS, plus the chromatic displacement, gives the continuous source location $(x_i,y_i)$. Its offset from the native pixel lattice sets the sampling phase; no rounding to quarter-pixel positions is performed.
3. **Sample within the local profile:** for a native pixel centered at $(x_p,y_p)$, evaluate the local grid at sixteen subpixel centers. For representation oversampling $O=4$,

$$
s_a=\frac{a-3/2}{4},\qquad a\in\{0,1,2,3\},
$$

$$
g_x=c_P+4(x_p-x_i+s_a),\qquad
g_y=c_P+4(y_p-y_i+s_b),
$$

where $c_P$ is the profile-array index corresponding to its origin. Distinguish $O$, the profile representation factor, from $F$, the earlier template sampling factor, even when both equal four.

4. **Interpolate below a quarter pixel:** set $j=\lfloor g_x\rfloor$, $k=\lfloor g_y\rfloor$, $\alpha=g_x-j$, and $\beta=g_y-k$. The bilinear value is

$$
\begin{aligned}
\widetilde P(g_x,g_y)={}&(1-\alpha)(1-\beta)P_{k,j}
+\alpha(1-\beta)P_{k,j+1}\\
&+(1-\alpha)\beta P_{k+1,j}
+\alpha\beta P_{k+1,j+1}.
\end{aligned}
$$

An evaluation point 0.10 native pixel beyond a profile sample is 0.40 grid intervals beyond it: along that axis the weights are 0.60 and 0.40. Quarter-pixel grid spacing constrains the represented profile detail, not the allowed precision of a fitted source position.

5. **Sum into the native pixel:** the unnormalized rendered response is

$$
R_{ip}=\sum_{a=0}^{3}\sum_{b=0}^{3}
\widetilde P_i\!\left(c_P+4(x_p-x_i+s_a),\ c_P+4(y_p-y_i+s_b)\right).
$$

Apply the renderer's documented finite-support normalization and pixel-validity mask to obtain $E_{ip}$, then form $m_p=\sum_i f_iE_{ip}$. Stored values are subpixel flux fractions, so this operation is a sum, not an extra average by sixteen. Handle out-of-support samples using the renderer's zero-support convention. Do not integrate the resulting native-pixel response again.

**Implementation reference:** [direct pixel block-sum renderer](../../forward_epsf_wcs/epsf_model.py) (`render_physical_pixels_blocksum`) specifies this operation; the packed banded renderer combines the same sampling weights efficiently with the core-recentering operation. Describe the mathematical operation in the paper rather than implying that every code path materializes sixteen samples separately.

**Figure 5 inference inset:** native pixel boundaries over the 4× local profile, source center at a non-quarter-pixel phase, the sixteen evaluation points for one pixel, and a zoom showing four-sample bilinear weights. This makes the distinction between spatial node blending and subpixel profile sampling visible. Validate interpolation accuracy with phase sweeps or finer-grid comparisons; continuity does not establish arbitrary astrometric accuracy.


**Planned presentation:** the model equation, a compact parameter/input inventory, and the Figure 5 inference inset. This subsection defines what is fitted; optimization stages and sample roles belong in Section 4.2.

### 4.4 Colour: the Gaia-XP colour and the chromatic ePSF

- **Colour.** Each star's colour is Gaia-XP $u=\sum_b s_b\lambda_b$: the PS1 band-share-weighted effective wavelength, mapped to BP−RP units per CCD (BP−RP fallback for about 1% of stars).
  - $s_b$: the star's flux share in PS1 band $b$ from XP synthetic photometry, with the label weights fixed by D18.
  - $\lambda_b$: the band wavelengths.
  - $\delta=c-c_{\rm ref}$: the colour offset.
- **Chromatic model A3 (17 coefficients per CCD), in two parts:**
  - A radial colour shift toward or away from the optical axis, $\Delta\boldsymbol x=[\delta(s_0+s_1r+s_2r^2)+(\delta^2-\langle\delta^2\rangle)(q_{s0}+q_{s1}r)]\hat{\boldsymbol n}$. Here $r$ is the distance to the tess-point optical axis in kpx, $\hat{\boldsymbol n}$ is the unit vector toward the axis, and $\langle\delta^2\rangle$ is the mean over fit stars.
  - Colour shape terms: kurtosis, dilation, blur and trefoil (radial), plus CCD-frame elongation planes. All are projected off $P_0$ (raw-P gauge) so flux cannot trade with colour.
  - Full equations and symbol definitions: Appendix B and `COLOUR_MODEL_V2_DECISION_20260929.md`.
- The linearisation $E(c)\approx P_0+\delta P_1$ used in §5.2 follows from this model.

**Adopted chromatic model, explained beside the evidence:**

- **Adopted 2026-09-29 (supersedes #14 below): chroma v2 arm A3** = #14 + quadratic-in-colour radial shift ($s_{q0}, s_{q1}$) + CCD-frame colour elongation planes, with the Gaia-XP band-mix colour $u$ in place of BP−RP (BP−RP fallback for stars without XP). 17 colour parameters. Stage-1 evidence: `dev_runs/chroma_v2_20260928`; generalisation to more CCDs (stage 2/2b) running. The §4.4 figures, the coefficient table and Appendix B must be regenerated for A3 on the F=4 image, out-of-fold. Known limits: F1 red-end shift +4.8 mpx (target ±2), F1 mid-distance radial−tangential width band, red−mid residual 7.4/5.5 vs targets 5.5/5.2.
- **Previously adopted (2026-09-24/25): colour model #14**, 9 global coefficients in ρ = distance to the tess-point optical axis: radial shift S0+S1ρ+S2ρ², plain kurtosis K0+K1ρ, blur B, dilation D0+D1ρ, radial trefoil T. Raw-P gauge, one global c_ref, L2, no halo. It supersedes the node-grid shift + dilation working candidate. The full bullets, numbers and figures (Figs 10/15/12 regenerated for #14, the coefficient table) are in `latex/main.tex` §4.1, §4.2, §4.4, §5.2 and Appendix B, which are authoritative for these sections. Figure sources: `figures/chromatic/model14/`.
- State the assumption of coefficients linear in $\delta_i$ and the calibrated color range. Rendering a displaced profile need not be linear in color.
- For shift plus dilation, reserve the profile-level schematic

$$
P_i = \mathcal{G}\!\left[P_0(\boldsymbol{r}_i)
 + \delta_i\,\epsilon(\boldsymbol{r}_i)D[P_0(\boldsymbol{r}_i)]\right],
\qquad
D[P]=2P+\boldsymbol{u}\cdot\nabla P.
$$

  Here $\mathcal{G}$ denotes the adopted flux/origin constraints; describe the actual gauging and rendering order from the selected implementation. This is a profile schematic, not a substitute for the native-pixel renderer in Section 4.1. Translation enters through $\boldsymbol{q}_i$ rather than being counted again as a shape term.
- Explain the sign convention: the code's ungauged dilation generator has positive coefficient corresponding to narrowing at small perturbation. Show normalized profile differences to make the coefficient physically interpretable.
- State the added parameter count for the adopted grid. On a 6×6 grid, two displacement components add 72 coefficients and dilation adds 36. Spatial sharing reduces per-star freedom but does not by itself establish identifiability.
- Explain only the degeneracies needed to interpret the adopted result: reference-color/base coupling, displacement versus profile recentering, and size versus other unresolved shape structure. Do not require a narrative of every rejected model.

### 4.5 Fitting and evaluation

**Question:** how are geometry, profile shape, and source flux inferred without letting one absorb arbitrary changes in another, and how is performance assessed?

- **Training and checks (D19, user 2026-09-30):** the delivered calibration is always trained on all stars; overfitting and held-out checks always use **K = 5** spatial folds.

**Calibration sample and parameter roles:**

- Explain bright ePSF-contributing groups and isolated faint WCS anchors, including magnitude, color, quality, crowding, support, and companion selection. State which populations constrain which parameters and whether anchor gradients are stopped for relevant shape terms.
- Account for source/group counts, overlapping pixels, rejected groups, masked detector regions, and usable color/phase coverage. Put the concise recipe in Table 1 and detailed selection thresholds in supporting material.
- Show spatially resolved color coverage for both populations. Define the weighting used for $c_{\mathrm{ref}}$; the development value near 0.987 is not a universal constant. Different populations can have different effective mean colors and couple base-profile and color-dependent parameters.
- State constraints and normalization consistently across fitting stages. Recentring the reference color requires a consistent base-model reparameterization; it is not automatically a remedy for all parameter coupling.

**Objective and fitting sequence:**

- State the pixel-data objective, noise convention, robust treatment if used, and regularization separately. Explain the single-star or grouped profiled-flux solve; fluxes are not independent optimizer leaves if solved conditionally.
- Describe the actual static sequence: initialize from Section 3; refine geometry and the spatial reference profile with the documented freeze/unfreeze policy; introduce the adopted chromatic terms; state whether geometry and the base profile remain free in the final stage.
- Explain initialization, convergence criteria, and brightness-conditioned single-frame group rejection. Do not import temporal/frame-axis rejection rules or temporal basis functions.
- Record masking and background provenance. Residual background uncertainty is assessed through preparation sensitivity, not fitted local backgrounds.
- Give the reproducible configuration and computational cost if measured. Move optimizer hyperparameters and long stage schedules to an appendix rather than making them the scientific narrative.

**Evaluation rules, stated before showing results:**

- Define development and held-out source/group samples, their spatial/color coverage, and allowed nuisance fits on test stars. Track whether evaluation stars influenced background/kernel preparation or initialization; describe any remaining dependence rather than calling all withheld optimizer stars fully independent.
- Compare the color-independent and adopted chromatic fits using identical evaluation stars, pixel support, weights, and rejection masks. Document shared initialization and fitting freedom. Do not let a model improve its score by rejecting a different population.
- Define positional bias/scatter, profile prediction error, and color-dependent residual statistics. Separate pixel-data loss from regularization; neither alone establishes astrometric accuracy.
- Specify uncertainty from appropriate star/group resampling and spatial checks. Independently fitted FFIs can assess reproducibility, but are not required to introduce temporal fitting; their scatter can include real temporal variation.
- Freeze development choices before final evaluation. Place actual validation plots beside the corresponding results in Sections 4.3 and 4.4, not in a separate validation section.

**Planned presentation:** a short fitting sequence and sample/metric definitions; optional coverage panel. No main-text catalogue of unsuccessful model variants.

### 4.6 Results: spatial ePSF, WCS and bright-star photometry

**Question:** what spatial calibration does the de-crowded image support, and how accurately does it predict positions and stellar pixels?

**Technique and recovered quantities:**

- Connect the shared spatial model to the detector's changing response. Show the reference-color ePSF across the CCD/region with common normalization, origin, scale, and source-support information.
- Show WCS corrections relative to initialization, clearly distinguished from residual errors against independent measurements. Model movement is not itself an accuracy gain.
- Establish the color-independent baseline needed for Section 4.4. Label baseline and adopted-model products explicitly: the final reference-color WCS/ePSF from a chromatic fit is not necessarily identical to the separately optimized color-independent solution.
- Define the delivered reference-color WCS, spatial response, valid detector region, and uncertainty/support information. Section 4.4 adds the color correction; a standard WCS header alone cannot encode source-dependent positions.

**Results with their validation:**

- Report positional bias and scatter on withheld stars, residual-vector maps, and dependence on magnitude, crowding, detector position, and pixel phase. Compare compatible centroid conventions and propagated catalog epochs.
- Compare archive/pass-1 calibration and the final solutions on the same evaluation sample. Distinguish relative agreement with Gaia from an absolute noiseless truth claim.
- Show observed/model/residual stamps and population-level profile errors, including representative difficult cases and failure fractions.
- Quantify what preparation contributes using the intermediate products in Section 3. Include an improved-WCS/$F=1$ control only if claiming a separate gain from finer template sampling. Use matched targets or controlled recovery; raw RMS across different products is not a fair comparison.
- Report retained-star signal preservation, sensitivity to refined background preparation, remaining neighbors, support boundaries, and interpolation/pixel-phase effects where they limit the stated accuracy.
- Use independently fitted FFIs as a compact reproducibility check if available, without requiring a temporal model or light curves.
- **Brightness-dependent width (added 2026-09-29, gate G2).** The core width depends on stellar brightness out of sample (2.3/1.4×10⁻³ px² from T8.5 to T12.5, ≈0.3–1% of flux; `dev_runs/epsf_width_decision_20260928`). If the global width($q$) term lowers out-of-fold flux error vs Gaia-XP synthetic TESS magnitudes on both fields, it is part of the adopted model and reported here. Otherwise report the measured trend as a limit (§5.2). A second bright-star ePSF is **not** an option: that evidence was in-sample memorisation.
- **Bright-star photometry (Figure 17; placement user 2026-09-30).** The delivered bright-star fluxes come from the joint fit. Accuracy is quoted only from held-out stars, whose flux is re-solved with the calibration fixed from the other folds, which is forced photometry with a fixed ePSF and WCS. Show $\Delta m$ against Gaia-XP synthetic $T$ by colour, brightness and detector position, the aperture-correction split, and the comparison with production photometry. The G5 colour slope is presented and attributed here.
- **All §4 metrics are out-of-fold** (K = 5 spatial folds, D19; leak-free initialisation). In-sample numbers appear only as the memorisation ratio.

**Figures:** Figure 14 (provisional identifier) for the final spatial ePSF, Figure 9 for astrometry, and Figure 11 for native-pixel predictions, shared with Section 4.4. Give the profile field its own figure rather than crowding it into the astrometric validation. Refer back to Figure 8 for preparation comparisons rather than repeating the same images. Table 2 is optional if independent-image validation is available.

**Endpoint:** a measured spatial calibration and a clearly defined baseline, with uncertainty and validity limits. Numerical claims remain pending until evaluation is complete.

### 4.7 Results: chromatic response


**Question:** what color dependence remains after spatial calibration, how is it represented, and how much does the adopted correction improve predictions?

**Evidence motivating the model:**

- Start with color-dependent residuals from the color-independent fit. Define registered fractional residuals $(\mathrm{data}-\mathrm{model})/\mathrm{flux}$ and the blue-minus-red stack, including weighting, registration, support, and actual color separation.
- Match color populations in detector position, magnitude, phase, and crowding as far as possible. Achromatic errors do not automatically cancel between different populations.
- Show the displacement-like pattern and remaining shape structure on common scales. A joint x/y displacement approximation can clarify the pattern; a complete decomposition into every candidate generator is not required.
- Estimate uncertainty or a repeated-split reference. A random-split stack is a sampling diagnostic, not the detector noise floor or an automatic significance threshold.

**Measured effects and validation:**

- Show displacement vectors, magnitude, and dilation/profile-change maps with color/sample support and uncertainty. Separate common-frame displacement from recentered shape changes.
- Measure displacement in independent color bins with uncertainty and free-intercept fits to assess linearity. Avoid treating crossing the reference color or a smooth fitted field as independent confirmation.
- Compare the color-independent and adopted chromatic model using common-scale residual stacks, held-out position errors versus color, and held-out pixel prediction metrics. State whether each plot is a training diagnostic or independent evaluation.
- Establish the contribution of any claimed shape correction through its uncertainty and, where needed, a compact shift-only control. A full ladder of higher-order alternatives is unnecessary for the main narrative.
- Test the material alternatives to an instrumental chromatic interpretation: PS1 bandpass/epoch mismatch, remaining neighboring light, color-position coupling, background preparation, and profile interpolation. Report remaining ambiguity rather than assigning every residual to optics.
- Quantify radial/tangential structure only if useful and supported. A fitted center is not automatically the physical optical axis; broader optical interpretation belongs in Section 5.
- Report remaining color-dependent structure and the range where the correction is useful. The paper need not eliminate every residual or establish a unique optical cause.

**Figures:** Figure 10 shows the motivating color signature and its correction; the dedicated Figure 15 measures displacement versus color; Figure 12 shows spatial chromatic fields. Reuse Figures 9 and 11 for positional and pixel-prediction gains. Table 3 is a compact baseline-versus-adopted performance summary. The former full model-ladder Figure 13 is optional supporting material, not a required main figure.

**Optional supporting analysis only:** anisotropy/shear, core/wing, blur, and free color-response experiments may justify a choice or diagnose a limitation if necessary. Preserve their development record without making failed variants a required result. Lower fitting loss alone does not justify adoption, and a loss-independent diagnostic on training stars remains in-sample.

**Method sources:** [static-fit implementation notes](../../forward_epsf_wcs/docs/SINGLE_FFI_STATIC_FIT.md), [model-form ablation](../../forward_epsf_wcs/docs/CHROMA_MODEL_FORM_ABLATION_20260916.md), [profile generators](../../forward_epsf_wcs/epsf_model.py), and [loss/rendering implementation](../../forward_epsf_wcs/loss.py). No experiments were rerun for this outline. Audit the development example's temporal-WCS initialization and older background processing before claiming that the demonstrated calibration is wholly standalone from a single FFI.

## 5. Final de-crowded image: per-band templates convolved with the calibrated chromatic ePSF (rewritten 2026-09-30; gated on G4)

**Purpose:** use the trained WCS and ePSF. First the template is remapped on the calibrated WCS; then each band is convolved with its own chromatic kernel; then stars are measured by forced photometry. Nothing new is fitted except an optional verification refit (§5.4). If G4 fails, the section becomes an outlook paragraph in §6.3.

### 5.1 Remapping the band templates on the calibrated WCS

- The mapping (§2.5) is re-run with the §4 WCS and the band templates are rebuilt at F=4.
- The template WCS must equal the ePSF WCS. Phase A showed a 0.047 px offset otherwise, and in Phase 1 the template and A3 WCS differed by 3 mpx median and 16 mpx maximum.

### 5.2 Per-band kernels from the chromatic ePSF

$$E_b(\boldsymbol u;\boldsymbol r)=E(\boldsymbol u;\boldsymbol r,c_b)\approx P_0(\boldsymbol u;\boldsymbol r)+\delta_b\,P_1(\boldsymbol u;\boldsymbol r)$$

- $c_b$: the effective colour of PS1 band $b$ in BP−RP units.
- $\delta_b=c_b-c_{\rm ref}$, recalibrated for the adopted weights (D18).
- $P_0$: the reference-colour ePSF.
- $P_1$: its colour derivative, per mag.
- $\boldsymbol u$: pixel offset.
- $\boldsymbol r$: detector position.

State the linearisation floor. **Figure 20** shows the per-band kernels.

### 5.3 The final de-crowded image

$$D_{\rm final}=(I-B)-\sum_b w_b\,(E_b\ast T_b)$$

- Relative to $D_{\rm prel}$ only the kernel changes, from one fitted $K$ to one calibrated $E_b$ per band, so each PS1 source is rendered with the PSF for its own colour.
- Compared on held-out stars and an unseen frame: pixel and per-star χ², red-star χ², red−blue dipole (**Figure 16**; Phase 1 stand-in: dipole +117 → +10 mpx, star χ² about −50%).
- **Figure 18** shows the final column next to the preliminary one.

### 5.4 Optional: re-running the scene model on the final image (verification)

- The §4 scene model can be re-run on $D_{\rm final}$ to refine the ePSF and WCS. Its main use is verification: the WCS must not shift (D14/G6).
- Dry run done 2026-09-30 (`dev_runs/d14_refit_20260930`): the F=1→F=4 input changes positions by 0.53 mpx rms at the same start, against 3.0–3.5 mpx training drift and 14 mpx split-half noise; flux changes by ≤0.3 ppt.
- Show this as a compact figure or a table.

### 5.5 Forced photometry and targeted de-crowding (name to confirm)

- **Forced photometry:** fluxes measured on the final image with the calibrated ePSF and WCS held fixed, at each star's Gaia position and colour.
  - It applies directly to the bright stars, which are never in the template (optional check that the final image does not bias them).
  - Fainter stars are inside the template and need targeted de-crowding first.
- **Targeted de-crowding** (the pipeline's "star" branch; working name):

$$M_t=\sum_b w_b\,(E_b\ast S_b^{(t)}),\qquad D_t=D_{\rm final}+M_t$$

- $S_b^{(t)}$: the PS1 band-$b$ image masked to target $t$'s segment (§2.3).
- $M_t$: the target's modelled TESS image at the PS1 epoch.
- $D_t$: the targeted de-crowded image, in which the target's TESS-epoch light is restored and its neighbours are still removed.

This is the §2.3 omission done after convolution: equivalent because convolution is linear, and it costs one small convolution per target. It is exact only with the kernels that built the template.
- **Figure 19:** targeted de-crowding stamps for hard cases (crowded, red, next to a bright star).
- **Figure 21:** joint photometry, held-out bright stars (§4.6 method) and targeted faint stars against Gaia-XP, over $T\approx8$–16.
- Evidence is being produced by a separate agent (user, 2026-09-30).

### 5.6 Limits

- **Camera 3 / CCD 3:** red stars are fainter than the v2.0 prediction in every product (Appendix C, S-weights). How to handle it is your decision.
- The per-band red-sharpening core is larger on C3K3 (−3.9…−5.6% vs −2.2…−3.4% on F1).
- PS1 segmentation failures.
- Neighbour variability between the PS1 and TESS epochs.

## 6. Discussion and conclusions

**Purpose:** explain what the demonstrated calibration enables, bound the interpretation, and close the paper without repeating a separate results section. Combine discussion and conclusions; no separate Section 6 or 7 is planned.

### 6.1 What PS1-assisted single-image calibration enables

- Synthesize the contribution of resolved-scene preparation, de-crowding, Gaia-constrained geometry, and color-dependent response. Tie each claimed benefit to Section 3 or 4 evidence.
- Explain why retaining observed bright-star TESS profiles while suppressing modeled neighbors makes calibration possible in crowded images. Distinguish the approximate PS1 reference scene from the measured TESS response.
- Relate demonstrated gains to existing TESS/Kepler calibration approaches and the untapped information in TESS images. Do not infer light-curve precision or science yields from pixel/astrometric improvements alone.
- State the practical calibration products: reference-color WCS, spatial profile representation, chromatic correction, valid region/color range, masks/support, and reproducible configuration.

### 6.2 Interpretation and limits

- Discuss the measured color dependence and plausible optical interpretation, keeping evidence separate from hypotheses. Apparent radial structure does not uniquely identify a mechanism or establish the optical-axis location.
- Bound detector, magnitude, color, crowding, and PS1-coverage applicability. Distinguish fitted regions from unsupported full-CCD extrapolation.
- Discuss template bandpass/epoch mismatch, remaining neighbors, background uncertainty, catalog errors, noise assumptions, and finite sampling where they materially affect conclusions.
- Explain the information limits of one exposure and which residuals remain unresolved. Summarize only the robustness evidence needed for these limits, referring back to Section 4.
- Keep development failures out of the main discussion unless they change the adopted model's interpretation or reliability.
- Limits to state (2026-09-29): F1's mid-distance radial−tangential colour-width band; role/brightness-dependent colour displacement (recheck on A3); the per-star residual floor (not shared by same-phase neighbours; time behaviour is Paper 2); the brightness width if G2 fails; the −3.5 %/mag Gaia flux colour slope until G5 attributes it. Figure 17 shows this slope, the aperture correction and the brightness term directly.

### 6.3 Temporal extension and concluding findings

- Briefly motivate Paper 2: evolution of the sky-to-detector mapping, including velocity aberration and pointing effects; thermal changes in the PSF and potentially optical distortion; consequences for time-series measurements.
- Distinguish aberration-induced apparent mapping changes from physical optical distortion. Define coordinate conventions so aberration is not applied twice; do not attribute all WCS evolution to aberration.
- Cite relevant prior motion treatments identified in the [coverage audit](../shared/VELOCITY_ABERRATION_PSF_AUDIT.md) without expanding this paper into temporal fitting.
- End with a short summary of verified findings: the reference-scene/de-crowding method, achieved spatial calibration, measured chromatic behavior, and validated benefit of the adopted correction. Leave numerical conclusions blank until supported.
- Paper 1 ends with a calibrated individual image and its validity limits. A solved light curve or temporal model is not a completion requirement.

**Writing-length decision:** these three labels organize the outline; they may become consecutive paragraphs under one “Discussion and conclusions” heading if the eventual section is short.

**Figure policy:** no new main figure is required in Section 5. Use Figure 8 for the preparation benefit, Figures 14 and 9 for the spatial calibration, and Figures 10–12 and 15 for chromatic behavior and predictive improvement. Section 5.2 may refer to a supporting sensitivity figure when discussing a quantified limitation. A new optical-interpretation figure is warranted only if independently established geometry adds evidence beyond the measured field in Figure 12. No temporal trend or light-curve figure belongs here.

## Figure plan by subsection (2026-09-30, second revision)

Main = main text; Supp = supporting or appendix. ✅ current · 🟡 draft or stand-in · ⬜ not made. Canvas: https://claude.ai/artifact/1ENTknYu81AwPRUAGmCbfx

| § | Main | Supp | Status and next action |
|---|---|---|---|
| 1 | optional teaser (FFI → final de-crowded image → targeted faint star) | — | ⬜ after §5.5 |
| 2.1 | Fig 1 coverage | — | ✅ (caption fixes) |
| 2.2 | Fig 2 bandpasses | App C: weights table, S-bandpass, S-response, S-weights | ✅ |
| 2.3 | Fig 3 skycell decomposition plus a segment-map zoom | excluded-light budget | 🟡 rebuild with D13 weights and add the zoom |
| 2.4 | — | seam/padding check | optional |
| 2.5 | Fig 4 PS1 / F=1 / F=4 sampling; Table 1 | — | 🟡 rebuild on real templates; ⬜ Table 1 |
| 3 | Fig 5 calibration loop (horizontal) | — | ✅ redrawn |
| 3.2 | **Fig 6** background removal | — | ⬜ inputs: F=1 lane kernel_fit products; confirm the F=4 run |
| 3.3 | **Fig 18** preliminary vs final cutouts; **Fig 7** why the preliminary products are not enough | — | 🟡 both can be made now from existing products |
| 4.3 | Fig 5 inset (rendering) | — | ✅ |
| 4.5 | — | fold coverage; K = 5 learning curves | 🟡 restyle harness figs |
| 4.6 | Fig 14 final ePSF; Fig 9 held-out WCS; Fig 11 pixel stamps; Fig 17 bright photometry | brightness width (G2); Table 2 | 🟡/⬜ wait on final fits |
| 4.7 | Fig 10, Fig 15, Fig 12 | Table 3 | 🟡 #14 versions; regenerate |
| 5.2 | **Fig 20** per-band kernels | linearisation error | 🟡 stand-in |
| 5.3 | Fig 16 per-band vs Hotpants | — | 🟡 Phase 1 stand-in |
| 5.4 | G6 refit comparison (figure or table) | — | 🟡 dry-run figure; restyle |
| 5.5 | **Fig 19** targeted de-crowding stamps; **Fig 21** joint photometry | bright stars on the final image | ⬜ extraction agent |
| App A | real F=1 vs F=4 comparison; Fig 8 demo | — | 🟡 cutouts exist; metrics to do |
| App B | A1–A3, B4–B7 | Table B2 | ✅ |
| App C | weights table, S-bandpass, S-response, S-weights | — | ✅ |

## Next steps (2026-09-30)

1. **Decisions (user)** that freeze the model:
   - D16 prior recipe: the bake-off is paused; fluxtruth favours moment-blind.
   - D9 brightness width: `bfwidth_prod_20260930` is running.
   - The local-background term: `localbg_20260930` step 3 is running.
   - C3K3 handling.
   - Where the delivered ePSF comes from. The D14 dry run shows the template choice moves the calibration far less than training or sampling noise, so the recommendation is one final fit per field on the F=4 preliminary image, so the text says exactly what was done, with D14 as confirmation.
2. **Final calibration fits** (F1, F2, S22; all stars plus K = 5 folds) with the frozen model (b914ef).
3. **Regenerate §4 figures and tables** from them: 14, 9, 11, 17 (§4.6), 10, 15, 12 (§4.7), Tables 2–3, Table B2 (figure agent).
4. **Per-band rebuild** from the final fit (9f71e7): Fig 16, S-weights, Fig 18 (final column), Fig 20.
5. **Star extraction** (new agent): Fig 19 and the faint half of Fig 21.
6. **Figures that can be made now** (figure agent):
   - Fig 18 and Fig 7 from existing products (stand-ins); Fig 6 (background loop);
   - App A.1 F=1 vs F=4 comparison;
   - Fig 3 rebuilt with the D13 weights and a segment zoom;
   - Table 1;
   - Fig 20 and the G6 figure restyled from the stand-ins;
   - §4.2 coverage and learning-curve figures.
7. **Records:** MODEL_FREEZE D15 and the §8 checklist to match this structure (paper-completion session).

## Figure specifications and plotting priorities

**Provisional inventory: fourteen main-figure concepts, one optional supporting model-comparison figure, and three table concepts.** Numbers remain planning identifiers until consolidation into reading order; this is not a fixed journal figure budget. Figures 1–8 cover preparation and initial calibration. Figures 9–12, the final-ePSF Figure 14, and the color–displacement Figure 15 present spatial and chromatic results with validation in Sections 4.3–4.4. Figure 13 remains optional supporting material. Renumber only after the panel plans are settled.

### Section 4–5 figure narrative and shared comparison rules

**Proposed reading order:** Figure 5 inference inset → optional sample-support panel → Figure 14 (final spatial ePSF) → Figure 9 (WCS accuracy) → Figure 10 (color signature and correction) → Figure 15 (displacement versus color) → Figure 12 (spatial chromatic response) → Figure 11 (native-pixel predictive benefit). Section 5 interprets these figures without requiring new results panels. Figure identifiers intentionally preserve earlier planning references.

| Location | Visual and purpose | Main-text status |
|---|---|---|
| 4.1 | Figure 5 inference inset: turn Gaia position and an oversampled profile into native pixels | Reuse the workflow figure; move a crowded inset to supporting material |
| 4.2 | Sample-support panel: where position, color, and profile information come from | Supporting by default; promote if coverage is central to interpretation |
| 4.3 | Figure 14: final reference-color ePSF across the detector | Main |
| 4.3 | Figure 9: position prediction errors on a common evaluation sample | Main |
| 4.4 | Figure 10: observed color dependence and remaining residual after correction | Main |
| 4.4 | Figure 15: independently measured displacement versus color, before and after correction | Main |
| 4.4 | Figure 12: size and direction of the fitted chromatic effects across the detector | Main |
| 4.4, referring back to 4.3 | Figure 11: what the correction improves in actual native pixels | Main; reduce stamp count before adding panels |
| 5.1–5.3 | Refer back to measured gains, support boundaries, and sensitivity checks | No additional main figure by default |

**Shared comparison contract:**

- Choose one primary FFI and freeze its source/group identifiers, evaluation roles, pixel masks, and detector coordinates for the main figures. Additional FFIs, if available, are fitted independently and summarized separately.
- Use consistent model labels: archive WCS; pass-1 calibration; final color-independent fit; adopted chromatic fit. The last two are the main chromatic comparison. Do not imply all four supply equivalent pixel models when only a WCS is available.
- Use the same test sources for paired comparisons. Record selection, exclusions, counts, catalog epoch propagation, preparation dependencies, and any nuisance flux/position fitting. Distinguish stars withheld from the final optimizer from stars independent of all learned preparation steps.
- Fix detector orientation, coordinate origin, angular conversion, color bins, and model colors/line styles across figures. Use separate visual encodings for stellar color and model identity.
- Report pixel units for positions, native-pixel offsets for profile coordinates, and explicitly defined flux-fraction or noise-normalized units for residuals. Profiles and residuals need different stretches; comparable residual panels need the same symmetric limits.
- Show uncertainty and sample support beside measured trends. Flag unsupported regions rather than filling them with visually convincing interpolation. A fitted profile map is a model product; held-out pixels and position measurements provide its validation.
- Caption checklist: question, FFI/detector, sample and count, model labels, units, normalization/origin, uncertainty definition, evaluation status, and the specific conclusion the panel supports. Leave numerical claims pending.

**Section 4.1 rendering inset — concrete layout:** three small panels: (a) detector position and four surrounding ePSF nodes with blending weights; (b) the local 4× profile with native-pixel boundaries and a source center at a non-quarter-pixel phase; (c) one native pixel's sixteen sampling locations plus a magnified bilinear cell showing four neighboring stored values. Annotate that the interpolated flux contributions are summed into one native pixel. Keep template factor $F$ distinct from profile factor $O$. This is a schematic of the operator, not a measured gain or an extra source of spatial resolution.

**Section 4.2 sample-support panel — supporting layout:** (a) detector map of ePSF-contributing groups and WCS anchors, with masks and node locations; (b) color–magnitude distributions for both populations with common bins and the reference color; (c) local color spread or effective sample count on the same detector map. Supply separate test-sample coverage if a combined plot hides gaps. Inputs are the actual frozen source-role table, flags, weights, and masks, not a generic Gaia field. This panel explains where the calibration is constrained; it does not establish accuracy.

### Figure 1 — Survey, sector, CCD, and skycell coverage

Use exactly the two-panel design in Section 2.1: `healpy.mollview` full sky on the left, and the adopted CCD's science-crop $x,y$ geometry on the right without an FFI image. Required inputs: sector pointing and all sixteen CCD boundaries; nominal PS1 survey extent; the complete mapping CSV with each skycell's WCS and dimensions; and the adopted FFI's TESS WCS. Project every skycell's four pixel-edge corners via PS1 WCS → sky → TESS WCS, then convert full-FFI coordinates to science-crop coordinates. Verify the polygon count equals the number of unique CSV skycells, and clip drawing to the science crop rather than selecting a small illustrative subset. Record whether sector-wide outlines use nominal focal-plane geometry or exposure-specific WCS.

### Figure 2 — How faithfully does the band combination approximate TESS?

- **Placement/question:** Section 2; could color-dependent template mismatch contaminate the chromatic measurement?
- **Main figure:** one response-curve plot at single-column width (approximately 3.4 inches), with PS1 r/i/z/y and TESS identified by an ordered compact legend above the axes. Preserve a clear response-normalization label and readable tick/legend text at final printed size.
- **Supporting diagnostic, separate from Figure 2:** synthesized-minus-reference TESS magnitude or fractional-flux residual versus color for a defined spectral library or independent validation sample, with counts/range or a residual summary identifying poorly constrained colors. Show the effect of any fitted global scale separately.
- **Inputs:** primary response-curve sources, coefficient provenance, calibrated flux units, spectral/sample selection, and uncertainty model. Record whether colors are synthetic or observed Gaia $\mathrm{BP}-\mathrm{RP}$ and how they are related.
- **Controls:** distinguish synthesis error from observed-image blending, extinction assumptions, variability, and PS1/TESS epoch differences. Do not interpret an arbitrary sum of peak-normalized filter curves as a calibrated synthetic response. A linear-versus-nonlinear recipe comparison can supplement, but cannot replace, validation against a TESS reference.
- **Readiness:** recover coefficient provenance and inputs first. The old draft's residual plot is a concept to reproduce, not a result to reuse.

### Figure 3 — High-resolution source removal and background decomposition

Use the full-skycell four-panel specification in Section 2.3, with logarithmic displays and actual stored dimensions. The full spatial extent is essential for examining chip-gap-associated background structure; do not substitute a cutout or a $6000\times600$ strip. Required inputs: pre-cleaning band-combined skycell, full-image segmentation and disjoint removal/background/retained masks, source catalog, and TESS-equivalent noise and flux conversion. Verify component reconstruction on valid pixels and document treatment of nonpositive values under `LogNorm`. A supporting close-up can test neighboring-source preservation; a separate background histogram or aperture-summed residual/noise ratio can quantify its significance for TESS.

### Figure 4 — The same scene at PS1, $F=1$, and $F=4$ sampling

Use the aligned three-panel design in Section 2.5. Required inputs: Gaussian-smoothed scene, matched-WCS template renderings, footprint/coverage maps, flux-unit conversion, and grid boundaries. Keep sampling and WCS changes distinguishable; no extra observational resolution is implied.

### Figure 5 — The calibration loop: preliminary, calibrate, rebuild, refit

**2026-09-30 (second revision):** redraw mostly horizontally, one column per section (§2 → §3 → §4 → §5), with the refit arrow looping back. Earlier: redrawn as a loop for the §2–§5 structure (F=4 preliminary Hotpants match → calibration → per-band template → final image, with star extraction → refit). The text below predates this revision.

- **Placement/question (2026-09-30: roadmap for §3–§5, opening of Section 3):** Opening of Section 3, before Section 3.1; how does one FFI proceed from PS1 scene preparation to calibrated geometry and chromatic response?
- **Panels:** the workflow above, with a compact legend distinguishing observed images, synthetic images, and fitted quantities. Show the initial background/kernel estimate and the de-crowded-background/kernel refinement within the native-scale stage, separate from the $F=1$/$F=4$ branches. A detail panel can compare masks/background maps as specified in Section 3.1; an optional final-fit inset shows the native-pixel prediction and comparison.
- **Required labels:** initial versus improved WCS; $F=1$ versus $F=4$ template; native-grid science and both outputs; Gaia's use in preparation where applicable and in final fitting. Explicitly separate template sampling from ePSF representation.
- **Inputs/readiness:** verified operator sequence and source-retention rule. Can be drawn before fitting; mark an unresolved handoff in the planning version instead of inventing it.

### Figure 6 — Background removal (RE-ASSIGNED 2026-09-30; spec in §3.2). The text below is the retired initial-ePSF spec (user: no initial-ePSF plot).

Use the 5×5 spatial display in Section 3.3 (initialization from the preliminary de-crowded image). Required inputs: pass-1 ePSF estimate, detector mapping, sample support, normalization/origin conventions, and uncertainty summaries. Label interpolation versus independently measured local profiles.

### Figure 7 — Why the preliminary products are not enough (RE-ASSIGNED 2026-09-30; spec in §3.3). The text below is the retired initial-WCS spec.

Use the before/after residual maps in Section 3.3 (initialization from the preliminary de-crowded image). Required inputs: initial and improved WCS, common source IDs and epoch-propagated positions, measured centroids, and sample roles. Retain identical map bins, vector scales, and error definitions.

### Figure 8 — What do the two de-crowding passes contribute?

**2026-09-30:** moved to Appendix A (F=1 vs F=4 performance). In §3 it is replaced by Figure 18 (real-data preliminary de-crowding cutouts).

- **Placement/question:** Section 3, quantified in Section 4.3; which improvements arise from preparation, better geometry, and finer sampling?
- **Panels:** primary columns for original TESS context, pass 1/$F=1$, and pass 2/$F=4$; keep the improved-WCS/$F=1$ control in a supporting panel or Section 4.3; rows for a preselected blend, a less crowded star, and a difficult region. Mark the calibration star retained in each image and show common masks. A compact lower panel summarizes contamination or held-out position error versus crowding for the matched cases.
- **Inputs:** same FFI, source IDs, both intermediate images, control rendering, matching settings, and source accounting. Comparable downstream fits must use the same evaluation stars and metrics.
- **Controls:** native grids and common physical stretches for comparable pixel products; separately labeled residual scales. Do not compare raw RMS across differently prepared images as though the target signal were identical. Define contamination from controlled recovery or an explicit neighbor model.
- **Readiness:** intermediate-image audit precedes rendering. If the control is unavailable, show the actual two-pass sequence and limit claims about why it improves.

### Figure 9 — Is the final WCS more accurate on withheld stars?

**Detailed layout:** a top row of two detector residual-vector maps (color-independent and adopted chromatic calibration), with identical spatial bins and vector scales. A bottom panel contains (c) the cumulative distribution of individual-star radial position errors for the main baselines. The dedicated Figure 15 carries the color-dependent displacement and remaining color bias; do not duplicate those panels here.

**Measurement definition:** reserve $\boldsymbol{e}_i=\widehat{\boldsymbol{q}}_{i,\mathrm{image}}-\boldsymbol{q}_{i,\mathrm{prediction}}$, in native pixels. Document how the image position is measured without using Gaia to force that same centroid. If image centroids depend on the fitted profile, disclose that dependence and test centroid-convention sensitivity. Gaia-fixed optimizer positions minus Gaia predictions are not an independent residual measurement. Report coordinate bias separately from scatter and from the radial-error distribution.

**Panel details:** arrows represent a stated robust mean or median residual per spatial bin, not every overlapping source. Omit or flag bins below a predeclared source-count threshold and provide uncertainty through resampling source groups. Annotate the common sample count, pixel-to-angle conversion, median radial error and a stated upper quantile; do not prescribe target numerical accuracy before measurement. Include archive/pass-1 curves only where a compatible error measurement is available. A small inset may show WCS correction relative to initialization, clearly labeled as a correction rather than accuracy.

**Evidence required:** an evaluation-source table containing image positions and uncertainties, epoch-propagated catalog positions, predictions for each solution, color, magnitude, crowding, group ID, and support flags. The interpretation is improved relative astrometric prediction within the tested region and sample. A residual trend with color is then explained in Section 4.4.

- **Placement/question:** Section 4.3; do geometry and color corrections improve position prediction beyond the initial and achromatic solutions?
- **Panels:** (a) detector residual-vector maps before/after on the same spatial bins and arrow scale; (b) radial-error cumulative distributions; (c) x/y residuals versus Gaia color, with uncertainty bands. Put magnitude/crowding stratification in supporting material if crowded.
- **Baselines:** archive WCS, pass-1 calibration, final achromatic fit, and final chromatic fit on identical held-out stars. Include the sampling control in a compact comparison table if plotting every curve becomes unreadable.
- **Inputs:** propagated catalog positions, independent image-position measurements, frozen test IDs, masks, crop-to-detector coordinates, and error estimates. Report pixels and angular conversion from the relevant geometry.
- **Controls:** test stars must not have calibrated the WCS, ePSF, or learned preparation operators used in their evaluation. Repeated star pixels are not independent samples. Report bias and scatter separately; catalog agreement is relative astrometric validation, not noiseless absolute truth.

### Figure 10 — What chromatic ePSF difference is actually measured?

**Detailed layout:** a 2×3 residual-image array. Columns show the blue population, red population, and blue-minus-red difference. Rows show the color-independent fit and the adopted chromatic fit. Use one flux-fraction scale across the six panels wherever readable; if the difference requires another scale, share that scale between its two rows and label it explicitly. Annotate mean color, effective count, and stack-amplitude statistic. Quantify the color–displacement relationship in the following dedicated Figure 15.

**Stack definition:** for each star use $r_{ip}=(D_p-m_p)/f_i$ on a stated common support, then register and combine with the documented weighting. The blue-minus-red image is a residual-stack difference, not automatically the direct difference between blue and red ePSFs. Preserve displacement in these common-coordinate panels; centering every residual on its own fitted peak would remove the signature being measured. Use the same registration rule for both model rows.

**Spatial choice:** begin with one predeclared detector region having adequate color overlap. If displacement directions vary across the CCD, a global stack can cancel them; use fixed spatial cells or a clearly defined coordinate rotation for an additional summary, and state whether that transformation depends on the fitted chromatic field. Do not select the region solely for the largest visible dipole.

**Handoff to Figure 15:** the residual images establish the color-associated signature; independent color-bin position measurements then quantify its amplitude and test the adopted color relation. A shift-only approximation and remainder can be a supporting panel if it helps distinguish displacement from shape.

**Evidence required:** registered residual stamps for both models, common star/group IDs, flux conventions, colors and matching weights, per-bin measurements, and resampling realizations. Add a null/sensitivity check in supporting material. The intended claim is a measured color-associated residual and its reduction; instrumental attribution also requires the preparation checks discussed in Section 5.

- **Placement/question:** Section 4.4; is there a reproducible within-FFI color-dependent displacement and/or shape difference?
- **Panels:** blue-minus-red fractional-residual stacks before and after the adopted chromatic correction on a common scale; a compact joint displacement approximation/remainder if needed to explain the signature; put independent color-bin displacement estimates in Figure 15. Put spatial coefficient fields in Figure 12; retain normalized-profile views as supporting context.
- **Inputs:** single-FFI star samples with color, magnitude, subpixel phase, crowding, detector position, and uncertainty; fitted profiles with a documented normalization/origin convention. Report color-bin means and actual color separation.
- **Controls:** match relevant covariates; retain displacement in common-frame panels and label any recentered shape-only panels. Include a resampling/null diagnostic and PS1-mismatch sensitivity. Model flexibility or a smooth interpolation is not independent evidence of a measured pattern.
- **Readiness:** establish single-image signal and sample overlap before choosing a visually dramatic example. Existing multi-frame stacks remain exploratory context.

### Figure 11 — Does the chromatic model predict stellar pixels better?

**Detailed layout:** three source rows by five columns: observed de-crowded native pixels; color-independent prediction; adopted chromatic prediction; color-independent residual; chromatic residual. Use a blue and a red example with comparable brightness and detector location where feasible, plus a representative difficult case. Choose examples by a declared rule, such as proximity to the median prediction error within each chosen population, rather than selecting the largest improvement. Mark saturated/bad pixels and fitted companions.

**Image display:** show native pixels without display smoothing. Normalize by a consistently defined source flux for comparisons between rows, with the same image stretch and the same symmetric residual stretch; state if each row instead has its own observed-flux normalization. The residual sign is data minus model. Add a population summary below the stamps so examples are not the sole evidence.

**Population summary:** plot paired change in held-out prediction error versus Gaia color, with median and uncertainty in each bin; a second compact panel may show the same quantity versus crowding if that addresses a central claim. Define improvement, for example $\Delta L_i=L_{i,\mathrm{color\text{-}independent}}-L_{i,\mathrm{chromatic}}$, so positive values mean improvement, using the same pixel-data loss, weights, support, and per-group normalization. Do not mix the regularization penalty into test-pixel loss or call a robust loss reduced chi-square. Report the distribution and fraction of groups improved, not only the aggregate mean.

**Prediction contract:** for the combined WCS/ePSF test, fix positions to the respective Gaia-constrained model prediction and permit the same conditional flux solve in both models; no local background. If positions are refitted to isolate shape quality, label that as a separate shape-only test because it absorbs positional error. State that flux estimated from test pixels makes this a conditional profile-prediction assessment, not a prediction of absolute stellar flux.

**Evidence required:** test pixel/noise/mask arrays, grouped scene models including companions, both predictions, nuisance-parameter rules, and per-group metrics. Claim improvement in native-image modeling on the evaluated stars; this figure supplies no light-curve performance measurement.

- **Placement/question:** Sections 4.3–4.4; does the measured response improve held-out image modeling as well as positions?
- **Panels:** observed versus predicted retained-star stamps and signed residuals for matched achromatic/chromatic fits; population summaries of held-out profile error or predictive loss versus color and crowding; optional null/injection interval supporting the interpretation.
- **Inputs:** identical background-subtracted preliminary-image pixels (§3), test stars, fitted stellar flux conventions, fixed background preparation, masks, model predictions, and noise treatment. Use a common pixel domain and document allowed nuisance parameters on test stamps; no local background is fitted.
- **Controls:** same display scales and noise model for comparisons; show representative and difficult cases using a predeclared selection rule. Track failure fractions and uncertainty rather than selecting only clean residuals. This is a single-image test, with no light-curve panel.

### Figure 15 — How does displacement depend on stellar color?

**Placement and provisional identifier:** a dedicated main figure in Section 4.4, immediately after Figure 10 and before Figure 12. Preserve the identifier until final renumbering. The sequence is observed residual signature → measured color relation → spatial variation of that relation.

**Question:** how much does the stellar image position change with Gaia $\mathrm{BP}-\mathrm{RP}$ within a fixed detector region, does a linear color dependence describe the measurements, and what color dependence remains after correction?

**Detailed layout:** a 2×2 set of axes plus a small detector locator map. Top left: measured x displacement versus $\mathrm{BP}-\mathrm{RP}$; top right: the corresponding y displacement. Bottom left and right: remaining x and y position residuals after applying the adopted chromatic correction. All panels share the same color axis. Use a few predefined, well-supported detector regions, with a consistent region color/symbol across panels and a matching locator-map key. If the region curves overlap excessively, use region columns with x/y subpanels rather than vertically offsetting measurements.

**Position and reference definition:**

- Define the upper-panel displacement relative to the color-independent Gaia-to-image prediction using the independently measured image positions described for Figure 9. State the residual sign, common core-origin convention, and native-pixel units.
- Show actual bin mean colors on the horizontal axis, uncertainty on the measured displacement, and color-bin extents or counts where useful. Mark $c_{\mathrm{ref}}$ with a vertical reference line and zero displacement with a horizontal line.
- Allow a free intercept in the empirical color relation; do not force the measured displacement to vanish at $c_{\mathrm{ref}}$. A region's achromatic position offset is distinct from its color slope.
- Overlay the adopted model relation, evaluated or averaged over the same spatial support as the measurements. If the reference-color WCS changes between the color-independent and chromatic fits, include that change when predicting the full upper-panel residual relation; do not compare it with $\delta\,\boldsymbol{d}$ alone without accounting for the offset.
- Use the same image-position measurements for the lower panels where their centroid convention remains applicable, subtracting the complete adopted position prediction. Disclose any profile-dependent remeasurement. Keep native-pixel units; a tighter lower-panel scale is acceptable if clearly labeled.

**Independent measurement of the relation:**

- Estimate positions/displacements separately in color bins, or use binwise fits with the relevant position freedom; do not generate the points by evaluating the imposed linear chromatic term. State which WCS/profile parameters are held fixed and which are measured.
- Use held-out stars where available and identify any preparation or profile-estimation dependence. If the bin measurements help select the color law, they are development evidence; use a separate sample for the final correction assessment.
- Choose color-bin boundaries from available support and adequate counts, not from the desired appearance of a straight line. Compare an empirical free-intercept slope with the adopted relation, reporting uncertainty and any curvature or departures supported by the data.
- Report slopes in native pixels per magnitude of $\mathrm{BP}-\mathrm{RP}$, color range, region counts, and residual trend after correction. A remaining slope consistent with zero within uncertainty is more informative than a visually flat fitted line.

**Spatial and population controls:** select regions before inspecting the strongest chromatic signal, require local color overlap, and match or account for magnitude, crowding, and pixel phase. Retain comparable within-region detector-position distributions across color bins; if the field varies appreciably even inside a region, use position matching or predictions averaged over each bin's actual support. Avoid a whole-CCD mean that can cancel differently directed shifts. Document excluded or unsupported regions.

**Uncertainty and inputs:** source/group-level resampling with the stated refitting policy; the frozen evaluation table containing Gaia color, measured position, both position predictions, detector location, magnitude, crowding, phase, region/bin membership, and flags. Shared profile or preparation uncertainties can correlate bins; state what the error bars include. Reuse Figure 9's position-measurement definition and Figure 10's population controls.

**Intended evidence:** an empirically measured color–displacement relationship, its supported functional form and spatial differences, and the remaining position trend after correction. This is not merely a visualization of the model's assumed color law and does not independently identify its optical cause.

### Figure 12 — Spatial chromatic displacement and shape

**Placement/question:** Section 4.4; how do the measured color response and its uncertainty vary across the detector?

**Detailed layout:** top row: (a) displacement vectors $\boldsymbol{d}(\boldsymbol{r})$ with a legend in native pixels per magnitude of $\mathrm{BP}-\mathrm{RP}$; (b) a dilation-coefficient map with the sign convention that positive coefficient narrows the profile. Mark the fitted node locations and unsupported regions in both maps. Put shift magnitude in arrow color only if readable; otherwise use a supporting magnitude map rather than adding another main panel.

Bottom row: (c) normalized blue/reference/red profile cuts at one or two predefined detector locations, with common scales; (d) recentered blue-minus-red shape images at those locations, or a profile-width difference with uncertainty if its finite-support definition is stable. These panels translate coefficients into an actual profile change. Choose the displayed colors within local data support and quote their values; do not extrapolate to extreme catalog colors for contrast. Mark the example locations on the upper maps.

**Displacement versus shape:** use a common reference frame to illustrate centroid motion in panel (c), and explicitly recenter panel (d) using the adopted core-origin convention to isolate shape. State whether these are evaluations of the fitted model or independently stacked measurements. Model evaluations illustrate the calibrated response; Figures 10 and 15 supply empirical residual and color–displacement evidence.

**Uncertainty/support:** resample at source/group level with the stated refitting policy; report which fitted and preparation quantities remain fixed. Use node uncertainty annotations or a small supporting map, plus local color spread/effective counts. Smoothly interpolated pixels are not independent measurements, and conditional uncertainties do not include unvaried template/background errors.

**Optional interpretation panel:** radial/tangential displacement versus field radius may replace a redundant profile panel only if an independently established optical-axis location and coordinate transformation are available. Otherwise keep radiality analysis in supporting material and distinguish an empirically fitted center from physical optical geometry. Section 5 can discuss this figure without adding a second field map.

**Evidence required:** adopted base-profile and chromatic fields, rendering configuration, reference color, actual color coverage, detector mapping, and uncertainty realizations. The intended claim is a spatially resolved calibrated color response over supported positions/colors, not a unique optical mechanism.

### Figure 14 — Final reference-color ePSF across the detector

**Provisional identifier:** insert before Figure 9 in reading order, then renumber at final consolidation. This separates a major paper product from the WCS-validation figure.

**Detailed layout:** a 5×5 mosaic of model profiles evaluated at fixed detector positions, echoing the initial ePSF display in Figure 6. Each tile uses the same native-pixel extent, profile normalization, origin, and stretch. Label the whole mosaic with detector orientation and coordinates; use a small locator/support map if a regular tile layout could be mistaken for contiguous image pixels. The 5×5 display is independent of the fitted node grid.

**Profile convention:** show the adopted model at $c_{\mathrm{ref}}$, explicitly distinguishing it from the separately fitted color-independent baseline. Render every tile at the same stated native-pixel phase, or show the stored 4× representation consistently; do not mix the two. Prefer a common-phase native response for direct instrumental interpretation, with fine-profile contours if needed and clearly identified. Do not imply that display interpolation adds measured resolution.

**Compact lower panel:** major/minor-axis cuts or encircled-flux curves at three predefined positions (for example a supported central region and two separated outer regions), using the same support and normalization. This makes spatial profile variation legible beyond the mosaic. If center/corner choices are outside supported coverage, choose documented alternatives.

**Comparison with initialization:** use Figure 6 as the main visual reference. If claiming a specific profile improvement, supply matched held-out errors in Figure 11 or a small supporting initial/final difference panel on identical grids and centroid conventions. Sharper or smoother-looking fitted profiles alone do not demonstrate a better calibration.

**Evidence required:** final reference-color node images, grid geometry, renderer/support settings, normalization/origin, source-support map, and selected positions. The figure shows the recovered spatial response; its predictive validity comes from Figures 9 and 11.

### Figure 17 — Photometric consistency: measured TESS flux vs Gaia-XP synthetic Tmag, by colour, brightness and detector position (added 2026-09-29, user request)

**2026-09-30:** Figure 17 is the held-out **bright-star** figure in §4.6. The faint targeted stars go in Figure 19 (stamps) and Figure 21 (joint panel) in §5.5.

**Question:** does the calibrated ePSF return the right flux for every star, whatever its colour, brightness or detector position? This is the aperture-correction figure. Its colour panel is the evidence for G5 (the −3.5 %/mag colour slope) and its brightness panel is the evidence for G2 (brightness width). Positions and profile residuals (Figs 9–11) never test this, because the conditional flux solve absorbs any profile normalisation error.

**Quantity:**
$$\Delta m_i = \left(-2.5\log_{10} F_i + Z\right) - T_i^{\mathrm{XP}}, \qquad A_i(r) = m_i^{\mathrm{psf}} - m_i^{\mathrm{ap}}(r),$$
where $F_i$ is star $i$'s fitted flux (e⁻ s⁻¹) from the conditional flux solve at fixed shared parameters; $Z$ is one fitted zero point per FFI (median over the evaluation set, not per bin); $T_i^{\mathrm{XP}}$ is the Gaia-XP synthetic TESS magnitude (XP spectrum × TESS response; G1 A4 "XP truth"; the `tess_mag` catalogue value is used only for stars without XP, and flagged); $m_i^{\mathrm{psf}} = -2.5\log_{10}F_i + Z$; and $m_i^{\mathrm{ap}}(r)$ is the magnitude in a background-subtracted circular aperture of radius $r$ native pixels on the same difference-image pixels. $A_i(r)$ is the aperture correction; $\Delta m_i$ splits as $A_i(r) + [m_i^{\mathrm{ap}}(r) - T_i^{\mathrm{XP}}]$.

**Panels:**
- (a) **Colour:** binned median $\Delta m$ vs colour $u$ (BP−RP on a secondary axis) with bootstrap errors, for the adopted A3 calibration, the colour-independent baseline and the production/pass-1 ePSF photometry. Overlay the aperture-only curve $m^{\mathrm{ap}}(r) - T^{\mathrm{XP}}$ at two radii, with a shaded band for the TESS response-curve choice (SVO vs POC v2, ±5% for M stars). The `bandpass_rizy_20260929` template band-weight prediction applies only to stars that are IN the template (T ≥ 13, held-out faint stars in the final image), not to the T < 13 fit stars, which the template omits; overlay it on that sample only. This separates model, response-curve/catalogue and bandpass contributions.
- (b) **Brightness:** $\Delta m$ vs $T^{\mathrm{XP}}$ over the fit range (T≈8–13) and the held-out faint range, same models. The width($q$) term (D9) belongs here if G2 passes; otherwise this panel shows the measured limit.
- (c) **Detector map:** binned $\Delta m$ over the CCD (common colour scale, optical-axis direction marked), for the production ePSF (left) and adopted A3 (right). On s25 the production ePSF showed ±0.05 mag, 55–60% of it aperture correction; forward_epsf_wcs was flat to ~3 mmag (`dev/forward_epsf_wcs/diagnostics/mag_residual_map/`, 09-17/22).
- (d) **Aperture correction:** $A_i(r)$ vs colour and vs radius $r$ = 2…6 px. Blue stars carry more wing light (+0.022–0.03 mag per mag at $r\ge5$ on s25/s52); a colour-blind ePSF shows an opposite-sign chromatic $A$ that cancels it in $\Delta m$. Showing both keeps the cancellation from hiding a real term.

**Sample and rules:** out of fold only (D8; G1 folds); one fixed star set for every model compared (isolated plus grouped, flagged); exclude |Δm|>0.5 mag outliers (known hp_d dead bands) and report how many; state N per bin. Label the zero point fitted, not absolute. Report slopes in mag per mag of colour and %/mag of Tmag with uncertainties.

**Inputs:** full_xsec + fold fits (`flux_solved.npz`, b914ef), XP synthetic T (G1 A4), fold file, difference-image pixels for apertures, production `centroids` photometry for the baseline, bandpass_rizy synthetic curves. Must be rebuilt after D13 (band weights) is applied.

**Readiness:** in-sample PREVIEW exists (2026-09-29, `figures/photometry/figure17_photometric_consistency_preview.*`, script alongside): F1, 10,493 fit stars, production / C0 / #14 / A3(plain prior). Colour slope of Δm: production −9±2, C0 +5±1, #14 +31±1, A3 +29±1 mmag/mag; A3's aperture split puts +25–26 of it in m_ap(r=5–6)−T^XP and only +3–6 in its own aperture correction → the G5 slope is mostly data-vs-XP-catalogue (response curve / instrument), not model normalisation (lead, in-sample). Tmag term −2.4 mmag/mag shared by all models (~15–20 mmag brighter-star deficit at T≈8.3). Map rms production 16.6 → A3 3.1 mmag. Final version waits on full_xsec (recipe may change, b914ef) + OOF fluxes. Replicate on F2 and S22 as a supporting panel or table.

### Figure 18 — Preliminary vs final de-crowding on real pixels (§3.3/§5.3; added 2026-09-30)

Real cutouts around 3–4 preselected $T<13$ stars with neighbours, on the adopted frame. Columns:
1. FFI (background subtracted).
2. Preliminary model $K\ast\sum_bw_bT_b$.
3. $D_{\rm prel}$.
4. $D_{\rm final}$ (preview of §5).

One stretch per row, native-pixel grid overlaid, calibration star marked. It demonstrates the claim that omitted stars keep their signal while neighbours are removed. It replaces the synthetic Figure 8 in the main text.

### Figure 19 — Targeted de-crowding (§5.5; added 2026-09-30; was "star extraction")

For a few $13\le T<16$ targets, from left to right:
- $D_{\rm final}$ (target absent);
- $M_t$;
- $D_t$;
- calibrated-ePSF model;
- residual.

A lower panel shows recovered flux against Gaia-XP synthetic $T$ for all extracted held-out stars, by colour. The inputs come from the extraction agent (user 09-30). Figure 17's faint-star sample reuses these fluxes.

### Figure 20 — Per-band kernels from the calibrated ePSF (§5.2; added 2026-09-30)

$P_0$ at three nodes (centre, the corner nearest the optical axis, the far corner), with $P_1$ and $E_b-P_0$ for $b=r,i,z,y$ on one symmetric scale per row. An arrow marks the direction to the axis. It shows that the red and the $y$ band get opposite dipoles (±0.6 of peak in the A3 stand-in), which is what one Hotpants kernel cannot do. Include a linearisation-error inset or a supporting panel. Stand-in source: `perband_ccd_20260929/kernels/figs/a3_band_epsf_node.png`.

### Figure 21 — Joint photometry, bright and faint (§5.5; added 2026-09-30)

$\Delta m$ against Gaia-XP synthetic $T$ versus $T$ (≈8–16), colour-coded by BP−RP. Held-out bright stars (forced, calibration from the other folds) and extracted faint stars appear together, with binned medians and scatter, and a vertical line at $T=13$ where the method changes. The second panel shows $\Delta m$ against colour for both populations, with the §2.2 synthesis-error curve overlaid for the faint (template-resident) stars only.

### Tables and supporting figures

**Figure 13 — Optional supporting model-complexity check:** retain this planning identifier for a compact comparison only if needed to justify the adopted model or explain a material limitation. A full development ladder, failed core/wing model, and loss-versus-residual story are not required main-paper content. Use identical masks, explicit metric definitions, uncertainty, and in-sample/held-out labels. The core main-paper comparison is color-independent versus adopted chromatic calibration in Figures 9–11.

- **Table 1 — data and calibration recipe:** exposure, detector/crop, bands/epochs, template scales, smoothing and matching configuration, sample/color coverage, masks, and frozen recipe reference. Move long parameter listings to an appendix.
- **Table 2 — independent-image validation:** per-FFI source counts, color coverage, held-out WCS bias/scatter, profile improvement, and failure fractions. Each FFI is fitted independently; no temporal coefficients or claims of stability.
- **Table 3 — calibration performance:** color-independent baseline versus adopted chromatic model, added parameters, positional bias/scatter, pixel prediction error, color-residual amplitude, uncertainty, and evaluation sample. Add a shift-only row only if needed to substantiate a separate shape gain. Full alternative-model results are optional supporting material; do not report pending validation as measured.
- **Supporting figures:** sample coverage and color/position overlap; seams and flux conservation; bright-source removal examples; missing-band/background/smoothing sensitivity; magnitude/phase controls; injections, color shuffles, and model-complexity tests. Reuse diagnostics from [FIGURE_PLAN.md](FIGURE_PLAN.md), which is the supporting inventory.

**Section 5.2 supporting sensitivity figure — only for material limits:** compare the adopted analysis against justified perturbations of background preparation, PS1 synthesis/neighbor treatment, and rendering phase/sampling choices. Use paired changes in positional bias/scatter and chromatic shift/shape amplitude with uncertainty, not a set of visually incomparable residual maps. Keep the evaluation sample fixed or explicitly account for changed membership. This figure tests sensitivity to preparation assumptions; it is not a return to the full optimizer/model ladder. Select perturbations from measured uncertainty or defensible configuration alternatives rather than arbitrary large changes. Discuss the dominant effects in Section 5 and leave minor checks in supporting material.

**Detailed readiness order for Sections 4–5:** (1) verify the fitted model, profile/rendering convention, and test-sample dependencies; (2) assemble one source/group-level evaluation table and both models' predictions; (3) inspect support and choose fixed display locations/color bins; (4) evaluate position and pixel metrics and uncertainties; (5) construct Figures 14 and 9, then 10, 15, 12, and 11; (6) decide which sensitivity checks materially change the discussion. No additional model variants are required just to populate figures, and no figure is treated as ready because a prior development screenshot exists.

**Order of work:** (1) freeze one FFI, recipe, and source accounting; (2) collect coverage and intermediate products for Figures 1–8 while recovering Figure 2 inputs; (3) audit the existing model-form evidence and its reference-color/metric conventions; (4) freeze development/evaluation splits and run matched controls; (5) produce Figures 14, 9–12, and 15 and the compact calibration-performance tables; produce supporting Figure 13 only if it resolves a specific interpretive need. This update records the plan; no plots or fits have been generated.

Every plot must record source paths/hashes, code/configuration, sample roles, masks, coordinate and flux conventions, uncertainty construction, and regeneration command. Missing archived figure files are not placeholders for current measurements.

## Appendix A — F=1 vs F=4 sampling (added 2026-09-30)

### A.1 F=1 vs F=4 performance

- Same frame, same background, same Hotpants settings: compare the F=1 and F=4 preliminary de-crowded images on held-out stars (star χ², neighbour residual around omitted stars).
- Explain why Paper 1 uses F=4 and why F=1 is kept for Paper 2's multi-frame work.
- Include the synthetic sampling demonstration (the current Figure 8) and, if useful, the F=1 preliminary image as a control for §4.

### A.2 Old §3 text (archive; background removal is now main-text §3.2)

The old §3 text follows verbatim for reference. Its pass-1/pass-2 language is superseded by the 2026-09-30 structure; the five-step background refinement and the Figure 5 detail-panel specification remain current.

<details><summary>Old §3 (pre-2026-09-30)</summary>

#### (old) 3. Two-pass template matching and calibration

**Purpose:** explain how an initial native-scale match supplies the calibration needed for a more finely sampled second match. These are two passes on the same FFI, not two observing epochs or a temporal WCS model.

**Two distinct refinements:** background → kernel → background → kernel is the background-refinement sequence described below; $F=1$ → improved WCS/ePSF → $F=4$ is the template-sampling sequence. Keep these separate in terminology and the diagram. The user supplies the background sequence; confirm from the chosen run whether it repeats at $F=4$ or the refined background is reused. Do not assume four background/kernel cycles.

**Figure 5:** two-pass method diagram with pixel scales labeled on every image branch and the improved-WCS feedback arrow into template construction.

```mermaid
flowchart TD
    P["PS1 reference scene"] --> T1["Template: native grid; initial WCS"]
    I["Selected low-background TESS FFI: native pixels"] --> B0["Mask all stars; initial TESSreduce background"]
    B0 --> H0["Initial Hotpants kernel on background-subtracted FFI"]
    T1 --> H0
    H0 --> D0["Preliminary de-crowded image"]
    D0 --> BR["Mask retained stars and residuals; refine TESSreduce background"]
    BR --> IR["FFI with refined background subtraction"]
    I --> IR
    IR --> H1["Rerun Hotpants: refined native-scale kernel"]
    T1 --> H1
    H1 --> D1["First de-crowded image: native pixels"]
    D1 --> C1["Initial ePSF and improved WCS"]
    P --> T4["Rebuilt template: 4× finer per axis"]
    C1 -->|"Improved WCS"| T4
    T4 --> H4["Pass 2: oversampling-aware Hotpants"]
    IR -->|"Background handoff at F=4 to verify"| H4
    H4 --> D4["Second de-crowded image: native pixels"]
    D4 --> F["Joint WCS and chromatic ePSF refinement"]
    C1 -->|"Calibration initialization; exact handoff to verify"| F
    G["Gaia positions and colors"] --> F
    G -->|"Catalog-guided source removal, where used"| P
```

##### (old) 3.1 Pass 1: Hotpants matching, the de-crowded image, and initial calibration

- Map the PS1 reference scene onto the native TESS grid using the available initial WCS.
- Explain Hotpants kernel matching explicitly: which image is convolved, the kernel basis and spatial variation, fitting stamps/masks and weights, and photometric normalization. Record actual settings, including whether any residual background term is enabled; do not assume a simultaneous polynomial background model. Template and science are both at native TESS sampling.

**Background and kernel refinement, before ePSF/WCS estimation:**

1. Select an FFI with minimal background using a stated selection criterion. Selecting one exposure does not mean averaging exposures or fitting a temporal background.
2. Mask all identified stars and estimate the initial image-level background with TESSreduce. Record source-mask extent, bad-pixel/strap treatment, interpolation/smoothing settings, and remaining sky coverage.
3. Subtract that estimate and run Hotpants to obtain a preliminary kernel. Apply it to the prepared PS1 template to construct a preliminary de-crowded image.
4. Re-estimate the background with TESSreduce on this preliminary de-crowded image. Because modeled faint sources have been subtracted, mask the retained bright stars and significant subtraction residuals; fewer pixels require masking. Show the resulting increase in usable area rather than assuming complete removal of contamination.
5. Apply the refined background correction and rerun Hotpants to obtain the refined kernel and de-crowded image used for initial ePSF/WCS estimation. Track whether the second estimate is a residual correction added to the first map or a replacement total map; do not subtract the same background twice.

Reserve the product relation

$$
D_1 = \left(I - B_{\mathrm{refined}}\right) - \mathcal{M}_1[T_1],
$$

with a stated subtraction sign. Here $B_{\mathrm{refined}}$ denotes the total image-level correction and $\mathcal{M}_1$ the template matching operator including photometric scale. Align it with the actual saved products. The final stellar optimizer does not fit a local background.

**Figure 5 detail panel (or supporting figure):** initial all-star mask and background estimate versus the reduced mask and refined estimate, on the same sky area and common background units. Annotate masked fractions and show the background correction difference. Pair this with the workflow arrows; do not confuse the preliminary background-loop image with the final $F=1$ product compared against $F=4$ in Figure 8.

- Present the de-crowded image as the key enabling product: the PS1 template models neighboring light while selected stars omitted in skycell space retain their TESS signal after subtraction. Explain why this makes their empirical profiles and centroids easier to measure. This is not a claim that every contaminant or background residual vanishes.
- Estimate an empirical ePSF and improved stellar positions/WCS from this first-pass image.
- State the actual order of ePSF, centroid, and WCS estimation, and which outputs initialize the next pass.

**Figure 6 — first-pass spatial ePSF:** a 5×5 grid of initial empirical ePSFs across the fitted CCD/region, with detector locations, identical normalization, pixel scales, and display convention. Indicate source counts/support; distinguish profiles reconstructed from local stars from evaluations of a spatial interpolation. A 5×5 display does not prescribe 25 fitted nodes. These are the pass-1 estimates, not the final chromatic profiles.

**Figure 7 — first-pass WCS residuals:** before/after residual-vector maps and a compact residual distribution on matched stars, labeled with the initial and improved WCS. State whether positions are training or withheld measurements, the reference epoch, units, and sample size. This demonstrates the geometric handoff to template rebuilding; final accuracy remains a separate result.

**Role:** establish the calibration needed to rebuild the template and show what the de-crowded product enables.

##### (old) 3.2 Pass 2: finer template, native science pixels

**2026-09-29 (D15):** the F=4 Hotpants product described here is kept as a comparison baseline (Figure 8, G4 context), not as the image the §4 calibration is fitted on. The paper's second pass is now §3.3.

- Explain the dependency on pass 1: use its improved WCS to construct the $F=4$ template from the PS1 scene, retaining finer scene sampling during matching. Report any additional iterations only if actually used. Evaluation of geometry-versus-sampling gains belongs in Section 4.3.
- Apply the oversampling-aware Hotpants implementation, which accepts a finer-sampled template and the native TESS science image.
- Describe template convolution and its mapping/reduction to native science pixels as one matching operator; the kernel is constrained by the observed native pixels.
- The second-pass convolved model and difference/de-crowded products remain on the native grid. This second-pass image supplies the pixels for the Gaia-constrained refinement in Section 4.
- Specify kernel/background/support choices and identify differences from the first-pass matching. Do not conflate oversampled matching with optional irregular-stamp modes.
- State explicitly whether this pass reuses the refined image-level background or repeats the background/kernel refinement. Whichever the run uses, deliver background-subtracted pixels to Section 4 without introducing local stellar-background fits.

**Suggested name:** “oversampling-aware Hotpants” or “Hotpants with a supersampled template,” followed by a precise description. “Special Hotpants” alone does not explain the method.

**Figure 8 — $F=1$ versus $F=4$ de-crowded images:** aligned native-grid outputs for the same retained stars, with the original TESS region as context and common display scales. Show neighboring-light residuals and retained stellar cores. Keep the improved-WCS/$F=1$ control as a supporting panel or Section 4.3 comparison so the main view clearly shows the two products.

**Source status:** the $F=1$ → calibration → $F=4$ sequence is supplied by the user for this paper. The generic sampling contract is supported by [oversampling documentation](../../../docs/markdown/oversampled_templates.md), [Hotpants wrapper](../../../syndiff_pipeline/difference_imaging/stages/hotpants.py), and [kernel reconvolution](../../../syndiff_pipeline/difference_imaging/stages/kernel.py). These support a high-resolution template with native science/output, but do not alone verify the exact run configuration or orchestration of this two-pass sequence. Record those from the selected experiment before drafting implementation details.

##### (old) 3.3 Final pass: F=4 per-band templates matched with the calibrated chromatic ePSF (added 2026-09-29, gated; the paper's second pass under D15)

**Status:** in Paper 1 only if gate G4 in [MODEL_FREEZE.md](MODEL_FREEZE.md) passes; otherwise it becomes a short outlook paragraph in §5.3 and moves to Paper 2. Evidence so far comes from one 256×256 section only: `dev_runs/perband_template_20260928` and `dev_runs/template_kernel_20260924` (FE docs `TEMPLATE_KERNEL_PHASE_A_20260924.md`).

- **Purpose:** close the loop. The chromatic ePSF measured in §4 is used to build the template, so each PS1 band $b\in\{r,i,z,y\}$ is convolved with its own kernel $E_b = P_0 + \delta_b P_1$ from the frozen colour model. This replaces the single Hotpants kernel, which cannot follow a colour-dependent PSF.
- Describe the kernel distillation (ePSF on the template grid → matching kernel) and why the template WCS must equal the ePSF WCS (Phase A: a 0.047 px offset otherwise).
- Kernels come from the **frozen** model (A3 in $u$ + moment-blind prior + width term if G2 passes), not the section test's linearised C5-in-BP−RP model.
- **Evaluation must be out of sample**: the ePSF was fit on this FFI's pass-2 image, so score pass 3 against pass 2 on held-out stars and on at least one FFI the ePSF never saw. Report pixel χ², per-star χ², red-star χ², and the red−blue dipole (section result: 6.84→5.17, 1.93→1.38, 5.2→1.56, ±0.9%→≤0.3%; in-sample, one section).
- **No open-ended iteration:** one pass-3 template. At most one ePSF refit on the pass-3 image, reported as a convergence check.
- Prerequisite: fix or rule out the `ps1_process` sparse-path cell-overlap bug found in `perband_template_20260928` for this CCD.

**Figure (new, 16):** pass-2 Hotpants vs pass-3 per-band difference image on a common scale, with red−blue residual stacks before/after. Held-out stars only.


</details>

## Appendix B — Selecting the chromatic ePSF model (added 2026-09-25)

Written out in `latex/main.tex`, Appendix B. It covers:
- B.1 the candidate generators and their collinearity;
- B.2 why the diagnostic is the colour-split stack rather than the loss;
- B.3 post-hoc selection with held-out colour halves, including the axis scan;
- B.4 the 16-variant trained ablation on S24 C2 K2 and S20 C3 K3, with convergence;
- B.5 the fitting choices: raw vs. mean gauge, global vs. per-population c_ref, L2 vs. Huber, halo and pedestal dropped;
- B.6 the replication-field figures.

Figures A1–A3 and Tables A1/A2 are in `figures/chromatic/model14/`. Decision record: `dev/forward_epsf_wcs/docs/COLOUR_MODEL_DECISION_20260924.md`.

**A3 update (2026-09-29, figure agent):** new B.6 "From #14 to A3" (current B.6 → B.7). Figures B4 (stage-1 arms), B5 (#14→A3 on 11 FFIs), B6 (per-CCD plane coefficients), B7 (camera-2/CCD-2 width band = D5 limit) and Table B2 (17 coefficients) in `figures/chromatic/a3/`; draft text, symbol definitions and captions in `figures/chromatic/a3/APPENDIX_B_A3_NOTES.md`. In-sample model-selection evidence; Table B2 to be refreshed with the A3 + moment-blind full_xsec values.

## Appendix C — Band-synthesis verification (added 2026-09-30)

This covers:
- the weights fit (NNLS on Gaia-XP spectra × TESS POC v2.0, colour-balanced, out of field);
- the weights table by spectral type (production, TESSreduce, adopted);
- **S-bandpass** (synthesis error vs colour);
- **S-response** (difference-image and bright-star evidence for v2.0 over SVO);
- **S-weights** (the per-band image with production vs adopted weights, three fields, including the C3K3 red deficit).

The detailed plan moved from §2.2 follows.


- Describe the weighted r/i/z/y image combination as an approximation to the TESS-band scene; reserve an equation and coefficient table for the verified recipe.
- The older draft and current `combine_rizy_bands` defaults both contain weights (0.238, 0.344, 0.283, 0.135). Record them as a code-supported starting point, not confirmation of the selected experiment's settings or of their accuracy.
- Explain how the weights were derived, the relevant flux convention, and the stellar spectra/color range used to assess them. Resolve the original calibration source before manuscript drafting.
- Separate bandpass synthesis from the overall photometric scale in Hotpants and the TESSreduce background estimation in Section 3. Document any residual Hotpants background term only if enabled in the actual run. Absolute AB calibration need not become a separate Paper 1 result.
- Quantify residual color dependence using synthetic photometry or an independent photometric check. The old linear-versus-nonlinear comparison tests two approximations, not agreement with true TESS flux by itself.
- Discuss inter-band PSF differences, missing bands, nonstellar spectra, and spatial color gradients as possible template errors. These matter because the final measurement is itself chromatic.

**Figure 2 — PS1 and TESS bandpasses, one journal column:** show PS1 filter responses and the TESS response, identifying the r/i/z/y bands used in synthesis. Design at approximately 3.4-inch width with readable type and a compact, ordered legend above the curves, avoiding overlaps with the responses. Keep the main figure to the response plot; any color-residual diagnostic is a separate supporting figure, not an added panel that forces Figure 2 to span two columns.

**§2.2 plan (user 2026-09-30, MODEL_FREEZE D13/D17/D18; evidence `dev_runs/bandpass_rizy_20260929`, canvas https://claude.ai/artifact/K9xoQjYHDCakkKutaT3He8).** Tell it in four steps:
1. **What the template does.** $C=\sum_b w_bF_b$ is a filter with effective response $R_{\rm eff}\propto\sum_b (w_b/Z_b)S_b$, compared with the TESS response $R_T$. $F_b$: PS1 band-$b$ flux image (AB, ZP 25). $S_b$: PS1 system passband. $Z_b=\int (c/\lambda)S_b\,d\lambda$ (with $c$ the speed of light). Use TESS **POC v2.0** everywhere (D17).
2. **Why it cannot be exact.** Four fixed filters cannot reproduce a different bandpass for all SEDs. PS1 curves include a 1.2-airmass atmosphere (O₂ 760 nm, H₂O 930–970 nm) that TESS does not see. r responds at 550–580 nm where TESS does not. The error is a smooth function of colour (±0.5 % at fixed colour on real XP SEDs), with a ~1–2 % floor for late M whatever the weights.
3. **How the weights were chosen and how good they are.** NNLS on Gaia-XP stars × v2.0, colour-balanced, fitted out of field (F2+S22 → F1, rotated). Contrast with TESSreduce (Calspec, $g-r<0.8$, SVO curves) and TGLC (per-star Stassun relation). A table gives the error by spectral type for production, TESSreduce and adopted weights. State once that the legacy weights leave M stars 5–9 % too bright.
4. **Consequences.** They enter §5 (per-band products are built with the adopted weights; the remaining red residual is A3's red-PSF sharpening beyond BP−RP≈2, a stated limit) and Fig 17 / G5 (the colour slope vs XP × v2.0 is not explained by the synthesis for T<13 fit stars, which are not in the template). The colour label $u$ keeps its current definition (D18).


Supporting figures: S-bandpass (synthesis error vs colour), S-response (evidence for v2.0 with residual plots), S-weights (per-band product with production vs adopted weights). Specs and status are in MODEL_FREEZE §8.


## Temporal context reserved for Paper 2

- Frame the final inference as forward modeling with Gaia positional/color constraints; specify whether these enter as fixed inputs or probabilistic priors.
- Include the requested link to Paper 2: velocity aberration changes the apparent sky-to-detector mapping and can enter fitted distortion terms; thermal changes may alter the PSF and potentially optical distortion. Distinguish apparent mapping changes from physical optical distortion, and leave their temporal characterization to Paper 2.

## Outline decisions still to resolve

- **(2026-09-30) Where the delivered ePSF comes from under the F=4 preliminary de-crowded image:** (a) refit on the F=4 preliminary de-crowded image; or (b) deliver the D14 refit on the final image. All current fits (full_xsec, folds, §4 figures) use the F=1 image. MODEL_FREEZE D15 needs updating either way.
- **(2026-09-30) Camera 3 / CCD 3 red deficit (§5.5):** per-CCD flux scale vs stated limit.
- **(2026-09-30) Star-extraction evidence (§5.3, Fig 19):** separate agent, user-started.


1. Exact definition of the de-crowded image and which stellar flux is retained.
2. ~~Exact calibration handoff between passes~~ superseded 2026-09-30 by the preliminary → calibration → per-band structure; the remaining open part is the first decision above (delivered ePSF source).
3. Fair common-domain comparison demonstrating the specific benefit of crowding reduction.
4. How template errors are prevented from masquerading as chromatic instrumental response.
5. Verified PS1 band-combination provenance and flux units; preprocessing width, source-removal settings, and overlap/remapping operator for the selected run.
6. Background-map accounting (residual versus total correction), TESSreduce settings, any Hotpants residual background term, and whether background refinement repeats at $F=4$ or its output is reused.

7. (2026-09-29) Gates G1–G5 in [MODEL_FREEZE.md](MODEL_FREEZE.md): cross-fit harness, brightness width, model freeze, pass-3 per-band templates, flux colour slope.
8. ~~Whether pass 3 changes which image §4's final fit uses~~ RESOLVED 2026-09-29 (D15), REOPENED 2026-09-30 by the F=4 preliminary de-crowded image (see first decision above): §4 fits the F=1 difference image; the F=4 per-band final image is used for the D14 refit (convergence, G6) and for G4.

These are the next method-planning questions; no manuscript drafting or new fit execution is implied.