# Final magnitude/colour ePSF investigation

**Bounded scientific investigation complete; numerical results and final figures independently reviewed.** Campaign: 2026-10-04, completed analysis on 2026-10-05 UTC. Manuscript writing remains paused.

## Decision

**Retain the existing full spatial/chromatic A3 recipe. Do not promote the added CCD colour shifts or the prescribed magnitude-width term. Noise-level closure has not been achieved.**

The investigation supports a bounded model-choice conclusion. It does not establish that all real magnitude/colour dependence has been captured, that the optimizer has reached a unique solution, or that absolute photometric errors are negligible. The conspicuous residual structure is not removed by either tested extension.

## The decisive reserved-exposure comparison

The exposure was reserved using metadata before its model outcomes were inspected. It received fresh header/epoch catalogue geometry, a fresh analytic/header seed followed by strictly training-only photutils initialization. All three models used the same fixed source population and physical evaluation cores. The fresh exposure uses a different catalogue-role population and initialization from development; this is a conditional recipe comparison, not a claim about fully converged model-family equivalence. The width predictor was rebuilt from this exposure's own training calibration and catalogue information.

| Addition to A3 | Change in held-core error score | Conditional 97.5% interval | Decision |
|---|---:|---:|---|
| Two learned CCD colour shifts (CCD19) | +0.375% | −0.225% to +0.905% | No demonstrated improvement |
| Fixed historical magnitude-width response (A3w) | +1.865% | +0.712% to +3.071% | Worse for this tested recipe |

![Reserved-exposure A3, CCD19 and fixed-width residuals, with separately scaled differences](crossfit/F1_confirmation/fold0/confirmation/confirmation_residuals.png)

The top three image rows share a residual scale. Every arm uses the same per-star A3 fitted-flux denominator; the lower difference rows share a separately magnified scale. Columns use catalogue magnitude and the actual fitted colour coordinate. Common central and asymmetric structure remains. These equal-star mean images are descriptive; the primary statistic is the median paired per-star score ratio.

![Absolute held-core error-score levels for all three reserved-exposure models](crossfit/F1_confirmation/fold0/confirmation/confirmation_score_levels.png)

Positive values mean worse prediction. The comparison retains all 2,249 pre-frozen cores, uses 50 dependence components, and shares 10,000 bootstrap draws across the two contrasts. The two nominal 97.5% component-bootstrap intervals implement the declared Bonferroni allocation for nominal conditional 95% family coverage. Bootstrap coverage is approximate, not a guaranteed finite-sample coverage theorem. An independent reviewer reproduced both bootstrap columns exactly. The intervals condition on the fitted profiles, common preparation and realized evaluation calibration; they do not propagate every learned-model or detector-noise uncertainty.

The score and contrast are

$$c_i(M)=\frac{1}{n_i}\sum_{p\in C_i}\frac{[D_p-\widehat D_p(M)]^2}{v_p},
\qquad
\Delta_M=\operatorname{median}_i\ln\!\left[\frac{c_i(M)}{c_i(\mathrm{A3})}\right].$$

Here $i$ indexes a fixed evaluation star; $M$ is the tested model; $C_i$ is its original-valid physical core within three native TESS pixels; $n_i=|C_i|\ge15$ is its pixel count; $p$ identifies a physical image pixel; $D_p$ is the prepared image in electrons per second; $\widehat D_p(M)$ is the complete fitted scene prediction in the same units, including neighbours; and $v_p$ is the renderer's supplied pixel variance, including its configured numerical floor, in squared image units. The dimensionless score $c_i$ is a mean normalized squared residual, **not** a reduced chi-square with an established number of independent degrees of freedom. The dimensionless contrast $\Delta_M$ is reported above as $100[\exp(\Delta_M)-1]$ percent.

Evaluation profiles/WCS are fixed, while fluxes and the global nuisance-prior calibration are re-estimated under the declared common evaluation rule. The upstream background and Hotpants preparation use the whole image, so this is independent forward-model validation conditional on that preparation, not a completely withheld-raw-pixel end-to-end test.

[Locked protocol](review/CONFIRMATION_RESERVATION.md) · [Exact result](crossfit/F1_confirmation/fold0/confirmation/summary.json) · [Independent reproduction](review/confirmation_outcome_review.json) · [Exact model equations and units](models/FINAL_MODEL_CONTRACT.md).

## What magnitude and colour tests actually establish

The existing 17 A3 colour coefficients contribute substantially: switching them off at otherwise fixed fitted parameters increases the F1 median per-star score by a factor of 1.4788. This is a conditional ablation, not a fully refitted achromatic-family comparison.

![Actual data/model/residual images for the colour ablation](models/colour_off/held_stacks.png)

The two extensions also fail to demonstrate improvement on the current-template F1 development sample. The fixed-width term changes the central profile slightly but leaves the larger common pattern. C4 provides a second development comparison with no CCD19 promotion case. These results reject promotion of these particular prescriptions; they do not show that physical brightness effects or additional chromatic effects are absent.

![Current-template A3 versus fixed-width residuals, common catalogue-flux denominator and shared scales](significance/current_F1/width_comparison_images.png)

All three current F1 models retain clear structure relative to their own fixed-profile fitted-noise reference. The tests account for repeated physical pixels, flux refitting, correlated measured modes and the three declared model checks. Robust images and influence checks show that a handful of stars does not explain the remaining central feature. Learned-profile/preparation uncertainty and all physical noise covariance are not included. The large standardized reference statistics must not be reported as physical ePSF detection sigmas.

[Current three-model significance, null simulations, injections and all underlying images](significance/current_F1/).

## Independent prediction and numerical limits

All 25 spatial fits and their scoring are complete across five fields: **49,541 fixed held cores**. One C4 star has a negative fitted amplitude and remains in every primary statistic; its unnormalized stamp is shown separately from positive-flux compatibility rasters.

![Actual held-out data, full-scene models and directly stacked residuals across all five fields](review/k5_data_model_review.png)

These descriptive five-field panels use the fixed nominal catalogue normalization stated on the figure. They are not an independent absolute-flux calibration.

![Five-field held-out residuals by catalogue magnitude, common normalization and scale](review/k5_residual_review.png)

![Five-field score levels and descriptive training/prediction gaps](review/k5_score_review.png)

The brightest bin has a median paired held-out/in-training score ratio ranging from about 6 to 16.6 across fields. Its comparator is the same star under the other four current-generation fold models, whose training included its pixels. This is a descriptive generalization gap. Spatial training coverage, initialization and the contributor/anchor role transition are confounded with magnitude; the contrast cannot isolate an intrinsic magnitude-dependent ePSF effect. The near-unity anchor-bin ratio does not mean those residuals are noise-like.

![Same F1 stars and native pixels under held-out and all four in-training models](crossfit/figures/F1_matched_stacks.png)

A separate training-only spatial-start control gives close aggregate scores but retains flux and WCS differences. The median absolute flux difference is 0.0592%, and its 95th percentile is 0.2946%; these are route differences, not measured astrophysical biases or equivalence bounds. Both development endpoints reach the final iteration cap. Confirmation endpoints stop under the prescribed scalar-plateau rule, with substantial native quasi-gradients; that stopping rule is not a convergence certificate.

![Underlying models and residuals from the two spatial starts](crossfit/start_control_report/figs/magnitude_images.png)

![Flux and score differences between spatial starts](crossfit/start_control_report/figs/endpoint_metrics.png)

[Five-fold evidence and inspected grids](crossfit/figures/) · [Start sensitivity and checkpoint curves](crossfit/start_control_report/) · [Independent audit](review/SIGNOFF.md).

## Why the remaining structure is not assigned a single physical cause

The RAW/prepared-image comparison is sensitive to the neighbour basis. Adding known neighbour profiles reduces some RAW-minus-prepared differences, but the same flexibility absorbs deliberately injected mismatch and reduces target-flux information. Faint external calibration is also insufficient to serve as fixed absolute flux truth. Those controls are attribution-inconclusive; they do not prove either missing PS1 light or intrinsic ePSF width as the sole cause.

The original C4 source-accounting ledger validates its original template generation. It does not automatically certify the different latest-v3 template or catalogue completeness. The reserved preparation also records unavailable TNS masking after an HTTP403 response. These limitations are retained in the provenance.

[Illustrated RAW and neighbour-basis controls](raw_control/) · [Faint-flux calibration](calibration/extension_C4/) · [Pixel-noise audit](noise/README.md) · [Template-generation scope](noise/ledger_review.md).

## Paper-ready conclusion and next boundary

A defensible conclusion is:

> Spatial and chromatic modelling captures substantial repeatable structure, but coherent held-out residuals remain. The targeted F1/C4 development comparisons and the prospectively reserved F1 comparison do not justify replacing the reference A3 recipe with either proposed extension. We therefore retain A3 with explicit residual and numerical-stability limits; noise-limited ePSF modelling and absolute photometric accuracy are not established.

This investigation is sufficient to close the comparison of the three declared recipes. It does not make the entire paper submission-ready or validate sector-length light curves. A stronger noise-level or precision claim requires evidence addressing scene/calibration ambiguity and numerical stability under a justified science-error budget. Expanding the model again is not supported by this experiment alone.

No production default was changed and no manuscript was drafted. Writing stays paused. Exact recipes, source/input hashes, archived attempts, inspection records and reproducible commands remain in the linked subreports.

**Conclusion:** A3 is the supported baseline for the tested recipe and domain. The original aspiration of almost noise-level magnitude/colour residuals remains unmet.