Purpose of this note

Manuscript v30.4 is a wording-only revision of v30.3: no numbers, equations, figures, or validation domains changed. Three short passages were added to strengthen the defense of the empirical $(\widehat{S},b)$ beyond-Fisher corrections (Eqs. mean-corr/sigma-corr) against the natural referee question, “why are these corrections empirical rather than derived?” This note expands each added passage with the full supporting evidence, which is intentionally kept out of the manuscript itself (referenced there only as “to be presented elsewhere”).

The short version: we tried to derive the corrections twice and tried to show they are a universal PSF-independent law once. All three attempts failed, and each failure is informative. Taken together they convert “we do not have a derivation” into “we have evidence for why no simple derivation exists, and evidence for exactly where the empirical law is valid.”

Passage 1: the residual is beyond any local second-order expansion

What the manuscript says

The empirical status of these corrections is itself a constrained result, not merely a modeling convenience. [...] a second-order expansion of $C_\mathrm{best}$ conditioned on the amplitude score, evaluated with no fitted coefficients, fails to reproduce the five coefficients in Equations (mean-corr)–(sigma-corr): its mean contribution is approximately a constant position-rank gain rather than the observed $\widehat{S}^{-0.40}$ decay, and its variance term does not yield the empirical $1/\widehat{S}$ scatter deficit. This places the low-signal residual beyond a local quadratic or Bartlett-type expansion, consistent with the position-argmax mechanism isolated in the reference-fitter decomposition.

The full picture

The empirical laws being defended are, from the production 17-position $\times$ 6-background reference-fitter run ($\sim 8\times10^5$ fits): $$\begin{aligned} \Delta\mu(\widehat{S},b) &= \big(a_\mu + c_\mu \log_{10} b\big)\,\widehat{S}^{-p_\mu}, \quad (a_\mu, c_\mu, p_\mu) = (0.59,\, -1.02,\, 0.40) \label{eq:target-mean}\\ \sigma_C^\mathrm{ratio}(\widehat{S},b) &= 1 - \frac{a_\sigma + c_\sigma \log_{10} b}{\widehat{S}}, \quad (a_\sigma, c_\sigma) = (1.49,\, 0.39) \label{eq:target-sigma} \end{aligned}$$

We attempted a first-principles derivation in two stages.

Stage A (unconditional Bartlett term). The first-order theory linearizes $\Delta C$ in the Poisson fluctuations and conditions on the fitted score. The next order in that expansion is the quadratic (Bartlett) score-space term, $Q = z^\top H z - \operatorname{tr}(H)$, whose unconditional variance is $2\,\operatorname{tr}(H)$ — a constant $+2$ (amplitude-only) or $+6$ (three-parameter) added to the variance regardless of $\widehat{S}$. This is the wrong sign and the wrong shape: it only adds variance, while the empirical law in Eq. [eq:target-sigma] needs a variance deficit that is large near threshold and vanishes at high $\widehat{S}$.

Stage B (score-conditioned expansion). We then derived the proper object: the quadratic score-space term projected onto the score-fixed (fixed-$\widehat{S}$) subspace, using the amplitude-score projector $P_S = I - a_S(a_S^\top a_S)^+a_S^\top$ and the position-block curvature $I_\mathrm{pos} = B^\top B$ with $B = P_S\,[\,S\,\partial p/\partial x/\sqrt{\mu},\ S\,\partial p/\partial y/\sqrt{\mu}\,]$. This gives a conditioned mean shift $\operatorname{tr}(I_\mathrm{pos}^+ I_\mathrm{pos}) + O(z^3)$ and a conditioned variance correction built from $U^\top I_\mathrm{pos}^+ R\, I_\mathrm{pos}^+ U$, with $R$ the curvature of the per-pixel Poisson mean. Evaluated numerically (Gaussian PSF, $\sigma = 1.5925$ px, $\widehat{S}\in[4,80]$, $b\in\{0.01,\dots,0.16\}$), with no coefficients fitted to the empirical law, the derived quantities were compared directly against Eqs. [eq:target-mean]–[eq:target-sigma]:

Independent re-derivation of the effective shapes from the raw evaluation grid confirms this is not a coefficient-tuning miss but a qualitative failure: the derived mean shift is nearly constant at $\approx +2$ across the full $\widehat{S}=4$–$80$ range (it does not decay at all, let alone as $\widehat{S}^{-0.40}$), and a constant $+2$ mean shift is actually inconsistent with the manuscript’s own validated high-$\widehat{S}$ closure ($\langle Z\rangle = +0.07$ at $\widehat{S}>20$, per the main-text reference-fitter validation table). Likewise the derived effective $A_\sigma(\widehat{S},b) \equiv \widehat{S}\,(1-\sigma_C^\mathrm{ratio})$ is not constant in $\widehat{S}$ at all — it changes sign across the grid — so there is no $1/\widehat{S}$ law to be found in this expansion at any order of local Taylor refinement.

Conclusion. Two structurally different local expansions (unconditional Bartlett, score-conditioned Bartlett) both fail, in different and specific ways. This is stronger evidence than a single failed attempt: it indicates the beyond-Fisher residual is not a local (interior-point Taylor) phenomenon at all. This is exactly consistent with the mechanism finding from the targeted paired-component decomposition (see the main text’s mechanism-decomposition section): the dominant low-$\widehat{S}$ term is centroid relocation / position argmax, i.e. the observed $\Delta C$ is (locally) a supremum over the position search grid, not a value at a single fitted point — and no expansion around one interior point can represent an argmax over a domain.

Passage 2: the corrections are PSF-family specific

What the manuscript says

[...] a cross-PSF transferability test using Gaussian $\sigma=1.59$ pixel, broad Gaussian $\sigma=2.79$ pixel, and King $r_c=2.5$ pixel, $\alpha=1.5$ toy configurations with the same $\mathtt{ecut}=15$ mask shows that the correction coefficients are PSF-family specific: the Gaussian-toy mean law differs from the XMM-psfgen-template production law at high significance in the most precise low-$\widehat{S}$ bins, and neither $bA_\mathrm{psf}$ nor $b\,n_\mathrm{pix}$ collapses the mean coefficients across PSF families. Consequently, the low-signal correction must be calibrated per PSF-template family. Within the XMM psfgen-template family used here, the 17-position off-axis stability demonstrated below is precisely the validated operational domain.

The full picture

Having failed to derive Eqs. [eq:target-mean]–[eq:target-sigma], we asked a weaker question: are they at least universal across PSF shape, once expressed in a suitable dimensionless background variable? The natural candidates, motivated by the position-argmax mechanism, are $b_\mathrm{eff} = b\,A_\mathrm{psf}$ (background per effective PSF footprint, $A_\mathrm{psf} = 1/\sum p_\mathrm{norm}^2$) and $b\,n_\mathrm{pix}$ (background per masked pixel).

We reran the same reference-fitter pipeline (identical fitting code, identical $\widehat{S}$-binning, identical circular $\mathtt{ecut}=15$ mask) for three PSF/mask configurations spanning a $3.4\times$ range in effective PSF area:

Two things stand out. First, the baseline Gaussian control does not reproduce the production XMM-template coefficients $(0.59, -1.02, 0.40, 1.49, 0.39)$ at all: $a_\mu$ is off by $5.2\sigma$, $c_\mu$ by $3.6\sigma$, $p_\mu$ (which came out $\approx 0$ for every configuration, i.e. no power-law decay was measurable in any toy at this precision) by $8\sigma$, and $a_\sigma$ by $2.6\sigma$; only $c_\sigma$ agrees. Since v30.3’s production coefficients were measured with pixelized XMM psfgen templates (King-like wings) rather than an idealized Gaussian, this control mismatch is itself evidence that PSF shape matters, not a bug: the most statistically precise low-$\widehat{S}$ bins reject the production mean law by $10$–$18\,\sigma$ when evaluated against a Gaussian PSF at $b\ge0.08$.

Second, no tested rescaling unifies the three configurations. We fit the same functional forms against raw $b$, $b_\mathrm{eff}=b A_\mathrm{psf}$, and $b\,n_\mathrm{pix}$, and checked whether the three configurations’ coefficients collapse onto one shared value within their bootstrap errors:

Conclusion. The King configuration — the one with the heaviest PSF wings — shows the largest variance deficit ($a_\sigma=1.63$ vs. $0.94$ for the narrow Gaussian), which is exactly the ordering a position-argmax mechanism predicts: heavier wings give the position optimizer more room to find spurious peaks. But the effect size does not collapse onto one universal curve under either dimensionless variable tried. The corrections are therefore treated as PSF-family-specific: valid within the XMM psfgen-template family (supported by the 17-position off-axis independence already in the main text), not claimed as instrument-independent.

Passage 3: the emldetect transfer is an implementation layer

What the manuscript says

Same-seed paired reference-fitter and standalone emldetect experiments show that the standalone $\Delta C$ tracks the reference fitter almost perfectly (correlation 0.996), with a small affine offset and an implementation jitter that grows with off-axis angle; this supports treating the transfer from the reference theory to the standalone pipeline as an empirical implementation layer rather than as a change to the observable-conditioning formalism.

The full picture

Using 21,000 same-seed paired rows (reference fitter vs. standalone emldetect, $b=0.04$, $S_\mathrm{true}\in[8,20]$, PN band-4-like configuration, 14–17 detector positions), the two pipelines’ $\Delta C$ values are related by a simple affine map plus a position-dependent jitter: $$\Delta C_\mathrm{eml} = a\,\Delta C_\mathrm{ref} + c + \varepsilon(\theta), \qquad a = 0.9857,\ \ c = 0.632,$$ with $\operatorname{corr}(\Delta C_\mathrm{ref}, \Delta C_\mathrm{eml}) = 0.996$ and jitter standard deviation growing from $\sigma_\varepsilon \approx 1.28$ on-axis to $\approx 2.07$ at $\theta\approx8.8'$ off-axis (a per-ring fit is statistically indistinguishable from a linear-in-$\theta$ fit, $\sigma_\varepsilon(\theta) \approx 1.32 + 0.046\,\theta$).

Propagating this transfer through the reference-fitter theory, $\mu_\mathrm{eml} = a\,\mu_{3p} + c$ and $\operatorname{Var}_\mathrm{eml} = a^2\operatorname{Var}_{3p} + \sigma_\varepsilon(\theta)^2$, and testing on held-out detector positions (leave-one-position-out), closes almost exactly the width gap noted for standalone emldetect at the detection threshold:

Critically, applying the (already-validated) low-signal mean correction of Eqs. [eq:target-mean] to $\mu_{3p}$ before the affine map leaves the transfer-layer width essentially unchanged and only shifts the mean residual — confirming the beyond-Fisher mean physics and the emldetect implementation layer are cleanly separable, as the manuscript’s framing assumes.

One caveat kept out of the manuscript (it is a next-step item, not a validated result): the affine-residual scatter is not constant in $\widehat{S}$ — it grows from $\approx1.6$–$1.8$ near threshold to $\approx3.0$ at $\widehat{S}\approx35$–$42$ — so the present $\theta$-only jitter model under-corrects well above the $\mathrm{DET\_ML}=6$–$10$ band. This does not affect the operational claim made in the manuscript, which is scoped to the detection threshold.

Summary