Poisson-count source detection fits a source model above a known background. A Poisson model describes random counts around an expected number. A likelihood ratio compares how well two models explain the same data. In the standard Cash-statistic formulation (Cash 1979), the improvement is $\Delta C = C_\mathrm{null} - C_\mathrm{best}$. Apart from a constant, $C$ equals twice the negative Poisson log-likelihood. Detection pipelines often report a monotone transform of $\Delta C$ as a “detection likelihood.” Here monotone means that the transform preserves the order of $\Delta C$ values. In the XMM-Newton Science Analysis System, emldetect fits one amplitude and two position coordinates (Cruddace et al. 1988; Watson et al. 2009). It reports $\mathrm{DET\_ML}$, an incomplete-gamma transform of $\Delta C$. This statistic is used for catalog entries, detection thresholds, and sensitivity maps.

Major X-ray surveys use sensitivity maps and source-count methods (Cappelluti et al. 2009; Georgakakis et al. 2008; Laird et al. 2009; Luo et al. 2017, e.g.,; Mateos et al. 2008; Puccetti et al. 2009; Wang et al. 2016; Xue et al. 2011). Poisson source detection and likelihood-ratio limits have a long history in high-energy astrophysics (Broos et al. 2010, e.g.,; Gehrels 1986; Kashyap et al. 2010; Kraft et al. 1991; Protassov et al. 2002; Starck et al. 2002). XMM-Newton source catalogs use emldetect (Carrera et al. 2007; Rosen et al. 2016; Traulsen et al. 2019; Watson et al. 2009; Webb et al. 2020).

We use $\mathrm{DET\_ML}$ only as the ordered threshold scale defined by SAS. We do not treat it as a calibrated frequentist false-alarm probability. The usual $\chi^2_3$ null reference assumes regular model parameters away from boundaries. This assumption fails here. Source amplitude lies on a boundary, and source position is undefined under the null (Chernoff 1954; Protassov et al. 2002; Self & Liang 1987). Monte Carlo experiments by Stewart (2009) tested the corresponding Cash-statistic reference. It was useful for representative searches after negative-amplitude solutions were excluded. The low-count Cash statistic can also differ from its high-count limit (Bonamente 2020). We do not try to settle this null-calibration question. We use the conversion between $\mathrm{DET\_ML}$ and $\Delta C$ for one purpose. It translates a chosen pipeline threshold into the statistic modeled below.

The practical threshold range has low signal. For many XMM-Newton applications, sensitivity maps use $\mathrm{DET\_ML}$ of order 6–10. Source counts are steep in this range, and Poisson fluctuations matter. A small threshold shift can therefore change the inferred limiting flux. Eddington bias also matters. Near the detection boundary, upward fluctuations are selected more often. The selected sample then differs from the full parent distribution (Eddington 1913; Wang 2004).

The analytical mean is already known in related forms. Stewart (2009) derived an amplitude-only Cash-statistic expectation for sensitivity calculations. That work extended an earlier matched-filter study (Stewart 2006). SAS esensmap also uses an emldetect-style analytic Cash model in the 3XMM and 4XMM pipelines (Rosen et al. 2016; Traulsen et al. 2019). The New-ANGELS XMM survey evaluated a related expectation at fitted flux instead of injected flux (Huang et al. 2025, appendix). Cash-statistic moments also exist for spectral goodness-of-fit problems (Kaastra 2017). Those moments describe a different quantity from the source-versus-background improvement studied here.

The unresolved quantity is the conditional width of the detection improvement. We find no earlier explicit pixel-sum variance for $\Delta C$ after fitting amplitude and two position coordinates. We also find no fitted-observable threshold calculation that carries this width forward. We therefore separate the problem into analytic location and scale terms plus an empirical low-count correction. The Asimov term is the expected-data likelihood contrast and gives the location. The related Kullback–Leibler quantity measures the expected log-likelihood difference. A Fisher/Wald projection gives the leading variance (Cowan et al. 2011; Wilks 1938). Fisher projection removes fluctuation directions absorbed by fitted parameters. Our result is the pixelized Poisson-imaging version of related Fisher analyses of joint photometry and astrometry (Mendez et al. 2014).

The projection identity is standard. Our new step is its explicit pixel-sum form for a free-position Poisson PSF fit. We also validate it near the detection threshold. The quantity $V_{3p}$ is the predicted three-parameter variance. It uses only the fitted amplitude, background, PSF, and mask. It has no fitted coefficient. The empirical functions $F_k$ and $G_k$ give small low-count corrections. They do not replace the analytic formula.

The resulting model has a common structure but not a universal set of empirical coefficients: $$\Delta C\mid\widehat S,b,\mathcal{P},\mathcal{M},k \simeq \mathcal{N}\!\left[ \mu_{3p}+\sqrt{V_{3p}}F_k(\mu_{3p}), V_{3p}G_k^2(\mu_{3p}) \right]. \label{eq:master-kernel}$$ Section 2 defines all symbols in Equation ([eq:master-kernel]). The functions $F_k$ and $G_k$ must be estimated separately. A correct mean removes gradients within a bin before the width is measured. It does not set the width. We apply the nonlinear incomplete-gamma transform from $\Delta C$ to $\mathrm{DET\_ML}$ only after constructing this distribution.

Section 2 gives the conditioning and selection rules. Section 3 derives the analytic location and Fisher-projected variance. Section 4 validates the reference-fitter kernel and its separate low-count corrections. Section 5 presents a pipeline-specific calibration case study with standalone emldetect. It also tests transfer across configurations. Section 6 gives fitted-amplitude threshold crossings and catalog uses. Sections 7 and 8 give the scope limits and conclusions.

The central quantity is $\Delta C$, the Cash-statistic improvement, conditioned on fitted data:

Here conditioning means holding the listed fitted quantities fixed. The symbol $\widehat{S}$ is the fitted source amplitude. The symbol $b$ is the local background in counts per pixel. The symbol $\mathcal{P}$ is the pixelized PSF, and $\mathcal{M}$ is the fitting mask. The index $k$ records the fitter and instrumental configuration. The terms $\mu_{3p}$ and $V_{3p}$ are the analytic three-parameter location and variance. The functions $F_k$ and $G_k$ correct the low-count location and width.

In simulations, $S_\mathrm{true}$ labels the injected source amplitude. It is the right variable for the separate generation model $p(\widehat S,\Delta C\mid S_\mathrm{true})$. However, catalog sources do not have an observed $S_\mathrm{true}$. It is therefore not the conditioning variable in our fitted-observable kernel. We use $S_\mathrm{true}$ only for generation checks. All $\Delta C$ moment estimates are conditioned on $\widehat S$. Conditioning on $S_\mathrm{true}$ tests behavior at a known injected flux. Conditioning on $\widehat S$ tests the kernel after fitting a source. Only the second case is used for our kernel and fitted-amplitude crossing maps.

For each source or fake-source trial, we evaluate the theory at the observed fitted quantities:

and the residual is

For a sample or bin,

The symbol $\mu_{3p}$ is the predicted mean for a three-parameter fit. The observed $\Delta C$ variance equals $\mathrm{Var}(R)$ only when $\mu_{3p}$ is nearly constant across the bin. Otherwise, mixing fitted amplitudes or positions adds variance outside the conditional theory. We therefore estimate a local conditional mean in the cells used for the width. We subtract this local trend before computing the standard deviation. The mean is needed for this step, but the remaining width is a separate target. Binning by $S_\mathrm{true}$ mixes fitted amplitudes and can increase the apparent position dependence. A cut on observed $\mathrm{DET\_ML}$, $\Delta C$, or the residual has a second problem. It removes part of the distribution whose moments are being measured. We use neither operation below.

The statistic $\mathrm{DET\_ML}$ is a nonlinear, order-preserving tail-probability transform of $\Delta C$:

with $\nu=3$. The incomplete gamma function then has shape parameter $\nu/2=1.5$ for a single-source three-parameter fit. The corresponding inverse is $$\Delta C_{\mathrm{th}} = 2\,Q_x^{-1}(1.5, \exp(-\mathrm{DET\_ML})), \label{eq:detml-inverse}$$ Here $\Delta C_{\mathrm{th}}$ is the threshold in Cash-improvement space. The function $Q$ is the regularized upper incomplete gamma function. It gives the upper-tail probability of a gamma distribution. The notation $Q_x^{-1}$ means inversion in its second argument. Thus $\mathrm{DET\_ML}=6,\,8,\,10$ correspond to $\Delta C_{\mathrm{th}}=14.3,\,18.6,\,22.8$, respectively. A Gaussian residual in $\Delta C$ is therefore not Gaussian after transformation to $\mathrm{DET\_ML}$. Selecting on observed $\mathrm{DET\_ML}$ also selects the noisy quantity being measured. This removes the lower part of the residual distribution and raises its mean near the threshold. We instead convert threshold ranges to $\Delta C$ with the three-parameter convention. We then use predicted conditions such as $\mu_{3p,\rm ref}\in[\Delta C_\mathrm{ML6},\Delta C_\mathrm{ML10}]$. An equivalent choice uses $\widehat{S}$ bins whose typical predicted $\Delta C$ lies in that interval. The conditioning then remains independent of the observed random fluctuation in $\Delta C_\mathrm{eml}$.

The model has one free amplitude and two free position coordinates. The Kullback–Leibler/Asimov construction and the Fisher/Wald projection identity are standard likelihood tools (Cowan et al. 2011; Wilks 1938). The Asimov construction evaluates a statistic on the expected data. The Fisher/Wald projection removes noise directions absorbed by fitted parameters. Our new result is their explicit pixel-sum form for a free-position Poisson PSF fit. We also validate this form. The projected variance, not the analytical mean alone, is the main theoretical contribution.

Consider a fitting region with pixels indexed by $i$. The observed counts are independent Poisson random variables:

For one source on a locally known background, the fitted model is

Here $b_i$ is the expected background in pixel $i$. The symbol $S$ is the source amplitude in the image-model units. The function $p_i(x,y)$ is the pixelized PSF template shifted to position $(x,y)$. We denote this pixelized PSF by $\mathcal{P}$. The PSF is normalized over the full template grid, $\sum_i p_i=1$. The mask value $M_i$ then selects the fitting footprint, denoted by $\mathcal{M}$. Thus $S$ is the total source amplitude in the template convention. It is not the source count inside the fitting mask. In the PN band 4 (2.0--4.5 keV) validation runs, $b_i$ is usually a constant $b$ within the fitting region. The notation also allows a background that varies by pixel.

A binary fitting mask $M_i$ selects the pixels in the fit. The main results use these binary masks. A weighted extension allows fractional weights. When the weights are exactly 0 or 1, it reduces to the binary convention used here.

After dropping constants that do not depend on the model parameters, the Cash statistic is

The null model contains only the background, $\mu_{i,0} = b_{i}$. The likelihood-ratio Cash improvement is

With this sign convention, a better source fit gives positive $\Delta C$. At a fixed fitted model $\mu_i$, the statistic becomes

The observable-conditioned theory evaluates its moments at

Here $\widehat{S}$ is the fitted source amplitude. The coordinates $\widehat{x}$ and $\widehat{y}$ are its fitted image position.

The Asimov image (the expected dataset where observed counts equal the model expectation; Cowan et al. 2011) uses $Y_i=\mu_i$ for this fitted model. Substitution into the Cash improvement gives the leading conditional location:

For a constant background $b_i = b$ and normalized PSF $p_i$, this gives the explicit pixel sum: $$\mu_{3p} = 2 \sum_i M_{i} \bigl[ (b + \widehat{S}\, p_{i}) \log\bigl((b + \widehat{S}\, p_{i})/b\bigr) - \widehat{S}\, p_{i} \bigr] \label{eq:mu3p-pixel-sum}$$

The symbol $\mu_{3p}$ is the predicted location for a three-parameter fit. This mean depends on the full pixel sum over the PSF and mask. A Gaussian width or a few low-order PSF moments cannot determine it. However, those summaries can still be useful checks.

If position is fixed and only amplitude is fitted, Equation ([eq:mu3p]) reduces to the expectation used by Stewart (2009). The notation and treatment of a small empirical offset differ. Our extension adds three features. It conditions on the reported fitted amplitude, includes the fitting mask, and carries both fitted position directions into the variance.

Writing $r_i = \mu_i / b_i$, Eq. [eq:mu3p] becomes $\mu_{3p} = 2\sum_i M_i b_i [r_i \ln r_i - (r_i - 1)]$. This is twice the per-pixel Poisson Kullback–Leibler divergence, summed over the fitting region. It is the Asimov location in a local likelihood-ratio approximation (Cowan et al. 2011). It does not mean that the finite-count, free-position distribution is exactly noncentral $\chi^2_3$. Here noncentral means a chi-square distribution shifted away from its null case. Because $r\ln r-(r-1)$ is never negative, $\mu_{3p}\geq0$.

The mean and variance have different approximation orders. The mean $\mu_{3p}$ is a leading Asimov quantity. It inserts the expected counts $Y_i = \mu_i$ into the Cash improvement. It does not include the Poisson fluctuations absorbed by the fit. This missing behavior includes the finite position search and its maximum. In contrast, the variance $\sigma^2_{3p}$ includes a first-order correction for this absorption. It uses the Fisher projection in Section 3.6. This difference motivates separate empirical location and scale functions at low fitted signal. Position parameters can move toward a local upward fluctuation. This maximum-search behavior is similar in structure to a finite look-elsewhere effect (Gross & Vitells 2010). Here this effect means the increase that can occur when many positions are searched. The paired test in Section 4 measures this mechanism only for the bounded integer-grid reference fitter. Section 5 keeps pipeline-specific effects separate.

The best-fit parameters satisfy the score equations. For any fitted parameter $\theta_a \in \{S, x, y\}$,

Linearizing around the fitted model, write

The first-order score constraint is

There are three such constraints:

These constraints make a three-parameter fit different from an unconstrained Poisson sum. The fit absorbs fluctuations along the amplitude and position directions. We must remove these directions when computing the residual variance.

At a fixed source model, $\Delta C$ is exactly a linear combination of independent Poisson counts. Define $a=2\sum_iM_i(b_i-\mu_i)$, $w_i=2M_i\ln(\mu_i/b_i)$, and $\lambda_i=\mu_i$. Then $$\begin{aligned} \Delta C &= a+\sum_iw_iY_i, \\ E[\Delta C] &= a+\sum_i\lambda_iw_i=\mu_{3p}, \\ \mathrm{Var}(\Delta C) &= \sum_i\lambda_iw_i^2 =4\sum_iM_i\mu_i\ln^2\!\left(\frac{\mu_i}{b_i}\right). \end{aligned} \label{eq:fixed-model-exact-moments}$$ These mean and variance expressions are exact for a fixed model and any fixed pixel weights $M_i$. Binary masks are the main convention used here. For a fixed model, these are the same mean and variance that Kaastra (2017) derived for the C statistic of the true model. The fitted-model case is what Section 4 measures.

We denote the exact fixed-model variance in Equation ([eq:fixed-model-exact-moments]) by $\sigma^2_\mathrm{unc}$. It is the first term of the projected variance $V_{3p}$ derived below. However, it is not the final variance of the fitted likelihood ratio. It ignores that $S$, $x$, and $y$ were chosen to maximize the likelihood. The Fisher projection below subtracts a squared term that cannot be negative. Thus $\sigma^2_\mathrm{unc}$ is an upper bound on the projected variance in the local Gaussian approximation.

Define the Fisher information matrix for the three fitted parameters:

For $\theta = (S,x,y)$, the entries are

The covariance between the $\Delta C$ residual and the score direction $\theta_a$ is encoded by

Explicitly,

Under the Gaussian/Fisher approximation, the score constraints remove the residual projection along the fitted parameters. This is the imaging form of the usual Fisher/Wald projection. It follows from the multivariate Gaussian conditional-variance identity: $$\mathrm{Var}(R\mid U=0)=\mathrm{Var}(R) -\mathrm{Cov}(R,U)\,\mathrm{Var}(U)^{-1}\,\mathrm{Cov}(U,R),$$ where $U$ denotes the vector of linearized score constraints. The resulting variance is

Equation ([eq:var3p]) is the central analytic result. The first term is the variance of the fixed-model linear residual. The second removes the fluctuation absorbed when amplitude and position are fitted again. Both terms use the fitted source model, pixelized PSF, background, and mask. They do not use an injected amplitude.

The projection $g^T I^{-1}g$ can be computed conveniently via the block structure of the Fisher matrix. Writing $$I = \begin{pmatrix} I_{SS} & H \\ H^T & K \end{pmatrix}, \label{eq:block-inverse-I}$$ where $H$ contains the $S$-position cross terms. The matrix $K$ is the $2\times2$ position block for $x$ and $y$. The projection becomes $$\begin{split} g^T I^{-1} g =&\; (g_{S} - H K^{-1} g_\mathrm{pos})^2 / (I_{SS} - H K^{-1} H^T) \\ &+ g_\mathrm{pos}^T K^{-1} g_\mathrm{pos}, \end{split} \label{eq:block-inverse-result}$$ Here $g_\mathrm{pos} = (g_x, g_y)^T$. This block form shows the position projection directly. It is also convenient for a large grid of $\widehat{S}$ and $b$ values. If only $S$ is fitted, the position directions disappear. The formula then reduces to the amplitude-only one-parameter expression.

The variance $V_{3p}$ is a local Gaussian/Fisher approximation around the fitted model. It should work when the likelihood surface is close to a parabola near its maximum. The fitted source must also be strong enough that position fitting does not absorb isolated Poisson noise features. The variance check conditioned on $\widehat{S}$ is presented in Section 4.3 and Figure [fig:deltac-scatter-variance]. That section also gives a global low-$\widehat{S}$ correction for the remaining width deficit. Fisher/Wilks approximations fail at the $S=0$ boundary. They are also incomplete in the low-signal threshold band. We therefore measure the remaining location and width corrections in Section 4. By itself, Equation ([eq:var3p]) is not a validated tail model at $\mathrm{DET\_ML}=6$–10. Section 4.5 shows that the corrected kernel is validated in this threshold band.

SAS emldetect is a closed program. Its internal pixelized PSF, optimizer, and search grid are unavailable for a term-by-term theory test. We therefore built a reference fitter that follows the analytic assumptions. This design separates two sources of error. The reference fitter measures theory-side error at low $\Delta C$. A comparison with emldetect then measures added implementation-side error.

Table [tab:assumption-checklist] compares the assumptions of each fitter with the theory. A “partial” entry means that the fitter has the named feature but does not match every mathematical detail. An interior optimum lies away from all parameter boundaries.

The reference fitter performs a maximum-likelihood search. However, it is not a fully continuous three-parameter optimizer. It tests a $7\times7$ integer position grid. At each position, it fits an amplitude within fixed bounds. Low-signal residuals can therefore contain grid selection and boundary effects. The matched experiment remains the main test of Equation ([eq:var3p]) and the low-count location–scale model. Section 5 measures the added standalone-pipeline difference.

The simulations span 17 detector positions, eight injected source strengths, and six background levels. Each grid cell has 1000 realizations. The design contains 816,000 fits. The location–scale analysis retains 810,987 clean rows. We require $\widehat S>0.5$, $V_{3p}>0.01$, finite moments, and successful fits. These rules remove 5,013 rows, or 0.6% of the full sample. Most removed rows have very low fitted amplitude, where the projected variance becomes degenerate. The $V_{3p}>0.01$ rule removes this unstable normalization.

We generated the pixelized PN band 4 PSF templates with psfgen under XMM-Newton SAS v20.0.0. The matched source fits do not call emldetect. They use an integer-grid position search, bounded scalar amplitude fitting, and a circular fitting mask with radius 15 pixels (60 arcsec). This mask corresponds to the emldetect parameter $\mathtt{ecut}=15$. The reference-fitter validation keeps every detector position. For standalone comparisons, we read the empirical statistic from the catalog $\mathrm{DET\_ML}$ column. We convert it using $\Delta C_\mathrm{eml} = 2\,Q_x^{-1}(1.5,\exp(-\mathrm{DET\_ML}))$ with $\nu=3$.

Each simulated source is fitted with a three-parameter Poisson likelihood. For every row, we evaluate the theory at the fitted source strength:

Here $\mu_{3p}$ is the predicted three-parameter location, and $V_{3p}$ is its Fisher-projected variance. We define the diagnostic residual as

This raw standardized residual is measured in units of the predicted standard deviation. The validation is conditioned on observables. The prediction uses the fitted amplitude $\widehat{S}$, not the injected amplitude $S_\mathrm{true}$.

Figure [fig:deltac-scatter-variance] is the main raw-data check for the matched experiment. It shows three facts required by the model. The pixel-sum location follows the main mean trend. The remaining low-signal shift is smooth. The Fisher-projected scale approaches the observed width at high signal. Rows show three representative detector radii. Colors show five backgrounds. The right column is the main variance check. It measures the standard deviation from $\Delta C-\mu_{3p}$, not from raw $\Delta C$. Thus a mean gradient within a bin does not appear as extra width. Dashed curves use the mean and width corrections in Equations ([eq:mean-corr]) and ([eq:sigma-corr]). We define their normalized $F(\mu_{3p})$ and $G(\mu_{3p})$ functions below.

Table [tab:ref-fitter-validation] gives low- and high-signal cases. At high signal ($\widehat S>20$), the standardized residual has mean $0.068\pm0.002$ and width $0.977\pm0.001$ under realization resampling. The position-resampling errors are $\pm0.006$ and $\pm0.002$, respectively. Thus the Asimov location and projected scale agree with the matched model. At $\widehat S<5$, the mean is $0.922\pm0.002$, in units of the predicted standard deviation ($\pm0.011$ by position). The width is $0.856\pm0.002$ ($\pm0.003$ by position). A low-count model must therefore correct location and scale separately.

Bootstrap resampling measures sampling uncertainty by repeatedly drawing from the available realizations or detector positions. We report both types in the table.

Define the raw normalized residual $$Z_0=\frac{\Delta C-\mu_{3p}}{\sqrt{V_{3p}}}.$$ The calibration estimates its location and scale as functions of $\mu_{3p}$, the observable expected likelihood strength. We use $\mu_{3p}$ because it is the simplest fitted-observable coordinate that captures the low-count trend. This choice is practical, not derived. The function $F$ gives the normalized mean shift. The function $G$ gives the normalized width. Our preferred compact forms are $$\begin{aligned} F(\mu) &= \frac{A_F}{1+(\mu/\mu_F)^{p_F}}, \\ G(\mu) &= 1-\frac{A_G}{1+(\mu/\mu_G)^{p_G}}, \label{eq:fg-kernel} \end{aligned}$$ with $$(A_F,\mu_F,p_F)=(1.530\pm0.042,4.944\pm0.266,0.9711\pm0.0132)$$ and $$(A_G,\mu_G,p_G)=(0.2484\pm0.0077,14.39\pm0.92,1.184\pm0.044).$$ Each function has three fitted coefficients. We fitted both functions to the same 749,863 rows. These fits are therefore in-sample descriptions. The out-of-sample threshold test in the next subsection is the independent check. The displayed coefficient errors are standard errors. Calculations use the full-precision coefficients in the data release.

The fit gives each bin a weight equal to the inverse squared standard error of that bin. We fit the compact functions only on $\mu_{3p}\geq3.0$. This excludes 61,124 rows, or 7.54% of the clean sample. Their $\Delta C$ values lie far below the $\mathrm{DET\_ML}=6$ threshold. Their residual is not Gaussian, as shown in Section 4.5. Including these rows changes $G$ by at most 0.00382 in the $\mathrm{DET\_ML}=6$–10 band and $F$ by at most 0.00450.

The conditional location and width are therefore $$\begin{aligned} \mu_{\Delta C}^\mathrm{corr} &=\mu_{3p}+\sqrt{V_{3p}}\,F(\mu_{3p}), \label{eq:mean-corr}\\ \sigma_{\Delta C}^\mathrm{corr} &=\sqrt{V_{3p}}\,G(\mu_{3p}). \label{eq:sigma-corr} \end{aligned}$$ The weighted root-mean-square error (RMSE) uses 32 $\mu_{3p}$ bins. Each bin is weighted by its number of rows. The RMSE is $0.00946\pm0.00104$ for $F$ and $0.00437\pm0.00065$ for $G$ under realization resampling (position-resampling errors: $\pm0.00092$ and $\pm0.00069$, respectively). Both functions approach the Fisher limits, $F\rightarrow0$ and $G\rightarrow1$, as $\mu_{3p}$ increases. The low-count location shift resembles the empirical additive offset discussed by Stewart (2009). However, our shift is not constant and is not interpreted as the same correction.

We use fixed $\Delta C$ values for $\mathrm{DET\_ML}=6$, 8, and 10. No row is selected by observed $\mathrm{DET\_ML}$, $\Delta C$, or a residual. In the realization-block split, the 1000 realizations in each grid cell are divided by their realization number into five blocks. Four blocks fit the kernel, and the remaining block tests it. Each block is held out in turn. Across the three thresholds, the mean absolute error is $0.01359\pm0.00010$ for the uncorrected Fisher kernel. It falls to $0.00243\pm0.00009$ after the location correction and $0.00199\pm0.00008$ after both corrections. The last value is close to the $0.00196\pm0.00008$ empirical-bin benchmark. Leave-one-position-out and leave-one-background-out tests give the same ordering (Table [tab:kernel-oos]).

The standard error is half the central 68% range of 1000 bootstrap replicates. Resampling with replacement duplicates some rows. The bootstrap mean of an absolute-error metric is therefore slightly above the point estimate. We quote the standard error, not that shifted mean.

The location correction gives the largest improvement. The width term gives a smaller but measured gain. The paired MAE decrease is $0.00044\pm0.00005$ under realization resampling ($\pm0.00007$ by position). Both gains exceed six standard errors. The corrected result nearly reaches the empirical cell benchmark, which uses no fitted curve. This result calibrates threshold-passing probabilities in the matched reference fitter. It does not show that transformed $\mathrm{DET\_ML}$ values are Gaussian. It is also not an injection-completeness measurement at fixed $S_\mathrm{true}$.

The corrected residual is $Z_\mathrm{corr}=(Z_0-F)/G$. Across all 810,987 clean rows, its mean is $+0.00510$, and its standard deviation is 1.002. Its skewness is 0.0158, where skewness measures asymmetry. Excess kurtosis measures tail weight and peak shape relative to a normal distribution. Across the three broad $\mu_{3p}$ bands, skewness ranges from $-0.0455$ to 0.0957. Excess kurtosis ranges from $-0.0980$ to 0.114.

Table [tab:percentile-calibration] gives all six tested percentiles. In the threshold band, the largest departure is 0.684 percentage points. The 2.3th and 10th percentiles differ by 0.408 and 0.108 percentage points. Across the three broad bands and the pooled sample, these two lower-tail departures never exceed 0.619 percentage points.

The broad-band skewness has magnitude below 0.096, and the excess kurtosis has magnitude below 0.114. The strongest non-Gaussian shape occurs at $\mu_{3p}<1$, where the skewness reaches about $-1.5$. This is far below the $\mathrm{DET\_ML}=6$ threshold, where $\mu_{3p}$ is about 14. Therefore, the lower-percentile crossings in Section 6 are supported for the matched PN band 4 kernel. Figure [fig:residual-shape-calibration] summarizes the shape and percentile checks.

Both $F$ and $G$ vary with $\mu_{3p}$. They therefore show a strong pooled association. We tested an affine, or straight-line, function of the fitted mean correction. The test asks whether it can replace the independently fitted $G(\mu_{3p})$. After removing the common $\mu_{3p}$ trend, the Spearman correlations between location and width are $-0.0465$ overall and $+0.0420$ for $5\leq\mu_{3p}\leq40$. This weak ranked association is the main evidence that the two corrections are separate.

Across 17 held-out whole-position groups, the independent and mean-linked models give width RMSE values of 0.0409 and 0.0455 in normalized $G$ units. Their ratio is 1.11, with position-wise ratios from 1.07 to 1.17. A ratio of one would mean equal held-out errors. We chose the limit of 1.10 before inspecting the result. It allows at most a 10% loss from the smaller model. This is a practical tolerance, not a law inferred from the data. The mean must be modeled before estimating the width because it removes gradients. However, the width needs its own correction function. This grouped test reuses the reference sample. It compares models but is not an independent confirmation data set.

A targeted paired rerun saved four likelihood components: oracle, amplitude-only, relocated-position, and full fits. The oracle component uses $S_\mathrm{true}$ only for this simulation diagnostic. It evaluates the fixed model at the injected amplitude and position. The rerun contains 96,000 fits across four detector positions. Among 49,193 clean rows with $\mu_{3p}<10$, amplitude fitting contributes $+0.838\pm0.005$ in $\Delta C$ units. Centroid relocation and position maximization contribute $+2.099\pm0.008$ in $\Delta C$ units. Negative oracle-noise and fitted-conditioning terms cancel much of this gain. The total fitted shift is therefore $+0.692\pm0.004$ in $\Delta C$ units. These are realization-cluster standard errors. The corresponding position-cluster errors are 0.012, 0.047, and 0.017.

The raw positive budget is $0.838+2.099=2.94$ in $\Delta C$ units. This is close to the Wilks expectation of 3 for three fitted parameters (Wilks 1938). The expected split is about 1 for amplitude and 2 for the two position coordinates. This is an interpretation, not a derived result. The nontrivial finite-count part is the $+0.692$ that remains after conditioning on fitted amplitude. No closed form is available for the width deficit. The result supports a finite-position maximum-search explanation in the integer-grid reference fitter. Its structure is similar to look-elsewhere behavior (Gross & Vitells 2010). It does not establish the same component budget for continuous fitting or standalone emldetect.

Within the matched PN band 4 configuration, the compact $\mu_{3p}$ kernel leaves RMS location trends of 0.0073 across positions. The trend is 0.0377 across backgrounds. This stability applies only within the tested configuration. It does not make the coefficients universal. The index $k$ in Equation ([eq:master-kernel]) includes PSF preparation, fitting mask, search range, and optimizer. Section 5 shows that changing these choices can change the sign of $F_k$ and the scale $G_k$.

High-signal agreement alone would not justify threshold use. For the current PN band 4 configuration, $\mathrm{DET\_ML}=6$–10 lies mostly below $\widehat S=20$. The held-out tests in Section 4.4 address this gap for the matched reference fitter. They validate the compact location–scale kernel over the tested threshold range. They do not establish an exact Gaussian tail at any smaller signal. They also do not establish the behavior of another optimizer or PSF convention.

End-to-end fake-source injection remains the correct method for absolute survey completeness. It includes candidate generation, source confusion, background-map construction, and every pipeline cut. Modern simulation-calibrated survey analyses use this approach (Brunner et al. 2022; Liu et al. 2022). The analytic formula has a different purpose. Equations ([eq:mu3p]) and ([eq:var3p]) keep the continuous dependence on background, PSF, and mask. A matched simulation then calibrates two dimensionless functions, $F_k$ and $G_k$. It need not tabulate a dense distribution at every position and flux. Section 5 tests the extra information needed for standalone emldetect.

The matched reference fitter isolates the statistical theory. However, emldetect uses a different optimizer, internal pixelized PSF, search surface, and mask convention. We use the standalone PN band 4 experiment as a calibration case study. The first result is a position-dependent calibration of the mean crossing. It is not an absolute completeness measurement. The second result is a transfer test. It shows that neither this calibration nor the reference-fitter $F$ and $G$ coefficients can be assumed for another camera, band, or search configuration.

The raw theory supplies the Asimov location $$\mu_{3p}(\widehat S,b,\mathrm{psf},\mathrm{mask}) \simeq E[\Delta C\mid\widehat S,b,\mathrm{psf},\mathrm{mask}]$$ The left side is the predicted three-parameter mean. It depends on the fitted source amplitude, local background, pixelized PSF, and fitting mask. If standalone emldetect exactly matched this model, the threshold limit would follow from $$\mu_{3p}(S_{\mathrm{lim}},b,\mathrm{psf},\mathrm{mask}) = \Delta C_{\mathrm{th}}$$ Here the right side is $\Delta C_{\mathrm{th}}$, the Cash-improvement threshold. In practice, the uncalibrated error varies with position. It is not one constant offset. Section 5.2 gives the numerical errors.

Figure [fig:emldetect-scatter-variance] is the main check for the standalone-emldetect chain. It has the same format as Figure [fig:deltac-scatter-variance]. The observable-conditioned structure remains visible. However, the standalone pipeline adds a position-dependent residual relative to the reference-fitter theory. The figure therefore motivates two parts in the final calibration: the analytic equation and an empirical $R_\mathrm{eml}$ correction.

We therefore define the standalone emldetect residual as

Here $\Delta C_\mathrm{eml}$ is the standalone emldetect Cash improvement. We convert it with the single-image $\nu=3$ convention. The quantity $\widehat{S}_\mathrm{eml}$ is the fitted emldetect source amplitude. The calibrated threshold equation is

The sign follows from the residual definition. If $\widehat{R}_\mathrm{eml}>0$, standalone emldetect gives a larger $\Delta C$ than the raw theory at the same fitted strength. The theoretical threshold needed for the sensitivity limit is then lower. If $\widehat{R}_\mathrm{eml}<0$, the required source amplitude is higher.

The calibration uses paired fake-source simulations. We analyze every realization with both the reference fitter and standalone SAS emldetect. After excluding the edge case, the grid has 14 PN band 4 positions. We define the target threshold band in $\Delta C$ space with Equation ([eq:detml-inverse]). Rows are selected by the reference-theory prediction: $$\mu_{3p,\mathrm{ref}} \in [\Delta C_{\mathrm{ML6}}, \Delta C_{\mathrm{ML10}}]$$ The threshold sample is not selected by observed $\mathrm{DET\_ML}_\mathrm{eml}$ or observed $\Delta C_\mathrm{eml}$. Both values contain the random fluctuation and implementation residual being calibrated. Instead, selection uses the predicted reference $\Delta C$ from the observable-conditioned theory. We build the baseline calibration grid at $b = 0.04\,\mathrm{counts\,pixel^{-1}}$. The uncertainty budget below includes transfer to $b=0.02$ and $b=0.08\,\mathrm{counts\,pixel^{-1}}$.

We estimate the residual $\widehat{R}_\mathrm{eml}$ at each detector position from the threshold-band sample. The interpolation coordinate combines off-axis radius with a four-quadrant direction label. This label is used only for interpolation. It is not a physical detector azimuth or a general PSF measure. We tested physical azimuth but did not use it. It gave poorer coverage on the sparse interpolation grid.

The paired standalone simulation used the same 17-position layout as the reference grid. Positions P6 and P10 were not generated because their input files were invalid at that time. We also exclude P3. It lies at an off-axis angle of 6.3 arcmin and an azimuth of 180 degrees. At the detector edge, its fitting-mask coverage is only about 0.55. This incomplete area breaks the constant-background assumption of the residual calibration. The remaining grid has 14 positions.

Validation leaves each position out in turn. The calibration uses only the remaining positions. We compare its predicted $S_\mathrm{lim}$ with the held-out position’s internal reference limit $S_\mathrm{ref}$.

The internal reference $S_\mathrm{ref}$ stays in the fitted-amplitude coordinate. At each held-out position, we measure $R_\mathrm{eml}$ from its threshold-band rows. We then solve $\mu_{3p}(S_\mathrm{ref})=\Delta C_\mathrm{th}-R_\mathrm{eml}$ at the same position and background. The prediction solves the same equation with the residual interpolated from the other positions. Thus both quantities use the fitted-amplitude coordinate. The held-out residual does not enter the interpolation model.

This test measures interpolation of the pipeline’s internal threshold map. It is not an absolute test against observations or injection-recovery fractions. Such a test requires dense fake-source injections at representative positions. We report the error as

Table [tab:unified-validation] summarizes this conditional interpolation test. At $\mathrm{DET\_ML}=6$, the preferred 2D interpolation has a 4.1% median error. Its maximum is 11.0% within the region covered by the other calibration points. This region contains 10 of 14 held-out positions. Nearest-position assignment covers all 14 positions. Its maximum error is 17.2%. Background transfer from $b=0.04\,\mathrm{counts\,pixel^{-1}}$ to $b=0.02$–$0.08\,\mathrm{counts\,pixel^{-1}}$ adds at most 5.7% error in $S_\mathrm{lim}$ at $\mathrm{DET\_ML}=6$. Both errors decrease at higher thresholds.