Scope of the v30.6 revision

v30.6 is a pure literature and wording revision of v30.5: no numbers, equations, figures, tables, validation domains, or conclusions were changed. All edits arise from a systematic novelty search against the published literature, intended to answer a question that must be self-answered before submission and that reviewers will inevitably ask— which parts of this paper are new, and which are not?

The search yielded two outcomes:

1. Good news (the majority): the distribution-level claims of this paper—conditioning on the fitted amplitude $\widehat{S}$, the three-parameter Fisher-projected variance, the beyond-Fisher low-signal correction law and its mechanism, the $R_\mathrm{eml}$ pipeline residual calibration, and the percentile threshold-crossing sensitivity map in $\Delta C$ space—are not found in the published literature.

2. One necessary correction: the conditional mean formula is not original to this paper. It has a direct precedent in the published literature (Stewart 2009), which v30.5 did not cite; and v30.5’s introductory statement that “prior work failed to treat $\Delta C$ as a sampling distribution” was too strong in light of that paper.

v30.6 therefore does two things: adds four (five-entry) previously omitted citations and narrowed the novelty claim from “we first did X” to “prior work stopped at the mean; we added the width and the threshold-crossing.” This narrowing sacrifices no substantive contribution; on the contrary, it makes the remaining claims irrefutable.

Search methodology and scope

The search targeted each of the paper’s five core claims separately, covering: X-ray survey sensitivity-map and selection-function methodology (XMM/Chandra/eROSITA), Cash-statistic distribution and moments, asymptotic likelihood theory in high-energy physics, and recent 2024–2026 catalog pipeline papers. The closest precedent (Stewart 2009) was read in full-text PDF section by section, not just by abstract.

Finding 1: Stewart (2009) is the direct precedent for the conditional mean

The fact

Maximum-likelihood detection of sources among Poissonian noise, I. M. Stewart, A&A 495, 989–1003 (2009), §4.1 Eq. (5) gives $$\langle U_\mathrm{Cash}\rangle \;\sim\; 1 + 2\sum_i\Big[(B_i+\alpha S_i) \ln\Big(\frac{B_i+\alpha S_i}{B_i}\Big)\Big] - 2\alpha\sum_i S_i .$$ Dropping the constant $1$, this is term-by-term identical to the single-amplitude limit of our Eq. (1) (eq:mu3p). Stewart explicitly writes that the expression “is considered to be useful for the construction of sensitivity maps for Cash source detection”—i.e., the intended use is also the same. The paper also:

• Inverts $\langle U\rangle(\alpha)$ to define detection sensitivity $\alpha_\mathrm{det}$, and explicitly discusses whether to use the mean or the median for this purpose (§2.3.1);

• Validates with $10^5$–$10^6$-member Monte Carlo that, after excluding negative-amplitude fits, the Cash-statistic null distribution is approximately $\chi^2$ at the corresponding degrees of freedom (3 when amplitude and position are free), and on this basis rebutted the critique of Protassov et al. (2002) of likelihood-ratio source detection (§2.4.3–2.4.4, Fig. 3);

• Explicitly states that the emldetect DET_ML column is computed correctly under their tests (§2.4.4, Fig. 4).

Why this does not threaten the paper

Stewart’s formula differs from our $\mu_{3p}$ in three respects, and those three differences are exactly where this paper stands:

Bonus: Stewart’s “+1” is the single-parameter predecessor of our beyond-Fisher shift

Stewart’s additive constant $1$, obtained “by a series of reasonable approximations,” which he notes degrades at low amplitude after discarding negative-amplitude fits—this is qualitatively the same phenomenon as the low-signal additive mean shift measured in this paper. v30.6 records this correspondence explicitly in the “Closed-form global corrections” section and notes the difference: in our three-parameter production setting the shift is not a constant but decays as $\widehat{S}^{-0.40}$ and depends on background. This connects a previously unexplained 2009 empirical fudge factor into our mechanism narrative, which is a net gain for both sides.

Finding 2: Cowan et al. (2011) is the correct methodological lineage

The derivation structure of our §3—Asimov dataset gives the conditional expectation, Fisher information gives the variance—is exactly the framework systematically established in high-energy physics by Cowan, Cranmer, Gross & Vitells (2011, EPJC 71, 1554), building on the results of Wilks and Wald for the asymptotic distribution of $q_0$ under the signal hypothesis. v30.5 used the term “Asimov image” without giving a source; the entry was in fact already in i044_references.bib (key Cowan_2011), just never \cited.

Citing it is advantageous: it positions this paper as the extension of a mature framework to pixelized Poisson imaging, where the genuinely difficult parts—pixelized PSF, fitting mask, fluctuation directions absorbed by free position parameters—are exactly our technical content.

Finding 3: Kaastra (2017) is the nearest neighbor for “Cash-statistic moments”

Kaastra (2017, A&A 605, A51) gives closed-form expressions for the expectation and variance of the C statistic. The key distinction: that is the C-statistic moment of the null model in a spectral goodness-of-fit context, whereas this paper treats the conditional moments of the likelihood-ratio improvement $\Delta C$ in a source detection context. Citing it draws a precise boundary around what has been done with “Cash-statistic moments” and what has not.

Finding 4: position-argmax is homologous with the look-elsewhere effect

The v30.4 companion note already argued that the beyond-Fisher residual is not a local phenomenon, because the observed $\Delta C$ is a supremum over the position search domain and an interior Taylor expansion cannot represent the argmax. This has a mature name in high-energy physics—the look-elsewhere effect (Gross & Vitells 2010, EPJC 70, 525). v30.6 adds this citation, giving the mechanism claim support from established theory rather than relying solely on this project’s empirical decomposition.

Finding 5: confirming what was not previously published

The following claims have no precedent found in the search and constitute the paper’s unique value:

1. Observable conditioning (on $\widehat{S}$ rather than $S_\mathrm{true}$). All published sensitivity-map / selection-function work— aperture photometry (Georgakakis et al. 2008), eSASS apetool, SAS esensmap, eROSITA digital twins (Liu, Brunner, Seppi, et al.), Chandra CSC2—conditions on the true injected flux.

2. Three-parameter Fisher-projected variance (pixel-sum formula including mask).

3. The beyond-Fisher correction law and its constraint evidence from two failed derivations, PSF-family specificity, and position-argmax mechanism.

4. The $R_\mathrm{eml}$ pipeline residual calibration: analytic backbone $+$ sparse-grid empirical residual $+$ LOPO error budget. Existing catalog practice has only the two extremes—pure analytic mean or pure end-to-end Monte Carlo; the middle route has no precedent.

5. Percentile threshold-crossing sensitivity map in $\Delta C$ space, and the bidirectional (catalog interpretation / catalog construction) use of the same forward kernel.

Line-by-line changes in v30.6

All changes are in i044_manuscript_v30.6_zh.tex, six body passages $+$ four new bib entries.

New bib entries (i044_references.bib, entries $37 \rightarrow 41$): Stewart2009, Stewart2006, Kaastra2017, GrossVitells2010. Cowan_2011 and wilks1938large were already in the bibliography but never cited; Cowan_2011 enters the body text from this version.

Compilation verification

v30.6 and v30.5 produce identical error counts under the same xelatex call (both 86), and the error types match line-by-line, with only line-number shifts from inserted text. That is: this revision introduced no new LaTeX errors. bibtex exited normally, with zero undefined references; all five new references entered the .bbl; the PDF was produced normally.

Two pre-existing issues (present in v30.5, not addressed here because they exceed the scope of a pure literature revision):

• §5.7 references appendix table tab:app-v15-dense-detx-dety, but that table’s \label is not in the file (the corresponding dense-grid table appears to have been deleted in some version), producing one undefined-reference warning. The table needs to be restored or the sentence rewritten.

• The math environments contain many \begin{aligned} closed by \end{array} mismatches (21 instances), two equation*/equation mismatches, and \DETML not defined in the preamble. These are legacy Markdown$\rightarrow$LaTeX conversion artifacts; nonstopmode still produces the PDF, but they should be cleaned up before submission.

Summary

One-sentence summary: v30.6 surrendered one priority claim that was never defensible (the analytic mean) and gained a post-search irrefutable positioning—prior X-ray sensitivity-map work stopped at the mean of the distribution; this paper adds the width, the position-dependent pipeline calibration, and the treatment of threshold selection as a distribution-crossing event.