XMM-Newton 点源检测似然比的零参数条件宽度：模拟基准中的机制与局限A Parameter-Free Conditional Width for the XMM-Newton Point-Source Detection Likelihood Ratio: Mechanism and Limits in a Simulation Benchmark

〈作者名单——由 PI 决定〉〈Author list — PI to decide〉

摘要Abstract

XMM-Newton 的点源检测把每个候选折成检测统计量 DET_ML，而目录的误差与完备性都依赖给定测量计数 \(\hat{S}\) 与背景 \(b\) 时似然比 \(\Delta C\) 的条件宽度。我们用可控模拟基准问：能否不用任何拟合参数写出它。高计数端，零参数 Fisher 宽度 \(V_{3p}\) 正确（位置固定，std z 0.98–1.00 于 DET_ML \(\gtrsim\) 20）。目录区间（DET_ML 6–20）真实 PSF 下系统性偏窄（0.840–0.984）：两个几乎相消大项之一的约 1 % 错配被相消放大成 8–16 % 方差高估，随 PSF 集中度与计数上升。同一机制也适用于 emldetect（L1）：错配 +1.19 % [+1.03, +1.35]，解释箱内缺口的 0.89 [0.84, 0.93]。在 emldetect 输出上，条件规律的四个候选被一支先在"恰好为真"数据上验证的评分器全部拒收（通过数 10/72、48/72、42/72、35/72 对完美规律约 62–63/72）。2026-10-21 的复核表明该评分器混用估计对象（宽度是位置内量、均值与覆盖率合并、零假设自适应），故"宽度可推广"只能读作同一位置内。分开两个目标后：未见位置总体最好候选通过 44/72（能段 4）与 36/72（能段 1），逐位置每条候选都接近 0/72，位置效应 \(\tau_{mean}\) 0.14 z / 0.18 z 且重尾。中心项的主导成分是最好候选自身的样本内表征误差（同位置内失败 24 \(\to\) 3、未见位置 21 \(\to\) 8；R119 限定：3 与 8 是 12 个种子里好的一端（同位置 3–15、未见 7–16）），不是位置泛化缺口。XMM-Newton point-source detection folds each candidate into the statistic DET_ML, whose conditional width given the measured counts \(\hat{S}\) and background \(b\) underlies catalogue errors and completeness; a controlled benchmark asks whether it is writable without fitted parameters. At high counts the Fisher width \(V_{3p}\) is correct (position fixed, std z 0.98–1.00 at DET_ML \(\gtrsim\) 20); in the catalogue interval (DET_ML 6–20) it is narrow under the real PSF (0.840–0.984) — a about-1 % mismatch in one of two nearly cancelling terms amplified into an 8–16 % variance over-statement. The same mechanism holds for emldetect (L1): mismatch +1.19 % [+1.03, +1.35], explaining 0.89 [0.84, 0.93] of the within-cell deficit. On emldetect output, four candidate forms of the conditional law are all rejected by a scorer validated on exactly-true data (passes 10/72, 48/72, 42/72, 35/72 against about 62–63/72 for a perfect law). A 2026-10-21 audit shows it mixed estimands (within-position width, pooled mean and coverage, an adaptive null), so "the width generalises" reads only as within one position. On the population of unseen positions the best passes 44/72 (band 4) and 36/72 (band 1); per position every candidate is near 0/72, with a heavy-tailed position effect \(\tau_{mean}\) 0.14 z / 0.18 z. The centre term's dominant component is an in-sample representation error (within position 24 \(\to\) 3 failures, unseen positions 21 \(\to\) 8; R119 qualifies: 3 and 8 are the favourable end of 12 seeds, within-position 3–15 and unseen 7–16), not a position-generalisation gap.

关键词：X 射线天文学；天文数据分析；似然比检验；XMM-Newton；模拟与基准Keywords: X-ray astronomy; Astronomy data analysis; Likelihood ratio test; XMM-Newton; Simulation and benchmarking

1. 引言1. Introduction

X 射线巡天把“这个源是不是真的”折成一个数：检测似然 DET_ML。它来自似然比检验——把“有源”模型相对“只有背景”模型的似然提升取对数。XMM-Newton 的 emldetect 用 Cash 统计量做这件事，输出目录里每个源的 DET_ML、计数与误差。^[s1] 目录用户拿到的完备性阈值（常用的 DET_ML = 6）与误差棒，最终都依赖一个更细的量：给定测量计数 \(\hat{S}\) 与背景 \(b\) 时，\(\Delta C\) 这个统计量围绕其条件均值散开多宽。宽度偏窄，误差就偏小、完备性就偏高；宽度偏宽，反之。所以条件宽度是目录可信度的底层量。An X-ray survey folds “is this source real” into one number: the detection likelihood DET_ML. It comes from a likelihood-ratio test — the logarithm of the likelihood gain of a “there is a source” model over a “background only” model. XMM-Newton's emldetect does this with the Cash statistic and outputs each source's DET_ML, counts and errors in the catalogue.^[s1] The completeness threshold a catalogue user relies on (the commonly used DET_ML = 6) and the error bars both rest, in the end, on a finer quantity: how widely the statistic \(\Delta C\) scatters around its conditional mean, given the measured counts \(\hat{S}\) and the background \(b\). If the width is narrow the errors are small and the completeness high; if it is wide, the reverse. The conditional width is therefore the quantity underneath catalogue reliability.

理论上有一条现成的路。对嵌套模型的似然比检验，Wilks 定理给出大样本极限：两倍的检验统计量渐近服从卡方分布，自由度等于新增参数的个数。对单幅图像、三个自由参数（振幅与二维位置）的情形，这给出 \(\nu = 3\) 的卡方，于是 \(\Delta C\) 的条件方差原则上可以由 Fisher 信息完全写出——零参数，不需要任何拟合。问题是它准不准：Wilks 定理是渐近的，而目录恰恰关心弱源——只有几十个光子、检测统计量在阈值附近的地方。那里似然面远离二次，渐近论证最不可靠。^[s2]There is a ready-made theoretical route. For a likelihood-ratio test of nested models, Wilks' theorem gives the large-sample limit: twice the test statistic is asymptotically chi-squared with as many degrees of freedom as added parameters. For a single image with three free parameters (amplitude and a two-dimensional position) this gives a \(\nu = 3\) chi-squared, so the conditional variance of \(\Delta C\) can in principle be written entirely from the Fisher information — parameter-free, no fitting required. The question is whether it is accurate: Wilks' theorem is asymptotic, and catalogues care precisely about faint sources — a few tens of photons, with the detection statistic near the threshold. There the likelihood surface is far from quadratic and the asymptotic argument is least reliable.^[s2]

已有的经验校准路线（用真实观测的重复指向或注入源）给出宝贵但混合的结果：它们同时改变仪器、背景、源群与流水线，所以“宽度差多少”有答案，“为什么差”没有。我们采取另一条路：可控模拟基准。同一套物理构造（真实 PSF 图、逐像素背景、抖动）生成图像，再分别用我们自己的拟合器（L0）与 emldetect（L1）去测，逐层比较。这样每个差异都能追到流水线里的具体环节。^[s3]Existing empirical-calibration routes (repeat pointings of real fields, or injected sources) give valuable but confounded results: they change instrument, background, source population and pipeline all at once, so “how much does the width differ” has an answer but “why” does not. We take a different route: a controlled simulation benchmark. The same physical construction (real PSF images, per-pixel background, jitter) generates the images, and our own fitter (L0) and emldetect (L1) then measure them, layer by layer. Every difference can then be traced to a specific step of the pipeline.^[s3]

本文做三件事。第 2 节给出模拟与两条流水线；第 3 节按计数分层给出条件宽度的检验结果——高计数锚处零参数公式通过、目录区间系统性偏窄、机制归因于 penalty 错配的相消放大、以及 emldetect 流水线不适用该机制；第 4 节讨论局限与实用建议。我们要回答的核心问题只有一个：零参数的条件宽度在哪个区间可信，在哪个区间必须校准，为什么。This paper does three things. Section 2 presents the simulations and the two pipelines; Section 3 gives the conditional-width tests layer by layer — the parameter-free formula passes at the high-count anchor, is systematically narrow in the catalogue interval, the mechanism is a penalty mismatch amplified by cancellation, and the mechanism does not apply to the emldetect pipeline; Section 4 discusses the limits and the practical recommendations. The core question is a single one: in which interval the parameter-free conditional width can be trusted, in which it must be calibrated, and why.

2. 模拟与数据2. Simulations and Data

本节给出全部输入，以便复现。所有图像由同一套构造生成：真实 PSF 栅格、逐像素背景图、曝光图与抖动；两组流水线在同一批图像上独立测量。This section gives all the inputs so that the work can be reproduced. Every image is generated by the same construction: a real PSF raster, a per-pixel background map, an exposure map and jitter; the two pipelines measure the same batch of images independently.

2.1 源群与参数区间2.1 Source population and parameter range

源的流量按观测的源计数分布抽样，\(dN/dS \propto S^{-2.5}\)，流量支撑取 [5.58, 51.20]（对应 L0 生产数据的 \(S_{true}\) 范围）。背景 \(b\) 与流量 \(S\) 在一个二维格子上联合采样。真实 PSF 来自 M31 的真实星表图像（ecut 15 px 掩膜），而不是解析高斯。^[s4]Source fluxes are drawn from the observed source-count distribution, \(dN/dS \propto S^{-2.5}\), with the flux support [5.58, 51.20] (the \(S_{true}\) range of the L0 production data). The background \(b\) and the flux \(S\) are sampled jointly on a two-dimensional grid. The real PSF comes from a real M31 catalogue image (an ecut 15 px mask), not from an analytic Gaussian.^[s4]

2.2 两条流水线2.2 The two pipelines

L0 是我们自己的拟合器：全局网格 + Nelder–Mead 搜索、位置自由（检测似然的自由度 \(\nu = 3\)，即一幅图像加振幅与二维位置）、用真实 PSF 模板族在拟合位置打分。L1 是 SAS 的 emldetect：用替代核在名义位置打分。^[s1] 两者的振幅估计量也不同：\(S_{true}\) = 2.5 时 \(E[\hat{S}]/E[S_{eml}]\) = 1.38，20 以上为 0.92。L0 is our own fitter: a global grid plus a Nelder–Mead search, position free (the detection likelihood has \(\nu = 3\) degrees of freedom — one image plus amplitude and a two-dimensional position), scored with the real PSF template family at the fitted offset. L1 is SAS's emldetect: scored with a surrogate kernel at the nominal position.^[s1] The two also use different amplitude estimands: \(E[\hat{S}]/E[S_{eml}]\) = 1.38 at \(S_{true}\) = 2.5 and 0.92 at 20 and above.

2.3 统计量与条件化2.3 The statistic and the conditioning

对每一行模拟，我们计算 \(z = (\Delta C - \mu(\hat{S}))/\sqrt{V(\hat{S})}\)，其中 \(\mu\) 与 \(V\) 都是给定可观测 \(\hat{S}\) 的条件量与条件方差。全程按 \(\hat{S}\) 分箱——不按模拟标签 \(S_{true}\) 分箱，因为后者不是目录用户能拿到的东西。^[s5] 检验的判据是每箱的 \(z\) 标准差与均值（带标准误）。For each simulated row we compute \(z = (\Delta C - \mu(\hat{S}))/\sqrt{V(\hat{S})}\), where \(\mu\) and \(V\) are the conditional mean and variance given the observable \(\hat{S}\). Throughout we bin by \(\hat{S}\) — not by the simulation label \(S_{true}\), because the latter is not something a catalogue user has.^[s5] The test statistic is the per-bin standard deviation and mean of \(z\), with standard errors.

2.4 复算与门控2.4 Replays and gating

每条记录性结论都由一次独立复算支撑：用任务自己的脚本重跑，输出与报告逐位相同。在共享集群上的作业在命令内部先检查守卫状态（READY）再启动。本页引用的每个数字都经过这一步。^[s6]Every record-grade conclusion is backed by an independent replay: the job's own script is re-run and its output matches the report exactly. Jobs on the shared cluster check the guard status (READY) inside the same command before starting. Every number quoted on this page has gone through that step.^[s6]

3. 结果3. Results

本节按证据递进给出三条结果：公式在高计数锚处是对的；在目录区间系统性偏窄，且偏窄的量级随 PSF 集中度与计数上升；偏窄的机制是相消放大，而它对 emldetect 流水线不适用。每条结果先给图或表，再给结论。This section gives three results in order of evidence: the formula is correct at the high-count anchor; it is systematically narrow in the catalogue interval, with the deficit growing with the PSF concentration and the counts; the mechanism is cancellation amplification, and it does not apply to the emldetect pipeline. Each result presents its figure or table first, then the conclusion.

3.1 高计数锚：零参数公式通过3.1 The high-count anchor: the parameter-free formula passes

固定位置、只让振幅自由时，Fisher 宽度 \(V_{3p}\) 在高计数端是正确的。合并标准差比（std z）在 DET_ML \(\gtrsim\) 20 且低背景时为 0.98–1.00；在 z_off 打分下，67 个位置的合并 std z 于 DET_ML \(\geq\) 50 处 mean \(\approx\) 0、std 0.989–1.003（所有位置组、两种先验）。^[s7] 这给出公式的锚区间：那里零参数宽度不需要任何校准。With the position fixed and only the amplitude free, the Fisher width \(V_{3p}\) is correct at high counts. The pooled standard-deviation ratio (std z) is 0.98–1.00 at DET_ML \(\gtrsim\) 20 and low background; under the z_off scoring the pooled std z over 67 positions is mean \(\approx\) 0 and std 0.989–1.003 at DET_ML \(\geq\) 50 (every position group, both priors).^[s7] This is the formula's anchor interval: there the parameter-free width needs no calibration at all.

3.2 目录区间：真实 PSF 下系统性偏窄3.2 The catalogue interval: systematically narrow under the real PSF

在 L0 生产数据（67 位置，离轴 0–13′）上，合并 std z 于 DET_ML \(\approx\) 4.6 / 7.9 / 11.1 / 22.8 / 36.8 为 0.840 / 0.894 / 0.909 / 0.959 / 0.984，直到 DET_ML \(\approx\) 36–60 才回到 1。^[s8] 偏窄不依赖源计数先验（\(\leq\) 0.6 SE），并且没有任何可观测的点修正能闭合它：按实测 MLE 偏差校准的振幅修正仍留 std z 0.952–0.972（DET_ML 6.8–18.6），带 Fisher 展布的 \(E[V_{3p} \mid \hat{S}]\) 更差（0.84–0.96）。^[s9]On the L0 production data (67 positions, off-axis 0–13 arcmin) the pooled std z is 0.840 / 0.894 / 0.909 / 0.959 / 0.984 at DET_ML \(\approx\) 4.6 / 7.9 / 11.1 / 22.8 / 36.8, returning to 1 only at DET_ML \(\approx\) 36–60.^[s8] The narrowing is independent of the source-count prior (\(\leq\) 0.6 SE), and no observable point correction closes it: an amplitude correction calibrated to the measured maximum-likelihood bias still leaves std z 0.952–0.972 (DET_ML 6.8–18.6), and \(E[V_{3p} \mid \hat{S}]\) with a Fisher spread is worse (0.84–0.96).^[s9]

3.3 机制：penalty 错配被相消放大3.3 The mechanism: a penalty mismatch amplified by cancellation

把 \(V_{3p}\) 展开成两项：\(V_{3p} = \mathrm{term1} - 4\,\mathrm{penalty}\)。这两项几乎相消：term1 是残差方差的 15–31 倍，方差只来自两项之差。Fisher penalty 与有效 penalty 只差 0.1–1.4 %，而这个约 1 % 的错配被相消放大成 8–16 % 的方差高估。一个算例：在 DET_ML \(\approx\) 3.5 的箱里，term1 = 63.6、残差方差 = 1.90、penalty 错配 \(\approx\) 1.4 %，于是 \(V_{3p}(\hat{S})/\mathrm{Var}(R)\) = 1.44——1 % 的一级误差变成约 44 % 的方差误差。^[s10] 错配随 PSF 的中心集中度上升（同二阶矩的高斯只有 \(\approx\) 0.05 %，真实 PSF 是 +1.0–1.8 %），也随计数上升（固定 \(\hat{S}\) 把背景减半，错配抬升 +0.80 pp 于 \(\hat{S}\) 2.77 与 4.09）。^[s11] 没有解析修正能闭合它：Bartlett 的 +2\(\cdot\)rank 项符号反了；beyond-Fisher 的位置曲率项过度修正 1.3–7.4 倍、在最低 \(\hat{S}\) 处给出负方差、在最高分箱处变号。^[s12]Expand \(V_{3p}\) into two terms: \(V_{3p} = \mathrm{term1} - 4\,\mathrm{penalty}\). The two nearly cancel: term1 is 15–31 times the residual variance, and the variance is only the difference of the two. The Fisher penalty matches the effective penalty to 0.1–1.4 %, and this mismatch of about 1 % is amplified by the cancellation into an 8–16 % variance over-statement. A worked example: in the DET_ML \(\approx\) 3.5 bin, term1 = 63.6, the residual variance is 1.90, the penalty mismatch is \(\approx\) 1.4 %, and \(V_{3p}(\hat{S})/\mathrm{Var}(R)\) = 1.44 — a 1 % first-order error becomes about a 44 % variance error.^[s10] The mismatch grows with the PSF's central concentration (a Gaussian of equal second moment has only \(\approx\) 0.05 %, the real PSF +1.0–1.8 %) and with the counts (halving the background at fixed \(\hat{S}\) raises the mismatch by +0.80 pp at \(\hat{S}\) 2.77 and 4.09).^[s11] No analytic correction closes it: the Bartlett +2\(\cdot\)rank term has the wrong sign, and the beyond-Fisher position-curvature term over-corrects by 1.3–7.4 times, gives a negative variance at the lowest \(\hat{S}\) and flips sign at the top bin.^[s12]

3.4 emldetect 流水线：同一机制同样成立3.4 The emldetect pipeline: the same mechanism holds as well

在 L1（emldetect）上，同一 penalty 错配同样成立：用合并估计量（而不是逐格比值的平均），L1 的错配在 DET_ML 6–10 是 +1.19 % [+1.03, +1.35]，解释了箱内宽度缺口的 0.89 [0.84, 0.93]——预测 std z 0.919 对实测 0.918，按行合并 0.940。^[s13] 所以 3.3 的机制同时决定两条流水线的宽度；剩下的 0.11 与按行合并宽度 0.940 [0.922, 0.958]（比预测宽 0.021）归位置/b 混合，错配随计数下降（+0.63 % 于 10–20，100 以上约 0）。早先的 −0.14 % [−0.91, +0.45] 是逐格比值的平均，已被推翻（07 R083，被 R086 反转）。两条流水线的宽度差（匹配比较下 L1 − L0 = +0.24 / +0.19 / +0.12 / +0.08 / +0.07 于 DET_ML \(\approx\) 2–7）仍不能直接归因于拟合器：两边连打分与振幅估计量都不同。^[s14] 决定性的配对实验（让 emldetect 跑在同一批 L0 图像上，只让拟合器不同）已完成（0b，已复算）：高计数锚两臂一致（L0 −0.008/0.978，emldetect −0.012/0.971），DET_ML ≲ 7 处 emldetect 中心比 L0 低 0.16–0.53 z、宽 0.05–0.14，所以 L0 的低计数中心项超额不是 emldetect 的性质。^[s16]On L1 (emldetect) the same penalty mismatch holds as well: with pooled estimators (not a mean of per-cell ratios) the L1 mismatch at DET_ML 6–10 is +1.19 % [+1.03, +1.35], explaining 0.89 [0.84, 0.93] of the within-cell width deficit — predicted std z 0.919 against the observed 0.918, row-pooled 0.940.^[s13] So the mechanism of Section 3.3 carries both pipelines' widths; the remaining 0.11 and the row-pooled width 0.940 [0.922, 0.958] (0.021 wider than predicted) belong to the position/b mixing, and the mismatch falls with counts (+0.63 % at 10–20, about 0 above 100). The earlier −0.14 % [−0.91, +0.45] was a mean of per-cell ratios and is reversed (07 R083, reversed by R086). The width difference between the two pipelines (in the matched comparison L1 − L0 = +0.24 / +0.19 / +0.12 / +0.08 / +0.07 at DET_ML \(\approx\) 2–7) still cannot be attributed to the fitter directly: the two sides differ in scoring and amplitude estimand as well.^[s14] The decisive paired experiment (emldetect on the same L0 images, so that only the fitter differs) is completed (0b, replayed): at the high-count anchor the two arms agree (L0 −0.008/0.978, emldetect −0.012/0.971) and at DET_ML ≲ 7 emldetect's centre is 0.16–0.53 z lower and its width 0.05–0.14 larger than L0, so the L0 low-count centre excess is not an emldetect property.^[s16]

3.5 低计数中心项：L0 流水线自己的性质，成因开放3.5 The low-count centre term: a property of the L0 pipeline, cause open

低计数端还出现中心项偏高。用一个与 L0 逐项匹配的对照（同样的 PSF 栅格、掩膜、背景律、抖动、位置）重放，超额仍然存在，说明它衡量的是 L0 流水线自己的性质、不是 emldetect 的物理；同图配对的 emldetect 复核了这一归属（0b，已复算）：同一批图像上 emldetect 与 L0 高计数一致，DET_ML ≲ 7 处 emldetect 的中心反而比 L0 低 0.16–0.53 z、宽 0.05–0.14（3 个试点位置，最低 Ŝ 箱 N_eff 6–55），所以这个超额不是 emldetect 的性质；但它的具体成因开放——把同一批对照图换用 FFT 模板族与 v2t1 重新拟合（配对，10,000 行），中心只移动 −0.003（精确 plug-in）或 +0.037 \(\pm\) 0.034（核 \(\delta\)），只补上 0–27 % 的差距，所以早先的 72/28 归因（模板族/拟合器对 plug-in 均值）已撤回。按可观测 \(\hat{S}\) 分箱的差距为 +0.389（\(\hat{S}\) < 4）、+0.18/+0.15/+0.15/+0.15/+0.15/+0.14（\(\hat{S}\) 4–11）、+0.058 \(\pm\) 0.007（\(\hat{S}\) \(\geq\) 11），在高计数锚处 −0.006 \(\pm\) 0.020。^[s15]At the low-count end the centre term is also biased high. Replaying against a control matched item by item to L0 (the same PSF raster, mask, background law, jitter and position), the excess survives — so it measures a property of the L0 pipeline itself, not emldetect physics; the paired emldetect run confirms the attribution (0b, replayed): on the same images emldetect and L0 agree at high counts and at DET_ML ≲ 7 emldetect's centre is 0.16–0.53 z lower and its width 0.05–0.14 larger than L0 (3 pilot positions, lowest-Ŝ bins N_eff 6–55), so the excess is not an emldetect property; but its specific cause is open: refitting the same control images with the FFT template family and v2t1 (paired, 10,000 rows) shifts the centre by only −0.003 (the exact plug-in) or +0.037 \(\pm\) 0.034 (the kernel \(\delta\)), closing 0–27 % of the gap, so the earlier 72/28 attribution (template family/fitter against plug-in mean) is withdrawn. The gaps in observable \(\hat{S}\) bins are +0.389 (\(\hat{S}\) < 4), +0.18/+0.15/+0.15/+0.15/+0.15/+0.14 (\(\hat{S}\) 4–11) and +0.058 \(\pm\) 0.007 (\(\hat{S}\) \(\geq\) 11), with −0.006 \(\pm\) 0.020 at the high-count anchor.^[s15]

3.6 条件规律的候选形式：宽度一半在同一个位置内可推广，中心一半不能3.6 Candidate forms of the conditional law: the width half generalises within one position, the centre half does not

上面的机制回答了"为什么"，下一步是把它变成目录可用的条件规律 \(p(\mathrm{DET\_ML} \mid \hat{S}, b, \mathrm{index})\)（\(\mathrm{index}\) 是源计数斜率，作为先验变量进入规律）。我们在 emldetect 输出（M2 相机、第 4 能段、67 个视场位置、约 10\(^{6}\) 行）上检验了四个候选形式，每个都在一半位置上拟合、在另一半上打分（72 个留出格）。打分前先验证评分器本身：在"规律恰好为真"的合成数据上，其 p 值分布均匀（K-S p 0.59，250 个重复）。结果是负的：四个候选全部被拒收——零参数基线通过 10/72 格，288 参数的制表修正通过 48/72，二次 index 形式 42/72，生成式混合 35/72；一条完美规律在同样噪声下平均也只通过约 62–63/72，所以 48/72 虽是四者最好（p = 0.0015），仍远低于零假设。^[s17]这些通过数自此作为历史诊断保留——2026-10-21 的复核（下段）表明该评分器混用了估计对象，其读数不能作为"规律在任意视场位置上成立"的证据。The mechanism above answers "why"; the next step is to turn it into a catalogue-usable conditional law \(p(\mathrm{DET\_ML} \mid \hat{S}, b, \mathrm{index})\) (with \(\mathrm{index}\), the source-count slope, entering the law as a prior variable). We tested four candidate forms on emldetect output (the M2 camera, band 4, 67 field positions, about 10\(^{6}\) rows), each fitted on half the positions and scored on the other half (72 held-out cells). Before scoring, the scorer itself was validated: on synthetic data where the law is exactly true, its p-values are uniform (K-S p 0.59 over 250 replicas). The result is negative: every candidate is rejected — the parameter-free baseline passes 10/72 cells, a 288-parameter tabulated correction 48/72, a quadratic-index form 42/72 and a generative mixture 35/72; a perfect law passes only about 62–63/72 under the same noise, so 48/72, though the best of the four (p = 0.0015), is far below the null.^[s17] Those pass counts are kept as historical diagnostics from here on — the 2026-10-21 audit (next paragraph) shows the scorer mixed estimands, so its readings cannot serve as evidence that a law holds at any field position.

把最好的候选拆开看，失败集中在中心而不是宽度：宽度格 68/72 通过，而 24 个失败全部在均值上，且全部为负、集中在中计数、随 index 增长；高计数锚 12/12 的通过完全取决于一个位置（位置 14）落在哪一半。位置不是缺失的变量：把平滑的离轴或 PSF 项加进中心，留出失败反而从 24 个升到 58–72 个；唯一能改善的形式是一个逐格中心截距（18 个参数），它把失败降到 11、负和从 −2.28 降到 −0.80，而位置斜率不加分。^[s17]2026-10-21 的专家复核把上述读法重新界定（R114，已采纳）：该评分器混用三个估计对象——宽度是每个位置各自减去自身加权均值后的标准差（一个位置内量），均值与覆盖率却把各位置的行合并计算，而零假设的位置散布还从候选自己的留出残差估计。因此"宽度可以推广"只能读作在同一个位置内成立。一次独立的数值复核（R117，已复算）把这一点做到了精确：位置内二阶矩对任何常数位置偏移恒等不变（差异 1e-15）——一条位置内校准精确、但带 0.30 z 逐位置偏移的规律仍通过宽度 70/72，而 68 % 覆盖率的失败从 4 个升到 29 个。于是新的评分器（SCORER_CONTRACT v2，R118，已复算）把两个目标分开：未见位置的总体上，D / H2 / MIX3d / C 通过 44 / 50 / 32 / 12 格（共 72，M2 能段 4），能段 1 上 D / H2 为 36 / 43，四个候选全部在聚合统计量 Q 上被拒（p \(\leq\) 0.009）；逐位置上每条候选都大幅失败（能段 4 为 1 / 0 / 1 / 0，能段 1 为 0 / 1），冻结的位置效应 \(\tau_{mean}\) 为 0.14 z（能段 4）与 0.18 z（能段 1），且重尾（df \(\approx\) 3）。中心项也有了新读数（R116，已复算）：主导成分是最好候选自身的样本内表征误差——在同一个位置内用新的独立重复打分，失败从 24 个降到 3 个；在未见位置上从 21 个降到 8 个（用一个 72 参数的逐格诊断修正）。因此"中心差距是位置泛化缺口"这句话不再成立。专家提出的一条先验混合恒等式作为代数成立（闭合到 1e-9），但其协方差项在失败格中比观测误差大 2–8 倍，且已被该候选自己拟合的 index 斜率抵消（一次独立的数值复核），所以它不是那个未解决的残差。宽度读数同时被限定（R115，已复算）：前述 0.976 [0.956, 0.993] 的留出宽度只是一次偏低的分半，在 40 个按半径分层的分半上平均为 0.998，而 6 参数修正只通过 25/72 个宽度格。所以已解决的只是宽度在同一个位置内的那一半（机制统一），仍开放的是中心与它对单个拟合位置的依赖。评分器的判定在多个替代零假设下全部成立，但其验证带限定：位置聚落的尺度参数由同一估计量恢复（循环论证），且零假设在格间独立抽取位置效应，而数据的格间相关为 +0.44——通过数零假设的标准差偏小约 30 %，最好候选的 p 值随之变动约一个量级（无判定翻转）。^[s17]Taken apart, the best candidate's failures concentrate on the centre, not the width: its width cells pass 68/72, while all 24 failures are on the mean — all negative, concentrated at mid count and growing with index; and the 12/12 pass of the high-count anchor hinges entirely on which half one position (position 14) falls into. Position is not the missing variable: adding a smooth off-axis or PSF term to the centre raises the held-out failures from 24 to 58–72 instead; the only form that improves matters is a per-cell centre intercept (18 parameters), which takes the failures to 11 and the negative sum from −2.28 to −0.80, with the position slope adding nothing.^[s17] An expert review on 2026-10-21 re-scoped that reading (R114, accepted): the scorer mixed three estimands — the width is the standard deviation after each position subtracts its own weighted mean (a within-position quantity), the mean and coverage pool rows across positions, and the null's position scatter is estimated from the candidate's own held residuals. "The width generalises" can therefore only be read as holding within one position. An independent numerical audit (R117, replayed) made this exact: the within-position second moment is identically invariant (to 1e-15) to any constant per-position offset — a law with exact within-position calibration but a 0.30 z per-position offset still passes width 70/72, while the 68 % coverage failures rise from 4 to 29. The new scorer (SCORER_CONTRACT v2, R118, replayed) therefore separates two targets: on the population of unseen positions, D / H2 / MIX3d / C pass 44 / 50 / 32 / 12 of 72 cells (M2 band 4) and band-1 D / H2 36 / 43, with all four candidates rejected on the aggregate statistic Q (p \(\leq\) 0.009); per position every candidate fails by a wide margin (band 4: 1 / 0 / 1 / 0; band 1: 0 / 1), with the frozen position effect \(\tau_{mean}\) 0.14 z (band 4) and 0.18 z (band 1), heavy-tailed (df \(\approx\) 3). The centre term also has a new reading (R116, replayed): the dominant component is an in-sample representation error of the best candidate itself — scored on new independent realisations at the same positions, the failures drop from 24 to 3; at unseen positions from 21 to 8 (with a 72-parameter per-cell diagnostic correction). The sentence "the centre gap is a position-generalisation gap" no longer holds. The expert's prior-mixing identity holds as algebra (closing to 1e-9), but its covariance term is 2–8\(\times\) the observed error in the failing cells and is already cancelled by the candidate's own fitted index slope (an independent numerical audit), so it is not the unresolved residual. The width reading is qualified at the same time (R115, replayed): the 0.976 [0.956, 0.993] held-out width quoted above was one low split, averaging 0.998 over 40 radius-stratified splits, and the 6-parameter correction passes only 25/72 width cells. So what is solved is only the width half within one position (the mechanism is unified), and what remains open is the centre and its dependence on a single fitted position. The scorer's verdicts stand under every alternative null we built, but its validation carries qualifications: the position-cluster scale is recovered by the same estimator (a circularity), and the null draws position effects independently across cells while the data show a cross-cell correlation of +0.44 — the pass-count null's standard deviation is about 30 % too small, and the best candidate's p-value moves by about an order of magnitude with it (no verdict flips).^[s17]

进度判据本身也被重新校准（2026-10-21 16:53 至 2026-10-22 11:35 EDT）。“每个条目都在容差 \(\varepsilon\) 内”的判据被证明噪声受限：一条完美规律在能段 4 上也只能让 21–27 % 的位置满足它，因为每个条目自身的抽样误差约 0.12 z，而位置效应本身是 0.14 z。判据因此改为每个位置一个合并偏移，PI 定下容差：比较规律用 \(\varepsilon\) 0.05、跟踪进度用 \(\varepsilon\) 0.10（\(\varepsilon_w\) 0.10），\(\varepsilon\) 0.20 因饱和撤掉；打分手冻结为 score_pooled_v1.py（18 个格、位置真值校准的 95 % 区间）。旧读数“规律就压在位置盲地板上”被地板改正推翻：原地板只注入了 0.634 倍的声明 \(\tau^2\)，改正后最好候选在 \(\varepsilon\) 0.10 处的 0.545 高于地板 0.469。位置效应的来源也收窄到像素级几何且两条能段共享（r = +0.936）：不是源表（每个位置注入同一份 54 源栅格），导出的相位项被驳回，而“把注入点移到像素中心会把偏移放大 2 倍”的读法被撤回——那是打分器的伪迹（同一格掩膜下两臂比值 1.16）。第一个拟合前位置项（拟合盒背景水平加一个相位指示量，\(\leq\) 4 个参数）把最好候选在 \(\varepsilon\) 0.10 处从 0.545 抬到 0.727（配对差 +0.182 [+0.030, +0.333]，全管线零假设下 p 0.003–0.004），但它尚未在未见位置上检验。宽度侧，判据本身也被校准：一条逐行精确的方差律通过“8 箱全在 1 \(\pm\) 0.02 内”的概率只有 66.8 %，所以“2 箱失败”不等于“规律错”；PI 批准新判据“8 箱允许 1 箱失败”（完美规律通过约 95 %）。在这条判据下，第一条可观测箱级宽度规律 \(\bar{V}(\hat{S}, b) = E[V2 \mid \hat{S}, b]\) 于 \(\beta\) \(-1.5\) 与 \(-1.8\) 通过（1/8），但带宽在测试半上选定（40 个随机分半下 P(\(\leq\) 1 失败) 0.79），且同一构造迁到 emldetect 行后在全统计量下仍失败 5/5 箱。^[s19]The progress criterion itself was also re-calibrated (2026-10-21 16:53 through 2026-10-22 11:35 EDT). The “every entry within the tolerance \(\varepsilon\)” rule was shown to be noise-limited: even a perfect law satisfies it at only 21–27 % of positions on band 4, because each entry's own sampling error is about 0.12 z while the position effect itself is 0.14 z. The criterion therefore became one pooled offset per position, and the PI fixed the tolerances: \(\varepsilon\) 0.05 to compare laws, \(\varepsilon\) 0.10 as the progress tracker (\(\varepsilon_w\) 0.10), with \(\varepsilon\) 0.20 dropped as saturated; the scorer is frozen as score_pooled_v1.py (18 cells, position-true-calibrated 95 % intervals). The old reading “the laws sit at the position-blind floor” is overturned by the floor correction: the old floor injected only 0.634\(\times\) of the declared \(\tau^2\), and after the correction the best candidate's 0.545 at \(\varepsilon\) 0.10 sits above the floor of 0.469. The position effect's origin is also narrowed to pixel-level geometry shared by both bands (r = +0.936): not the source list (every position injects the same 54-source grid), the derived phase term is rejected, and “moving the injection points to pixel centres amplifies the offsets 2\(\times\)” is withdrawn — a scorer artefact (ratio 1.16 under the same cell mask). A first pre-fit position term (the fit-box background level plus a phase indicator, \(\leq\) 4 parameters) lifts the best candidate at \(\varepsilon\) 0.10 from 0.545 to 0.727 (paired difference +0.182 [+0.030, +0.333], p 0.003–0.004 under the full-pipeline null), but it has not yet been tested on unseen positions. On the width side the criterion itself was also calibrated: a row-exact variance law passes “all 8 bins inside 1 \(\pm\) 0.02” only 66.8 % of the time, so “2 failing bins” does not mean “the law is wrong”; the PI approved the new rule “at most 1 failing bin of 8” (a perfect law passes about 95 %). Under it the first observable bin-level width law \(\bar{V}(\hat{S}, b) = E[V2 \mid \hat{S}, b]\) passes at \(\beta\) \(-1.5\) and \(-1.8\) (1 of 8), but the bandwidth was chosen on the test half (P(\(\leq\) 1 failing bin) 0.79 over 40 random splits), and the same construction still fails 5/5 bins on emldetect rows at full statistics.^[s19]

2026-10-23 更新：能段/仪器普适性在冻结合并口径下首次打分。把冻结的能段 4 定律零新参数地拿到留出的能段与仪器块上：M2 能段 1 的中心通过（D+SS 于 \(\varepsilon\) 0.10 为 24/33 = 0.7273，配对差 0.000 [−0.121, +0.121]），M1 能段 1 是一个已分辨的中心失败（8/21 = 0.381，配对差 −0.333 [−0.619, −0.048]），M1 其余能段在现有设计下不能分辨（MDS 约 0.30）。第一次打分（A0-SCORE）曾把 M2 能段 1 记为 FAIL（0.606，低于线 0.627）；红队复算发现那是打分手的口径错误——位置项被按行除了进去，而冻结口径把它从合并偏移中减掉；按冻结口径重打后通过（见来源对照表 [s20]）。宽度侧，细锚方差律在 emldetect 行上达到可观测神谕地板（2/9 失败箱，神谕自身也失败 2/9），经验合成的簇自助 95 % CI 为 [1, 7] %；"8 箱允许 1 箱失败"的判据按实际箱数（9–10）重标定为 \(k^*\) 2–3，零假设的选择留给 PI。目录侧，冻结目录律在 \(\mu_0 \lesssim 250\) 通过；250 以上 \((f\cdot\mu)^2\) 恢复 MOS 覆盖率但其 PSF 标定依据已撤回，逐相机 \(c\cdot\mu^p\) 在不相交 \(\mu_0\) 区域之间不迁移。位置项在 27 个新位置上的检验为 AMBIGUOUS（配对差 +0.148 [−0.037, +0.333]），且 8/27 个新位置受源拥挤污染，因此不能读作位置项的检验；一个 139 个隔离位置（NN \(\geq\) 45 px）的设计已就绪，等 PI 批准。Update 2026-10-23: band/instrument universality scored for the first time under the frozen pooled convention. Taking the frozen band-4 law to held-out band and instrument blocks with zero new parameters: M2 band 1's centre passes (D+SS at \(\varepsilon\) 0.10 is 24/33 = 0.7273, paired difference 0.000 [−0.121, +0.121]), M1 band 1 is a resolved centre failure (8/21 = 0.381, paired difference −0.333 [−0.619, −0.048]), and the remaining M1 bands cannot be resolved under the present design (MDS about 0.30). The first scoring run (A0-SCORE) had recorded M2 band 1 as a FAIL (0.606, below the bar 0.627); a red-team replay found that to be a scorer convention error — the position term had been divided in row by row, whereas the frozen convention subtracts it from the pooled offset — and under the frozen convention the block passes (see the source map, [s20]). On the width side, the fine-anchor variance law reaches the observable-oracle floor on emldetect rows (2/9 failing bins, the oracle itself failing 2/9), with the empirical composition's cluster-bootstrap 95 % CI at [1, 7] %; the "at most 1 failing bin of 8" rule, re-calibrated to the actual bin count (9–10), gives \(k^*\) 2–3, with the choice of null left to the PI. On the catalogue side, the frozen catalogue law passes for \(\mu_0 \lesssim 250\); above 250 the \((f\cdot\mu)^2\) term restores MOS coverage but its PSF-calibration justification is withdrawn, and the per-camera \(c\cdot\mu^p\) form does not transfer across disjoint \(\mu_0\) regions. The position term's test at 27 fresh positions is AMBIGUOUS (paired difference +0.148 [−0.037, +0.333]), and 8 of the 27 fresh positions are source-crowded, so it cannot be read as a test of the position term; a design with 139 isolated positions (NN \(\geq\) 45 px) is ready and awaits the PI.^[s20]

2026-10-24 更新：Tier 1 第一臂已打分，背景通道给出一条未决的新依赖。留出能段块打分（T1-e）在唯一携带判定的块 M2 能段 1 上给出三条分开的判定：中心 D+SS 于 \(\varepsilon\) 0.10 是 24/33 = 0.727，配对差区间 [−0.152, +0.061]，按预注册中心规则确认；宽度 4 个失败箱对 \(k^*\) 2，红队复核后旗标记 AMBIGUOUS（超出真实且单侧，约 2 %，但箱边移 −1/3 个箱即翻成 PASS）；锚 8/12；参照块 M2 能段 4 为 25/33、宽度 1/2、锚 12/12。执行方的"整体 REFUTED"不是预注册组合，因此不记总体判定。背景通道（T1-c）：冻结定律在 DR14 估计背景下不普适，配对窗口偏移 +0.174 z（527 次重复，簇区间 [0.165, 0.260]，不开窗口 0.151，故非选择伪影），机制 \(z_B - z_A \approx 48.2 \times \Delta b_{\rm PSF}\)；候选参数 C2（\(b_{\rm eff} = \gamma \cdot \mathrm{bgm}_B\)，\(\gamma = 0.921 + 0.0041(\mathrm{off} - 3')\)）把留出偏移压到 −0.052 z（区间 [−0.073, −0.032]），暂定于一个观测、9 个留出位置。代价锚实测 0.97 CPU-s/行（区间 0.86–1.3），旧 0.4 作废；T1-a 按 PI 选项 A 只跑 M2（N = 115，149–192 CPU-h）并已发射；PN b1/b4 不重跑。星表侧：DET_ML 有序残差确认为标注同义反复（实/零之比约 0.93）；断幂律过 5/6 主格（M2 低 \(\mu_0\) 按簇自助稳健失败、其后补丁为事后选择）；eta 振幅修好合并宽度但逐格形状失败，下一步是尾部项。Update 2026-10-24: Tier 1's first arm is scored, and the background channel adds a new, unresolved dependency. The held-out band-block scoring (T1-e) gives three separate verdicts on the only verdict-carrying block, M2 band 1: the centre D+SS at \(\varepsilon\) 0.10 is 24/33 = 0.727, paired-difference interval [−0.152, +0.061], confirmed under the pre-registered centre rule; the width has 4 failing bins against \(k^*\) 2, re-flagged AMBIGUOUS after the red-team re-check (the excess is real and one-sided, about 2 %, but a −1/3-bin edge shift flips the verdict to PASS); the anchor is 8/12; the reference block M2 band 4 gives 25/33, width 1/2 and anchor 12/12. The executor's "overall REFUTED" is not a pre-registered combination, so no overall verdict is recorded. The background channel (T1-c): the frozen law is not universal under the DR14-estimated background, with a paired window offset of +0.174 z (527 repeats, cluster interval [0.165, 0.260]; 0.151 with no window, so not a selection artefact) and the mechanism \(z_B - z_A \approx 48.2 \times \Delta b_{\rm PSF}\); the candidate parameter C2 (\(b_{\rm eff} = \gamma \cdot \mathrm{bgm}_B\), \(\gamma = 0.921 + 0.0041(\mathrm{off} - 3')\)) brings the held offset to −0.052 z (interval [−0.073, −0.032]), provisional on one observation and 9 held positions. The cost anchor is measured at 0.97 CPU-s/row (band 0.86–1.3), superseding the broken 0.4; T1-a runs M2 only under the PI's option A (N = 115, 149–192 CPU-h) and has been launched; PN b1/b4 are not re-run. On the catalogue side: the DET_ML-ordered residual is confirmed a labelling tautology (real/null about 0.93); the broken power law passes 5/6 primary cells (M2's low \(\mu_0\) a robust cluster-bootstrap fail, its later patch a post-hoc choice); the eta amplitude restores the pooled width while the per-cell shape fails, so the next candidate is a tail term.^[s21]

4. 讨论4. Discussion

回到核心问题：零参数的条件宽度在哪个区间可信？答案分三层。高计数锚（DET_ML \(\gtrsim\) 20，固定位置；\(\geq\) 50，位置自由）——公式可信，不需要校准。目录区间（DET_ML 6–20）——真实 PSF 下公式系统性偏窄 8–16 %，偏窄的机制已定位（penalty 错配被相消放大），但没有解析修正，实用路线是经验校准。emldetect 流水线——同一机制同样成立：合并估计量下其错配 +1.19 % [+1.03, +1.35]，解释箱内宽度缺口的 0.89 [0.84, 0.93]（预测 std z 0.919 对实测 0.918），所以宽度侧的失因问题已解决。规律阶段把这一点做成了判决（R103），随后被专家复核重新界定（R114，已采纳）：在 emldetect 输出上，条件规律的四个候选形式被一支先在"恰好为真"数据上验证过的评分器全部拒收（通过数 10/72、48/72、42/72、35/72，对零假设均值约 62–63/72；最好的 48/72 也只有 p = 0.0015）。把最好的候选拆开看，宽度这一半在同一个位置内可以推广，中心这一半不能（宽度格 68/72 通过；24 个均值失败全部为负、集中在中计数、随 index 增长；高计数锚取决于一个位置的分半；平滑的位置项反而把留出失败从 24 个推到 58–72 个）。但"宽度可以推广"这句话必须带"在同一个位置内"：该评分器的宽度是每个位置各自减去自身加权均值后的标准差，而一次精确复核（R117）证明这个量对任何常数位置偏移恒等不变（1e-15）——一条带 0.30 z 逐位置偏移的规律仍通过宽度 70/72。新的评分器（R118）因此分开两个目标：未见位置的总体上 D / H2 / MIX3d / C 通过 44 / 50 / 32 / 12 格（共 72，能段 4）、能段 1 的 D / H2 为 36 / 43，全部被 Q 拒（p \(\leq\) 0.009）；R124 限定：这是一次不稳定的抽样——14 个位置分半上 D 22–46、H2 26–52（能段 4），能段 1 的 D 14–53、H2 30–52；H2 优于 D 在固定分半上不显著（McNemar p 0.18 / 0.25），所以不能作"H2 最好"的排名，任何规律排名之前先要 scorer v3；逐位置上每条候选都大幅失败（能段 4 为 1 / 0 / 1 / 0，能段 1 为 0 / 1），冻结位置效应 \(\tau_{mean}\) 0.14 z（能段 4）/ 0.18 z（能段 1），重尾 df \(\approx\) 3。中心项的主导成分也改判为样本内表征误差（R116：同一位置内失败 24 \(\to\) 3、未见位置 21 \(\to\) 8；R119 限定：3 与 8 是好的一端，12 个重抽样种子里同位置为 3–15、未见位置为 7–16，须按种子区间引用），不是位置泛化缺口。一句话：公式的适用区间是锚区间；目录区间要校准；两条流水线的宽度由同一个机制决定；而单图 emldetect 规律的宽度一半在同一个位置内已有机制与可用校准（40 个分半的留出宽度平均 0.998），中心一半仍开放，位置不是缺失的变量。PI 于 2026-10-21 定下进度指标：不是每个位置，而是大多数位置逐位置准确，位置域越大越好（如离轴 14′ 内全部位置），允许 MASKFRAC 之类的质量截断；当天 11:42 EDT 修订为位置上没有通过/不通过阈值——进度指标是已解决位置的比例（与面积），可以从 0 开始，任何增长都算进展，90 % 默认值撤回。进度曲线的第一个点（R120）已经在：能段 4 的 D / H2 各只有 4/33 个位置的条目通过比例 \(\geq\) 0.90（均值 0.643 / 0.651），能段 1 为 1/33，离轴三分位 0.767 / 0.605 / 0.542，失败位置散布而非视场外缘效应。宽度机制本身也在同一天被推进（R121）：亏缺被归因为条件方差 \(\mathrm{Var}[\Delta C \mid \hat{\theta}] < V_{3p}(\hat{\theta})\)，新的 \(O(n^{-1})\) 闭式因子在 \(b = 0.01\) 精确、在 \(b = 0.04/0.16\) 欠修，预注册判据 FAIL（8 个箱里 3 个超出 2\(\sigma\)）。2026-10-22 的后续（R125–R152）把判据与位置项都推进了一步：判据改为每个位置一个合并偏移（\(\varepsilon\) 0.05 比较规律 / 0.10 跟踪进度），打分手冻结；地板改正后最好候选在 \(\varepsilon\) 0.10 处高于地板；位置效应收窄到像素级几何且两能段共享；第一个拟合前位置项（拟合盒背景水平）把最好候选从 0.545 抬到 0.727（p 0.003–0.004，尚未在新位置检验）；宽度判据校准为“8 箱允许 1 箱失败”后，第一条可观测箱级规律 V̄ 通过（1/8），但迁到 emldetect 全统计量下仍失败 5/5 箱。Back to the core question: in which interval can the parameter-free conditional width be trusted? The answer has three layers. The high-count anchor (DET_ML \(\gtrsim\) 20 with the position fixed; \(\geq\) 50 with the position free) — the formula is trustworthy and needs no calibration. The catalogue interval (DET_ML 6–20) — under the real PSF the formula is systematically narrow by 8–16 %; the mechanism is located (a penalty mismatch amplified by cancellation) but no analytic correction exists, so the practical route is empirical calibration. The emldetect pipeline — the same mechanism holds as well: with pooled estimators its mismatch is +1.19 % [+1.03, +1.35], explaining 0.89 [0.84, 0.93] of the within-cell width deficit (predicted std z 0.919 against the observed 0.918), so the width side's open-cause question is resolved. The law stage turns this into a verdict (R103), and an expert review then re-scoped it (R114, accepted): on emldetect output, four candidate forms of the conditional law are all rejected by a scorer first validated on exactly-true data (passes 10/72, 48/72, 42/72, 35/72 against a null mean of about 62–63/72; the best 48/72 has only p = 0.0015). Taken apart, the best candidate's width half generalises within one position while its centre half does not (the width cells pass 68/72; the 24 mean failures are all negative, concentrated at mid count and grow with index; the high-count anchor hinges on one position's split; a smooth position term pushes the held-out failures from 24 to 58–72 instead). But the sentence "the width generalises" must carry the words "within one position": the scorer's width is the standard deviation after each position subtracts its own weighted mean, and an exact audit (R117) proves that quantity is identically invariant (1e-15) to any constant per-position offset — a law carrying a 0.30 z per-position offset still passes width 70/72. The new scorer (R118) therefore separates two targets: on the population of unseen positions D / H2 / MIX3d / C pass 44 / 50 / 32 / 12 of 72 cells (band 4) and band-1 D / H2 36 / 43, all rejected on Q (p \(\leq\) 0.009); R124 qualifies: these are one draw of an unstable statistic — over 14 position splits D 22–46 and H2 26–52 (band 4), band-1 D 14–53 and H2 30–52; H2 above D is not significant on the fixed split (McNemar p 0.18 / 0.25), so no "H2 is best" ranking can be made, and scorer v3 is needed before any law ranking; per position every candidate fails by a wide margin (band 4: 1 / 0 / 1 / 0; band 1: 0 / 1), with the frozen position effect \(\tau_{mean}\) 0.14 z (band 4) / 0.18 z (band 1), heavy-tailed with df \(\approx\) 3. The centre term's dominant component is also reclassified as an in-sample representation error (R116: within position the failures go 24 \(\to\) 3, at unseen positions 21 \(\to\) 8; R119 qualifies: 3 and 8 are the favourable end — across 12 resampling seeds within-position runs 3–15 and unseen 7–16, so quote them as seed ranges), not a position-generalisation gap. In one sentence: the formula's applicable interval is the anchor interval; the catalogue interval must be calibrated; the two pipelines' widths are carried by one mechanism; and the single-image emldetect law's width half has a mechanism and a usable calibration within one position (a held-out width averaging 0.998 over 40 splits), while its centre half remains open, with position not the missing variable. The PI set the progress metric on 2026-10-21: not every position, but most positions accurate per position, over the largest possible position domain (e.g. all positions within 14′ off-axis), with quality cuts such as MASKFRAC allowed; at 11:42 EDT the same day it was amended to no pass/fail threshold on positions — the progress metric is the solved fraction (and solved area) of positions, which may start at 0, where any increase is progress, and the 90 % default is withdrawn. The first point of the progress curve (R120) is already in: on band 4 only 4/33 positions have an entry pass fraction \(\geq\) 0.90 for D and for H2 (means 0.643 / 0.651), 1/33 on band 1, off-axis terciles 0.767 / 0.605 / 0.542, with the failing positions scattered rather than at the outer field of view. The width mechanism itself also advanced that day (R121): the deficit is re-attributed to the conditional variance \(\mathrm{Var}[\Delta C \mid \hat{\theta}] < V_{3p}(\hat{\theta})\), and a new \(O(n^{-1})\) closed-form factor is exact at \(b = 0.01\) and under-corrects at \(b = 0.04/0.16\), so the pre-registered verdict is FAIL (3 of 8 bins beyond 2\(\sigma\)). The 2026-10-22 follow-up (R125–R152) advanced both the criterion and the position term: the criterion became one pooled offset per position (\(\varepsilon\) 0.05 to compare laws / 0.10 as the tracker) with the scorer frozen; after the floor correction the best candidate sits above the floor at \(\varepsilon\) 0.10; the position effect narrowed to pixel-level geometry shared by both bands; a first pre-fit position term (the fit-box background level) lifts the best candidate from 0.545 to 0.727 (p 0.003–0.004, not yet tested on fresh positions); and after the width rule was calibrated to “at most 1 failing bin of 8”, the first observable bin-level law V̄ passes (1 of 8) but still fails 5/5 bins on emldetect rows at full statistics.

局限。其一，所有结论来自模拟基准，尚未在真实观测的重复指向上复核；模拟的 PSF、背景律与抖动都取自单一场（M31），外推到其他场需要重新验证。其二，emldetect 一侧的宽度缺口虽已被机制解释（0.89 [0.84, 0.93]），但按计数修正的 \(V_{3p}\) 尚未做出来并通过检验，本文不能给出其误差的修正方案。其三，低计数中心项超额的成因开放：72/28 的归因已被配对重拟合撤回（只补 0–27 %），位置间的因子 2 差异提示它还依赖位置。其四，配对实验（同图同打分）已完成（3 个试点位置），其结论已写入 3.4 与 3.5：L0 超额不是 emldetect 的性质，但 67 位置的推广尚未做。Limitations. First, all conclusions come from a simulation benchmark and have not been cross-checked on repeat pointings of a real field; the simulated PSF, background law and jitter are taken from a single field (M31), and extrapolating to other fields needs re-validation. Second, although the emldetect width deficit is explained by the mechanism (0.89 [0.84, 0.93]), the count-corrected \(V_{3p}\) has not been built and tested yet, so this paper cannot offer a correction for its errors. Third, the low-count centre excess's cause is open: the 72/28 attribution is withdrawn by the paired refit (closing only 0–27 %), and the factor-2 spread across positions suggests it also depends on position. Fourth, the paired experiment (same images, same scoring) is completed (3 pilot positions) and its outcome is written into Sections 3.4–3.5: the L0 excess is not an emldetect property, but the 67-position extension has not been done.

对目录用户的实用建议。在 DET_ML \(\geq\) 50 的区间，直接用零参数公式给出的误差；在 DET_ML 6–20，把宽度按实测偏差经验校准（例如按 \(\hat{S}\) 分箱的 std z 表），并报告校准的来源与区间；跨流水线比较时，不要把一条流水线的校准搬到另一条上，先做同图同打分的配对比较。Practical recommendations for catalogue users. At DET_ML \(\geq\) 50, use the errors given by the parameter-free formula directly; at DET_ML 6–20, calibrate the width empirically against the measured deficit (for example with a per-\(\hat{S}\)-bin std z table), reporting the calibration's source and interval; when comparing across pipelines, do not carry one pipeline's calibration to the other — run a paired comparison on the same images with the same scoring first.

5. 总结5. Summary

我们用一套可控的模拟基准检验了 XMM-Newton 点源检测似然比的零参数条件宽度，得出：We tested the parameter-free conditional width of the XMM-Newton point-source detection likelihood ratio with a controlled simulation benchmark, and found:

1. 高计数锚处公式通过：固定位置时 std z 0.98–1.00（DET_ML \(\gtrsim\) 20）；位置自由的 z_off 打分下 std 0.989–1.003（DET_ML \(\geq\) 50，67 位置、两种先验）。^[s7]The formula passes at the high-count anchor: std z 0.98–1.00 at DET_ML \(\gtrsim\) 20 with the position fixed; std 0.989–1.003 at DET_ML \(\geq\) 50 under the position-free z_off scoring (67 positions, both priors).^[s7]
2. 目录区间系统性偏窄：合并 std z 0.840–0.984（DET_ML \(\approx\) 4.6–36.8），与先验无关，且没有任何可观测的点修正能闭合。^[s8][s9]The catalogue interval is systematically narrow: pooled std z 0.840–0.984 (DET_ML \(\approx\) 4.6–36.8), independent of the prior, and no observable point correction closes it.^[s8][s9]
3. 机制是相消放大：\(V_{3p}\) 的两大项几乎相消（term1 是残差方差的 15–31 倍），约 1 % 的 penalty 错配被放大成 8–16 % 的方差高估，且错配随 PSF 集中度与计数上升；无解析修正。^{[s10][s11][s12]}The mechanism is cancellation amplification: the two large terms of \(V_{3p}\) nearly cancel (term1 is 15–31 times the residual variance), a penalty mismatch of about 1 % is amplified into an 8–16 % variance over-statement that grows with the PSF concentration and the counts; no analytic correction exists.^{[s10][s11][s12]}
4. 同一机制同样适用于 emldetect：合并估计量下其错配 +1.19 % [+1.03, +1.35]，解释箱内宽度缺口的 0.89 [0.84, 0.93]（预测 std z 0.919 对实测 0.918）；同图配对实验（0b）已完成，确认 L0 低计数超额不是 emldetect 的性质。^[s13][s14]The same mechanism applies to emldetect as well: with pooled estimators its mismatch is +1.19 % [+1.03, +1.35], explaining 0.89 [0.84, 0.93] of the within-cell width deficit (predicted std z 0.919 against the observed 0.918); the paired same-images experiment (0b) is completed and confirms the L0 low-count excess is not an emldetect property.^[s13][s14]
5. 低计数中心项偏高是 L0 流水线自己的性质、成因开放（72/28 归因经配对重拟合撤回，只补 0–27 %；\(\hat{S}\) 分箱 +0.389 / +0.14–0.18 / +0.058 \(\pm\) 0.007；同图配对的 emldetect 无此超额，0b）。^[s15]The low-count centre excess is a property of the L0 pipeline itself with its cause open (the 72/28 attribution is withdrawn by the paired refit, closing only 0–27 %; \(\hat{S}\)-bin gaps +0.389 / +0.14–0.18 / +0.058 \(\pm\) 0.007; the paired emldetect run shows no such excess, 0b).^[s15]
6. 条件规律的四个候选形式全部被拒收，且失败集中在中心：留出格通过数 10/72、48/72、42/72、35/72（零假设均值约 62–63/72，评分器先在"恰好为真"数据上验证）；宽度格 68/72 通过，24 个均值失败全部为负、集中在中计数、随 index 增长，位置不是缺失的变量。2026-10-21 的复核把上述读法重新界定：该评分器混用三个估计对象——宽度是位置内量，均值与覆盖率跨位置合并，零假设还跟着候选自适应；因此"宽度可以推广"只能读作在同一个位置内成立（精确复核证明宽度对逐位置偏移恒等不变，1e-15：一条带 0.30 z 逐位置偏移的规律仍通过宽度 70/72）。新评分器分开两个目标：未见位置的总体上 D / H2 / MIX3d / C 通过 44 / 50 / 32 / 12 格（共 72，能段 4）、能段 1 的 D / H2 为 36 / 43，全部被 Q 拒（p \(\leq\) 0.009）；R124 限定：这是一次不稳定的抽样——14 个位置分半上 D 22–46、H2 26–52（能段 4），能段 1 的 D 14–53、H2 30–52；H2 优于 D 在固定分半上不显著（McNemar p 0.18 / 0.25），所以不能作"H2 最好"的排名，任何规律排名之前先要 scorer v3；逐位置上每条候选都大幅失败（能段 4 为 1 / 0 / 1 / 0，能段 1 为 0 / 1），冻结位置效应 \(\tau_{mean}\) 0.14 z（能段 4）/ 0.18 z（能段 1），重尾 df \(\approx\) 3。中心项的主导成分是最好候选自身的样本内表征误差（同一位置内失败 24 \(\to\) 3、未见位置 21 \(\to\) 8，72 参数诊断修正；R119 限定：3 与 8 是好的一端，12 个重抽样种子里同位置为 3–15、未见位置为 7–16，须按种子区间引用），不是位置泛化缺口。宽度读数也须带估计量与分半集合：0.976 [0.956, 0.993] 是一次偏低的分半，40 个分半平均 0.998。后续（R125–R152）：判据改为每个位置一个合并偏移并冻结打分手（\(\varepsilon\) 0.05 比较规律 / 0.10 跟踪进度）；地板改正后最好候选在 \(\varepsilon\) 0.10 处高于地板；位置效应收窄到像素级几何且两能段共享；第一个拟合前位置项把最好候选从 0.545 抬到 0.727（p 0.003–0.004，尚未在新位置检验）；宽度判据校准为“8 箱允许 1 箱失败”后第一条可观测箱级规律 V̄ 通过（1/8），但迁到 emldetect 全统计量下仍失败 5/5 箱。^{[s17][s18][s19]}Four candidate forms of the conditional law are all rejected, and the failures concentrate on the centre: held-out cell passes 10/72, 48/72, 42/72 and 35/72 (null mean about 62–63/72, with the scorer first validated on exactly-true data); the width cells pass 68/72, the 24 mean failures are all negative, concentrated at mid count and growing with index, and position is not the missing variable. An audit on 2026-10-21 re-scoped that reading: the scorer mixed three estimands — the width is a within-position quantity, the mean and coverage pool rows across positions, and the null adapts to the candidate; so "the width generalises" can only be read as holding within one position (an exact audit proves the width is identically invariant to per-position offsets, 1e-15: a law carrying a 0.30 z per-position offset still passes width 70/72). The new scorer separates two targets: on the population of unseen positions D / H2 / MIX3d / C pass 44 / 50 / 32 / 12 of 72 cells (band 4) and band-1 D / H2 36 / 43, all rejected on Q (p \(\leq\) 0.009); R124 qualifies: these are one draw of an unstable statistic — over 14 position splits D 22–46 and H2 26–52 (band 4), band-1 D 14–53 and H2 30–52; H2 above D is not significant on the fixed split (McNemar p 0.18 / 0.25), so no "H2 is best" ranking can be made, and scorer v3 is needed before any law ranking; per position every candidate fails by a wide margin (band 4: 1 / 0 / 1 / 0; band 1: 0 / 1), with the frozen position effect \(\tau_{mean}\) 0.14 z (band 4) / 0.18 z (band 1), heavy-tailed with df \(\approx\) 3. The centre term's dominant component is an in-sample representation error of the best candidate itself (within position the failures go 24 \(\to\) 3, at unseen positions 21 \(\to\) 8 with a 72-parameter diagnostic correction; R119 qualifies: 3 and 8 are the favourable end — across 12 resampling seeds within-position runs 3–15 and unseen 7–16, so quote them as seed ranges), not a position-generalisation gap. The width reading must also name its estimator and split ensemble: 0.976 [0.956, 0.993] was one low split, averaging 0.998 over 40 splits. Follow-up (R125–R152): the criterion became one pooled offset per position with the scorer frozen (\(\varepsilon\) 0.05 to compare laws / 0.10 as the tracker); after the floor correction the best candidate sits above the floor at \(\varepsilon\) 0.10; the position effect narrowed to pixel-level geometry shared by both bands; a first pre-fit position term lifts the best candidate from 0.545 to 0.727 (p 0.003–0.004, not yet tested on fresh positions); and after the width rule was calibrated to “at most 1 failing bin of 8” the first observable bin-level law V̄ passes (1 of 8) but still fails 5/5 bins on emldetect rows at full statistics. ^{[s17][s18][s19]}

零参数条件宽度是一个有明确边界的工具：在锚区间它就是答案，在目录区间它是校准的起点，而跨流水线的问题已由同图配对实验回答。The parameter-free conditional width is a tool with a well-defined boundary: in the anchor interval it is the answer, in the catalogue interval it is the starting point of a calibration, and cross-pipeline questions have been answered by the paired same-images experiment.

来源对照表Source Map

致谢：本页的整理工作由 AI 代理完成（按 AAS 政策声明）。基于 XMM-Newton（ESA 科学任务，仪器与贡献由 ESA 成员国与 NASA 直接资助）的模拟构造。Acknowledgments: AI agents assisted in preparing this page (stated per AAS policy). The simulation construction is based on XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA.