{"id":"1155b259-cd52-4680-9be4-167b009a7a11","arxiv_id":"2607.24250","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":3.5,"correctness_risk":"low","formal_verification":"none","parameter_count":6,"one_line_summary":"Fitting an SPE response model to the charge spectrum recovers PMT gain to better than ~1% and outperforms the model-independent occupancy method, even when a soft under-amplified component is present.","lead":"This technical note derives and compares standard mathematical models for calibrating photomultiplier tubes under faint Poisson light, including occupancy and SPE-fitting methods. It is a pedagogical reference that also tests precision on toy Monte Carlo and Hamamatsu R1408 data and checks the effect of a soft charge component.","discovery_kind":"extension","skeptic_critique":{"model":"moonshotai/kimi-k3","headline":"The <1% gain-bias claim rests on circular toys plus a stability test that checks consistency, not accuracy; the one non-circular robustness check (Sec. 12) probes a single ad-hoc soft-component point, so the claim's generality is unproven.","rationale":"The reader already identified the parametric-family adequacy of Eq. 7.1 as the weakest assumption and already conditioned acceptance on a broader soft-component scan; my analysis lands in the same place and merely sharpens it: the circularity of the Sec. 10 toys means all the evidentiary weight for \"unbiased gain\" sits on the Sec. 12 single-point test and on the real-data stability argument, and the latter tests consistency rather than accuracy. This does not overturn the paper — the mathematics is standard and correct, the DFT code is public, and the first-moment absorption mechanism makes the Sec. 12 result physically plausible rather than lucky. But it does mean the <1% claim should be read as demonstrated only within the probed regime (soft component well separated below the SPE peak), which is exactly the CONDITIONAL posture the reader took. Hence verdict UNCHANGED: the reader's CONDITIONAL with the requested broader soft-component scan is the right call, and the concrete test above is essentially the condition the reader already imposed, made specific.","tokens_in":34307,"tokens_out":2128,"duration_ms":49109,"concrete_test":"Scan the (w2, α2) plane for Eq. 12.1: w2 ∈ {0.05, 0.10, 0.20, 0.30, 0.40} and 1/α2 ∈ {0.25, 0.5, 1, 2, 4} × (1/α), holding w, α, λ, θ at Sec. 7 values. For each point generate 200 toys at μ=2, fit with the single-exponential model (Eq. 7.1), and record ΔG and χ²/NDOF. If ΔG exceeds 1% wherever 1/α2 ≳ 1/λ while χ²/NDOF stays near 1, the robustness claim needs a domain restriction; if ΔG stays <1% everywhere, the claim is materially strengthened. Supplement with an analytic check: verify whether best-fit w'/α' tracks w/α + w2/α2 across the scan, confirming the first-moment-absorption mechanism.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The strongest claim has three legs. (1) Toy MC: toys are generated from Eq. 7.1 and fitted with Eq. 7.1 (Secs. 9–10), so recovery of the gain to <0.5% is guaranteed by construction and only validates the DFT machinery and fitter, not the physics. (2) R1408 data: the evidence offered is parameter stability vs μ and good χ²/NDOF (Table 1, Fig. 15). But stability across light level demonstrates internal consistency of the assumed model, not closeness to the true gain — a systematically wrong SPE family that is self-similar across μ would pass the same test. The only external anchor is the occupancy method, which the paper itself shows is biased at 3–6%, so the ~5% fit-vs-occupancy offset (Fig. 16) cannot adjudicate which is right; there is no independent ground truth for the real gain. (3) The soft-component stress test (Sec. 12) is the single genuinely non-circular check: toys from Eq. 12.1 fitted with the wrong model (Eq. 7.1) still recover G to <1%. But it is one parameter point — w2=0.20, α2=46, i.e. w2=w/2 and 1/α2 = (1/α)/2 — chosen by fiat, not from the IceCube values that motivated the study. The mechanism of the recovery is that the fit absorbs the combined soft component into inflated w', α' such that w'/α' ≈ w/α + w2/α2 (first-moment matching of the low-charge region, with the gamma mean 1/λ' barely moving). That absorption is plausible precisely because both exponentials are soft relative to the SPE peak; nothing in the note shows it survives when the second component's mean approaches the gamma peak (1/α2 → 1/λ ≈ 0.13 nVs) or when w2 grows large. If absorption fails in that region, the headline \"<1% unbiased gain\" holds only for soft components that are sufficiently soft — a qualifier the paper does not state. The authors themselves flag this: \"Ideally, we would like to try more cases\" (Sec. 12).","agreement_with_reader":"agree"},"referee_report":{"model":"moonshotai/kimi-k3","summary":"This technical note presents a pedagogical exposition of the standard theory of PMT charge response to faint Poissonian light: photoelectron statistics (Poisson thinning through QE and CE), the Bellamy-type compound-Poisson charge response S_R(x), its mean/variance, and two numerical evaluations (recursive numerical convolution and a DFT/FFT method based on the closed-form summed characteristic function, Eq. 6.11). Two gain-calibration methods are compared — the model-independent occupancy method (Saldanha et al.) and the conventional S_R(x) fit — using toy Monte Carlo generated from an exponential-plus-gamma SPE model (Eq. 7.1) and 15 datasets from a Hamamatsu R1408 PMT at mu ≃ 0.5–3. The authors report gain recovery with fractional bias <~0.5–1% for the fit versus ~3–6% for occupancy, parameter stability of the SPE model versus light level on real data (Table 1), and a stress test (Sec. 12) in which a second soft exponential component added to the toy SPE model degrades occupancy (~6%) but not the fit (<1%). A public C++/ROOT implementation is provided.","tokens_in":34736,"tokens_out":3742,"duration_ms":116108,"significance":"As a pedagogical reference the note succeeds: the derivations (Poisson thinning, Sec. 4.1; moments of S_R, Sec. 5; convolution identities, Appendix A; DFT identity, Sec. 6.2) are standard but cleanly and self-containedly written, and the note collects material otherwise scattered across Bellamy, Smirnov, and Saldanha et al. It ships a public, reusable implementation of the DFT method and both calibration techniques, a genuine asset for students. The R1408 dataset (15 light levels, χ²/NDOF ≃ 1, stable SPE parameters) is a useful worked example, and the Sec. 12 wrong-model stress test is, in principle, a falsifiable robustness check. The physics content is incremental over Refs. [24, 30] by the same group, but the comparative fit-vs-occupancy study and the soft-component investigation add modest new value appropriate for a JINST technical note.","major_comments":[{"comment":"The headline accuracy claim — 'the fit can determine the gain in an unbiased way, with an accuracy of better than 1.0%' (Sec. 11.2) — is not supported at that strength by the evidence presented. The toy-MC leg (Sec. 10) generates and fits with the same model (Eq. 7.1), so sub-percent recovery is guaranteed by construction and validates only the DFT machinery and fitter, as the authors themselves note. The real-data leg shows χ²/NDOF ≃ 1 and parameter stability versus μ (Table 1), but these test internal consistency of the assumed SPE family, not closeness to the true gain: a systematically wrong but self-similar SPE model would pass the same tests. The only external anchor, the occupancy method, is shown to be biased at 3–6%, so the ~5% fit-vs-occupancy offset in Fig. 16 cannot adjudicate which method is closer to the truth. The claim should be restated as precision/consistency at the ~1","section":"Secs. 10–11.2, Eq. (8.16), Figs. 12, 16"},{"comment":"The soft-component study is the only genuinely non-circular robustness check, but it probes a single ad-hoc parameter point (w2 = 0.20, α2 = 46, i.e. w2 = w/2 and 1/α2 = (1/α)/2) rather than values motivated by the IceCube measurement (Ref. [43]) that prompted the section. The recovery mechanism visible in the text — the fit absorbs the second exponential into inflated w′, α′ while the gamma mean 1/λ′ barely moves — is first-moment matching of the low-charge region, which can be expected to work precisely because both exponentials are soft relative to the SPE peak. Nothing shown demonstrates survival when the second component is harder (α2 comparable to λ) or heavier. Either a modest scan over (w2, α2), or an explicit statement that the conclusion holds only for soft components of this type, is needed; the sentence 'it is safe to say that the fit is the most precise method for gain deter","section":"Sec. 12, Eq. (12.1), Fig. 18"},{"comment":"The origin of the occupancy bias (~3% at f = 0.2, rising to ~6% with the soft component) is never analyzed. Since the method is presented as the model-independent alternative, the note should explain whether the bias is the expected leakage of underamplified 1-PE charge below the threshold cut (inflating I_T and hence μ′), quantify its scaling with w and α, and state the associated systematic uncertainty. Without this, the reader cannot judge whether the 3% is an intrinsic limitation of the truncated-integral estimator or specific to the R1408 SPE shape used here. This matters because Sec. 13 recommends the fit over occupancy partly on the basis of this number.","section":"Secs. 9.2, 11.2, Figs. 10, 16"}],"minor_comments":[{"comment":"The numerator product ends in '(N−n−1)/N'; it should read (N−n+1)/N as in Eq. (3.5).","section":"Eq. (3.6)"},{"comment":"The variance formula uses 'a' in two places (w/a² and (1/λ − 1/a)²); this should be α for consistency with the rest of the note.","section":"Eq. (8.17)"},{"comment":"The caption says 'Gamma distribution for λ=20 and some values of θ', but the legend shows θ=20 with λ=1, 5, 10. Caption and legend are interchanged; also the axis label 'A.U.' is unexplained.","section":"Fig. 6 and caption"},{"comment":"'Fig. (bottom right)' appears twice; the pedestal panel should be '(bottom left)'. The red threshold line is described as cutting at 10% but Sec. 9.2 adopts f = 0.2; clarify.","section":"Fig. 8 caption"},{"comment":"The χ² uses Neyman's variance (D_i in the denominator), which biases low-count bins; a sentence justifying this choice (or noting Pearson/alternative) would help, since the valley region driving w and α is exactly where counts are low.","section":"Eq. (10.1)"},{"comment":"'one gets the gain G in an almost unbiased way' is contradicted by the ~3% bias shown in Fig. 10; rephrase as 'model-independent'.","section":"Sec. 9.1"},{"comment":"Typos/grammar: 'form a uniform distribution' → 'from' (Sec. 3); 'smaller that 0.01' → 'than' (Sec. 3); 'dumping of the Poisson factors' → 'damping' (Sec. 4.4); 'the underline assumptions' → 'underlying' (Sec. 6); 'one has to deal a single quantum efficiency' → 'deal with' (Sec. 11.1); the sentence 'Of course, and we would have gotten the same results with DFT, albeit with a larger amount of time' (Sec. 7) is garbled — presumably 'the same results with numerical integration'. Informal asides ('For us the battle is won !', 'there is a synergy in action !') could be toned down for a journal version.","section":"Various"},{"comment":"State in the caption or text that the occupancy points use f = 0.2, and quote the two means and standard deviations numerically (currently only ~0.0942 nVs and 'better than 1%' are given).","section":"Fig. 16"},{"comment":"The uncertainties on w and α at μ = 3.074 roughly double relative to lower-μ rows; a brief comment on whether this is statistical or signals the onset of the pedestal-suppression regime would be useful.","section":"Table 1"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is largely a pedagogical consolidation of the first author's prior work (Refs. [13, 24, 27, 30]), with a heavy self-citation pattern; the genuinely new material is the fit-vs-occupancy comparison on R1408 data and the Sec. 12 soft-component study. This is acceptable for a JINST technical note but the editor may wish to confirm the incremental content meets the journal's threshold. The comparative claim (fit superior to occupancy) is demonstrated; the absolute <1% accuracy claim is not, and my requested revisions are aimed at aligning the wording with what the evidence supports rather than demanding new data-taking."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This is a technical note, not a discovery paper. What you get is a clear walk-through of the standard PMT charge-response math (Poisson PE stats, SPE convolution series, pedestal, mean/variance identities), the two usual numerical routes to S_R(x), and a head-to-head of occupancy versus SPE fitting on toys plus one R1408 bench campaign. The DFT pipeline and the public code are the practical bits people will actually use.\n\nWhat is new is modest but real: a controlled scan of fractional gain bias versus μ and occupancy threshold, a re-fit of R1408 data with the gamma+exponential model showing parameter stability across μ ≈ 0.5–3, and a soft-component stress test. The appendix proofs and the Poisson/binomial derivations are clean. χ²/NDOF near 1 and stable λ, θ, w, α on real data are what you want to see. They also flag the lab geometry (fiber at photocathode center) and the in-situ problem honestly in the outlook.\n\nThe soft spot is the strength of the “fit recovers G to <1%” headline. Toys generated from Eq. 7.1 and fitted with Eq. 7.1 mainly validate the fitter and DFT, not the physics. On real data, stability versus light level shows internal consistency of the assumed SPE family, not absolute accuracy; the occupancy cross-check is itself biased at a few percent, so the ~5% fit–occupancy offset does not crown a winner. The one non-circular check (Sec. 12) is a single ad-hoc second exponential; the fit absorbs it by inflating w and α (first-moment matching). That is interesting and useful, but it is one point, and the authors say they would like more cases. I would not over-read the generality.\n\nCitation pattern is heavy on the authors’ own earlier notes, which is fine given the continuity of the code and the R1408 work; the classic Bellamy/Smirnov/Saldanha lineage is properly present.\n\nWho it is for: students and postdocs commissioning PMTs or writing SPE fitters, and anyone who wants one place that lays out the formulae without hunting five papers. Not for people looking for a new physics result or a solved in-situ recipe.\n\nI would send it to peer review as a JINST-style technical note. Ask for the raw charge histograms and a slightly broader soft-component scan if the authors can manage it; do not desk-reject. Worth engaging if you or your students touch PMT calibration.","headline":"Clean pedagogical synthesis with a useful MC/method comparison; the <1% unbiased-gain claim is real for the lab geometry and model class they tested, but rests partly on circular toys and one soft-component point.","tokens_in":31262,"tokens_out":647,"would_cite":true,"duration_ms":21583,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"Fitting the single-photoelectron model recovers PMT gain to better than 1%, while the model-independent occupancy method is biased by several percent.","keywords":["photomultiplier tubes","single photoelectron response","gain calibration","occupancy method","charge response function","discrete Fourier transform","gamma distribution","soft component"],"falsifier":"Repeat the same R1408 (or equivalent) runs with an independent, higher-resolution charge measurement or a second soft-component model whose parameters are fixed by pulse-shape selection; if the fitted gain then shifts by more than one percent while the occupancy result stays fixed, the claim fails.","tokens_in":30832,"feed_emoji":"📡","tokens_out":939,"duration_ms":19228,"temperature":0.7,"pith_summary":"This technical note lays out the full mathematical chain that turns faint Poisson light into the measured charge spectrum of a photomultiplier tube, then compares the two standard ways of extracting the tube's gain. The ideal response is a Poisson-weighted sum of convolutions of the single-photoelectron charge distribution with electronic noise; that sum is evaluated either by recursive numerical integration or by a discrete Fourier transform. On both toy Monte Carlo samples and real Hamamatsu R1408 data the conventional fit to a gamma-plus-exponential single-photoelectron model recovers the true gain with fractional bias below one percent across mean photoelectron numbers from roughly 0.5 to 3. The simpler occupancy method, which uses only the pedestal fraction, is systematically high by three to six percent. Adding an extra soft exponential component degrades the occupancy result further but leaves the fit essentially unbiased, because a single exponential still approximates the low-charge tail well enough for the gain. The authors therefore recommend the fit when a stable parametric model can be validated by parameter constancy versus light level, and they caution that lab geometry with a fiber at the photocathode center does not automatically transfer to in-situ illumination of large detectors.","feed_headline":"PMT gain fit beats occupancy by several percent","feed_subtitle":"A gamma-plus-exponential model recovers true gain to <1% on Monte Carlo and R1408 data","key_machinery":"The realistic charge response SR(x): the Poisson-weighted sum of n-fold convolutions of the single-photoelectron density S(x) with the pedestal noise B(x), evaluated either recursively or via the closed Fourier product exp(−μ) B̃(k) exp(μ S̃(k)).","core_discovery":"On both controlled Monte Carlo and real R1408 spectra the conventional single-photoelectron-model fit extracts the gain with fractional accuracy better than about one percent over the practical working range of mean photoelectron number 0.5–3, while the model-independent occupancy method remains biased at the several-percent level; a second soft exponential does not spoil the fit but further degrades occupancy.","pith_inferences":["The same Fourier product formula immediately supplies the full likelihood for multi-PMT energy reconstruction once each tube’s SPE parameters are known.","If the second soft exponential is itself voltage- or temperature-dependent, periodic refits of the single-exponential effective model may still track gain drifts without needing the full two-exponential form.","Occupancy bias of a few percent is large enough to matter for sub-percent energy-scale goals such as those of next-generation neutrino mass-ordering experiments."],"forward_implications":["Laboratories can prefer the SPE-model fit over occupancy whenever a stable parametric family can be validated by parameter constancy across light levels.","A single exponential remains an adequate effective description of the low-charge tail for gain extraction even when a second softer exponential is physically present.","Discrete-Fourier-transform evaluation of SR(x) makes the multi-parameter fit fast enough for routine use on large samples.","In-situ calibration of large monolithic detectors will inherit additional geometric biases not present in the centered-fiber laboratory geometry used here."],"fun_headline_variants":["SPE fit recovers PMT gain to <1% as occupancy lags several percent","Gamma SPE model beats occupancy for PMT gain on MC and R1408","Single-photoelectron fit yields sub-percent gain accuracy","Occupancy trails SPE fit by several percent in gain recovery","Soft exponential spares SPE gain fit but degrades occupancy"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"That the chosen parametric family for the single-photoelectron charge shape is close enough to the true shape that a good fit and stable parameters versus light level guarantee an unbiased gain.","fun_headline_variants_meta":{"raw":{"variants":["SPE fit recovers PMT gain to <1% as occupancy lags several percent","Gamma SPE model beats occupancy for PMT gain on MC and R1408","Single-photoelectron fit yields sub-percent gain accuracy","Occupancy trails SPE fit by several percent in gain recovery","Soft exponential spares SPE gain fit but degrades occupancy"]},"model":"grok-4.5","effort":"low","cost_usd":0.003883,"raw_usage":{"total_tokens":1255,"prompt_tokens":803,"num_sources_used":0,"completion_tokens":72,"cost_in_usd_ticks":38828000,"prompt_tokens_details":{"text_tokens":803,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":380,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":803,"tokens_out":72,"duration_ms":7961,"temperature":1.0,"reasoning_tokens":380,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T19:59:46.222802+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Repeat the same R1408 (or equivalent) runs with an independent, higher-resolution charge measurement or a second soft-component model whose parameters are fixed by pulse-shape selection; if the fitted gain then shifts by more than one percent while the occupancy result stays fixed, the claim fails.","supporting_citations":[],"review_version":1}