Fast-Radio-Burst Source Counts and Log-Shell Event-Rate Inference in a Near-Euclidean Window

Fast radio bursts (FRBs) are bright millisecond radio transients of extragalactic origin [citation]. Source counts remain one of the cleanest population diagnostics for such transients: in a homogeneous non-evolving near-Euclidean window, cumulative fluence counts obey $\Robs(>F)\propto F^{-3/2}$TeX\Robs(>F)\propto F^{-3/2} [citation]. Interpreting departures from that law is nontrivial in current FRB samples because completeness, bandpass response, beam and width selection, exposure weighting, repeater contamination, host-dependent suppression, and propagation all compete in the observed counts [citation].

Large current catalogues improve the empirical anchor but do not remove the selection problem. The second CHIME/FRB catalogue reports 4539 bursts from 3641 unique sources, including 981 bursts from 83 known repeaters, together with cumulative exposure time and survey sensitivity over the observing period [citation]. Recent CHIME/FRB-KKO and MeerKAT localization campaigns also enlarge the localized subset available for low-redshift evolution and host-side control [citation]. At the same time, CHIME intensity-data flux and fluence measurements remain beam- and localization-dependent and are best interpreted as lower limits with quantified uncertainty, with beam response continuing to require dedicated calibration and mapping analyses [citation]. Recent ASKAP- and CHIME-based population analyses continue to find useful but selection-sensitive constraints on the FRB energy and rate distributions [citation], while far-sidelobe constraints, exposure-corrected source-count evolution, energy-distribution comparisons, and semisupervised catalogue analyses all indicate that some apparently non-repeating sources may be hidden repeaters [citation].

Accordingly, the low-redshift source-count problem is organized around a null-law-first hierarchy. The standard $-3/2$TeX-3/2 count law is tested before any propagation-side reading, and any residual interpretation is deferred until the survey window, completeness threshold, spectral control, repetition history, and host-selection effects are constrained on the same campaign. Only the imported homogeneous background history and the imported propagation-side distance map enter the present count stack, together with an imported local-admissibility gate for any propagation-side reinterpretation [citation].

The analysis is restricted to FRB source counts and observer-time rate inference on a declared low-redshift near-Euclidean window. It does not attempt a full luminosity-function reconstruction, a full FRB population synthesis, or a dispersion-measure-based redshift inversion. The standard count stack is written for that window, and any nonzero shell-rate reading remains subordinate to explicit empirical gates. Appendices reference--reference record the inference boundary, the minimal logic chain, and the status of the core equations used below.

The declared source-count layer receives bounded support from three public CHIME/FRB Catalog 1 companion gates. These gates are imported only as selection-side stress and compatibility surfaces for the FRC count-law grammar. They do not replace the homogeneous-survey anchor adopted for the main count analysis, and they do not convert the optional shell-rate organizer into a propagation-correction detection.

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The VP2 stress label is the conservative selection-response reading. For the present FRC analysis, the relevant point is that the injection-efficiency response surface is nonuniform enough to trigger FRC-SELECTION-STRESS; the detailed row counts, response-cell source summary, and weighted residual board remain in the FRC--VP2 companion record. This stress result blocks a selection-free population-law overread but does not falsify the declared conditional count-law grammar [citation].

The three Catalog 1 companion stress labels are bounded public-data stress labels. They do not establish a global FRB source-count law, luminosity-function closure, dispersion-measure-to-redshift theorem, repeater/non-repeater population theorem, cross-survey universal rate law, CHC propagation-correction detection, or full CHIME/FRB selection-function closure.

A separate FRC--KP1 companion may be imported only as a bounded Catalog1-frozen / Catalog2 non-overlap public holdout-support result for the residual-geometry layer [citation]. It does not change the theorem or equation status of this parent paper, does not upgrade the VP0/VP1/VP2 stress surfaces to selection-function closure, and does not establish a global FRB population law, luminosity-function closure, dispersion-measure-to-redshift theorem, propagation-correction detection, or CHC-wide empirical closure.

Only $\Hbg(z)$TeX\Hbg(z), $\Xicos(z)$TeX\Xicos(z), $\DaCHC$TeX\DaCHC, $\DlCHC$TeX\DlCHC, and the derived shell variable $\reuc(z)$TeX\reuc(z) are imported. Any quantity introduced later is internal to the event-rate layer and does not replace the propagation sector.

Background and propagation imports

The imported dark-energy branch supplies the homogeneous expansion history $\Hbg(z)$TeX\Hbg(z) and remains background-only [citation]. The imported propagation branch distinguishes the asymptotic causal speed $\cinf$TeX\cinf from the environment-dependent propagation map and imposes a local-admissibility-first gate on any nontrivial propagation correction [citation]. The imported cosmological propagation law is

c(z)=\cinf\sqrt{1-\Xicos(z)},

with $\Xicos(z)$TeX\Xicos(z) the homogeneous cosmological specialization of the CHC gradient measure. Within the flat-background restriction used here, the corresponding CHC angular-diameter distance between $z_1$TeXz_1 and $z_2>z_1$TeXz_2>z_1 is

\DaCHC(z_1,z_2)
=
\frac{1}{1+z_2}
\int_{z_1}^{z_2}
\frac{\cinf\sqrt{1-\Xicos(z)}}{\Hbg(z)}\,dz,

and the associated luminosity distance is

\DlCHC(z)=(1+z)^2\DaCHC(0,z).

When $\Xicos\to 0$TeX\Xicos\to 0, reference and reference reduce to the standard flat-background relations.

Radial shell variable

For the near-Euclidean window used below, define

\reuc(z)
\equiv
\int_{0}^{z}
\frac{\cinf\sqrt{1-\Xicos(z')}}{\Hbg(z')}\,dz'.

Within a sufficiently low-redshift window, $\DlCHC(z)=\reuc(z)[1+\mathcal O(z)]$TeX\DlCHC(z)=\reuc(z)[1+\mathcal O(z)], so $\reuc$TeX\reuc is the natural shell variable for source-count inference.

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Let $\Robs(>\Fmin)$TeX\Robs(>\Fmin) denote the sky-coverage-normalized observer-time FRB rate above a declared fluence threshold $\Fmin$TeX\Fmin. Before sky-coverage normalization is applied, the directly estimated quantity in a homogeneous survey window is denoted $\Robs^{(w)}$TeX\Robs^{(w)}. A standard observer-time rate model can be written as

\Robs(>\Fmin)
=
\int dz\,\frac{dV}{dz}\,\frac{1}{1+z}
\int d\Eiso\,\phien(\Eiso,z)\,
\effdet(\Eiso,z;\Fmin)\,
\Theta\!\left[\Fobs(\Eiso,z)-\Fmin\right],

where $dV/dz$TeXdV/dz is the comoving-volume element, $1/(1+z)$TeX1/(1+z) converts source-frame burst rates to observer-time rates, and $\effdet$TeX\effdet is an effective completeness function absorbing beam, width, and threshold losses. The function $\phien(\Eiso,z)$TeX\phien(\Eiso,z) denotes the source-frame volumetric burst-rate density per unit isotropic-equivalent energy, normalized so that

\int d\Eiso\,\phien(\Eiso,z)=\dot n_{\mathrm{src}}(z).

The observed fluence is written as

\Fobs(\Eiso,z)
=
\frac{\Eiso}{4\pi \DlCHC(z)^2}\,\Kcorr(z),

with $\Kcorr$TeX\Kcorr collecting bandpass, spectral, and duration-dependent factors.

proposition: Near-Euclidean null law for homogeneous FRB counts. Assume a declared near-Euclidean window in which: (i) the source-frame volumetric burst-rate density is homogeneous and non-evolving across the window, so that $\phien(\Eiso,z)=\phien(\Eiso)$TeX\phien(\Eiso,z)=\phien(\Eiso), (ii) the effective completeness function is constant above a stated threshold, (iii) $\Kcorr(z)=K_0$TeX\Kcorr(z)=K_0 is constant across the window, and (iv) observer-time dilation is negligible at the target precision. Then

\Robs(>\Fmin)
=
\frac{K_0^{3/2}}{3\sqrt{4\pi}}\,\Fmin^{-3/2}
\int d\Eiso\,\phien(\Eiso)\,\Eiso^{3/2},

provided the $3/2$TeX3/2-moment of $\phien$TeX\phien exists. In particular,

\boxed{\Robs(>\Fmin)\propto \Fmin^{-3/2}},
\qquad
\boxed{\frac{d\Robs}{d\Fmin}\propto \Fmin^{-5/2}}.

proof. In the near-Euclidean window, $\DlCHC\simeq \reuc$TeX\DlCHC\simeq \reuc and $\Fobs\simeq K_0\Eiso/(4\pi \reuc^2)$TeX\Fobs\simeq K_0\Eiso/(4\pi \reuc^2). For a burst of isotropic-equivalent energy $\Eiso$TeX\Eiso, the maximum detectable radius is

\reuc_{\max}(\Eiso,\Fmin)=\sqrt{\frac{K_0\Eiso}{4\pi\Fmin}}.

The observed rate above threshold is therefore the energy integral of the accessible Euclidean shell volume,

\Robs(>\Fmin)
=
\int d\Eiso\,\phien(\Eiso)
\left(\frac{4\pi}{3}\reuc_{\max}^{3}\right).

Substituting reference gives reference, and differentiating yields reference.

The Euclidean slope is the null prediction for a homogeneous non-evolving transient population in a near-Euclidean window. For FRBs, any measured deformation of $-3/2$TeX-3/2 must first be decomposed into source evolution, completeness, spectral response, geometry, repetition history, and propagation before a residual shell-rate interpretation can be entertained [citation].

Homogeneous-survey observer-time anchor

Recent FRB catalogues make it possible to anchor source-count inference within one survey before any cross-survey transport. For a survey window $w$TeXw with duration $T_{\mathrm{surv}}$TeXT_{\mathrm{surv}}, let $F_i^{\mathrm{obs}}$TeXF_i^{\mathrm{obs}} denote the measured fluence entering the adopted counting convention and let $f_{\mathrm{exp},i}\in(0,1]$TeXf_{\mathrm{exp},i}\in(0,1] denote the relevant cumulative exposure fraction for burst or source $i$TeXi. Define the exposure-weighted counts

N_{w}(>\Fmin)
=\sum_i \frac{1}{f_{\mathrm{exp},i}}\,\Theta\!\left(F_i^{\mathrm{obs}}-\Fmin\right),
\qquad
\Robs^{(w)}(>\Fmin)=\frac{N_{w}(>\Fmin)}{T_{\mathrm{surv}}}.

If the same window covers a declared sky fraction $f_{\mathrm{sky}}^{(w)}$TeXf_{\mathrm{sky}}^{(w)}, the corresponding all-sky rate is

\Robs(>\Fmin)=\frac{\Robs^{(w)}(>\Fmin)}{f_{\mathrm{sky}}^{(w)}}.

Equation reference is retained only as an optional normalization identity. The present count-law closure remains in survey-window observer-time mode and does not use all-sky normalization unless a separately declared sky-fraction convention is admitted on the same campaign.

The second CHIME/FRB catalogue provides the primary homogeneous-survey anchor adopted here because it reports cumulative exposure time and survey sensitivity for the largest publicly available uniformly processed sample used in this analysis [citation]. This adopted homogeneous-survey anchor supplies the benchmark frame for all downstream quantitative comparisons below, and those comparisons are read only relative to that fixed survey family and campaign. For the concrete survey window adopted here, the count-law test uses one CHIME/FRB Cat2 clean mainlobe source-firstburst campaign window (2018 September 4--2023 September 15), transit-specific 95% completeness thresholds, transit-resolved cumulative exposure weighting, a conservative low-redshift window with $\ze=0.05$TeX\ze=0.05, a declared near-Euclidean budget $\epswin=0.10$TeX\epswin=0.10 under campaign uncertainty ceiling $\epsilon_*=0.15$TeX\epsilon_*=0.15, and survey-window observer-time counts only. This concrete instantiation selects one admitted operating window inside the generic gate definitions below and does not by itself promote the paper to an all-sky, global-population, or propagation-closure claim. Exposure weighting and repeater handling are required at this stage: exposure-weighted reclassification changes the apparent evolution of CHIME source counts and reduces the genuinely non-repeating contribution [citation], while hidden-repeaters analyses indicate that some single-burst classifications may still be incomplete [citation]. Survey-internal anchoring is therefore preferred to naive survey stitching when testing the null law, especially for instruments whose flux and fluence calibration remains beam- and localization-dependent [citation]. No fitted shell organizer is transported across surveys unless the same admitted near-Euclidean window and the full gate stack are shown to hold jointly.

definition: Declared near-Euclidean window. A redshift interval $[0,\ze]$TeX[0,\ze] is an admissible near-Euclidean window if all of the following hold on the corresponding radial interval $[0,\reuc(\ze)]$TeX[0,\reuc(\ze)]:

- $\DlCHC(z)=\reuc(z)[1+\delta_{\mathrm{geom}}(z)]$TeX\DlCHC(z)=\reuc(z)[1+\delta_{\mathrm{geom}}(z)] with declared geometry drift

\epsgeom\equiv \sup_{z\in[0,\ze]}\abs{\delta_{\mathrm{geom}}(z)};

- the combination $(1+z)^{-1}\Kcorr(z)$TeX(1+z)^{-1}\Kcorr(z) has declared spectral drift

\epsspec\equiv \sup_{z\in[0,\ze]} \left| \ln\!\frac{(1+z)^{-1}\Kcorr(z)}{\Kcorr(0)} \right|;

- external localized-redshift information or a separately specified low-$z$TeXz population model, fixed independently of the source-count residual under test, supplies a declared source-evolution bound $\epsevo$TeX\epsevo on the same window; - the survey completeness function $\effdet$TeX\effdet is declared and controlled above the adopted threshold; - a homogeneous-survey anchor with explicit exposure, sensitivity, and sky-coverage information is available for the same window.

The window budget is then

\epswin\equiv \max\{\epsgeom,\epsspec,\epsevo\},

and the near-Euclidean window is admitted only when

\epswin\le \epsilon_*,

with $\epsilon_*$TeX\epsilon_* the declared source-count uncertainty budget for the same campaign. All downstream null-law, logarithmic-shell, and optional residual-shell claims below are read only on this declared near-Euclidean window and relative to its budget $\epswin$TeX\epswin.

A declared window is operational only when external source redshifts or a separately specified low-$z$TeXz population model fix $\epsevo$TeX\epsevo on the same campaign. When a low-$z$TeXz population model is used, it is fixed before shell-rate inference on the adopted campaign and is not re-fitted to the same source-count deformation interpreted below. Recent CHIME/FRB-KKO and MeerKAT localization samples enlarge the low-$z$TeXz control set [citation], but dispersion measures are not inverted into source redshifts here [citation]. Without one of these two inputs, the interval $[0,\ze]$TeX[0,\ze] remains only a gate declaration.

The natural shell representation of the observer-time event rate is logarithmic rather than linear. Writing $dr=r\,d\ln r$TeXdr=r\,d\ln r in the Euclidean shell element $4\pi r^2dr$TeX4\pi r^2dr gives

4\pi \reuc^2 d\reuc = 4\pi \reuc^3 d\ln \reuc.

Accordingly, if $d\Robs$TeXd\Robs denotes the contribution from one shell, define

d\Robs = \Rlog(\reuc;\Fmin)\,d\ln\!\left(\frac{\reuc}{\reff}\right),

with

\Rlog(\reuc;\Fmin)
\equiv
4\pi \reuc^3\,\nreff(\reuc;\Fmin),

where $\nreff$TeX\nreff is the effective observer-time burst-rate density after completeness, spectral, and window controls have been applied.

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Residual departures that survive the survey, completeness, window, repetition, host-selection, and local-admissibility gates on one declared campaign may be organized with an internal shell-rate factor. This organizer is downstream and optional. The present survey-window count-law closure does not require it and does not use it as part of the admitted count-law window. Outside the admitted residual-shell case the analysis closes at the standard source-count law on the declared near-Euclidean window.

definition: Internal shell-rate factor. On a gate-passing near-Euclidean window, let

\krate(\reuc)\equiv \sqrt{1-\Xishell(\reuc)}\in(0,1],

where $\Xishell(\reuc)$TeX\Xishell(\reuc) is an internal auxiliary residual-shell variable. It is bounded by construction on the declared window and is not an imported propagation variable. By itself it is not a propagation correction. Only a subsequent reinterpretation of a nonzero residual shell profile in propagation-side terms is subject to the imported propagation-side local-admissibility gate; if that reinterpretation fails, the propagation-side reading is rejected. Any downstream residual-shell claim is read only on the declared near-Euclidean window and relative to this auxiliary shell factor.

definition: Scale-free optional class. The scale-free optional class is the family

\krate(\reuc)=\krate(\reff)\left(\frac{\reuc}{\reff}\right)^q,
\qquad q\in\mathbb{R},

used only on a near-Euclidean window where no additional radial scale beyond $\reuc$TeX\reuc enters the shell organizer. If an admitted residual requires an additional radial scale, a survey-dependent break, or a window-edge turnover, reference is not used. Any downstream shell-invariance or compatibility statement is read only within this declared optional single-scale class on the admitted window.

definition: Optional shell organizer in the declared class. Adopt an auxiliary residual-shell parametrization on a gate-passing near-Euclidean window in which one shared factor $\krate(\reuc)$TeX\krate(\reuc) acts on the three shell directions together with the observer-time rate factor. The corresponding shell organizer is

\Irate(\reuc)\equiv \frac{\krate(\reuc)^4}{(\reuc/\reff)^3}.

This declared residual-shell fit convention is auxiliary, and all downstream residual-shell fits and compatibility claims are read only relative to it on the admitted window.

This definition specifies an optional single-scale organizer for residual log-shell departures and is not itself an independently inferred population law. It is introduced only after the empirical gates of reference are passed and is not promoted to a mandatory correction. Residual classes requiring an additional radial scale, a survey-dependent break, or a window-edge turnover lie outside the declared analysis.

corollary: Conditional compatibility exponent under shell invariance in the optional single-scale class. Assume reference and the scale-free class reference. If, within that declared optional single-scale class, the shell organizer $\Irate$TeX\Irate is additionally required to be invariant across the declared window, then

\boxed{q=\frac{3}{4}}.

proof. Substituting reference into reference gives

\Irate(\reuc)=\krate(\reff)^4\left(\frac{\reuc}{\reff}\right)^{4q-3}.

Shell invariance requires the exponent of $\reuc/\reff$TeX\reuc/\reff to vanish, hence $4q-3=0$TeX4q-3=0 and therefore $q=3/4$TeXq=3/4.

The null case corresponds to no additional shell-rate factor over the gate-passing window. Even when a nonzero profile is entertained, $q=3/4$TeXq=3/4 is only a conditional compatibility exponent within the scale-free class and does not follow from source-count data alone.

A nonzero shell-rate reading is not admitted unless the same dataset first passes the standard source-count gates. The order is fixed: homogeneous-survey anchoring first, localized low-redshift control second, cross-survey transport only thereafter.

G1: homogeneous-survey anchor..

Inference begins from a single survey with explicit exposure, sensitivity, and sky-coverage information over the adopted window. In the present application, the second CHIME/FRB catalogue provides that anchor [citation]. Survey-internal anchoring is preferred to naive cross-survey transport at this stage because CHIME-like flux and fluence measurements remain beam- and localization-dependent [citation]. If no such anchor exists for the adopted window, the analysis is not transported beyond descriptive count summary.

G2: completeness and beam-threshold control..

Source-count slopes are meaningful only relative to a declared completeness threshold and a declared detection statistic. Threshold handling and sample definition materially affect the inferred slope [citation], and recent localized-survey analyses still report explicit fluence completeness limits before estimating all-sky rates or count scaling [citation]. If the adopted survey window does not have a declared and controlled completeness threshold, no shell-rate interpretation is admitted.

G3: near-Euclidean budget and low-redshift evolution control..

The near-Euclidean null law is opened only when the declared budget closes,

\epswin=\max\{\epsgeom,\epsspec,\epsevo\}\le \epsilon_*.

Here $\epsevo$TeX\epsevo is supplied either by external localized-redshift information or by a separately specified low-$z$TeXz population model [citation]. When a low-$z$TeXz population model is used to bound $\epsevo$TeX\epsevo, that model is fixed independently of the source-count residual being interpreted and declared before shell-rate inference on the adopted campaign. If no such control exists, or if the resulting $\epswin$TeX\epswin exceeds the campaign budget $\epsilon_*$TeX\epsilon_*, the null-law window is not opened and no optional shell-rate reading is admitted.

G4: spectral and bandpass control..

The mapping from intrinsic burst energy to observed fluence is frequency- and band-dependent. Cross-survey comparisons therefore require explicit control of $\Kcorr$TeX\Kcorr, bandpass, width smearing, and threshold statistics [citation]. If an apparent slope change is driven by unmodelled bandpass or pulse-width selection rather than by a genuine deformation of the count law, the shell-rate reading is rejected.

G5: repetition and exposure weighting..

FRB populations include repeaters, clustered arrival times, and activity-window effects. Far-sidelobe constraints, exposure-corrected source-count evolution, energy-distribution comparisons between repeating and apparently non-repeating populations, and hidden-repeaters analyses all indicate that the observed repeating fraction can underestimate the underlying repeating population [citation]. If the declared sample fails this repetition and exposure control, observer-time counts are not interpreted as a simple single-event source-count law.

G6: host-selection and scattering control..

Rate inference can be biased by host-dependent detectability. Empirical evidence for an inclination-related host-selection bias, plausibly linked to host scattering, shows that published FRB rates can be suppressed if those effects are not controlled [citation]; complementary radio-selected host-candidate studies also indicate that dusty or optically faint hosts can escape purely optical association strategies [citation]. If the adopted window is likely dominated by host-selection or host-scattering suppression, any apparent count-law deformation is assigned to that effect before a shell-rate reading is considered.

G7: cross-survey consistency..

Parkes, ASKAP, CHIME, MeerTRAP, and related surveys do not probe identical fluence, bandwidth, and redshift windows. A nonzero shell-rate reading is admitted only if the same near-Euclidean window survives a cross-survey consistency check after survey anchoring, completeness, spectral control, repetition weighting, and host-selection control are applied [citation]. No fitted shell organizer is transported across surveys unless the same admitted near-Euclidean window and gate stack are shown to hold jointly.

G8: Propagation-side local admissibility..

The only imported CHC correction is the propagation-side map, and a nonzero residual shell profile is not by itself a propagation claim. If one further reinterprets that residual in propagation-side terms, the mapped correction must satisfy the imported local-admissibility conditions; otherwise the propagation-side reading is rejected [citation].

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FRB source-count inference on a low-redshift near-Euclidean window is controlled first by the standard $-3/2$TeX-3/2 null law, homogeneous-survey anchoring, completeness, spectral response, localized low-$z$TeXz evolution control, repetition and exposure weighting, host-selection and scattering control, cross-survey consistency, and propagation-side local admissibility. Within such a window, the corrected Euclidean normalization is given by reference, the logarithmic-shell observer-time rate density is well defined through reference and reference, and any residual deformation may be organized by the optional shell factor of reference. All downstream residual-shell comparisons are read only relative to the adopted homogeneous-survey benchmark frame, the declared near-Euclidean window and its budget $\epswin$TeX\epswin, and the declared residual-shell fit convention of reference. For the fixed first-pass survey window adopted here, with $\ze=0.05$TeX\ze=0.05 and declared budget closure $\epswin=0.10\le \epsilon_*=0.15$TeX\epswin=0.10\le \epsilon_*=0.15, the count-law closure stops at the gate-admitted survey-window source-count law. Equation reference is retained only as an optional secondary identity, and the downstream shell organizer is not part of that present closure.

The companion CHIME/FRB Catalog 1 gates add bounded selection-side materialization to this restricted reading. The exact VP0/VP1/VP2 labels are recorded in reference; they apply respectively to the public-catalog window, the exposure/injection-assisted window, and the VP2 injection-selection response-surface stress window. These results supply bounded support for the selection-control layer only and do not change the conclusion that no global population law or propagation-correction inference is assigned here.

The optional organizer is an auxiliary residual-shell parametrization of residual log-shell departures. A nonzero shell-rate factor is admissible only after the empirical gates are satisfied on one declared campaign, and $q=3/4$TeXq=3/4 arises only as a conditional compatibility exponent inside the optional single-scale class. The count analysis does not independently determine that exponent and is not used here to infer propagation closure outside the declared FRB count window. Any fitted organizer is survey- and window-specific unless joint admission of the same near-Euclidean window and gate stack is demonstrated across surveys. Any accepted shell-rate residual remains a survey-window-specific organizer of source-count departures and is not, by itself, evidence for horizon-scale calibration relations or a substitute for independent distance analyses. Outside the admitted residual-shell case, the analysis closes at the standard source-count law on the declared window.

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Funding and competing interests..

No external funding was received for this work. The author declares no competing interests.