Replay a conditional JADES volume for three real sources

Pin the release and original target class

Start with version two of the Curtis-Lake and colleagues DR4 selection paper: the area in Section 2.4, target classes in Section 3.3 and the volume/success treatment in Section 4.5. Use the frozen Combined_DR4_external_v1.2.1.fits catalog, GOODS-N mediumjwst original priority 1a, excluding promoted oddballs. NIRSpec’s microshutter array (MSA) opens selected tiny shutters to admit target light for spectroscopy. Allocation records who actually received a spectrum.

The small public CSV retains twelve original IDs, spectral flags, redshifts, spectrum-derived UV magnitudes, allocation probabilities and both gold flags. The roughly ninety-four-megabyte full FITS URL and SHA256 allow an optional external cross-check. Running the calculation needs only the small extract.

Reading: Curtis-Lake et al. · JADES DR4 · Sandbox Physics

Account for all twelve observations

Thirteen candidates belonged to this original class and twelve were observed. Seven have robust A, B or C redshifts above nine; four have robust redshifts outside the target range; one has no robust redshift. Its E flag remains unknown. A photometric redshift is not substituted for spectral confirmation.

Declare a bright slice: redshift greater than nine and at most 10.5, with spectrum-derived UV magnitude from minus twenty-two up to, but excluding, minus 19.75. Three actual sources qualify, listed below. This exploratory slice was chosen to lie entirely inside the subsequent model flux cut after inspecting the catalog; it is not a blind hypothesis test.

Original source ID suffix	Spectral flag	Redshift	Spectral UV magnitude
619	A	9.06815	−20.0002
19715	A	9.30259	−19.9739
55757	A	9.74211	−19.7987

Reading: JADES DR4 · Sandbox Physics

Calculate geometry from area and distance

Table 2 gives a usable MSA union area of 25.886 square arcminutes, of which 93.5 percent had NIRCam coverage at original selection. Their product is 24.20341 square arcminutes. A union avoids counting overlaps between pointings more than once.

Adopt the paper’s flat cosmology: Hubble constant 67.66, matter fraction 0.310 and no radiation. The earlier synthetic image example used seventy and 0.3. Each is explicit; every distance and volume in this replay uses the former.

\Omega=24.20341\left(\frac{\pi}{180\times60}\right)^2\,\mathrm{sr},\qquad V_{\rm geom}=\frac{\Omega}{3}\left[D_C(10.5)^3-D_C(9)^3\right]

Comoving distances in megaparsecs give about 63905.614 cubic megaparsecs. The checker also directly integrates the volume element and compares Astropy at identical parameters.

Reading: Curtis-Lake et al. · Hogg (1999) · Sandbox Physics

Check how far each luminosity can enter the model

An apparent-brightness cut can shorten a source’s contributing redshift range. Specify a flat frequency spectrum, insert its spectrum-derived UV magnitude and solve where it reaches AB 27.9. The smaller of that luminosity horizon and the bin’s upper redshift determines the usable range.

m(z)=M_{\rm UV}+5\log_{10}\!\left(\frac{D_L(z)}{\mathrm{Mpc}}\right)+25-2.5\log_{10}(1+z)

Solve this declared flat-spectrum relation at apparent magnitude 27.9. Keep spectral UV luminosity distinct from original selection-aperture photometry.

Among all seven original successes	Spectral-model apparent magnitude	Relative to 27.9 cut
17858	27.9948	Fainter; fails
59720	27.9008	Fainter; fails
80088	28.4677	Fainter; fails

Reading: Curtis-Lake et al. · JADES DR4 · Sandbox Physics

A spectral luminosity can cross the original cut

These three sources are genuine original-class successes yet exceed the original cut under this spectrum-based model. Investigate which aperture supplied the original selected flux and which data normalized the spectrum. Their values remain in the result; the replay does not shift magnitudes or drop inconvenient rows.

All three sources in the separately declared bright slice satisfy the model cut, as does its faint edge at redshift 10.5. Each can therefore use the same geometric redshift-bin volume. That check solves the luminosity boundary within the specified spectral model, without reconstructing original imaging-selection probabilities.

Reading: JADES DR4 · Sandbox Physics

State two assumptions about the unknown redshift

Assume allocation and spectral success are exchangeable between the full original class and its bright subset. The subset inherits the class-average allocation probability and the class ratio of successful to unknown genuine high-redshift targets. This condition allows the class counts to be applied to the bright slice.

If the unknown target is truly outside the high-redshift range, the conditional spectral-success factor is one. Multiply geometric volume by twelve thirteenths. If the unknown is in range, seven successes correspond to eight genuine high-redshift observed targets, adding a factor of seven eighths.

V_1=V_{\rm geom}\frac{12}{13},\qquad V_2=V_{\rm geom}\frac{12}{13}\frac78

The volumes are about 58989.797 and 51616.073 cubic megaparsecs. They are two specified unknown-target scenarios, not a confidence interval or bounds enclosing all selection systematics.

n_1=\frac{3}{V_1}\simeq5.09\times10^{-5}\,\mathrm{Mpc}^{-3},\qquad n_2=\frac{3}{V_2}\simeq5.81\times10^{-5}\,\mathrm{Mpc}^{-3}

These are conditional class densities for the bright slice, with no per-magnitude unit. No arbitrary magnitude width is added to rename them a published luminosity-function point.

Reading: Curtis-Lake et al. · Sandbox Physics

Retain the conditions on pointing and candidate selection

The MSA pointings were optimized to include rare high-redshift targets. Their locations can favour target-rich regions. Even perfect allocation and redshift-success corrections would not make this conditional class density a random cosmic sample. This limitation goes beyond three-source Poisson fluctuations or ordinary field-to-field variance.

Parent imaging detection, visual inspection, photometric-redshift selection and luminosity-dependent spectral success remain uncalibrated by this package. A more general population result must model those steps and the pointing optimization. The preceding four-band simulation supplies probabilities only for its controlled experiment.

Old and updated gold flags are retained to expose membership changes. A sample selected from updated photometry cannot automatically inherit original-class allocation statistics. Current class counts also give twelve thirteenths and twenty-one twenty-sixths, which do not reproduce the two printed Table 10 factors. The limited replay and published aggregate are identified separately; no data adjustment or claim of an arithmetic error resolves that discrepancy.

Reading: Curtis-Lake et al. · Sandbox Physics

Run from twelve fact rows to the result

Download the conditional-volume package and inspect its README and conditions.json: release, area, boundaries, assumptions and permission. Read the twelve CSV rows, then run replay.py after extraction. It reclassifies the sources, solves seven luminosity horizons and returns volumes and the three selected IDs while retaining all mismatches and checks.

An actual fresh-directory cold run passed sixty-two checks. The original result has a separate forty-seven-gate acceptance; the public package adds rejection cases for duplicate IDs, bad flags, wrong classes and nonfinite data. MIT covers original programs only. Catalog and table facts retain JADES and author attribution.

Reading: Sandbox Physics · JADES DR4

Try the observation

Compare simulated probabilities with original observing conditions

Read the fixed synthetic survey’s redshift range, area and size.
Open the real package’s conditions.json and identify its different area, cosmology and target class.
Distinguish known-profile noise recovery from original-class allocation and spectral success.

The Lab supplies controlled pixels and a fixed synthetic survey. A separate package recomputes the real sample’s conditional volume with its own declared inputs.

Exercises and explained answers

Keep only two successes and also reduce area to two thirds. Does the unchanged ratio reproduce the original three-source result?

No. Count and area are distinct declared inputs. Verify the three original IDs, class and area before computing a density.

What changes if the E-flag object is directly counted as a high-redshift success?

It turns an unknown into a claimed measurement and changes successes/counts. The two conditional volumes vary an assumption about its true range; the three confirmed bright-slice sources remain three.

Why does an accepted synthetic injection test not supply the real survey’s missing selection probabilities?

It uses a prescribed spectrum, shape, noise and supplied position. Original observations also involve different images, visual candidate inspection, photometric redshifts and optimized pointings. Their probabilities must be evaluated in their own procedure.

Primary sources and calculation notes

Hogg (1999), Distance measures in cosmology
Luminosity-distance section: bolometric flux, angular/luminosity distance and spectral-density transformations. The numerical examples use the explicitly adopted radiation-free model.
Curtis-Lake et al., JADES DR4 Paper I, v2
Version dated 7 May 2026. Target-read Sections 2.4/3.3/4.5/5, Tables 2/4/10 and equations 1–3. Retain original classes, optimized pointings and the distinction between original selection and updated gold samples.
JADES DR4, twelve original-class fact rows
Small attributed numerical extract from the exact Combined_DR4_external_v1.2.1.fits Obs_info audit. Original catalog URL/SHA, table conditions and source terms are inside the conditional replay package. No full FITS redistribution.
Sandbox Physics, conditional JADES volume cold replay
Twelve fact rows, attributed Table 2/4 conditions, original volume functions, all seven horizon checks and retained model-cut/Table 10 mismatches. Actual cold replay 62 checks; conditional target-class scope, not cosmic-average LF.