FRC-KP1: A Catalog1-Frozen CHIME/FRB Catalog2 Public Holdout Prediction Test for CHC-FRC

The purpose of FRC--KP1 is to test holdout transfer rather than retrospective fit. The preceding FRC public-data companions establish bounded CHIME/FRB Catalog 1 selection, exposure, and injection-selection stress surfaces. In particular, FRC--VP2 states a public Catalog 1 injection-selection response surface using Catalog 1, the live-injection data set, the full synthetic-injection HDF5 data set, and CHIME/FRB exposure maps. FRC--KP1 asks a different question: quote Can a scalar phase-load / finite-access residual geometry frozen on Catalog 1 and the public VP2 response surface support the residual organization of a non-overlap Catalog 2 holdout without refitting the residual geometry? quote This companion note records the bounded result of that test.

The allowed public reading is strictly bounded: quote Under the declared Catalog1-frozen scoring convention, the non-overlap CHIME/FRB Catalog2 holdout satisfies the stated rank-one-collapse and phase-load-alignment diagnostics. This is a bounded holdout-support result, not a global FRB population law, luminosity-function closure, or propagation-correction detection. quote The following readings are explicitly forbidden:

- full CHIME/FRB selection-function closure; - global FRB population law; - luminosity-function closure; - dispersion-measure-to-redshift theorem; - repeater/non-repeater population theorem; - propagation-correction detection; - parent FRC theorem closure; - CHC-wide empirical proof.

FRC--KP1 is not named FRC--VP3 because it is not simply another support companion in the same selection-response chain. The intended lineage is

\text{FRC-VP0/VP1/VP2 selection-response support}

\quad\longrightarrow\quad
\text{FRC-KP1 holdout-support test}.

The VP companion records define and stress the public selection/exposure/injection surfaces. KP1 uses those surfaces to freeze a holdout support object and then tests it on a later public holdout.

The existing VP2 stress result is not erased by KP1: the Catalog 1 injection-selection surface is nonuniform, so a selection-free count-law interpretation is inadmissible. KP1 adds a separate held-out predictive test, whose negative proper-score result constrains the parent residual-shell construction.

The public input surfaces are listed in Table reference. Catalog 1 and the injection-selection response surface are inherited from the FRC--VP2 declared public-source summary. Catalog 2 is taken from the public CHIME/FRB storage objects made visible through the CHIME/FRB public database, direct Catalog 2 storage route, and Catalog 2 paper surface [citation].

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The non-overlap holdout rule is conservative. Catalog 2 rows are excluded if public Catalog 2 flags identify them as Catalog-1-origin or Catalog-1-parameter rows, and they are also excluded if their tns_name or previous_name appears in Catalog 1. A unit convention note is essential: the public Catalog 2 CSV has 5,045 rows because some FRB names appear in more than one table row, while the official Catalog 2 count of 4,539 FRBs is recovered by counting distinct tns_name values in the same CSV. KP1 therefore uses ``row'' only for rows of the public machine table and uses tns_name when referring to the official FRB event-name unit. The primary scoring pipeline is row-level after the declared exclusions, positive-fluence requirement, non-repeater policy, and VP2 binning; no claim is made that 5,045 is the official FRB event count. The resulting holdout accounting is: center

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The frozen object is a Catalog1-derived scalar phase-load vector. Events are filtered by the declared population-style policy: public excluded rows are removed, positive fluence is required, and the primary board uses non-repeaters. Events are binned by the FRC--VP2 public fluence and dispersion-measure proxy bins. Excess dispersion measure is used only as a proxy coordinate; it is not inverted to redshift.

For each cell $b$TeXb, define

N_b = \text{event count},

W_b = \text{selection/exposure weighted count},

E_b = \text{effective exposure/selection proxy},

F_b = \text{representative fluence coordinate},

D_b = \text{dispersion-measure proxy coordinate}.

A fixed cumulative near-Euclidean shape is used as the secondary count-law organizer,

S_b = E_b\left(\frac{F_b}{\mathrm{median}(F)}\right)^{-3/2}.

Only a single amplitude is fit on Catalog 1:

A_{\CatOne}=\frac{\sum_b W_b S_b}{\sum_b S_b^2}.

The Catalog1 standardized residual is

r_b^{\CatOne}=\frac{W_b-A_{\CatOne}S_b}{\sqrt{\max(A_{\CatOne}S_b,1)}}.

The frozen phase-load vector is the mean-centered unit vector

u_b^{\CatOne}=\frac{r_b^{\CatOne}-\langle r^{\CatOne}\rangle}{\left\|r^{\CatOne}-\langle r^{\CatOne}\rangle\right\|_2}.

No Catalog 2 residual, sign choice, domain tuning, redshift inversion, or luminosity-function reconstruction is allowed in the freeze step.

The primary Catalog 1 freeze board has 464 prepared non-repeater table rows, corresponding to 434 distinct tns_name identifiers after the same population-style filters, and 31 cells. The frozen amplitude is

A_{\CatOne}=5.536031002639719,

and the phase-load vector has unit norm.

Catalog 2 holdout cells are scored by matching the Catalog1-frozen phase-load cells to the Catalog2 non-overlap holdout cells. The primary prepared holdout contains 2,676 non-repeater table rows, corresponding to 2,534 distinct tns_name identifiers, and 33 cells; 31 cells overlap the Catalog 1 freeze board.

For each matched Catalog-2 cell $b$TeXb, let $W_b^{\CatTwo}$TeXW_b^{\CatTwo} be the holdout weighted count and let $S_b^{\CatTwo}$TeXS_b^{\CatTwo} be the count-law organizer computed with the same frozen response, fluence-normalization, and cell-construction convention used in the Catalog-1 freeze board. No Catalog-2 amplitude is fitted. The Catalog-2 standardized residual is

z_b^{\CatTwo}
=
\frac{W_b^{\CatTwo}-A_{\CatOne}S_b^{\CatTwo}}
{\sqrt{\max(A_{\CatOne}S_b^{\CatTwo},1)}} .

The primary alignment metric is the correlation between $z_b^{\CatTwo}$TeXz_b^{\CatTwo} and $u_b^{\CatOne}$TeXu_b^{\CatOne}. The residual matrix is arranged over fluence and dispersion-measure proxy bins:

M_{ij}^{\CatTwo}=z^{\CatTwo}(\mathrm{fluence\ bin}=i,\mathrm{DM\ proxy\ bin}=j).

Rank-one collapse is measured by

\rho_{\mathrm{rank1}}=\frac{\sigma_2(M)}{\sigma_1(M)}.

Additional primary metrics are sign agreement, Spearman rank preservation, and a shuffled-load permutation null.

The declared strong-primary thresholds are reporting/classification gates used by the KP1 scoring procedure and recorded in the supplementary ranking summary. They should not be read as an independently published universal statistical standard, and the result does not rest on threshold crossing alone: the continuous metric values, shuffled-load null, bootstrap interval, jackknife record, and stress variants are reported so that the evidence can be read threshold-free as well. The Catalog2-blind freeze record retains the frozen metric list, random seed, forbidden operations, and Catalog1-only freeze surface; the threshold table below is the deterministic classification layer attached to that scoring rule.

The declared strong-primary thresholds are: center

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center

The primary holdout result is given in Table reference. All primary phase-load / rank-collapse metrics pass the strong-primary reporting gates. Because those gates are not treated as an external discovery convention, the interpretation remains bounded and is tied to the continuous scores and robustness checks rather than to a binary threshold label alone.

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The decisive predictive count-law comparison is the holdout log-predictive density. Its scores are center

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proposition: Predictive-density rejection of the frozen CHC count transfer. On the common Catalog 2 holdout and the fixed scoring convention displayed above, the frozen CHC count-transfer model is rejected relative to both comparators: its LPD differences are

\Delta\mathrm{LPD}_{\rm CHC-exp}=-340.887183885731,
\qquad
\Delta\mathrm{LPD}_{\rm CHC-PL}=-835.253487802791.

The rank-one and correlation diagnostics cannot reverse this predictive verdict because they are not proper scores for the held-out count distribution.

proof. All three LPDs are evaluated on the same holdout, and a larger log predictive density is preferred. Direct subtraction gives $-1369.5517979448648-(-1028.6646140591338)=-340.8871838857310$TeX-1369.5517979448648-(-1028.6646140591338)=-340.8871838857310 and $-1369.5517979448648-(-534.2983101420742)=-835.2534878027906$TeX-1369.5517979448648-(-534.2983101420742)=-835.2534878027906. Both differences are negative. The other diagnostics measure matrix shape or association after the chosen residual construction and are not alternative likelihoods for the held-out counts; they therefore describe residual geometry but do not establish predictive count performance.

The permutation value is computed as

p_{\mathrm{perm}}=\frac{1+\#\{|r_{\mathrm{perm}}|\ge |r_{\mathrm{obs}}|\}}{5000+1},

where the null shuffles the Catalog1-frozen phase-load values over the scored overlap cells. This is an absolute/two-sided cell-label shuffle diagnostic for alignment, not a generative null for the CHIME selection function. It preserves the observed Catalog2 residual vector and cell occupancy pattern but assumes exchangeability of the frozen load labels across the matched cells. Consequently the small value is exploratory and cannot substitute for the held-out predictive density.

Therefore the frozen count-transfer prediction fails its proper-score comparison. The residual-geometry calculation is retained as a descriptive stress diagnostic. The primary label is

\boxed{\code{FRC-KP1-PREDICTIVE-COUNT-TRANSFER-REJECTED}}.

Association diagnostics do not supply predictive support for this result.

A separate robustness pass tests the stability of the residual-geometry diagnostics under changes in event policy, exposure policy, binning, bootstrap resampling, and one-cell jackknife removal. Its diagnostic label is

\boxed{\code{FRC-KP1-RESIDUAL-GEOMETRY}
\code{ROBUSTNESS-DESCRIPTIVE}}.

The decision reasons are:

- core stress count: 0; - support-or-strong variants: 9/12; - strong variants: 7/12; - lower-only exposure stress caps confidence; - high-fluence-only subset remains stress; - overcoarsened six-cell binning stress is recorded but non-primary; - bootstrap alignment lower tail is nonfatal.

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The bootstrap stability record is: center

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center The one-cell jackknife result is 33/33 support-or-strong and 0/33 stress.

After the robustness pass, two additional bounded auxiliary stress checks were evaluated as diagnostics. These analyses do not change the primary hypothesis, do not refit the Catalog2 holdout, and do not introduce a second prediction claim. Their purpose is narrower: to test whether the same Catalog1-frozen CHC--FRC phase-load direction is specific relative to non-CHC controls and stable across declared holdout slices.

The specificity analysis is an adversarial-null board. It compares the target CHC--FRC frozen phase-load direction against physical, policy, random-orthogonal, cell-permuted, sign-shuffled, and bootstrap Catalog1 load families under the same Catalog2 matched-cell scoring surface. The target direction, denoted D0, ranks first on all declared primary specificity metrics: absolute Pearson alignment, absolute Spearman alignment, best-orientation sign agreement, and absolute projection. It also beats the declared random-orthogonal, cell-permuted, and sign-shuffled adversarial families at their family thresholds. The analysis label is

\boxed{\code{FRC-KP1-SPEC-ADVERSARIAL-NULL-COMPATIBLE}}.

One policy decoy was blocked because it became constant or zero-norm under the frozen board; this is recorded as a blocked control rather than as supporting evidence.

The multi-slice analysis tests whether the primary Catalog2 alignment is concentrated in only one convenient region of the holdout. The scored slices cover the declared time, exposure, fluence, and dispersion-measure axes. Of 11 scored slices, 10 have the positive declared direction and 9 are classified strong. All four declared axes have positive slice counts. The analysis label is

\boxed{\code{FRC-KP1-MULTISLICE-STRESS-COMPATIBLE}}.

The global slice reproduces the primary board values: phase-load correlation 0.889326696895743, fixed sign agreement 0.9032258064516129, Spearman correlation 0.6713709677419356, and exposure-scale transfer 0.7940469731995563.

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SPEC and MULTISLICE show that the constructed residual direction is not confined to one convenient slice under the stated diagnostics. They do not remove the predictive-density rejection and do not imply a CHIME selection-function result, a luminosity-function result, a global source-count law, or CHC empirical confirmation.

The Catalog2 non-overlap surface was not used to choose the Catalog1 residual direction, and the row/event-name convention is explicitly separated from the official Catalog 2 count. The resulting alignment is a reproducible descriptive feature of the constructed residual matrices. It is not evidence that the frozen CHC model predicts held-out counts, because the common-data predictive score strongly favors both comparison models.

The result is also limited. The secondary count-law transfer does not beat simple count baselines. The high-fluence-only subset weakens the rank-one collapse. Lower-only exposure and extreme overcoarsening produce stress diagnostics. These limitations are part of the reported stress profile; the resulting classification is robust bounded support with stress limitations rather than strong support.

FRC--KP1 therefore supplies a negative predictive test of the frozen count-transfer model together with exploratory residual-geometry diagnostics. It does not support a universal FRB theory or a CHC propagation correction.

The companion source summaries identify the primary holdout result, the robustness summary, and the public-source boundary records without relying on any local filesystem root. These records support the bounded Catalog2 holdout reading and do not introduce a second prediction claim. The companion source summaries are:

- the primary Catalog2 holdout support summary, which fixes the public Catalog1-derived geometry, the Catalog2 holdout surface, the scoring rule, and the reported prediction-layer summary; - the robustness-family support summary, which reports the declared perturbation families and the associated stability summaries.

The SPEC and MULTISLICE support-check records are summarized as companion source summaries:

- SPEC adversarial-null support board; - MULTISLICE stress-panel support board.

The companion source summary identifies the public-data source surfaces, freeze records, scoring summaries, result tables, public-source summaries, and bounded stress checks used for the bounded holdout reading.

The Catalog 2 prediction is the appropriate place to evaluate the compatibility manifold fixed by Catalog 1. Once the source-count and selection parameters are frozen, the combined held-out observable vector must lie, within declared uncertainty, in the image of the lower-dimensional response map. At first order this is equivalent to testing the left-null residuals of the Catalog 1 sensitivity matrix.

Re-estimating a parameter from Catalog 2 changes the manifold and invalidates the holdout interpretation. Likewise, adding one nuisance coefficient per new bin can make the image locally open and remove all overidentifying restrictions. The strongest result of this protocol is therefore not agreement of a refitted slope but survival of predeclared null combinations across the catalog boundary with the original parameter count, covariance model, and selection surface unchanged.

The closure test for the two-catalog predictive holdout is applied to a dimensionless observable vector $y\in\mathbb R^m$TeXy\in\mathbb R^m formed from fixed reference scales and the declared basket of Catalog 2 counts and distributional contrasts. Let $a$TeXa range over the independent constitutive inputs comprising Catalog 1 population and selection fit, transfer nuisance, and Catalog 2 observation map.

proposition: Functional saturation, finite closure, and sector admissibility. Suppose the unrestricted prediction map $F:a\mapsto y$TeXF:a\mapsto y is continuously differentiable on a Banach space of constitutive inputs. If $D_aF$TeXD_aF is surjective and has a bounded right inverse at the calibration point, the unrestricted family is locally open in observable space and supplies no nonzero local equality restriction on $y$TeXy. Suppose instead that a single microscopic closure replaces $a$TeXa by finite parameters $\theta\in\mathbb R^p$TeX\theta\in\mathbb R^p, with profiled nuisance coordinates $\eta\in\mathbb R^q$TeX\eta\in\mathbb R^q. If

J=D_\eta F_{\rm cl}D_\theta F_{\rm cl},
 \qquad \operatorname{rank}J=r<m,

then there are $m-r$TeXm-r independent first-order restrictions

w^{\mathsf T}\delta y=0,
 \qquad w\in\ker J^{\mathsf T}.

If the rank is constant locally, these restrictions are tangent to a compatibility manifold of codimension $m-r$TeXm-r. For this sector, the finite closure is admissible only if Catalog 1 alone freezes the finite parameter vector and Catalog 2 is evaluated only through covariance-weighted transverse residuals.

proof. Split surjectivity gives a bounded right inverse $R$TeXR with $D_aF\,R=I_m$TeXD_aF\,R=I_m. The Banach-space submersion theorem then makes $F$TeXF locally onto a neighborhood of the calibrated observable vector. Any smooth equality holding throughout that image must therefore vanish on an open set and contributes no model-specific local restriction. Under finite closure, the attainable first-order variations are exactly the column space of $J$TeXJ. Its orthogonal complement is $\ker J^{\mathsf T}$TeX\ker J^{\mathsf T}, whose dimension is $m-r$TeXm-r by rank--nullity, which proves reference. The constant-rank theorem supplies the stated local manifold. The sector condition is necessary because any Catalog 2 refit moves the test point along a newly enlarged tangent space and destroys the holdout. Failure of that condition therefore rejects the proposed microscopic closure before parameter estimation can be counted as evidence for it.

FRC--KP1 is a completed public-data holdout test. The Catalog1-frozen residual direction retains rank-one and alignment features under several diagnostic perturbations, but the held-out count prediction performs substantially worse than both exposure-only and free-power-law comparators. Because log predictive density evaluates the predictive distribution on the common holdout, this rejects the frozen CHC count-transfer model under the declared protocol. The final label is

\boxed{\code{FRC-KP1-PREDICTIVE-COUNT}
\code{TRANSFER-REJECTED}}.

The residual-geometry diagnostics remain available for method development but do not constitute predictive validation.

Data and code availability..

This companion manuscript uses public CHIME/FRB data products and reproducibility materials as described in the text. Cited public references and companion statements, where provided, are cited only as support summaries for the declared holdout calculation. FRC--KP1 is a bounded public predictive-holdout companion and is not promoted to a full CHIME selection-function analysis, global FRB population law, luminosity-function closure, DM--redshift theorem, propagation-correction detection, or CHC-wide empirical closure.

Funding and competing interests..

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