CMB-VP0: Primary-Sector Linked-Ruler Public Gates in CHC

The CMB paper declares a restricted primary-sector object: a last-accessibility width envelope, a fixed-kernel acoustic-angle comparison, a linked-ruler relation between the recombination ruler and the drag-era ruler, and an external later-transfer consistency layer. The present companion note implements those declared gates on public source surfaces. It does not infer a new background expansion, does not re-run a full CMB likelihood, and does not construct a complete polarization or lensing transfer theory.

The public result is intentionally named a primary diagnostic partial. The word ``partial'' marks that the companion record evaluates the declared public surfaces and declared algebraic gates without attempting a full CMB parameter fit.

The companion record uses public source surfaces from Planck PR4/NPIPE, ACT DR6, and DESI DR2. Planck PR4 is the public NPIPE reprocessing of LFI and HFI data using a common pipeline, with public full-sky maps, half-ring maps, low-resolution maps, effective beams, and single-channel maps [citation]. The Planck 2018 parameter paper supplies the acoustic-angle benchmark comparison used here, reporting the angular acoustic scale at high precision [citation]. ACT DR6.02 provides maps, beams, passbands, NILC products, PSPIPE products, likelihood-related material, and public MCMC chains [citation]. DESI DR2 provides cosmology chains, posterior maximization products, and BAO supplementary data products, including a Zenodo archive for reproducing numerical results in the DR2 BAO paper [citation].

The evidence record identifies the required public surfaces and their public-source basis. It also identifies the DESI DR2 BAO and ACT DR6 LCDM source surfaces through cited public references and companion statements. Large public inputs are represented in the manuscript only by source name, diagnostic role, and provenance statement.

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The ledger is a public-source basis board. It is not a claim that the companion record performs a full CMB parameter inference. Public-source summaries are used only to identify source surfaces and result summaries.

The companion record follows five bounded steps.

- Public-source construction: read the Planck PR4/NPIPE, ACT DR6.02, DESI DR2 release, and DESI DR2 BAO supplementary surfaces named in reference. - Public-source check: record the declared public-source basis and large-source resource status where applicable. - Gate computation: evaluate the width, acoustic-angle, linked-ruler, and later-transfer diagnostic rules from the declared gate configuration. - Classification assignment: assign only the four declared sub-gate labels and the combined CMB-VP0-PRIMARY-DIAGNOSTIC-PARTIAL classification. - Non-claim confirmation: verify that the result remains a restricted primary-sector diagnostic and does not become a likelihood closure, Boltzmann replacement, drag-transfer inference, late-transfer theorem, or DE/LBD/DM closure.

G2: width gate For a representative peak window, retained oscillatory fraction one half, acoustic peak scale $\ell_{\rm peak}=2500$TeX\ell_{\rm peak}=2500, and $\bar c\in[0.45,0.47]$TeX\bar c\in[0.45,0.47], the companion record evaluates the temperature-sector width envelope. The comparison is

\theta_* = 0.010411,\qquad
\sigma_*/r_* \in [0.0962492636,0.1005270086].

This lies within the declared expected range $[0.096,0.100]$TeX[0.096,0.100] under tolerance $0.005$TeX0.005. The assigned label is

\mathrm{CMB\text{-}WIDTH\text{-}GATE\text{-}SATISFIED}.

G3: acoustic-angle gate The acoustic-angle gate uses the benchmark comparison

100\theta_* = 1.0411,\qquad \sigma(100\theta_*)=0.0003,

with $\alpha_*=0.0499$TeX\alpha_*=0.0499 and declared $|\epsH|\le 0.012$TeX|\epsH|\le 0.012. The assigned label is

\mathrm{CMB\text{-}ACOUSTIC\text{-}ANGLE\text{-}GATE\text{-}SATISFIED}.

This is a declared benchmark comparison, not a full CMB likelihood evaluation.

G4: linked-ruler gate The linked-ruler gate evaluates the fixed-kernel relation between the recombination ruler and the drag-era ruler. The companion record uses

\alpha_*=0.0499,
\qquad
\alpha_{\rm d}=0.0951,
\qquad
|\epsH|\le 0.012,

and returns

|\delta\ln(r_{\rm d}/r_*)|\le 2.712\times10^{-4},

against a declared expected bound $2.7\times10^{-4}$TeX2.7\times10^{-4}. The assigned label is

\mathrm{CMB\text{-}LINKED\text{-}RULER\text{-}GATE\text{-}SATISFIED}.

This is not a complete drag-transfer inference.

G5: later-transfer diagnostic The later-transfer gate verifies the presence of declared external consistency surfaces. The companion record lists Planck PR4 and ACT DR6 landing surfaces as required-present surfaces, and lists ACT lensing derived products and the ACT DR6 chain readme as optional later-transfer source bases. The assigned label is

\mathrm{CMB\text{-}LATER\text{-}TRANSFER\text{-}DIAGNOSTIC\text{-}CHECK\text{-}SATISFIED}.

This is a source-basis diagnostic. It is not a late-transfer theorem.

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The combined classification is

\boxed{\mathrm{CMB\text{-}VP0\text{-}PRIMARY\text{-}DIAGNOSTIC\text{-}PARTIAL}}.

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Failure of any declared sub-gate would have excluded the corresponding primary-sector family on the declared window. Passage of the present gates means only that the declared restricted primary-sector diagnostics survive the public-source comparison.

The public-source summary is reproducible from the declared source surfaces and diagnostic summaries stated above. The principal diagnostic-summary groups are: center

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center The companion source summary identifies the public Planck PR4/NPIPE, ACT DR6, and DESI DR2 source surfaces, the declared diagnostic summaries, and the public-source boundary notes used for the width, acoustic-angle, linked-ruler, and later-transfer gates. The scientific classification is read only from those declared public-source summaries and result summaries.

A linked-ruler construction is predictive only if one primary-sector parameter vector determines every retained ruler and width summary. Let $S$TeXS be the common response Jacobian. The combinations in $\ker S^{\mathsf T}$TeX\ker S^{\mathsf T} are first-order residuals that cannot be absorbed by admissible parameter shifts, and their number is $m-\operatorname{rank}S$TeXm-\operatorname{rank}S. These transverse residuals should be reported alongside the individual ruler fits.

Primary-spectrum, lensing, and distance calibrations that receive independent CHC response coefficients can raise $S$TeXS to full row rank and eliminate the linkage test. The public test therefore freezes the coupling map and covariance before evaluating the retained summaries. Compact target topology does not add a free calibration direction: the homogeneous background is zero-spatial-winding unless a solved inhomogeneous branch is explicitly introduced.

The closure test for the CMB primary linked-ruler gates 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 primary CMB and BAO ruler summaries with their null contrasts. Let $a$TeXa range over the independent constitutive inputs comprising primary-summary map, linked-ruler transfer, covariance, and calibration parameters.

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 the calibration is frozen on a declared subset and the linked ruler is tested in the remaining covariance-weighted subspace.

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 using the linked ruler to set its own transfer coefficient converts the comparison into a consistency fit. Failure of that condition therefore rejects the proposed microscopic closure before parameter estimation can be counted as evidence for it.

CMB-VP0 supplies a source-table diagnostic for the restricted CMB analysis. Planck PR4/NPIPE, ACT DR6, and DESI DR2 products provide the external numerical surfaces, from which the width, acoustic-angle, linked-ruler, and later-transfer quantities are transcribed or recomputed under the stated conventions. These checks verify source consistency and implementation of standard ruler relations. They do not identify the CHC amplitude function, determine the Gaussian width independently of that function, or compare a CHC likelihood with a standard Boltzmann model. The declared reading is therefore CMB-VP0-PRIMARY-DIAGNOSTIC-PARTIAL, not empirical validation of a CHC primary-sector model.

Data and code availability..

This companion manuscript uses public cosmic-microwave-background data products, reference quantities, and cited public references and companion statements as described in the text. Cited public references and companion statements, where provided, are identified by the companion source summaries cited in the text.

Funding and competing interests..

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