CHC-PFW Public Scalar, Covariance, Contour, and ROOT Support Gates for Precision-Observable Windows

The parent -PFW paper defines a precision-observable basket on one stipulated family. The present paper asks only whether the quoted public inputs were transcribed consistently: quote Do the quoted covariance, contour, ROOT, and scalar-source objects satisfy their stated elementary consistency checks? quote The checked objects satisfy those source-integrity conditions. This result is not a validation of the CHC response map.

This paper does not claim any of the following:

- a global precision-electroweak likelihood; - a global leptonic-flavor or oscillation fit; - an ATLAS profile-likelihood reproduction; - a Daya Bay spectral/covariance reconstruction; - a T2K internal likelihood reproduction; - a new world-average combination; - an empirical proof of .

The only admitted output is a reproducible public validation board for selected declared windows.

The public validation board has four analyses.

- Scalar-source reproduction. Published scalar values and one algebraic transform are checked against the declared CHC-PFW windows. - LEP/SLD covariance subblock. The public arXiv source of the LEP/SLD final $Z$TeXZ-resonance report is identified, and the line-shape covariance / correlation subblock with lepton universality is identified from the TeX table. - KamLAND contour membership. Public KamLAND $\Delta\chi^2$TeX\Delta\chi^2 map files are parsed, and the declared $\ttwelve$TeX\ttwelve window is tested against KamLAND-only and global public map surfaces. - T2K ROOT source support comparison. The public electronic ROOT source surface for the T2K oscillation-parameter measurement is identified and inspected for $\stwo$TeX\stwo-containing objects and contour-range membership.

The board is intentionally incomplete as a global fit. It is a public support witness for the declared windows only.

For reader reproducibility the board is recorded as a non-floating support summary rather than as a wide table:

- LEP/SLD covariance subblock: public arXiv source for the LEP/SLD final $Z$TeXZ-resonance report [citation]; covariance matrix symmetry / positive-semidefinite check and declared $m_Z,\Gamma_Z$TeXm_Z,\Gamma_Z interval inclusion pass. - KamLAND contour membership: public KamLAND $\Delta\chi^2$TeX\Delta\chi^2 map files [citation]; declared $\ttwelve$TeX\ttwelve window intersects the public 1$\sigma$TeX\sigma, 90%, and 2$\sigma$TeX\sigma surfaces. - T2K ROOT support comparison: public electronic ROOT source for T2K [citation]; public objects identified, object list inspected, and $\stwo$TeX\stwo contour-range membership supported on the declared public surface. - Scalar-source reproduction: ATLAS, LEP/SLD, Daya Bay, KamLAND, and T2K public scalar surfaces [citation]; declared scalar intervals and benchmark transforms pass.

Each analysis is a public-source summary gate, not a replacement for the collaboration analyses.

The LEP/SLD final $Z$TeXZ-resonance combination provides public line-shape pseudo-observables and their correlations [citation]. The support summary identifies the public arXiv source document set and states the lepton-universality line-shape subblock from the source TeX table. The subblock uses

(\mZ,\GZ,\sigma^0_{\rm had},R_\ell,A^{0,\ell}_{\rm FB})
  =
  (91.1875,2.4952,41.540,20.767,0.0171),

with one-sigma uncertainties

(0.0021,0.0023,0.037,0.025,0.0010).

The identified correlation matrix is

\rho=

 1.000  -0.023  -0.045  0.033  0.055

 -0.023  1.000  -0.297  0.004  0.003

 -0.045  -0.297  1.000  0.183  0.006

 0.033  0.004  0.183  1.000  -0.056

 0.055  0.003  0.006  -0.056  1.000
.

The covariance matrix is then

C_{ij}=\rho_{ij}\sigma_i\sigma_j.

The gate checks symmetry, positive semidefiniteness, and inclusion of the parent-paper $\mZ$TeX\mZ and $\GZ$TeX\GZ central values in their declared intervals. The observed maximum symmetry error was $2.65\times10^{-23}$TeX2.65\times10^{-23}, and the numerical eigenvalues were positive:

(9.92\times10^{-7},\ 4.32\times10^{-6},\ 4.88\times10^{-6},\ 5.88\times10^{-4},\ 1.41\times10^{-3}).

The local label is therefore center PFW-LEP-SLD-PUBLIC-COVARIANCE-SUBBLOCK-CHECK-SATISFIED. center This is a public-table covariance subblock. It is not a LEP/SLD raw-data refit and not a full electroweak global likelihood.

The support summary identifies the public KamLAND $\Delta\chi^2$TeX\Delta\chi^2 map files and parses rows of the form

(\log_{10}\ttwelve,\log_{10}\Delta m^2,\Delta\chi^2).

For the declared CHC-PFW solar-angle window

\ttwelve\in[0.49,0.66],

we test whether the public map contains points inside the declared window below standard two-parameter contour thresholds. The KamLAND-only map has its public minimum at

\log_{10}\ttwelve=-0.26,
  \qquad
  \log_{10}\Delta m^2=-4.12,
  \qquad
  \Delta\chi^2=0,

which corresponds to $\ttwelve\simeq0.5495$TeX\ttwelve\simeq0.5495. This lies inside the declared window. The KamLAND+solar global public map has a minimum slightly below the lower declared edge, but the best point inside the declared window occurs at

\log_{10}\ttwelve=-0.30,
  \qquad
  \log_{10}\Delta m^2=-4.12,
  \qquad
  \Delta\chi^2=0.3,

which also passes the 1$\sigma$TeX\sigma, 90%, and 2$\sigma$TeX\sigma public contour thresholds used by the gate. The local label is center PFW-KAMLAND-PUBLIC-CHI2-CONTOUR-MEMBERSHIP-SATISFIED. center This is a public contour-membership check, not a new KamLAND or solar-neutrino oscillation fit.

The T2K collaboration provides public electronic contour information for the oscillation-parameter measurements reported in Ref. [citation]. The companion source summary identifies the public Bayesian and frequentist contour surfaces and records the public-source basis for the declared $\sin^2\theta_{23}$TeX\sin^2\theta_{23} window. The manuscript uses the following public contour classes:

- Bayesian ROOT-format posterior object; - Frequentist ROOT-format contour object.

The public source presents one-dimensional posterior / $\Delta\chi^2$TeX\Delta\chi^2 objects and two-dimensional contour graphs for oscillation parameters. The gate verifies that $\stwo$TeX\stwo-containing objects are present and extracts the $\stwo$TeX\stwo-$\Delta m^2$TeX\Delta m^2 frequentist normal-hierarchy contour ranges with reactor constraint. The declared CHC-PFW window

\stwo\in[0.529,0.593]

intersects the extracted contour ranges, and the extracted best-fit marker lies inside the declared interval. The local label is center PFW-T2K-ZENODO-ROOT-CONTOUR-EXTRACTION-SATISFIED. center This is a public ROOT source surface and contour-range extraction. It is not a T2K internal likelihood reproduction and not a global oscillation analysis.

The scalar-source analysis retained from the VP1 support summary checks direct scalar intervals and one algebraic transform. The checked rows include

\mW=80.3665\pm0.0159\ \mathrm{GeV},

  \mZ=91.1875\pm0.0021\ \mathrm{GeV},

  \seff=0.23153\pm0.00016,

  \GZ=2.4952\pm0.0023\ \mathrm{GeV},

  \Gl=83.984\pm0.086\ \mathrm{MeV}.

For Daya Bay, the public scalar value is transformed by

\sin^2\theta_{13}
  =\frac{1-\sqrt{1-\sin^2(2\theta_{13})}}{2},

which maps $\sin^2(2\theta_{13})=0.0856\pm0.0029$TeX\sin^2(2\theta_{13})=0.0856\pm0.0029 to a one-sigma range approximately

\sin^2\theta_{13}\in[0.021121,0.022637],

inside the declared CHC-PFW interval. The retained scalar-source label is center PFW-VP1-PUBLIC-SCALAR-SOURCE-READBACK-SATISFIED. center

The combined public source basis has two declared VP1-local support-analysis labels: center PFW-VP1-PUBLIC-SCALAR-AND-CONTOUR-SUPPORT-SATISFIED center for the manuscript-level scalar/contour analysis, and center PFW-VP1-PUBLIC-COVARIANCE-CONTOUR-ROOT-SUPPORT-SATISFIED center for the deeper covariance, contour, and ROOT support comparison. The public manuscript sequence remains CHC-PFW-VP1. The scalar--contour comparison and the deeper covariance--contour comparison are distinct bounded support comparisons, not separate parent-paper claims. Companion source summaries record the public input surfaces, the scalar, covariance, and contour comparisons, the diagnostics, and the public-source conditions. These classifications remain local to the declared support comparisons and do not promote the parent paper to a global fit claim.

Covariance and contour checks validate a common electroweak--leptonic family only in directions that cannot be generated by refitting its frozen parameters. If $S$TeXS is the response Jacobian of the retained ROOT and scalar summaries, the column space of $S$TeXS is the tangent space of admissible calibration shifts, while $\ker S^{\mathsf T}$TeX\ker S^{\mathsf T} contains the first-order predictions. The covariance-weighted projection onto that null space supplies the appropriate public residual.

This projection and its numerical rank must be fixed before contour inspection. Adding a nuisance coefficient for each retained summary can make $S$TeXS full row rank and reduce every transverse residual to zero by construction. The analysis should therefore report the number of surviving restrictions and validate them on quantities not used to choose the family or covariance model.

The closure test for the public covariance and ROOT 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 reproduced intervals, contours, and transverse residuals. Let $a$TeXa range over the independent constitutive inputs comprising public contours, ROOT objects, covariance blocks, and coordinate transformations.

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 source reproduction is completed before one frozen covariance is propagated into the null-space statistic.

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 matching published contours validates extraction but does not validate the CHC parameterization. Failure of that condition therefore rejects the proposed microscopic closure before parameter estimation can be counted as evidence for it.

The public LEP/SLD covariance subblock, KamLAND $\Delta\chi^2$TeX\Delta\chi^2 maps, T2K ROOT source, and selected scalar transformations pass the stated source-integrity checks. This establishes reproducibility of the inputs, not empirical confirmation of CHC-PFW.

Because the parent parametrization is rank deficient and target-interpolating, interval and contour membership cannot discriminate it from alternatives. A stronger claim requires an identifiable restricted model, a joint likelihood with nuisance correlations, and genuinely held-out predictions fixed before inspection.

The companion source summaries record the public scalar-source, covariance-subblock, contour-membership, and public contour-source support comparisons used in this companion paper, including declared source boundaries, diagnostic summaries, and the VP1-local support-analysis labels. Primary public inputs should be consulted at the cited public sources. These materials are bounded public support summaries only; they are not a global electroweak, collider, or flavor likelihood and not an unpublished covariance reconstruction.

Funding and competing interests..

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