Public Branch-Law Pilot Gates for CHC-CBL

The CBL branch law separates an absorbed carrier budget into branch-resolved channels of the schematic form

\Pabs = \Pext + \Pled + \Psink,
\qquad
\etaext + \etaled + \etasink = 1,
\qquad
\eta_i = \frac{P_i}{\Pabs}.

This identity is exact only after a declared carrier family, interface family, calibration convention, and observation window have been fixed. A public-data pilot can test whether the branch-law algebra and estimator formalism can be evaluated on available data surfaces, but it cannot supply same-instance branch fractions unless the branches are measured or exported for the same sample, interface, geometry, wavelength or energy grid, and calibration window.

The present public-pilot record therefore uses a deliberately toned classification ladder: center

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& Optical-constants predictor is evaluated from $n,k$TeXn,k records.

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& XCOM component tables close as normalized branch fractions.

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& No same-instance branch manifest is supplied.

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& Public branch-law pilot succeeds without empirical closure. tabular center

The following overreads are excluded: same-instance empirical branch closure, sealed calibration-instance certificate, Maxwell replacement, QED replacement, universal detector closure, and post-hoc assignment of missing branch fractions.

The companion record uses three public reference surfaces. EKHI supplies a curated open database of optical and thermal radiative properties of solid materials, including emittance, reflectance, transmittance, and absorptance curves digitized from the TPRC Data Series [citation]. The refractiveindex.info dataset supplies an open-source optical-constants repository with YAML-based metadata and material records [citation]. NIST XCOM supplies photon cross sections for coherent and incoherent scattering, photoelectric absorption, pair production, and total attenuation coefficients for elements, compounds, and mixtures over the 1 keV--100 GeV range [citation].

The support summary identifies five public source surfaces for the CBL-VP0 companion record and three real XCOM tables in the XCOM analysis. The public surfaces are used as public pilot inputs only. They are not treated as a same-instance interface-tomography dataset.

The public pilot is tied to the following source surfaces and diagnostic summaries. These entries state the public-source basis and diagnostic role of each analysis. center adjustboxmax width=

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The construction record is: public source surfaces are identified; source tables or curves are normalized under their declared conventions; the EKHI curve-surface, Fresnel predictor, XCOM component-fraction, and same-instance boundary gates are evaluated; the declared labels are assigned; and the non-claim boundary is checked. Public availability is summarized by supplementary public-source basis and result summaries. Large source tables remain external to the manuscript claim, and the public-pilot classification uses only the declared support summaries.

The R/T/A analysis asks whether public radiative-property curves can be ingested into the branch-budget formalism. The acquired EKHI-side pilot curve is a nickel hemispherical total-emittance curve. It contains an emittance branch surface, but not a same-window reflectance, transmittance, and absorptance triple. The analysis is therefore classified as

\boxed{\texttt{RTA-PARTIAL}}.

The classification is intentionally partial. A full same-window $R/T/A$TeXR/T/A budget gate would require, on the same sample and same measurement window,

R(\lambda)+T(\lambda)+A(\lambda)=1

within a declared tolerance and with a covariance or uncertainty object. The declared public surface does not supply that object.

The $n,k$TeXn,k analysis is a deterministic predictor consistency check. For a normal-incidence interface with complex refractive index

\tilde n = n+ik,

the predictor computes

R = \left|\frac{\tilde n-1}{\tilde n+1}\right|^2.

Representative records from the refractiveindex.info archive are parsed and evaluated. The resulting analysis classification is

\boxed{\texttt{CBL-NK-PREDICTOR-GATE-SATISFIED}}.

This is not empirical branch tomography. It confirms that public optical-constants records can be used to instantiate a deterministic branch predictor under a declared interface convention.

The XCOM analysis reads component photon-interaction cross sections as branch-like terms. For a given energy $E$TeXE, let $\mu_j(E)$TeX\mu_j(E) denote tabulated component coefficients, such as coherent scattering, incoherent scattering, photoelectric absorption, and pair-production components when present. The normalized component fraction is

f_j(E)=\frac{\mu_j(E)}{\sum_k \mu_k(E)}.

A closure residual is computed by

\Delta(E)=\left|1-\sum_j f_j(E)\right|.

The evaluated public analysis uses real NIST XCOM tables for C, Al, and SiO$_2$TeX_2. The resulting table contains 244 rows. The maximum absolute closure residual is

\max_E \Delta(E)=0.0007123287671233,

which is below the declared source-rounding-aware tolerance

\tau_{\rm XCOM}=0.001.

The classification is therefore

\boxed{\texttt{CBL-XCOM-BRANCH-PILOT-GATE-SATISFIED}}.

This analysis is a photon-interaction branch-algebra pilot. It is not an interfacial branch-tomography experiment and does not assign CBL carrier branches to a same-instance device or detector.

The same-instance analysis could not be evaluated. The public data contain no manifest tying externalized, internal, and sink-like branches to the same sample, incident carrier family, geometry, wavelength or energy grid, calibration map, and uncertainty or covariance object. Consequently,

\boxed{\texttt{CBL-SAME-INSTANCE-NOT-ADOPTED}}.

A future same-instance record would require at least:

- sample or interface identifier; - incident carrier or absorbed-power reference; - branch-resolved reflected, transmitted, emitted, internal, and sink-like readouts as appropriate; - common axis definitions and units; - calibration map and uncertainty or covariance object; - a declared pass/fail residual gate.

The final analysis board is center

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R/T/A public curve analysis & RTA-PARTIAL Optical-constants predictor analysis & CBL-NK-PREDICTOR-GATE-SATISFIED XCOM photon-interaction branch analysis &

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Same-instance branch tomography &

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The combined public-pilot classification is

\boxed{\texttt{CBL-VP0-PUBLIC-PILOT-PARTIAL}}.

The label remains partial because the strongest empirical object, a same-instance branch-tomography manifest, is absent. The XCOM analysis supplies bounded public branch-algebra support but does not change the same-instance status.

center

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is a source-backed public branch-law and estimator-formalism pilot on EKHI, refractiveindex.info, and NIST XCOM objects. & Do not read the pilot as same-instance empirical branch tomography, a sealed calibration/covariance certificate, a Maxwell/QED replacement, a Planck-law replacement, or a detector-microdynamics closure. tabular center

The result should be re-evaluated if any of the following occurs:

- an EKHI, refractiveindex.info, or XCOM public-source basis cannot be identified under the declared source-surface convention; - the XCOM component fractions fail the declared rounding-aware tolerance on the public real tables; - a full same-window $R/T/A$TeXR/T/A triple is added and fails closure; - a same-instance branch manifest is supplied, which would define a distinct same-instance empirical object rather than automatically change the existing public-pilot label; - any public wording reads the pilot as a same-instance empirical closure, a calibration-instance certificate, or a Maxwell/QED replacement.

A branch budget is admissible only if its event channels form a completely positive trace-preserving instrument. For a no-event sink $K$TeXK, declared operators $L_a$TeXL_a must satisfy $K=\sum_aL_a^\dagger L_a$TeXK=\sum_aL_a^\dagger L_a, and the summed branch probabilities must equal the trace lost from the no-event trajectory. A missing budget fraction requires an explicit unresolved channel rather than renormalization after selection.

The pilot becomes predictive when one frozen material/interface parameter vector controls several branch fractions and optical responses. Its sensitivity matrix must have rank below the retained observable dimension, leaving left-null compatibility residuals. Branch-specific efficiencies that saturate the rank can reproduce the budget but do not test a common conversion mechanism.

The closure test for the public branch-law pilot 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 branch ratios and control-condition contrasts. Let $a$TeXa range over the independent constitutive inputs comprising branch-law parameters, instrument efficiencies, calibration samples, and covariance.

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 one lower-rank branch map and one complete instrument are frozen before the pilot null contrasts are evaluated.

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 per-branch efficiencies otherwise make agreement automatic and obscure missing probability. Failure of that condition therefore rejects the proposed microscopic closure before parameter estimation can be counted as evidence for it.

CBL-VP0 establishes a reproducible public-data pilot for the branch-law formalism. The companion pilot acquires public radiative-property and optical-constants surfaces, evaluates an $n,k$TeXn,k Fresnel predictor, and converts real NIST XCOM component tables into normalized branch-fraction pilot boards. The XCOM analysis reaches CBL-XCOM-BRANCH-PILOT-GATE-SATISFIED; the same-instance branch-tomography analysis remains CBL-SAME-INSTANCE-NOT-ADOPTED. The correct combined classification is CBL-VP0-PUBLIC-PILOT-PARTIAL. No stronger empirical or foundational claim is assigned.

The companion source summary contains source summaries, EKHI and refractiveindex.info pilot outputs, XCOM parsed tables, XCOM branch-fraction tables, same-instance boundary checks, and the combined gate classification. External source surfaces remain attributed to their original providers.

Funding and competing interests..

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