Public Computational and Analytic Benchmark Gates for CHC-QTT Support-Registration Factorization

The CHC-QTT sector reads barrier-mediated tunneling signals through a support--registration factorization of the schematic form

I_\alpha(V,\lambda)
=
\int \Rcal_\alpha(E,V,\lambda)\,
\exp\!\big[-2\Acal_{B,\alpha}(E,V,\lambda)\big]~\dd E
+
\delta I_\alpha .

Here $\exp[-2\Acal_B]$TeX\exp[-2\Acal_B] is the barrier-mediated phase-link support factor and $\Rcal$TeX\Rcal is the registration factor carrying interface, lead, density-of-states, bias-window, or readout availability. The present companion benchmark record does not attempt a same-instance experimental closure of reference. It supplies a first public benchmark with two declared components:

- a computational STM registration-feature test using public JARVIS-STM image data; - analytic support-action consistency gates for Fowler--Nordheim, rectangular-barrier, and Simmons-style families.

The result is therefore a computational/analytic benchmark certificate for the registered checks. It is not a measurement of a hidden traversal path, a QED replacement, a detector microdynamics model, or a tunneling-time theory.

The STM benchmark uses the JARVIS-STM computational database. Choudhary et al. introduced a systematic database of STM images calculated using density functional theory and the Tersoff--Hamann method for $716$TeX716 exfoliable two-dimensional materials, and made the generated computational STM images available through JARVIS-STM and Figshare [citation]. The Tersoff--Hamann formalism provides the STM theory anchor in which the tunneling current is connected to the surface local density of states at the tip position [citation]. The older Bardeen matrix-element formulation supplies the standard many-particle tunneling anchor [citation]. The Fowler--Nordheim, Simmons, and rectangular-barrier analytic gates are used only as support-action extractor consistency checks, with Fowler--Nordheim and Simmons serving as field-emission and MIM tunneling anchors [citation].

The public-source summary identifies two official source surfaces: center

Figure or table content is omitted from the web reader; use the canonical manuscript for the exact object.

center The public benchmark record identifies the official STM source surfaces and the declared feature tables used for the benchmark summaries. The computational STM objects are not treated as experimental STS current--voltage curves.

The public record identifies the computational image source, lattice-label metadata, declared feature-extraction route, and resulting feature summaries. Public source is retained in the companion source summary and is not part of the physical claim.

STM registration-feature gate

Let $I_+$TeXI_+ and $I_-$TeXI_- denote the positive- and negative-bias computational STM images associated with a declared material identifier. Images are normalized by

I_{\rm norm}=\frac{I-\min(I)}{\max(I)-\min(I)+\epsilon},
\qquad
\epsilon=10^{-12}.

For a paired positive/negative image on the common cropped shape, the pair-distance feature is

d_{+-}=\frac{\|I_{+,\rm norm}-I_{-,\rm norm}\|_2}{\sqrt{N}},

where $N$TeXN is the number of pixels in the common image domain. A high-frequency Fourier proxy is also computed using a fixed one-third Nyquist-radius cutoff. The final feature table is stored in float64. Flat-image cases are handled by explicit flags; no flat-image rows occur in the present evaluation.

Analytic support-action gates

The Fowler--Nordheim sanity gate uses

J(F)=C F^2\exp(-B/F).

The extractor regresses $\log(J/F^2)$TeX\log(J/F^2) against $1/F$TeX1/F; the recovered slope is $-B$TeX-B. The registered cases use $B\in\{6.83,8.25,9.75\}$TeXB\in\{6.83,8.25,9.75\}.

The rectangular-barrier gate compares finite positive values of the exact transmission coefficient

T_{\rm rect}(E)=
\left[
1+\frac{U_0^2\sinh^2(\kappa a)}{4E(U_0-E)}
\right]^{-1},
\qquad
\kappa=\frac{\sqrt{2m(U_0-E)}}{\hbar},

with its WKB envelope $\exp(-2\kappa a)$TeX\exp(-2\kappa a) on the declared parameter grid. The gate is a positivity and finite-value check, not a new tunneling law.

The Simmons-style thickness-decay gate uses the synthetic form

J(s,V)=C V\exp(-2\kappa s).

The extractor regresses $\log(J/V)$TeX\log(J/V) against thickness $s$TeXs and returns $\kappa=-\frac12\,\partial_s\log(J/V)$TeX\kappa=-\frac12\,\partial_s\log(J/V). The registered cases use $\kappa\in\{1.25,1.75,2.25\}$TeX\kappa\in\{1.25,1.75,2.25\}.

The JARVIS-STM public source set contains $716$TeX716 declared JIDs. All $716$TeX716 have positive- and negative-bias pairs. The deterministic feature extraction produces $716$TeX716 finite feature rows in $15$TeX15 completed feature groups. The analytic layer produces $9$TeX9 completed chunks: three Fowler--Nordheim cases, three rectangular-barrier cases, and three Simmons-style cases.

center

Figure or table content is omitted from the web reader; use the canonical manuscript for the exact object.

center

For the Fowler--Nordheim cases, the relative errors in $B$TeXB recovery are $2.08\times10^{-15}$TeX2.08\times10^{-15}, $6.46\times10^{-16}$TeX6.46\times10^{-16}, and $1.28\times10^{-15}$TeX1.28\times10^{-15}. For the Simmons-style cases, the relative errors in $\kappa$TeX\kappa recovery are $2.49\times10^{-15}$TeX2.49\times10^{-15}, $1.14\times10^{-15}$TeX1.14\times10^{-15}, and $9.87\times10^{-16}$TeX9.87\times10^{-16}. Rectangular-barrier exact and WKB values are finite and positive on all declared cases. The declared construction summary records $15$TeX15 feature groups and $9$TeX9 analytic chunks, and the merge-consistency check returns declared chunk presence, non-overlap, non-gap, source-identity match, and no NaN/inf conditions.

The declared classification rule returns

\boxed{\mathrm{QTT\mbox{-}VP0\mbox{-}COMPUTATIONAL\mbox{-}GATES\mbox{-}SATISFIED}} .

The classification means that the public computational STM acquisition, pair reconstruction, deterministic registration-feature extraction, and analytic support-action checks are satisfied on the declared benchmark records.

remark: Non-claim boundary. The result does not close same-instance experimental STM/MIM/Fowler--Nordheim validation. It does not identify a separated detector microdynamics, does not replace QED, does not infer a hidden path through the barrier, and does not decide tunneling-time questions. Experimental literature benchmarks and same-sample cross-window closure remain separate verification stages.

The support--registration factorization is strengthened by two independent tests. A proposed phase-loading term must first be locally variational: after pullback to a scalar functional $\lambda[\mathcal H]$TeX\lambda[\mathcal H], its Fr\'echet derivative must equal its formal adjoint. Alignment or numerical agreement without this Helmholtz property cannot establish an action-level loading law. Registration must separately be represented by a CPTP instrument, so any no-event sink satisfies $K=\sum_aL_a^\dagger L_a$TeXK=\sum_aL_a^\dagger L_a and all missing norm appears in declared event branches.

These conditions are computationally decidable on the finite benchmark. Discretized response Jacobians can test formal symmetry, Kraus completeness can test probability conservation, and a shared sensitivity matrix across support and registration observables can reveal left-null compatibility relations. A fit that passes only after adding one coefficient per output has full nuisance rank and carries no overidentifying content.

The closure test for the computational and analytic tunneling 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 STM images and Fowler--Nordheim, rectangular-barrier, and Simmons benchmark residuals. Let $a$TeXa range over the independent constitutive inputs comprising barrier parameters, simulated-image registration map, analytic normalization, and numerical tolerances.

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 benchmark formulas and image observables are evaluated with a frozen bridge whose sensitivity retains a nonzero transverse 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 formula reproduction tests implementation but cannot identify a freely refactorable support--registration decomposition. Failure of that condition therefore rejects the proposed microscopic closure before parameter estimation can be counted as evidence for it.

QTT--VP0 reproduces the stated data-handling pipeline and standard analytic targets. It does not identify the barrier action separately from registration or provide an empirical CHC test.

The companion source summary summarizes the public source, processed feature tables, analytic benchmark tables, protocol-compliance checks, merge-consistency checks, declared computational procedures, and result summaries for the declared computational/analytic VP0 gates. Raw computational STM public sources remain attributable to the official JARVIS-STM/Figshare sources. The record is a bounded computational/analytic companion source summary only; it is not same-instance experimental STM/MIM/Fowler--Nordheim validation and not an empirical tunneling-time claim.

Funding and competing interests..

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