Barrier-Mediated Phase-Link Transmission and Tunneling Spectroscopy in the CHC Framework

Quantum tunneling is commonly described as transmission through a classically forbidden barrier, a reading already central to early nuclear tunneling work [citation]. That description is operationally effective, yet it easily suggests an over-literal picture in which a small material object secretly traverses the forbidden region. The surface phenomena are more precise. Field emission measures a barrier reshaped by an electric field [citation]. Narrow junctions and tunnel diodes show current controlled by spectral overlap as well as barrier transparency [citation]. Superconducting tunnel experiments use the current--voltage curve to extract an energy gap and density-of-states structure [citation]. Scanning tunneling microscopy uses a controllable vacuum gap to turn exponentially sensitive vacuum tunneling into atomic-scale surface images [citation]. Bardeen's matrix-element formulation and the Tersoff--Hamann theory of the scanning tunneling microscope show that the registered current is determined jointly by the barrier coupling and by the spectral/interface states available for registration [citation].

The imported CHC phase-link construction separates propagation, accessibility, local commit, durable readout, and wavefunction semantics. The propagation layer supplies coherent phase-link support; the accessible-event layer supplies Born-type event statistics on declared effect domains; detector-side papers separate local opening, boundary response, and durable readout; and the wavefunction semantics layer rejects the reading of the wavefunction as material substance [citation]. The present construction uses those imported objects but does not absorb their analyses. Its owned object is narrower: barrier-mediated phase-link transmission and the registered tunneling signals that arise when suppressed support is closed at an interface, lead, or detector.

The central distinction is

\text{evanescent support}\neq \text{durable event}\neq \text{hidden path}.

A low-admittance barrier can carry a decaying phase-link support. That support is not yet a detector event. Registered current appears only when the exit-side structure admits a channel, lead state, density-of-states overlap, or detector-interface registration. The same distinction turns the paper from a purely interpretive statement into a measurable factorization: tunneling current is split into a barrier support action and an interface registration factor.

The standard formulas are retained rather than replaced. A rectangular barrier gives the exact textbook coefficient. Smooth barriers give the WKB exponential envelope. STM, metal--insulator--metal junctions, Fowler--Nordheim emission, Esaki tunneling, and Giaever superconducting spectroscopy are treated as surface windows in which different parts of the support--registration split dominate. The resulting support-action extractor and cross-window same-family gate provide direct rejection conditions.

Stationary one-dimensional branch

Let $\Bcal_{1D}$TeX\Bcal_{1D} be the family of single-channel, nonrelativistic, time-independent barrier problems

-\frac{\hbarred^2}{2m}\frac{\dd^2\psi}{\dd x^2}+U(x)\psi=E\psi,
\qquad m>0,

with real $U(x)$TeXU(x), asymptotic free regions, and flux-normalized incoming/outgoing amplitudes. The branch is a coherent scattering branch only. It is not a detector Hamiltonian, a many-body transport theory, or a claim about a literal path inside the barrier.

When $U(x)>E$TeXU(x)>E on an interval $[x_-,x_+]$TeX[x_-,x_+], define the barrier support action

\Acal_B(E)=\int_{x_-}^{x_+}\frac{\sqrt{2m\{U(x)-E\}}}{\hbarred}\,\dd x.

The associated support envelope is

\Scal_B(E)=\exp[-2\Acal_B(E)].

This object is dimensionless, bounded by unity when $\Acal_B\ge0$TeX\Acal_B\ge0, and carries only barrier-mediated support attenuation. It is not by itself an event probability unless an interface family supplies the registration layer.

Rectangular barrier

For

U(x)=

0,x<0,

U_0,0\le x\le a,

0,x>a,

\qquad 0<E<U_0,

one has

\kappa=\frac{\sqrt{2m(U_0-E)}}{\hbarred},
\qquad
\Acal_B(E)=\kappa a.

Matching $\psi$TeX\psi and $\psi'$TeX\psi' at the two interfaces gives

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

In the opaque limit $\kappa a\gg1$TeX\kappa a\gg1,

T_{\rm rect}(E)
=
\frac{16E(U_0-E)}{U_0^2}\,\ee^{-2\kappa a}
\left[1+O(\ee^{-2\kappa a})+O\!\left(\frac{16E(U_0-E)}{U_0^2}\ee^{-2\kappa a}\right)\right].

Thus the exponential part of the exact coefficient is precisely the CHC barrier support envelope. The prefactor belongs to interface matching on this branch.

Smooth opaque barriers

For a smooth barrier with two real turning points, the leading WKB transmission family is written as

T_{\rm WKB}(E)=C_{\rm WKB}(E)\,\exp[-2\Acal_B(E)]\,[1+\eps_{\rm WKB}(E)],

where $C_{\rm WKB}$TeXC_{\rm WKB} denotes the connection/prefactor contribution and $\eps_{\rm WKB}$TeX\eps_{\rm WKB} records declared smoothness and non-opaque residuals. The support envelope is common, while the prefactor and residual remain family dependent.

proposition: Barrier support recovery on the declared stationary branch. On $\Bcal_{1D}$TeX\Bcal_{1D}, the exact rectangular-barrier coefficient reference has the opaque-envelope factor $\exp[-2\Acal_B]$TeX\exp[-2\Acal_B], and the smooth opaque WKB branch has the same exponential factor in reference. Therefore $\exp[-2\Acal_B]$TeX\exp[-2\Acal_B] is the barrier-mediated phase-link support factor common to the declared one-dimensional comparison family.

proof. For the rectangular barrier, $\Acal_B=\kappa a$TeX\Acal_B=\kappa a and reference follows from $\sinh^2 z=(\ee^{2z}-2+\ee^{-2z})/4$TeX\sinh^2 z=(\ee^{2z}-2+\ee^{-2z})/4. For the smooth opaque family, reference is the declared leading WKB form. In both cases the same exponential attenuation $\exp[-2\Acal_B]$TeX\exp[-2\Acal_B] appears.

Registration factor

A registered tunneling signal is not exhausted by the barrier action. A current or conductance also requires available states, a bias window, lead occupation factors, interface matrix-element prefactors, detector gain, or surface density of states. These non-exponential objects are grouped into an interface-side registration factor.

definition: Weak-tunneling support--registration family. A weak-tunneling support--registration family $\Fcal_\alpha$TeX\Fcal_\alpha consists of a barrier model, an interface or lead geometry, a bias convention, and a registration factor $\Rcal_\alpha(E,V,\lambda)\ge0$TeX\Rcal_\alpha(E,V,\lambda)\ge0 such that the registered signal can be written as

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

Here $\lambda$TeX\lambda denotes the declared geometric or material controls, and $\delta I_\alpha$TeX\delta I_\alpha contains the stated finite-temperature, high-bias, multi-channel, inelastic, non-opaque, or strong-coupling residuals of that family.

theorem: Non-identifiability of an unrestricted support--registration split. Ignore the residual term for clarity and suppose $(\Acal_{B,\alpha},\Rcal_\alpha)$TeX(\Acal_{B,\alpha},\Rcal_\alpha) satisfies reference. For every bounded real function $f(E,V,\lambda)$TeXf(E,V,\lambda), the transformed pair

\Acal'_{B,\alpha}=\Acal_{B,\alpha}+f,
\qquad
\Rcal'_\alpha=\Rcal_\alpha e^{2f}

produces the same signal. Hence $I_\alpha$TeXI_\alpha alone does not identify the barrier action and registration factor separately.

proof. Pointwise in $(E,V,\lambda)$TeX(E,V,\lambda), $\Rcal'_\alpha e^{-2\Acal'_{B,\alpha}} =\Rcal_\alpha e^{2f}e^{-2\Acal_{B,\alpha}-2f} =\Rcal_\alpha e^{-2\Acal_{B,\alpha}}$TeX\Rcal'_\alpha e^{-2\Acal'_{B,\alpha}} =\Rcal_\alpha e^{2f}e^{-2\Acal_{B,\alpha}-2f} =\Rcal_\alpha e^{-2\Acal_{B,\alpha}}. Integration over energy therefore leaves reference invariant. Since infinitely many nonzero bounded functions $f$TeXf are admissible, the inverse problem has an infinite-dimensional equivalence class unless independent information constrains one factor.

proposition: Bardeen-type factorization. Consider a weak-coupling transfer-Hamiltonian window in which the squared interface matrix element admits the factorization

|M(E,V,\lambda)|^2=M_0(E,V,\lambda)^2\exp[-2\Acal_B(E,V,\lambda)]

with $M_0$TeXM_0 nonnegative and slowly varying on the declared energy window. If the registered current has the Bardeen form

I(V)=C\int \dd E\; |M(E,V,\lambda)|^2\rhoL(E)\rhoR(E+eV)
\{f(E)-f(E+eV)\}+\delta I,

then reference holds with

\Rcal(E,V,\lambda)=C\,M_0(E,V,\lambda)^2\rhoL(E)\rhoR(E+eV)
\{f(E)-f(E+eV)\}.

proof. Substitute reference into reference and collect all non-exponential factors into $\Rcal$TeX\Rcal. This is an algebraic factorization on the declared weak-coupling window.

remark. Bardeen's many-particle tunneling formulation explicitly treats tunneling current between two conductors separated by an oxide barrier through matrix elements of the two systems [citation]. In CHC language, the matrix element supplies the interface bridge: its exponential part is barrier support, while its prefactor and density-of-states factors are registration data.

CHC reading

Equation reference is the technical core of the CHC reinterpretation. The barrier interior carries suppressed support. The registered signal is produced only after that support meets an interface family that can register it. Consequently, tunneling is not a direct proof of material traversal inside the barrier. It is a support--registration phenomenon.

\boxed{
\text{barrier support }\ee^{-2\Acal_B}
+\text{ interface registration }\Rcal

\longrightarrow\ \text{registered current or event}
}

This does not weaken the standard formulas. It assigns their factors to different operational layers.

Scanning tunneling microscopy

STM is the primary surface witness because the control parameter is geometric and the registered current is exponentially gap sensitive. Binnig, Rohrer, Gerber, and Weibel first demonstrated tunneling through a controllable vacuum gap and then atomic-scale surface microscopy using vacuum tunneling [citation]. Tersoff and Hamann showed that in the small-bias locally spherical tip model, the tunneling current is proportional to the surface local density of states at the tip position [citation].

In the declared STM window,

\Ist(d,V)
=
C_{\rm tip}\,V\,\rhoLDOS(\mathbf r_0,\EF)
\ee^{-2\kappa d}
[1+\eps_{\rm STM}(d,V,T,\mathrm{tip},\mathrm{surface})],

with

\kappa=\frac{\sqrt{2m\Phi_{\rm eff}}}{\hbarred}.

The exponential factor is barrier support through the vacuum gap. The product $C_{\rm tip}V\rho_s$TeXC_{\rm tip}V\rho_s is registration: it records tip geometry, bias window, and surface spectral availability. The current is not a count of already completed events inside the vacuum barrier; it is the interface-side registration of suppressed support.

For two distances $d_1,d_2$TeXd_1,d_2 in the same fixed STM family, define the support-slope estimator

\widehat{\Delta\Acal}_{\rm STM}(d_1,d_2)
=
-\frac12\log\!
\left[
\frac{\Ist(d_2,V)\,\Rcal_{\rm STM}(d_1,V)}{\Ist(d_1,V)\,\Rcal_{\rm STM}(d_2,V)}
\right],

where $\Rcal_{\rm STM}=C_{\rm tip}V\rho_s$TeX\Rcal_{\rm STM}=C_{\rm tip}V\rho_s under the declared correction convention. If the registration ratio is fixed or corrected, then

\widehat{\Delta\Acal}_{\rm STM}(d_1,d_2)=\kappa(d_2-d_1)+O(\eps_{\rm STM}).

The STM window therefore supplies a directly measured barrier-support slope.

Metal--insulator--metal tunnel junctions

For a metal--insulator--metal junction with film thickness $s$TeXs and voltage $V$TeXV, Simmons derived a generalized formula for tunneling through a barrier of arbitrary shape in a thin insulating film and applied it to rectangular and image-force-corrected barriers [citation]. In the present notation, the admitted low-transparency junction family is written as

J_{\rm MIM}(V,s)
=
\Rcal_{\rm MIM}(V,s)\,
\ee^{-2\Acal_{B,{\rm MIM}}(V,s)}[1+\eps_{\rm MIM}(V,s)],

where $\Rcal_{\rm MIM}$TeX\Rcal_{\rm MIM} carries electrode densities of states, voltage window, temperature convention, and non-exponential barrier-shape factors. On a near-rectangular effective barrier,

\Acal_{B,{\rm MIM}}(V,s)\simeq
s\frac{\sqrt{2m\Phi_{\rm eff}(V)}}{\hbarred}.

A thickness-slope extractor is therefore

-\frac12\frac{\partial}{\partial s}\log\!
\left(\frac{J_{\rm MIM}(V,s)}{\Rcal_{\rm MIM}(V,s)}\right)
=
\frac{\partial\Acal_{B,{\rm MIM}}}{\partial s}+O\!\left(\frac{\partial\eps_{\rm MIM}}{\partial s}\right).

This window extends QTT from STM distance dependence to junction thickness and voltage dependence without opening bulk conductor transport.

Fowler--Nordheim field emission

For field emission from a metal surface, the barrier is lowered and tilted by an applied field $F$TeXF. Fowler and Nordheim treated electron emission from cold metals in intense electric fields as tunneling through such a field-shaped barrier [citation]. In the elementary triangular model

U(x)-E=\phi-eFx,
\qquad 0\le x\le \frac{\phi}{eF},

the support action is

\Acal_{\rm FN}(F)
=
\int_{0}^{\phi/(eF)}\frac{\sqrt{2m(\phi-eFx)}}{\hbarred}\,\dd x
=
\frac{2\sqrt{2m}\,\phi^{3/2}}{3e\hbarred F}.

Thus the exponential part is

\ee^{-2\Acal_{\rm FN}(F)}
=
\exp\!\left[-\frac{4\sqrt{2m}\,\phi^{3/2}}{3e\hbarred F}\right].

The registered current density is written in the support--registration form

J_{\rm FN}(F)=\Rcal_{\rm FN}(F,\phi,\mathrm{surface})\,F^2\,\ee^{-2\Acal_{\rm FN}(F)}[1+\eps_{\rm FN}],

where image corrections, field-enhancement factors, surface state effects, and temperature corrections belong to the declared residual or registration model. The Fowler--Nordheim plot is therefore a support-action plot: after registration correction, the slope in $1/F$TeX1/F reads the field-shaped barrier action.

Tunnel spectroscopy: Esaki and Giaever windows

In tunnel spectroscopy the barrier support may be slowly varying while registration varies sharply with voltage because the available states change. This makes spectroscopy the clearest case where barrier transparency alone is not the phenomenon.

The declared spectral family is

I_{\rm spec}(V)
=
C\int \dd E\;\rhoL(E)\rhoR(E+eV)
\{f(E)-f(E+eV)\}
\ee^{-2\Acal_B(E,V)}+\delta I_{\rm spec}.

Esaki's narrow-junction effect is read as a registration-window phenomenon: band overlap and occupancy determine where current can register, and negative differential response can arise when the spectral overlap decreases with bias even if the barrier support has not disappeared [citation]. Giaever tunneling is read similarly: superconducting density-of-states structure and the BCS gap enter the registration factor, allowing the gap to be inferred from the current--voltage curve [citation].

If $\Acal_B(E,V)$TeX\Acal_B(E,V) is approximately constant over the narrow spectral window, then

\frac{\dd I_{\rm spec}}{\dd V}
\quad\text{primarily probes}\quad
\frac{\dd}{\dd V}
\int \dd E\;\rhoL(E)\rhoR(E+eV)
\{f(E)-f(E+eV)\},

up to the fixed support factor and declared residuals. In CHC terms, spectroscopy images registration structure, not an interior path.

Tunnel conductance interface and bulk-transport exclusion

Linear tunnel conductance can be read through a transmission-conductance interface. In a declared phase-coherent low-temperature window,

G_{\rm tunnel}
=
\frac{2e^2}{h}\sum_n T_n,
\qquad
T_n=\eta_n\ee^{-2\Acal_{B,n}}[1+\eps_n],

which is compatible with Landauer--B\"uttiker conductance language for phase-coherent transport [citation]. This interface is used only for barrier-limited conductance. It does not claim a theory of bulk metallic conduction, phonon scattering, disorder transport, Kubo response, or full device modeling.

The support--registration split becomes scientific only if it can fail. For a declared window $\alpha$TeX\alpha, define the corrected support-action estimator

\widehat{\Acal}_{B,\alpha}(V,\lambda)
=
-\frac12\log\!
\left[
\frac{I_\alpha(V,\lambda)-\delta I_\alpha(V,\lambda)}{\int \dd E\;\Rcal_\alpha(E,V,\lambda)}
\right]

when the support action is effectively constant on the declared energy window. In the fully energy-resolved case the fitting version is used:

\widehat\theta_\alpha
=
\arg\min_{\theta\in\Theta_\alpha}
\left\|
I_\alpha(V,\lambda)-
\int\dd E\;\Rcal_\alpha(E,V,\lambda;\theta)
\ee^{-2\Acal_{B,\alpha}(E,V,\lambda;\theta)}
\right\|_{W_\alpha}.

Here $W_\alpha$TeXW_\alpha is the fixed weighting/covariance convention for that window. A common barrier-support family over a set of windows $\mathcal S$TeX\mathcal S is admitted only if a single parameter point $\theta_*$TeX\theta_* satisfies

\Delta_{\mathcal S}(\theta_*)
:=
\max_{\alpha\in\mathcal S}
\left\|
I_\alpha-
\int\dd E\;\Rcal_\alpha\ee^{-2\Acal_{B,\alpha}(\theta_*)}
\right\|_{W_\alpha}
\le \eta_{\rm support}.

theorem: Same-family support consistency gate. Fix a set of tunneling windows $\mathcal S$TeX\mathcal S, a shared barrier-support parameter space $\Theta$TeX\Theta, a registration model $\Rcal_\alpha$TeX\Rcal_\alpha, a residual convention $\delta I_\alpha$TeX\delta I_\alpha, and weights $W_\alpha$TeXW_\alpha for every $\alpha\in\mathcal S$TeX\alpha\in\mathcal S. If there exists $\theta_*\in\Theta$TeX\theta_*\in\Theta satisfying reference, then the windows admit a common support-action reading on that declared family. If no such $\theta_*$TeX\theta_* exists after the registration factors and residual budgets are fixed, the common support-action reading fails for that family.

proof. This is the definition of the declared same-family fit gate. The data are compared to one common support-action family after fixing registration and residual conventions. Failure occurs when the shared parameter point does not exist.

This theorem is deliberately modest. It does not assert that STM, MIM, and Fowler--Nordheim data must always share one barrier model. It states what must be true if they are claimed to be the same barrier-support family. The gate converts the CHC interpretation into a testable residual condition.

The technical content can be summarized as a formally specified layer separation.

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

The construction is aligned with the broader CHC stance that propagation is phase-link structure rather than material spreading, and that local commit or durable record formation belongs to a later interface layer. It adds a new declared object: barrier-mediated phase-link transmission with registered tunneling spectroscopy. It does not re-open wavefunction ontology, detector microdynamics, electromagnetic field theory, or bulk transport.

The following exclusions are load-bearing:

- no QED or Maxwell replacement is introduced; - no detector micro-Hamiltonian, durable-readout theorem, or objectivity theorem is introduced; - no bulk conductor transport theory is introduced; - no Josephson phase-dynamics theory is introduced, although Josephson tunneling remains a neighboring comparison class [citation]; - no universal tunneling-time theorem is asserted; - no hidden material trajectory is inferred from evanescent support.

The declared support--registration branch fails under any of the following conditions.

F1. No barrier support action..

If a proposed window cannot define a nonnegative barrier action $\Acal_B$TeX\Acal_B or an equivalent transmission support factor on its declared branch, the CHC barrier-support reading is unavailable.

F2. Registration retuning..

If a comparison requires changing $\Rcal_\alpha$TeX\Rcal_\alpha, the bias convention, the tip/surface model, the junction model, or the weighting rule after inspecting the result, it leaves the declared family.

F3. Cross-window inconsistency..

If no common $\theta_*$TeX\theta_* satisfies reference after the registration model and residuals are fixed, the same-family support-action claim fails.

F4. Spectroscopy overread..

If a density-of-states or band-overlap feature is described as a barrier-transparency feature without the registration factor, the interpretation fails its own factorization.

F5. Transport overreach..

If the barrier-limited conductance interface is used to claim a full bulk conductor theory, the construction exceeds its declared analysis.

F6. Event overread..

If evanescent support inside the barrier is identified with a completed detector event, durable record, or hidden material path, the CHC reading fails.

F7. Public-comparison overread..

If a companion VP0, VP1, or VP2 benchmark record is described as QED replacement, detector microdynamics theory, unrestricted tunneling-time theory, same-instance STM/MIM/Fowler--Nordheim validation, same-device validation, or raw/export source-data recovery beyond its bounded label, the construction exceeds its analysis. In particular, the VP2 figure-derived Tier-C partial label may not be promoted without raw/export windows and an explicit identity-bearing ledger.

The support--registration construction above is the owned object of this paper. Three companion records instantiate parts of that construction on public computational, literature, and figure-derived surfaces without changing the theorem or equation status of the present paper.

center

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center

These records may be used as bounded support for the public readability of the QTT gates [citation]. They do not replace the standard tunneling formulae recovered above, do not add a detector Hamiltonian, do not close QED or Maxwell electrodynamics, and do not decide tunneling-time questions. The same-family cross-window gate in reference remains a declared-family failure condition, not a claim that unrelated STM, MIM, and Fowler--Nordheim data must share one barrier family.

Barrier transmission, phase loading, and detector registration obey different structural conditions. A phase-dependent barrier law represents an action-level loading only if the extracted functional has a formally self-adjoint Fr\'echet derivative. A finite completely monotone detector memory has minimal positive-pole order equal to the stabilized rank of its moment Hankel matrices; the one-pole case requires $\mu_0\mu_2-\mu_1^2=0$TeX\mu_0\mu_2-\mu_1^2=0. A registration sink is probability conserving only after a factorization $K=\sum_aL_a^\dagger L_a$TeXK=\sum_aL_a^\dagger L_a supplies the complementary CPTP event branches.

These tests make the factorization operational. Transmission data constrain the Hamiltonian barrier sector, lag data constrain the memory rank, and counts constrain the instrument. A common phase mechanism becomes predictive only if one frozen parameter map spans all three observable groups with lower rank than the combined basket. The corresponding left-null residuals, rather than separate within-group fits, provide the cross-sector rejection test.

The closure test for the barrier-mediated transmission and tunneling spectroscopy 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 currents, spectra, dwell-time proxies, transients, and event counts. Let $a$TeXa range over the independent constitutive inputs comprising barrier action, interface density of states, transient memory, and registration instrument.

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 microscopic barrier--lead Hamiltonian fixes support, response, memory, and registration across every declared group.

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 the support--registration reparametrization otherwise permits an arbitrary function to move between factors without changing the data. Failure of that condition therefore rejects the proposed microscopic closure before parameter estimation can be counted as evidence for it.

Barrier-mediated tunneling is fixed as a support--registration problem. The barrier supplies the evanescent support factor $\exp[-2\Acal_B]$TeX\exp[-2\Acal_B]; the interface supplies registration through density of states, bias windows, lead structure, tip geometry, surface states, or detector availability; the measured object is a current, conductance, count, or event only after that registration layer closes.

The standard rectangular-barrier coefficient, the WKB opaque envelope, the Tersoff--Hamann STM relation, Simmons-type junction behavior, Fowler--Nordheim field emission, and tunnel-spectroscopy current integrals are reproduced on their standard approximation domains. By reference, the measured current does not identify the support action without independently fixed registration data or a restrictive joint parametrization. Agreement with these established formulas therefore demonstrates consistency of the representation, not a CHC-specific tunneling prediction.

The resulting CHC statement is not that tunneling is less real. It is that what is real is layered: suppressed phase-link support exists across the barrier, while eventhood and current belong to exit-side registration. This preserves the standard formulas while removing the hidden-path ontology.

The following statements isolate the algebraic pieces intended for Arb/FLINT, SymPy/SageMath, Lean, and Rocq checking.

lemma: Dimensionless support action. Let $m>0$TeXm>0, $U(x)-E\ge0$TeXU(x)-E\ge0 on $[x_-,x_+]$TeX[x_-,x_+], and $\hbarred>0$TeX\hbarred>0. Then

\Acal_B(E)=\int_{x_-}^{x_+}\sqrt{2m(U(x)-E)}\,\dd x/\hbarred

is dimensionless in SI units.

proof. The integrand numerator has units $\sqrt{M\cdot ML^2T^{-2}}\,L=ML^2T^{-1}$TeX\sqrt{M\cdot ML^2T^{-2}}\,L=ML^2T^{-1}, which are the units of action. Division by $\hbarred$TeX\hbarred gives a dimensionless quantity.

lemma: Support boundedness. If $\Acal_B\ge0$TeX\Acal_B\ge0, then $0<\exp[-2\Acal_B]\le1$TeX0<\exp[-2\Acal_B]\le1. If $\Acal_B>0$TeX\Acal_B>0, then $0<\exp[-2\Acal_B]<1$TeX0<\exp[-2\Acal_B]<1.

proof. This follows from monotonicity and positivity of the real exponential.

lemma: Rectangular opaque expansion. For $z=\kappa a\gg1$TeXz=\kappa a\gg1,

\left[1+Q\sinh^2z\right]^{-1}
=Q^{-1}4\ee^{-2z}\left[1+O(\ee^{-2z})+O(Q^{-1}\ee^{-2z})\right]

provided $Q\ee^{2z}\gg1$TeXQ\ee^{2z}\gg1. With $Q=U_0^2/[4E(U_0-E)]$TeXQ=U_0^2/[4E(U_0-E)], this gives reference.

proof. Use $\sinh^2z=(\ee^{2z}-2+\ee^{-2z})/4$TeX\sinh^2z=(\ee^{2z}-2+\ee^{-2z})/4, factor $(Q/4)\ee^{2z}$TeX(Q/4)\ee^{2z} from the denominator, and expand the reciprocal.

lemma: Triangular barrier action. For $\phi>0$TeX\phi>0 and $F>0$TeXF>0,

\int_0^{\phi/(eF)}\frac{\sqrt{2m(\phi-eFx)}}{\hbarred}\,\dd x
=
\frac{2\sqrt{2m}\,\phi^{3/2}}{3e\hbarred F}.

proof. Let $u=\phi-eFx$TeXu=\phi-eFx. Then $\dd x=-\dd u/(eF)$TeX\dd x=-\dd u/(eF), the integration limits become $u=\phi$TeXu=\phi and $u=0$TeXu=0, and the integral is $\sqrt{2m}\,(eF\hbarred)^{-1}\int_0^\phi u^{1/2}\dd u$TeX\sqrt{2m}\,(eF\hbarred)^{-1}\int_0^\phi u^{1/2}\dd u.

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Funding and competing interests..

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