De Broglie Recovery and Accessible Wavefunction Semantics: Phase-Link Kinematics on a Global Phase Field

The de Broglie relation

\lambdadB = \frac{h}{p}

remains one of the most stable kinematical laws in quantum physics. Its empirical range now spans large organic molecules [citation], native polypeptides [citation], positronium diffraction [citation], nanoparticle interferometry [citation], and continuously trapped atom interferometers with explicitly characterized dephasing control [citation]; a broader cluster- and molecule-scale overview is given in [citation]. Wave mechanics ties momentum to phase gradient [citation], while the interpretation of the wavefunction remains contested [citation]. Decoherence accounts for the loss of accessible interference but does not by itself provide durable record formation or a complete measurement theory [citation]. Detector tomography, self-characterization, spatially resolved detector tomography, generalized-measurement implementations, repeated-measurement benchmarking, and recent precision bounds for measurement characterization sharpen operational measurement semantics without by themselves supplying durable-readout architecture or shared-fact objectivity [citation].

The analysis is restricted to the upstream semantic/kinematical layer. Propagation is organized by phase-link structure induced by the global phase field, and detector-side closure enters here only through the boundary-marker notion of local commit on a declared finite observation domain. Within that boundary, the standard de Broglie law is recovered in a phase-response eikonal limit, the wavefunction is treated as an accessible phase-statistics representation on a finite observation domain rather than as either a material substance or mere bookkeeping ignorance, and finite-domain/gate/phase-gradient corrections are organized only after the recovery regime has been explicitly fixed. Global phase-field organization, finite accessibility, event closure, and recovered statistics remain distinct layers rather than collapsing into a single symbol $\Psi$TeX\Psi.

The analysis is limited to the recovery law reference in the declared eikonal regime, the role assignment of the wavefunction as an accessible phase-statistics representation on a finite observation domain, and the admissible fixed-family correction and residual-visibility organizers introduced below. The local-commit notion is used only as a boundary marker between accessible propagation and detector-side analyses outside the declared scope. No claim is made here about detector architecture, durable-readout construction, shared-fact objectivity, or a full Born-rule derivation.

Scope and boundary

The analysis is restricted to the semantic/kinematical layer in which ordinary de Broglie kinematics is recovered and finite-domain phase-response corrections are organized. Detector readout architecture, durable-record criteria, and shared-fact objectivity are not derived here. Full Born-rule derivations, global resolutions of $\psi$TeX\psi-ontic/$\psi$TeX\psi-epistemic disputes, and Bell- or Kochen--Specker-type analyses also lie outside the declared scope. The analysis is limited to accessible propagation, the recovery meaning of $\lambdadB=h/p$TeX\lambdadB=h/p, and controlled visibility/coherence tests on declared comparison families.

remark: Recovery discipline. Ordinary quantum kinematics is retained as the declared recovery regime. Equation reference is treated as a recovery law valid in a declared low-$\Xiph$TeX\Xiph, gate-weak, finite-domain regime. The proposed correction class is meaningful only after that standard regime has been explicitly recovered.

Observation domain and local commit

definition: Finite observation domain. A declared observation domain is the tuple

\Dobs = (\Odom_{\rm geom},T,B,\Ggate),

where $\Odom_{\rm geom}$TeX\Odom_{\rm geom} denotes the admitted geometric/path sector, $T$TeXT the observation window, $B$TeXB the accepted bandwidth or resolution band, and $\Ggate$TeX\Ggate the gate architecture relevant to event registration.

definition: Local commit. A local commit is the event-local closure marker by which previously accessible phase-link structure is reallocated into a recordable outcome class within the declared gate architecture $\Ggate$TeX\Ggate. In this role, the notion is used only as a boundary marker between accessible propagation and detector-side analyses outside the declared scope. Persistence, amplification, durable readout, and shared-fact status are not part of this definition.

Layer separation

Four layers remain distinct in the discussion of the wavefunction and measurement:

- Global phase-link structure: global phase-field organization on which propagation is sustained; - Finite accessibility: the restriction of operationally available phase information to the declared domain $\Dobs$TeX\Dobs; - Local commit: event-local closure into a recordable outcome class; - Recovered statistics: stable frequencies obtained under repeated experiments carried out within the same declared domain.

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Ontology, accessibility, event closure, and recovered frequencies are assigned to distinct layers rather than to a single mathematical object. Here $\Psi$TeX\Psi carries the second layer; the third enters only through the notion of local commit, and the fourth only through a limited recovery bridge. This assignment keeps de Broglie recovery and visibility corrections separate from detector microphysics and shared-fact criteria.

principle: Phase-link propagation. Free propagation is represented as the maintenance of a phase-link structure organized by the global phase field $\Hbg$TeX\Hbg. When a cosmological reading is involved, this global phase field may encode an expansion-response branch; it is not a fixed material background and not a literal spreading medium.

For electromagnetic propagation, the same convention treats light not as an additional primitive substance, but as the observable name for a commit-capable electromagnetic energy-information excitation on the phase-link structure induced by the global phase field and conditioned by local phase-field response where matter, media, boundary, or detector conditions are present. Photon language remains the standard quantum recovery description of the corresponding electromagnetic field excitation or detected energy quantum.

principle: Finite-domain accessibility. Only phase information accessible within the declared observation domain $\Dobs$TeX\Dobs contributes to the operational wave description relevant to that domain.

definition: Effective phase-response ansatz. A free material state is represented by

\Psiacc(x)=A(x)e^{i\theta(x)},
\qquad
\theta(x)=\theta_0+\alpha\,S(x;\Hbg),

where $A(x)$TeXA(x) is a slowly varying amplitude on the relevant transport scale and $S(x;\Hbg)$TeXS(x;\Hbg) is an action-like phase-response functional determined by the declared global phase-field configuration $\Hbg$TeX\Hbg.

definition: Recovery-limit momentum map. Define the local phase covector and the recovery-limit momentum map by

k_\mu=\nabla_\mu\theta,
\qquad
p_\mu=\hbar k_\mu .

Equation reference is used as the momentum map of the accessible propagation layer in the recovery regime; it is not introduced as a new microscopic postulate about all hidden or detector-side structure.

definition: Momentum-conditioned phase-link period. The spatial de Broglie period is defined by

\lambdadB = \frac{2\pi}{|\bm{k}|}.

In the accessible-phase semantics, $\lambdadB$TeX\lambdadB is the spatial period of a momentum-conditioned phase link, not the wavelength of a material substance.

proposition: De Broglie recovery in the declared phase-response eikonal regime. Under Principles reference and reference, the ansatz reference, and the recovery-limit momentum map reference, the eikonal regime yields

\lambdadB = \frac{h}{p}

as the leading-order phase-link period.

proof. In the eikonal regime the amplitude varies on scales much larger than the local phase period, so translations are governed at leading order by the phase increment $\Delta\theta=k_\mu\Delta x^\mu$TeX\Delta\theta=k_\mu\Delta x^\mu. A spatial translation along the propagation direction that changes the phase by $2\pi$TeX2\pi therefore defines one full phase-link period, yielding reference. Using reference, one has $|\bm{p}|=\hbar|\bm{k}|$TeX|\bm{p}|=\hbar|\bm{k}|, and therefore

\lambdadB = \frac{2\pi}{|\bm{k}|}=\frac{2\pi\hbar}{p}=\frac{h}{p}.

Hence the ordinary de Broglie law is recovered as the leading-order spacing of the accessible phase link.

remark: Why $1/p$TeX1/p is natural here. In the accessible-phase semantics, larger momentum means a larger local phase gradient and therefore a shorter spatial interval over which the phase can advance by $2\pi$TeX2\pi. The inverse proportionality $\lambdadB\propto 1/p$TeX\lambdadB\propto 1/p is thus the natural spacing law of a momentum-conditioned phase link.

definition: Accessible wavefunction semantics. The wavefunction $\Psiacc$TeX\Psiacc is the effective representation of accessible phase statistics on the declared domain $\Dobs$TeX\Dobs. It is not taken to be a literal material substance, and it is not reduced to bookkeeping ignorance. Its role is to compress the part of the phase-link organization that remains operationally accessible in the declared observational regime. It is family-relative and domain-relative rather than a platform-independent record or objectivity variable.

remark: Relation to $\psi$TeX\psi-ontic / $\psi$TeX\psi-epistemic debates. The broader ontology debate surveyed by Harrigan and Spekkens [citation] and the constraints highlighted by Pusey, Barrett, and Rudolph [citation] lie outside the declared scope. With finite accessibility and local commit kept explicit, $\Psiacc$TeX\Psiacc is used here only as the representation of accessible phase statistics on the declared domain.

proposition: Within the present accessible phase-link semantics, two-path interference does not require material splitting. Two-path interference does not require a material substance to split into two parts. It requires only that the relevant phase-link structure remain coherent across the two accessible paths until a local commit occurs.

proof. At the propagation stage, the relative phase between the two accessible paths is encoded in $\Psiacc$TeX\Psiacc, so the interference term depends on the superposed phase response rather than on a literal partition of material substance into two separate objects. A detector-side closure on the declared domain is represented here only through the boundary-marker notion of local commit, which closes the previously accessible phase-link organization into one recordable outcome class without supplying detector microphysics, durable readout, or shared-fact status. Propagation and event closure therefore belong to different layers.

remark: Decoherence is not yet commit. The semantic layer separates loss of accessible interference from event formation. Decoherence can suppress or scramble accessible phase relations, but suppression alone is not yet a recordable event [citation]. A local commit is the event-local closure marker by which one outcome class becomes recordable within the declared gate architecture. Persistence, amplification, durable-readout realization, and shared-fact status remain outside the declared scope, and the detector-side mechanism that realizes them is not derived here.

remark: Consequence of the layer separation. The layer separation keeps global phase-field organization, operational accessibility, event closure, and recovered frequencies in distinct roles. With that split explicit, de Broglie recovery, wavefunction semantics, and visibility corrections can be discussed without attributing detector-side recorded-outcome structure to propagation itself.

definition: Correction controls. The correction class introduced below depends on three declared controls: the finite observation domain $\Dobs$TeX\Dobs, the gate architecture $\Ggate\subset\Dobs$TeX\Ggate\subset\Dobs, and a dimensionless phase-gradient hierarchy parameter $\Xiph$TeX\Xiph that orders departures from the recovery regime. $\Xiph$TeX\Xiph functions only as a correction-ordering control variable in the analysis; it is not itself identified with a detector or readout observable.

definition: Observed-wavelength correction class. The observed wavelength is written as

\lambdaobs
=
\lambdadB\,F(\Xiph,\Dobs)
=
\frac{h}{p}\,[1+\dOmega+\dXi+\dG+\mathcal O((\dOmega,\dXi,\dG)^2)],

where

- $\dOmega$TeX\dOmega abbreviates combined finite-domain corrections from geometry, observation time, and accepted bandwidth; - $\dXi$TeX\dXi encodes hierarchy-controlled phase-gradient corrections; - $\dG$TeX\dG encodes gate-conditioned corrections in the inferred wavelength on the declared observation domain and does not modify the intrinsic recovery-limit phase-link period $\lambdadB=h/p$TeX\lambdadB=h/p.

Here $\mathcal O((\dOmega,\dXi,\dG)^2)$TeX\mathcal O((\dOmega,\dXi,\dG)^2) denotes terms that are at least quadratic in the declared first-order correction controls.

remark: Operational status of the gate term. The gate-conditioned contribution $\dG$TeX\dG belongs to the inference layer on the declared observation domain. It is not a redefinition of the intrinsic recovery-limit phase-link period and it does not extend detector architecture beyond the boundary object of local commit.

remark: Admissibility clauses for $F$TeXF. The correction factor $F$TeXF is admissible only if $F>0$TeXF>0 on the declared comparison window and $F\to 1$TeXF\to 1 in the low-$\Xiph$TeX\Xiph, gate-weak, finite-domain recovery limit. The linearized decomposition in reference is first-order bookkeeping, not a claim that all correction sources are exactly separable beyond that window.

proposition: Admissible finite-domain corrections preserve de Broglie recovery on the declared recovery window. The correction class reference is admissible only if the standard de Broglie law is recovered in the declared recovery regime. The correction class therefore supplements rather than denies ordinary wave kinematics.

proof. By construction, the correction factor $F$TeXF is normalized to $1$TeX1 in the low-$\Xiph$TeX\Xiph, gate-weak, finite-domain limit. Any admissible correction therefore vanishes in that regime, leaving $\lambdaobs\to h/p$TeX\lambdaobs\to h/p. If no such limit exists, the correction class is inadmissible rather than a replacement for ordinary kinematics.

definition: Accessible coherence ceiling. The accessible coherence length is written as

\Lcoh \sim \lambdadB\,\Phicoh(T,B,\Ggate,\Xiph),

with $\Phicoh>0$TeX\Phicoh>0 a dimensionless function of the observation window, accepted bandwidth, gate burden, and hierarchy control. On declared comparison families, shorter de Broglie periods or stronger gate burden can reduce the scale on which phase-link coherence remains experimentally accessible within the declared domain. Any directional monotonicity invoked for $\Phicoh$TeX\Phicoh is understood only on the declared comparison window and is not asserted here as a universal law beyond that window.

remark: Why short $\lambdadB$TeX\lambdadB is a natural leverage variable. Because the key dimensionless control often involves $L/\lambdadB$TeXL/\lambdadB, short de Broglie wavelengths provide natural leverage for correction searches. Large molecules, biomolecular matter waves, nanoparticles, and trapped matter-wave platforms therefore provide useful comparison classes for this program [citation].

Declared comparison family

definition: Declared comparison family. A declared comparison family is the tuple

\mathcal F_{\mathrm{cmp}}
=
(\mathcal P,\mathcal O_V,\mathcal B_{\mathrm{env}},\mathcal B_{\mathrm{gate}},
\mathcal U_{\Xi},\mathsf{Est},\mathcal W),

where $\mathcal P$TeX\mathcal P is the fixed platform class and geometry, $\mathcal O_V$TeX\mathcal O_V the fixed visibility/coherence observable and extraction rule, $\mathcal B_{\mathrm{env}}$TeX\mathcal B_{\mathrm{env}} and $\mathcal B_{\mathrm{gate}}$TeX\mathcal B_{\mathrm{gate}} the declared environmental and gate baselines, $\mathcal U_{\Xi}$TeX\mathcal U_{\Xi} the externally declared monotone control or proxy family used to order $\Xiph$TeX\Xiph on the admitted window, $\mathsf{Est}$TeX\mathsf{Est} the common estimator and uncertainty rule, and $\mathcal W$TeX\mathcal W the admitted comparison window. The tuple $\mathcal F_{\mathrm{cmp}}$TeX\mathcal F_{\mathrm{cmp}} is the declared comparison family, and all downstream residual-visibility statements are read only relative to this declared family and its admitted response window $\mathcal W$TeX\mathcal W.

Declared visibility law

The empirical handle adopted here is a residual-visibility program once the comparison family $\mathcal F_{\mathrm{cmp}}$TeX\mathcal F_{\mathrm{cmp}} and the conventional environmental and gate-related contributions have been declared. We write

V = V_0\exp[-\Cenv-\Cgate-\Cxi],
\qquad
\Cxi = \kappaxi\,\Xiph\left(\frac{L}{\lambdadB}\right)^m,

with $V_0$TeXV_0 the declared baseline visibility of the comparison family after the admitted preparation/readout normalization, $\sigV$TeX\sigV the admitted residual-visibility uncertainty band used for upper-limit reporting on the same comparison window, $m>0$TeXm>0, $\kappaxi\ge 0$TeX\kappaxi\ge 0, and $\Cenv,\Cgate$TeX\Cenv,\Cgate reserved for conventional environmental and gate-induced coherence losses. Here $L$TeXL denotes the declared interferometric transport scale or effective path-separation scale used for the same comparison family; it is fixed on the admitted comparison window and is not retuned run by run. Known matter-wave decoherence mechanisms already populate $\Cenv$TeX\Cenv, including collision-, thermal-, internal-state-, dipole-interaction-, and platform-specific contributions [citation]. The residual term $\Cxi$TeX\Cxi is used only as a declared comparison-family organizer for any additional visibility suppression whose leverage increases with $L/\lambdadB$TeXL/\lambdadB.

remark: Comparison-family fixing. Equation reference is admitted only relative to a declared comparison family in the sense of Definition reference. All elements of $\mathcal F_{\mathrm{cmp}}$TeX\mathcal F_{\mathrm{cmp}} are fixed before residual data are inspected, and post hoc replacement of the baseline model, visibility extractor, control proxy, fit window, or estimator is inadmissible. The decomposition $\Cenv+\Cgate+\Cxi$TeX\Cenv+\Cgate+\Cxi is first-order bookkeeping on the declared window, not a claim of exact separability at arbitrary scale. Within a fixed $\mathcal F_{\mathrm{cmp}}$TeX\mathcal F_{\mathrm{cmp}}, the exponent $m$TeXm and coefficient $\kappaxi$TeX\kappaxi are held fixed across the declared dataset or platform subset and are not re-fitted independently run by run.

remark: Detector-facing non-role of the visibility law. Equation reference is not a detector-click counting law. Detector dead time, afterpulsing, window-to-window memory, and durable-readout accumulation belong to downstream detector-facing architecture and calibration analyses rather than to the semantic/kinematical layer treated here [citation]. The present use of $\Cgate$TeX\Cgate is restricted to the declared inference layer on the comparison family.

proposition: Fixed-family upper-limit reporting from null residuals on a declared comparison window. If no residual visibility loss beyond declared environmental and gate baselines is observed, the experiment yields an upper limit on $\kappaxi$TeX\kappaxi rather than a null interpretational statement.

proof. Define the baseline-subtracted residual

\Visres := -\ln\!\left(\frac{V_{\rm meas}}{V_0e^{-\Cenv-\Cgate}}\right).

Together with the admitted tolerance band $\sigV$TeX\sigV, $\Visres$TeX\Visres defines the declared residual witness convention on the same comparison family and admitted response window; all downstream structural upper-limit statements are read only relative to that fixed convention. If the data are consistent with $|\Visres|\le \sigV$TeX|\Visres|\le \sigV, then reference implies

|\kappaxi| \lesssim \frac{|\Visres|+\sigV}{\Xiph\,(L/\lambdadB)^m},

for the declared experimental window. Thus a null residual becomes an upper bound on the residual coefficient rather than an undisciplined absence claim.

Structural reading of the empirical interface

No benchmark comparison family, benchmark recovery window, or benchmark residual-evaluation policy is certified by the structural recovery statements alone. Its empirical interface is therefore structural only: a later benchmark-facing comparison record would have to freeze one comparison family, one recovery window, one residual witness convention, and one residual-evaluation policy before any positive-residual or release-facing quantitative claim could be attached. Contemporary matter-wave platforms are cited only as examples of the kinds of declared comparison families on which the structural recovery and upper-limit framework could later be instantiated [citation].

Empirical testing protocol

The empirical program is stated with explicit exclusion and upper-limit rules. It proceeds in three steps. First, one demonstrates a recovery window in which $\lambdaobs\approx h/p$TeX\lambdaobs\approx h/p and the observed visibility is controlled by declared $\Cenv$TeX\Cenv and $\Cgate$TeX\Cgate baselines. Second, one declares a comparison family in the sense of Definition reference, fixing the visibility extractor, environmental and gate baselines, control/proxy ordering for $\Xiph$TeX\Xiph, estimator, comparison window, and any platform-independent parameters before residual data are inspected. Third, one varies the leverage variable $L/\lambdadB$TeXL/\lambdadB and the declared domain controls while maintaining stated control over $\Cenv$TeX\Cenv and $\Cgate$TeX\Cgate. If no residual remains, the result is the explicit upper limit reference for the declared domain rather than a redefinition of the recovery law. A positive residual claim requires a separately frozen benchmark record.

remark: External observable. The most natural external observable here is the scaling of fringe visibility and accessible coherence length with $L/\lambdadB$TeXL/\lambdadB, mass, momentum, and gate burden. This observable provides a practical interface with current interferometric platforms while preserving the semantic/kinematical scope of the paper.

- Failure of de Broglie recovery. If a declared low-$\Xiph$TeX\Xiph, gate-weak recovery regime fails to reproduce $\lambda=h/p$TeX\lambda=h/p, the recovery claim fails. - No admissible correction class. If experimental residuals cannot be organized into a finite-domain/environment/gate hierarchy with $F\to 1$TeXF\to 1 in the declared recovery limit, then the correction class reference is not admissible. - No residual scaling. If declared matter-wave platforms show no reproducible residual dependence on $L/\lambdadB$TeXL/\lambdadB beyond known $\Cenv$TeX\Cenv and $\Cgate$TeX\Cgate, then the analysis supports only upper limits such as reference and no positive $\Cxi$TeX\Cxi claim; conversely, any future positive-residual claim remains outside the declared comparison-family statement unless a separately frozen benchmark record is attached. - Propagation/event conflation. If the semantic/kinematical layer cannot be separated from detector microphysics, durable-readout architecture, or shared-fact objectivity even at the level of role assignment, then the present decomposition fails and the paper becomes overextended.

remark: Scope limits. The paper does not derive the Born rule, solve the full measurement problem, decide the $\psi$TeX\psi-ontic/$\psi$TeX\psi-epistemic dispute in general, or provide a relativistic quantum-field-theoretic completion. The analysis is limited to semantic/kinematical recovery and to a disciplined correction program for matter-wave visibility and coherence.

The recovery relation reference identifies $\lambdadB=h/p$TeX\lambdadB=h/p as the leading-order spatial period of a momentum-conditioned phase link in the phase-response eikonal regime. The wavefunction then serves as an accessible phase-statistics representation on a declared finite observation domain, not as a literal material substance or a detector/objectivity variable.

On fixed comparison families, finite-domain corrections are organized by reference, accessible coherence by reference, and residual visibility by reference with upper-limit reporting through reference. Global phase-field organization, finite accessibility, local phase-field response, local commit, and recovered statistics remain distinct. Within that separation, departures from the recovery regime remain a matter for controlled coherence and visibility tests rather than for reinterpretation of the recovery law itself. The fixed-family residual organizer $\Cxi$TeX\Cxi does not define a detector-threshold law, a Born-rule derivation, or a shared-fact criterion. The local-commit notion remains only a boundary marker between accessible propagation and detector-side analyses outside the declared scope. No benchmark family, recovery window, or residual-evaluation policy is treated as frozen into a sealed positive-residual comparison class. If the declared recovery window or residual-separation conditions fail, the analysis supports only fixed-family upper limits on residual visibility.

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Funding and competing interests..

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