Bound-State Vibrational Phase Spectra, Electronic Transitions, and Quantum Discreteness in the CHC Framework

Discrete line spectra, spectrally resolved absorption and emission, channel-selective suppression, and quantum-jump records are standard surface signatures of bound quantum systems. Precision hydrogen spectroscopy continues to resolve line frequencies at extraordinarily high accuracy [citation]. Direct time-domain records of quantum-jump-like detector events also show that observed discreteness can close at the readout layer while remaining embedded in a longer coherent dynamics [citation]. Cavity and solid-state platforms exhibit the same basic surface pattern---discrete mode splitting, channel-dependent line weights, and sideband structure---under very different microscopic realizations [citation]. The analysis below assigns these phenomena to the global/local phase-field ontology of by separating propagation phase, local commit, and durable readout.

Common propagation classes on the global phase field separate phase-linked propagation from local event closure [citation]. Accessible-event statistics on admitted effect domains and detector-local opening/commit formalism then separate propagation, local commit, and durable record into distinct layers [citation]. De Broglie recovery and accessible wavefunction semantics further restrict wavefunction language to accessible phase-link kinematics rather than to a material substance [citation]. Mass as phase rigidity in stable composite matter already provides the imported rigidity language needed to read bound structures through their internal phase response rather than through an ontic orbit picture [citation]. These imported objects fix the admissible reading of an electronic transition: the transition cannot be a literal motion of an autonomous particle between ontic paths, because both wavefunction and propagation language are already formally specified more narrowly in the root and de Broglie layers.

Fix one declared bound-state family and one declared carrier-channel set. Within that admitted family, an electronic transition is read as a relocking of a phase-cohesive bound-state configuration between admissible rigidity modes, and the observable line spectrum is the carrier-export map of the corresponding mode-separation differences. On one declared family with a discrete spectral sector and admitted weak carrier coupling, the line catalog and detector-facing line profile therefore admit a formally specified formulation. No claim is made here about a full QED derivation, detector microdynamics closure, all-atom precision spectroscopy fit, compactification closure, selected-family completion, ultraviolet completion, or string-theory identity. The supporting microscopic seat introduced below is not promoted to a compactification theorem.

A declared bound-state family is specified by an admitted branch label $b$TeXb, a Hilbert space $\Hbind^{(b)}$TeX\Hbind^{(b)}, a self-adjoint bound-state Hamiltonian $\Hint^{(b)}$TeX\Hint^{(b)}, a declared carrier-channel set $\mathfrak C^{(b)}$TeX\mathfrak C^{(b)}, and a detector domain $\Dobs=(\Ogeom,T,B,\Ggate)$TeX\Dobs=(\Ogeom,T,B,\Ggate) used only at the readout layer. The same family may be supported by an auxiliary microscopic seat $\Kint^{(b)}$TeX\Kint^{(b)} carrying an internal response operator

\Lint^{(b)}\chi^{(b)}_\nu(y)=\lambda^{(b)}_\nu\chi^{(b)}_\nu(y),
\qquad y\in \Kint^{(b)},

but that supporting seat is used only as a microscopic basis organizer. No compactification or selected-family closure theorem is asserted from reference.

The admitted effective Hamiltonian family is written as

\Hint^{(b)}
=
\sum_a \frac{(\hat p_a-\mathcal A_a^{(b)})^2}{2m_{a,\mathrm{eff}}(\Xibind)}
+
V_{\mathrm{coh}}^{(b)}(\mathbf q;\HH),

where $m_{a,\mathrm{eff}}(\Xibind)$TeXm_{a,\mathrm{eff}}(\Xibind) is a declared rigidity-sensitive effective inertial parameter imported from the composite-matter rigidity analysis, and $V_{\mathrm{coh}}^{(b)}$TeXV_{\mathrm{coh}}^{(b)} denotes the bound-state coherence potential on the admitted branch. Equation reference is used only on a nonrelativistic declared bound-state family. No universal microscopic Hamiltonian across all atoms, molecules, solids, or detector platforms is claimed.

definition: Bound-state phase-cohesive mode family. A bound-state phase-cohesive mode family on a declared branch $b$TeXb is a tuple

\mathfrak B^{(b)}=(\Hbind^{(b)},\Hint^{(b)},\{\psi_n^{(b)}\}_{n\in\mathcal I_b},\mathfrak C^{(b)},\Dobs)

with the following properties:

- $\Hint^{(b)}$TeX\Hint^{(b)} is self-adjoint and bounded from below on its declared domain; - the declared spectral sector on the family is pure point, equation ^(b)\psi_n^(b)=E_n^(b)\psi_n^(b), n I_b, equation with $E_n^{(b)}<E_{n+1}^{(b)}$TeXE_n^{(b)}<E_{n+1}^{(b)} after multiplicities are resolved by the declared quantum numbers; - each admitted carrier channel $\alpha\in\mathfrak C^{(b)}$TeX\alpha\in\mathfrak C^{(b)} is represented by a densely defined coupling operator $\Cchan_\alpha^{(b)}$TeX\Cchan_\alpha^{(b)} on the declared spectral sector; and - the detector domain $\Dobs$TeX\Dobs is used only to define local opening and local commit closure of observed events and does not enter the spectral-sector ontology.

remark: Subordinate microscopic vibrational basis only. When the auxiliary seat $\Kint^{(b)}$TeX\Kint^{(b)} is compact or effectively finite in the declared microscopic sector, the resulting mode basis may be read as a string-like or vibrational microscopic basis. That language is subordinate and purely supportive. It is not promoted here to a string ontology, a compactification theorem, or a containment claim between and string or M-theoretic frameworks.

The phase-rigidity reading imported from stable composite matter is turned here into a bound-state spectral object. For each mode in the declared family, define the rigidity frequency and rigidity scale by

\omega_n^{(b)}:=\frac{E_n^{(b)}-E_{\mathrm{ref}}^{(b)}}{\hbar},
\qquad
\mathcal{R}_{\mathrm{ph},n}^{(b)}:=\hbar\omega_n^{(b)},

where $E_{\mathrm{ref}}^{(b)}$TeXE_{\mathrm{ref}}^{(b)} is a declared family reference energy. The spectral separation between two modes is

\Delta E_{m\to n}^{(b\to b')}:=E_m^{(b)}-E_n^{(b')},
\qquad
\omega_{m\to n}^{(b\to b')}=\frac{\Delta E_{m\to n}^{(b\to b')}}{\hbar}.

In the common case $b'=b$TeXb'=b, reference is the line frequency of the family-conditioned relocking.

A supportive toy realization tied to the existing rigidity language is

E_n(\Xi)=\hbar\omega_0\sqrt{1-\Xi}\left(n+\frac12\right),
\qquad
\Rph^{(\mathrm{HO})}\propto\hbar\omega_0,

retained only as a witness that the rigidity scale can be read directly from a discrete mode family. Equation reference is not promoted to a universal microscopic law.

proposition: Admissible rigidity-spectrum proposition on one declared bound-state family. Let $\mathfrak B^{(b)}$TeX\mathfrak B^{(b)} be a bound-state phase-cohesive mode family in the sense of reference. Then the declared line catalog

\Supp^{(b)}:=\left\{\omega_{m\to n}^{(b\to b)}:\ m,n\in\mathcal I_b,\ m>n\right\}

is a countable subset of $\mathbb R_{>0}$TeX\mathbb R_{>0}, and every element of reference is a difference of rigidity scales,

\omega_{m\to n}^{(b\to b)} = \frac{\mathcal{R}_{\mathrm{ph},m}^{(b)}-\mathcal{R}_{\mathrm{ph},n}^{(b)}}{\hbar}.

proof. By reference, the declared spectral sector is pure point and indexed by a countable set. Its ordered pair set is therefore countable, so the image of that pair set under $(m,n)\mapsto(E_m-E_n)/\hbar$TeX(m,n)\mapsto(E_m-E_n)/\hbar is countable. For $m>n$TeXm>n, the separation $E_m^{(b)}-E_n^{(b)}$TeXE_m^{(b)}-E_n^{(b)} is strictly positive, and reference follows directly from the definition of $\mathcal{R}_{\mathrm{ph},n}^{(b)}$TeX\mathcal{R}_{\mathrm{ph},n}^{(b)}. Pure-point support alone does not imply that the difference set is locally finite: transition frequencies may have finite accumulation points. A spectrally discrete line catalogue in the stronger, locally finite sense therefore requires an additional gap or finite-window local-finiteness condition.

Equations reference--reference are the first exact declared result of the paper. They do not say that every observed line is already recorded or that every spectral difference is physically exported through every channel. They say only that, on one declared family, the line catalogue is the countable difference set of admissible rigidity modes; local finiteness must be checked separately on the spectral window.

The next object is the channel-dependent relocking amplitude between declared modes. The carrier is not the ontic cause of the transition; it is the admitted export or intake channel of a bound-state phase reconfiguration.

definition: Carrier-coupled relocking amplitude. For an admitted channel $\alpha\in\mathfrak C^{(b)}$TeX\alpha\in\mathfrak C^{(b)} and declared initial/final modes $\psi_m^{(b)}$TeX\psi_m^{(b)}, $\psi_n^{(b')}$TeX\psi_n^{(b')}, define the carrier-coupled relocking amplitude

\Mamp_{m\to n}^{(\alpha)}
:=
\mel{\psi_n^{(b')}}{\Cchan^{(b,b')}_\alpha}{\psi_m^{(b)}}.

A channel is admissible on the declared family if reference is well defined on the spectral sector and if the channel spectral density $\rho_\alpha(\omega)$TeX\rho_\alpha(\omega) is finite on the corresponding mode-separation window.

Under weak channel coupling and a declared Markovian export window, the transition rate is the standard first-order spectral rate of the declared channel,

\Gammaexp_{m\to n}^{(\alpha)}
=
\frac{2\pi}{\hbar}
\abs{\Mamp_{m\to n}^{(\alpha)}}^2
\rho_\alpha\!\left(\frac{\Delta E_{m\to n}^{(b\to b')}}{\hbar}\right).

The formula is adopted only on the weak-channel family just stated and is not promoted to a full microscopic radiation theory.

proposition: Relocking amplitude and export-law proposition. Let $\mathfrak B^{(b)}$TeX\mathfrak B^{(b)} be a declared bound-state phase-cohesive mode family and let $\alpha$TeX\alpha be an admitted weak carrier channel in the sense of reference. Then the channel-resolved exported or absorbed spectral weight on that family is supported only on the line catalog reference, with rate given by reference.

proof. Because $\Cchan_\alpha$TeX\Cchan_\alpha is evaluated only between discrete spectral states in the declared family, every nonzero contribution to the first-order spectral rate must occur at a mode-separation frequency of the form reference. The weak-coupling transition formula then yields reference. No contribution appears away from the line catalog because the initial and final mode labels are discrete on the declared family.

The operative point is that reference is a carrier law, not an orbit-jump ontology. The channel weight is computed from a relocking amplitude between phase-cohesive modes. The carrier only exports or imports the corresponding mode separation.

Selection rules are retyped by the same logic. In the standard electric-dipole approximation, one says that a transition is allowed or forbidden according to whether a transition matrix element vanishes. The same structure survives here, but its meaning changes. The vanishing no longer says that a particle is prohibited from jumping; it says that the declared channel geometry does not supply an admissible phase-coupling path between the two modes.

corollary: Selection-channel suppression corollary. Let $\mathfrak B^{(b)}$TeX\mathfrak B^{(b)} and $\alpha\in\mathfrak C^{(b)}$TeX\alpha\in\mathfrak C^{(b)} be as above. If the channel spectral density is strictly positive at the transition frequency, then

\Gammaexp_{m\to n}^{(\alpha)}=0
\iff
\Mamp_{m\to n}^{(\alpha)}=0.

Without strict positivity of the spectral density, only $\Mamp_{m\to n}^{(\alpha)}=0\Rightarrow \Gammaexp_{m\to n}^{(\alpha)}=0$TeX\Mamp_{m\to n}^{(\alpha)}=0\Rightarrow \Gammaexp_{m\to n}^{(\alpha)}=0 follows. Therefore an allowed channel requires both an admitted nonzero phase-coupling path and nonzero carrier spectral support at the transition frequency.

proof. At the transition frequency, strict positivity of the spectral density makes the prefactor in reference positive. The rate then vanishes if and only if the squared matrix element vanishes. If the density is merely nonnegative, a zero density can instead suppress a nonzero matrix element, which proves only the stated one-way implication.

A hydrogenic one-photon $2p\to1s$TeX2p\to1s line is then an admitted channel because the corresponding coupling path is nonzero on the declared electric-dipole carrier geometry. A one-photon $2s\to1s$TeX2s\to1s channel is suppressed on that same geometry because the corresponding matrix element vanishes. The suppressed channel is not forbidden because nature disallows branch switching; it is suppressed because that specific one-carrier path does not open. Higher-order, multiphoton, or different carrier pathways may still exist, as shown experimentally for electric-dipole-forbidden transitions [citation].

The discrete spectral support of Proposition reference is not yet a detector record. The propagation layer, the local commit layer, and the durable-record layer remain distinct. This separation is already fixed by the imported propagation-class formalism, accessible-event statistics, detector-local opening, boundary insertion, and irreversible readout constructions [citation]. Once those imported distinctions are respected, discreteness appears at the detector-facing layer through local commit closure rather than through an ontic jump law.

definition: Detector-separated observed line map. Let $p_m$TeXp_m denote the occupation weight of the initial mode $m$TeXm on the declared detector window and let $\Rdet(\cdot)$TeX\Rdet(\cdot) be the admitted detector response kernel on the same window. The detector-separated observed line map is

I_{\mathrm{obs}}(\omega)
=
\sum_{m>n} p_m\,\Gammaexp_{m\to n}\,
\Rdet\!\left(\omega-\frac{\Delta E_{m\to n}}{\hbar}\right).

The kernel in reference is detector separated by construction and is not imported back into the spectral ontology of the bound state.

A local commit window $\Omega\subset T$TeX\Omega\subset T induces the event-closure probability

P^{\Omega}_{m\to n}
=
1-
\exp\!\left[-\int_{\Omega} \Gammaexp_{m\to n}(t)\,\dd t\right],

with no commitment here to any specific detector microdynamics beyond the imported opening/commit formalism.

proposition: Detector-facing discreteness as local-commit closure. On a declared detector window with detector response kernel $\Rdet$TeX\Rdet and local commit closure probability reference, the observed discrete line structure is a detector-facing closure of the mode-separation support reference; it is not an ontic statement that the propagation layer itself consists of literal discontinuous particle jumps.

proof. By Propositions reference and reference, the carrier-side support is discrete on the line catalog. Equation reference shows that every observed spectral contribution is a detector-separated image of that catalog. Equation reference then closes the corresponding channel into an event on the detector window. No step in reference or reference requires the propagation layer to be ontically discontinuous. The discreteness resides in the mode-separation support and in its local commit closure on the declared detector window.

This is the second exact declared result of the paper. It explains why one can observe jump-like detector events and line-like spectral peaks without promoting jump ontology to the propagation layer.

Standard textbook picture

In the standard Coulomb picture, the hydrogen atom is described by a bound electron in a Coulomb potential with discrete eigenstates labeled by principal and orbital quantum numbers. A line is emitted or absorbed when the system moves between two such eigenstates. At the one-photon electric-dipole level, the $2p\to1s$TeX2p\to1s channel is allowed, while the $2s\to1s$TeX2s\to1s channel is suppressed on the same one-photon channel geometry. Precision measurements of hydrogenic transition frequencies, including the $1S$TeX1S--$2S$TeX2S witness and its isotope-shift variants, establish the experimental sharpness of the discrete spectral structure [citation].

CHC-native relocking rewriting

In the present language, the hydrogenic bound state is a declared phase-cohesive mode family. The labels $1s$TeX1s, $2s$TeX2s, and $2p$TeX2p do not denote ontic orbits traversed by a pointlike particle. They denote admissible rigidity modes of the bound electron-proton configuration on the declared Coulomb-family witness. The exported frequency of the $2p\to1s$TeX2p\to1s line is not the kinematic trace of a particle jump, but the carrier-side image of the rigidity-mode separation between those two bound-state configurations. The suppression of the one-photon $2s\to1s$TeX2s\to1s line on the same channel means only that the electric-dipole carrier geometry does not open that specific coupling path. The branch switching itself is not thereby forbidden.

For the declared hydrogenic witness, one may use the usual reduced-mass Coulomb spectrum as the leading imported line anchor,

E_n^{\mathrm{H}}
=
-\frac{\mu_{\mathrm{eff}}\alpha_{\mathrm{em}}^2c^2}{2n^2}+\delta_n^{\mathrm{res}},

where $\delta_n^{\mathrm{res}}$TeX\delta_n^{\mathrm{res}} collects fine-structure, Lamb-type, and higher-order residuals not treated here. The binding frequencies are then

\Omega_n^{\mathrm{H}}:=\frac{|E_n^{\mathrm{H}}|}{\hbar},
\qquad
\omega_{m\to n}^{\mathrm{H}}=\Omega_n^{\mathrm{H}}-\Omega_m^{\mathrm{H}}+\frac{\delta_n^{\mathrm{res}}-\delta_m^{\mathrm{res}}}{\hbar}.

For the witness channel $2\to1$TeX2\to1,

\Delta E_{2\to1}^{\mathrm{H}}
=
\frac{3}{8}\mu_{\mathrm{eff}}\alpha_{\mathrm{em}}^2c^2+\delta_{21}^{\mathrm{res}}.

The detector-separated observed line profile is therefore

I_{\mathrm{obs}}^{\mathrm{H}}(\omega)
=
\sum_{m>n} p_m^{\mathrm{H}}\,\Gammaexp_{m\to n}^{\mathrm{H}}\,
\Rdet\!\left(\omega-\omega_{m\to n}^{\mathrm{H}}\right),

with the leading visible witness carried by the $2p\to1s$TeX2p\to1s term and the one-photon $2s\to1s$TeX2s\to1s suppression encoded by reference.

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

The point of reference is not to out-perform precision hydrogen theory. It is to show that the same line data can be read, on one declared witness family, without ontic orbit-jump language and without collapsing propagation, local commit, and durable record into one layer.

The same formalism extends to vibrational-rotational and condensed-matter witnesses. A rovibrational family may be written schematically as

E_{vJ}=E_{\mathrm{el}}+\hbar\omega_e\left(v+\frac12\right)+B_eJ(J+1)+\delta_{vJ}^{\mathrm{anh+rot}},

so that rovibrational line positions are again mode-separation differences on a declared family [citation]. Likewise, excitonic phonon sidebands and vacuum-Rabi-like doublets can be read as exported signatures of relocking between coupled microscopic branches rather than as evidence for an autonomous wave substance or a literal ontology of jumps [citation]. These secondary witnesses are retained only to show the portability of the relocking formalism across bound-state families. No universal microscopic closure across all molecules or solids is asserted.

The declared results above do not establish a full QED derivation, an all-atom precision spectroscopy fit, a detector microdynamics theory, or a universal microscopic completion of quantum theory. They do not identify the wavefunction with a material substance. They do not identify light with an autonomous causal particle in the ontic sense used by the standard jump picture. They do not establish any compactification or selected-family completion theorem, and they do not identify with string theory or M-theory. The auxiliary microscopic seat reference, when used, remains subordinate and supportive only.

The discrete spectrum of a bound-state Hamiltonian is a local spectral statement and does not follow from compactness of the root scalar target. Compact topology can impose winding or holonomy boundary conditions, thereby changing the allowed global mode sectors, but it supplies neither the material potential nor the numerical transition scale. Any phase-dependent vibrational loading must descend from a declared action; the extracted loading functional must pass the formal self-adjointness condition required by the inverse variational problem.

Transition probabilities require an independent instrument condition. If a detector sink is used, $K=\sum_aL_a^\dagger L_a$TeXK=\sum_aL_a^\dagger L_a must hold and the jump branches must restore trace preservation. Thus three layers remain distinct: the Hamiltonian fixes spectral lines, compact topology fixes admissible global sectors, and a CPTP instrument fixes registered outcomes. A common phase model becomes testable only when shared parameters predict several line shifts or branching observables and satisfy the associated left-null compatibility relations.

The closure test for the bound-state spectra and registered transitions 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 level spacings, line strengths, transition rates, and registered counts. Let $a$TeXa range over the independent constitutive inputs comprising bound-state Hamiltonian, phase-dependent coefficients, transition coupling, and detector 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 Hamiltonian fixes spectra and transition operators, and one normalized instrument maps transitions to records.

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 compact phase sectors and detector outcomes are additional structures not derivable from spectral discreteness alone. Failure of that condition therefore rejects the proposed microscopic closure before parameter estimation can be counted as evidence for it.

Exact three-line restriction..

Paper CHC-MHB supplies a bound finite-spectrum example in which three phase-dependent lines are not fitted independently. After centering the line frequencies and dividing out their common quadratic spread, their signed cubic symmetric invariant equals $\cos\theta$TeX\cos\theta. The identity is unchanged by line permutation, a common frequency offset, or a positive calibration scale. It is therefore an explicit example of spectral discreteness supplemented by a microscopic relation that has positive predictive codimension; discreteness alone would not supply that relation.

On one declared bound-state family, a self-adjoint bound-state Hamiltonian, a pure-point spectral sector, and an admitted weak carrier-channel set are sufficient to define an admissible rigidity spectrum, a carrier-coupled relocking amplitude, and a detector-separated spectral export map. The candidate line catalogue is the countable difference set of rigidity modes on that family; its local finiteness and the visibility of each line require, respectively, a spectral gap/local-finiteness condition and nonzero channel spectral density. The observed discreteness of the line profile is a detector-response statement on the declared window. Within that restricted scope, an electronic transition is represented as a change between admitted bound-state modes rather than as a literal orbit of a point particle.

The hydrogenic $n=2\to1$TeXn=2\to1 witness, together with allowed and suppressed channel structure, already shows the practical content of that retyping. Molecular ladders, excitonic phonon sidebands, and cavity mode splitting provide secondary witnesses of the same formalism. No stronger reading is claimed here. In particular, no full QED replacement, compactification closure, ultraviolet completion, universal quantum-foundation theorem, or string-theory identity is asserted.

The auxiliary internal seat reference is used only to organize a discrete microscopic basis. If a compact or effectively finite internal response space supports a self-adjoint operator $\Lint^{(b)}$TeX\Lint^{(b)} with discrete spectrum, then the corresponding basis $\{\chi_\nu^{(b)}\}$TeX\{\chi_\nu^{(b)}\} may be used to expand the declared bound-state family. This is only a supporting microscopic seat. It does not by itself define a selected compact family or a compactification theorem.

The rate formula reference is the first-order spectral rate on the declared weak carrier window. It connects nonzero relocking amplitudes to channel-resolved exported or absorbed weight. Any stronger microscopic radiation theory lies outside the declared scope.

Funding and competing interests..

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