Liouville–Continuity Derivation of the Scretching–Schrödinger Equation: A Probability-Conservation and Phase-Space-Conservation Route to the V _SQCConstraint Term
This paper derives the Scretching–Schrödinger Equation (SSE) from a Liouville–continuity foundation by showing how classical phase-space conservation leads naturally into quantum probability conservation and then into spectroscopic wavefunction constraint. The derivation begins with the Liouville equation, which preserves the phase-space density F(r,p,t)F(\mathbf r,\mathbf p,t)F(r,p,t), and reduces it to the configuration-space continuity equation for the probability density ρ(r,t)\rho(\mathbf r,t)ρ(r,t). In the quantum limit, the identification ρ=∣ψ∣2\rho = |\psi|^2ρ=∣ψ∣2 requires unitary time evolution generated by a self-adjoint Hamiltonian, giving the ordinary Schrödinger equation and, for stationary states, the standard eigenvalue equation: H^0ϕ=Eϕ.\hat H_0 \phi = E\phi .H^0ϕ=Eϕ. The SSE is then obtained by imposing the Scretching Quantum Chain (SQC) and Maxwell–Scretching Chain (MSC) as additional spectroscopic admissibility conditions on the wavefunction. In this formulation, the SSE is not merely a modified Schrödinger equation. It is a probability-conserving and phase-space-conserving eigenvalue theory in which the allowed states are constrained by oscillator strength, transition dipole moment, Einstein coefficients, spectral frequency, and the SQC invariant KνK_\nuKν. The resulting Hamiltonian is: H^SSE=H^0+V^SQC,\hat H_{\mathrm{SSE}}=\hat H_0+\hat V_{\mathrm{SQC}},H^SSE=H^0+V^SQC, where V^SQC\hat V_{\mathrm{SQC}}V^SQC is the spectroscopic constraint operator. Therefore, the SSE wavefunction is a Schrödinger-type quantum amplitude supplemented by SQC/MSC spectroscopic closure, rather than a wavefunction whose spectra are computed only after the eigenvalue problem has already been solved. This places spectroscopy inside the admissibility structure of the quantum state itself. Keywords: Liouville equation, continuity equation, Scretching–Schrödinger Equation, SSE, SQC, MSC, probability conservation, phase-space conservation, eigenvalue equation, oscillator strength, transition dipole moment, Einstein coefficient.
This paper derives the Scretching–Schrödinger Equation (SSE) from a Liouville–continuity foundation by showing how classical phase-space conservation leads naturally into quantum probability conservation and then into spectroscopic wavefunction constraint. The derivation begins with the Liouville equation, which preserves the phase-space density F(r,p,t)F(\mathbf r,\mathbf p,t)F(r,p,t), and reduces it to the configuration-space continuity equation for the probability density ρ(r,t)\rho(\mathbf r,t)ρ(r,t). In the quantum limit, the identification ρ=∣ψ∣2\rho = |\psi|^2ρ=∣ψ∣2 requires unitary time evolution generated by a self-adjoint Hamiltonian, giving the ordinary Schrödinger equation and, for stationary states, the standard eigenvalue equation: H^0ϕ=Eϕ.\hat H_0 \phi = E\phi .H^0ϕ=Eϕ. The SSE is then obtained by imposing the Scretching Quantum Chain (SQC) and Maxwell–Scretching Chain (MSC) as additional spectroscopic admissibility conditions on the wavefunction. In this formulation, the SSE is not merely a modified Schrödinger equation. It is a probability-conserving and phase-space-conserving eigenvalue theory in which the allowed states are constrained by oscillator strength, transition dipole moment, Einstein coefficients, spectral frequency, and the SQC invariant KνK_\nuKν. The resulting Hamiltonian is: H^SSE=H^0+V^SQC,\hat H_{\mathrm{SSE}}=\hat H_0+\hat V_{\mathrm{SQC}},H^SSE=H^0+V^SQC, where V^SQC\hat V_{\mathrm{SQC}}V^SQC is the spectroscopic constraint operator. Therefore, the SSE wavefunction is a Schrödinger-type quantum amplitude supplemented by SQC/MSC spectroscopic closure, rather than a wavefunction whose spectra are computed only after the eigenvalue problem has already been solved. This places spectroscopy inside the admissibility structure of the quantum state itself. Keywords: Liouville equation, continuity equation, Scretching–Schrödinger Equation, SSE, SQC, MSC, probability conservation, phase-space conservation, eigenvalue equation, oscillator strength, transition dipole moment, Einstein coefficient.
This paper derives the Scretching–Schrödinger Equation (SSE) from first principles by unifying the standard probability-amplitude foundation of quantum mechanics with experimentally anchored spectroscopic closure. Beginning with the Born wavefunction Ψ(r,t), the Planck–de Broglie relations E=ℏωE=\hbar\omegaE=ℏω and p=ℏkp=\hbar kp=ℏk, probability conservation, and the requirement that Hamiltonian evolution be self-adjoint, the paper reconstructs the ordinary Schrödinger equation as the baseline unitary wave equation for real-energy quantum systems. The derivation then extends the ordinary Hamiltonian by imposing the Scretching Quantum Chain (SQC) as a transition-level constraint connecting wavefunction matrix elements, transition dipole moments, oscillator strengths, Einstein radiative coefficients, molar absorptivity, and Maxwell–Scretching optical closure. In this formulation, the Scretching–Schrödinger Hamiltonian is not merely a phenomenological energy shift but a constrained operator equation, HSSE=H0+VSQC,H_{\mathrm{SSE}} = H_0 + V_{\mathrm{SQC}},HSSE=H0+VSQC, with the operator domain D(HSSE)=D(H0)∩D(VSQC).D(H_{\mathrm{SSE}})=D(H_0)\cap D(V_{\mathrm{SQC}}).D(HSSE)=D(H0)∩D(VSQC). Here, VSQCV_{\mathrm{SQC}}VSQC is treated as a symmetric or self-adjoint spectroscopic correction operator that preserves the mathematical structure required for physical quantum evolution while enforcing experimentally measurable spectroscopic constraints. Hydrogen is used as the zero-correction benchmark because its Coulomb Hamiltonian already provides an exact analytic reference for bound-state spectra and transition moments. The corrected 1s→2p1s\rightarrow 2p1s→2p analytic dipole agrees with the SQC-derived dipole to approximately 0.03%, supporting the interpretation that the SSE reduces to ordinary Schrödinger theory when no additional spectroscopic correction is required. The resulting framework therefore presents the SSE as a rigorously constrained extension of quantum mechanics: ordinary wave mechanics supplies the self-adjoint dynamical core, while the SQC and Maxwell–Scretching chains impose measurable optical, radiative, and molecular-spectroscopic closure conditions on allowed transitions.
This paper reformulates the Scretching–Schrödinger Equation (SSE) as a spectroscopically constrained quantum and gauge-field formalism that connects ordinary Schrödinger wave mechanics, Dirac relativistic quantum mechanics, quantum electrodynamics, Glashow–Salam–Weinberg electroweak theory, and quantum chromodynamics. The study begins with the standard requirement that quantum dynamics be governed by a self-adjoint Hamiltonian, ensuring real energy eigenvalues, unitary time evolution, and probability conservation. Within this structure, the Scretching Quantum Chain (SQC) is introduced as a transition-level closure condition linking wavefunction matrix elements, oscillator strength, Einstein radiative coefficients, transition frequency, and Maxwell–Scretching macroscopic optical observables. The central operator of the paper is the constrained Hamiltonian H^SSE=H^0+V^SQC,\hat{H}_{\mathrm{SSE}}=\hat{H}_{0}+\hat{V}_{\mathrm{SQC}},H^SSE=H^0+V^SQC, where H^0\hat{H}_{0}H^0 is the ordinary quantum Hamiltonian and V^SQC\hat{V}_{\mathrm{SQC}}V^SQC is the spectroscopic constraint operator. In the zero-correction limit, where V^SQC→0\hat{V}_{\mathrm{SQC}}\to 0V^SQC→0, the theory reduces to the ordinary Schrödinger equation. In the constrained case, admissible transitions must satisfy the measurable SQC residual condition RSQC=0.R_{\mathrm{SQC}}=0.RSQC=0. This residual expresses the requirement that the microscopic quantum transition, its radiative probability, its oscillator strength, and its macroscopic optical observables agree within a single deterministic closure. The paper then extends this SSE constraint architecture beyond nonrelativistic wave mechanics. In the Dirac formalism, the SSE is recast as a covariant spinor equation in which the usual relativistic wave operator is supplemented by an SQC admissibility term. In QED, the closure is interpreted through photon exchange, current insertions, radiative amplitudes, transition dipole moments, and modified propagator constraints. In electroweak theory, the same logic is promoted to the SU(2)L×U(1)YSU(2)_L \times U(1)_YSU(2)L×U(1)Y gauge structure, producing closure conditions for neutral-current and charged-current transitions. In QCD, the framework is extended to the non-Abelian SU(3)CSU(3)_CSU(3)C color sector, where color currents, running coupling behavior, and spectroscopic hadronic constraints are interpreted through an SQC-compatible admissibility condition. New equations are derived for the SSE–Dirac operator, SSE-modified photon and gauge-boson propagators, SQC current insertions, electroweak neutral-current closure, and color-spectroscopic QCD constraint terms. Numerical verification is provided through the corrected hydrogen Lyman-alpha benchmark, the corrected KSSEK_{\mathrm{SSE}}KSSE constant, electroweak coupling reconstruction from α\alphaα and sin2θW\sin^{2}\theta_Wsin2θW, and one-loop QCD running. These examples show how the SSE framework can be tested against established quantum, spectroscopic, and field-theoretic quantities. The principal conclusion is that the SSE does not replace ordinary quantum mechanics, relativistic quantum mechanics, QED, electroweak theory, or QCD. Rather, it adds a deterministic spectroscopic admissibility condition to them. In this interpretation, the SSE functions as a unifying constraint framework that connects quantum wavefunctions, radiative amplitudes, gauge currents, transition probabilities, and macroscopic optical observables within a single measurable closure structure.
This paper derives the Scretching–Schrödinger Equation (SSE) in full theoretical detail from the ordinary Schrödinger equation, transition dipole theory, oscillator-strength formalism, Einstein coefficient theory, and the Maxwell–Scretching experimental closure. The derivation begins with the time-dependent Schrödinger equation, separates the stationary time-independent form, and then connects quantum-state evolution to spectroscopic observables through the transition dipole moment, oscillator strength, spontaneous-emission coefficient, and absorption-emission closure relations. Within this framework, the SQC spectroscopic invariant is introduced as Kν=2πe2ε0mec3K_{\nu}=\frac{2\pi e^{2}}{\varepsilon_{0}m_{e}c^{3}}Kν=ε0mec32πe2, which links electromagnetic coupling, electron mass, light speed, and frequency-dependent spectroscopic behavior. The paper then derives the SSE closure constant, KSSE=3ℏ(4.32×10−9)2meK_{\mathrm{SSE}}=\frac{3\hbar(4.32\times10^{-9})}{2m_{e}}KSSE=2me3ℏ(4.32×10−9), as the coefficient that transfers the Maxwell–Scretching experimental closure into the quantum wave-equation formalism. The resulting Scretching–Schrödinger Equation is formulated as a constrained wave equation in which the Hamiltonian contains an additional spectroscopic correction operator, VSQCV_{\mathrm{SQC}}VSQC. This operator represents the SQC-imposed correction that connects ordinary quantum-mechanical evolution with experimentally recoverable spectroscopic closure behavior. The derivation also corrects two important formal issues. First, the Hamiltonian is treated properly as an unbounded self-adjoint operator acting on a suitable dense domain in Hilbert space, rather than as an ordinary finite matrix or unrestricted algebraic operator. Second, the hydrogen 1s→2p1s\rightarrow2p1s→2p radial transition integral is written in its corrected form, 276 a035\frac{2^{7}\sqrt{6}\,a_{0}}{3^{5}}35276a0, rather than the incorrect expression 282 a035\frac{2^{8}\sqrt{2}\,a_{0}}{3^{5}}35282a0. Together, these corrections strengthen the mathematical foundation of the SSE and ensure that the spectroscopic derivation remains consistent with standard quantum mechanics, hydrogenic transition theory, and the Maxwell–Scretching closure framework [1–5].
This paper describes, explains, and compares the wavefunction in three related quantum frameworks: nonrelativistic quantum mechanics (NRQM), relativistic quantum mechanics (RQM), and the Scretching–Schrödinger Equation (SSE). In NRQM, the wavefunction is treated as a complex probability amplitude whose squared modulus gives the probability density for finding a particle in a specified region of space. For bound atomic systems, it is governed by the Schrödinger equation and is commonly labeled by the quantum numbers nnn, lll, mlm_lml, and msm_sms, which specify the principal energy level, orbital angular momentum, magnetic projection, and spin projection. In RQM, the wavefunction must be compatible with special relativity. For spin-0 particles, this leads to the Klein–Gordon scalar wavefunction, while for spin-12\frac{1}{2}21 particles, the relativistic wavefunction becomes the four-component Dirac spinor. These relativistic forms preserve probability-amplitude interpretation but require Lorentz-compatible energy–momentum relations, relativistic covariance, and, in the Dirac case, explicit treatment of spin, antimatter solutions, and relativistic current conservation. The SSE framework extends the ordinary Schrödinger wavefunction by imposing a deterministic spectroscopic closure condition through the Scretching Quantum Chain (SQC) and Maxwell–Scretching Chain (MSC). In this formulation, the wavefunction retains the standard quantum-number structure of atomic quantum mechanics, including radial, angular, and spin components, but it is additionally constrained to reproduce measurable spectroscopic quantities such as oscillator strength, transition dipole moment, Einstein AAA and BBB coefficients, transition frequency, absorbance-linked optical quantities, and the SQC invariant Kν=2πe2ε0mec3.K_\nu=\frac{2\pi e^2}{\varepsilon_0 m_e c^3}.Kν=ε0mec32πe2. Thus, the SSE wavefunction is not merely a probability amplitude. It is a probability amplitude constrained by experimentally anchored spectroscopic closure. The referenced SSE formalism defines the SSE Hamiltonian as H^SSE=H^0+V^SQC,\hat{H}_{\mathrm{SSE}}=\hat{H}_0+\hat{V}_{\mathrm{SQC}},H^SSE=H^0+V^SQC, where H^0\hat{H}_0H^0 is the ordinary quantum Hamiltonian and V^SQC\hat{V}_{\mathrm{SQC}}V^SQC is the spectroscopic constraint operator. The SSE wavefunction may therefore be written schematically as ΨSSE(r,θ,ϕ,s)=RSSE(r) Ylm(θ,ϕ) χs [fconstraint],\Psi_{\mathrm{SSE}}(r,\theta,\phi,s) = R_{\mathrm{SSE}}(r)\,Y_l^m(\theta,\phi)\,\chi_s\,[f_{\mathrm{constraint}}],ΨSSE(r,θ,ϕ,s)=RSSE(r)Ylm(θ,ϕ)χs[fconstraint], where RSSE(r)R_{\mathrm{SSE}}(r)RSSE(r) is the radial component modified or constrained by SSE closure, Ylm(θ,ϕ)Y_l^m(\theta,\phi)Ylm(θ,ϕ) is the spherical harmonic angular component, χs\chi_sχs is the spin function, and [fconstraint][f_{\mathrm{constraint}}][fconstraint] represents the SQC/MSC spectroscopic constraint that forces agreement between the wavefunction matrix elements and measured or computed optical transition quantities. The central comparison developed in the paper is that NRQM describes the wavefunction as a probability amplitude, RQM generalizes the wavefunction into Lorentz-compatible scalar or spinor fields, and SSE adds a deterministic spectroscopic layer to the quantum wavefunction. In this sense, SSE does not discard Schrödinger or Dirac quantum mechanics; instead, it proposes an additional closure architecture in which allowed wavefunctions must satisfy both the usual quantum dynamical equation and the SQC/MSC spectroscopic consistency conditions.
Introduction The Scretching–Schrödinger Equation (SSE) is developed as a quantum-mechanical framework that imposes bidirectional spectroscopic closure between the wavefunction and the measured spectrum, ψ⟷spectrum,\psi \longleftrightarrow \mathrm{spectrum},ψ⟷spectrum, through the Scretching Quantum Chain (SQC) invariant, Kν=7.42165600145×10−22 s.K_{\nu}=7.42165600145\times10^{-22}\ \mathrm{s}.Kν=7.42165600145×10−22 s. In its original non-relativistic form, the SSE modifies the ordinary Schrödinger framework by requiring that the wavefunction not only solve an eigenvalue problem, but also satisfy a deterministic spectroscopic closure condition. This condition links transition matrix elements, oscillator strengths, Einstein coefficients, absorption parameters, and measured transition frequencies. The result is a constrained quantum theory in which ψ\psiψ generates spectra, while experimental spectra feed back as admissibility conditions on ψ\psiψ. This paper systematically recasts the SSE across the full hierarchy of modern quantum theory: ordinary quantum mechanics, relativistic quantum mechanics, quantum electrodynamics, electroweak theory, and quantum chromodynamics. The objective is to show that the SSE closure principle can be expressed not merely as a correction to non-relativistic wave mechanics, but as a generalized spectroscopic constraint compatible with the major operator, field, and gauge-theoretic structures of twentieth- and twenty-first-century physics. The first stage reformulates the SSE within ordinary quantum mechanics in both the Schrödinger wave-mechanical formalism and the Heisenberg matrix-mechanical formalism. In the Schrödinger picture, the SSE is written as a constrained wave equation, iℏ∂ψ∂t=(H^0+V^SQC)ψ,i\hbar\frac{\partial \psi}{\partial t} = \left(\hat{H}_0+\hat{V}_{\mathrm{SQC}}\right)\psi,iℏ∂t∂ψ=(H^0+V^SQC)ψ, where H^0\hat{H}_0H^0 is the ordinary Hamiltonian and V^SQC\hat{V}_{\mathrm{SQC}}V^SQC is the spectroscopic closure operator. In the Heisenberg picture, the same closure is recast into operator evolution and matrix-element constraints, showing that the SSE condition is independent of representation. The wavefunction, operator algebra, transition amplitude, and spectrum therefore become mutually constrained expressions of the same physical closure. The second stage extends the SSE into relativistic quantum mechanics (RQM) through the Klein–Gordon and Dirac equations. The Klein–Gordon–SSE form imposes SQC closure on relativistic scalar fields, while the Dirac–SSE form imposes the same closure on four-component spinor states. These extensions preserve relativistic covariance, introduce proper mass-shell structure, and connect the SSE to spin, antimatter, fine structure, and relativistic transition amplitudes. In this form, the SSE becomes a spectroscopic constraint on relativistic wave equations rather than a purely non-relativistic correction. The third stage recasts the framework within quantum electrodynamics (QED). Here the SSE closure principle is applied to photon exchange, electron self-energy, vertex corrections, Ward identity structure, and the running electromagnetic coupling. The QED–SSE formulation treats the SQC closure functional as a constraint on radiative processes, linking Feynman amplitudes to spectroscopic observables. The photon propagator, electron propagator, interaction vertex, and renormalized coupling are therefore interpreted not only as field-theoretic objects, but also as carriers of measurable spectroscopic closure. The fourth stage extends the formalism into the Glashow–Salam–Weinberg (GSW) electroweak theory, whose gauge group is SU(2)L×U(1)Y.SU(2)_L\times U(1)_Y.SU(2)L×U(1)Y. In this setting, the SSE closure principle is applied to weak and electromagnetic transitions after spontaneous symmetry breaking. The framework is developed through the electroweak gauge fields, Higgs mechanism, weak mixing angle, and the resulting W±W^{\pm}W±, Z0Z^0Z0, and photon fields. The masses of the weak bosons and the structure of charged and neutral currents are incorporated into an SSE weak-transition closure law, allowing the SQC framework to be tested against electroweak-scale processes. The fifth stage applies the SSE to quantum chromodynamics (QCD), based on the color gauge group SU(3)c.SU(3)_c.SU(3)c. Because QCD includes gluon self-interaction, asymptotic freedom, confinement, and non-Abelian color dynamics, it provides the most demanding field-theoretic extension of the SSE. The QCD–SSE formulation introduces a color-closure functional designed to connect quark and gluon transition amplitudes with spectroscopic or hadronic observables. In this context, the SSE is not presented as a replacement for QCD, but as a proposed closure constraint that may organize measurable color-sector transitions, confinement-scale behavior, and hadronic spectra. For each theoretical level, this paper derives a corresponding SSE-extended equation, lists the relevant Feynman rules, presents representative Feynman diagrams, and provides numerical examples with verified solutions. The analysis traces the mathematical progression from wave mechanics to matrix mechanics, from non-relativistic quantum theory to relativistic wave equations, from single-particle dynamics to QED radiative corrections, from Abelian gauge theory to electroweak unification, and finally from electroweak theory to the non-Abelian color structure of QCD. The primary textbook sources abstracted for the gauge-theory and particle-physics development are Aitchison and Hey’s Gauge Theories in Particle Physics, Volumes 1 and 2, including the third and fourth edition treatments of quantum electrodynamics, electroweak theory, and quantum chromodynamics [1,2]. All numerical constants used in the worked examples are taken from CODATA 2022, ensuring that the calculations are anchored to contemporary reference values. The central claim advanced in this paper is that the SSE may be written as a representation-independent and scale-extendable spectroscopic closure principle. In ordinary quantum mechanics, it constrains wavefunctions and operator matrix elements. In RQM, it constrains relativistic scalar and spinor amplitudes. In QED, it constrains radiative amplitudes and renormalized electromagnetic interactions. In GSW theory, it constrains weak and electroweak transitions. In QCD, it proposes a color-sector closure functional for non-Abelian gauge dynamics. Thus, the SSE is developed as a candidate bridge between measured spectra, quantum amplitudes, and gauge-field theory across the full hierarchy of modern quantum physics.
This paper develops an advanced unified description of the hydrogen atom wavefunction across three major theoretical levels: nonrelativistic quantum mechanics, relativistic quantum mechanics, and the Scretching–Schrödinger Equation framework. The study begins with the standard Schrödinger hydrogen wavefunction, where the electron state is represented as a complex probability amplitude labeled by the quantum numbers nnn, lll, mlm_lml, and msm_sms. It then advances to the relativistic Dirac treatment, where the scalar wavefunction is replaced by a four-component spinor labeled by nnn, κ\kappaκ, and mjm_jmj, thereby incorporating spin, relativistic energy corrections, and the natural appearance of fine-structure splitting. The central contribution of the paper is the extension of the hydrogen atom into the Scretching–Schrödinger Equation (SSE) framework, where the ordinary Schrödinger structure is preserved but constrained by a deterministic spectroscopic closure condition from the Scretching Quantum Chain (SQC) and Maxwell–Scretching Chain (MSC). In this formulation, a valid hydrogen wavefunction must not only satisfy the wave equation but must also reproduce the full physical chain ε→f→∣μ∣2→A21→Kν,\varepsilon \rightarrow f \rightarrow |\mu|^2 \rightarrow A_{21} \rightarrow K_{\nu},ε→f→∣μ∣2→A21→Kν, linking molar absorptivity, oscillator strength, transition dipole moment, Einstein spontaneous-emission coefficient, and the SQC invariant. The invariant Kν=2πe2ε0mec3=7.42165600145×10−22 sK_{\nu}=\frac{2\pi e^2}{\varepsilon_0 m_e c^3} =7.42165600145\times 10^{-22}\ \text{s}Kν=ε0mec32πe2=7.42165600145×10−22 s is derived from first principles and treated as the fixed spectroscopic time-scale anchor of the hydrogen SQC closure. The paper verifies the SQC formulation against NIST hydrogen spectral data for the Lyman series (n=2–4)(n=2–4)(n=2–4) and Balmer series (n=3–7)(n=3–7)(n=3–7), obtaining relative errors no larger than 2.7×10−42.7\times 10^{-4}2.7×10−4, with the remaining discrepancy attributed to rounding limits. It also derives the corrected radial transition integral for the 1s→2p1s\rightarrow 2p1s→2p transition, Ir0=1.2902662 a0,I_{r0}=1.2902662\,a_0,Ir0=1.2902662a0, and uses this value to support the deterministic connection between hydrogen wavefunctions, transition moments, oscillator strengths, and Einstein coefficients. The manuscript further develops the hydrogen SSE Hamiltonian in C*-algebraic form, extends the wavefunction treatment to isotope-corrected protium, deuterium, and tritium, and demonstrates that KνK_{\nu}Kν remains invariant under isotope substitution. Fine-structure effects—including Lamb-shift corrections, hyperfine splitting, and Zeeman perturbations—are examined as higher-order refinements within the SSE context rather than violations of the underlying SQC invariant. Finally, the paper maps the full SQC-to-MSC projection, showing how quantum oscillator strength projects upward into macroscopic electromagnetic observables such as absorption, refractive behavior, dielectric response, and radiative transition strength. The result is a deterministic, parameter-free quantum–electromagnetic hydrogen framework in which NRQM, RQM, SSE, SQC, and MSC are treated as connected layers of the same physical closure chain.