We formulate a Lorentz-covariant action-principle foundation for a quantum substrate by representing the conserved substrate variable asa non-negative scalar field \(S\) on a bundle \(E=M\times F\), where \(M\) is aLorentzian spacetime manifold and \(F\simeq\mathbb{R}_{\ge0}\) encodesCausality-Interval structure. The fundamental substrate action is defined on\(E\), while matter and gravitational observables couple only to theCollapse-Boundary-projected field\(S_{\mathrm{obs}}^{(\nu)}(x)=\int_F K^{(\nu)}(x,\xi;u^\mu)S(x,\xi)\,d\xi\).This separation preserves Lorentz covariance at the bundle level whileallowing sector-dependent projected dynamics after projection.The projected sector structure is expressed through a modal response operator\(\mathcal{O}^{(\nu)}_{\rm QSD}=\mathcal{T}_{c_t}+\mathcal{B}_\nu\mathcal{C}^{(\nu)}_{c_s}\),where \(c_t\) is the transverse substrate mode-rate and \(c_s(\nu)\) is thescalar recovery or clearance mode-rate of a mass-conditioned sector. Thus\(c_s(\nu)=\nu c_t\) is a substrate mode-ratio relation, not a second speed oflight. From the transverse coherence length\(L_{\rm coh}=\hbar/(mc_t)\) and the recovery interval\(\tau_\nu=L_{\rm coh}/c_s(\nu)\), we obtain the cross-modal recovery energy\(E_{\rm rec}^{(\nu)}=\hbar/\tau_\nu=m c_t c_s(\nu)=\nu m c_t^2\).For the fermionic branch, \(\nu=2\) and \(c_t=c\) give\(E_{\rm rec}^{(2)}=2mc^2\), matching the inverse angular-frequency scale ofDirac zitterbewegung without interpreting \(c_s\) as transverse propagation.We derive the substrate Euler--Lagrange equation, state the associatedconservation and quantization structures, and organize matter coupling, Dysonmass dressing, \(C\ell(6)\)-compatible representations, coherence-envelopebranches, and weak-field gravitational recovery as admissibility conditions ordownstream solution branches. The result is a compact Lorentz-covariantLagrangian base for systematically developing QSD.
We formulate a Lorentz-covariant action-principle foundation for a quantum substrate by representing the conserved substrate variable asa non-negative scalar field (S) on a bundle (E=M\times F), where (M) is aLorentzian spacetime manifold and (F\simeq\mathbb{R}{\ge0}) encodesCausality-Interval structure. The fundamental substrate action is defined on(E), while matter and gravitational observables couple only to theCollapse-Boundary-projected field(S{\mathrm{obs}}^{(\nu)}(x)=\int_F K^{(\nu)}(x,\xi;u^\mu)S(x,\xi),d\xi).This separation preserves Lorentz covariance at the bundle level whileallowing sector-dependent projected dynamics after projection.The projected sector structure is expressed through a modal response operator(\mathcal{O}^{(\nu)}{\rm QSD}=\mathcal{T}{c_t}+\mathcal{B}\nu\mathcal{C}^{(\nu)}{c_s}),where (c_t) is the transverse substrate mode-rate and (c_s(\nu)) is thescalar recovery or clearance mode-rate of a mass-conditioned sector. Thus(c_s(\nu)=\nu c_t) is a substrate mode-ratio relation, not a second speed oflight. From the transverse coherence length(L_{\rm coh}=\hbar/(mc_t)) and the recovery interval(\tau_\nu=L_{\rm coh}/c_s(\nu)), we obtain the cross-modal recovery energy(E_{\rm rec}^{(\nu)}=\hbar/\tau_\nu=m c_t c_s(\nu)=\nu m c_t^2).For the fermionic branch, (\nu=2) and (c_t=c) give(E_{\rm rec}^{(2)}=2mc^2), matching the inverse angular-frequency scale ofDirac zitterbewegung without interpreting (c_s) as transverse propagation.We derive the substrate Euler--Lagrange equation, state the associatedconservation and quantization structures, and organize matter coupling, Dysonmass dressing, (C\ell(6))-compatible representations, coherence-envelopebranches, and weak-field gravitational recovery as admissibility conditions ordownstream solution branches. The result is a compact Lorentz-covariantLagrangian base for systematically developing QSD.
We formulate a Lorentz-covariant action-principle foundation for a quantum substrate by representing the conserved substrate variable as a non-negative scalar field S on a bundle E=M×F, where M is a Lorentzian spacetime manifold and F≃R≥0 encodes Causality-Interval structure. The fundamental substrate action is defined on E, while matter and gravitational observables couple only to the Collapse-Boundary-projected field Sobs(ν)(x)=∫FK(ν)(x,ξ;uμ)S(x,ξ)dξ. This separation preserves Lorentz covariance at the bundle level while allowing sector-dependent projected dynamics after projection. The projected sector structure is expressed through a modal response operator OQSD(ν)=Tct+BνCcs(ν), where ct is the transverse substrate mode-rate and cs(ν) is the scalar recovery or clearance mode-rate of a mass-conditioned sector. Thus cs(ν)=νct is a substrate mode-ratio relation, not a second speed of light. From the transverse coherence length Lcoh=ℏ/(mct) and the recovery interval τν=Lcoh/cs(ν), we obtain the cross-modal recovery energy Erec(ν)=ℏ/τν=mctcs(ν)=νmct2. For the fermionic branch, ν=2 and ct=c give Erec(2)=2mc2, matching the inverse angular-frequency scale of Dirac zitterbewegung without interpreting cs as transverse propagation. We derive the substrate Euler--Lagrange equation, state the associated conservation and quantization structures, and organize matter coupling, Dyson mass dressing, Cℓ(6)-compatible representations, coherence-envelope branches, and weak-field gravitational recovery as admissibility conditions or downstream solution branches. The result is a compact Lorentz-covariant Lagrangian base for systematically developing QSD.
Standard quantum mechanics and quantum field theory provide highly successful statistical and dynamical descriptions of emission, measurement, and detection, but they do not specify a substrate-level mechanism by which hidden pre-observational structure becomes a discrete recorded outcome. We develop such a mechanism within Quantum Substrate Dynamics (QSD) while leaving the predictive machinery of QM and QFT intact. The framework organizes local quantum evolution into finite Causality Intervals (CIs) terminated by Collapse Boundaries (CBs), where scalar-sector structure is re-locked, filtered, and, when admissible, serialized into transverse propagation. We distinguish a \emph{Quantum Emission Opportunity} (QEO), defined as a pre-collapse scalar admissibility condition, from a \emph{Quantized Emission} (QE), defined as the post-CB serialized output that survives re-locking and projects into an allowed transverse channel. In this account, most QEOs do not become emissions: they may lose alignment, fail CB filtering, lack sufficient recovery capacity, or be reabsorbed into the scalar recovery process. We formulate the QEO/QE pipeline in terms of CB-local admissibility, projection coefficients, environmental shaping, and detector-envelope conversion. The Born-rule form is retained operationally and interpreted as the normalized squared projection of surviving scalar structure onto the locally admissible transverse-mode family at the CB. Apparent probability is then understood as a coarse-grained description over unresolved scalar-sector configurations and CI histories, rather than as an additional stochastic ingredient at the substrate level. The framework also gives a limited treatment of entanglement as conditional two-region serialization under shared preparation, while deferring full Bell-factorization and CI-space ontology to companion work. Finally, we identify structured experimental signatures suggested by the pipeline, including mode-selective suppression, detector-envelope asymmetry, post-boundary spectral shaping, interferometric residuals, and CI-pacing effects. The contribution is structural and interpretive: QSD supplies a causal substrate account beneath the established quantum formalism, clarifying how QEO formation, CB filtering, serialization, environmental shaping, and detection may underlie the discrete outcomes described by standard theory.
Q Quantum Theory is formulated here at the foundational and quantum-mechanical level. Its basic postulate is that the physical quantum entity, denoted by Q, is the whole quantum state rather than an independently real material point whose unknown position is described by a secondary probability cloud. In a one-Q sector admitting a local amplitude representation,\begin{equation} Q(x)=\sqrt{\rho(x)}\,e^{i\theta(x)}Z(x),\end{equation}where x is a spacetime point, ρ is the Born response weight, θ is the phase, and Z is a normalized internal structural direction. A dagger denotes the Hermitian adjoint and ∂μ differentiates with respect to xμ. The decomposition has a local rephasing redundancy. This is a redundancy of the local factorization of the complete Q amplitude, not the physical electromagnetic U(1) gauge symmetry. The associated decomposition-invariant phase-orientation one-form is\begin{equation} \kappa_\mu=\partial_\mu\theta-iZ^\dagger\partial_\mu Z =\frac{\operatorname{Im}\langle Q,\partial_\mu Q\rangle}{\rho}, \qquad \rho>0.\end{equation}In Schr\"odinger correspondence sectors this structure yields exact probability-current and momentum identities, including the standard Abelian gauge-covariant current. In that audit the electromagnetic connection is a correspondence variable; its Q-level interpretation as a local representation of inter-Q relational comparison geometry is developed in the companion manuscript Paper II-A. Klein-Gordon and Dirac audits show that relativistic currents and stress-energy depend on the Lorentz representation: κμ carries phase-orientation translation structure, while amplitude gradients and internal orientation can contribute independently to energy and stress. The standard Hilbert-space probability calculus is retained. Interference, tunnelling, entanglement, localized detection and cloud-chamber tracks are interpreted as properties or records of whole-Q states rather than as evidence for hidden point trajectories. Entanglement is represented by a nonfactorizable joint Q state; standard Bell correlations and no-signalling are retained without identifying correlation with a superluminal physical signal. Boundaryless state support is distinguished from the propagation of a newly generated disturbance, which is required to obey retarded relativistic causality. For stable relativistic sectors, invariant mass is treated as a derived characterization of the translation spectrum: timelike Q states possess a rest frame and rest energy, whereas null Q states do not. The foundational structure does not specify a unique interaction action or establish ultraviolet finiteness. Finite-response dynamics requires a separate construction.
The substrate field theory, developed in the first three papers of this series, posits a universal physical medium whose rigidity K and effective density ρ₀ determine the speed of light and whose dissipation governs cosmological redshift and orbital decay. In this paper, we show that the same substrate field naturally gives rise to all quantum phenomena as emergent properties of its microscopic structure. The substrate is assumed to possess a discrete granularity at a fundamental scale l₀, leading to quantization of energy and momentum. Linear excitations of the substrate—wave packets—exhibit wave–particle duality. The linearity of the wave equation allows coherent superpositions, giving rise to superposition and entanglement. The substrate's inherent dissipation provides a physical mechanism for decoherence, resolving the measurement problem without invoking a conscious observer. The Heisenberg uncertainty relation emerges as a Fourier property of wave packets in the substrate. Thus quantum mechanics is not a fundamental theory but an effective description of substrate dynamics at microscopic scales. This framework unifies quantum phenomena with classical physics, cosmology, and thermodynamics, eliminating the need for quantum postulates and explaining the apparent "weirdness" of the quantum world as natural consequences of a physical medium.Part IV of the Substrate Field Theory series
In previous work, the master action of the null-directional kinematics (NDK) framework was constructed from Fisher-information transport on the sphere bundle over spacetime, yielding a real amplitude field ψ(k^;x) with a quadratic kinetic sector inherited from the information geometry of probability densities. This paper derives the emergent Schrödinger dynamics of that framework in a controlled low-energy limit. The derivation proceeds in four steps. First, a controlled angular reduction justified by the spectral gap of the sphere-bundle mode tower truncates the full field to a spacetime envelope R(x) times a frozen angular profile. Second, the resulting reduced action is shown to be a real Fisher-density theory in spacetime. Third, the requirement that this density evolve locally and reversibly forces a canonically conjugate transport potential S(x,t) via a continuity constraint. Fourth, the resulting (ρ,S) system is shown to be exactly the Madelung form of a Schrödinger equation. The complex field is therefore not grafted onto the real theory but emerges as the compact packaging of a conjugate pair forced by reversible transport. Independent arguments from the dipole-current sector and the isotropic rest-mode sector pin the effective mass and the Fisher coefficient in terms of master-action couplings, yielding an emergent quantum scale of the form ℏeff2∼24βΛ∗/c2, with the precise numerical coefficient fixed up to order-unity convention-dependent factors. Stress-energy matching of the canonically normalized positive-frequency envelope fixes the density-normalization bridge at n∗=1. A systematic analysis then shows that all internal one-particle consistency conditions—free-particle dispersion, probability current, Galilean covariance, and circulation quantization—close without contradiction but do not generate an additional equation fixing the numerical value of ℏeff. The program therefore derives the structure of quantum mechanics from the NDK directional substrate, while the absolute quantum action scale remains open and is isolated to a single product of master-action couplings.