The relationship between quantum mechanics and general relativity is typically pursued via quantizing the spacetime metric or modifying the Standard Model. This paper proposes an alternative structural approach: treating both frameworks as local differential theories whose admissible global solutions are restricted by the geometry of their domains. The Schrödinger equation, expressed in Madelung form, makes this explicit: probability is conserved, and the resulting current must route around regions where the wavefunction is zero. This same structure of current routing around obstructions explains double-slit interference, Bell correlations, gravitational lensing, and the geometric role of phase in coherent evolution. Here, "phase" is used in its geometric sense as a connection on the space of states, while "density" is the conserved quantity whose zeros define the topological obstructions. The incompatibility therefore dissolves not by quantizing gravity or modifying quantum mechanics, but by recognizing their common structure. This paper initiates a series (Unitary Scale Theory, UST) examining scale-independent consequences of covering-space constraints on conserved currents. Version history v1 (January 2026): Initial publication. v2c (February 2026): Title refined to clearly reflect scope. Holonomy language updated throughout for consistency with covering-space / lift-projection framework developed in subsequent papers in the series. No changes to mathematical content or results.
The relationship between quantum mechanics and general relativity is typically pursued via quantizing the spacetime metric or modifying the Standard Model. This paper proposes an alternative structural approach: treating both frameworks as local differential theories whose admissible global solutions are restricted by the geometry of their domains. The Schrödinger equation, expressed in Madelung form, makes this explicit: probability is conserved, and the resulting current must route around regions where the wavefunction is zero. This same structure of current routing around obstructions explains double-slit interference, Bell correlations, gravitational lensing, and the geometric role of phase in coherent evolution. Here, "phase" is used in its geometric sense as a connection on the space of states, while "density" is the conserved quantity whose zeros define the topological obstructions. The incompatibility therefore dissolves not by quantizing gravity or modifying quantum mechanics, but by recognizing their common structure. This paper initiates a series (Unitary Scale Theory, UST) examining scale-independent consequences of covering-space constraints on conserved currents. Version history v1 (January 2026): Initial publication. v2c (February 2026): Title refined to clearly reflect scope. Holonomy language updated throughout for consistency with covering-space / lift-projection framework developed in subsequent papers in the series. No changes to mathematical content or results.
Quantized observables are usually attributed to quantum mechanics. We argue their origin is topological. Integer quantum numbers arise wherever transport around a nontrivial loop requires equivalence after projection in the associated phase bundle. Using Spacetime Algebra, we show that the rotor closure condition R_γ = ±1 provides a common covering-space criterion for integrality in atomic orbitals, superconducting rings, and black hole photon rings, spanning 23 orders of magnitude. Dynamics sets the characteristic scale. Topology enforces integrality. This is the second paper in a series examining scale-independent consequences of covering-space constraints on conserved currents. Version history v1 (February 2026): Initial publication as "Holonomic Closure as the Geometric Origin of Quantization Across Scales." v2 (February 2026): Title revised to reflect rotor transport and covering-space framework. Lift/projection language standardized throughout for consistency with series. SU(2)→SO(3) double cover explicitly named. No changes to mathematical content or results.
Standard quantum mechanics introduces Hilbert space, unitary evolution, the Born rule, and tensor-product composition as foundational elements of its formalism. General relativity introduces curved spacetime geometry and Einstein field equations. This work develops a conserved informational-throughput framework in which normalized quantum-state structure and Newtonian-limit gravitational time dilation arise from a common set of relations connecting local energy density to finite informational capacity. Starting from the informational demand relation κ = ρ_E V A₀/(ħc), the throughput field χ(r) = √(1 − 2κ/r), and the derived Capacity-Conservation Law (CCL), χ² + L = 1, we develop a unified informational route to a normalized Hilbert-compatible state structure, Schrödinger-type evolution, Born-style probability weighting, Tsirelson-limited shared-capacity correlations, and gravitational time dilation via χ(r) ≡ proper time rate k(r). In this framework, quantum coherence, superposition, entanglement, and probability are interpreted as consequences of conserved informational throughput rather than as independent postulates. The same scalar χ links quantum-state normalization with gravitational clock suppression.
Standard quantum mechanics introduces Hilbert space, unitary evolution, the Born rule, and tensor-product composition as foundational elements of its formalism. General relativity introduces curved spacetime geometry and Einstein field equations. This work develops a conserved informational-throughput framework in which normalized quantum-state structure and Newtonian-limit gravitational time dilation arise from a common set of relations connecting local energy density to finite informational capacity. Starting from the informational demand relation κ = ρ_E V A₀/(ħc), the throughput field χ(r) = √(1 − 2κ/r), and the derived Capacity-Conservation Law (CCL), χ² + L = 1, we develop a unified informational route to a normalized Hilbert-compatible state structure, Schrödinger-type evolution, Born-style probability weighting, Tsirelson-limited shared-capacity correlations, and gravitational time dilation via χ(r) ≡ proper time rate k(r). In this framework, quantum coherence, superposition, entanglement, and probability are interpreted as consequences of conserved informational throughput rather than as independent postulates. The same scalar χ links quantum-state normalization with gravitational clock suppression.
Important status notice: This version has been superseded by v2.0. Subsequent work shows that the single-domain three-diagonal construction used here produces an experimentally excluded common-mode photon-speed anisotropy. Readers should consult the revised version and 10.5281/zenodo.21464971. We propose that conservation of probability — the single axiom A=1 — is sufficient to derive the principal results of quantum mechanics without additional postulates. On a bipartite three-dimensional causal lattice T³_diamond, each particle is represented as a causal session: an independent phase oscillator that propagates amplitude across the lattice subject to |ψ_R|² + |ψ_L|² = 1 at every tick. No Hamiltonian and no measurement postulate are assumed; a Hilbert-space structure with unitary evolution emerges from the U(1) oscillator and the per-session amplitude constraint rather than being postulated a priori. From this single constraint, the paper derives: (i) the Dirac equation in the continuum limit a→0, with the gamma matrices identified as geometric encodings of the bipartite RGB/CMY sublattice geometry; (ii) wave-particle duality and the Born rule as consequences of amplitude redistribution under A=1, with Gleason's theorem securing the |ψ|² uniqueness from the framework's emergent Hilbert space; (iii) quantization of atomic orbitals as Arnold tongue attractors of the two-body phase dynamics, recovering the hydrogen Bohr radius to four significant figures (k_min = 0.0970 vs k_Bohr = 0.0971) without fitting; (iv) gravity as a clock-density gradient rather than spacetime curvature — a refraction of causal paths through regions of higher session density, reproducing the 1/r² force law with an explicit weak-field metric tensor g_μν as a functional of the clock-density field; (v) photon emission as a structural consequence of A=1: the Coulomb interaction is a probability source that a single session cannot accommodate, mandating creation of a new photon session at orbit lock-in; (vi) the structural form of the induced gauge action (Sakharov / Zeldovich), whose leading-order term has the Maxwell form on O_h-averaged observables, with a closed-form gauge-sector birefringence sharing the same optical axis (1,1,−1) as the kinematic-sector prediction. Energy and momentum conservation are not independent inputs — they are consequences of A=1 applied to closed session sets. The numerically tractable results are verified or partially verified by a suite of more than fifteen experiments with no free parameters beyond the Planck-scale calibration of the lattice spacing; an audit table records the status of every claim. The paper conjectures that the Standard Model gauge group SU(3) × SU(2) × U(1), together with Lorentz SO(3,1), is the automorphism group of the framework's extended per-site amplitude — a precise 18-dimensional Lie-algebra equality. The existing framework is shown to carry SO(3,1) × U(1) (dim 7) on the per-site (ψ_R, ψ_L) amplitude; the remaining factors require structural extensions (per-site C² isospin and C³ colour indices) that are the subject of a companion paper. Six computational scaffolding scripts establish the conjecture's components. This is Paper I of the A=1 Discrete Causal Lattice series. Companion papers in preparation address the Standard Model gauge derivation, the Farey-sequence mass spectrum (and its Veneziano-amplitude limiting case), plasma and gradient-ionization phenomenology, the proton-interior structure, quantum measurement theory, and a Hilbert-6th-style combinatorial foundation for continuum physics. What changed in v1.0 (since v0.98-RC): * Three new audit-table rows: long-horizon escape mechanism (PART), lock-in resonance grid-independence across {65, 81, 97, 113}³ grids (PART), and the Standard Model gauge-group conjecture as a precise 18-dimensional Lie-algebra equality (STUB).* A Gleason-uniqueness subsubsection in the Born-rule discussion, addressing the necessity question for the |ψ|² measure.* An explicit metric-tensor formula in the gravity section, identifying g_μν as a functional of ρ_clock in the weak-field limit.* Refined automorphism conjecture with explicit decomposition into established and structural-extension factors.* New Acknowledgments section. Source code, paper sources, and data files: https://github.com/JackDMenendez/discrete-causal-latticeLicense: Paper text and figures CC BY 4.0; source code MIT.
Abstract: We present a model-theoretic framework that derives the mathematical structures of physics from operational coherence constraints. Uniquely, we employ a “nominalist air-gap,” extracting these structures from a first-order language containing no mathematical objects in its ontology. Our construction has three main components: (1) We show how any minimal coherence model M |= T_min generates a canonical effect algebra E(M ) via a constructive free completion. (2) We demonstrate that the apparent conflict between Quantum Mechanics and General Relativity can be expressed as a difference in the scope of the Riesz Decomposition Property (RDP). Global RDP (T_rich) forces the algebra to embed into an ordered real vector space, while Local RDP (T_grav) gives rise to a presheaf of local algebras over a derived context topology. (3) We analyze a Path Independence property: global RDP is naturally associated with path-independent transport in context space, whereas non-trivial holonomy obstructs any such global linear structure. Motivated by this obstruction, we formulate a Logical Preservation Principle according to which physical evolution must restrict to locally coherent sectors when global RDP would lead to inconsistency, and we interpret this restriction as objective collapse.
The conserved probability current of the Schrodinger field admits a dimension-dependent gauge representation that provides a convenient setting for comparing hydrodynamic, topological and infrared structures. We use this representation to develop a unified treatment of several deformations of the Madelung momentum. BF couplings incorporate electromagnetic interactions, Berry and spin connections, projected non-abelian adiabatic data, and intrinsic phase holonomy within the same framework. In (2+1) dimensions, the Chern-Simons deformation can be reduced to a nonlocal functional of the wavefunction. Its Coulomb-gauge form separates into a phase-sensitive geometric contribution and a density-dependent dynamical contribution; we show explicitly how this separation changes under a general gauge transformation and recover the topological braiding phase in the well-separated adiabatic regime. For systems with a spatial boundary, we derive the equal-time symplectic structure, the associated surface charges, and their boundary algebra. Finally, after a nonlinear interaction produces the Bogoliubov acoustic branch, the radiative memory can be expressed in the dual two-form variables. Its zero-average component is generated by a residual large gauge transformation, whereas the monopole component remains independent global data. This provides a gauge-theoretic description of the memory/asymptotic-symmetry sector of the acoustic infrared structure.