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.
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.
The Schrödinger equation is normally introduced as a foundational postulate of quantum mechanics. In this article, we show that Schrödinger evolution arises naturally from the informational throughput framework defined by three relations connecting energy density to a bounded update capacity χ. Without assuming Hilbert space axioms, wavefunction ontology, or spacetime geometry, conservation of informational throughput forces norm-preserving evolution, which uniquely leads to complex phase rotation and the Schrödinger equation. The result suggests that quantum dynamics may emerge from finite informational processing capacity of physical reality.
The Problem of Time — the apparent incompatibility between quantum mechanics' time parameter t and general relativity's proper time τ — has been treated as a fundamental obstacle to quantum gravity. I argue that this is a level-confusion: both theories define time within their own entity-theoretic frameworks, generating an apparent opposition that disappears when time is relocated to a meta-structural layer. Drawing on the meta-level temporal structure T_SDT := (V_G, ≺) (Zeng 2026), the paper formalizes QM and GR time as two projections of a single Universal Time Formula, parameterized by entity-theoretic tokens. Four token specializations recover the corresponding time concepts. A Bridge Theorem establishes mutual compatibility of all token projections. A bidirectional QM-GR channel is constructed (forward: GR geometry → QM phase; reverse: discrete chain-pair → Lorentzian spacetime), composing into a self-consistent closed loop. Three further results move beyond dissolution toward unification. First, a Madelung-bridge construction derives the Schrödinger equation — Hilbert space, unitarity, Hamiltonian, quantum potential, Born rule — from the determination-graph axioms together with the continuum embedding (CL1), with no independent QM input; the derivation extends to Klein-Gordon. Second, a Planck-scale token μ = t_P / |A_{v'}| is constructed on the discrete 𝒢_D, where |A_{v'}| is the admissibility cardinality; under CL1, this single step-measure recovers QM (|A| ≫ 1) and GR (|A| = 1) as its two continuum limits, with Einstein's equation proved in the classical regime (|A| = 1) via a two-stage argument (Benincasa-Dowker action convergence + standard scalar-tensor variation). The exact quantum-gravitational correction is derived as a scalar-tensor equation with φ = 1/|A|. Third, the discrete determination graph without continuum embedding constitutes a self-contained Planck-scale quantum gravity theory: it has a well-defined path integral, a finite-dimensional S-matrix (unitary in the κ = 0 sector, isometric in the κ > 0 sector; no UV divergences), preserves information across black hole horizons, resolves singularities, derives the Past Hypothesis, and provides structural origins for gauge symmetry, Hawking radiation, spectral dimensionality, and the spin-statistics connection. The Problem of Time is not merely dissolved — it is replaced by a constructive framework in which QM and GR are unified as two limits of a single discrete structure.
We propose a self-consistent axiomatic framework that unifies quantum unitary evolution, thermal dissipation, and spacetime geometry based on three fundamental principles: the \emph{complex-time principle}, the \emph{information-geometry equivalence principle}, and the \emph{maximal entropy gradient principle}. Starting from these axioms, we rigorously derive three core equations: a complex-time quantum-thermal master equation, an information-induced gravitational field equation, and a unified gradient-flow dynamics that couples quantum states, thermodynamics, and curvature. From these, we further derive four key derived equations: a generalized Einstein field equation in the low-energy limit, a quantum-thermodynamic uncertainty relation, a relation between entropy production and curvature scalar, and a conservation law for quantum information current. The framework naturally recovers standard quantum mechanics, thermodynamics, and general relativity in appropriate limits, while offering new predictions about the relation between quantum entanglement and spacetime structure. All derivations are carried out in full mathematical detail, and the resulting equations are original, closed-form, and internally consistent.
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 This article does not claim to present a successful theory of quantum gravity. Instead, it undertakes a systematic and self-aware survey of the research program aiming to derive both quantum mechanics and general relativity from a purely relational network, based on a set of minimal ontological postulates. The framework rests on three axioms: (1) physical reality consists of nodes without intrinsic properties and their symmetric relations; (2) the relation strengths are non-negative and symmetric, with the total information capacity conserved as an ensemble expectation; (3) local evolution is restricted to redistribution of relations within triads. We honestly display and rigorously distinguish the conceptual achievements from the essential obstacles of this framework. On the quantum side, we obtain solid conceptual progress: the Born rule, unitary evolution, and topological nonlocality of entanglement all emerge naturally as representation-theoretic consequences of the statistical description of a finite information capacity network. This strongly suggests that the mathematical structure of quantum mechanics may be a generic feature of the statistical representation of any complex informational system satisfying “finite information + local relational evolution.” On the gravitational side, however, we encounter chasms that current mathematical and conceptual tools cannot bridge. A central contribution of this article is to clearly label these obstacles as fundamental open scientific problems: the conflict between a positive-definite relational ontology and the target of pseudo-Riemannian signature (signature problem); the missing coarse-graining procedure from a discrete network to a smooth manifold (coarse-graining problem); and the disharmony between external algorithmic time and emergent physical time (time problem). We argue that these are not technical details unique to this framework, but core challenges that any “spacetime from relations” program must confront. The article positions itself as a “conceptual cartography”—it precisely delineates the known boundaries of this research program, exposes the fatal rift often bypassed in mainstream discourse, and transforms all unresolved issues into a formal invitation for future investigation.