We present a machine-verified formalization, in Lean 4 with Mathlib, of the hypothesisthat matter is an emergent invariant of computation. The framework models a quantumcellular automaton as a unitary operator on a finite-dimensional Hilbert space equippedwith a conserved energy functional and a physical-subspace projector. Matter is identifiedwith the quotient of the fixed-point subspace of the evolution by a global-phase equivalencerelation. We prove: (1) unitarity preserves inner products and norms; (2) the energy func-tional is conserved under iterated evolution; (3) nontrivial fixed-point subspaces exist; and(4) coarse-grained dynamics converge to a stable effective theory. All proofs compile withzero sorry statements and use only the standard axioms of dependent type theory (propext,Classical.choice, Quot.sound).
We present a machine-verified formalization, in Lean 4 with Mathlib, of the hypothesisthat matter is an emergent invariant of computation. The framework models a quantumcellular automaton as a unitary operator on a finite-dimensional Hilbert space equippedwith a conserved energy functional and a physical-subspace projector. Matter is identifiedwith the quotient of the fixed-point subspace of the evolution by a global-phase equivalencerelation. We prove: (1) unitarity preserves inner products and norms; (2) the energy func-tional is conserved under iterated evolution; (3) nontrivial fixed-point subspaces exist; and(4) coarse-grained dynamics converge to a stable effective theory. All proofs compile withzero sorry statements and use only the standard axioms of dependent type theory (propext,Classical.choice, Quot.sound).
This work introduces a minimal variational framework in which localized energy basins and their fixed points form the foundation for emergent quantum dynamics. Rather than imposing quantization axiomatically, discrete spectra arise naturally from geometric regularity, topological invariants, and adjacency-driven transport within a continuous relational configuration space. Key contributions include: Emergent Schrödinger Dynamics: Oscillatory modes near basin fixed points admit a complex structure, leading to a Schrödinger-type evolution derived from a quadratic variational principle rather than canonical quantization. Topology-Induced Quantization: Integer-valued invariants such as winding and linking numbers enforce discrete excitation spectra without probabilistic postulates. Hydrogenic Consistency: The framework reproduces the hydrogen spectrum with emergent parameters, matching experimental wavelengths to within 0.1 nm. Conceptual Foundations: Spacetime is treated as an emergent descriptor of basin adjacency and deformation, not as a pre-imposed manifold. This approach provides a conservative and mathematically rigorous route to quantum-like behavior grounded in geometry and topology, while remaining compatible with classical correspondence in the coarse-grained limit. Extensions to relativistic regimes, gauge structures, and gravitational coupling are deferred to future work. Keywords: Emergent quantum dynamics, fixed-point geometry, basin framework, topology-induced quantization, Schrödinger evolution, variational principles.
This preprint is the second installment in a research program that develops a geometric ontic framework for finite-dimensional quantum theory. Paper I (“A Geometric Ontic Framework for Quantum States Based on SU(2) Symmetry and Non-Factorizable Projections”) introduced the kinematic layer of the theory: quantum states were represented as points on an underlying ontic manifold equipped with SU(2) symmetry, and standard single-particle quantum structure emerged from geometric projection rules. That first step reconstructed the full Hilbert-space picture for a single degree of freedom and identified how non-factorizable geometry underlies entanglement. The present paper extends this framework by introducing a dynamical layer. We postulate that the ontic metric G depends on the global configuration of the system, and we study its evolution under a general metric flow. Two principles guide this construction. The Quantum Equivalence Principle (QEP) requires reversible operational dynamics to preserve local distinguishability, recovering unitary evolution in symmetric limits. The Quantum Mach Principle (QMP) assigns dynamical content to the metric by allowing mild back-reaction from the ontic distribution. Together with a coarse-graining map RN(λ), these ingredients generate operational states and lead to an emergent Schrödinger-type dynamics. Standard quantum mechanics appears as the symmetric fixed point of the metric flow, while small departures from symmetry lead to controlled deviations from unitary evolution and the Born rule. This provides a geometric explanation for quantum kinematics and suggests testable signatures beyond the standard theory. This work lays the foundation for Paper III, where composite systems and curvature-induced entanglement dynamics are developed.
This work presents a rigorous, first-principles audit of unitarity in quantum mechanics, treating unitarity not as a foundational axiom but as a hypothesis whose validity depends on explicit physical and mathematical assumptions. The analysis is conducted entirely within the operational framework of open quantum systems, using completely positive trace-preserving (CPTP) maps to describe the dynamics of reduced subsystems. By systematically isolating and auditing every prerequisite required for unitary evolution—including perfect isolation, Hilbert-space factorization, linearity, the Born rule, and time parameterization—the manuscript demonstrates that unitarity is neither fundamental nor universal, but instead an emergent, scale-dependent consistency condition. The paper distinguishes clearly between mathematical bookkeeping identities and physically meaningful conservation laws, showing that probability conservation does not require unitarity, and that reversibility and energy conservation are not guaranteed by unitary evolution alone. Explicit, mathematically consistent non-unitary evolution models are constructed, including Lindblad dynamics, non-Hermitian Hamiltonians, stochastic Schrödinger equations, non-Markovian evolution, and quantum Darwinism. The work further identifies the precise regimes in which unitary dynamics emerges as an effective approximation—such as weak coupling, short timescales, or limited observational resolution—and delineates the conditions under which it fails. Empirical evidence from atomic spectroscopy, quantum information processing, decoherence experiments, and measurement theory is used to anchor the analysis without assuming fundamental unitarity. The central conclusion is unambiguous: no physically accessible open or effective subsystem undergoes strictly unitary evolution. Unitarity is shown to be an emergent, conditional, and scale-dependent property rather than a fundamental law of nature. This manuscript is intended as a foundational clarification of the status of unitarity in quantum theory and is suitable for researchers in quantum foundations, open quantum systems, quantum information, and quantum gravity. FUSION OS INNOVATIONS
We present a fully formal derivation of quantum-theoretic structure from a single primitive: the free tree T generated by all finite distinctions. Taking T as the sole ontological substrate, we construct the additive group via a universal algebraic quotient, prove that internal complementary branches supply additive inverses, and show that finite cyclic relabelling symmetries force the scalar field to be C. The Hilbert space is shown to be the unique ℓ^2-completion satisfying translation invariance, disjoint additivity, and unitarity of internal symmetries—conditions forced by the tree’s combinatorics. On coherent subspaces, approximate isometries yield exact unitaries via the polar decomposition; the Riesz functional calculus then extracts self-adjoint generators without external assumptions. Finally, we eliminate the external time parameter by defining the continuum limit relative to an observer’s finite record resolution, using the Trotter product formula to recover the Schrödinger equation as an effective coarse-graining theorem. All constructions remain strictly internal to T; no background spacetime, external fields, or prior Hilbert-space structure is postulated.
We propose a categorical framework for physical emergence in which the renormalizationgroup is identified with the iteration of left Kan extensions along coarse-graining functors.Motivated by Grothendieck’s theory of motives, which extracts the universal cohomologicalcontent of algebraic varieties by considering all possible realizations simultaneously, we con-struct an analogous theory for physical systems by considering the category of all possiblecoarse-grainings. The “motive” of a physical system is the structure invariant under allsuch coarse-grainings: the fixed point of the renormalization group flow. We show that thisframework (i) gives a categorical derivation of universality classes as isomorphism classesin a motivic quotient, (ii) characterizes phase transitions as failures of representability ofpro-systems of Kan extensions, and (iii) recovers the Connes–Marcolli Cosmic Galois Groupas the automorphism group of the motivic quotient. The identification of the partition func-tion with a Kan extension is made rigorous through enriched category theory and weightedcolimits, where the Boltzmann factor enters as the weight functor. As a detailed workedexample, we analyze the two-dimensional Ising model, identifying the critical “stacky point”as the c = 1/2 Virasoro minimal model with automorphism group Virc=1/2 ⋊ Z/2Z, andconnecting the logarithmic heat capacity singularity to the obstruction to representability.We formulate precise conjectures relating this framework to established results in Wilsonianrenormalization, conformal field theory, and algebraic quantum field theory. A careful dis-ambiguation of the polysemous terms “motive” (pure/Chow, Nori, mixed) and “emergence”(thermodynamic, epistemic, ontological) is provided, identifying which interpretation is op-erative in each section and separating th
Relational Actualism (RA) proposes that the transition from quantum potentia to classical actuality is grounded in a precise physical criterion: an irreversible increase in quantum relative entropy ΔS(ρ∥σ0) > 0 with respect to the vacuum. The companion paper Relational Actualism: The Topology of Matter (RATM) has now established, via 67 Lean-verified theorems, that the BDG-stable causal topology types in 4D correspond exactly to the Standard Model particle spectrum—and that electrons and photons, the particles from which qubits are built, propagate at the minimum BDG actualization robustness score of 1. Qubit fragility is structurally grounded: no stable particle with lower BDG score exists. In the perturbative regime this coincides with the on-shell dispersion relation pμ pμ = m2c2, at which a mediating boson crosses the kinematic threshold from virtual to real and an irreversible vertex is inscribed into the growing causal directed acyclic graph (DAG) of the universe. This paper develops the implications of this framework—established in the companion papers RAQM, RAGC, RACI, and RAHC—for two fields that have lacked a precise physical account of irreversibility at the quantum level: quantum computing and quantum thermodynamics. For quantum computing, the Kinematic Snap provides a principled physical floor on decoherence, distinguishes genuinely off-shell (decoherence-immune) interaction regimes from on-shell ones, and gives the threshold theorem a physical rather than purely operational interpretation. For quantum thermodynamics, RA grounds Landauer’s principle in the causal DAG, resolves the Maxwell’s demon paradox without informationtheoretic postulates, gives the fluctuation theorems a precise actualization-event interpretation, and establishes that the thermodynamic arrow of time is structural rather than emergent. A key technical result: the quantum relative entropy—the central quantity in both quantum information theory and RA’s actualization criterion—is frame-independent by a compiled Lean 4 theorem (frame_independence in RA_AQFT_Proofs_v10.lean), proved using the continuous functional calculus unitary-conjugation lemma from the Lean-QuantumInfo library. The RA suite now has 101 Lean-verified results (zero sorry tags) across RA_AQFT_Proofs_v2.lean and the new RA_D1_Proofs.lean, which proves the BDG particle classification including colour confinement and the particle universe closure theorem. The RA suite now has 93 Lean-verified results across two proof files (RA_AQFT_Proofs_v2.lean: 26 results; RA_D1_Proofs.lean: 67 results), all with zero sorry tags and no axioms beyond the Mathlib standard library. A key quantitative connection: the Kinematic Coherence Bound threshold pth = αEM ≈ 1/137 ≈ 7.3°ø10−3 is shown to be consistent with the galactic-scale coupling constant ξ of the companion paper RADM to within ∼2% using the full baryonic disk mass, at the Planck scale, via the thermodynamic field equations established in RAGC. The Kinematic Coherence Bound threshold pth = αEM ≈ 1/137 ≈ 7.3°ø10−3 also enters the Poisson-CSG generative measure of RAGC (equation for cRA k ): it is the per-link actualization probability that determines whether the causal DAG is locally 3D or 2D, connecting quantum error correction thresholds to galactic-scale dimensional reduction and the Tully-Fisher relation. The fault-tolerance threshold, the speed of light c = lP/tP, and the biological Causal Firewall are all the same Erd˝os-R.nyi percolation transition at different scales of the causal graph. The Einstein field equations are shown to be the unique consistent macroscopic description of the RA-CSG by Lovelock’s theorem, with GR exact at μ ≫1 and RA’s departures from GR (rotation curves, Hubble tension, Planck corrections) arising precisely where μ ∼ 1. Throughout, implications are stated conditionally on the correctness of the RA framework, which is currently under peer review.