Quantum field theory unifies special relativity and quantum mechanics yet relies on externally imposed prescriptions—quantization, renormalization, and probabilistic interpretation—that lack a common algebraic foundation. This work derives these features from a single structural principle: the finite-resolution realization of the additive group [Formula: see text] under four internal consistency constraints. These constraints—finite operational resolution (FR1), multiplicative linearity (FR2), phase coherence (FR3), and statistical typicality (FR4)—generate, without additional postulates, the Hilbert-space formalism of quantum theory, the Born probability rule, and the Kolmogorov structure of statistical mechanics as limiting projections of one additive algebra. When extended to spacetime translations and their automorphisms, the same framework reproduces the kinematic and dynamical structure of quantum field theory while resolving ultraviolet divergences through intrinsic finiteness rather than external renormalization. Quantization and unitarity appear as consequences of additive symmetry, while the gravitational field emerges as the algebraic kernel that transports boundary information into bulk observables without requiring a separate quantization of the metric. The result is a unified, finite, and strictly algebraic foundation for physical law.
Quantum field theory unifies special relativity and quantum mechanics yet relies on externally imposed prescriptions—quantization, renormalization, and probabilistic interpretation—that lack a common algebraic foundation. This work derives these features from a single structural principle: the finite-resolution realization of the additive group [Formula: see text] under four internal consistency constraints. These constraints—finite operational resolution (FR1), multiplicative linearity (FR2), phase coherence (FR3), and statistical typicality (FR4)—generate, without additional postulates, the Hilbert-space formalism of quantum theory, the Born probability rule, and the Kolmogorov structure of statistical mechanics as limiting projections of one additive algebra. When extended to spacetime translations and their automorphisms, the same framework reproduces the kinematic and dynamical structure of quantum field theory while resolving ultraviolet divergences through intrinsic finiteness rather than external renormalization. Quantization and unitarity appear as consequences of additive symmetry, while the gravitational field emerges as the algebraic kernel that transports boundary information into bulk observables without requiring a separate quantization of the metric. The result is a unified, finite, and strictly algebraic foundation for physical law.
Translator This paper reformulates quantum entanglement within the framework of Finite-Resolution Physics (FRP), a finite-coherence version of quantum theory derived from the Reflexive Monoid Algebra (RMA) hierarchy.All experimental predictions of entanglement—joint probabilities, Bell-inequality violations, and Born frequencies—remain identical to those of standard quantum mechanics.The difference lies in structure: entanglement is no longer a separate postulate about tensor-product Hilbert spaces but an algebraic phase-consistency relation (FR3) and typicality constraint (FR4) linking additive subsystems under finite resolution.This reinterpretation explains nonlocal correlations as boundary-level coherence, not superluminal influence, and describes decoherence as the loss of FR3 coherence, marking the transition from quantum (Qμ) to statistical (Sμ) regimes.The work provides a unified algebraic account of entanglement, nonlocality, and decoherence within a finite-resolution operational framework.
Framework position. This paper is a foundational contribution to the internal-symmetry branch of the Closure–Stabilization Framework. It identifies internal symmetry with the stabilizer of a Simultaneously Interlocked Boundary structure (SIB) on an already represented sector. In linear dualizable realizations, the SIB is represented by its annihilator Kμ⊥, recovering the earlier SILM formulation as a special case. The paper therefore places internal symmetry downstream of stabilization, representation, and boundary compatibility, while keeping projective/spinorial external representation sectors conceptually distinct from the internal symmetry acting on their multiplicity or species degrees of freedom. This manuscript develops the internal-symmetry layer of Finite-Resolution Physics (FRP). In FRP, mathematical representation is taken to arise only after operational closure relations stabilize under refinement. The paper asks where internal symmetry enters this architecture and what determines its admissible form. The central claim is that, on a stabilized finite-resolution plateau, admissible representation channels must satisfy joint boundary-coherence constraints across finite-resolution interfaces. These constraints define a finite-dimensional boundary constraint space. The dual annihilator of this space is identified as the algebraic space of Simultaneously Interlocked Lagrange Multipliers, or SILMs. The physical internal symmetry group is then defined as the stabilizer of the SILM space inside the group of finite-resolution-preserving unitary automorphisms that commute with the external carrier action. Thus internal symmetry is not introduced as a primitive kinematical input, nor as an arbitrary gauge group. It is a plateau-relative stabilizer of finite-resolution boundary coherence. The paper also explains how symmetry plateaux correspond to refinement ranges over which the SILM space is canonically identified, while symmetry transitions correspond to changes in the SILM structure. In the finite-dimensional Hermitian case, the resulting stabilizers reduce to products of unitary groups determined by SILM multiplicity data. A simple three-plus-two-plus-one example is discussed as an illustration of the stabilizer mechanism. The paper does not derive the Standard Model gauge group. Rather, it identifies the structural position that any Standard-Model-like internal symmetry would occupy within the FRP architecture: a compact internal stabilizer plateau compatible with a real-carrier, quantum-field-theory-like external structure.
(This work presents an initial conceptual formulation of the renormalization-based approach to quantum theory within the TQF framework. A subsequent version 2 develops formulation of the Quantum reconstruction based on a linear state space, providing derivation of quantum-mechanical kinematics—including the emergence of the Hilbert space structure and the Born rule—from resolution independence. Readers primarily interested in the formal derivation are encouraged to consult the later versions, while the present work may be viewed as outlining the guiding ideas and initial physical intuition behind the quantum reconstruction.) The mathematical structure of quantum mechanics is traditionally introduced axiomatically, with the linearity of the state space, the Hilbert space structure, and the Born rule postulated independently. In this work, we investigate whether these features can instead emerge from a fundamental symmetry principle: the absence of a preferred scale, implemented as invariance ofphysical observables under coarse-graining transformations. We formulate a framework in which physical states are represented as functionals over configuration space and are subject to a family of coarse-graining transformations forming a renormalization group (RG) semigroup. Physical observables are required to be invariant under these transformations, while dynamical operators are allowed to transform covariantly withinthe induced RG flow. Under general consistency conditions—including associativity, symmetry, continuity, and invariance under refinement—we show that the admissible composition law of states admits an additive representation up to an invertible reparametrization, leading to a linear structure of the state space in reparametrized variables. Imposing consistency of weight assignments under coarse-graining further constrains the norm, selecting a quadratic form and inducing an inner product structure. As a result, the space of states acquires the structure of a complex Hilbert space without being postulated a priori. Within this framework, probability emerges as a measure induced by the state functional. By analyzing the transformation of weight assignments under coarse-graining, we identify an effective renormalization group flow in the space of admissible probability rules and show that the quadratic rule arises as a distinguished and heuristically stable fixed point. This provides adynamical origin for the Born rule. We further analyze the semiclassical regime, in which coarse-graining suppresses interference between configuration sectors, leading to decoherence and the emergence of effective classical behavior. In this limit, classical dynamics and an effective notion of time arise from the phase of the state functional, and an approximate Schr¨odinger equation is recovered for fluctuations. These results suggest that the core structural features of quantum mechanics can be understood as universal fixed-point properties of an underlying renormalization symmetry acting on the space of states. This perspective provides a conceptual bridge between quantum mechanics and scale-invariant quantum field theories, including the Timeless Quantum Field framework.
Modern theoretical physics employs several fundamental mathematical structures, including Green functions, Hilbert spaces, operator algebras, Gibbs ensembles, gauge fields, and spacetime geometry. These structures are traditionally introduced independently and serve as separate foundations for quantum mechanics, quantum field theory, statistical mechanics, and general relativity. In the present work, we propose an alternative mathematical organization based on a single Reconstruction Principle. Beginning with one primitive mathematical architecture, successive reconstruction layers generate transport operators, Green functions, Diagram Hilbert Spaces, complex quantum operators, Gibbs ensembles, entropic projections, and finally macroscopic spacetime together with gauge fields. Each layer is reconstructed uniquely from its predecessor without introducing additional primitive mathematical structures. Consequently, Hilbert space, Green functions, and spacetime are interpreted as successive mathematical representations of one underlying architecture rather than independent foundations. The resulting reconstruction hierarchy provides a unified conceptual framework connecting quantum mechanics, statistical mechanics, gauge theory, and emergent gravity while substantially reducing the number of independent assumptions.
The mathematical structure of quantum mechanics is traditionally introduced axiomatically, with the linearity of the state space, the Hilbert space structure, and the Born rule postulated independently. In this work, we investigate whether these features can emerge from a fundamental symmetry principle: the absence of a preferred scale. We formulate a framework in which physical states are represented as functionals over configuration space and are subject to a family of coarse-graining transformations forming a renormalization group (RG) semigroup. Physical observables are required to be invariant under these transformations, expressing scale neutrality at the level of state descriptions. Under general consistency conditions—including associativity, symmetry, continuity, and invariance under refinement—we show that the admissible composition law of states is uniquely additive, leading to a linear structure of the state space. Imposing consistency of weight assignments under coarse-graining further constrains the norm, selecting a quadratic form and inducing an inner product structure. As a result, the space of states acquires the structure of a complex Hilbert space without being postulated a priori. Within this framework, probability emerges as a measure induced by the state functional.By analyzing the transformation of weight assignments under coarse-graining, we define a renormalization group flow in the space of admissible probability rules and show that the quadratic rule arises as a distinguished and heuristically stable fixed point. This provides a dynamicalorigin for the Born rule. We further analyze the semiclassical regime, in which coarse-graining suppresses interference between configuration sectors, leading to decoherence and the emergence of effective classical behavior. In this limit, classical dynamics and an effective notion of time arise from the phase of the state functional, and an approximate Schr¨odinger equation is recovered for fluctuations. These results suggest that the core structural features of quantum mechanics can be understood as universal fixed-point properties of an underlying renormalization symmetry. This perspective provides a conceptual bridge between quantum mechanics and scale-invariant quantum field theories, including the Timeless Quantum Field framework.
The companion paper (Tang, 2026) showed that the Self-Consistent Reference Frame Principle (RCC), applied to local Hilbert-space fibres, yields the Born rule, the Yang--Mills action, and the Einstein--Hilbert action as consequences of reference-frame self-consistency and causality. (In that derivation, "quantum foundations" refers to the Born rule and the emergence of Hilbert-space structure; unitary dynamics, the tensor-product rule for composite systems, and the measurement postulate lie outside its scope.) The present paper removes the linearity assumption. It reformulates RCC on nonlinear state manifolds---smooth manifolds without a preferred linear or metric structure, on which symmetry groups act by diffeomorphisms. The core object is the nonlinear comparator, a G-equivariant diffeomorphism between fibres that translates reference frames. Each result of the linear paper---the Born-rule probability formula, the gauge connection, the gravitational metric, and the phase unity identity---is reconstructed in this more general setting and reduces to its linear counterpart precisely when the state manifold carries a Kähler structure, the geometric condition under which P=|⟨φ|ψ⟩|² is recovered. (Unitary dynamics, the tensor-product rule, the observable algebra, and the measurement postulate remain external inputs.) The nonlinear framework reveals structures inaccessible in the linear case: the Born rule generalises to a family of G-invariant transition functions; gauge theory arises as a special case of fibre-bundle geometry with a nonlinear G-action; nonlinear comparator corrections---torsion and anholonomy effects from the curvature of the fibre---generate additional SMEFT operators, most notably O₃B with C₃B≠0; and the gravitational sector receives subleading corrections from fibre-valued geometric invariants. The nonlinear framework yields a strictly larger set of falsifiable predictions: the linear results are recovered in the Kähler limit, while the deviations constitute a new experimental window into the geometry of state space.