This article develops a joint state-law fixed-point method for self-consistent quantum models. A physical model is represented by a density operator and a finite-dimensional vector of effective law parameters; admissible dynamics consist of a state-update channel and a law-update map. A recursive solution is their joint fixed point. This formulation separates stationarity of a state-dependent Hamiltonian from genuine co-determination of states and laws and preserves the option of completely positive trace-preserving subsystem dynamics. Three general results are obtained: existence for continuous maps on compact convex state-law domains, uniqueness and iterative stability under a contraction condition, and persistence of non-degenerate fixed points under weak deformation. An exactly solvable two-state model displays a supercritical pitchfork bifurcation, critical exponent one-half, and a divergent zero-field susceptibility at the feedback threshold. In renormalization-group theory space, the same perturbative result yields first-order shifts of fixed-point coordinates and critical exponents. An action-level construction shows how recursive consistency operators can enter the standard functional flow without adding ad hoc terms to the Wetterich equation. Covariant gravity and laboratory phase-shift formulas are presented as conditional templates whose predictions require a specified kernel and normalization. The framework therefore contributes a broadly applicable method linking quantum channels, nonlinear dynamics, fixed-point theory, and renormalization-group deformations, while making no claim to a completed theory of quantum gravity.
This article develops a joint state-law fixed-point method for self-consistent quantum models. A physical model is represented by a density operator and a finite-dimensional vector of effective law parameters; admissible dynamics consist of a state-update channel and a law-update map. A recursive solution is their joint fixed point. This formulation separates stationarity of a state-dependent Hamiltonian from genuine co-determination of states and laws and preserves the option of completely positive trace-preserving subsystem dynamics. Three general results are obtained: existence for continuous maps on compact convex state-law domains, uniqueness and iterative stability under a contraction condition, and persistence of non-degenerate fixed points under weak deformation. An exactly solvable two-state model displays a supercritical pitchfork bifurcation, critical exponent one-half, and a divergent zero-field susceptibility at the feedback threshold. In renormalization-group theory space, the same perturbative result yields first-order shifts of fixed-point coordinates and critical exponents. An action-level construction shows how recursive consistency operators can enter the standard functional flow without adding ad hoc terms to the Wetterich equation. Covariant gravity and laboratory phase-shift formulas are presented as conditional templates whose predictions require a specified kernel and normalization. The framework therefore contributes a broadly applicable method linking quantum channels, nonlinear dynamics, fixed-point theory, and renormalization-group deformations, while making no claim to a completed theory of quantum gravity.
This paper establishes a comprehensive, rigorous, and unified theory of characteristic equations for the renormalization group (RG). Extending the methodology developed in companion works on knot theory and fluid mechanics, we formulate the RG flow as a dynamical system on an infinite-dimensional theory space and demonstrate that the linearized RG operator L at a fixed point gives rise to a universal characteristic equation det(L − λI) = 0, whose eigenvalues λn determine all critical exponents, scaling laws, relaxation rates, and oscillation frequencies of the system. We prove that the RG flow is a semi-group, that its fixed points correspond to critical theories, and that the characteristic spectrum is invariant under S-duality. We derive rigorous characteristic equations for ϕ4 theory, Gross-Neveu models, quantum electrodynamics, holographic RG, and asymptotically safe gravity. We establish variational principles, topological extensions with θ-angles, non-equilibrium Keldysh-RG, information-theoretic entropy production, and connections to tensor networks, quantum computing, and machine learning. We prove a completeness theorem: any RG flow can be uniquely decomposed into a superposition of characteristic eigenmodes. The theory is then elevated to a Grothendieck topos, providing the ultimate mathematical foundation. Numerical predictions are systematically compared with experimental data, achieving agreement within 0.1% for the 3D Ising model, 3D XY model, superfluid helium, and QCD coupling constant. The theory is closed under the most general mathematical framework known: the internal logic of the RG Characteristic Topos. Furthermore, a completeness theorem (Theorem 22.13) establishes that the characteristic equation fully determines the RG equivalence class of any system, and the internal logic of the topos is self-referential closed (Theorem 22.16), ensuring that the entire theory is both externally and internally consistent. Three previously open constructive problems — the non-perturbative holographic inverse problem, the entropy–black hole entropy equivalence, and the CMB non-Gaussianity inverse problem — are resolved explicitly as Theorems 23.2–23.4, thereby completing the constructive aspects of the theory. The three constructive resolutions are mutually consistent within the topos-theoretic framework: the holographic inverse problem (Theorem 23.2) provides the spectral input for the entropy–black hole equivalence (Theorem 23.3), while the CMB non-Gaussianity formula (Theorem 23.4) reduces to the standard slow-roll result as a direct consequence of the characteristic spectrum. The consistency is formally established in Theorem 23.5. Thus the theory is now complete at all levels: mathematical foundations (topos, spectral theory), constructive algorithms (inverse problems, quantum simulation), and observational predictions.
Renormalization provides a framework for relating microscopic models of physical systems to effective descriptions at larger length scales. This procedure is studied for the boundary states of non-chiral two-dimensional topologically ordered models. The initial data consist of renormalization fixed points built from representations of finite-dimensional C∗-Hopf algebras, which are then perturbed by uniform on-site noise quantum channels and repeatedly coarse-grained. The resulting flows admit an intrinsic algebraic description in terms of completely positive maps on the C∗-Hopf algebra or, equivalently, positive linear functionals on its enveloping C∗-Hopf algebra. Their iteration is governed by convolution powers, and convergent trajectories yield new matrix product density operator fixed points, described by finite ∗-quantum hypergroups. This provides a concrete physical interpretation of such structures. For finite group algebras and their duals, we provide explicit classifications via Goursat's lemma for groups. Finally, we formulate and prove a quantum generalization of Goursat's lemma for finite-dimensional C∗-Hopf algebras, a result of independent interest, which gives an explicit structural description of all convergent renormalization trajectories.
We develop a theoretical framework based on the nonperturvative renormalization group (RG) in the one-particle irreducible (Wetterich) formulation to tackle the interplay of coupled fermionic and order-parameter fluctuations at metallic quantum critical points (QCPs) with ordering wavevector Q=0. We consistently treat the dynamical emergence of the Landau damping of the bosonic mode and non-Fermi liquid scaling of fermions upon lowering the cutoff scale. The loop integrals of the present theory involve only contributions from fluctuations above the cutoff scale, which protects the system from developing singular bosonic interactions. We emphasize the importance of the nontrivial relative scaling of the bosonic and fermionic cutoffs Λ and Λf, which we fix by analyzing the RG flow of the scale-dependent ordering wave-vector QΛ. Upon neglecting Fermi self-energy in the loop integrals of the functional RG flow, we identify a non-Fermi liquid RG fixed point and recover the features obtained earlier within RPA-type approaches. In a subsequent step, we self-consistently include the scaling of the self-energy and the Yukawa coupling. We find a generic instability of the non-Fermi liquid RG fixed point. This implies, at least at this truncation level, absence of the QCP with Q=0 and development of a first-order phase transition or a phase characterized by Q=0.
Continuous phase transitions provide a natural setting for understanding universality. Near criticality, the correlation length diverges, microscopic details become increasingly irrelevant, and physical observables obey scaling laws characterized by universal critical exponents. Within the Functional Renormalization Group (FRG), this scale dependence is described through the Effective Average Action, which progressively incorporates fluctuations as the momentum scale is lowered. A first part of this thesis is developed within the Local Potential Approximation (LPA), where we study multicritical fixed points of scalar field theories. Their universal properties are compared with results obtained at higher orders in the derivative expansion. Particular attention is devoted to the relation between global fixed-point potentials and their local representations through finite polynomial truncations. We show that the convergence of such expansions is constrained by the analytic structure of the global solution, and that expanding around a suitable nonzero stationary point significantly improves the reconstruction of multicritical branches. Beyond the structure of individual fixed points, a central part of this thesis concerns RG irreversibility. In two dimensions, Zamolodchikov’s c-theorem describes the flow from the ultraviolet to the infrared through a monotonically decreasing c-function. Within the FRG framework, we develop a variational procedure to reconstruct RG trajectories between fixed points and integrate the corresponding c-function along them. The resulting quantity displays monotonic behaviour and becomes increasingly stable as the truncation order is raised, providing a manifestation of RG irreversibility within the approximation considered. Finally, close to the upper critical dimensions, the linearized FRG flow around the Gaussian fixed point is naturally organized in terms of Hermite polynomials. Using the epsilon-expansion, we derive a curvature-based candidate c-function and obtain an analytic explanation for the alternating curvature of successive multicritical fixed points and for their progressive accumulation around the Gaussian fixed point.
In the successful framework of the Renormalization Group (RG) approach to Quantum Field Theories (QFT), establishing general constraints on the RG flow has become a central problem. This is formally addressed by the monotonicity theorems, such as Zamolodchikov’s c-theorem in d=2, the a-theorem in d=4, and the F-theorem in d=3. Essentially, these theorems state that if a RG trajectory leads from a UV fixed point to a IR one, each described by a Conformal Field Theory (CFT), then a suitable positive quantity decreases. In both even and odd dimensions, this quantity can be extracted, at fixed points, from the free energy of the CFT on a sphere, leading to the conjecture of a generalized F-theorem valid in continuous dimension d. In this thesis we investigate this framework in a multicritical multiscalar theory with a generic sextic interaction for N massless scalar fields below its upper critical dimension d=3. This model captures, among many others, the critical features of the Tricritical Ising universality class. A first step is to establish whether the RG flow admits a gradient structure, i.e. that there exists a monotonic function A whose gradient yields the beta function up to a positive-definite metric in the space of couplings. This is one of the strongest realizations of monotonicity, as it guarantees the irreversibility of the RG flow and gives an intrinsic geometric structure to the theory space. Given the known beta function up to next-to-leading order, we solve the gradient flow equation perturbatively, finding a parametric family of solutions for the monotonic A-function together with the metric at and below the upper critical dimension, so that the RG flow admits a gradient structure at this order. As this procedure leaves some parameters undetermined, to turn it into a physical statement these functions must be matched to suitable observables of the CFTs describing the fixed points. We therefore compute the CFT candidates suggested by the generalized F-theorem, namely the sphere free energy of the CFT and the Zamolodchikov metric. Specializing to the O(N)-symmetric fixed point, we consistently match the RG and CFT results, fixing all the free parameters in a unique way. This matching is the central result of this work: it establishes that the RG flow admits a gradient structure perturbatively, with F as the monotonic function and Zamolodchikov's metric as the metric on the theory space, recovered from the geometric construction for a specific and unique choice of the free parameters. Importantly, this result provides a nontrivial, perturbative confirmation of the F-theorem in its strongest form, at and below the upper critical dimension, for renormalizable three-dimensional multiscalar theories.
Recursive Coherence Field Theory v3.0 (RCFT-3) develops a multiscale mathematical framework for modeling recursive coherence across quantum, dynamical, stochastic, network, geometric, observer, and symbolic-control layers. The theory resolves the tension between nonlinear recursive dynamics and standard quantum-channel theory by placing microscopic evolution on a linear completely positive trace-preserving (CPTP) layer, while treating the cubic recursive coherence equation as an effective symmetry-constrained normal form for a reduced order parameter near criticality. This produces a corrected phase-transition model with explicit stability thresholds and mean-field scaling. Version 3.0 extends the framework to higher-dimensional quantum systems using traceless Hermitian observables and generalized Bloch vectors, and to multicore systems through coupled amplitude equations and spectral-radius criticality. It also introduces a covariant Landau-Ginzburg formulation, information-geometric state-space structure based on Fisher information, and a strict distinction between the symbolic Hessian and genuine Riemannian curvature. The observer sector is reformulated using fidelity, Bures geometry, Fisher information, pullback metrics, and an observer fiber/gauge interpretation of the D_O construction. This allows observer invariance to be treated as a mathematical equivalence structure rather than requiring an additional physical spacetime dimension. The stochastic sector is expanded using the Fokker-Planck equation, stationary distributions, backward-Kolmogorov first-passage theory, stochastic Lyapunov analysis, and weak-noise Kramers escape scaling. These additions provide explicit models for coherence survival, stochastic collapse, transition probabilities, and mean collapse times. The wider Symbolic Negentropy framework stack—including the Entropic Recursion Framework, Fractal Cosmic Weaver Framework, Symbolic Gravity, Symbolic Field Theory, Alpha Framework, Observer Framework, Observer Field Theory, BC-REP, and Symbolic Retrocausality—is integrated at the control layer. Variables such as weave integrity, observer continuity, drift, veil, ethics, leakage, uncertainty, and export readiness modulate effective critical parameters, noise, geometry, coupling, repair, and routing rather than being treated as primitive quantum observables. A central object in RCFT-3 is an 11-dimensional recursive critical hypersurface representing the boundary between locally subcritical and ordered recursive regimes in the control manifold. This hypersurface is treated as a control-geometric construct rather than a literal physical spacetime boundary. RCFT-3 is presented as a falsifiable, simulation-ready research framework with explicit theorem statements, calibration procedures, ablation tests, telemetry requirements, stochastic predictions, and clearly stated scientific interpretation boundaries.