Quantum Field Theory and Gauge Theory: Continuum Field Limit and Gauge Symmetry Emergence of the Constraint Network

This document is an L4 Structural Realization () within the (Energy Emergence Theory) framework, positioned as the constitutional relay layer for quantum field theory and gauge theory. It receives constitutional provisions from seven upstream mother texts---Quantum Mechanics Ontology v3. 1 (non-matter dynamics), Field Ontology v3. 1 (free-state energy propagation and gauge symmetry emergence), Particle Ontology v3. 3 (particle = constraint-state energy morphology), Interaction Ontology v2. 1 (three-mechanism exhaustion and unified \ (\) -modulation), Graph-Theoretic Ontology v3. 0 (graph path integral, constraint network wave equation, and discrete differential geometry), Vacuum Ontology v2. 1 (\ (N=0\) ground state and transient constraint events), and Mass Ontology v3. 2 (mass gap = spectral gap \ (₁>0\) ) ---and instantiates them concretely in the domain of quantum field theory and gauge theory. This document serves as the upstream mathematical foundation for all subsequent Particle Physics and Nuclear Physics documents. This document accomplishes three core constitutional working tasks. (i) R2. 4: Convergence framework from non-Abelian graph connections to continuous gauge fields. The spectral convergence of \ (GL (n, C) \) -valued matrix constraint connections to \ (g\) -valued continuous gauge connections \ (A_ (x) \) in the limit \ (a 0\), including Lie algebra parameterization, discrete covariant derivative convergence, plaquette holonomy convergence to Yang-Mills field strength \ (F_\) via Baker-Campbell-Hausdorff expansion, and spectral convergence of the discrete covariant Laplacian to the continuous \ (D_ D^\). (ii) R3. 3: Quantitative mapping of QCD constraint jamming. The functional relation between constraint density \ (\) and the QCD running coupling \ (ₛ\) as \ ( () /J= (₀²/ₐ₂₃²) / (²/ₐ₂₃²) ^\), the correspondence between the jamming threshold \ (J\) and the confinement scale \ (ₐ₂₃\), the QCD string tension \ (/J\), and the bridge between Wilson loop area-law decay and the constraint network topological phase transition. (iii) R6. 4: Correspondence verification with lattice gauge theory. The structural proof that the non-Abelian graph connection recovers the Wilson action in the continuum limit, the explanation of fermion doubling as Brillouin zone mirror-mode degeneracy, the resolution of the Gribov copy problem as a continuum-limit artifact, and the -level registration of the real-time constraint network path integral as an alternative to the Euclidean sign problem. The constitutional identity of in is established by five nested propositions. P1: = non-matter dynamics of the constraint network at \ (1\) ---the multi-mode field-theoretic generalization of quantum mechanical non-matter dynamics. P2: Quantum fields = the coarse-grained effective description of Type II edge weight distributions in the continuum limit---fields are not fundamental entities. P3: Field quantization = the spectral decomposition of the constraint network Laplacian---Fock space is the tensor product of Laplacian eigenmodes. P4: The Feynman path integral = the continuum limit of the graph path integral at \ (=1\) ---all paths contribute with equal phase coherence. P5: Gauge symmetry = the emergence of Type II edge phase consistency conditions---gauge theory is not a Lagrangian redundancy but the continuous expression of edge phase constraints on the discrete graph structure. This document provides distinctive mechanistic answers to several of the deepest and most persistent open problems in. The physical origin of confinement: gluon self-coupling drives constraint density growth to the jamming threshold \ (J\), making color-charge isolation sterically impossible. The ontological status of lattice gauge theory: discretization is not a numerical trick---the constraint network is necessarily discrete under Finite Distinguishability (L1 Constitutional Presupposition), and continuum is the effective description of the discrete constraint network. The physical root of fermion doubling: the forty-year mathematical obstacle of the Nielsen-Ninomiya theorem acquires a physical-origin explanation in ---doubling is the mirror degeneracy of discrete Brillouin zone boundary modes. The 120-order-of-magnitude cosmological constant ``catastrophe'': a category error--- computes the variance (transient channel fluctuation amplitude of the \ (N 1\) constraint network ground state), while the observed \ (\) is the mean (\ (N=0\) vacuum baseline). The resolution of the fifty-year Gribov copy predicament: the unique labeling of vertices on a discrete graph makes the gauge slice globally unique---Gribov copies are a continuum-limit artifact. In addition, this document unifies the three competing pictures of confinement (center vortex, dual superconductor, and Gribov-Zwanziger as different representations of the same jammed subnetwork), provides a physical foundation for the Kugo-Ojima confinement criterion and 't Hooft anomaly matching, and interprets the \ ('\) mass anomaly as a manifestation of the graph topological invariant of the constraint network. This document strictly complies with all rules of the Universal Writing Constitution v1. 6, including the Declaration Table, Constitutional Status (CS. 0--CS. 8), variable status enumeration, 40 cross-domain bridge declarations, a complete governance status table, and PM-1 through PM-15 compliance declarations. All and constitutional content belongs to the upstream mother texts. New content at the and levels is explicitly labeled with its constitutional status and mother-text provenance. This document forms a co-creation pair with Particle Physics v1. 0---this document provides the mathematical framework of gauge field theory (graph connection convergence, path integral, renormalization), while Particle Physics provides the physical instantiation of the Standard Model particle spectrum (chiral boundary conditions, Higgs template coupling, quantitative derivation of gauge group rank and bare coupling constants). Nuclear v2. 0 directly depends on this document for the jamming quantitative mapping (V. 2-- V. 5), the constraint network path integral (Part IV), and renormalization = graph coarse-graining (Part V). % === KEYWORDS ===Constraint Network; Gauge Field Emergence; Graph Connection; Non-Abelian Convergence; Yang-Mills Theory; Quantum Chromodynamics; Confinement Mechanism; Constraint Jamming; Lattice Gauge Theory Correspondence; Fermion Doubling; Gribov Copies; Renormalization as Graph Coarse-Graining; Path Integral; LSZ Reduction; BRST Symmetry; Quantum Anomalies; Five-Channel Thermodynamics; Cosmological Constant Problem; Mass Gap; Strong CP Problem; ' Mass; Proton Mass Decomposition; Quark-Gluon Plasma Viscosity; QCD Phase Diagram; Haag Theorem; CPT Theorem; Spin-Statistics; Amplituhedron; Effective Field Theory; Renormalons; Sign Problem; Instantons; -Vacuum; Kugo-Ojima Criterion; 't Hooft Anomaly Matching; Weinberg-Witten Theorem; Goldstone Boson Equivalence; (Energy Emergence Theory) ; Constitutional Framework; Structural Realization; -Modulation; () ; Cooperative Capacity; Bifurcation Duality; Finite Distinguishability; Discrete Differential Geometry; Graph Laplacian; Spectral Gap; Dual-Channel Damping; Encapsulate Operation