When is 4D-Yang-Mills-Theorie possible

Core Objective: To construct pure quantum Yang-Mills theory with compact simple gauge group SU (N) in four-dimensional Euclidean space, prove the strict positivity of its mass gap, and reduce the mass-gap problem to a small set of explicit, identified inputs. Methodology: Uses lattice regularisation, renormalisation-group control of the continuum limit, and Osterwalder-Schrader reconstruction of a relativistic quantum field theory on Minkowski space. Expresses the clustering rate of the reflection-positive two-point function as the mass gap. Factorizes the mass-gap problem into three independent aspects: Existence (forced by reflection positivity and Fekete's lemma), Value (governed by an anomalous dimension), and Positivity (carried by the discrete centre of the gauge group and uniform control of confining flux). Key Innovations the non-analytic dimensional-transmutation factor cancels in the ratio, leaving a value governed by an anomalous dimension. (P) Positivity: This is the sole non-analytic content of the problem. It is shown to be equivalent to the persistence of centre symmetry (Z/NZ) along the renormalization trajectory. Construction and Methodology The construction proceeds through three main phases: Lattice Regularization: The theory is defined using the Wilson lattice action, which is proved to be well-defined, gauge-invariant, and reflection positive at every spacing. Renormalization-Group (RG) Control: The continuum limit is controlled by the RG flow. The ultraviolet stability is taken from the programme of Bałaban, while this work supplies the infrared completion, controlling the flow as the confining flux organizes. Osterwalder–Schrader Reconstruction: The Euclidean correlation functions are verified against the axioms of temperedness, reflection positivity, Euclidean invariance, and symmetry. The reconstruction theorem then yields a relativistic Wightman quantum field theory on Minkowski space with a unique vacuum and a mass spectrum. Key Technical Frameworks Flow-Time Coordinate: Throughout the construction, the author uses a "flow-time" coordinate where the dimensional-transmutation scale is an entire function and the marginal coupling region is a compact interval. Tail Equation: The scale generated by dimensional transmutation is represented as an exponential (carrying positivity), a prefactor (carrying value), and a formal series. Pseudospectral Gap: Because the transfer operator along the marginal trajectory is non-normal, the physical mass gap is identified as a resolvent-norm quantity (pseudospectral gap) rather than a simple eigenvalue gap. The Discriminator: A crucial requirement of the proof is that it must fail for the abelian U (1) theory, which is gapless. The construction achieves this by tying the gap to the discrete centre of SU (N), whereas the centre of U (1) is continuous. Status of the Main Theorem The Main Theorem (16. 2) establishes the existence of a quantum Yang–Mills theory with a mass gap > 0, but it remains conditional on identified inputs: Uniform positivity of the confining scale across the marginal window as the lattice spacing a 0. Uniform correlation bounds, combining Bałaban’s UV programme with the IR flux bounds provided in this work. The uniqueness of the continuum limit and the restoration of rotational invariance. Beyond the high-level objective of proving the existence of a mass gap, the monograph identifies several deep, structural nuances that are often overlooked in "first-view" summaries but are essential to the construction's validity and the difficulty of the four-dimensional problem. 1. The Pseudospectral Nature of the Gap A common misconception is that the mass gap is simply the distance between the first and second eigenvalues of a Hamiltonian. However, the sources reveal that while the transfer operator is self-adjoint at any fixed lattice spacing, the renormalization-group (RG) pace map is non-normal. The Distinction: In a non-normal system, the physical mass gap is a pseudospectral/resolvent-norm quantity, not a spectral one. The Consequence: A finite lattice will always return a positive spectral gap via Perron–Frobenius, but this is a "lattice artefact" that does not represent the physical gap in the continuum. Ignoring this distinction leads to "empty" identifications of the gap that do not survive the continuum limit. 2. The Analyticity Obstruction It is often assumed that more precise computations or higher-order perturbation theory could eventually "solve" the mass gap. The sources establish that this is mathematically impossible due to the analyticity obstruction. Essential Singularity: The generated scale (a) has an essential singularity at g² = 0; every derivative in its Taylor series is zero, yet the function itself is not zero. Structural Failure: Any quantity calculated from a finite region or to a finite order in the coupling is analytic and therefore cannot capture the non-analytic scale. The proof must be trajectory-global rather than local to the coupling. 3. The "Harmonic Borderline" of Marginality The mass gap is described as an infinite product of per-pace increments along the RG trajectory. Delicate Balance: Because the theory is marginal in four dimensions, the series governing this product sits exactly at the harmonic borderline between convergence and divergence. Fragility: The gap's positivity is not a "robust" or automatic result; it depends on sub-leading structures because the leading harmonic term alone is insufficient to guarantee a non-zero limit. 4. The U (1) Discriminator A vital but subtle requirement is that any valid proof for SU (N) must fail when applied to an abelian U (1) theory. Why it Matters: U (1) in four dimensions is gapless and possesses a massless photon. The Mechanism: The construction identifies the discrete center (Z/NZ) of SU (N) as the anchor for positivity; because U (1) has a continuous center, the "twist" used to prove a gap in SU (N) averages to zero in U (1), correctly predicting the absence of a gap. 5. Instability as the Confining Mechanism While "instability" often suggests a failure in a physical theory, here it is the origin of the gap. Nielsen–Olesen Mode: The constant-field perturbative vacuum is unstable. Vortex Condensate: This instability is exactly what allows the theory to "roll" into a structured configuration—a condensate of center vortices—which constitutes the true confining vacuum. Without this self-interaction-driven instability, the theory would remain in a gapless, Maxwell-like state. 6. The Dimensional Origin of Difficulty The monograph clarifies that the mechanism for the gap (the screening of a topological gas) was already proven in three dimensions. The d=4 Problem: In three dimensions, the coupling is super-renormalizable, meaning the topological gas is "dilute" and easily controlled. Marginality: In four dimensions, the coupling is marginal (g=0), meaning the "corridor" between weak and strong coupling is infinitely long and dense, leaving no small parameter to control the gas at any scale. This change in scaling—not a change in physics—is the true reason the 4D problem remained open. The most critical element of the monograph is the "Junction" of the five roads and the resulting "Band Condition" on the odd energy fraction. While previous summaries noted that the construction reduces to a few inputs, the sources specify that all diverse technical approaches—the invariant cone, the ’t Hooft flux sectors, the zero-set geometry, the window map, and the exact dualities—converge at a single mathematical identity. 1. The Junction and the "Odd Energy Fraction" The construction establishes that the entire mass-gap problem can be collapsed into a single scalar variable called the odd energy fraction (q). This variable represents the fraction of "energy" in the centre-charged (odd) sector of the theory's weight. The Band Condition: The "Main Theorem" of the work is proved if this single value qₙ stays within a specific "band" (qₙ q_, 1/2 - q_) uniformly as the lattice is refined. Division of Labor: The sources reveal that the "lower half" of this band (keeping q from vanishing) is guaranteed by reflection positivity and a "per-pace floor" identity, while the "upper half" (keeping q away f