Extract a static energy from long Euclidean rectangles

Let a spatial rectangle have separation R=ra and Euclidean duration T=na on an isotropic lattice. The normalized fundamental trace is averaged over the gauge ensemble. In a transfer-matrix setting it decomposes into weighted exponentials of energies in the static pair sector, with amplitudes determined by the chosen loop operator. Excited contamination can remain important even when the loop is geometrically large. Smearing or a variational operator basis may improve overlap without changing the target spectrum.

An adjacent-time ratio gives a dimensionless effective energy aVeff. It approaches the lowest static energy only when excited contributions and temporal wraparound effects are controlled. The energy includes an additive source self energy that diverges as the regulator is removed; differences, forces, or an explicit subtraction have cleaner physical comparisons. A quoted absolute potential offset is therefore not universal across lattice spacings or static-action choices.

In a pure gauge theory with external sources that cannot be screened by available dynamical matter, an asymptotic area law indicates a linearly rising static energy. A strong-coupling expansion of the Wilson action can demonstrate such behavior in its convergence regime: the leading contribution tiles a loop’s minimal area with plaquettes. This is controlled information at strong bare coupling, not a proof that every coupling lies in the same phase or that the continuum mass-gap problem is solved.

With dynamical fundamental quarks, the flux tube can break by producing screening pairs. The true large-separation ground-state energy then saturates relative to the appropriate hadronic threshold, even if a simple Wilson-loop operator has poor overlap onto that state. Absence of an obvious plateau in a limited operator basis is not evidence against string breaking. The gauge group, matter representations, temperature, and order of limits all affect the relevant confinement diagnostic.

Loop expectations become small at long time, increasing statistical difficulty. Correlated estimates at adjacent times require a joint uncertainty analysis, and apparent ratio plateaus should be checked against operator changes and fit windows. Finite spatial size can alter flux configurations and interactions with images. A successful fit establishes a result within its lattice ensemble and systematic checks; it does not alone establish continuum confinement as a mathematical theorem.

The heat lab isolates exponential decay in a different system. Reset and set mode to 1 and diffusivity to 0.5, then Step twice to time 1; predict amplitude e⁻⁰·⁵. Reset and set mode to 1 and diffusivity to 1, then Step twice; predict e⁻¹. This is an analytic periodic heat mode with decay rate κn². It illustrates reading a decay constant, but has no Wilson-loop ensemble, source self energy, static potential, or Monte Carlo fluctuations.