Learn by changing the assumptions
An equation becomes easier to understand when you can change one assumption and see what happens. These laboratories are deliberately small: each isolates a single conceptual issue that matters when reading mass-gap research.
01 · Read a correlator
Section titled “01 · Read a correlator”Two states contribute to a Euclidean correlator. Change their energies and the overlap with the lighter state; observe the effective mass. Set the light-state overlap to zero and explain why the measured decay then misses that state entirely.
The model uses exact exponentials. It contains no Monte Carlo noise, so you can first understand the spectral question without statistical complications. You can export the generated samples as CSV.
Open the correlator laboratory →
Background: Euclidean correlators and operator overlap.
02 · Take a continuum limit
Section titled “02 · Take a continuum limit”Hold a physical mass fixed and decrease the lattice spacing. Watch the dimensionless lattice mass shrink and the correlation length in lattice units grow. Change the box length separately to distinguish removal of the short-distance regulator from the large-volume limit.
This is a scale-accounting model. It does not simulate Yang–Mills configurations or demonstrate that the continuum theory exists.
Open the continuum laboratory →
Background: the finite-volume trap, the continuum limit, and uniform estimates.
03 · Follow the running coupling
Section titled “03 · Follow the running coupling”Explore the one-loop running coupling of pure SU(N) Yang–Mills theory. Move the renormalization scale and vary N. Identify the region in which the weak-coupling expansion loses its usefulness.
The appearance of a dynamically generated scale is a key clue. Turning that clue into a proof of a positive physical gap is another problem.
Open the running-coupling laboratory →
Background: dimensional transmutation and the proof map.
04 — Distinguish two positivity tests
Section titled “04 — Distinguish two positivity tests”A cosine kernel is an ordinary positive covariance, yet a reflected one-time test gives a negative answer. Compare it with a massive exponential, vary the test times and coefficients, and inspect both forms. A single counterexample can disprove a universal condition; finitely many successful examples cannot prove it.
Open the positivity laboratory →
Background: reflection positivity and Gaussian measures.
05 — Enlarge the box
Section titled “05 — Enlarge the box”For a free mode with Dirichlet endpoints, the massless lowest energy is positive at every finite interval length but tends to zero as the interval grows. Add an input mass to see the distinct limiting behavior. Keep the boundary condition visible while comparing several modes and interval sizes.
Open the finite-volume laboratory →
Background: the finite-volume lesson and uniform estimates.
All five labs include a worked static explanation and exercises with solutions. Graphs are deterministic educational calculations, not new Yang–Mills measurements. Controls work with the keyboard; reset restores the stated default. CSV exports identify the model and its selected parameters where a data table aids the exercise.