Run a mass-gap reading group

A productive group can finish each meeting with one calculation and one carefully bounded statement. The aim is to learn how to read evidence about the mass gap, not to promise mastery of constructive field theory in twelve weeks.

Prerequisites: the course entry contract and access to the linked primary sources. No measure-theory course is assumed.

After this page: organize a sequence, assign preparation that produces a visible result, and distinguish a source statement from the group’s own assessment.

A meeting format that leaves a record

Reserve roughly a quarter of the meeting for a learner’s explanation of the main question, half for a worked exercise or source statement, and the rest for identifying an unresolved assumption. These are scheduling suggestions, not measured completion times.

Rotate three roles. A presenter develops the argument on the board. An assumption reader checks dimension, gauge group, geometry, matter and limits. A source reader checks the exact version and locator. All participants should attempt the exercise; roles are aids to discussion rather than badges of expertise.

Keep a short shared note with the date, pages read, source version, calculation, conclusion and one remaining question. Do not write “paper verified” when the group read its introduction or one theorem.

Twelve meetings

Meeting	Read before meeting	Work together	Leave with
1	The problem and the official statement below	Separate an action, a constructed theory and a spectral assertion	A precise target paragraph
2	Physical states	Prove a closed lattice loop is gauge invariant	A list of physical probes and their restrictions
3	Dimensional transmutation	Integrate the one-loop beta function and inspect its domain	An explanation of why a scale is not a gap proof
4	Euclidean correlators	Derive a two-state spectral sum	A correlator with its units and vacuum subtraction
5	Operator overlap and the lab	Hide a light state by setting its overlap to zero	A counterexample to a one-channel gap claim
6	Lattice definition	Write the finite compact-group integral	A regulator ledger
7	Finite volume and the box lab	Compute the massless Dirichlet lowest mode	A finite-box gap that vanishes with box size
8	Continuum limit	Keep physical mass fixed while refining the lattice	A correct $a$, $am$, $L$ and $N_s$ table
9	Spectroscopy	Follow one mass ratio to its original source table	A statement with channel, scale and uncertainty
10	Reflection positivity and the lab	Compare ordinary and reflected covariance matrices	An explicit reason probability positivity is insufficient
11	A rigorous result	Read Definition 1.4 and Theorem 1.6 of Cao–Nissim–Sheffield v2	A theorem scope card
12	Reading research	Compare two cards with different evidence types	A sourced comparison and a next reading route

The main external readings are Jaffe and Witten’s official section 4 and section 5, and Cao, Nissim and Sheffield, arXiv:2509.04688v2. The latter is a fixed-lattice area-law statement in a restricted coupling regime. It must not be presented as a construction of the four-dimensional continuum vacuum theory.

Six-meeting alternative

Combine consecutive pairs of lessons: 1–2, 3–4, 5–6, 7–8, 9–10 and 11–12. Keep one calculation per combined meeting and move the second exercise to individual preparation. This reduces discussion time; it does not reduce the source assumptions or substitute attendance for understanding.

If the group mainly wants numerical work, use spectral extraction after meeting 4. If it mainly wants analysis, insert the tightness bridge before meeting 5. Both groups return to the same final comparison.

A reusable source-audit assignment

Choose one claim, not an entire research program. Write six short entries:

1. Identity: authors, title, exact version, theorem/equation/table locator and link.
2. Setting: dimension, gauge group and representation, regulator, geometry, matter and boundary conditions.
3. Hypotheses: list the quantitative assumptions rather than replacing them with “sufficiently nice.”
4. Conclusion: state what quantity is controlled, with units and quantifiers.
5. Remaining step: identify one specific missing implication toward the target.
6. Reading record: distinguish abstract, statement, proof sections and any calculation actually reproduced.

This is an editorial learning exercise. A card may accurately report a theorem without its author having verified the proof. The research atlas uses the same distinction between a source claim and an assessment.

Worked assignment: find the unsupported step

A fictional report says: “At every lattice spacing the measured mass is positive, so the limiting physical gap is positive.” Suppose its data are $am(a)=a^2$ with physical $a$ measured relative to a fixed reference length.

Discussion: the reported physical mass is $m(a)=(am(a))/a=a$, which tends to zero. Positivity at each cutoff is insufficient. Nor would one channel alone control the full physical spectrum.

Exercise: replace the data by $am(a)=2a$. What has improved, and what remains?

Solution: the inferred channel mass now stays at 2 in the chosen units. The example no longer loses its mass simply through scaling. It still supplies no construction of limiting observables, no proof of all reconstruction hypotheses, no bound on unseen channels, and no control of volume or systematic errors. A useful report names the improvement without promoting it to a solution.

Choose a continuation

Use the approach guides to choose a method, the open-obstacle index to formulate a precise question, and the notation crosswalk before comparing equations from different papers. Record unanswered questions for later reading; do not assign the unsolved full problem as a routine homework exercise.