Open obstacles and precise next questions

An obstacle is useful when it says which implication is missing and in which setting. “The continuum problem” is too broad to guide a reading decision. This index turns the site’s seven proof-obligation coordinates into concrete questions.

Prerequisites: the exact target and the proof map.

After this page: locate the limiting step relevant to a result and distinguish an author’s stated open question from an editorial inference. This is a research-navigation index, not a list of independently sufficient conditions or a ranking by difficulty.

Questions tied to the target

Jaffe and Witten’s official sections 4–5 specify the target theory, the relevant gauge-invariant observables and the physical gap. The questions in this table are editorial decompositions of that target.

Coordinate	A precise question	Start reading
qft	Do the limiting observables satisfy the positivity, symmetry, locality and regularity hypotheses needed for physical reconstruction?	Reflection positivity
UV	How is the limiting short-distance behavior identified with the intended gauge theory, rather than merely some limiting measure?	Renormalized observables
continuum	Which smeared observables converge as the cutoff is removed, in which topology and along which parameter trajectory?	Limits and uniformity
volume	Which estimates remain uniform as the physical box grows, and how are boundary dependence and vacuum sectors treated?	Finite-volume lesson
nontrivial	What observable information survives the limit and rules out a vacuous or inappropriate trivial theory?	Axioms and target
gap	Is there a positive physical spectral threshold after the necessary limits, covering the excited physical spectrum rather than one channel?	Gap and decay
group	Which compact simple groups and global forms are covered, and which hypotheses restrict a proposed extension?	Groups and representations

The coordinates overlap. A direct construction need not pass through a lattice, and a single estimate can bear on several questions.

What a specific source leaves open

A lattice criterion needs its premise. Chatterjee v5, Definition 2.3 and the discussion before Theorem 2.4 distinguishes boundary-uniform decay from weaker familiar decay statements. Applying the criterion requires establishing its actual premise. The additional passage from a lattice theorem to the continuum physical target is an editorial next-step question, not an extra conclusion of that theorem. Dossier.

Local stochastic dynamics needs more than a local solution. Chevyrev–Shen v2, introduction, printed page 2 explicitly identify the invariant-measure construction as open in their stated three-dimensional setting. Even a completed result there would retain its dimension. Stochastic guide.

A controlled mechanism needs a justified continuation. The small-circle deformation in Ünsal–Yaffe v2, sections 2–3 changes the model and exposes a useful hierarchy. Extending its conclusion to undeformed $\mathbb R^4$ requires further arguments. This formulation is the site’s inference from the change of setting, not a claim that the paper silently proves such an extension. Semiclassical guide.

A finite constraint system needs limiting control. The hierarchy described in Guo et al. v2, sections 2.2–3 constrains selected lattice observables. The inference “these finite constraints establish the full continuum spectrum” is not available. Ask separately about solver certification, hierarchy convergence and physical limits. Bootstrap guide.

A conditional construction needs its coupling premise. The Bałaban dossier now traces the recovered Theorem 1 and supporting setup. Distinguish its conditional conclusion from establishing the required trajectory; then ask which observable and spectral estimates survive the physical limits. The constructive guide explains the theorem’s historical qualification.

Worked diagnostic: a bound that loses the gap

Consider a hypothetical family with

$$C_a(t)\leq A e^{-a t},\qquad a>0,$$

where $t$ and $a$ are dimensionless after choosing fixed reference units. There is a positive rate in the bound for every member, but the displayed rate tends to zero as $a\to0$. The bound alone therefore supplies no uniform positive threshold. It also concerns only the observable defining $C_a$.

Exercise: what extra information would make the bound useful for a limiting gap claim?

Solution: one needs a lower decay threshold bounded away from zero in fixed physical units, convergence of the appropriate correlation objects, and control strong enough to pass the inequality to a reconstructed physical theory. One must also connect the class of observables to the entire relevant excited state space. Replacing “positive for each cutoff” by “uniformly positive” fixes one issue, not all of them.

Turn an obstacle into a reading task

Write a sentence of the form: “For this model and source version, estimate X is controlled in parameter Y; to infer Z, the missing step is W.” Attach a theorem, section or table locator and state whether W is source-authored or your own inference. Then follow the atlas to the relevant result rather than treating this index as an announcement of a new open theorem.