  Kochen-Specker (KS) theorem reveals the inconsistency between quantum theory
and any putative underlying model of it satisfying the constraint of
KS-noncontextuality. A logical proof of the KS theorem is one that relies only
on the compatibility relations amongst a set of projectors (a KS set) to
witness this inconsistency. These compatibility relations can be represented by
a hypergraph, referred to as a contextuality scenario. Here we consider
contextuality scenarios that we term KS-uncolourable, e.g., those which appear
in logical proofs of the KS theorem. We introduce a hypergraph framework to
obtain noise-robust witnesses of contextuality from such scenarios.
  Our approach builds on the results of R. Kunjwal and R. W. Spekkens, Phys.
Rev. Lett. 115, 110403 (2015), by providing new insights into the relationship
between the structure of a contextuality scenario and the associated
noise-robust noncontextuality inequalities that witness contextuality. The
present work also forms a necessary counterpart to the framework presented in
R. Kunjwal, Quantum 3, 184 (2019), which only applies to KS-colourable
contextuality scenarios, i.e., those which do not admit logical proofs of the
KS theorem but do admit statistical proofs.
  We rely on a single hypergraph invariant, defined in R. Kunjwal, Quantum 3,
184 (2019), that appears in our contextuality witnesses, namely, the weighted
max-predictability. The present work can also be viewed as a study of this
invariant. Significantly, unlike the case of R. Kunjwal, Quantum 3, 184 (2019),
none of the graph invariants from the graph-theoretic framework for
KS-contextuality due to Cabello, Severini, and Winter (the "CSW framework",
Phys. Rev. Lett. 112, 040401 (2014)) are relevant for our noise-robust
noncontextuality inequalities.
