role	date_checked	primary_source	contribution	selection_status
selected frontier	2026-09-07	https://arxiv.org/html/2608.19324v1	Fazzi--Krawczyk define the BAE-to-untwisted-CM limit map, state bijectivity on the CM sector for isolated solutions, verify rank two, and flag continuous families as the next test	selected
formal influence	2026-09-07	https://arxiv.org/html/2608.19324v1#S2.SS4	The paper credits an unpublished Aharony argument communicated by Mamroud for the type-A third-derivative mechanism; Fazzi--Krawczyk extend it to arbitrary semisimple gauge algebra	credited, not a separate public source
runner-up	2026-09-07	https://arxiv.org/html/2608.29217v1#S5	Kobayashi's complex-Fourier-zero rigidity conjecture for convex bodies remains open even for arbitrary planar bodies	not selected: reduced zero sets lose multiplicity and the first route lacks a bounded structural gate
runner-up	2026-09-07	https://arxiv.org/html/2608.27834v1	Geisser isolates p-primary motivic base-change rigidity outside weights 0, 1, and dim X	not selected: the first missing bridge enters broad cycle-complex and de Rham--Witt territory
runner-up	2026-09-07	https://arxiv.org/html/2608.13551v1#S1	Park reduces a route toward PPT-cubed to whether 2-positive and 2-copositive maps are decomposable	not selected: the natural attack is a high-dimensional cone and witness search
runner-up	2026-09-07	https://arxiv.org/html/2608.06209v1#S4.SS4	Continuous K-theory A1-invariance for regular analytic adic spaces is a canonical open map with strong partial results	not selected: technical dependency depth obscures a decisive first obstruction
runner-up	2026-09-07	https://arxiv.org/html/2609.04073v1#S5	Shimizu's categorical Takeuchi questions ask for an ind-completion extension and a nonnormal coideal correspondence	not selected: the formulation still requires choosing the suitable modifications
attack dependency	2026-09-07	https://dlmf.nist.gov/20.2	Standard Jacobi theta oddness and quasi-periodicity used to prove ellipticity and Cartier divisibility	formal dependency
attack dependency	2026-09-07	https://dlmf.nist.gov/20.7	Standard Jacobi theta modular transformations used to prove modular equivariance of the residual ideal	formal dependency
attack scope	2026-09-07	https://arxiv.org/html/2608.19324v1#S3	Fazzi--Krawczyk state bijectivity for isolated solutions and treat A2 as thin evidence for a continuous extension	frontier delimited; the present result proves local flatness, not global bijectivity
attack dependency	2026-09-07	https://stacks.math.columbia.edu/tag/0BVV	The Kähler different of a square complete-intersection presentation is generated by its Jacobian determinant	formal dependency
attack dependency	2026-09-07	https://stacks.math.columbia.edu/tag/0BWB	Tate's formula identifies the trace element with the complete-intersection Jacobian under a generator of the relative dualizing module	formal dependency
attack dependency	2026-09-07	https://stacks.math.columbia.edu/tag/0BVH	The finite-locally-free discriminant is the determinant of the trace pairing	formal dependency
attack calibration	2026-09-07	https://arxiv.org/html/2608.18209v1	The companion rank-two analysis tests isolated Bethe solutions for nondegeneracy using the Bethe Jacobian	context only; no generic-simplicity theorem imported
attack result	2026-09-07	internal exact derivation	The normalized root force is a triple centered average; the local trace discriminant is the norm of the averaged-Hessian determinant; a corank-one first splitting gate is dV2(q)(v)	not externally reviewed; no priority claim
attack scope	2026-09-07	https://arxiv.org/html/2608.19324v1#A2	Fazzi--Krawczyk identify the fully rational B2 extrema used as the square-lattice collision branch	external solution family credited; local singularity analysis is new synthesis
attack calibration	2026-09-07	https://arxiv.org/html/2608.18209v1	The companion rank-two study verifies nonzero generic Bethe Jacobians near the B2 solutions	context for generic splitting, not a formal proof dependency
attack result	2026-09-07	internal exact derivation	At j=1728 a regular fully rational B2 extremum has zero Hessian, local Jacobian length 9, and a joint modular/flavor smoothing into nine Morse points with discriminant order 18	not externally reviewed; no priority claim
stopping scope	2026-09-07	https://arxiv.org/html/2608.19324v1#S2.SS5	Fazzi--Krawczyk state that continuous families occur generically above rank two and identify them as a next test; they do not assert that every degenerate isolated extremum lies on such a family	frontier delimited
stopping context	2026-09-07	https://arxiv.org/html/1003.4689v4	Etingof--Felder--Ma--Veselov construct commuting elliptic Calogero--Moser Hamiltonians for crystallographic reflection groups	integrability context only; no reducedness theorem imported
stopping result	2026-09-07	internal exact derivation	The family V_t=x1^4/4+t*x1^5/5+(x2-t)^2/2+sum_{i>=3}xi^2/2 has completed finite-flat critical algebra k[t][[epsilon]]/epsilon^3 and a separated Liouville-integrable Hamiltonian realization, while every reduced critical fiber is one point	proves the proposed implication does not follow from the abstract hypotheses; no claim about CM-specific potentials