Three anti-invariant classes on a nonintegrable four-manifold

We construct smooth nonintegrable almost complex structures on the closed real four-manifold given by a genus-two curve times an elliptic curve, with at least three independent real closed anti-invariant de Rham classes. The structures equal the product complex structure outside one coordinate ball and converge to it in the smooth topology. This gives a negative answer to the unrestricted Draghici-Li-Zhang integrability conjecture. No exact value of the anti-invariant dimension, compatibility with a fixed symplectic form or real analyticity is asserted. The complete proof uses two holomorphic two-forms on the product, a Stokes argument for three real cohomology classes, and a compactly supported exact deformation preserving a variable-coefficient relation among their representatives. A positive real two-plane defines the new almost complex structure; an explicit differential of a complex coframe proves nonintegrability. The earlier generic-vanishing theorem is credited to all four authors Tan, Wang, Zhang and Zhu. Li-Zhang and Draghici-Li-Zhang cohomology, Donaldson positive planes, Bande-Kotschick closed pairs, Hind-Tomassini examples and Chen-Zhang nonintegrable products provide material prior context. Independent bounded primary-source priority audits through October16,2026 found no verified earlier qualifying counterexample. They do not certify novelty, first priority, present openness or exhaustive absence. Version and access gaps, including accepted versus journal-final texts and incomplete coverage of older product models, are disclosed. The report cited as OWR33/2020 has bibliographic year2020 and actual EMS publication date September14,2021. Withdrawn claims are not used as theorems. The deposit includes the five-page research note and a source-free verification archive containing editable LaTeX, all24 original PR948 files unchanged, curated mathematical and priority audit evidence, and a separately bounded strict finite verification profile. The original PR declares4/5 author turns; native earlier per-turn history and total were not authenticated, and this audit adds zero substantive proof-search responses. Fixed symbolic replay was tested with CPython3.14.6 and SymPy1.14.0. Byte integrity and finite identities do not replace the global proof or certify historical priority. Raw third-party source bodies, private fixtures and credentials are excluded. AI tools were used extensively in construction, research, source search, symbolic computation, writing and adversarial review. This preprint is unrefereed; no independent human peer review, journal acceptance or formal proof-assistant certification is claimed.