Jenny Lorraine Nielsen, Conne's Rigidity Theorem: Disproof of Open AI's Counterexample and Proof of the Conjecture - PhilArchive
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Conne's Rigidity Theorem: Disproof of Open AI's Counterexample and Proof of the Conjecture
Jenny Lorraine Nielsen
Abstract
We first disprove the recent OpenAI "disproof" of Connes' rigidity conjecture, then prove the conjecture itself.
**Part I** shows that the claimed Lean-formalized counterexample is invalid. The crux of the error is that D ⋊ K (a nontrivial semidirect product) is **not** equivalent to D × K (a trivial direct product), and neither is equivalent to D ×_c K (a cocycle extension with the `shiftedCarry` cocycle). The formalisation constructs groups of the third type but verifies the ICC property using orbit arguments valid only for the second. In a semidirect product, conjugation acts on the normal factor through the action φ; in a cocycle extension, conjugation acquires five additional terms from the cocycle, which can collapse conjugacy classes and create central elements. We trace the construction through 37,000 lines of published Lean code and identify two independent paths to failure. First, the orbit lemmas underpinning the ICC certificates operate on quotient objects and do not account for the cocycle's contribution to conjugation. Second — and independently dispositive — the zero-cocycle group is a direct product Z^{2n} × Sp(2n, Z) whose centre contains the entire lattice (ruling out ICC) and which surjects onto Z^{2n} (ruling out property (T)). A single failure suffices; this group fails on both counts. The claimed disproof is invalid.
**Part II** proves the conjecture: every countable discrete ICC group G with Kazhdan's property (T) is W*-superrigid, L(G) ≅ L(H) ⟹ G ≅ H. The classical barrier — the absence of Cartan subalgebras in L(G) — is bypassed by lifting to Bernoulli crossed products A_G = L^∞(X) ⋊ G, which possess canonical Cartans. The semidirect structure ⋊ encodes a nontrivial principal G-bundle; this nontriviality is the engine of the proof. We introduce the twisted comultiplication Θ(fu_g) = fu_g ⊗ Φ(u_g), a unital *-homomorphism from A_G into A_G ⊗̄ A_H functioning as a connection form between measurable principal bundles. Ioana's classification theorem for Bernoulli actions of property-(T) groups — which classifies arbitrary unital *-homomorphisms, with group morphisms as output, not hypothesis — decomposes Θ into morphisms δ₁: G → G and δ₂: G → H. The Bernoulli deformation α_t fixes Θ(L(G)) pointwise, so spectral-gap and absorption arguments apply without hypotheses on the Fourier decomposition of Φ(u_g). Cases (I) and (II) are eliminated by weak convergence and Cartan non-intertwining. Injectivity of δ₂ follows directly from factoriality: a nonzero *-homomorphism from a II₁ factor is injective, so (δ₂)_*: L(G) ↪ L(H) is injective, which forces δ₂ to be injective. Surjectivity follows from the relationship between Φ and (δ₂)_*: they differ by inner automorphism and character twist, so since Φ is surjective, δ₂(G) = H. As corollaries: the forgetful functor from...