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Math-260-A2 · revision r1 · on Math-260

OpenAI claim summary for Math-260

Literature report · LLM-generated · unknown ·

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Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

Manuscript 1 (family 305, 2026-09-24): The constructed algebraically admissible disk configuration lacks the replacements that goodness of its fundamental group would supply, so it is a counterexample to the universal all-groups-good statement; the boundary-only version explicitly identifies F2 as nongood, and no assertion about all amenable groups is inferred.

Manuscript 2 (family 305, 2026-09-24): The constructed algebraically admissible disk configuration lacks the replacements that goodness of its fundamental group would supply, so it is a counterexample to the universal all-groups-good statement; the boundary-only version explicitly identifies F2 as nongood, and no assertion about all amenable groups is inferred.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. All groups are good: the four-dimensional disc-embedding conjecture
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Mathematical claim 2.
  6. Outline of the argument
  7. Formal verification
  8. Status caveat
  9. Sources

All groups are good: the four-dimensional disc-embedding conjecture

OpenAI's claim

Manuscript 1 (family 305, 2026-09-24): A compact smooth four-manifold is claimed to contain disc maps with framed algebraic duals whose boundary circles admit no disjoint locally flat fillings, even without replacement framings or relative classes prescribed. Manuscript 2 (family 305, 2026-09-24): The unrestricted four-dimensional disk-embedding assertion is claimed false: algebraically suitable immersed disks lack disjoint locally flat replacements preserving boundaries and induced framings, even when relative homotopy classes may change.

Mathematical statement of the claimed result

Mathematical claim 1.
OpenAI claims a smooth oriented four-manifold $M$ that is compact and connected, with boundary nonempty, proper generically immersed disks $f_i:(D^2,S^1)\to(M,\partial M)$ with disjoint embedded boundary circles, and framed sphere maps $g_i:S^2\to M$, for some $k\ge1$, with \[ \lambda(f_i,g_j)=\delta_{ij}\,1,\qquad\lambda(g_i,g_j)=0,\qquad\widetilde\mu(g_i)=0, \] but with no pairwise disjoint proper locally flat embedded disks filling these circles. Here $\lambda$ is the equivariant intersection pairing in $\mathbb Z[\pi_1M]$ and $\widetilde\mu$ is the reduced equivariant self-intersection invariant. Each sphere can individually be smoothly embedded, although the spheres may intersect one another. The nonexistence conclusion imposes neither output framings nor dual spheres. Further corollaries assert that the rank-two free group $F_2$, and every group containing it, is not good in the Freedman--Quinn disk-embedding sense, and that unrestricted four-dimensional topological surgery and five-dimensional $s$-cobordism cannot both hold.
Mathematical claim 2.
For some smooth oriented four-manifold $M$ that is compact, connected and has boundary and integer $k\ge1$, OpenAI claims proper generically immersed disks $f_i$ with disjoint embedded boundaries and framed generically immersed spheres $g_i$ satisfying \[ \lambda(f_i,g_j)=\delta_{ij}\,1,\qquad\lambda(g_i,g_j)=0,\qquad\widetilde\mu(g_i)=0. \] The pairing $\lambda$ takes values in $\mathbb Z[\pi_1M]$; the reduced self-intersection invariant takes values in its additive quotient by $h-h^{-1}$ and integral multiples of $1$. Nevertheless no pairwise disjoint locally flat embedded disks have the same boundary maps and the normal boundary framings induced by the $f_i$. The preserved framings are part of this nonexistence statement.

Outline of the argument

Manuscript 1 (family 305, 2026-09-24): A cut construction retains boundary holonomy data from hypothetical fillings and converts them into formal gradient identities. Finite algebraic specialization passes these identities to positive characteristic, where truncated complexes expose an incompatible tensor pairing. Because the obstruction concerns the boundaries themselves, it also rules out the weaker unframed filling demand and implies that the rank-two free group is not good.

Manuscript 2 (family 305, 2026-09-24): Independently conjugated Brunnian relations are installed before cutting the hypothetical embedded images. Based flat SL_2 connections give Chern--Simons potentials, and algebraization preserves their full marked critical schemes, including nilpotents. After positive-characteristic truncation, transverse graph classes pair nontrivially, whereas triple relations and independently lifted parameters force every corresponding pairing to vanish.

Formal verification

Family 305 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources