we-solve (beta)

Math-2260-A1 · revision r1 · on Math-2260

OpenAI claim summary for Math-2260

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: partially solved. Claims here do not change the source status.

Claim: literature report

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

Manuscript 1 (family 318, 2026-09-25): Corollary 1.2 excludes even an underlying-spectrum height-three strong wedge equivalence at every prime at least five, so one of these instances refutes the universal strong chromatic-splitting prediction.

Manuscript 2 (family 318, 2026-09-27): A nonzero sphere-kernel class in the canonical weak splitting map excludes a homotopy retraction at height p for every p>=5; the strong conjecture requires that unit summand, so these counterexamples refute its universal assertion.

Manuscript 3 (family 318, 2026-09-25): The p=n=3 counterexample rules out finite assembly by lower spheres in the specified derived-complete-sphere module category, a narrower setting than the problem’s underlying p-local sphere spectrum and unspecified maps.

Manuscript 4 (family 318, 2026-09-25): The ordered filtration has the classical local-sphere cofibers in the stated generic-prime range, but a filtration with potentially nonzero attachments is not the wedge equivalence and canonical split unit required by strong chromatic splitting.

Manuscript 5 (family 318, 2026-09-27): The ordered filtration has the classical local-sphere cofibers in the stated generic-prime range, but a filtration with potentially nonzero attachments is not the wedge equivalence and canonical split unit required by strong chromatic splitting.

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

Manuscript

Contents
  1. Major problems 5 — The chromatic splitting conjecture, which is considerably more complicated to state.
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Mathematical claim 2.
  6. Mathematical claim 3.
  7. Mathematical claim 4.
  8. Mathematical claim 5.
  9. Outline of the argument
  10. Formal verification
  11. Status caveat
  12. Sources

Major problems 5 — The chromatic splitting conjecture, which is considerably more complicated to state.

OpenAI's claim

Manuscript 1 (family 318, 2026-09-25): At primes at least five, the K$3$-local sphere is claimed to retain a nonzero degree-minus-three class under rationalized K$2$-localization, obstructing unmarked strong-splitting wedge equivalences; Manuscript 2 (family 318, 2026-09-27): The canonical weak-splitting unit is claimed to have no retraction at height p for p at least five and height p+1 for p at least seven, witnessed by $\beta_1^{(p-1)^2}$. Manuscript 3 (family 318, 2026-09-25): At prime three and height three, the overlap is claimed to lie outside the ordinary thick subcategory generated by lower local spheres, even allowing arbitrary finite cofibers and retracts of derived-complete sphere modules; Manuscript 4 (family 318, 2026-09-25): For n at least one and p greater than n+1, lower-height localization of the K(n)-local sphere is claimed to have an ordered $2^n$-stage filtration with the classical chromatic-splitting cofibers and canonical initial unit. Manuscript 5 (family 318, 2026-09-27): At height three and every prime at least five, the overlap is claimed to have an explicit eight-stage sphere-layer filtration with identified signed attachments; its first height-one attachment is nonzero.

Mathematical statement of the claimed result

Mathematical claim 1.
For a prime $p\ge5$, let $S=S_p^\wedge$ be the completed sphere, $L_j$ localization at Johnson--Wilson $E(j)$, $L_{K(j)}$ localization at Morava $K(j)$, and $L_0$ rationalization. OpenAI claims that the canonical map \[ L_0L_{K(3)}S\longrightarrow L_0L_{K(2)}L_{K(3)}S \] is nonzero on $\pi_{-3}$. Consequently it claims that $L_2L_{K(3)}S$ is not equivalent, even without compatibility with the unit summand, to \[ L_2(S\vee\Sigma^{-1}S)\vee L_1(\Sigma^{-3}S\vee\Sigma^{-4}S) \vee L_0(\Sigma^{-5}S\vee\Sigma^{-6}S\vee\Sigma^{-8}S\vee\Sigma^{-9}S). \] This is an obstruction to the proposed strong height-three wedge formula at every prime in the stated range.
Mathematical claim 2.
Let $p\ge5$, $S=S_p^\wedge$, $L_j=L_{E(j)}$ and $L_{K(j)}$ be chromatic localizations. Put \[ d_p=2p(p-1)-2,\qquad k_p=(p-1)^2,\qquad z_p=\beta_1^{k_p}\in\pi_{d_pk_p}S, \] where $\beta_1$ is the first classical beta element. OpenAI claims that the image of $z_p$ in $\pi_{d_pk_p}L_{p-1}S$ is nonzero but is killed by the unit-induced map \[ i_{p,S}:L_{p-1}S\longrightarrow L_{p-1}L_{K(p)}S. \] Hence this map has no homotopy retraction. The finite input is the sphere and the height equals the prime; this does not assert such failure at every height.
Mathematical claim 3.
At prime $3$, with completed sphere $S=S_3^\wedge$, OpenAI claims \[ L_2L_{K(3)}S\notin\operatorname{Thick}\{L_0S,L_1S,L_2S\} \] in the category of $E(2)$-local $S$-modules. Here $L_j$ is $E(j)$-localization, $L_{K(3)}$ is Morava-$K(3)$ localization, and the thick subcategory allows finite cofibers, suspensions and retracts. Thus arbitrary finite attachments, including Moore corrections, cannot assemble this chromatic overlap from the indicated lower-height localized spheres.
Mathematical claim 4.
For $n\ge1$ and prime $p>n+1$, put $S=S_p^\wedge$, $D=L_{K(n)}S$ and $X_{n,p}=L_{n-1}D$, where $L_j=L_{E(j)}$. OpenAI claims a filtration in $E(n-1)$-local $S$-modules with $2^n$ cofibers indexed by subsets $I\subseteq\{1,\ldots,n\}$: \[ C_\varnothing=L_{n-1}S,\qquad C_I=\Sigma^{-d(I)}L_{n-\max I}S\ (I\ne\varnothing),\qquad d(I)=\sum_{i\in I}(2i-1). \] The empty subset is first, the remaining maxima are nondecreasing, and the first-layer map is the canonical unit. It claims a single family $y_i\in\pi_{1-2i}D$ whose ordered products $y_I$ give, for every $1\le t<n$, equivalences \[ L_{K(t)}\left(\bigvee_{I\subseteq\{1,\ldots,n-t\}}\Sigma^{-d(I)}S\xrightarrow{(y_I)}D\right). \] A filtered extension is asserted, not a split wedge decomposition before localization.
Mathematical claim 5.
For every prime $p\ge5$, put $S=S_p^\wedge$ and $X=L_2L_{K(3)}S$. OpenAI claims an eight-layer filtration $0=F_0\to\cdots\to F_8\simeq X$ of $E(2)$-local $S$-modules whose ordered cofibers are \[ L_2S,\ \Sigma^{-1}L_2S,\ \Sigma^{-3}L_1S,\ \Sigma^{-4}L_1S,\ \Sigma^{-5}H\mathbb Q_p,\ \Sigma^{-6}H\mathbb Q_p,\ \Sigma^{-8}H\mathbb Q_p,\ \Sigma^{-9}H\mathbb Q_p. \] Here $L_j=L_{E(j)}$, $L_{K(3)}$ is Morava-$K$ localization and $H\mathbb Q_p$ is the Eilenberg--Mac Lane spectrum. The inclusion $F_1\simeq L_2S\to X$ is the unit localization map. The displayed layers do not assert that the attaching maps vanish.

Outline of the argument

Manuscript 1 (family 318, 2026-09-25): Marked connected deformations produce a genuine K$2$-local detector receiving the height-three sphere, forcing factorization through the localization unit. Division-algebra and linear trace classes are compared on a determinant-lattice modification space, retaining integral cohomology before rationalization. Their nonzero rank-two restriction conflicts with the vanishing natural map that the proposed wedge would have in degree minus three. Manuscript 2 (family 318, 2026-09-27): The named beta power survives under the height-$p-1$ unit by an ordinary homotopy-fixed-point detection calculation. At each upper height, a separated multiplicative filtration puts beta_1 in degree at least two and vanishes beyond a dimension-controlled bound, killing that same power. Completion and naturality identify a nonzero lower-localized class in the unit's kernel, obstructing a retraction.

Manuscript 3 (family 318, 2026-09-25): Integral connected and etale markings retain an exceptional norm-character line on the height-two stratum. Full principal-unit action separates that line from every periodicity power carried by the permitted inputs. Comparisons through ordinary finite-stage coefficients and actual homotopy preserve the character across the whole deformation, giving a representation obstruction to any finite assembly. Manuscript 4 (family 318, 2026-09-25): Finite marking and transfer calculations first establish coefficient finiteness. A separately constructed integral family of sphere-cochain classes has normalized exterior products acting as bases under lower-height tests. Removing blocks in decreasing chromatic height yields the ordered cofibers; independent rational dimension counts supply the final block while retaining the extensions between stages.

Manuscript 5 (family 318, 2026-09-27): Integral frame comparisons construct actual bases at heights two and one, while a trace-class calculation fixes their nonzero rational degree-three comparison. A chosen compatibility homotopy feeds two fracture squares, giving the filtration and the signs of its connecting maps. The separate canonical-map obstruction then detects a nontrivial first middle attachment for these same choices.

Formal verification

Family 318 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