Weak chromatic splitting fails at n = p for every odd prime p

Agnes Beaudry; Tomer Schlank

Weak chromatic splitting fails at n = p for every odd prime p

Primarily human-written textHuman understanding: all partsmath.AT — Algebraic Topology

Contributed by Agnes Beaudry ↗, Tomer Schlank

Version 1 / Oct 07, 2026 / CC BY 4.0

Abstract

We prove that for every odd prime

p

there is a height

n

for which the map $L_{n-1}S_p \to L_{n-1}\L_{K(n)}S_p$ is not injective on homotopy groups, hence not the inclusion of a wedge summand. In particular, for such pairs

(n,p)

the weak chromatic splitting conjecture is false. More precisely, we prove the conjecture fails for

(n,p) = (3,3)

and for all pairs

(n,p)

with

p \le n < \sqrt{2}(p-1)

. In particular, the conjecture fails for

(n,p) = (p,p)

for every odd prime

p

. This disproof was produced autonomously by Claude in a session working on Freyd's generating hypothesis with no human in the loop. This note is a human write-up of the disproof. We did not contribute any ideas.

Provenance statement

The disproof of the weak chromatic splitting conjecture given in this note was produced autonomously by Claude. We did not contribute any ideas and Claude was not prompted on this problem but on Freyd's generating hypothesis in stable homotopy theory. Although the relationship between Freyd's generating hypothesis and the chromatic splitting conjecture is well established, for example, in work of Hopkins-Devinatz, we did not expect Claude to find a counter example to the weak chromatic splitting conjecture, which we both believed to be true. The result arose in the context of paid consulting with Anthropic. Claude was run internally by Anthropic, using its own computing resources. The transcript was shared with us September 8, 2026. The mathematical exposition was drafted by us, Agnes Beaudry and Tomer Schlank, after we verified the details of Claude's disproof, on September 12, 2026. The result was announced at 2026 Clay Research Workshop on Telescopic Homotopy Theory and Algebraic K-theory, on September 21, 2026. This text is not AI generated. It was produced by us, using Claude only for copyediting.

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Version history

v1Submitted by Agnes BeaudryInitial depositCurrentOct 07, 2026