Chromatic splitting: filtrations and counterexamples

What OpenAI claims

Disproves strong chromatic splitting at height three for primes p ≥ 5, and weak splitting for the derived p-completed sphere at heights p (p ≥ 5) and p+1 (p ≥ 7). Nevertheless, for n ≥ 1 and p > n+1, the overlap L_n-1L_K(n)S_p^∧ admits a 2ⁿ-stage filtration by the predicted localized-sphere pieces. At height three and prime three, even finite assembly from such pieces fails in the category of E(2)-local modules over the derived completed sphere.

Papers (5)

1. Filtered chromatic splitting at generic primes

   25 September 2026 · Preprint, not peer reviewed · PDF

   For every n ≥ 1 and prime p > n+1, we construct a 2ⁿ-stage ordered filtration of L_n-1L_K(n)S_p^∧ with the classical chromatic-splitting cofibers. The map from the first stage to the target is the canonical localization unit.

2. The height-three chromatic overlap: an explicit filtration and its attachments

   27 September 2026 · Preprint, not peer reviewed · PDF

   For every prime p ≥ 5, we construct an explicit eight-stage filtration of L₂L_K(3)𝕊_p^∧ by the local-sphere layers in the height-three chromatic-splitting pattern. The map from its first stage to the overlap is the canonical unit, and two signed fracture formulas identify all attachments for the chosen local maps and compatibility homotopy. A companion canonical-map theorem further implies that the first height-one attachment is nonzero.

3. A rational obstruction to strong chromatic splitting at height three

   25 September 2026 · Preprint, not peer reviewed · PDF

   For every prime p ≥ 5, the canonical map L₀L_K(3)S → L₀L_K(2)L_K(3)S for the sphere spectrum S is nonzero on π−3. Consequently, the height-three strong chromatic splitting formula is false in this range, even as an equivalence of underlying E(2)-local spectra without specified summand maps.

4. Failure of finite assembly for a chromatic overlap at the prime three

   25 September 2026 · Preprint, not peer reviewed · PDF

   At the prime three and height three, the chromatic overlap cannot be constructed from the rational, height-one, and height-two local spheres by finitely many sums, shifts, cofibers, and retracts in the category of E(2)-local modules over the derived 3-complete sphere. This gives a negative answer to the ordinary finite-assembly question for chromatic overlaps.

5. Counterexamples to weak chromatic splitting: sphere kernels and descent exponents

   27 September 2026 · Preprint, not peer reviewed · PDF

   For the derived p-completion of the sphere, the canonical weak chromatic splitting map has no homotopy retraction at height p for every prime p ≥ 5, and at height p+1 for every prime p ≥ 7. The classical sphere product β₁^(p-1)2 gives a nonzero kernel class in both ranges.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.