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Math-2409 · mathematics · Topology

Chromatic Fiber Structure at Bad Primes

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
At primes p that are bad relative to chromatic level n, give a plausible systematic description of the homotopy type of $\Sigma F_n$, where $F_n$ is the common fiber in the chromatic splitting square for the p-local sphere.

Formulation references: [S1], [S2].

Short English statement

At primes p that are bad relative to chromatic level n, give a plausible systematic description of the homotopy type of $\Sigma F_n$, where $F_n$ is the common fiber in the chromatic splitting square for the p-local sphere.

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2409-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2409

Manuscript 1 (family 318, 2026-09-27): The primary manuscript establishes nonzero sphere-kernel classes refuting the canonical weak chromatic splitting retraction at specified bad-prime heights. The bad-prime sphere kernels at n=p and n=p+1 give specific obstruction data for the overlap but not the requested systematic homotopy-type description of the entire bad-prime fiber.

Chromatic Fiber Structure at Bad Primes

OpenAI's claim

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 $\beta_1^{(p-1)^2}$. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
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.

Outline of the argument

Manuscript 1 (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.

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

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