Eta's coefficients indexed by n versus by m — is one coordinate trivial and the other near-impossible?

Status: open · laid 2026-09-27 · type: lead

Statement. The coefficients of the product

∏ (1 − q^k) = Σ a(n) q^n

are zero except at generalized pentagonal n, and the nonzero positions have an exact arithmetic description: a(n) ≠ 0 exactly when 24n + 1 is a perfect square, m², and the sign is then given by a Dirichlet character mod 12 evaluated at m. That is η written as a theta series — a sum over m of a character times q^((m²−1)/24) — rather than as a product.

So the same sequence has two natural indexings:

• indexed by n: a(n) is almost always 0, and the nonzero positions are a thin quadratic set. To produce a(n) one must effectively decide whether 24n + 1 is a square and, if so, extract m.
• indexed by m: the answer is a character value at m, a periodic function of m mod 12, with no squares to detect and nothing thin about the support.

The probe. Learn the coefficients in each coordinate and compare. Is the task trivial in the m coordinate and near-impossible in the n coordinate? The m version asks for a period-12 pattern, which is among the easiest things a small network can represent. The n version asks for square-detection on an arithmetic progression, which is not a local function of the digits of n in any obvious basis.

Why it is filed as a control rather than as a curiosity. lab/runs/037-grokking-multiplicative-characters.md asks which coordinate system a network's learned representation lives in, and answers it by reading the representation. This probe asks the prior question from the other end: when a task is easy in one coordinate and hard in another, does the network find the easy coordinate, or does it fail in the hard one? If it fails in the n coordinate, that bounds what "finds the right coordinates" can mean — a network reading a representation in the wrong coordinates is not merely inefficient, it is stuck — and it gives run 037's question a measured floor. Both coordinates here are known, exact, and cheap, which is what makes it usable as a control.

A second reading available for free. The sign is a Dirichlet character, so the m-indexed task is itself a character-learning task — on a tiny group, (Z/12)^x — and can be read with the same character-projection instrument the multiplicative work uses. That makes the two coordinates comparable with one tool rather than two.

Why a probe and not a case. "Trivial versus near-impossible" is a comparison, not a differential: there is no competing account of why the n coordinate would be hard that a measurement would separate from another. It becomes a case when the branches are written — for instance, whether failure in the n coordinate is about square-detection specifically or about the sparsity of the target, which are separable by a control task with the same sparsity and no quadratic structure.

No prediction and no falsifier are filed here — both are the human's.

Cross-refs. P-032 and P-033 are the same theorem as a bijection and as a recurrence. The coordinate question is run 037's, and P-031 is where it turns into a question about which instrument can see the coordinates at all.

Back-links (2026-09-29)

Dated pointers added by the codifier from lab/backlinks.py; nothing above this heading was changed.

• Run 037 — confirmed (2026-09-27, measured). lab/runs/037-grokking-multiplicative-characters.md: CONFIRMED 2026-09-28 by the repo owner, scored on the prediction's intent.