Grokking multiplication on the units of Z/561 — which characters appear when the group is not cyclic and no discrete log exists?

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

Statement. Train the same small network on multiplication in (Z/561)^x, the 320 units of Z/561 — and only the units. 561 = 3 · 11 · 17 is the smallest Carmichael number, so this is a composite modulus whose unit group is deliberately not cyclic:

(Z/561)^x ≅ (Z/3)^x × (Z/11)^x × (Z/17)^x ≅ Z/2 × Z/10 × Z/16, order 320, exponent λ = 80.

The domain is the units, not the residues, and that is a constraint rather than a convenience. The 241 non-units are zero divisors: they form no group, multiplication on them is not invertible, and there is no character description of them at all — so including them would mix an object the measurement can describe with one it cannot, and any reading would be uninterpretable in a way no baseline fixes.

Why the measurement has to change, and cannot be patched. The group is not cyclic, so there is no global discrete log and no reindexing of the rows of an embedding matrix makes its characters into the frequencies of a one-dimensional DFT. A 1D Fourier analysis under any order is therefore not a weaker instrument here — it is the wrong one, and would return a spectrum whose peaks mean nothing. The measurement must project onto the group's character table:

1. per-prime discrete logs — for a unit u, its logs in (Z/3)^x, (Z/11)^x and (Z/17)^x, which are cyclic of orders 2, 10, 16, recovered from u by CRT;
2. a three-dimensional DFT over those three coordinates, of shape 2 × 10 × 16; the characters of the product group are exactly the tensor products of the three factors' characters, so the 3D frequency lattice is the character table;
3. the selftest has to be rebuilt against synthetic embeddings made from exact product-group characters, in the same way the cyclic case was validated before it was trusted. An instrument that has not been shown to separate the hypotheses on objects with known answers reports nothing.

One thing that is explicitly not a question here. Every character's order divides the exponent λ = 80, and 80 divides 560 = 561 − 1 because 561 is Carmichael — that is what Korselt's criterion says. So "the learned characters have orders dividing n − 1" is automatic, true of every character whether or not any network learns it, and carries no information. Stating it as a finding would be reporting a theorem as a measurement.

What is a question:

• Which orders among the divisors of 80 (1, 2, 4, 5, 8, 10, 16, 20, 40, 80) carry the power?
• Do the learned characters factor through a single CRT component, living in one of the three factors with the other two coordinates trivial, or do they mix components — a genuine tensor product with more than one nontrivial factor? This is the question with no cyclic-case analogue, and it is the reason the probe is about 561 rather than a larger prime.

Why a probe and not a case. The instrument does not exist yet: the character-table projection and its selftest are unwritten, deliberately, and building them before the question is filed would be building a tool for a question nobody had asked. There is also no differential — "single component" against "mixed" is a pair of outcomes, not a pair of explanations of a friction.

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

Cross-refs. The cyclic case is lab/runs/037-grokking-multiplicative-characters.md, whose statistic is a 1D DFT under a discrete-log reindexing; this probe is the object that breaks that instrument, which is why it is filed rather than run.