Under weight decay, does a partition-recurrence readout become sparse exactly on the generalized pentagonal lags, with alternating-pair signs?

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

Statement. Euler's pentagonal number theorem, read as a recurrence, computes the partition function from its own earlier values:

p(n) = p(n−1) + p(n−2) − p(n−5) − p(n−7) + p(n−12) + p(n−15) − …

The lags are the generalized pentagonal numbers 1, 2, 5, 7, 12, 15, 22, 26, … — the pair k(3k−1)/2, k(3k+1)/2 for k = 1, 2, 3, … — and the signs run + + − − + + − −, one sign per k, two lags per sign. Every other lag has coefficient exactly zero.

The probe. Train a linear or near-linear readout on the task: input the window p(n−1), …, p(n−W) reduced mod a small prime, target p(n) mod that prime, with the window W chosen large enough that the exact rule fits inside it. Working mod a small prime keeps the task finite and exact — no float drift, no dynamic range problem from p(n)'s growth — and makes "got it right" a discrete statement.

The question is what weight decay does to the readout. The rule is a sparse integer linear functional of the window, so a correct solution can be sparse; nothing forces it to be, because many dense combinations agree with the rule on the training range. Under weight decay, does the readout become sparse with support exactly on the generalized pentagonal lags — and with the + + − − pattern in the signs — or does it settle on a dense solution that is equally correct on the data and carries no visible trace of the theorem?

Why this is the sharpest of the partition probes. The answer is a set of integers and a set of signs, both known in advance and neither reachable from the data by accident: the support is checkable lag by lag, the signs are checkable pair by pair, and there is no summary statistic to argue about. The measurement is the weight vector itself.

What makes it not automatic. Sparsity under weight decay is a statement about which of many exact solutions the optimiser prefers, and the window contains the pentagonal lags among W−1 others. The recurrence is also not the only exact rule available inside a bounded window — the values satisfy other relations on a finite range — so a dense readout is not necessarily a failure to learn, it may be a different correct rule. Distinguishing those two is part of what a run here would have to do.

Why a probe and not a case. The branches are not written: "sparse on the pentagonal lags", "sparse on something else exactly correct in-window", and "dense" are three outcomes, and telling the second from the third requires a test of in-window degeneracy that has not been specified. The window size W and the prime also have to be chosen so the rule fits and the degeneracy is countable, which is a design step, not a detail.

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

Cross-refs. P-032 is the same theorem as a bijection and P-034 as a generating function in two coordinate systems. P-027 is the nearest methodological sibling: both read a learned object's structure off the weights rather than off the loss.