Do a learned Franklin involution's errors concentrate exactly on the pentagonal-number shapes where the map is undefined?

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

Statement. Franklin's involution is the combinatorial heart of Euler's pentagonal number theorem. On partitions of n into distinct parts it pairs each partition with another of opposite parity — by moving either the smallest part or the top staircase of the Ferrers diagram, whichever move is legal — so the parities cancel and the generating function collapses. It is an involution except on a thin set of shapes where the two moves collide and neither is defined, and the count of survivors is exactly the coefficient the theorem reports.

Those exceptional shapes are not scattered. They occur precisely when the smallest part and the top staircase overlap in the one configuration that blocks both moves, and that happens exactly at

n = k(3k − 1)/2 and n = k(3k + 1)/2,

the generalized pentagonal numbers — one unpaired shape at each, of sign (−1)^k. The fixed points of the involution are the terms of the pentagonal number theorem.

The probe. Train a model to compute the Franklin image: input a partition of n into distinct parts, target its image under the map. Then ask where the residual errors are. The claim to test is that they concentrate on the boundary shapes above — the configurations where the map is undefined and its neighbours are the hardest cases — rather than spreading over partitions by size, or by number of parts, or by any other coarse feature.

Why this is worth a measurement rather than a shrug. A model that has learned the rule should fail where the rule fails, and the rule's failures are a named, sparse, exactly enumerable set with an independent arithmetic characterisation. That is a much stronger fingerprint than "errors on hard examples": the error set is predicted before the model is trained, by a theorem, and its locations are pentagonal numbers rather than anything the network could reach from the input distribution. If instead the errors sit on large n, or on partitions with many parts, the model found something else and the shape of the something else is the next question.

Why a probe and not a case. The target has a subtlety that has to be settled first and is not settled here: on the exceptional shapes the map has no image, so the training target is undefined exactly where the interesting behaviour is, and how those examples are presented — excluded, given a sentinel, or given themselves as a fixed point — changes what an "error" there means. Until that is decided the discriminating test is not writable, which is the line between a probe and a case.

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

Cross-refs. P-033 is the same theorem approached through its recurrence rather than its bijection, and P-034 through its generating function's coordinates. The three are one object seen from the combinatorial, the recursive and the analytic side.