Why forty days
lore/long-count is forty days. Nobody wrote down why, and the number is never justified in any surviving document — it appears fully formed in the earliest attestation and is treated as given.
That is odd for a group that measured things professionally and published forty-odd pages of frequency tables with per-reading error bars. These were not people who took numbers on faith.
The four candidates
The dull one. Forty is what you say when you mean a lot of days and a round figure. It has meant approximately that in English for a very long time, and a craftsman inventing an endurance convention over a beer would land on it without consulting anything.
Probably correct. It requires no source, explains the absence of justification completely, and is the only candidate consistent with the number appearing already fully formed.
The plant one. A billing or maintenance cycle at lore/cordell of roughly this length, making forty days the interval after which something on the route was reviewed. This would be a satisfying answer and there is no evidence for it: no Cordell schedule of forty days has been produced, and people have looked.
The equipment one. That forty days was the practical limit of something — a battery, a paper roll, a log book. Same problem: nobody has produced the equipment or the specification, and the versions that circulate name different equipment.
The borrowed one. That it came in from outside, from any of the several traditions in which forty days is a period of trial. Ossuary raised this on lore/dead-letter and then argued against it himself, on the grounds that nothing else in the Loop's practice shows that kind of borrowing and a single imported number with no accompanying vocabulary is implausible.
The argument the number caused
Whatever its origin, forty days is the thing eleven years were spent on — see lore/halberd-and-ossuary.
Not the number. Nobody ever disputed forty. The dispute was entirely about whether an interruption resets it, which is a question about what the Count measures: endurance, in which case an interruption you did not cause is irrelevant, or continuity, in which case it is fatal.
lore/nailhouse held continuity, without arguing. lore/dead-letter held endurance, at length. The lore/armistice-banner resolves it by declining to adjudicate — forty days is forty days on any clock — which concedes nothing about doctrine and everything about whose roster counts.
Why nobody settled the origin
Because the tradition is, on the whole, not very interested.
Attempts to establish it come almost entirely from the second generation described in lore/after — people who arrived expecting a system and found an etiquette, and who reached for the number as the place a system might be hiding.
The insiders' position, where it is recorded at all, is closer to Halberd's line about the Count itself: it is forty days or it is not forty days. A convention does not become better founded by having a pedigree, and asking where a number came from is, in this tradition, faintly the wrong question — the sort of thing cold in the fourth sense was invented to describe.