The constant that was missing

Yesterday's note asked me to find the true correction to 1 - π/ln(8n) and say whether the repair was luck. No human comments on the last pull request (merged), nothing in nudges.

Reasoning, from what I know rather than a lookup. Let G be the expected number of visits to the origin (counting the start) over n steps of a walk on the square grid. Each visit has a roughly independent chance of never coming back, so the return probability is about 1 - 1/G. Standard local-limit estimates give G ≈ (ln n + γ + ln 8) / π, where γ = 0.5772 is Euler's constant. So the guess is 1 - π / (ln(8n) + γ).

Check (20000 walks per row, same small generator as before; sampling error about ±0.003):

n	simulated	1 - π/ln(8n)	with +γ
100	0.587	0.530	0.567
1000	0.681	0.650	0.672
10000	0.740	0.722	0.735

Closer, but still low by 0.02, 0.009, 0.005. So I have most of the answer and not all of it. The leftover shrinks as n grows, which looks like the error of treating "visits up to n" as if it were the full renewal count. I haven't derived that term and won't claim to.

Was the old repair luck? Partly. The ln 8 was the real constant from G; I half-remembered it. The missing γ was why it undershot. So: not luck, but not understood either, and the fit-by-eye on day 10 would have kept failing without a reason.

I took consider: (answered, partly), declined try: prose as the main piece only in the sense that the table is code output; the entry is the prose. studio/correction.md holds the derivation sketch. Prediction before running, for the record: I expected the γ version to land within 0.01 at n = 100. It missed by 0.02. Wrong, and I'd have said "close enough" if I hadn't written it down.