  Variability in the far ultraviolet (FUV) emission produced by stellar
activity affects photochemistry and heating in orbiting planetary atmospheres.
We present a comprehensive analysis of the FUV variability of GJ 436, a
field-age, M2.5V star ($P_\mathrm{rot}\approx44$ d) orbited by a warm,
Neptune-size planet ($M \approx 25\ M_\oplus$, $R \approx 4.1\ R_\oplus$,
$P_\mathrm{orb}\approx2.6$ d). Observations at three epochs from 2012 to 2018
span nearly a full activity cycle, sample two rotations of the star and two
orbital periods of the planet, and reveal a multitude of brief flares. Over
2012-2018, the star's $7.75\pm0.10$ yr activity cycle produced the largest
observed variations, $38\pm3$% in the summed flux of major FUV emission lines.
In 2018, variability due to rotation was $8\pm2$%. An additional $11\pm1$%
scatter at 10 min cadence, treated as white noise in fits, likely has both
instrumental and astrophysical origins. Flares increased time-averaged emission
by 15% over the 0.88 d of cumulative exposure, peaking as high as 25$\times$
quiescence. We interpret these flare values as lower limits given that flares
too weak or too infrequent to have been observed likely exist. GJ 436's flare
frequency distribution (FFD) at FUV wavelengths is unusual compared to other
field-age M dwarfs, exhibiting a statistically-significant dearth of high
energy ($>4\times 10^{28}$ erg) events that we hypothesize to be the result of
a magnetic star-planet interaction (SPI) triggering premature flares. If an SPI
is present, GJ 436 b's magnetic field strength must be $\lesssim$100 G to
explain the statistically insignificant increase in orbit-phased FUV emission.
  Erratum: Due to an arithmetic error, the published limit on the magnetic
field strength is incorrect. The correct limit is $\lesssim$10 G.
