الْبَقَرَة

Picture the tree from the parable you already know: a seed goes into the ground, nothing happens for
a while, and then — all at once — seven spikes, a hundred grains each. That is Qur’an 2:261’s picture of
sadaqah, and a previous paper built an entire growth model on it: a single yearly (or decade-long, depending
on the seed) harvest, one dramatic multiplication, repeat.
But three surahs later, the Qur’an describes a tree that does not work that way at all. Its fruit does not
arrive once a season — it arrives kulla hın, “every hın,” at every turn, all the time (Q. 14:25). No single
harvest day. No bare months followed by abundance. Just a tree that is always giving something.
Two trees, one Book, and — it turns out — one honest question a model of compounding sadaqah cannot
dodge: does it matter how often the return arrives, if the total for the year is the same either way?
Most people’s intuition says yes, obviously, more often is better. The mathematics in this paper says
something sharper and slightly uncomfortable: harvesting continuously buys you no extra growth at all.
What it buys you instead is protection — against the one moment, at some point during your life, when the
reckoning arrives and finds you empty-handed because your harvest hadn’t come yet. That protection turns
out to be worth almost exactly ln 700 ≈ 6.55: the natural logarithm of the very multiplier the seed parable
already promised you.
Figure 1. sketches the two trees as the paper actually treats them: not as botany, but as two different
rhythms laid over the same timeline.

A seed yielding a fixed 700-fold return per growth cycle, without specifying the seed or its cycle length. This omission is modelled as a free parameter T in a compound-growth function N(t) = N0 700^t/T, with representative cycle lengths anchored to reported growth cycles of real seed-bearing species and the model’s instantaneous rate of growth derived in closed form. The model is then extended to the verses that condition the multiplier itself on how the giving is done: a collapse to zero for insincere giving, a bounded range for sincere giving, and a late-life recurrence of that same collapse tied to a giver’s remaining time and dependents. A structurally different promise in Surah al-Qadr – that a single night can be worth more than a thousand months – is examined separately and modelled as an open-ended floor on the starting value of a deed rather than an extension of the seed model’s own rate.