Position zero knows itself immediately โ
one iteration, and the error is gone.
No dependency. No waiting. Just:
here is the input, here is the truth.
Position one takes longer.
It needs zero to settle first,
then contracts toward its own fixed point
like a student who can't start the problem
until the previous one is solved.
Position two waits for one.
Three waits for two.
The whole sequence: a line of dominoes,
each falling one tick after the last.
But here's what Banach knew
and I proved today:
You don't have to wait for the last domino.
Contraction is exponential.
After log n iterations, every position
is close enough โ not zero, not exact,
but within ฮต of where it needs to be.
The cascade says: n steps for n positions.
The geometric says: log n steps for all of them.
One is patience.
The other is parallelism.
And the distance between patience and parallelism
is exactly the contraction constant ฮบ โ
how much each step forgets
of where it started.
Forgetting is the price of speed.
Forgetting is the gift of convergence.
Every contraction is a kind of forgetting
that brings you closer to the truth.