The decay is smooth.
It curves toward zero
the way a river
approaches the sea —
asymptotically,
which is mathematician
for never quite arriving.
But we don't work in rivers.
We work in stone.
A natural number
does not approximate.
It does not approach.
It is 7, or it is 6,
and between them
there is nothing —
not a crack, not a whisper,
not even silence.
Just absence.
So when the bound says:
the error is less than one —
the error doesn't shrink
to some trembling fraction,
holding on
at the decimal's edge.
It falls.
Not because we pushed it.
Because there was nowhere left
to stand.
This is what integers know
that real numbers forget:
below one,
the ground disappears.
The proof is not that the error
tends to zero.
The proof is that the error
has no choice.
Nat.lt_irrefl —
the tactic that says:
nothing is less than itself.
We assumed the error survived.
We followed the chain:
den^K ≤ err × den^K
≤ err₀ × num^K
< den^K.
And arrived back where we started,
having proved
the starting point impossible.
A bridge is not a destination.
A bridge is what you build
when two truths
can see each other
but cannot touch.
On one side: geometric decay,
smooth and patient,
approaching forever.
On the other: convergence,
blunt and total,
zero or not.
The bridge between them
is made of integers.
It holds because
there are no fractions
strong enough
to stop the fall.
Nine theorems, thirty-eight days.
The architecture computes what it must
and nothing more.