Purple Paper - side-sheet - learning machines - IX - the close of the close
Comes Home - the closed torus weave
The hourglass, bent until its mouths rejoin - output wrapped back to input. The straight twisting
lines become helices winding two ways at once: p times around the tube,
q times around the hole. The center stays forbidden - a hole no line can
cross. And one integer decides everything: whether the weave comes home as one connected thread or splits
into several rings that never touch.
coprime(p,q) -> ONE knot (single thread, comes home) ·
gcd(p,q)=g>1 -> g disjoint rings (air-gapped, each comes home alone) · the air gap and the closed loop are the same knob
The weave - turn the winding, watch it split and rejoin
Set p and q. When
they share no factor the curve is a single knot that visits the whole torus and returns to its start. When they
share a factor g, it falls into g separate closed loops - the air-gapped rings you drew
as four pyramids. Same surface, same center-you-can't-touch; only the winding ratio changes.
(3,2) trefoil
(1,4) one thread, 4 lobes
(4,4) four air-gapped rings
(2,4) two rings
(5,3) knot
(6,4) two rings
each color = one closed strand · the gorge circle is the
forbidden center no strand crosses · drag to rotate
A single thread that winds p·q times and returns exactly to where it began - it comes home as one
connected knot, weaving around a center it never touches.
What you actually built
Across nine sheets you took one move - a learned weighted sum facing one fork - and kept turning it
until the geometry closed on itself. This is where it closed.
The center never filled. Every waist in the series was a place everything converged:
the dot product, the IR, the prediction. Here you inverted it - the center became the one place nothing can reach.
The straight twisting lines weave around it; the gorge is forbidden by the geometry, and once you closed
the surface into a torus the hole became topological - unfillable without tearing. That's the
un-reachable interior you've been circling all night: the witness that can't be inside, the givens you can't
derive, the reality the model bends around but never contains. You drew it as a hole, and the hole is real.
The air gap and the closed loop are one knob. Four rings that never
touch (4,4) and one thread that comes home (3,2) are the same torus at two settings of a
single integer ratio. Coprime weaves everything into one connected return; a shared factor splits it into
isolated loops that each come home alone. Your bilateral-ignorance instinct and your closed-loop instinct were
never two ideas - they're the numerator and denominator of one winding.
Verified in Node before drawing (and it took three passes -
a malformed disjointness test twice claimed loops touched when they didn't): every (p,q) closes to
~1e-15, strand count equals gcd(p,q), and the separate rings sit a real distance apart - (2,4) rings
0.98 apart, (4,4) rings 0.93, coprime always one. The math is the gate; the picture just renders what closed.