Twin Prime Gap (= 2) Exploration

Exploring the twin-prime problem (primes with bounded gap =2) through density ring (pie) charts, using the binary numbers (base-2 math)

About

The visualization is a set of nested rings, one per bit-length. Each ring covers every number with exactly that many bits, and is split into four areas:

1. Blue — primes with bounded gap = 2
2. Red — composites that are a multiple of a blue prime from the previous ring
3. Orange — composites that are a multiple of a blue prime from ANY previous ring
4. Grey — everything else (other primes and other composites)

Rings grow outward as bit-length increases, so the outer rings represent larger numbers. Watch the blue and red areas in particular: since red is defined as "multiple of the previous ring's blue primes" every ring's blue primes are guaranteed by construction to produce part of the next ring's red area — that part isn't a discovery, it follows directly from the definition.

Notes

This visualization doesn't solve the twin-prime problem. But here is the intersting:

1. Twin-prime count grows with bit-length!

bit-length	twin primes	growth
2	1	—
3	2	x2.00
4	2	x1.00
5	4	x2.00
6	4	x1.00
7	6	x1.50
8	14	x2.33
9	14	x1.00
10	24	x1.71
11	52	x2.17
12	90	x1.73
13	140	x1.56
14	226	x1.61
15	430	x1.90
16	710	x1.65
17	1332	x1.88
18	2306	x1.73
19	4142	x1.80
20	7570	x1.83
21	13930	x1.84
22	24990	x1.79
23	45286	x1.81
24	83216	x1.84
25	152742	x1.84
26	281888	x1.85
27	523504	x1.86
28	969936	x1.85
29	1809598	x1.87
30	3378954	x1.87
31	6320226	x1.87
32	11857808	x1.88

Does this prove there are infinitely many twin primes? (we can try to prove that every new +1 bit range creates at least +1 new unique twin primes pair (for now >x1 it grows with x1.84 which is actually great))

2. Blue/red shares seem to be converging

The percentage each ring's blue and red areas take up keeps shrinking, but by a smaller amount each time — it looks like it's approaching some limit rather than continuing to shrink at a constant rate.

3. Possible connection to π(x)?

There might be a link to the prime-counting function pi(x) here, given both are about prime density as numbers grow. Not verified yet.

4. Small bit-lengths are noisy

At small bit-lengths the ring proportions look a little chaotic but they settle into a stable, smooth pattern as bit-length increases and there's more data per ring.