2025-12-16

Today I implemented the simpler approach described on 2025-12-15 and got integer coordinates.

From a high level, my starting complex is a 4-ball $D^4$. I then cut out a 4-dimensional solid cylinder $D^3\times I$. Resulting in

$$D^4\setminus (D^3\times I).$$

More precisely, the following region is cut out:

$$\bigcup _{w=1}^6[3,4{]}^3\times \{w\}\cup \bigcup _{w=1}^5[3,4{]}^3\times [w,w+1]=[3,4{]}^3\times [1,6].$$

For the input 2-cycle, I take a face of the 4-ball, $F=\{w=0\}$, and compute its boundary

$$\pi =\partial F\cong S^2$$

which wraps around the cut cylinder $D^3\times I$. The resulting optimal 2-chain snaps around the hole from the cylinder and is homologous to $S^2$.

Code output:

Sanity: ∂² == 0? true
Complex Info:
 - Dimension: 4
 - Euler Characteristic: 2
 - Valid: true

Input chain (length = 150): (5, 7, 10, 0) + (9, 0, 9, 0) - (9, 5, 0, 0) - (1, 10, 5, 0) - (3, 1, 0, 0) + (5, 9, 10, 0) - (9, 9, 0, 0) - (5, 3, 0, 0) + (10, 9, 9, 0) - (7, 5, 0, 0) + (5, 0, 1, 0) + (10, 1, 1, 0) + (9, 3, 10, 0) + (7, 1, 10, 0) + (10, 5, 7, 0) - (1, 9, 0, 0) + (3, 1, 10, 0) + (1, 5, 10, 0) - (3, 5, 0, 0) + (10, 7, 7, 0) + (10, 7, 9, 0) - (0, 1, 1, 0) + (10, 7, 3, 0) - (0, 7, 1, 0) + (10, 1, 3, 0) + (10, 5, 3, 0) + (1, 0, 7, 0) + (1, 0, 1, 0) - (0, 1, 9, 0) + (5, 0, 7, 0) - (5, 7, 0, 0) - (5, 9, 0, 0) + (3, 9, 10, 0) + (10, 3, 9, 0) - (3, 10, 5, 0) + (7, 0, 5, 0) + (7, 3, 10, 0) - (0, 1, 7, 0) - (7, 7, 0, 0) + (3, 0, 9, 0) + (1, 3, 10, 0) + (1, 7, 10, 0) - (1, 10, 3, 0) - (9, 7, 0, 0) + (10, 3, 1, 0) - (5, 1, 0, 0) + (5, 0, 3, 0) + (1, 1, 10, 0) + (1, 0, 3, 0) - (0, 5, 1, 0) + (3, 0, 5, 0) - (7, 1, 0, 0) + (10, 9, 3, 0) + (9, 7, 10, 0) + (10, 1, 7, 0) - (0, 3, 1, 0) - (0, 5, 9, 0) - (5, 10, 5, 0) + (9, 0, 7, 0) - (0, 7, 9, 0) + (7, 9, 10, 0) - (9, 3, 0, 0) + (10, 9, 5, 0) - (1, 5, 0, 0) - (0, 5, 3, 0) - (0, 9, 7, 0) - (0, 1, 3, 0) - (0, 1, 5, 0) + (3, 5, 10, 0) - (0, 9, 3, 0) + (5, 3, 10, 0) + (10, 1, 5, 0) + (1, 0, 9, 0) - (9, 10, 1, 0) + (10, 5, 1, 0) - (0, 5, 7, 0) - (7, 9, 0, 0) - (9, 10, 5, 0) - (3, 10, 7, 0) + (10, 3, 5, 0) - (5, 10, 3, 0) + (5, 1, 10, 0) - (9, 10, 9, 0) + (9, 9, 10, 0) - (3, 10, 3, 0) + (3, 7, 10, 0) - (1, 3, 0, 0) + (9, 1, 10, 0) + (3, 0, 7, 0) + (3, 0, 1, 0) - (5, 10, 9, 0) - (7, 10, 7, 0) - (1, 1, 0, 0) - (7, 10, 3, 0) + (3, 3, 10, 0) + (10, 3, 7, 0) + (9, 0, 5, 0) + (5, 0, 5, 0) + (10, 7, 1, 0) + (7, 0, 1, 0) + (10, 3, 3, 0) + (9, 0, 3, 0) - (0, 5, 5, 0) - (7, 3, 0, 0) - (3, 10, 1, 0) - (0, 9, 1, 0) - (1, 10, 1, 0) + (10, 5, 5, 0) - (0, 7, 7, 0) - (1, 10, 9, 0) - (0, 3, 5, 0) - (7, 10, 9, 0) - (0, 3, 3, 0) + (3, 0, 3, 0) - (0, 3, 9, 0) + (1, 9, 10, 0) + (7, 0, 9, 0) + (7, 0, 7, 0) - (0, 9, 9, 0) - (3, 10, 9, 0) + (10, 9, 7, 0) - (3, 9, 0, 0) - (3, 7, 0, 0) - (9, 1, 0, 0) + (5, 5, 10, 0) + (7, 7, 10, 0) - (0, 7, 3, 0) - (0, 7, 5, 0) + (10, 9, 1, 0) + (10, 1, 9, 0) - (1, 7, 0, 0) - (9, 10, 7, 0) - (5, 5, 0, 0) - (7, 10, 5, 0) + (10, 7, 5, 0) - (3, 3, 0, 0) + (5, 0, 9, 0) + (9, 0, 1, 0) + (1, 0, 5, 0) + (10, 5, 9, 0) - (5, 10, 7, 0) - (7, 10, 1, 0) - (1, 10, 7, 0) + (9, 5, 10, 0) - (5, 10, 1, 0) + (7, 5, 10, 0) + (7, 0, 3, 0) - (0, 3, 7, 0) - (9, 10, 3, 0) - (0, 9, 5, 0)
Sanity: is 2-chain? true
Sanity: is 2-cycle? true
Subcomplex Info:
 - Dimension: 2
 - Euler Characteristic: 2
 - Valid: true

Building model...
 - Computing boundary matrix...
 - Variables...
 - Objective...
 - Constraints...
Solving model...

Solution:
x = (6, 5, 5, 10) - (5, 6, 5, 10) + (5, 4, 5, 10) - (5, 5, 4, 10) - (4, 5, 5, 10) + (5, 5, 6, 10)
 (Length: 6)

Sanity: is 2-chain? true
Sanity: is 2-cycle? true
Subcomplex Info:
 - Dimension: 2
 - Euler Characteristic: 2
 - Valid: true