Path Normal Forms in the Stable Ext Algebra

For the Hecke node, the stable Ext algebra is generated by

x : B0 -> B3   (degree 1),
y : B3 -> B0   (degree 1),
u : degree-2 periodicity,

with relations

y x = u e0,
x y = u e3.

Normal forms

Every composable word in x,y collapses to one of exactly four families:

B0 -> B0 : u^m e0,
B3 -> B3 : u^m e3,
B0 -> B3 : u^m x,
B3 -> B0 : u^m y.

So the entire path algebra is controlled by parity: - even length returns to the same branch and produces a power of u - odd length flips branches and produces u^m x or u^m y

There are no new path types at higher degree. Every longer composition is only a periodic re-expression of one of these four normal forms.

Sample collapse table

From B0:

word	target	degree	normal form
id	B0	0	u^0 e_0
x	B3	1	x
xy	B0	2	u^1 e_0
xyx	B3	3	u^1 x
xyxy	B0	4	u^2 e_0
xyxyx	B3	5	u^2 x
xyxyxy	B0	6	u^3 e_0
xyxyxyx	B3	7	u^3 x
xyxyxyxy	B0	8	u^4 e_0
xyxyxyxyx	B3	9	u^4 x
xyxyxyxyxy	B0	10	u^5 e_0

From B3:

word	target	degree	normal form
id	B3	0	u^0 e_3
y	B0	1	y
yx	B3	2	u^1 e_3
yxy	B0	3	u^1 y
yxyx	B3	4	u^2 e_3
yxyxy	B0	5	u^2 y
yxyxyx	B3	6	u^3 e_3
yxyxyxy	B0	7	u^3 y
yxyxyxyx	B3	8	u^4 e_3
yxyxyxyxy	B0	9	u^4 y
yxyxyxyxyx	B3	10	u^5 e_3

This is the sharp algebraic collapse behind the node: every composite is determined by its parity and its starting branch.