Interaction-System Certificate for the Level-163 Hecke Node

This note separates three layers that should not be conflated:

1. CM skeleton: the mediator is real

For the pair factor

R = Z_3[eta]/(eta^2 - 3 eta),

the three CM indecomposables are B0, R, B3. Using the explicit branch maps

i0(x)=(3x,0),   q0(a,b)=a,
i3(x)=(0,3x),   q3(a,b)=b,
eta(a,b)=(0,3b),

all quiver relations were verified exhaustively modulo 81.

Verified relations:

q0 i0 = 3 id_B0,     q3 i3 = 3 id_B3,
i0 q0 = 3 - eta,     i3 q3 = eta,
q0 i3 = q3 i0 = 0,
eta^2 = 3 eta.

So every round-trip through the singular mediator carries the arithmetic cost 3 already in the CM quiver, before any stable quotient is taken.

The absence of direct cross-branch arrows is exact over Z_3:

Hom_R(B0,B3) = Hom_R(B3,B0) = 0,

because an R-linear map is multiplication by lambda with 3 lambda = 0 in Z_3, hence lambda = 0. Finite truncations mod 3^k do show torsion ghost maps, but these disappear in the inverse limit:

modulus solutions to 3 lambda = 0 mod 3^k
3 [0, 1, 2]
9 [0, 3, 6]
27 [0, 9, 18]
81 [0, 27, 54]

2. Residue machine: the 3-state flag is intrinsic

In the cyclic basis

[w, T_3 w, T_11 w],

every Heegner operator preserves the flag

0 < <T_11 w> < <T_3 w, T_11 w> < <w, T_3 w, T_11 w>.
operator preserves <T11w> preserves <T3w,T11w> top middle bottom
T_3 True True 0 1 0
T_7 True True 2 2 2
T_11 True True 0 0 0
T_19 True True 0 2 0
T_43 True True 1 2 1
T_67 True True 1 2 1
T_163 True True 2 1 2

The scalar pattern is exactly pair -> v7 -> pair from bottom to top, so the residue machine is a genuine 3-step filtered Hecke object, not just a decorative basis choice.

3. Invertible residue shadow: projective unit group C6

Reducing modulo 3 gives

A = F_3 x F_3[eps]/(eps^2).

Normalizing projective classes so the first coordinate is 1, the six classes are

[[1, 1, 0], [1, 1, 1], [1, 1, 2], [1, 2, 0], [1, 2, 1], [1, 2, 2]]

and the class of T_43 generates all of them.

Hecke operator residue coords [e7,epair,n] normalized projective class order
T7 [2, 2, 2] [1, 1, 1] 3
T43 [2, 1, 1] [1, 2, 2] 6
T67 [2, 1, 1] [1, 2, 2] 6
T163 [1, 2, 0] [1, 2, 0] 2

The powers of the generator class [T_43] are

[[1, 1, 0], [1, 2, 2], [1, 1, 2], [1, 2, 0], [1, 1, 1], [1, 2, 1]]

so in particular

[T_43]^3 = [T_163],   [T_43]^4 = [T_7],

which is a concrete cyclic order-6 shadow of the singular node.

4. Stable quotient: four normal forms

In the stable two-object quotient, use generators

e0, e3, x, y, u

with

yx = u e0,   xy = u e3,
e0 x = x = x e3,   e3 y = y = y e0,
e0 e3 = e3 e0 = 0.

The pairwise multiplication table is already a finite-state reducer:

left right result
e0 e0 {'u_power_added': 0, 'symbol': 'e0'}
e0 e3 0
e0 x {'u_power_added': 0, 'symbol': 'x'}
e0 y 0
e3 e0 0
e3 e3 {'u_power_added': 0, 'symbol': 'e3'}
e3 x 0
e3 y {'u_power_added': 0, 'symbol': 'y'}
x e0 0
x e3 {'u_power_added': 0, 'symbol': 'x'}
x x 0
x y {'u_power_added': 1, 'symbol': 'e3'}
y e0 {'u_power_added': 0, 'symbol': 'y'}
y e3 0
y x {'u_power_added': 1, 'symbol': 'e0'}
y y 0

Every nonzero word in e0,e3,x,y of length at most 8 reduced to one of the four normal forms

{ u^m e0, u^m e3, u^m x, u^m y }.

Sample reductions:

word reduction
e0 u^0 e0
e3 u^0 e3
x u^0 x
y u^0 y
e0 e0 u^0 e0
e0 x u^0 x
e3 e3 u^0 e3
e3 y u^0 y
x e3 u^0 x
x y u^1 e3
y e0 u^0 y
y x u^1 e0

So the stable category is the reduced interaction algebra, while the CM skeleton is the unreduced mediator graph. The arithmetic node contains both levels at once; the stable quotient is not the whole story.