3-adic Thickening of the Level-163 Hecke Fiber

Setup

• Use the 3+4+6 mod-3 primary decomposition of T_2.
• The lifted change-of-basis matrix has determinant 80; in particular it is a 3-adic unit.
• So the mod-3 splitting basis lifts canonically to a Z_3 basis, although the higher-order thickening need not stay block diagonal.

Off-block leakage

• modulo 9: all off-block entries for T_1,...,T_13 are divisible by 3? True
• modulo 9: all off-block entries for T_1,...,T_13 are divisible by 9? False
• modulo 27: all off-block entries for T_1,...,T_13 are divisible by 3? True
• modulo 27: all off-block entries for T_1,...,T_13 are divisible by 9? False
• modulo 81: all off-block entries for T_1,...,T_13 are divisible by 3? True
• modulo 81: all off-block entries for T_1,...,T_13 are divisible by 9? False

So the mod-3 factorization survives as a first-order filtration in the 3-adic thickening, but it does not split block-diagonally modulo 9 or 27.

Canonical root-lift data from the three orbit factors of T_2

The exact characteristic factors of T_2 over Z on the 1, 5, and 7 orbit pieces are: - V1: x - V5: x^5 + 5*x^4 + 3*x^3 - 15*x^2 - 16*x + 3 - V7: x^7 - 3*x^6 - 5*x^5 + 19*x^4 - 23*x^2 + 4*x + 6

modulo 9

• V1 root lifts reducing to 0 mod 3: [0]
• V5 root lifts reducing to 0 mod 3: [3]
• V7 root lifts reducing to 0 mod 3: [3]

modulo 27

• V1 root lifts reducing to 0 mod 3: [0]
• V5 root lifts reducing to 0 mod 3: [12]
• V7 root lifts reducing to 0 mod 3: [3]

modulo 81

• V1 root lifts reducing to 0 mod 3: [0]
• V5 root lifts reducing to 0 mod 3: [12]
• V7 root lifts reducing to 0 mod 3: [30]

Interpretation:

• modulo 9, the three roots are 0,3,3
• modulo 27, they are 0,12,3
• modulo 81, they are 0,12,30

So the V_5 and V_7 contributions remain collided modulo 9 and only split at the 27-level.

Lifted Q-block evidence

modulo 9

• lifted Q-block for T_11:

[3 0 0]
[0 6 0]
[3 8 3]

• square:

[0 0 0]
[0 0 0]
[0 0 0]

• cube:

[0 0 0]
[0 0 0]
[0 0 0]

• fourth power:

[0 0 0]
[0 0 0]
[0 0 0]

modulo 27

• lifted Q-block for T_11:

[12  0  0]
[18 24  0]
[ 3 26 21]

• square:

[9 0 0]
[0 9 0]
[0 9 9]

• cube:

[ 0  0  0]
[ 0  0  0]
[ 0 18  0]

• fourth power:

[0 0 0]
[0 0 0]
[0 0 0]

modulo 81

• lifted Q-block for T_11:

[66  0  0]
[45 51  0]
[30 53 21]

• square:

[63  0  0]
[ 0  9  0]
[54  9 36]

• cube:

[27  0  0]
[ 0 54  0]
[27 18 27]

• fourth power:

[0 0 0]
[0 0 0]
[0 0 0]

This gives the basis-dependent pattern:

• modulo 9, the lifted T_11 block is still square-zero
• modulo 27, its square and cube are nonzero but the fourth power vanishes
• modulo 81, the same fourth-power vanishing persists

Honest conclusion

The canonical part is the root-collision pattern: the mod-3 singular factor does not split immediately; the V_5 and V_7 zero roots stay merged modulo 9 and separate only modulo 27.

The lifted Q-block nilpotency pattern is suggestive but not yet canonical, because the mod-3 splitting does not remain block diagonal modulo higher powers of 3; the off-block couplings are only divisible by 3, not zero.

So the safe statement is:

• the mod-3 singular factor has a genuine second-order 3-adic thickening
• the first canonical splitting of the collided V_5/V_7 roots occurs at modulus 27
• any claim about a canonical nilpotent of order 3 or 4 still needs a local-factor construction, not just a lifted basis block