Let E/Q be the elliptic curve 163a1 (y² + y = x³ − 2x + 1) and f = Σ anqn its weight-2 newform. The six small Hecke operators have eigenvalues a₃ = 0, a₇ = 2, a₁₁ = −6, a₁₉ = −6, a₄₃ = 7, a₆₇ = −2. Their gate types on the 3-adic formal-group quotient are:
| p | ap | v₃(ap) | u₃(ap) mod 3 | Type |
|---|---|---|---|---|
| 3 | 0 | -- | -- | Annihilator |
| 7 | 2 | 0 | 2 | Swap |
| 11 | −6 | 1 | 1 | Shift (+1) |
| 19 | −6 | 1 | 1 | Shift (+1) |
| 43 | 7 | 0 | 1 | Identity |
| 67 | −2 | 0 | 1 | Identity |
The transition rule on a unit state (r, k) in F₃x x Z≥0 under a non-annihilator gate Tp is:
The shell coordinate k is a deterministic counter, additively driven by v₃(ap) of the applied gates.
Let X₁, X₂, ... be i.i.d. uniform on the five non-annihilator gates Σ* = {T₇, T₁₁, T₁₉, T₄₃, T₆₇}. Define ξi = v₃(ap(Xi)) in {0, 1}. Reading off the gate table: exactly two of the five gates have v₃(ap) = 1 (namely T₁₁ and T₁₉), the remaining three have v₃(ap) = 0. So each ξi is Bernoulli(2/5).
More generally, for any finite gate set S* of non-annihilator primes:
For E = 163a1: |S*| = 5 and Σv₃ = 0 + 1 + 1 + 0 + 0 = 2, giving μ = 2/5. The multiset {v₃(ap)} = {0, 1, 1, 0, 0} determines everything.
Proof. By the Bernoulli lemma, ξ₁, ..., ξn are i.i.d. Bin(1, 2/5). Independence of state follows because the increment ξi depends only on the gate Xi, not on (ri−1, ki−1). Linearity of expectation gives E[Kn − k₀] = n · (2/5), and independence gives Var = n · (2/5)(3/5) = 6n/25.
Under sampling proportional to ap² on S* = {7, 11, 19, 43, 67}, with weights {4, 36, 36, 49, 4} summing to 129 = 3 · 43:
The two natural drift rates are μunif = 2/5 and μa² = 24/43, with ratio 60/43.
The automaton A₁₆₃ was defined and studied in the companion paper on the Hecke automaton at conductor 163. The shell coordinate k is a deterministic counter on each trajectory, additively driven by v₃(ap) — this paper extracts the probabilistic consequences. The trace Gram matrix G(d, d') = Tr(TdTd' | S₂(Γ₀(163))) satisfies |G(d, d')| ≤ 2 min(G(d,d), G(d',d')), but no closed-form link between the drift rate μ = 2/5 and the trace constants is proved or conjectured. The drift is a property of the formal-group automaton; the trace inequality is a property of S₂(Γ₀(163)).