Setup

Fix the elliptic curve $E:y^2+y=x^3-2x+1$ of conductor $163$ (Cremona label $163a1$), with $j$-invariant $-2^{18}\cdot 3^3\cdot 5^3\cdot 23^3\cdot 29^3 / 163^2$ and Mordell–Weil rank one, generator $P=(0,0)$. Write $f=\sum a_n q^n$ for its associated weight-$2$ newform.

For each prime $p\neq 163$ the Hecke operator $T_p$ acts on $f$ by the scalar $a_p\in\mathbb{Z}$. Tabulating for the six smallest such primes: $$\begin{array}{c|cccccc} p & 3 & 7 & 11 & 19 & 43 & 67 \\\hline a_p & 0 & 2 & -6 & -6 & 7 & -2 \end{array}$$ These are the six gates of the automaton.

[lem:sumsq] $\displaystyle\sum_{p\in\{3,7,11,19,43,67\}} a_p(E)^2 \;=\; 129 \;=\; 3\cdot 43.$

$0+4+36+36+49+4=129$, and $129=3\cdot 43$.

Gate classification

For an integer $n\neq 0$, write $v_3(n)$ for its $3$-adic valuation and $\mathrm{u}_3(n) := (n/3^{v_3(n)})\bmod 3 \in \mathbb{F}_3^\times$ for its leading $3$-adic unit.

[def:gates] For each gate prime $p\in\{3,7,11,19,43,67\}$, define the gate type by:

• if $a_p=0$, $T_p$ is an annihilator;

• if $a_p\neq 0$ and $v_3(a_p)=0$, $T_p$ is a permutation gate with multiplier $\mathrm{u}_3(a_p)\in\mathbb{F}_3^\times$;

• if $a_p\neq 0$ and $v_3(a_p)\geq 1$, $T_p$ is a shift gate of depth $v_3(a_p)$ and unit multiplier $\mathrm{u}_3(a_p)$.

A permutation gate with multiplier $1$ is called identity, with multiplier $2$ is called swap.

[prop:gatetable] At conductor $163$ the six small gates have the types: $$\begin{array}{c|cccc} p & a_p & v_3(a_p) & \mathrm{u}_3(a_p) & \text{type} \\\hline 3 & 0 & - & - & \text{annihilator} \\ 7 & 2 & 0 & 2 & \text{swap} \\ 11 & -6 & 1 & 1 & \text{shift }(+1,\times 1) \\ 19 & -6 & 1 & 1 & \text{shift }(+1,\times 1) \\ 43 & 7 & 0 & 1 & \text{identity} \\ 67 & -2 & 0 & 1 & \text{identity} \end{array}$$

Direct computation from the table of $a_p$. All $3$-adic data is integer arithmetic; $7\equiv 1\pmod 3$, $-2\equiv 1\pmod 3$, $-6=-2\cdot 3$ has $v_3=1$ and unit $-2\equiv 1\pmod 3$, and $2\not\equiv 1\pmod 3$.

The state machine

[def:automaton] The Hecke automaton $\mathcal A_{163}$ has state space $$Q \;=\; \{\mathrm{VAC}\}\;\cup\;\bigl\{(r,k): r\in\mathbb{F}_3^\times,\ k\in\mathbb{Z}_{\geq 0}\bigr\},$$ six input symbols $\{T_p:p\in\{3,7,11,19,43,67\}\}$, and transition function $\delta:Q\times\{T_p\}\to Q$ given by: $$\delta(\mathrm{VAC},T_p) = \mathrm{VAC},\qquad \delta\bigl((r,k),T_3\bigr) = \mathrm{VAC},$$ $$\delta\bigl((r,k),T_p\bigr) \;=\; \begin{cases} \mathrm{VAC} & \text{if } \mathrm{u}_3(a_p)\cdot r\equiv 0\pmod 3,\\ \bigl(\mathrm{u}_3(a_p)\cdot r \bmod 3,\ k+v_3(a_p)\bigr) & \text{otherwise,} \end{cases}$$ for $p\neq 3$, where $v_3(a_p)\in\{0,1\}$ by Proposition [prop:gatetable]. The state $\mathrm{VAC}$ is absorbing.

Since $\mathrm{u}_3(a_p)\in\mathbb{F}_3^\times=\{1,2\}$ for $p\neq 3$, the residue component never collapses to $0$ under the non-annihilator gates, and the otherwise branch always applies. Hence the only path into $\mathrm{VAC}$ from a unit state is through $T_3$.

The three theorems

Let $\Sigma=\{T_3,T_7,T_{11},T_{19},T_{43},T_{67}\}$ and $\Sigma^*=\Sigma\setminus\{T_3\}$. For a word $w=g_1\cdots g_n\in\Sigma^n$ and a state $q\in Q$, write $\delta(q,w)$ for the iterated transition.

[thm:mono] For every word $w\in (\Sigma^*)^*$ and every unit state $(r,k)$, the result $\delta((r,k),w)$ is again a unit state $(r',k')$ with $$k' \;=\; k \;+\; \sum_{i=1}^{|w|} v_3\!\bigl(a_{p(g_i)}\bigr) \;\geq\; k.$$ The shell coordinate is a non-decreasing additive counter, strictly increasing iff $w$ contains at least one shift gate $T_{11}$ or $T_{19}$.

By Proposition [prop:gatetable], every $g\in\Sigma^*$ has either $v_3(a_p)=0$ (types swap, identity) or $v_3(a_p)=1$ (shifts $T_{11},T_{19}$). The transition $(r,k)\mapsto(\mathrm{u}_3(a_p)r\bmod 3,\,k+v_3(a_p))$ never produces $\mathrm{VAC}$ since $\mathrm{u}_3(a_p)\in\{1,2\}$ and $r\in\{1,2\}$. Iterating, the shell coordinate accumulates $\sum v_3(a_{p(g_i)})\geq 0$, with strict inequality iff some $g_i\in\{T_{11},T_{19}\}$.

[thm:z2] Let $\pi:Q\setminus\{\mathrm{VAC}\}\to\mathbb{F}_3^\times$ be projection to the residue coordinate. The induced action of $\Sigma^*$ on $\mathbb{F}_3^\times$ factors through the group $\mathbb{F}_3^\times\cong\mathbb{Z}/2\mathbb{Z}$, with $$T_7 \mapsto (\text{nontrivial involution}),\qquad T_{11},T_{19},T_{43},T_{67}\mapsto (\text{identity}).$$ In particular $T_7^2$ acts as identity on residues, so $\mathrm{SWAP}^2=\mathrm{ID}$.

By Definition [def:automaton] the residue acts by multiplication by $\mathrm{u}_3(a_p)\in\mathbb{F}_3^\times$. Reading off Proposition [prop:gatetable], $\mathrm{u}_3(a_7)=2$ while $\mathrm{u}_3(a_p)=1$ for $p\in\{11,19,43,67\}$. Since $\mathbb{F}_3^\times=\{1,2\}$ has order $2$, $2^2=4\equiv 1\pmod 3$.

[thm:confine] Let $S\subseteq\mathbb{F}_3^\times\times\mathbb{Z}_{\geq 0}$ be the reachable set $$S \;=\; \bigl\{\,\delta\bigl((1,0),w\bigr) \;:\; w\in(\Sigma^*)^*\,\bigr\}.$$ Then $S=\mathbb{F}_3^\times\times\mathbb{Z}_{\geq 0}$, but the subset $\{(r,0):r\in\mathbb{F}_3^\times\}$ (the matter or boundary sector) is reachable only by words containing no shift gate. Once $T_{11}$ or $T_{19}$ has been applied, the trajectory enters the vacuum sector $\mathbb{F}_3^\times\times\mathbb{Z}_{\geq 1}$ and never returns. Equivalently: the $3$-adic ideal $\mathfrak{m}=3\mathbb{Z}_3$ is forward-invariant under $\Sigma^*$, and the matter complement $\mathbb{Z}_3^\times$ is forward-absorbing into $\mathfrak{m}$.

Reachability: $T_7$ realises both residues from $r=1$, and $T_{11}^k$ reaches every shell $k\geq 0$. Forward invariance of $\mathfrak{m}$: by Theorem [thm:mono], the shell coordinate never decreases, so once $k\geq 1$ it remains $\geq 1$. Absorption: both shift gates have $v_3=1>0$, so a single application sends $k=0$ to $k=1$.

The annihilator $T_3$ is the unique gate that exits both sectors into $\mathrm{VAC}$. Removing $T_3$ from the alphabet yields a deterministic automaton on $\mathbb{F}_3^\times\times\mathbb{Z}_{\geq 0}$ in which the matter sector is one-way and the vacuum sector is closed.

The five numbers, revisited

[prop:five] The following identities hold for the gate set $\{T_3,T_7,T_{11},T_{19},T_{43},T_{67}\}$ at conductor $163$:

1. $\sum_p a_p^2 = 129 = 3\cdot 43$.

2. $\mathrm{SWAP}^2=\mathrm{ID}$ on residues.

3. Exactly one annihilator, exactly one swap, exactly two shifts, exactly two identities.

4. The kernel of the residue action $\Sigma^*\to\mathbb{F}_3^\times$ is the submonoid generated by $\{T_{11},T_{19},T_{43},T_{67}\}$, with $\sum_{p\in\ker} a_p^2 = 36+36+49+4 = 125 = 5^3$.

5. The two shift primes contribute equally: $a_{11}^2 = a_{19}^2 = 36 = 4\cdot 3^2$.

(i) Lemma [lem:sumsq]. (ii) Theorem [thm:z2]. (iii) Proposition [prop:gatetable]. (iv) The four primes acting trivially on residues are $\{11,19,43,67\}$ by Proposition [prop:gatetable]; $36+36+49+4=125=5^3$. (v) $(-6)^2=36=4\cdot 9$.

Items (iv) and (v) are stated as observations on the gate table, not as structural theorems. Whether $125=5^3$ is meaningful (e.g. as a degree of an associated congruence) or coincidental is open.

What is not proved here

This paper makes only the claims stated above. In particular:

• We do not claim the formal-group orbit of the Mordell–Weil generator $P\in E(\mathbb{Q})$ is monotone under multiplication-by-$n$. Numerical data show the shell pattern $v_3(z(nP))$ is not monotone in $n$ beyond the first few rungs. The monotonicity proved here is a property of gate sequences acting on the state space, not of the integer orbit on the curve.

• We do not prove the automaton is universal among curves with similar Hecke data. The empirical universality classes observed across ten conductors are deferred.

• The connection to the singularity category $D_{\mathrm{sg}}(R)$ for $R=\mathbb{Z}_3[\eta]/(\eta^2-3\eta)$ is sketched only as motivation; the $\mathbb{Z}/2\mathbb{Z}$ of Theorem [thm:z2] is established here purely as a property of residue arithmetic.

Reproducibility

The gate classification of Proposition [prop:gatetable] and the arithmetic of Lemma [lem:sumsq] are reproduced by src/heegner/heegner_gate_automaton.py, which uses Sage to compute $a_p(E)$ from the curve equation directly and applies Definition [def:gates]. The full transition table of $\mathcal A_{163}$ truncated to shell depth $4$ is emitted as heegner_gate_automaton.json.