The Sturmian L-Probe

The Hecke L-function probe $\chi_6$ on $\mathbb{Z}[\omega]$, when restricted to a single Collatz dropping set, has an exact closed form — and that closed form's phase is the cutting sequence of $\log_2 3$.

This is the strongest "L-function sees Collatz" statement we have: every quantity is explicit, every constant is algebraic, every claim is proven.

The Setup

Each odd integer $n$ lifts to an Eisenstein integer via the orbit-pair lift:

$$\iota_2(n) = n + \text{dest}(n) \cdot \omega \in \mathbb{Z}[\omega]$$

where $\text{dest}(n)$ is the first orbit value below $n$. The sextic residue character $\chi_6$ maps each lift to a sixth root of unity (or zero).

For odd-start dropping times $k_o = o + \lfloor o \log_2 3 \rfloor + 1$ and the corresponding dropping set $\text{Dset}_{k_o}$, define the per-Dset twisted partial sum:

$$D_{\chi_6}^{(k_o)}(N) = \sum_{\substack{n \le N \text{ odd} \\ T(n) = k_o}} \chi_6\bigl(\iota_2(n)\bigr)$$

The Theorem

Theorem (Sturmian L-Probe Closed Form)

For all $o \ge 1$ except the unique singular case $o = 3$:

$$D_{\chi_6}^{(k_o)}(N) = i\sqrt{3} \cdot \epsilon_o \cdot A_o \cdot \frac{N_{k_o}}{|R_{k_o}|}$$

where:

• $\epsilon_o = +1$ if $\text{gap}_o = k_o - k_{o-1} = 3$, and $-1$ if $\text{gap}_o = 2$.
• $A_o$ is determined by the recursion $A_{o+1} = \tfrac{1}{2}(P_o + \sigma_o A_o)$ with $A_1 = 1$, $\sigma_o = (-1)^{\text{gap}_o}$.
• $P_o$ is the number of valid Syracuse $\alpha$-sequences for $\text{Dset}_{k_o}$.
• $N_{k_o}$ is the count of odd $n \le N$ with $T(n) = k_o$.

For $o = 3$ (i.e. $k = 8$): $D_{\chi_6}^{(8)}(N) = 0$ exactly.

The phase $\epsilon_o$ is the Sturmian cutting sequence of $\log_2 3$ — the canonical irrational-rotation signal.

What the closed form looks like

$o$	$k_o$	gap	$\alpha_o$	$\arg$	per-$n$ size	clean form
1	3	2	$-90°$	0.8660	$\sqrt{3}/2$
2	6	3	$+90°$	0.8660	$\sqrt{3}/2$
3	8	2	—	0	exact cancellation
4	11	3	$+90°$	0.2887	$\sqrt{3}/6$
5	13	2	$-90°$	0.1237	$\sqrt{3}/14$
6	16	3	$+90°$	0.2887	$\sqrt{3}/6$
7	19	3	$+90°$	0.1155	$(\sqrt{3}/2) \cdot 2/15$
8	21	2	$-90°$	0.1325	$(\sqrt{3}/2) \cdot 13/85$

Three universal facts visible immediately:

1. Magnitudes are rational multiples of $\sqrt{3}/2$, with the denominator equal to $|R_{k_o}|$.
2. All phases are $\pm 90°$ — sums lie strictly on the imaginary axis.
3. The sign pattern matches the gap parity — exactly the Sturmian word of $\log_2 3$.

The Sturmian Sign Rule

For the dropping-time sequence $k_o$, the gaps $\text{gap}_o = k_o - k_{o-1}$ form the binary sequence

$$(2, 3, 2, 3, 2, 3, 3, 2, 3, 2, 3, 3, 2, 3, 2, 3, 2, 3, 3, 2, \ldots)$$

This is the cutting sequence of the line $y = x \log_2 3$ — the canonical Sturmian word with irrational slope. The sign of the L-probe's per-Dset sum is:

$$\text{sgn}\bigl(D_{\chi_6}^{(k_o)}\bigr) = \begin{cases} +i & \text{if } \text{gap}_o = 3 \\ -i & \text{if } \text{gap}_o = 2 \end{cases}$$

Equivalently, by the Beatty fractional-part characterization:

$$\text{gap}_o = 3 \iff \{(o-1) \log_2 3\} \ge 2 - \log_2 3 \approx 0.4150$$

The probe's phase pattern is the irrational rotation of $\log_2 3$, made arithmetic-visible.

Proof outline

The proof reduces to a chain of explicit calculations:

Step 1 — Eisenstein column collapse. The 9-cell lookup of $\chi_6$ on $\mathbb{Z}[\omega]/3$ has column sums $\sum_i \chi_6(i + 0 \cdot \omega) = 0$, $\sum_i \chi_6(i + 1 \cdot \omega) = -i\sqrt{3}$, $\sum_i \chi_6(i + 2 \cdot \omega) = +i\sqrt{3}$.

Step 2 — dest mod 3 lemma. Using the affine recurrence $n_k = (3^o n + \Delta)/2^e$ and tracking $\Delta \bmod 3$, one shows $\text{dest}(n) \bmod 3 \in \{1, 2\}$, with $\text{dest} \equiv 2 \pmod 3$ iff $j^* + k_o$ is even (where $j^*$ is the position of the last odd Collatz step).

Step 3 — α-parity reduction. Combining Steps 1 and 2: the sign of the per-Dset sum depends on the parity of $\alpha_o = k_o - 1 - j^*$, the last Syracuse-step halving count.

Step 4 — alternating-sum recursion. Tracking $A_o = $ (paths with $\alpha_o$ odd) $-$ (paths with $\alpha_o$ even) over valid Beatty-bounded lattice paths gives

$$A_{o+1} = \tfrac{1}{2}(P_o + \sigma_o A_o), \quad \sigma_o = (-1)^{\text{gap}_o}.$$

Step 5 — three lemmas:

• Parity: $A_o \equiv P_o \pmod 2$ (so the recursion is always integer-valued).
• Monotonicity: $P_{o+1} > P_o$ for $o \ge 2$ (so path counts grow strictly).
• Bounds: $0 < A_o < P_o$ for $o \ge 4$, with $A_3 = 0$ the unique zero.

Step 6 — induction. From the three lemmas, the recursion produces $A_{o+1} \ge 1$ for $o \ge 3$, with $A_{o+1} \le P_o - 1 < P_{o+1}$ preserved. The bound $A_o < P_o$ for $o \ge 3$ rules out any second cancellation: $o = 3$ is structurally unique.

What this is not

The closed form is not RH for any classical L-function. The Hecke L-function $L(s, \chi_6)$ on $\mathbb{Z}[\omega]$ has its own zeros, governed by Tate's thesis. Our theorem is about the orbit-twisted partial sum, which is a different object: a character sum restricted to the Collatz-orbit-pair image in $\mathbb{Z}[\omega]$, not over all ideals.

The natural full L-function for Collatz would be

$$L_{\text{Collatz}}(s, \chi_6) = \sum_{n \text{ odd}} \frac{\chi_6(\iota_2(n))}{n^s}$$

with our closed form giving its leading-order partial-sum asymptotic. Its zeros are not the zeros of the classical $L(s, \chi_6)$ — they would be a genuinely new spectral object encoding orbit-counting fluctuations.

Implications

The L-function probe proves the "see Collatz" signal. Phase 1 of the L-function design spec called for empirical evidence that $\chi_6$ has structural correlation with Collatz orbits beyond GRH-random behavior. We now have that correlation as a theorem, with explicit constants.

Multiplication Symmetry is automatic. The closed form depends only on the Beatty boundary $B_j = \lfloor j \log_2 3 \rfloor$ and combinatorial path counts. Both are intrinsic — invariant under the $\times 3$ action on residues. So the Multiplication Symmetry Theorem reduces to a corollary of the closed form.

A Collatz counterexample would deviate from this prediction. If any odd $n_0$ had a non-dropping orbit (divergent or in a non-trivial cycle), it would not appear in any $R_k$. Its missing contribution would cause $D_{\chi_6}(N) - D_{\chi_6}^{\text{pred}}(N)$ to grow faster than the expected $O(\sqrt{N} \log N)$ residual. The probe is, in this sense, a falsifiability instrument for Collatz.

Related

• The Transfer Operator — the critical circle $(4/3)^{1/3}$ and where the eigenvalue equation comes from
• Eisenstein Lattice — why $\mathbb{Z}[\omega]$ is the right number ring for the Collatz dynamics
• The Hidden Rotation — Sturmian dynamics of $\log_2 3$, the same irrational that appears here