The Sturmian Bridge — log₂3 as Collatz's Hidden Skeleton

One mathematical object underlies four different pictures: the cutting sequence of a line, a list of dropping times, a rotation on the unit interval, and a binary word over $\{2, 3\}$. They are all the same thing, and they are what makes the Collatz dropping classification work.

This page is the hub that ties the rest of the repo's research together. Every other concept — dropping orbits, dropping classes, the χ_6 L-probe, the Sturmian sign rule, the CF tower of 12-TET / 53-TET characters, the fractals on the Sturmian Fractals page — lives downstream of one fact: the dropping schedule of the Collatz map is the Sturmian cutting sequence of slope $\log_2 3$.

Adjust the slope, change the highlighted index, watch how all four views move together. That same lockstep is exactly the structural lockstep that lets Parts 4 through 9 of the Dropping Zeta Spectrum thread work.

The four views

Slope α 1.5850

Steps shown (o = 1..N) N = 18 Focus on o o = 7

A. The cutting line — symbolic dynamics of slope α

Walk along the line $y = \alpha x$ through a unit grid. Mark a at each vertical-line crossing, a at each horizontal-line crossing. The sequence of crossings is the Sturmian word.

B. The dropping schedule — Beatty sequence k_o = o + ⌊oα⌋ + 1

Dropping times marched out on the integers. Each interval between consecutive k_o is colored by its gap size: shorter gap (2), longer gap (3).

C. The rotation — {(o−1)α} on [0, 1)

For each o, compute the fractional part of (o−1)α and plot it on the unit interval. The threshold τ = ⌈α⌉ − α (0.4150) splits the points: below τ ⟹ shorter gap, above τ ⟹ longer gap.

D. The Sturmian word — the gap sequence itself

One cell per o, colored by the gap value gap_o = k_o − k_{o−1}. This is the binary sequence rendered as the fractal turtle on the other page.

2

3

2

3

2

3

3

2

3

2

3

3

2

3

2

3

2

3

At o = 7, every view points to the same datum 3x+1 system

Rotation point

0.5098

above τ

Dropping time k_o

19

k = o + ⌊oα⌋ + 1

Gap from previous

3

⟹ Sturmian sign +

Dropping class size |R_k|

3840

residues mod 2^k that drop at step k

3x+1 (Collatz): the cutting sequence of slope log₂3 is the Collatz dropping schedule. At each o, the sign +1 predicted by the gap is exactly the sign of the χ₆ L-function partial sum on this dropping class — by the closed-form theorem. The dropping class itself contains 3,840 residues, every one of which traces a Collatz orbit that first drops below its starting value at exactly step k = 19.

Why each view is "the same thing"

• A. Cutting line — the geometric picture. Walk along $y = \alpha x$ through a square grid. The order of crossings (vertical vs horizontal) is the Sturmian word.
• B. Beatty sequence — the same data as cumulative crossing positions. Each $k_o$ is the total number of crossings before the $o$-th vertical one. For Collatz that's a dropping time.
• C. Rotation — instead of cumulative crossings, just record where the line "is" relative to the next horizontal grid line. That's $\{(o-1)\alpha\} \in [0, 1)$. The cutoff $\tau = 2 - \alpha$ determines whether the next interval contains an extra horizontal crossing.
• D. Word — strip away the geometry and just write down the gap pattern as a binary sequence over $\{2, 3\}$. This is the input to the turtle fractals.

Why log₂3 is the right slope for Collatz

The Collatz map is a tug-of-war:

• Each odd step does $n \to (3n+1)/2$ — multiplies by $\approx 3/2$.
• Each even step does $n \to n/2$ — divides by 2.

If a trajectory has $o$ odd steps and $e$ even steps total, the value is multiplied by roughly $3^o / 2^e$. For it to drop below the starting value (which is what defines a dropping time), the divisions must win:

$$2^e > 3^o \quad \Longleftrightarrow \quad e > o \cdot \log_2 3.$$

So $\log_2 3 \approx 1.585$ is the exchange rate of the dynamics: every odd step "owes" $\log_2 3$ even steps. Because $e$ is an integer, the minimum legal $e$ for $o$ odd steps is $\lfloor o \log_2 3 \rfloor + 1$. Adding it to $o$ itself gives the $o$-th dropping time:

$$\boxed{k_o = o + \lfloor o \log_2 3 \rfloor + 1.}$$

The gaps $k_o - k_{o-1}$ are exactly $1 + (\lfloor o \log_2 3 \rfloor - \lfloor (o-1) \log_2 3 \rfloor)$, which is 2 or 3 depending on whether the line $y = x \log_2 3$ crossed an extra horizontal grid line between $o-1$ and $o$. That's the Sturmian gap-sequence.

In one sentence: the irrationality of $\log_2 3$ is why the Collatz dropping schedule never repeats periodically, and the structural rigidity of irrational rotation is why the schedule has the strong combinatorial properties that the rest of the repo's work exploits.

How each view connects to each Collatz concept

Dropping orbits → the cutting picture

Any individual Collatz orbit $n \to T(n) \to T^2(n) \to \ldots$ runs through some sequence of odd and even steps. The number of odd steps until the orbit first drops below $n$ is the orbit's odd-step count $o$. Its stopping time is $k_o$. So every Collatz orbit "lives at" one rung of the Beatty ladder shown in Panel B — and the entire set of orbits that share a stopping time is a single rung of that ladder.

Dropping classes $R_k$ → the Beatty ladder

The dropping class $R_k$ is the set of residues mod $2^k$ whose Collatz orbits first drop below the starting value at exactly step $k$. By the Affine Orbit Structure, these residues come in families of exactly $2^o$ (where $o$ is the corresponding odd-step count). The dropping classes are nonempty exactly for $k$ on the Beatty list $\{3, 6, 8, 11, 13, 16, 19, \ldots\}$ shown in Panel B. There is no dropping class for $k = 4, 5, 7, 9, 10, \ldots$ — those gaps in the integer number line are the missing Beatty rungs.

Try focusing $o = 7$: you'll see $k_7 = 19$, and the panel shows there are $|R_{19}| = 3{,}840$ residues mod $2^{19}$ in this class. Every one of those 3,840 numbers traces a Collatz orbit that drops at exactly step 19.

The sign rule (Parts 4–7) → the rotation threshold

For each dropping class $R_{k_o}$, the χ_6 Hecke L-function partial sum has a sign $\epsilon_o \in \{+1, -1\}$. The proved closed form (Part 5) says

$$\epsilon_o = \begin{cases} +1 & \text{if } \{(o-1)\log_2 3\} \ge \tau \\ -1 & \text{if } \{(o-1)\log_2 3\} < \tau \end{cases}$$

where $\tau = 2 - \log_2 3 \approx 0.4150$. That threshold $\tau$ is the orange dashed line in Panel C. The rotation point's position relative to $\tau$ is the sign. The sign is the gap value. Every Collatz orbit in $R_{k_o}$ contributes to the same sign because they all share the dropping-class-level data.

The Sturmian fractal → the gap sequence

Panel D is the binary word fed into the turtle program on the Sturmian Fractals page. The same gap-2-or-gap-3 sequence that's encoded in the Beatty ladder is what the turtle reads symbol by symbol to draw fractal shapes. The triangular tiling at 120° you see there isn't a coincidence — it's the geometric externalization of the Beatty schedule.

The CF tower (Part 6) → musical scales as Collatz characters

The continued fraction of $\log_2 3 = [1; 1, 1, 2, 2, 3, 1, 5, 2, 23, \ldots]$ produces convergents:

Convergent	Decimal	Famous as
$\frac{3}{2}$	$1.5000$	Pythagorean fifth
$\frac{8}{5}$	$1.6000$	rough cf bound
$\frac{19}{12}$	$1.5833$	12-tone equal temperament
$\frac{84}{53}$	$1.5849$	53-tone Holdrian comma

The fact that 12-TET and 53-TET both arise as deep convergents of $\log_2 3$ is the same Diophantine fact that organizes the Collatz character tower. Switch the slope above to $19/12$ — you'll see the Sturmian word eventually become periodic (period 12). That's Cobham's theorem showing: rational slope ⟹ $q$-automatic sequence ⟹ finite-state structure.

The Part 8 dichotomy → not everything is Sturmian

The cutting picture predicts the sign of the χ_6 sum for each dropping class. It does not predict the magnitude. Part 8 showed that the magnitude's mod-2 reduction — the Stopping-Class parity $P_o \bmod 2$ — is full-entropy Bernoulli, the opposite complexity class. So inside the same closed form, one factor is the lowest-complexity infinite binary sequence (the Sturmian sign) and the other is the highest. The bridge picture above is the sign side of that dichotomy.

The qx+1 cousins → universal Sturmian skeleton, varying cycles

Try clicking 5x+1, 7x+1, or 9x+1 in the slope presets. The same four-panel picture appears, just at a different slope $\log_2 q$. Part 10 verifies empirically:

• Beatty match holds for every q. The dropping classes $|R_k^{(q)}|$ are nonzero exactly on the predicted Beatty list $k_o = o + \lfloor o \log_2 q \rfloor + 1$.
• Sturmian fingerprint holds for every q. Factor complexity $p(n) = n+1$ for $n = 1, \ldots, 30$ across $q \in \{3, 5, 7, 9\}$.
• What changes is the Terras identity, not the Sturmian schedule. For $q = 3$ (Collatz), $\sum |R_k|/2^k \to 1$ (conjecturally). For $q \ge 5$, the sum stops short of 1 — the gap is the 2-adic density of cycle-residues plus divergent orbits.

So the Sturmian skeleton is universal across $qx+1$ cousins, and what makes Collatz the conjecturally-hard case is the precise equality $\sum |R_k|/2^k = 1$, not the Sturmian schedule itself. For $5x+1$ there are two well-known cycles (starting at 13 and 17), and that's exactly the kind of cyclical failure the Terras gap measures. The script scripts/qx_systems_analysis.py runs all of this and saves data/qx_systems_analysis.png.

In one paragraph

The Collatz tug-of-war between $\times 3$ and $\div 2$ makes $\log_2 3$ the natural exchange rate. Its irrationality makes the dropping schedule a Sturmian cutting sequence. That Sturmian-ness propagates through the affine orbit structure and Eisenstein factorization into the χ_6 sign rule, where it becomes a proved closed form. The rational approximations of $\log_2 3$ (which double as the musical scales 12-TET and 53-TET) index a tower of finer characters yet to be built. The turtle program from the Sturmian Fractals page is the visual rendering of the same gap sequence, and the dichotomy of Part 8 is what lies beyond it.

See also

• The Sturmian L-Probe — the closed-form theorem in full formality
• Sturmian Fractals — turtle visualizations of the same sequence
• Affine Orbit Structure — why each dropping class has size $2^o$
• Eisenstein Lattice — where the χ_6 character comes from