The Wobble

The Hidden Rotation showed that in base-6 log coordinates, every Collatz step is the same rotation: $\theta \mapsto \theta + \log_6 3 \pmod 1$, because $-\log_6 2 \equiv \log_6 3 \pmod 1$. That picture has exactly one imperfection — the "+1" in $3n+1$. This page is about what that imperfection is, harmonically: a complete dissection of the perturbation, with every visual mystery of the polar plots reduced to a named mechanism.

Writing $\theta_k = \{\log_6 x_k\}$, the orbit satisfies exactly

$$\theta_k = \theta_0 + k\,\alpha + W_k \pmod 1, \qquad \alpha = \log_6 3 \approx 0.613147,$$

where the wobble $W_k$ collects one positive increment per odd step:

$$W_k = \sum_{\substack{j < k \\ x_j \text{ odd}}} \log_6\!\left(1 + \frac{1}{3x_j}\right).$$

The wobble is the +1. Everything else is rigid rotation. Telescoping gives the exact identity $W_{\text{total}} = \log_6\!\big(2^E / (3^O n)\big)$ for an orbit reaching 1 — the wobble budget is literally the excess of halvings over triplings, the quantity every stopping-time argument is about.

For the 987-point orbit of 670617279, 92% of the total wobble comes from odd values below 100. The wobble is a record of the orbit's visits to small numbers.

Carrier × envelope: the two arithmetics factorize

The wobble is a pulse train, and its two components live in different arithmetics:

• Envelope (how much): the increment is fully determined by position: $\delta = \log_6(1 + 6^{-a}/3)$ where $a = \log_6 x$ is the altitude. Thousands of orbit points collapse exactly onto this one master curve — deterministic, archimedean clockwork.
• Carrier (when): the gaps between odd steps are $g = 1 + v_2(3x+1)$. Across ~17k gaps from unbiased large seeds: geometric gap law, zero lag correlation, periodogram flat against the renewal null. The timing is pure 2-adic noise — no fingerprint of the rotation survives in it.

When the wobble ticks is multiplicatively invisible; how much it ticks is additively invisible. The +1 decomposes across the two completions of $\mathbb{Q}$ it touches — and the decomposition is statistically exact (mutual information between head and tail invariants: $0.6\sigma$ from zero).

The crystal that doesn't melt

Treat $W_k$ as phase disorder on the rotation. The coherence factor $D(m) = |\langle e^{2\pi i m W_k}\rangle|$ measures how the disorder damps the $m$-th harmonic of the Weyl spectrum — formally identical to the Debye–Waller factor that measures how thermal vibration damps Bragg peaks in a crystal.

With $\sigma_W = 0.0082$, Gaussian phase noise would halve coherence by $m \approx 23$ and destroy it by $m \approx 60$. Observed: $D(m) \ge 0.915$ out to $m = 150$. The wobble is shot noise — long plateaus, rare discrete kicks — and discrete disorder preserves coherence that continuous jitter would destroy (the same reason the Mössbauer effect gives recoil-free emission). The quasicrystal structure of the orbit survives the +1 at every harmonic we can measure.

Why 44 beats 31: a one-sided lens

The peaks sit at the continued-fraction denominators of $\alpha = [0; 1,1,1,1,2,2,3,1,5,\dots]$: 13, 31, 106, 137 are convergents; 44 = 31+13 is a semiconvergent. By pure Diophantine quality, 31 ($\epsilon = +0.0076$) beats 44 ($\epsilon = -0.0215$). So why do real orbit plots favor 44?

Because the wobble is always positive, it can only push a near-return one way: it cancels misses that close from below ($\epsilon < 0$: 13, 44, 75, 106) and worsens misses that close from above ($\epsilon > 0$: 31, 137). Over 8 long orbits, $\epsilon < 0$ peaks are enhanced with median ratio 2.08; $\epsilon > 0$ peaks are damped.

And the cancellation is resonant. The local wobble rate falls as the orbit's altitude falls, so each $\epsilon < 0$ closure passes through exact cancellation at its own altitude. Binning the $q$-step closure miss by $r = \Delta W_{\text{window}} / |\epsilon_q|$ over 500 orbits:

Every $\epsilon < 0$ harmonic dips at exactly $r = 1$ — no free parameters — while 31 and 137 only climb. A falling orbit chirps through the resonances in order: 106 rings near altitude $6^{6}$, then 75, then 44 rings in the few-hundreds band that every orbit's visible tail occupies, then 13. The 44-cycles in the original Proportional Power Ratios article were never an approximation artifact — they are the +1 perturbation itself, caught at resonance. 31, the "better" approximation, never closes anywhere on a real orbit.

The wobble budget is a new invariant

$W_{\text{total}}(n)$ looks like a per-orbit quantity, but it's 99% a per-gateway quantity: defining the landmark $s(n)$ = first odd orbit value below 100, the 50 tail budgets $\{W_{\text{total}}(s)\}$ explain 99.0% of the variance over all odd $n < 10^5$.

And this gateway classification is provably transverse to the dropping-time framework: landmark fibers show zero residue structure mod $2^j$ for all $j \le 14$ (purity = shuffled null), and zero mutual information with dropping time. Dropping time is a head invariant — 2-adic, determined by $n \bmod 2^k$. The gateway is a tail invariant — archimedean, residue-free. Every integer carries both coordinates, and they are independent.

The sunflower test: predicting 137

What the eye sees in a polar render of the orbit (angle $2\pi\theta_k$, radius $k$) is a parastichy count — the same mathematics as the spiral arms of a sunflower head. The visible arm count at radius $k$ is

$$q^*(k) = \arg\min_q \; q^2 + (2\pi k\,\epsilon_q)^2,$$

which predicts: 13 arms inside $k \approx 160$, 31 arms to $k \approx 2840$, then 137 arms — with 106 skipped entirely. Orbits that long need seeds around $10^{140}$. A 145-digit seed gives a 3449-step orbit, and the observed arm count matches the prediction in all 12 radial bands:

A falsifiable prediction, made before the orbit was computed, confirmed on a number $10^{130}$ times larger than anything in the original article.

Physics dictionary

Collatz object	Physics counterpart
Rotation by $\log_6 3$	Integrable system; the unperturbed flow
Weyl peaks at 13/31/44/106/137	Bragg diffraction of a 1D quasicrystal
Wobble $W_k$	Disorder field / phase noise
Coherence $D(m)$	Debye–Waller factor; shot-noise analogue of the Mössbauer recoil-free fraction
Odd-step train × increment law	AM signal: renewal carrier, deterministic envelope
2-adic timing ⊥ archimedean amplitude	Adelic factorization (the Freund–Witten lesson from p-adic string theory)
Resonance at $\Delta W / |\epsilon_q| = 1$	Phase-locking of a driven oscillator; mode pulling
Arm counts 13 → 31 → 137	Phyllotaxis / parastichy

These are structural correspondences — the same mathematics on a different substrate — not claims of physical mechanism.

Where this lives in the repo

• Research note: Log-6 Rotation Duality — 17 numbered findings
• Scripts: scripts/collatz_log6_wobble*.py (five passes: dynamics, mechanism, bands, plateaus, resonances)
• The companion proof-side view of the same obstruction: The Countdown and the $\varepsilon$ term in the conservation law