Molecules that move: tandems, trains, rotors

Post 4 — the A=τ·D_v law, wake-locked tandems, the pair-only zone, and spontaneous rotation.

The traveling bond

Builds on the bond of post 3 and the drift bifurcation of post 2.

Round-1 left a tension: binding was certified at (D_v=2.0, τ=2.5), motility at (D_v=0.65, τ>4.78) — no overlap. The resolution is a one-line theorem: the steady states depend on τ and D_v only through the product A = τ·D_v (set ∂v/∂t=0: u = v − τD_v∇²v). So the whole static world — blob shape, tails, bond wells — is frozen along any path with A fixed, while τ alone dials the drift instability. Binding lived at A=5, motility at A≈3.1. The M4 family holds A=4 (bond exists, slightly shallower: static d*≈15.4) and walks τ up with D_v=4/τ:

The traveling bond is a wake-locked tandem: the pair moves along its own axis, follower surfing the leader's wake at discrete shell distances (14.78 or 25.68 — spaced by the tail wavelength ~10.9). Trains of three go faster than pairs, which go faster than singles (0.143 > 0.141 > 0.123): each blob deepens the collective wake.

Traveling-bond certification. Pair speed vs τ with the √-law fit (onset τ_c=5.636, r²=0.996); bond length under motion (contracts 15.4→14.76); the out-of-window audit point at τ=6.1 (predicted 0.161, measured 0.152).

The wake ladder. Tandem shells at 14.78 and 25.68 px (spacing = tail wavelength 10.9), and the 3-blob train — longer is faster: c(train) 0.143 > c(pair) 0.141 > c(single) 0.123.

M4 — the pair-only zone. At τ=5.70 (window 5.636–5.748), the same kick is given to a lone blob (left) and to a bound pair (right). The single dies back to rest — below its personal drift threshold. The pair travels steadily at c≈0.06. The molecule can move before its atoms can. This audited zone becomes the machine's parking brake below.

The heterodimer rotor: rotation as an attractor (rotating bound states of dissipative solitons: known — Moskalenko et al. EPL 63:361, incl. rotation-before-drift ordering; our additions: the nonreciprocal heterodimer design, ω(τ₁) dial law, and the single-static < pair-travel < rotor-spin hierarchy w/ transport use)

Nonreciprocity framing: with η only M→S the rotor still spins — the coupling is effectively nonreciprocal, connecting this object to the non-reciprocal phase-transition literature (Fruchart et al. Nature 592:363; Ivlev et al. PRX 5:011035).

M4's honest negative said symmetric pairs refuse to rotate — kicks either lock or convert to translation. The hypothesis: rotation of a symmetric pair is wake-frustrated; the fix is structural symmetry breaking — bind a motile blob to an anchored one, so the tangential push and the pivot are different objects. That needs a bond between species, which the shared-w architecture cannot make (w is monotone-repulsive — quantified no-go). The new xv architecture: each species gets fully private fields (u_i, v_i, w_i) and the only coupling is a weak cross-drive in the slow channel:

Why v? The slow inhibitor's halo is the unique sign-changing mediator: its radial response (measured from the certified stamp) is +1.04 in the core, crosses zero at r=6.1, and has a negative ring over r∈(6,15), minimum at r≈8. Cross-wiring imprints that ring into the partner's own propulsion field: repulsive core, attractive well at r≈8 — a cross-species bond at d*≈7–9 was predicted from this profile before the first run, and measured at 7.5–8.0 (η=0.05: d*=7.976, audited from d₀=12 to the third decimal). The same math predicts the rotor: the anchor's v-halo ring is radially confining and azimuthally flat — a self-assembled circular rail. Put a motile M (τ₁ past drift) on that rail around an anchored S (τ₂=2.5):

Rotor certification panel. ω(τ₁) dial curve (0.0035→0.0259 rad/tu, monotone), the 10k-tu longrun (17.4 revolutions, separation drift ±0.004px), and the kick-angle basin (±20° kicks converge to the same |ω| within 0.03%).

The first cross-species bond. The stamp-math prediction (v-halo negative ring, well at r≈8) vs measurement: d*(η=0.05)=7.976 two-sided exact; and the same ring acting as the rotor's self-assembled circular rail.

M7 — spontaneous rotation. No kick, no noise needed: the static heterodimer is linearly unstable to rotation — round-off alone picks CW or CCW (3 noise seeds: −,−,+). Right: the orbiter's track in the anchor frame — a circle at r=8.44 (the static bond dilates from 7.6 under rotation; the traveling bond of M4 contracts — opposite signs, both measured). ω is dialed by τ₁: 10 locked points, 0.0035→0.0259 rad/tu over τ₁∈[5.52, 6.10], monotone; 17.4 revolutions in the 10k-tu longrun with sep drift ±0.004 px; survives noise σ=0.04; grid-refinement 0.009%. Audit: fresh seed ω within +0.5%, onset-region point on-curve. This is the rotation-as-attractor the program demanded: radius pinned by the bond well, speed by the propulsion attractor — the only soft mode is orbit sense.

Three structural findings around the rotor: (1) a rotor-only zone — it spins at τ₁=5.52, below even the pair-travel threshold (5.636), extending the composite hierarchy: single-static < pair-travel < rotor-spin — each bound structure unlocks motion its parts cannot afford; (2) rotation stabilizes the bond: the static η=0.1 dimer is metastable (balloons at ~1700 tu) while the rotating one is immortal (10k tu) — motion as a stabilizer, the program's recurring theme inverted; (3) mechanism certified by decomposition: with η only in the M→S direction the rotor still spins (S is a passive pivot); with η only S→M it is static. Honest negatives: same-species 3-rotors exist but are knife-edged (±20° kicks fall back to translation — the frustration hypothesis confirmed as a basin property); the engineered ring-valley orbit lost to the bond (at machine-safe ε the cross-bond outcompetes the background ring — the "planet" prefers its tether).

The A=τ·D_v family map (parameters)

Phase-2 dials (adds to the section-7 table)