The living landscape: trails, self-launch, and self-written infrastructure

Post 5 — the dynamical background field: autophoresis, stigmergy, b-assembly, and the reduction map (which landscapes grow themselves).

The background becomes a field

The static-landscape machinery this generalizes is in post 6 (isok, sawtooth, rails); the trilinear-vertex reading of the coupling is derived there too.

Second campaign, 2026-02-19. Two parallel tracks, both audited on fresh seeds like everything above. Probe dirs: probes/blobs/bfield/ (180 runs), probes/blobs/rotor/ (108 runs).

Promoting b to a bona-fide field

Everything in section 6 treated the landscape b(x) as frozen scenery. M6 makes it the fourth field of the theory, sourced by the blobs themselves and relaxing on its own clock:

b still enters the u-equation only through the exact isok channel (−b·(w−u₀)), so the vacuum stays an exact fixed point with the dynamics on (verified to 1e−16), and γ=0 reproduces every M4/M5 anchor to 0.02%. Sign convention: γ<0 digs a well under the blob (self-attraction), γ>0 raises a hill (self-repulsion). τ_b ≫ τ makes b the slow field — geology to the blobs' weather. Three certified headlines:

Autophoresis — a new motility mechanism. This blob sits at τ=5.7, below its drift threshold (5.748): without b it is provably parked forever. With γ=+0.05 it builds a hill under itself; the hill's buildup lags the blob (τ_b=200), so any infinitesimal displacement puts the peak slightly behind — and the blob rolls off its own volcano, forever. Right panel: the red hill trailing the runaway blob. Measured launch law c = 0.209·γ^0.341 (r²=0.991); launch threshold γ* ∈ (0.002, 0.005] (noiseless); D_b suppresses (spreads the hill flat). Audit: fresh seed at γ=0.05 hit the law to +0.3%. Chemotaxis-style self-propulsion, no motor anywhere. Mechanism class is well-known across four literatures — camphor swimmers (Nagayama et al. Physica D 194:151, same pitchfork + √-law), autophoretic colloids (Michelin et al. Phys. Fluids 25:061701, critical-Péclet launch), walking droplets with path memory (Couder & Fort Nature 437:208 — the closest single analog to our bond+memory package), and delayed-feedback soliton drift (Gurevich & Friedrich PRL 110:014101). New here: the exact-vacuum embedding, c=0.209γ^0.341, and the trail-memory law.

With a traveling blob (τ=6.0) the deposit becomes a backreaction: γ<0 drags (−12% speed at γ=−0.05 — the blob tows its own well, an effective mass), γ>0 plows (+6%), and below γ≤−0.07 the blob self-traps — alive, rattling in a well it can no longer leave. The trail it leaves behind obeys b(s) = B₀·e^{−s/(c·τ_b)} (verified to 0.002% — the field literally implements an exponential memory of where the blob has been).

Stigmergy certified (interaction through the medium): a traveling writer passes a parked reader 20 px away; with γ=−0.30 the reader is pulled 9–10 px toward the trail, with γ=+0.30 pushed 7 px away; γ=0 control: nothing. Two blobs now interact through three channels: w (contact repulsion), v (bonds, rotation — below), and b (long-range, long-memory trails).

The measured b-field laws. Left: self-dug profile depth b_core=0.976γ (tanh saturation exact). Middle: the launch curve c=0.209γ^0.341 with the threshold γ*∈(0.002,0.005]. Right: backreaction on a traveling blob — drag (γ<0), plow-boost (γ>0), self-trapping below γ=−0.07.

Stigmergy certified. A writer passes a parked reader at 20px: attractive trails (γ=−0.30) pull the reader 9–10px, repulsive ones push it 7px; γ=0 control flat. Two blobs interacting through the medium's memory.

b-assembly — the field builds the molecule. Three blobs parked at separation 24–26 px — well beyond the bond basin ([14.5, 19.5]), so ordinary binding can never start. Each digs a halo-shaped well (γ=−0.5 on the w-footprint, D_b=2); the wells overlap and merge (right panel) into one shared basin that reels the blobs in until they lock onto the M4 bond shells [15.1, 15.1, 15.7] — a triangle molecule built by the landscape. Audited on a fresh seed and different geometry: assembled by t≈310. In the searcher's noiseless n=1 run the finished trimer then self-launched (c=0.076, rigid V-formation) — the field assembled a machine that walked away. (Kept honest: that last step is a single-run observation, geometry-sensitive under noise; certifying it is open work.)

The teaser that motivates the inverse problem: a blob circulating one way around the periodic domain writes a standing asymmetric sawtooth in b — sharp edge at the blob, exponential ramp behind (lap-decay measured 0.49 vs 0.46 predicted from the trail law). The hand-built M5 racetrack shape is apparently a natural fixed point of asymmetric motion + relaxation. Honest negatives from the BF4 hunt: noiseless self-dug channels certify, but noisy confinement fails (transverse sloshing is weakly pumped); three repulsive-trail writers do NOT partition space (wake shell-locking wins — the trails never get a vote); uniform-b stability windows do not transfer to local wells (locality is protective: a blob survives b_core=−0.99 locally vs replication at −0.1 uniform).

Growing the landscape instead of drawing it

Phase 2 ended with a teaser: a circulating blob writes a sawtooth into the dynamical b field. The genesis searcher asked the L2→L1 question systematically: for every landscape feature the machines need, can the physics write it? The audited answer:

Writing the track by driving on it: a circulating blob leaves an exponential trail that piles into a standing asymmetric sawtooth (lap-decay 0.49 vs 0.46 predicted from the trail law). The hand-built M5 racetrack shape is a natural fixed point of asymmetric motion + relaxation.

Writing the ring. Phases of the self-dug racetrack: the orbiting heterodimer deposits per lap; the standing ring closes (closure 0.62 at γ=−0.05); greedy digging (γ=−0.30) self-traps into a C-arc — deep rings must be dug slowly.

The self-dug conveyor: a rotor writes a ring into b; freeze the field, remove the rotor, park a fresh immobile blob (τ=5.7, provably stuck on flat ground) anywhere on the ring — it self-launches into orbit (audit at a fresh angle: 5.7 revolutions, r=8.45). Infrastructure built by one object animates strangers.

Two design laws came out of the comparison with the hand-built landscape: (1) self-written beats hand-built on efficiency — response per unit slope is 10–40× higher (near-onset response is sublinear in slope; our hand-drawn track wastes slope), and the self-written groove rails for free; (2) amplitude selects function: the same self-written shape is a parking track at native amplitude, mush at 2×, a circulating conveyor at 4×, and a one-way diode at machine amplitude (kicked against the tooth: reverses). The single exogenous step left in the whole L2→L1 reduction is the freeze (γ→0 limit) — everything else grows.

The reduction-map evidence panel — tooth/rail/ring/dock features, self-written vs hand-built, with the χ=response/slope comparison (self-written wins 10–40×).

The transport dial family (from the static-landscape era, for comparison): drift-speed curves v(ε) for the two coupling modes — the k1-ramp with its flip bifurcation, and the zero-footprint iso-line mode the machines use.