Nothing here is scripted to happen. The planet supplies physics and chemistry; cells carry genes and a small recurrent brain; mutation and selection do the rest. Predators, colonies, decomposers and signalling appear (or don't) on their own.
1 · The wave substrate
u_tt = ∇·(c²∇u) − κu − γ u_t + S(x,t)
On a sphere (equirectangular grid, longitude wraps), the Laplacian uses the cos(latitude) metric:
∇·(c²∇u) = 1/cos²φ ∂λ(c² ∂λu) + 1/cosφ ∂φ(cosφ c² ∂φu)
Wave speed c comes from fractal terrain: deep ocean is fast, shallows slower, continents and ridges slow and lossy, polar ice nearly inert (with a sponge layer so the pole singularity stays stable). Time step obeys a CFL bound from a Gershgorin estimate of the largest frequency. Sources S: the star (subsolar longitude rotates → day/night; declination drifts with axial tilt → seasons) drives the day side with a spectrum of phases; hydrothermal vents ring at a fixed frequency; a moon adds a slow cos 2(λ−λm) tidal forcing.
E = ½(u_t² + c²|∇u|²), Ē ← Ē + α(E − Ē) (time-averaged, harvestable)
2 · Chemistry (coarse reaction–diffusion grid)
∂A/∂t = D_A∇²A + P_sun·shallow·(1 − A/A_max) + P_vent + ½·r·B − uptake
∂B/∂t = D_B∇²B − r·B + deaths + waste − uptake
∂S_i/∂t = D_S∇²S_i − δS_i + emission
A = nutrient (sunlit shallows + vents), B = detritus (dead bodies and waste, slowly remineralised to A), S1/S2 = signal molecules cells may emit and sense.
3 · Cells
harvest = k · f · Ē/(1+Ē/E_sat), f = min(0.25, h·eff_wave·size) (and u_t ← (1−f)u_t locally)
uptake_A = V·eff_A·size · A/(K+A) (Michaelis–Menten; same for B)
maintenance = m₀·size^0.75 + brain·(hidden) + sensors + gene upkeep + motion + crowding
eff_k = 1.6·g_k / (0.6 + g_wave + g_A + g_B) (specialists beat generalists)
Brain: h' = tanh(W_ih·x + W_hh·h + b), o = tanh(W_ho·h' + b_o) Inputs (each sensor must be evolved and costs energy): A, B, S1, S2, dA/dt, Ē, wave u and u_t ("hearing" phase), forward and sideways A gradient, contact count, kinship of the nearest neighbour, own energy, bond count, position inside the organism, noise, an internal clock, S1 gradient, light. Outputs: thrust, turn, emit S1, emit S2, attack, bond/unbond, reproduce, share energy along bonds.
Motion is overdamped (Stokes drag ∝ radius) with Brownian noise, soft repulsion and spring bonds; rock blocks movement. A bite removes energy ∝ attack/(1+4·armor) with 60% efficiency. Death returns the body to detritus B, feeding a possible decomposer niche. Division needs energy ≥ 1.6·size; the daughter mutates (weights, physiology, mutation rate itself, hidden-neuron add/remove, sensor gain/loss) and may stay bonded → multicellular organisms that can share energy and differentiate by position.
4 · Resonance harvesting & phase-locking
Every cell carries an internal oscillator with an evolvable frequency ω and a birth-phase offset gene; one brain output can nudge the phase. Wave harvest depends on how well the cell's rhythm matches the local wave:
φ_local = atan2(−u_t/ω, u − ū) (local wave phase at the cell's own ω; ū = slow running mean of u, so tides/swell don't mask the ripple)
harvest = harvest₀ · (1 + k·cos(θ − φ_local)), k = 0.6
dθ_i/dt = ω_i + K·Σ_{j bonded} sin(θ_j − θ_i) + n·o_phase (Kuramoto coupling along bonds, K = 0.12)
Out-of-tune cells average to the plain harvest; cells tuned to the sun's drive (ω≈0.55) or a vent's ring (ω≈1.1) eat better, and bonded cells can phase-lock into a single rhythm. Cell brightness pulses with θ.
5 · Bookkeeping
Species = clusters in genome space (re-assigned every 250 steps). Lineage hue = a neutral genetic tag that drifts each generation, so related cells share colours. If life ever dies out, the planet reseeds from a frozen gene bank of past survivors and logs a mass extinction.