RADIATIVEMASTER EQUATION

The Supertransfer Kernel

One Lindblad cell, from two sodium atoms to a black-hole Dyson sphere.

A Dyson sphere is a superabsorber. Every rung of the energy ladder is the same cell: a donor population that collectively absorbs, supertransfers the excitation, pushes it through a symmetry-breaking ratchet, and re-emits directed, band-shifted output. The kernel computes the dimensionless physics; only the rescaling (λ, Γ, N, P) changes per rung. This page runs that kernel live — the 3.4 kB WebAssembly module is embedded below.

1 · The physics, in five claims

Superradiance turns on at λ/2π, and it is a crossover, not a transition

Two emitters share a decay channel at rate Γ₁₂ = Γ·sinc(kr). Full Dicke cooperation needs kr ≲ 1, i.e. r ≲ λ/2π — the same reduced wavelength that divides near-field from radiation zone (Γ₁₂ and the coherent dipole shift are the imaginary and real parts of one Green's function). It degrades smoothly: at r = λ/4 you still keep sinc ≈ 0.64. For finite N there is no nonanalyticity — the sharp version is an emergent many-body effect of ordered arrays, absent at N = 2.

Supertransfer moves energy as n·m, not n+m

Donor group (n atoms) pumps acceptor group (m atoms) through the shared field. The symmetric-ladder matrix element carries √n·√m amplitude, so the rate scales as n·m. For the minimal 2+2 cell that is a ×4 early-time enhancement — measured 3.93 in the kernel.

Superabsorption is time-reversed superradiance — so it needs a ratchet

A good absorber is an equally good emitter; detailed balance refuses to give you net storage. You must break the symmetry: dump the acceptor's top state into a shelf faster than it can re-radiate. In sodium this shelf is pair pooling — |3p,3p⟩ → |3s,4d⟩ (Δ = +602 cm^-1, dipole–dipole, enormous at sub-λ/2π spacing) — which decouples from the 589 nm field and cascades to the 819 nm output (3d→3p), the band GaAs actually wants.

There is no temperature anywhere

The cell's state is the density matrix ρ — populations and coherences. The kernel has no variable that could hold a temperature; offense by equilibrium-smuggling is structurally impossible. This is the discipline that makes the $/W and Dyson-efficiency arguments correct: route energy through narrowband light with chemical potential μ→hν, never through a thermal reservoir. Brightness is not Planck-bounded once you are on the coherent branch.

The cell is scale-free

The dimensionless kernel K(s, κ) → (transfer efficiency, collective enhancement) is identical at every rung. A Dyson sphere is N copies of the lab cell. That is the whole 60-orders-of-magnitude talk in one sentence.

2 · The kernel

Exact Lindblad master equation, integrated by RK4. Hilbert space D = 12: donor Dicke ladder (3: |gg⟩,|sym⟩,|ee⟩) ⊗ acceptor ladder + pooling shelf (4: |gg⟩,|sym⟩,|ee⟩,|S⟩).

Jump operators

channel	rate
symmetric mode (A+B)/√2	Γ(1+s)
antisym mode (A−B)/√2	Γ(1−s)
pooling ratchet \|S⟩⟨ee\|	κ

s = sinc(kd) is the inter-group collective coupling; Ω = −½Γcos(kd)/(kd) the coherent exchange. Γ ≡ 1.

Exports (C ABI → wasm)

st_init(s, omega, kappa)
st_step(n_steps, dt)
st_shelf()        -> pooled-pair yield
st_photon_rate()  -> collective 589 rate

Pure Zig, no std import. Same source compiles native (validator) and wasm32-freestanding (this page).

3 · Run it

collective coupling s = sinc(kd) 0.990 ratchet κ/Γ 0.40 coherent exchange Ω/Γ 0.000

pooled-pair yield

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collective rate (t=0)

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supertransfer ×

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4 · Validation

The wasm reproduces the Python expm reference to four decimals (computed live on this page at load — see the banner). Horizon for convergence: T = 500/Γ, ~100k RK4 steps.

regime	s	Ω	κ	kernel yield	python ref

free-space kd=1 → n·m early-rate enhancement = 4 (measured 3.93)

pooled-pair yield: 6% free-space → 22% shared-mode

optimal ratchet κ* ≈ 0.4–1.6 Γ

5 · The 60-orders-of-magnitude ladder

The same cell, rescaled. Kardashev K = (log₁₀P_W − 6)/10.

rung	mass (kg)	power (W)	K	kernel role
1 Na atom	3.8e−26	1e−11	−1.7	N=1 baseline, σ=3λ²/2π
1000 Na atoms	3.8e−23	1e−5	−1.1	Dicke burst, rate∝N²
lab Lightcell / HPS	~0.5	1e3	−0.3	Holstein trap, pooling, 819 nm
jet Lightcell	~2e2	1e6	0.0	combustion pump, coherent branch
power plant	~5e5	1e8	0.2	grid cell array, $/W
superheavy / orbital	~5e6	1e11	0.5	beamed coherent output
Moon-ISRU swarm + shade	~1e11	2e14	0.83	superabsorber array, K≈civ
Dyson sphere (Sun)	~3.3e23	3.8e26	2.0	full-N capture, K-II
blue/red giant sphere	~3e31	8e30	2.4	geometric μ factor
neutron-star sphere	~2.8e30	1e31	2.5	spin-down, relativistic
SMBH sphere	~2e38	1e40	3.4	accretion + Blandford–Znajek, K-III+

Mass span: 3.8e−26 → 2e38 kg ≈ 64 orders of magnitude. Power span: 1e−11 → 1e40 W ≈ 51 orders. Ordering follows ambition, not strict mass (a neutron star is lighter than a blue-supergiant system but harvested later).

6 · Build

cd radiative/supertransfer-kernel
# native validator (reproduces all anchors):
zig build-exe main.zig -O ReleaseFast -femit-bin=validate && ./validate
# browser kernel (3.4 kB):
zig build-exe kernel.zig -target wasm32-freestanding \
  -O ReleaseSmall -fno-entry -rdynamic -femit-bin=supertransfer.wasm

Zig 0.16: std.time.Timer was removed — do not use it.

7 · Open threads

Rung 1 (WebGPU): M-box bucket brigade as 2nd-order cumulants {⟨σ^z_i⟩, ⟨σ⁺_iσ⁻_j⟩}, O(M²) ODEs, one WGSL invocation per (i,j) pair. Must regress to these rung-0 numbers at M=2. Truncating the ⟨σ⁺σ⁻⟩ matrix is equilibrium-smuggling — keep it.
Gain/cm on 4s→3p at burner density — the single number that decides product-vs-poster for the coherent (Lightsaber) branch.
Na 4p branching ratios (4s vs 3d vs 330 nm UV) from NIST — decides whether 819 nm or the coherent IR lines win the cascade flux.
Rung 2 (Holstein voxel grid): spectrum-vs-device-size, the white→reddish curve as a falsifiable prediction.

Kernel: kernel.zig · validator: main.zig · module: supertransfer.wasm (embedded in this file).
Companion docs: supertransfer-kernel.md · ../master-eqn-handoff/supertransfer-handoff.md · ../README.md
Radiative / Lightcell — Manifest "Dyson sphere across 60 orders of magnitude".