The replay invariant
The reason one walk can answer every acquisition is a single physical fact about how the walk is built.
Walker positions r(t) and boundary events depend only on the geometry, the intrinsic diffusivity, the permeability, and the seed — not on the gradient waveform, the RF pulses, T₂, T₁, ρ, or the susceptibility orientation.
Those latter quantities only gate the phase and the per-walker weight. So they split cleanly into two groups: what the walk fixes, and what it leaves as a replay knob.
| Fixed at walk time (the substrate) | A replay knob (the acquisition) |
|---|---|
| geometry (the mesh) | gradient waveform G(t) — PGSE, OGSE, STE, arbitrary |
| intrinsic diffusivity | real RF / B₁ pulses (refocusing, saturation) |
| permeability (membrane crossing) | B₀ and susceptibility χ, any orientation |
| the random seed | T₂, T₁, surface relaxivity ρ, MT |
walk(geometry, intrinsic-D, permeability, seed) → r(t), boundary contact # once, expensive
replay(r(t), G(t), RF, B₀, χ, T₂, T₁, ρ, …) → signal # many times, cheap
Why intrinsic D and permeability are fixed
They shape where the walkers go — the diffusion distance per step, and whether a walker crosses a membrane. Change them and you have a different walk. Everything that only affects phase (gradients, RF, fields, relaxation) can be applied afterwards, so those are free to sweep at replay.
Two consequences that make the Commons possible
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Additive over time
A Monte-Carlo walk is Markov, so it is additive over time: a walk to 40 ms is the exact leading prefix of a walk to 100 ms. You can resume an existing pack and extend it, rather than re-walking from zero.
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Additive over walkers
Independent walks of the same substrate (disjoint seeds) are additive over walkers: pool them and the noise floor falls as 1/√ΣN. This is what turns a substrate into a shared resource.
Why it's lossless — and fast
Replay isn't a re-simulation. The pack stores a compressed basis of the walk, and a decode contracts against that basis directly — without ever rebuilding the original trajectories, and faster than if it had to. Fidelity is a build-time certificate: within the declared envelope the decode reproduces the walk's signal to the Monte-Carlo floor, and any lossy channel must clear that floor before it may ship.