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What it means to solve a substrate

"Solved" is a promise with a number behind it. A substrate is solved when its replay is converged to a known precision, over a walk long enough that any realistic acquisition fits inside it — and then it is frozen, so no one ever has to solve it again.

Two conditions, both measurable: hit the target noise floor, and reach the target duration. Clear both, and the substrate is frozen, given a DOI, and done — forever.

1 · A target precision, σ*

A Monte-Carlo signal carries a noise floor that falls as 1/√N with the number of walkers N. We fix a target σ* — the precision at which the replay is converged enough that its residual noise sits below anything a downstream fit would care about (a fraction of a percent). A substrate isn't solved until its measured floor — a split-half estimate on the pooled ensemble, not an optimistic guess — is at or below σ*.

How many walkers that takes is not a fixed number — it depends on the substrate:

  • Small & simple

    A compact geometry with light restriction converges to σ* with relatively few walkers.

  • Large & complex

    A big, heterogeneous substrate — strong restriction, crossings, a heavy-tailed susceptibility field — needs far more walkers to reach the same σ*.

That's precisely why solving is a commons: you can't know the walker budget up front, so contributors keep adding shards until the measured floor crosses σ*. Each shard only lowers it; the substrate tells you when it's done.

2 · A target duration, 200 ms

Precision alone isn't enough if the walk is too short. A stored walk to time T answers every acquisition whose timings fit inside T — and only those. So every substrate is solved out to a standard 200 ms — long enough that even long-Δ, long-TE protocols fit within the pack.

Extending, not re-walking

Because a walk is additive over time, a substrate already solved to (say) 100 ms can be resumed and carried to 200 ms without re-walking from zero — the first 100 ms is the exact prefix of the longer walk.

Solved forever

  • Converged

    Measured floor ≤ σ* — the signal is precise to a stated tolerance, not "looks about right."

  • Complete

    Walked to 200 ms — any realistic acquisition fits inside the stored walk.

  • Permanent

    Frozen, hashed, and DOI'd — the answer never has to be recomputed by anyone, ever.

This is the point of the whole exercise. Simulating a realistic substrate is expensive and, done privately, thrown away after one paper. A solved substrate in the Commons is the opposite: converged to a stated precision, complete to 200 ms, frozen with a permanent identifier, and credited to the people who built and converged it. Anyone, ever after, decodes any acquisition from it — at analytical-model speed, without paying the Monte-Carlo cost again.

The promise

Solve it once — to σ*, and to 200 ms — and it stays solved. That is what forever means here: a permanent, citable, reusable answer, not a result that evaporates when the cluster job ends.