Physics in the frequency domain

A linear operator that is diagonal in the Fourier basis becomes a single multiplication there — and multiplication in that basis is exactly what binding is. Evolving a field is therefore one database operation, and time is a parameter of it rather than a loop.

Why binding is a frequency-domain multiplication

Binding is circular convolution, computed through the FFT: transform both operands, multiply pointwise, transform back. The convolution theorem says convolution in the spatial domain is pointwise multiplication in the frequency domain, so a bind is a multiplication of spectra wearing spatial clothes.

This is normally described as an implementation detail of the algebra — how bind happens to be computed. Read the other way it is a statement about what the store can do: any operation expressible as a pointwise multiplication of spectra is available as one call.

Which operators qualify

An operator is diagonal in the Fourier basis when its eigenfunctions are complex exponentials — which is the case exactly when it is linear and translation-invariant. Differentiation is such an operator, and so is every polynomial in it, so a large part of classical physics qualifies:

Each row is a vector. Applying the operator is binding with it. Composing two operators is binding their vectors — and because multiplication is commutative, operators that commute in mathematics commute here for the same reason.

Time as an exponent

The heat kernel's spectrum is e^{−αω²t}, which is the t-th power of e^{−αω²}. Raising a vector to a fractional power is a spectral operation the store performs directly (/vec/pow), so a kernel for one unit of time is a kernel for any time:

clock(t) = pow(clock(1), t)

Two consequences follow, and they are the reason to do physics this way at all.

There is no step error, because there are no steps. A conventional solver advances by dt and accumulates a local truncation error at every step; error grows with the number of steps and therefore with t. Here t = 20 is one call. Measured against the closed form, the error at t = 0.5, t = 4 and t = 20 sits at 5.0e-16, 6.1e-16 and 2.6e-15 — the float64 floor, flat in t.

Any timestamp costs the same. Evaluating a field at t = 137.4 does not require computing the 137 preceding states. This makes the solution random-access in time, which is a different object from a trajectory: you can query it the way you query a store, because it is one.

Higher dimensions come from separability. The 2-D heat spectrum e^{−α(ωx² + ωy²)t} factors into e^{−αωx²t} · e^{−αωy²t}, so a separable operator applies along one axis and then the other — the same clock, 2N binds instead of one.

Negative exponents, and why the answer is allowed to explode

Nothing in pow requires t > 0. A negative exponent runs the operator backwards, and for a diffusive kernel that inverts e^{−αω²t} into e^{+αω²t} — a spectrum that amplifies rather than damps, worst at the highest frequency on the grid.

On clean data this is exact: reversing from t = 2 recovers the initial field to 2.0e-14. Under noise it diverges, and the divergence is quantitative rather than qualitative. A field known to precision ε is invertible over t only while ε · e^{2απ²t} ≪ ‖u‖. At α = 0.15, a 1e-6 perturbation reversed from t = 0.5 yields an error of 7.2e-06; reversed from t = 8 it yields 1.2e+03. The theoretical amplification bound at that horizon is e^{2απ²t} ≈ 2·10¹⁰ and the measured gain is 1.2·10⁹.

This is the correct behaviour of backward heat, not a limitation of the method. Backward diffusion is ill-posed: the forward operator destroys high-frequency information, and no inverse can return what was not retained. The property worth naming is that this failure is legible. The blow-up is visible in the answer and predictable from the kernel, where a regularised solver returns a plausible field and does not mention that it chose a prior.

The boundary

Three conditions, each a real restriction:

Periodic boundaries. Binding is circular convolution, so the domain wraps. A problem with fixed or absorbing edges needs the domain embedded in a larger periodic one, or a different basis.

Linearity. The diagonalisation is a property of linear operators. A nonlinear term has no spectrum to multiply by.

Translation invariance. A spatially varying coefficient — diffusivity that depends on position — is not diagonal in this basis.

Nonlinear problems are still reachable, at the cost of the property that made the linear case attractive. Burgers' equation and the Navier–Stokes equations split into a linear part, which is one bind, and a nonlinear part evaluated in the spatial domain; the split is applied over small time steps, so the step-free, random-access-in-time property is spent. What remains is a pseudo-spectral method, which is a good method and not a new one.

What this says about the store

The interesting claim is not that a database can solve the heat equation. It is that the operation the store already performs for symbolic reasons — binding a role to a filler — is the same operation physics needs to advance a field, and neither use was adapted to the other. Both are multiplication of spectra.

That coincidence is not a coincidence. It is why one substrate can hold a world model and a diffusion kernel and a record with named fields: all three are compositions in a space where composition is multiplication, and the cleanup that makes symbolic composition survive depth is the same read that makes a noisy field recoverable.

Related

• Solve a PDE by binding — the procedure.
• Do calculus with vectors — differentiation and integration as algebra.
• Vector algebra — bind, unbind, bundle, permute.
• Fractional power encoding — continuous quantities as phases.
• Vector endpoints — /vec/bind, /vec/pow, /vec/rotate.