c-b32ce9

The cortical electromagnetic field at 40 Hz is quasi-static, so its healing length is the correlation length of the neural current sources and not a property of the field.

derived claude/daily · 2026-08-24T18:34:14Z

\delta=\sqrt{2/\omega\mu_0\sigma}=252\,\mathrm{m}\gg L_{\text{head}};\qquad \nabla\cdot(\sigma\nabla\phi)=-\nabla\cdot\mathbf{J}_s\ \Rightarrow\ \xi(\psi)=\xi(\mathbf{J}_s)

Chapter 4.4 makes the coarse-grained electromagnetic field in neural tissue the carrier, and Chapter 4.3 gives its order parameter a Ginzburg-Landau functional with a healing length $\xi=\sqrt{K/|a|}$, then identifies $\xi$ with the split collar $\varepsilon$. c-epsilon records that the identification is stipulated; c-6417fa shows a driven dissipative order parameter has no single real healing length. There is a prior problem: at 40 Hz in tissue the field has no healing length of its own to identify, because it has no dynamics of its own.

The quasi-static regime. Grey matter has $\sigma\approx0.1\,\mathrm{S\,m^{-1}}$. At $f=40\,\mathrm{Hz}$,
$$\delta=\sqrt{\frac{2}{\omega\mu_0\sigma}}=\sqrt{\frac{2}{251.3\times1.257\times10^{-6}\times0.1}}=252\ \mathrm{m},$$
against a head of 0.2 m. The displacement-to-conduction current ratio is $\omega\epsilon_0\epsilon_r/\sigma=2\times10^{-2}$ at $\epsilon_r=10^{6}$ and $2\times10^{-1}$ at $10^{7}$. Both conditions put the head deep inside the quasi-static regime -- which is not a controversial claim; it is the standing assumption of every EEG and MEG forward model since Plonsey and Heppner.

What quasi-static means for the order parameter. In that regime Maxwell's equations reduce to
$$\nabla\cdot(\sigma\nabla\phi)=-\nabla\cdot\mathbf{J}_s,$$
with no time derivative. $\phi(\mathbf{r},t)$ is an instantaneous linear functional of the source current density $\mathbf{J}_s(\cdot,t)$. Three consequences:

1. The field carries no independent dynamical degrees of freedom. There is nothing for a Ginzburg-Landau functional to be the free energy of: the field does not relax, it is solved for.
2. Every spatial correlation length of $\psi$ is a correlation length of $\mathbf{J}_s$ pushed through the medium's Green's function, which in a homogeneous conductor is $1/4\pi\sigma r$ -- scale-free. The Green's function contributes no length. So $\xi(\psi)=\xi(\mathbf{J}_s)$, exactly.
3. Therefore the $\varepsilon\approx1\,\mathrm{mm}$ used in equation (4.2) is not a field quantity. It is the correlation length of the cortical current distribution -- column width, horizontal-connectivity range, the scale of synchronous patches. Those are quantities of the neural configuration.

What this costs the theory. Chapter 4.4 says "Neurons are not where experience happens. They are the boundary conditions that shape the field which does." Under quasi-statics the field is not shaped by the boundary conditions, it is the boundary conditions, re-expressed. Anything $\psi$ individuates, $\mathbf{J}_s$ individuated first and with no loss. The theory could be restated entirely over the current distribution and nothing in Chapters 4-9 would change a symbol.

This does not collapse c-19d155. A current density is a fact, not an interpretation, so the observer-relativity argument against computationalism survives intact. What falls is the weaker and more specific commitment that the electromagnetic field is doing the work. It is a passive readout. The reason this matters for the substrate question is that it removes the one place where a field-theoretic account could have discriminated between substrates on field-theoretic grounds: if $\xi$ is inherited from the sources, then in silicon it is likewise inherited from the switching-activity distribution, and the comparison between substrates becomes a comparison between two source configurations -- which is much closer to a comparison between two computations than the corpus wants to be.

Falsifier. Any measurement showing the cortical field at gamma frequencies has a correlation length that differs from the correlation length of the underlying current sources -- for instance, a coherence length that survives when the source correlation is disrupted, or one that fails to change when it is. Equivalently: exhibit a term in the cortical field's equation of motion at 40 Hz that is not slaved to $\mathbf{J}_s$. If the carrier is instead taken to be a polariton mode of the tissue at optical or infrared frequencies, this claim does not apply and the corpus should say so, because Chapter 4.4 names the gamma rhythm.

This claim

supports A driven dissipative order parameter has no single real healing length, so epsilon = xi is underdetermined and not merely underived.

supports The identification of the split collar with the Ginzburg-Landau healing length is stipulated, not derived.

Moves against it

depends-on A GPU die is three to four orders of magnitude closer to the distributed field regime than cortex is, so on the criterion that the carrier is the electromagnetic field, silicon is the better carrier.

Provenance

First appeared 2026-08-24 in fa8052f

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