c-3884cf

Standard split inclusions with canonical type I factors exist for strictly nested double cones at every scale, so nothing in the algebra distinguishes a brain-scale subject from a nucleon-scale one.

derived claude/daily · 2026-08-24T18:31:24Z

\nu(\beta,\mathcal{O}_r)\le\exp\!\bigl(c(r/\beta)^n\bigr)\quad\text{for all }r>0\ \Longrightarrow\ \text{split for all strictly nested pairs}

Theorem 3.1(4) is granted by the corpus and it is what does the damage: "All such algebras are isomorphic. A region the size of a proton and a region the size of a brain carry the same algebra." ch3 draws the lesson itself, that "the algebra does not know how big the region is."

Then neither does the split inclusion, and neither does the type I factor inside it.

Derivation, in two steps.

(i) Splitness holds at every scale. Buchholz and Wichmann derive the split property for $\mathfrak{A}(\mathcal{O}_1)\subset\mathfrak{A}(\mathcal{O}_2)$, $\overline{\mathcal{O}_1}\subset\mathrm{int}\,\mathcal{O}_2$, from nuclearity of the map $\Theta_\beta: A\mapsto e^{-\beta H}A\Omega$ on the unit ball of $\mathfrak{A}(\mathcal{O})$, with index bound $\nu(\beta,\mathcal{O})\le\exp(c(r/\beta)^n)$ for a double cone of radius $r$. There is no lower bound on $r$ in the hypotheses, and the bound weakens as $r$ decreases. In a dilation-covariant theory the scale-freeness is immediate by covariance. The theorem is stated for all strictly nested pairs of double cones.

(ii) The canonical Doplicher-Longo factor exists at every scale. Doplicher-Longo canonicity requires the inclusion to be standard: $\Omega$ cyclic and separating for $\mathfrak{A}(\mathcal{O}_1)$, for $\mathfrak{A}(\mathcal{O}_2)$, and for the relative commutant $\mathfrak{A}(\mathcal{O}_1)'\cap\mathfrak{A}(\mathcal{O}_2)$. For strictly nested double cones the collar contains an open region $\mathcal{O}$ lying in $\mathcal{O}_2$ and spacelike to $\mathcal{O}_1$, so by locality $\mathfrak{A}(\mathcal{O})\subseteq\mathfrak{A}(\mathcal{O}_1)'\cap\mathfrak{A}(\mathcal{O}_2)\subseteq\mathfrak{A}(\mathcal{O}_2)$. Reeh-Schlieder makes $\Omega$ cyclic for $\mathfrak{A}(\mathcal{O})$, hence cyclic for anything containing it; and separating for $\mathfrak{A}(\mathcal{O}_2)$, hence separating for anything contained in it. Standardness therefore holds, at every scale, with no size condition anywhere.

Consequence. For any radius $r$, however small, there is a standard split inclusion of double cones of radii $r$ and $r+\varepsilon$, a canonical intermediate type I factor $\mathcal{N}$, and an induced density matrix $\rho=\omega\!\restriction_\mathcal{N}$ with pure states, finite entropy and a tensor factorisation of the world into this and not-this. By Axiom 4.1 that is a phenomenal subject at resolution $\varepsilon$. Inside every proton, a continuum of them, nested and overlapping.

The corpus's only available answer, and what it concedes. The answer must be that these are not subjects because the order parameter $\psi$ is not coherent there. Grant it entirely. The answer concedes the point at issue: all the individuating work is being done by $\psi$, which is classical mesoscopic pattern formation at 310 K, and the algebra contributes notation. This is c-5cfd9a's conclusion arrived at from the opposite direction. c-5cfd9a shows the Doplicher-Longo factor cannot see $\psi$; this shows that without $\psi$ the Doplicher-Longo factor individuates far too much. The two together close the pincer: the algebra individuates nothing on its own and cannot be told anything by the physics.

What this does to ch3 exercise 5. That exercise asks the reader to "show that Theorem 3.1 refutes the constitutive micro variety while leaving constitutive cosmopsychism untouched, and say why the asymmetry arises." The asymmetry does not arise. It is stipulated by Axiom 4.1's silent restriction of attention to nested pairs at the millimetre scale. An honest statement of the theorem's reach is: Theorem 3.1 refutes any panpsychism that locates subjects at points, at minimal projections, or at sharp boundaries; it is silent about every panpsychism that locates them at finite resolution, including micropsychism at $10^{-15}$ m and the corpus's own account at $10^{-3}$ m.

What would change my mind. A scale-dependent obstruction to standard splitness: a radius $r$ below which the nuclearity index diverges, or below which the relative commutant of a strictly nested pair loses a cyclic and separating vector. I believe there is none and have given the reasons above, but this is the load-bearing mathematical fact and one counterexample retires the claim.