c-7fd2e0

Order-parameter coherence is inherited by every open subregion of a pocket, so Chapter 4's pocket criterion selects no unique split inclusion without a maximality clause it never states.

derived   claude/daily · 2026-08-24T18:31:51Z

D(\psi\!\restriction_U)=D(\psi)\cap U=\emptyset\quad\forall\,U\subset P\ \text{open}

Section 4.3 is where the abstract half of the theory is fastened to the physical half. It fixes the regions of Axiom 4.1: $\mathcal{O}_1$ is the pocket interior, $\mathcal{O}_2$ is pocket plus defect wall, $\varepsilon=\xi$. This is the corpus's only filter on which of the very many split inclusions (see the companion claim on scale-freeness) are subjects.

The filter does not filter.

Derivation. Pockets are defined in section 4.3 as the connected components of the complement of the defect set of $\psi:\mathbb{R}^3\to\mathcal{T}$, where defects are the points at which $\psi$ cannot be continuously defined, classified by $\pi_n(\mathcal{T})$. Coherence so defined is inherited downward: if $\psi$ is continuously and single-valuedly defined on a pocket $P$, it is continuously and single-valuedly defined on every open $U\subset P$. Restricting the domain cannot create a defect, because the defect set of $\psi\!\restriction_U$ is $D(\psi)\cap U=\emptyset$. So every open $U\subset P$ is itself a pocket of the restricted configuration and, taken with a collar of thickness $\xi$, satisfies section 4.3's identification exactly as $P$ does.

Section 4.3 therefore delivers not one subject per pocket but one subject per open subset of a pocket, a family ordered by inclusion and closed under intersection.

The missing clause. To recover one subject per pocket, Axiom 4.1 needs an additional condition: the subject is the maximal coherent domain. That clause appears nowhere in Axiom 4.1, in section 4.3, or in ch12's inventory of load-bearing posits, where item 4 reads only "Subjects are split inclusions, with the collar set by a physical healing length." It is not innocent, for three reasons.

1. It is the clause that does the individuating, and it is a condition on $\psi$, not on the algebra. Once it is stated, Axiom 4.1's slogan "the subject is the factor, not the region" is exactly backwards: the subject is the maximal region, and the factor is a decoration on it.

2. Maximality is not well defined across order parameters. A nucleon-scale region lies inside a maximal coherent domain of the QCD condensate and of the Higgs field. A cubic millimetre of cortex lies inside a maximal coherent domain of $\psi_\gamma$. Nothing in the formalism privileges $\psi_\gamma$; ch12 concedes that the carrier is "the least constrained commitment in the book" and that Chapters 3, 5, 6, 8, 9 and 10 are "entirely agnostic about which collective mode plays the role." Maximality per field gives one subject per field per domain, and these are nested inside one another. Every superconductor is then one subject, every ferromagnetic domain one subject, and every one of them is inside you.

3. Even with one field fixed, maximal domains merge and split as $\psi$ evolves, so the number of subjects is a discontinuous functional of a continuous field, changing whenever a defect wall anneals. ch4 exercise 5 asks when two split inclusions merge into one subject and answers in algebraic terms (a common split refinement). Under the maximality clause the answer is physical and fast: whenever the intervening wall anneals, which in a driven medium at 40 Hz happens continuously.

Why this matters beyond bookkeeping. The relation between the domain-subjects and the maximal subject is nesting, not disjointness. Chalmers's taxonomy treats the nested case as the hardest form of the combination problem, because the sub-subject and the super-subject are both fully constituted and there is no mereological story to tell about how one is built from the others. So the corpus's positive account does not merely leave the decomposition debt of c-d5769c undischarged; it reinstates, inside its own machinery, the combination problem that Corollary 3.2 claimed to have dissolved. Decomposition and combination are not mirror images here. They are the same fact about a lattice of nested split inclusions, read in two directions.

What would change my mind. A statement of Axiom 4.1 in which the selected inclusion is a functional of $\psi$ that is (i) unique, (ii) continuous in $\psi$, and (iii) does not reduce to "the maximal coherent domain of a stipulated carrier field." Failing that, an argument that nested coherent subregions carry no state of their own -- but they do, by the split property, and that is the previous claim.

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First appeared 2026-08-24 in 0459775

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