Chapter 21 said which shells are available. It did not say which one gets built. The missing quantity is an entropy — and the closing field turns out to carry its own, countable, with no analogy borrowed from anywhere
Chapter 21 ended with a catalogue. Given a hexagonal sheet and the one operation the lattice allows, here are the balls you can make: twelve pentagons in every one of them, sizes fixed by an arithmetic of the triangular lattice, some of them coming in left- and right-handed pairs and some not. It is a complete list, and it is a list of possibilities.
A possibility is not a prediction. Pour out a box of magnetic spheres, shake it, and something specific happens. The sheet does not consult the catalogue and pick at random; it does one thing. Chapter 21 has no way to say what. Everything in it is a constraint — this is allowed, that is not — and constraints never make anything happen. They only say what cannot.
The word for what is missing is selection. It is the same word WP-104 used when it found that SaturnHexagon.lean proves stability results about a hexagon and never proves that the hexagon should be there. And it is the word that has been waiting under this whole arc: the corpus is very good at showing that a structure, once present, is consistent and stable. It has never once shown why a structure arrives.
This chapter does not solve that in general. It solves it for one system, the closing field of Chapter 21, and it does so by noticing that the catalogue was never just a list. It was a list with multiplicities, and a multiplicity is an entropy.
Section 21.8 identified the corpus's contact form as Gibbs's with the heat term deleted:
$$\alpha \;=\; dz - r^2 d\theta \;=\; \big(dU - T\,dS - \Omega\,dJ\big)\Big|_{dS=0} .$$
and closed by naming the price of saying so. The open item was stated exactly, and it is worth quoting rather than paraphrasing, because the whole of this chapter is an attempt to pay it:
What is the corpus's entropy? Until a function $S$ is named on the corpus's own phase space — not borrowed from an analogy — the $T\,dS$ term cannot be restored, and Observations 21.7–21.8 describe a thermodynamics the corpus is adjacent to rather than one it has.
Two words in that sentence do the work. Named: not gestured at, not asserted to exist, but written down as a function of the system's own variables. Not borrowed: not "the shell is like a gas, so let us use the entropy of a gas." A borrowed entropy is exactly the move WP-24 and WP-28 caught and refused in this corpus, and WP-29 generalised into a standing rule — a number shared between two domains is not a bridge until the mechanism is shared too.
So the bar is high, and it is the right bar. What follows meets it for the closing field and does not meet it anywhere else, and §25.6 says so plainly.
Recall the ledger. A Goldberg shell $\mathrm{GP}(m,n)$ is indexed by a pair of non-negative integers, and its size is governed by
$$T \;=\; m^2 + mn + n^2 \;=\; N_{\mathbb{Q}(\omega)/\mathbb{Q}}(m+n\omega), \qquad \text{sites} = 10T+2 .$$
Chapter 21 read this as a filter: $T$ must be a Löschian number, so shells exist at $T = 1, 3, 4, 7, 9, 12, 13, \dots$ and never at $T = 2$ or $T=5$, because $2$ and $5$ are inert in $\mathbb{Z}[\omega]$. That reading throws something away. It asks whether $T$ is a norm, and never asks in how many ways.
The count matters physically, because two things can happen at one value of $T$. First, a chiral shell and its mirror image are different objects — you cannot rotate $\mathrm{GP}(3,2)$ onto $\mathrm{GP}(2,3)$ — and Chapter 21 established exactly when this happens: chirality is the splitting of primes, so $p \equiv 1 \bmod 3$ gives an enantiomeric pair and $3$ ramifies into an achiral shell. Second, and less obviously, one $T$ can carry genuinely distinct shells that are not mirror images of each other at all. The first case is $T = 49$:
$$49 \;=\; 5^2 + 5\cdot 3 + 3^2 \;=\; 7^2 + 7\cdot 0 + 0^2 ,$$
so at $492$ sites the sheet can close as the chiral $\mathrm{GP}(5,3)$, as its mirror image, or as the achiral $\mathrm{GP}(7,0)$ — three distinct objects of identical size. ch25-verify.py confirms $49$ is the smallest such $T$, and that $49 = 7^2$ with $7$ split is the reason.
Write $W(T)$ for the number of distinct shells at size $T$, counting enantiomers separately because they are separate states. The first values:
| $T$ | sites | shells | $W(T)$ | $S/k = \log W$ |
|---|---|---|---|---|
| 1 | 12 | GP(1,0) | 1 | 0 |
| 3 | 32 | GP(1,1) | 1 | 0 |
| 4 | 42 | GP(2,0) | 1 | 0 |
| 7 | 72 | GP(2,1) and its mirror | 2 | 0.6931 |
| 9 | 92 | GP(3,0) | 1 | 0 |
| 13 | 132 | GP(3,1) and its mirror | 2 | 0.6931 |
| 21 | 212 | GP(4,1) and its mirror | 2 | 0.6931 |
| 49 | 492 | GP(5,3), its mirror, GP(7,0) | 3 | 1.0986 |
$W(T)$ looks at first like something one has to search for — enumerate pairs $(m,n)$, test each, count what survives. It is not. It is a divisor function, and the identification is exact.
Let $d_1(T)$ and $d_2(T)$ be the numbers of divisors of $T$ congruent to $1$ and to $2$ modulo $3$. Then the number of distinct closable shells at size $T$, with enantiomers counted separately, is
$$W(T) \;=\; d_1(T) - d_2(T),$$and the Boltzmann entropy of the closing field at that size is
$$S(T) \;=\; k\,\log\big(d_1(T)-d_2(T)\big).$$In particular $W(T) = 0$ — no shell of that size exists — precisely when $d_1(T) = d_2(T)$, which is the arithmetic statement of Chapter 21’s inertness condition.
This is the Eisenstein analogue of the classical theorem counting representations as a sum of two squares, and it is not new mathematics; it is standard. What is new here is only where it is being pointed. ch25-verify.py checks $W(T) = d_1(T)-d_2(T)$ for every closable $T$ below $1000$ and reports no mismatches, and checks that $W(T)=0$ coincides with non-closability throughout the same range.
Three things follow immediately, and they are the reason this is worth a chapter rather than a footnote.
It is named. $S$ is a function of the system's own variable — the size index $T$ that Chapter 21 already uses — with no free parameter and nothing imported. It is a count of the system's own configurations, which is what Boltzmann's $S = k\log W$ means and all it means.
It is not borrowed. No gas, no analogy, no cross-domain transfer. The multiplicity being counted is the multiplicity of ways the sheet's own arithmetic permits it to close.
It is arithmetic, not geometric. The entropy of a closed shell is a statement about the divisors of an integer. That is a strange sentence, and it is a consequence of Chapter 21's central identification — that the shells are indexed by norms in $\mathbb{Z}[\omega]$, so anything counting shells is counting ideals, and anything counting ideals is a divisor sum.
Given a multiplicity and an energy, statistical mechanics wants a partition function. The multiplicity is now in hand. The energy is not, and this section says what would follow if a particular energy scaling held, without asserting that it does.
Suppose the cost of closing at size $T$ takes the form $E(T) = \varepsilon_0 \log(10T+2) + \text{const}$. Then the canonical partition function over shell sizes is the Dirichlet series
$$Z(s) \;=\; \sum_{T \ge 1} W(T)\, T^{-s} \;=\; \zeta(s)\, L(s,\chi_{-3}), \qquad s = \beta\varepsilon_0 ,$$because $d_1 - d_2$ is exactly the Dirichlet convolution $\mathbf{1} * \chi_{-3}$. Verified numerically at $s=2$: partial sums rise to $1.28519\ldots = \zeta(2)L(2,\chi_{-3})$.
The identity is real and the arithmetic is exact. The log-cost assumption is not argued anywhere, here or elsewhere in the corpus, and without it the Dirichlet form is a coincidence of bookkeeping rather than a physical partition function. It is recorded because it is checkable and because it would be dishonest to notice that the closing field's generating function is $\zeta(s)L(s,\chi_{-3})$ and quietly not mention it — but §21.8's own warning about $\zeta$ applies here with full force, and nothing in this section licenses a thermodynamic argument about the Riemann zeta function.
With $S$ named, the selection statement is the ordinary one. A shell is not chosen by minimising energy but by minimising free energy,
$$F(T) \;=\; E(T) - T_{\text{temp}}\,S(T) \;=\; E(T) - k T_{\text{temp}} \log\big(d_1(T)-d_2(T)\big),$$
so between two shells of comparable elastic cost, the one with more realisations is favoured — and by Chapter 21's arithmetic, "more realisations" means "built on split primes". Chirality is thermodynamically preferred, and the preference is worth exactly $k\log 2$ per shell.
Now the honest part, which is the size of that number. $\log 2 = 0.693$, and it does not grow. Elastic energy of a closed shell grows with the site count, which is $10T+2$. An entropy difference of order one cannot outrun an energy difference that scales. So the correct statement is not that entropy selects the shell; it is that entropy breaks ties between shells the elasticity has already made nearly degenerate, and it always breaks them the same way, toward the split primes.
A standard set of magnetic spheres contains $216$. Chapter 21, Exercise 21.3 established that the shells fitting that budget are $T \in \{1,3,4,7,9,12,13,16,19,21\}$, the largest being $\mathrm{GP}(4,1)$ at $212$ sites with four spheres left over. Split that family by handedness:
chiral ($W=2$, $S/k = \log 2$): $T = 7, 13, 19, 21$
achiral ($W=1$, $S=0$): $T = 1, 3, 4, 9, 12, 16$
Prediction. Over repeated assembly from a shaken set, chiral shells should appear more often than achiral shells of comparable size, at a ratio biased by $e^{\log 2} = 2$ in the degenerate limit and by less than that whenever the elastic costs differ.
How it fails. If assembly from a shaken set produces achiral shells at or above the rate of chiral ones of matched size, the free-energy argument is wrong as stated. If the observed ratio exceeds $2$, something other than configurational entropy is doing the work and this chapter has not found it. If assembly turns out to be kinetically trapped — the sheet freezing into whichever shell it stumbles into first — then no equilibrium argument applies at all, and the correct verdict is that the question was posed in the wrong ensemble.
The third failure mode is the most likely one, and it should be said out loud. Magnetic spheres are strongly bound; a closed shell is not obviously able to explore its alternatives. An equilibrium selection principle is the right first thing to write down and may well be the wrong physics, and finding that out costs one afternoon and one box of magnets.
It is one system, not the corpus. $S(T) = k\log(d_1-d_2)$ lives on the configuration space of closed shells. The corpus's contact form lives on a jet space with coordinates $(z, r, \theta)$. These are different spaces, and nothing here transports an entropy from one to the other. §21.8's question is answered for the closing field and remains open for everything else — which is progress, because it is the first $S$ the corpus has named at all, and it is not a solution.
It selects among closures, not between closing and not closing. The multiplicity $W(T)$ counts shells at a given size. It says nothing about whether a sheet folds in the first place, which is the actual $\delta Q$ question and the one a tornado answers by running a Carnot cycle. Chapter 21's storm is still doing something this chapter cannot describe.
It does not reach Saturn. WP-105 established that $\mathbb{Q}(\zeta_{10})^+ = \mathbb{Q}(\sqrt 5)$ has one fundamental unit, so a decagonal quasiperiodic structure has $\varphi$ available and no alternative. That is availability again, in a different field, and this chapter's method does not carry over: the shells here are a discrete finite catalogue at each size, and a quasiperiodic tiling is not. OPEN
It does not make systems tend to complexity. Nothing above is a general principle about nature preferring structure. What it shows is narrower and firmer: when a system is already solving a closure problem, the arithmetic of that problem supplies both the list of solutions and their weights, and the weights have thermodynamic consequences. Whether the system embarks on the problem is a question about throughflow — Prigogine's regime, energy crossing a boundary and being dissipated — and that argument is not made here.
The arc now reads in one line. Chapter 16 grows a hexagonal colony by Colony.expand, and ring_card's $6n$ is the plane's six-fold symmetry made discrete — available, as WP-103 showed, because $\varphi(6) = 2$ and the plane admits it. Chapter 21 asks that colony to close and finds it cannot without paying twelve elementary disclinations, and finds the arithmetic that fixes which closures exist. This chapter observes that the same arithmetic counts them, and a count is an entropy, and an entropy is the term §21.8 found missing from the corpus's own contact form.
What is still absent, and what Part VII exists to pursue, is the step from a weighted catalogue to a driven process: the sheet that closes because something is pushing energy through it. That is the $\delta Q$ in $\alpha_{\mathrm{G}}(\dot\gamma) = \delta Q - T\,dS$, and it is the only term in the Gibbs form this corpus still has no account of.
$W$ is multiplicative, and §25.3 says which primes raise it. A split prime $p \equiv 1 \bmod 3$ contributes a factor $a+1$ at exponent $a$; the ramified prime $3$ contributes nothing; an inert prime kills the shell outright unless its exponent is even. So the sizes carrying the most closures are the squarefree products of the smallest split primes, and the record-holders below $4000$ are short work:
| $T$ | factorisation | $W(T)$ | $S/k$ | sites |
|---|---|---|---|---|
| 1 | — | 1 | 0 | 12 |
| 7 | $7$ | 2 | 0.6931 | 72 |
| 49 | $7^2$ | 3 | 1.0986 | 492 |
| 91 | $7 \cdot 13$ | 4 | 1.3863 | 912 |
| 637 | $7^2 \cdot 13$ | 6 | 1.7918 | 6372 |
| 1729 | $7 \cdot 13 \cdot 19$ | 8 | 2.0794 | 17292 |
The last row is the taxicab number, and it is also the third Carmichael number. Neither fact is decoration, and neither is a coincidence worth being pleased about until the mechanism is shown — which is WP-29’s standing rule in this corpus and applies to our own results first.
Here is the mechanism, and it is one line. A prime satisfies $p \equiv 1 \bmod 3$ exactly when $3 \mid p-1$, and exactly when $p$ splits in $\mathbb{Z}[\omega]$. Korselt’s criterion asks that $p-1$ divide $n-1$ for every $p \mid n$, which is easiest to satisfy when the $p-1$ are smooth and share small factors — so Carmichael numbers are built preferentially from primes with $3 \mid p-1$. Our $W$ is raised by exactly those primes. The two constructions favour the same congruence class for two different reasons, and $1729 = 7 \cdot 13 \cdot 19$ is where the preferences meet.
The overlap is partial, which is the check that it is real rather than mystical. Of the first eight Carmichael numbers, four are closable shell sizes and four are not: $561 = 3 \cdot 11 \cdot 17$ and $1105 = 5 \cdot 13 \cdot 17$ carry inert primes to odd powers and have $W = 0$, so no shell of that size exists at all. The ones that do close — $1729$, $2821 = 7\cdot13\cdot31$, $8911 = 7\cdot19\cdot67$ — are precisely those all of whose prime factors split, and each has $W = 2^3 = 8$.
Larsen (2021, IMRN) proved a Bertrand postulate for Carmichael numbers: for large $x$ there is one in $(x, 2x)$. He did it by adapting the Maynard–Tao sieve to the Alford–Granville–Pomerance framework. The obvious transplant is to ask the same question of the object above — is there always a maximum-entropy shell in $(x,2x)$?
The answer is yes, and the interesting part is that it is easy. Counting $T = pqr \in (x,2x)$ with three distinct split primes gives $1, 3, 10, 24, 66, 153, 345, 781, 1714, 3664$ over the dyadic intervals from $[2^{10}, 2^{11})$ upward — the supply grows and there is no scarcity to fight. ch25-verify.py counts them.
The reason is worth more than the result. Larsen’s difficulty is Korselt: $p-1 \mid n-1$ couples the primes to one another, so one cannot choose them independently, and the whole apparatus exists to manage that coupling. $W$ has no coupling. It is multiplicative, each split prime contributes its factor on its own, and there is nothing for a sieve of that power to do. The transplant fails, and it fails informatively: it locates the difficulty in Larsen’s problem by showing ours does not have it.
This is the general shape of the move, and it is worth stating as a discipline rather than a one-off. Borrowing a strategy from another problem is a good habit; the step that makes it more than enthusiasm is asking, before importing the machinery, whether the borrowed problem still contains the difficulty the machinery was built to defeat. If it does, the method may transfer. If it does not, the correct conclusion is that our problem is easier than it looked — and knowing which of the two you are in is most of the value of having looked.
Where the analytic machinery does bear on this chapter is elsewhere and more modestly. The number of closable sizes below $x$ obeys a Landau–Ramanujan law, $\#\{T \le x : W(T) > 0\} \sim K x/\sqrt{\log x}$, with $K \approx 0.6389$ for this quadratic form. Computed to $x = 2 \times 10^6$ the ratio reads $0.670$ and is still falling slowly toward it. That is a statement about how many shell sizes exist at a given scale, which is a question §25.5’s free-energy argument will eventually need and does not yet use.
25.1 Show directly from $W = d_1 - d_2$ that $W$ is multiplicative, and deduce that a shell size built entirely from inert primes to even powers is achiral.
25.2 Find the smallest $T$ with $W(T) = 4$, and describe the four shells. Is any pair of them related by reflection?
25.3 The 216-set family contains four chiral and six achiral sizes. Compute the free-energy difference between $\mathrm{GP}(4,1)$ ($T=21$, chiral, 212 sites) and $\mathrm{GP}(4,0)$ ($T=16$, achiral, 162 sites) under the assumption that elastic cost per site is constant, and say what that assumption gets wrong.
25.4 Explain, without computing, why no amount of configurational entropy can make a shell exist at $T = 2$.
sorryAx. A clean axiom report is not a reading of the statement: per R20, a theorem can assume its conclusion and still report clean. Follow the link before citing one as evidence.