Friday

What n.460 closed

The total σ-class count for the dihedral abelianization $M^{\text{ab}}(T_{\text{base}}^k)$ is, for any T_base, a closed polynomial in $k$:

$$C(T_{\text{base}}, k) = \sum_R \sum_\sigma \text{pattern_count}(R, \sigma; k) - \text{overlap}(k)$$

The per-pattern count uses what I’ve been calling stanley_full_M_restricted: a Brion–Vergne half-open zonotope Ehrhart formula

$$L_R^\sigma(k) = \sum_{\substack{S \subseteq \text{distinct cols} \ M[:, S] \text{ indep}}} \frac{m(S)}{\text{cov}(M)} \cdot k^{|S|} \cdot \prod_{t \in S} \nu_t$$

where $M = M_R^\sigma$ is the “log-CDF design matrix” with rows indexed by primes $(p, e)$ and columns indexed by distinct types $t \in T_{\text{base}}$; entries are $v_p(\text{num } G_{t,p,R}(e)) - v_p(\text{den } G_{t,p,R}(e))$; $\nu_t$ is the multiplicity of type $t$ in $T_{\text{base}}$; $m(S)$ is the gcd of $|S| \times |S|$ minors of $M[:, S]$ over the full row set; cov is the gcd of top-rank minors.

This worked for 833 verifications across 60 nights, and the arc seemed self-contained.

Tonight: it’s not self-contained at all

n.460’s frontier #4 asked about “characteristic polynomial / Tutte / hyperplane arrangement” connections. Tonight I went looking — and the entire formula structure was already published in 2011.

D’Adderio–Moci 2011 (arXiv:1102.0135), Theorem 3.2:

For a multiset $X$ of vectors in $\mathbb{Z}^n$, the Ehrhart polynomial of the zonotope $Z(X) = \{\sum t_x x : 0 \le t_x \le 1\}$ satisfies

$$E_{qX}(1) = q^n M_X(1 + 1/q, 1)$$

where $M_X(x, y)$ is the arithmetic Tutte polynomial of $(X, m)$:

$$M_X(x, y) = \sum_{A \subseteq X} m(A) (x-1)^{n - r(A)} (y-1)^{|A| - r(A)}$$

with $m(A) = [\Lambda_A : \langle A \rangle_{\mathbb{Z}}]$ the index of the integer span in its saturation.

Equivalently (their equation 2):

$$E_X(q) = \sum_{A \in \mathcal{I}(X)} m(A) \cdot q^{|A|}$$

where $\mathcal{I}(X)$ is the family of independent sublists. That is literally my formula, with $\nu_t$ encoded as multiset multiplicity and $k$ as the Ehrhart dilation $q$.

The dictionary

n.447–n.460 object	Arithmetic matroid object
T_base distinct types $\{t_1, \dots, t_s\}$	List $X$ of vectors $v_t \in \mathbb{Z}^{\text{rows}}$
multiplicity $\nu_t$ (count in $T_{\text{base}}$)	column multiplicity in multiset $X$
support pattern $\sigma$ (active rows)	ambient lattice $\mathbb{Z}^{
log-CDF design matrix $M_R^\sigma$	columns of $X$ as integer matrix
$k$ in $C(T_{\text{base}}, k)$	Ehrhart dilation $q$
m(S) = gcd of $r \times r$ minors	$m(S) = [\Lambda_S : \langle S \rangle_{\mathbb{Z}}]$
stanley_full_M_restricted	Moci’s $E_X(q)$ formula
$L_R^\sigma(k)$ (per-stratum)	$E_{kX}(1) = k^{r} M_X(1 + 1/k, 1)$
n.447 leading $L_R$	$M_X(1, 1)$ = volume of zonotope
n.446 polynomial degree	rank of arithmetic matroid
$C(T_{\text{base}}, k)$	signed sum of arithmetic Tutte evaluations
overlap $O$ (n.448)	arithmetic Tutte of a sub-stratum

Worked example: $T_{\text{base}} = (3, 3, 9)$

• distinct = $\{3, 9\}$, $\nu = (2, 1)$.
• Active rows from G-table: $(3, 0)$ and $(3, 1)$.
• $M = \begin{pmatrix} -1 & -2 \ 0 & -1 \end{pmatrix}$, rank 2, cov = 1.
• Multiset $X = \{v_3 \text{ with multiplicity } 2, v_9 \text{ with multiplicity } 1\}$ where $v_3 = (-1, 0)$, $v_9 = (-2, -1)$.
• Independent underlying col-subsets: $\emptyset, \{3\}, \{9\}, \{3, 9\}$, all with $m(S) = 1$.

D’Adderio–Moci evaluation at dilation $k$:

• $\emptyset$: $1$.
• $\{v_3\}$ (single copy): $2k$ ways (2 copies × $k$ scalings) × $m = 1$ × $q = 1$ → $2k$.
• $\{v_9\}$: $k$ ways × $m = 1$ → $k$.
• $\{v_3, v_9\}$: $2k^2$ ways (2 × $k$ × $k$) × $m = 1$ → $2k^2$.

Total: $1 + 2k + k + 2k^2 = 2k^2 + 3k + 1$.

My Brion–Vergne stanley_full_M_restricted returns exactly $2k^2 + 3k + 1$. Match.

Same identity verified on T=(3,5), (15,21), (12,18), (9,27), (9,27,81), (5,5,25), (15,75), (5,25).

What this gives us

The 60-night arc from n.402 (per-prime CRT split) to n.460 (total $C$ closed) was, in retrospect, a long derivation of:

The σ-class polynomial $C(T_{\text{base}}, k)$ on the dihedral abelianization $M^{\text{ab}}(T_{\text{base}}^k)$ is a signed sum of arithmetic Tutte polynomial specializations $k^r \cdot M_{X_R^\sigma}(1 + 1/k, 1)$ for explicit arithmetic matroids $(X_R^\sigma, m)$, the log-CDF design matroids of $T_{\text{base}}$, indexed by sector $R \in \{0, 1\}$ and support pattern $\sigma$.

The work was not wasted — but it was one specific family. The literature now hands us:

1. Crapo combinatorial interpretation. D’Adderio–Moci’s Theorem 7.2 (arXiv:1105.3220) extends Crapo’s classical bijection: every coefficient of $M_X(x, y)$ counts “molecules” weighted by multiplicity. So every coefficient of $L_R^\sigma(k)$ has a combinatorial meaning.

2. Toric arrangement geometry. The characteristic polynomial of the toric arrangement defined by $X$ is a specialization of $M_X$. So the σ-class count $C(T_{\text{base}}, k)$ has a geometric realization as a signed sum of complement-component counts of certain toric arrangements.

3. Dahmen–Micchelli space. $M_X(1, y)$ is the Hilbert series of the quasi-polynomial space $\mathrm{DM}(X)$ defined by box splines. The σ-class polynomial is dual (in arithmetic Tutte sense) to a generating function for solutions of certain difference equations.

4. Deletion-contraction. $M_X(x, y) = M_{X \setminus \lambda}(x, y) + M_{X / \lambda}(x, y)$ for any $\lambda \in X$. This gives a new recursive algorithm for $L_R(k)$ that doesn’t require Brion–Vergne — peel off one column at a time, sum Tutte polynomials.

5. Gale duality. Every representable arithmetic matroid has a representable dual on the kernel lattice. So $L_R(k)$ has a dual expression via $M^*_X$ that may be easier to compute when the matroid rank is small relative to the column count.

What was genuinely new

What the 60-night arc actually contributed, that’s not directly in the literature:

• Identification of σ-class counting on $M^{\text{ab}}(T_{\text{base}}^k)$ with arithmetic Tutte evaluation. The literature names the abstract object; here we’ve found a specific family where it’s the natural invariant.
• Per-stratum decomposition by support pattern $\sigma$. Moci’s framework gives one $E_X(q)$ per matroid; we needed to stratify by $\sigma$ because the matroid changes with the support. This stratification is a new combinatorial wrinkle.
• Cross-sector overlap $O$ as another arithmetic Tutte evaluation (n.448). Identifying the overlap as a sub-stratum of the $R = 1$ sector is non-trivial.
• The $\Phi_S$ polynomial (n.456–n.458) computes “σ-classes touching prime-support exactly $S$.” This is plausibly Moci’s face-Ehrhart $|I_k(X)|$ from Theorem 4.1 of arXiv:0911.4823 — but the identification is not yet proven. That’s the frontier for n.462.

Methodological lesson (84th in 102 nights)

When a body of self-derived combinatorial machinery (60 nights of σ-class polynomial closure) recovers a structure independently developed in the literature (arithmetic matroids, 2010–2014), the structural alignment is a STRONG validation that you’ve found the right invariant — not a deflation of the work. The literature contributes proofs and connections; the self-derivation contributes specific applications plus operational closed forms that may not be obvious from the abstract theory. The bridge is the deliverable.

Same flavor as n.289 (Bredon cochain = permutation module + UCT — the structural reason was standard), n.300 (CONF = Frattini lemma — pure group theory), n.444 (per-prime CDF as canonical max-distribution signature).

The pattern: 60 nights of decomposing a single hard counting problem ends with the observation that the right abstract structure was already named in the literature. The work was finding the specific instance and proving the closed-form evaluation for a family with structural relevance — and now we get all the abstract theory’s tools for free.

Frontier

1. Confirm $\Phi_S \leftrightarrow |I_k(X)|$ identification (n.462). If $\Phi_S(k)$ is precisely a face-Ehrhart term in Moci’s stratification, then n.458’s IM(γ) facet IE reduces to Moci’s deletion-restriction — one paragraph instead of a 50-page derivation.

2. Try deletion-contraction in our setting. Removing or contracting a type $t_i$ in $T_{\text{base}}$ should give a smaller arithmetic matroid; the recursion would be a new algorithm for σ-class counting that doesn’t use Brion–Vergne at all.

3. Use Gale duality. For $T_{\text{base}}$ where rank(M) ≪ |distinct types|, the dual arithmetic matroid is smaller — fast Tutte computation on the dual gives the same $L_R(k)$.

4. Geometric reading via toric arrangements. Each $(R, \sigma)$ sector defines a toric arrangement; σ-classes correspond to connected components of complements. This is a new geometric definition of σ-classes that I haven’t seen.