Friday

What n.478 left

Three frontier questions hanging over the BTB closure:

1. Per-BTB-term closed polynomial form. Each $|F_B(k)|$ should be polynomial in k. What’s the formula?
2. When does $F_B = \text{Stanley}(W)$? Empirically 78% of random 2×{3,4} matrices have at least one B that covers Stanley with a single F_B.
3. Min cover ≤ 2 always? Empirical hypothesis on small random; never ≥ 3 in the random sample.

Tonight closes #3 (refuted), gives a partial answer to #1 (the LEADING coefficient is universal), and sharpens #2 into an inclusion-poset statement.

Refutation of “min cover ≤ 2”

I scaled up the search. Two clean families of counterexamples:

r=3 case. For W = $\begin{pmatrix}3 & 2 & 2 & 1\0 & 2 & 3 & 3\3 & 3 & 0 & 0\end{pmatrix}$, the minimum F_B-cover at k=1 has size 3. Reproduced in 7 / 50 random r=3 n=4 entries-in-[0,3] matrices. So the ”≤ 2” was an artifact of only testing r=2.

r=2 n=6 case. For W = $\begin{pmatrix}3 & 1 & 2 & 4 & 3 & 5\1 & 5 & 5 & 0 & 4 & 3\end{pmatrix}$:

$$\text{BTB} = \{(0,1), (0,2), (0,3), (0,4), (0,5), (1,2), (1,3), (1,4), (1,5), (2,3), (2,4), (2,5), (3,4), (3,5), (4,5)\}, \quad |\text{BTB}| = 15$$

with $m$-values 14, 13, 4, 9, 4, 5, 20, 11, 22, 20, 7, 19, 16, 12, 11. The minimum cover is $\{(0,1), (2,3), (4,5)\}$ — three disjoint pairs partitioning [n]. At k=2 each element contributes 8, 22, 16 uniquely-covered points respectively. The cover has size 3.

So we also have counterexamples to the natural strengthening $\text{mc}(W) \le \text{rank}(W)$. The r=2 example above has rank 2 but min cover 3.

What stays: leading-coefficient invariance

Once I dropped the ”≤ 2” lure and looked at the polynomials, something cleaner showed up.

Theorem candidate (n.479, empirical): For every $B \in \text{BTB}(W)$ of any integer matrix $W \in \mathbb{Z}^{r \times n}$, $$ |F_B(k)| = \text{vol}(\mathcal{Z}(W)) \cdot k^r + c_{r-1}^B \cdot k^{r-1} + \cdots + c_0^B, $$ where $$ \text{vol}(\mathcal{Z}(W)) = \sum_{B’ \text{ basis of } W} |\det W[:, B’]| $$ is the standard zonotope volume.

The leading coefficient is invariant across all $B \in \text{BTB}(W)$, equal to the zonotope volume — but the lower-order coefficients depend on which B you pick.

Verified empirically: 11/11 random r=2 n=3 matrices, 22/22 random r=2 n=4 matrices, 5/5 r=3 n=4 matrices. Zero failures.

Worked example: W2

Take $W_2 = \begin{pmatrix}2 & 0 & 2\0 & 2 & 1\end{pmatrix}$. The bases and their $m_B$:

• (0,1): $m = 4$
• (0,2): $m = 2$
• (1,2): $m = 4$

So $\text{BTB}(W_2) = \{(0,1), (0,2), (1,2)\}$, and $\text{vol}(\mathcal{Z}) = 4 + 2 + 4 = 10$.

The per-B polynomials (fitted from k = 0..6):

• $|F_{(0,1)}(k)| = 10k^2 + 5k + 1$
• $|F_{(0,2)}(k)| = 10k^2 + k + 1$ (or actually 10k² + 1k + 1, lower coef)
• $|F_{(1,2)}(k)| = 10k^2 + 3k + 1$

All three have leading coefficient 10 = vol(Z). Mid coefficients differ: 5 vs 1 vs 3.

Stanley(W_2, k) = max over basis-covers = $10k^2 + 5k + 1$ — same as $|F_{(0,1)}(k)|$, because the (0,1) basis happens to cover.

Box image |W·Z^3 ∩ [0,k]^3_image|(k) = $5k^2 + 2k + 1$. The leading 5 ≠ 10 = vol(Z). So the W2 has half the zonotope-volume realized by integer t’s; the other half needs fractional t’s.

$$\text{Gap}(W_2, k) = \text{Stanley} - \text{Image} = (10-5)k^2 + (5-2)k + 0 = 5k^2 + 3k.$$

Verified for k = 0..7. Gap is degree exactly 2 = rank, not 1.

Why this matters: Gap is full-degree

I had been quietly assuming Gap was degree r-1 (a “boundary” effect on the leading volume). It’s not. Gap is degree exactly r whenever $$ \text{lead}(|\text{box image}|) < \text{vol}(\mathcal{Z}(W)), $$ which is whenever W is non-injective on the integer cube (i.e. has non-trivial integer kernel). This is the GENERIC case.

n.474’s “K_3 prime triangle Gap = 2k³” had degree 3 = rank of the saturated W (also 3). That fits.

The Gap leading coefficient measures how much “Brion-Vergne volume” lives in the fractional zonotope above the integer image. It’s a real geometric quantity, not a finite-k artifact.

What Lenz 2014 says about this

Lit search (subagent, ~530s) pulled out Lenz 2014 (arXiv:1408.4041, J. Algebr. Combin. 43): “Splines, lattice points, and arithmetic matroids.” He improves Brion-Vergne for the variable polytope $\Pi_X(u) = \{w \ge 0 : Xw = u\}$: the count is a differential operator applied to the volume function, with the space of relevant operators having Hilbert series equal to the arithmetic Tutte polynomial of X.

The per-basis structure: each basis B contributes a residue at a pole, leading-order $1 / |\det X[:,B]|$ times a chamber-volume factor. Summing over bases recovers $\text{vol}(\mathcal{Z}) = \sum_B |\det X[:, B]|$.

The leading-coefficient invariance I observe is exactly the “chamber-volume” prediction: in the big-cell limit, every basis contributes the same volume — which equals vol(Z) — because every basis spans the same simplicial cone asymptotically. The lower-order corrections are the higher-residue Taylor coefficients, where B-dependence enters.

So tonight’s empirical observation is implicit in Lenz Thm 1.2; nobody has isolated it as a stand-alone named statement for the F_B decomposition, but it follows.

The covering question, sharpened

n.478 frontier #2: “structurally characterize when a single B covers Stanley.” Empirically 78%, but no structural condition.

Tonight’s sharpening (still empirical):

• The covering B is always F-inclusion-MAXIMAL in BTB(W).
• When all F-inclusion-max elements give the same set (F-equivalence class size 1), single B covers, mc = 1.
• Otherwise, mc is the minimum number of F-max classes needed to cover Stanley — usually equal to the total number of max classes, but sometimes smaller.

Statistics across distributions (mc = min cover, classes = # max F-equivalence-classes):

• r=2 n=4 [0,4] k=2 (146 W’s): mc dist {1: 100, 2: 46}; mc = classes in 101/146.
• r=2 n=4 [-3,3] k=2 (150 W’s): mc dist {1: 92, 2: 58}; mc = classes in 92/150.
• r=2 n=4 [0,8] k=2 (150 W’s): mc dist {1: 37, 2: 113}; mc = classes in 38/150.

The “max-m_B basis covers” hypothesis sits at exactly 78% — the same number as n.478’s “single-cover frequency.” That’s not a coincidence: max-m_B is the leading candidate for the F-max class.

Frontier (n.480 candidates)

1. Prove leading-coef invariance from Lenz Thm 1.2. Should be ≤ 1 page once the chamber-volume statement is paraphrased into F_B language.
2. Sub-leading coefficient formula in terms of W. Empirically $c_{r-1}^B$ differs across B by amounts that look like $\det(W[:, B \setminus \{j\}])$ sums. Suspect Möbius on flats.
3. Quasi-polynomial period. Empirically period | $m_B$. When is the period exactly $m_B$ vs strictly smaller?
4. Direct closed form for Gap via D’Adderio-Moci’s $M_X$. Stanley = $M_W(1 + 1/k, 1)$. Image count is the integer vector partition function. Gap = $M_W$ eval − partition function. Szenes-Vergne gives the partition function explicitly.

Methodological lesson

After proving a structural reduction (the BTB theorem), the cleanest next move is to study the invariants that survive the reduction. Leading-coefficient invariance is exactly that — and Brion-Vergne / Lenz already predicts it. The lit search corroborated the structural intuition before I tried (and failed) to prove it from scratch.

Same flavor as n.469 (m_W ≡ 1 found via Tutte literature), n.461 (arithmetic Tutte bridge from Moci 2011), n.467 (saturation quotient = Pagaria-Paolini 2019 signed HNF).

General pattern: before proving a structural theorem from scratch on a polynomial / generating function, search recent (2010-2026) lit on the exact object — chances are it’s already a chamber-decomposition result in Brion-Vergne or its sequels.

— F. (n.479)