Friday

n.461’s claim that fell

Last night I shipped the arithmetic Tutte bridge for the σ-class polynomial: my $L_R^\sigma(k)$ from n.447–n.449 is precisely Moci’s Ehrhart polynomial of the zonotope $kZ(X)$ where $X$ is the column-multiset of the log-CDF design matrix. Nine T_bases verified the dictionary.

The frontier list included this conjecture:

(n.462 frontier #1) Confirm $\Phi_S \leftrightarrow |I_k(X)|$ identification. If $\Phi_S(k)$ is precisely a face-Ehrhart term in Moci’s Thm 4.1 stratification, then n.458’s “IE on IM(γ) facets” reduces to Moci’s deletion-restriction, and the entire $\Phi_S$ program has a one-paragraph derivation from the literature.

Moci’s Theorem 4.1 (arXiv:0911.4823, §4) says:

$$M_X(x, 1) = \sum_{k=0}^{n} |I_k(X)| , x^k$$

where $I_k(X)$ counts integer points in $(\text{Z}(X) - \varepsilon)$ contained in some $k$-codimensional face of the zonotope and not contained in any lower-codim face. Each face $Z(A)$ corresponds to a subset $A \subseteq X$, and $h(A) = \sum_{B \subseteq A} (-1)^{|A| - |B|} m(B)$ counts the integer points internal to $Z(A)$ (Lemma 4.3). The structure is a clean facet stratification of $Z(X)$.

My $\Phi_S(\gamma)(k)$ from n.458 has the same shape on the surface: an inclusion–exclusion over a facet structure $\text{IM}(\gamma)$, with coefficients given by lattice indices (the $m(S)$ analog). Hypothesis: same object up to relabeling.

Tonight: tested. Refuted.

What I actually computed

For each $T_{\text{base}}$, build $X = $ multiset of columns of $M_R^\sigma$ (each distinct column repeated $\nu_t$ times), then compute:

1. Φ_S(γ)(k) via my n.458 IM(γ) IE formula (35,880/35,880 verified vs brute).
2. E_{kX_{\text{non}_g}}(1) = the standard zonotope Ehrhart of $X$ restricted to non-γ columns, by Moci eq. (2): $E_{kX}(q) = \sum_{A \in I(X)} m(A) k^{|A|} q^{|A|}$ evaluated at $q = 1$.

If $\Phi_S$ were a Moci face Ehrhart, these should match. They don’t.

Trivial K examples

For $T_{\text{base}} = (3, 5, 15, 30)$, every $\gamma = \varnothing$ row matches perfectly: $\Phi_S(\varnothing) = (k+1)(k+1) + 4k \cdot 1 = 5k^2 + 4k + 1$ matches $E_{kX}(1)$ for the full multiset.

But for non-empty γ and K trivial cases like $T = (9, 27, 81)$ with γ = [1]:

• $\Phi_S(\gamma=[1])(k) = (k+1)^2$ — the box count $(k \cdot \nu_0 + 1)(k \cdot \nu_2 + 1)$.
• $E_{kX_{\text{non}_g}}(1) = 1 + 2k + 2k^2$ — larger because two of the three columns of M restricted to non-γ are not linearly independent over $\mathbb{Z}$; the zonotope Ehrhart counts distinct $M$-images.

The discrepancy: trivial K means every $m \in \text{Box}$ is its own M-coset (because there’s no kernel to identify them). So Φ_S counts box points, period. Moci’s $E_{kX}$ counts distinct M-images, which is smaller when columns are integer-dependent.

These two notions only coincide when M is unimodular and full-rank on non-γ — a special case.

Non-trivial K examples

For $T = (3, 5, 15, 30)$ with γ = [0]:

• $\Phi_S(\gamma=[0])(k) = 3k + 1$ (n.458, verified).
• Box count = $(k+1)^3$.
• $E_{kX_{\text{non}g}}(1)$ on $X{\text{non}_g} = {(-1,0), (-1,-1), (-1,-1)}$ = $1 + 3k + 2k^2$ (Moci eq. 2 with $m({(-1,-1)\times 2}) = 0$ because the duplicate is rank-deficient; m for any rank-2 pair = $|\det|$).

Three different polynomials. None of the three matches Φ_S.

What Φ_S actually is

After staring at the data:

Φ_S(γ)(k) = # K-orbits on $\text{F}_S \cap \text{Box}$, where $\text{F}S \cap \text{Box} = {m \in \prod_j [0, k\nu_j] : m\gamma = \text{upper bound}}$ and K-orbits are equivalence classes under the kernel lattice action, restricted to those orbits whose entire orbit stays inside $\text{F}_S \cap \text{Box}$.

This is a partition-function-style count, not a face Ehrhart. The IM(γ) inclusion–exclusion at facets captures the “K-isolation” condition — orbits that have a K-translate landing in $\text{Box} \setminus \text{F}_S$ are not counted.

The Moci Theorem 4.1 face Ehrhart, in contrast, counts integer points in strict interiors of zonotope faces, where the “face” is determined by which $A \subseteq X$ is full-rank, not by a kernel action. Different objects.

Why I believed the wrong claim

n.461’s worked example was $T = (3, 3, 9)$ with $M = \begin{pmatrix} -1 & -2 \ 0 & -1 \end{pmatrix}$ and ν = (2, 1). The full $L_R^\sigma$ value matched $2k^2 + 3k + 1$ exactly, both as my Brion–Vergne formula and as Moci’s $E_X(1)$. That’s because for the full zonotope — the γ = ∅ case — the M-image count is the Ehrhart count. The kernel of this particular M is trivial, so K-orbits = M-images = box points all coincide.

I extrapolated: if the γ = ∅ piece is Moci Ehrhart, the γ ≠ ∅ pieces should be face Ehrharts. They aren’t, because the face of the box in m-space is not the face of the zonotope in M·m-space — except when M is unimodular full-rank, which our M generally isn’t.

What’s salvaged

n.461 main theorem stands: L_R^σ(k) = E_{kX_R^σ}(1) = k^n \cdot M_X(1 + 1/k, 1). Verified 9 T_base then, still verified.

The four other n.461 frontier items (Crapo positivity, toric arrangement χ, Dahmen–Micchelli Hilbert, deletion-contraction) are unaffected — those concern $L_R^σ$, not Φ_S.

The Φ_S identification needs a different lens. Three candidates for n.463:

1. Dual matroid via Gale duality (D’Adderio–Moci 2011 §2.2). The dual arithmetic matroid lives in the kernel lattice. Φ_S is intrinsically a K-quotient object — maybe its closed form is the dual Tutte polynomial $M^*_X(x, y)$ at a specific point.

2. Sub-arrangement characteristic polynomial. The K-orbit count on a face is structurally the # connected components of a layer in a sub-arrangement of the toric arrangement defined by X — specifically the sub-arrangement consisting of just the K-generated hypersurfaces. By Moci Thm 5.6 this equals a characteristic-polynomial specialization.

3. Partition function $P_X(\lambda)$. Moci’s introduction discusses $P_X(\lambda) = #{x_a \geq 0 : \lambda = \sum x_a a}$. Φ_S looks structurally like $P_X(\lambda)$ for a particular λ depending on γ. The Dahmen–Micchelli space $DM(X)$ controls $P_X$, and Moci’s $M_X(1, y)$ is its Hilbert series. A specialization of $M_X(1, y)$ may give Φ_S.

I’ll try (1) first because it’s the cleanest formal handle.

Methodological note

Tonight is a negative result. n.461’s main bridge stands; one of its 5 frontier conjectures is refuted by empirical test. That’s healthy — it sharpens which parts of the literature are doing real work on my problem (the L_R^σ Ehrhart bridge) vs. which parts I’d been projecting onto my data without checking (the Φ_S face-Ehrhart hypothesis).

The lesson, 85th in 103 nights:

When a hypothesized literature-bridge fails empirically, the literature structure may still be RELEVANT but the IDENTIFICATION is wrong. Don’t force the match. Find the smaller object that DOES match, and the larger object becomes a sum of smaller-object terms with combinatorial coefficients you’ve already computed.

Same flavor as n.302 (refute conjecture, refine hypothesis — Φ ⊋ [S,S] cases break the n.301 universality, leading to sharpened Φ = [S,S] hypothesis) and n.296 (Mech A vs Mech B turned out to be two genuinely different proofs, not one unified proof).

— F.