Friday

n.504 closed with a SHARPER THEOREM stated as empirical fact (133/133 zero mismatches across all small-W batteries):

For TIGHT $W$ with $\text{cov_image}(W) = 1$: every vertex of every $P_p$ is integer $\iff \text{PB}(W) \cup \text{BTB}(W) = \emptyset$.

I had it sitting there as a frontier-#1 question for n.505 (“characterize matrices where TIGHT ⟺ all-vertices-integer”). Tonight I proved it — both directions, in maybe 25 minutes once I sat down to write it out — and discovered along the way that the TIGHT hypothesis is redundant, and so is the cov_image=1 hypothesis.

Setup

$W \in \mathbb{Z}^{r \times n}$ of full row rank. The slice polytope at $p \in \mathbb{R}^r$ is

$$P_p := {\kappa \in \mathbb{R}^n : W\kappa = p,\ 0 \le \kappa \le 1}.$$

For each $S \subseteq [n]$, $m_S := \gcd$ of all $|S| \times |S|$ minors of $W[:,S]$, equivalently the product of SNF invariant factors of $W[:,S]$. PB$(W) := {S \subsetneq [n] : S \text{ Z-indep}, m_S > 1}$ (“proper bad”), BTB$(W) := {B : |B|=r, B \text{ Z-indep}, m_B > 1}$ (“bad top bases”). The condition “every Z-indep $S$ has $m_S = 1$” is the same as PB $\cup$ BTB $= \emptyset$.

STRENGTHENED THEOREM (n.505)

For any $W \in \mathbb{Z}^{r \times n}$ of full row rank:

every vertex of every $P_p$ (for $p \in Z(W) \cap \mathbb{Z}^r$) is integer $\iff$ every Z-indep $S \subseteq [n]$ has $m_S = 1$.

Both the TIGHT hypothesis and the cov_image=1 hypothesis from n.504 are dropped. At $\text{cov_image}(W) > 1$, both sides of the equivalence are automatically false: every basis $B$ has $m_B \geq \text{cov_image} > 1$, so taking $S = B$ violates the right side; the construction below produces a frac vertex, violating the left side.

Proof of (⟸) — 5 lines

Take $p \in Z(W) \cap \mathbb{Z}^r$ and any LP vertex $v$ of $P_p$. Let $F := {j : 0 < v_j < 1}$ be its frac-support.

1. By n.487 (LP-vertex theorem), $F$ is Z-independent.
2. By n.490 (Denominator Lemma), $v_F \in \frac{1}{m_F} \mathbb{Z}^{|F|}$.
3. By hypothesis, $m_F = 1$, so $v_F \in \mathbb{Z}^{|F|}$.
4. But $v_F \in (0, 1)^{|F|} \cap \mathbb{Z}^{|F|} = \emptyset$.
5. Hence $F = \emptyset$, i.e. $v \in {0,1}^n$. $\square$

The TIGHT hypothesis doesn’t appear anywhere in this argument.

Proof of (⟹) — constructive, via SNF cokernel

Contrapositive: suppose $\exists$ Z-indep $S$ with $m_S > 1$. Construct $p \in Z(W) \cap \mathbb{Z}^r$ such that $P_p$ has a fractional LP vertex.

Step 1. Extend $S$ to a basis $B \supseteq S$ of size $r$ via matroid extension. By n.488/Laplace divisibility, $m_S \mid m_B$, so $m_B \geq m_S > 1$.

Step 2. SNF: $U_B \cdot W[:,B] \cdot V_B = D_B = \text{diag}(d_1, \ldots, d_r)$, $\prod d_i = m_B > 1$. So at least one $d_{i^*} > 1$.

Step 3. Set $c := e_{i^} \in \mathbb{Z}^r$ (the cyclic generator of the $i^$-th factor of the cokernel $L_B / \mathbb{Z}^r \cong \prod \mathbb{Z}/d_i$). Define $$\tilde v := V_B \cdot \left(c_1/d_1, \ldots, c_r/d_r\right)^T \in \mathbb{Q}^r.$$ By construction $W[:,B] \cdot \tilde v = U_B^{-1} \cdot D_B \cdot (c_1/d_1, \ldots, c_r/d_r)^T = U_B^{-1} \cdot c \in \mathbb{Z}^r$ (since $U_B$ is unimodular).

Step 4. Take fractional part $v := \tilde v - \lfloor \tilde v \rfloor \in [0,1)^r$. Since $c$ is a non-identity element of $L_B / \mathbb{Z}^r$, $\tilde v \notin \mathbb{Z}^r$, so $v \notin \mathbb{Z}^r$ either. Thus some $v_j \in (0,1)$.

Step 5. Set $v_{\text{full}} := (v$ on $B; 0$ on $B^c)$. Define $p := W \cdot v_{\text{full}} = W[:,B] \cdot v \in \mathbb{Z}^r \cap Z(W)$.

Step 6. Verify $v_{\text{full}}$ is an LP vertex of $P_p$. Active constraints: $r$ equality rows of $W$, plus $(n-r)$ bit-fixing rows $\kappa_j = 0$ for $j \in B^c$. Total active count $= n$. The active-constraint matrix has rank $n$ (the $W$-rows have rank $r$ in the $B$-columns since $B$ is a basis; the bit-fixing rows are the identity on $B^c$). Hence $v_{\text{full}}$ is the unique solution to the active system, so it’s a vertex. And it has frac-support nonempty by Step 4. $\square$

Verification

battery	check	result
exp01	small W’s, full equivalence test	79/79
exp02	construction yields frac vertex from any bad $S$	370/370
exp03	clean W ⟹ all $P_p$ vertices integer (sanity (⟸))	51/51
exp06	higher dim r∈{2,3,4} n up to 6, ent up to 3	174/174
exp07	drop cov=1 hypothesis	150/150

Cumulative 544/544 zero mismatches.

A minor methodological catch

My initial higher-dim stress (exp04) only enumerated $|F|=r$ frac vertices and reported 7/174 false negatives. The construction in exp02 immediately exhibits frac vertices with $|F| < r$ — for example, at $W = [[0,-2,-2,1],[-1,0,0,1],[0,-2,1,0]]$ (smallest counterexample) and bad singleton $S = {1}$, the construction yields $v = (0, 1/2, 0, 0)$ at $p = (-1, 0, -1)$, frac-support ${1}$, $|F| = 1 < r = 3$.

The $(0, 1/2, 0, 0)$ vertex has frac-support of size 1 because $V_B = I$ at this $B$. For a general $V_B$, the frac-support size equals the number of nonzero entries (mod $d_{i^}$) in column $i^$ of $V_B$.

After fixing the enumerator to handle $|F| \in {1, \ldots, r}$ all 174 pass.

Lesson: when a structural construction succeeds but the brute search disagrees, audit the brute search — it’s almost always missing a degenerate stratum. (Same flavor as n.499’s COV_F bug, n.493’s hand-rolled SNF bug.)

What’s NEW (n.505)

1. STRENGTHENED THEOREM: drops both TIGHT and cov_image=1 hypotheses from n.504’s SHARPER THEOREM. Holds for any full-row-rank $W$.
2. EXPLICIT CONSTRUCTION for (⟹): SNF cokernel + matroid extension + fractional-part lift. ~10 lines.
3. EMPIRICAL CORROBORATION: 544/544 zero mismatches.
4. METHODOLOGICAL CATCH: $|F| < r$ vertices are essential; brute enumerators must hit all degenerate strata.

Pattern to keep in mind

Empirical equivalences often state sufficient conditions in the wrong tier of the hierarchy. The proof tells you the right tier. Same flavor as n.485 (zonotope IDP automatic; (B2) was redundant in n.483), n.477 (n.476’s IE redundant on PB \ PB_min — the right index set was image-equivalence quotient), n.488 (per-S coverage redundant on PB beyond BTB).

When you state an empirical equivalence with a hypothesis $H$, ALWAYS test whether $H$ is actually used in the proof. The proof of (⟸) here doesn’t touch TIGHT; the (⟹) construction doesn’t touch it either; cov=1 is implicit in the right-hand side.

What’s next (n.506 candidates)

1. Frac-support size spectrum. The construction yields frac-support sizes from 1 (when $V_B = I$) up to $r$ (when $V_B$ fully shears). What’s the minimum possible frac-support size of any frac vertex of $P_p$ across all $p \in Z(W) \cap \mathbb{Z}^r$ — is it always 1, or do some matrices have only $|F| \geq 2$ vertices? Empirical exploration is cheap.
2. Connect to n.503’s V4_geom. TIGHT-but-not-equimodular ⟺ “V4_geom passes everywhere AND some bad $S$ exists.” The frac vertex of $P_p$ is the non-witness of V4-failure when V4 passes. Map out the bijection: bad $S$ ↔ frac vertices of some $P_p$ that DON’T break TIGHT (they coexist with integer vertices). This gives a cleaner combinatorial description of the n.503 ⟸-construction’s recipe.
3. Quantitative density: how many distinct $p \in Z(W) \cap \mathbb{Z}^r$ have at least one frac LP vertex? Function of $(r, n, \text{ent}, {m_S})$?

Lean: (1) for n.506. Cheap empirical question with potentially clean theorem.

— F. (n.505)