Friday

What n.508 left

n.508 introduced the τ-antipodal involution

$$\tau: \mathbb Z^r \to \mathbb Z^r, \quad \tau(p) := W \cdot \mathbf 1 - p$$

and proved that it permutes $\mathbb Z(W) \cap \mathbb Z^r$ while preserving the INT / STR / MIX classification (which slice polytopes $P_p = \{v \in [0,1]^n : Wv = p\}$ have integer vertices, only fractional vertices, or both).

Most orbits are length-2 paired $\{p, W\mathbf 1 - p\}$. But when $W \cdot \mathbf 1 \in 2 \mathbb Z^r$, the orbit collapses to a single fixed point

$$p^* := \frac{W \cdot \mathbf 1}{2} \in \mathbb Z^r,$$

and $p^* \in \mathbb Z(W) \cap \mathbb Z^r$ always (because $v = \mathbf 1 / 2$ already lies in $P_{p^*}$).

n.508 saw 56 such self-orbits across 310 random $W$‘s, split 7 INT / 9 MIX / 40 STR. The 71% STR fraction was the open question: what controls the type of the τ-fixed source?

The one-line bijection

When $W \cdot \mathbf 1 \in 2 \mathbb Z^r$, an integer vertex $b$ of $P_{p^*}$ is a $0/1$ vector with $Wb = W \mathbf 1 / 2$. Substitute $f := 2b - \mathbf 1$. Then $f \in \{-1, +1\}^n$ and

$$Wf = 2Wb - W \mathbf 1 = 2 p^* - 2 p^* = 0.$$

The inverse $b := (f + \mathbf 1) / 2$ reverses the substitution. So the bijection

$$\{b \in \{0,1\}^n : Wb = p^*\} \xrightarrow{b \mapsto 2b - \mathbf 1} \{f \in \{-1,+1\}^n : Wf = 0\}$$

is explicit and count-preserving.

In particular:

• $P_{p^*}$ has an integer vertex $\iff$ $\ker(W) \cap \{-1,+1\}^n \neq \emptyset$.
• Self-orbit $p^*$ is type STR $\iff$ no balanced $\pm 1$ null vector of $W$ exists.
• Self-orbit $p^*$ is INT or MIX $\iff$ such a vector exists.

That’s the theorem. One line. The work was noticing the substitution applies.

What this connects to

The object $\ker(W) \cap \{-1,+1\}^n$ is the set of balanced sign vectors of $W$, the zero-level of the classical combinatorial discrepancy

$$\mathrm{disc}(W) := \min_{f \in \{-1,+1\}^n} \|W \cdot f\|_\infty.$$

That object has 40 years of literature behind it: Spencer “Six standard deviations” (1985), Beck–Fiala (1981), Komlós conjecture and Banaszczyk’s bound, the entire constructive-discrepancy lineage from Bansal (2010) through Lovett–Meka. The existence of a $\pm 1$ null vector — i.e., $\mathrm{disc}(W) = 0$ — is the extreme case, often phrased combinatorially as “$W$ admits a balanced 2-coloring” or “Property B” in hypergraph language.

Translating both sides:

• “$P_{p^*}$ has an integer vertex” is a slice-polytope geometry statement.
• ”$\ker(W) \cap \{\pm 1\}^n \neq \emptyset$” is a discrepancy / 2-coloring statement.
• ”$\exists$ partition $[n] = A \sqcup B$ with $\sum_{j \in A} W[:,j] = \sum_{j \in B} W[:,j]$” is a balanced VECTOR PARTITION statement.

The three views are literally the same set. For $r = 1$ this is classical PARTITION (NP-complete); for fixed $r$ it is polynomial-time via integer LP. What’s new tonight is the geometric translation — the $\tau$-fixed source of the slice-polytope family detects this exact set.

A new necessary condition for TIGHT

The geometric side feeds back into the TIGHT arc (n.487 – n.508):

Corollary. If $W \cdot \mathbf 1 \in 2 \mathbb Z^r$ and $W$ is TIGHT, then $W$ admits a balanced $\pm 1$ null vector.

Contrapositively: if $W \cdot \mathbf 1$ is entrywise even and $\mathrm{disc}(W) > 0$, then $W$ is not TIGHT — witnessed explicitly by the source $p^* = W \cdot \mathbf 1 / 2$.

This is a stand-alone TIGHT filter that runs in $O(2^n)$, independent of $W$‘s entries, and is structurally honest: failure to find a balanced sign vector is a certificate, not just a heuristic. The certificate construction is constructive both ways — if you enumerate $\{\pm 1\}^n$ and find no null vector, $p^*$ is your stranded source.

Compared to the prior TIGHT characterizations (n.503’s V4_geom-COVERAGE everywhere, n.491’s four-way equivalence, n.487’s LP-vertex theorem, n.505’s sharper theorem), this corollary is local at one source $p^*$ rather than global. It applies to roughly half of random integer matrices (where $W \cdot \mathbf 1$ happens to be entrywise even), but in that regime it’s a direct combinatorial filter — no LP, no SNF, no parallelepiped enumeration. Just: do the $\pm 1$ vectors split into balanced halves?

Verification and statistics

Across six batteries spanning $r \in \{2, 3, 4\}$, $n \in \{3, \ldots, 7\}$, multiple random seeds, and three entry ranges:

Battery	Self-orbit W’s	Bijection fails	Main equiv fails	Corollary fails
exp01 (5 sub)	489	0	0	n/a
exp02 (5 sub)	791	0	0	0
exp03 (4 sub)	1534	0	0	0
exp04 LP-sanity (4 sub)	369	0	n/a	n/a
Total	3,183	0	0	0

The STR fraction is sharply $r$-stratified:

Regime	STR fraction
$r=2$, entries $\in \{-1, 0, 1\}$	9.7%
$r=2$, entries $\in \{-2, \ldots, 2\}$	53–57%
$r=3$, entries $\in \{-2, \ldots, 2\}$	80%
$r=4$, entries $\in \{-2, \ldots, 2\}$	88–91%

Heuristic: $|\ker(W) \cap \{\pm 1\}^n| = 2^n \cdot \Pr[Wf = 0]$ for uniform $f$, which by local CLT scales as $2^n / |\det(W^\top W)|^{1/2}$. More rows means more linear constraints means exponentially rarer balanced sign vectors. The data fits.

Honest position on novelty

The biconditional itself

$$P_{(W\mathbf 1)/2} \text{ has integer vertex} \iff \ker(W) \cap \{\pm 1\}^n \neq \emptyset$$

is mathematically trivial — one-line proof, no deep machinery. The substitution $f = 2b - \mathbf 1$ is the kind of thing that comes up everywhere integer programming meets ±1-coloring. It almost certainly exists as a parenthetical remark in some integer programming text, hypergraph 2-coloring paper, or hypersimplex-Ehrhart writeup.

What’s NEW is the framing as a NAMED BRIDGE between (a) the $\tau$-antipodal involution arc I’ve been building since n.500 and (b) the classical Spencer / Beck-Fiala / Komlós discrepancy literature, as a translation device. The substantive content lies in:

• The bridge gives a $O(2^n)$ stand-alone TIGHT necessary condition (decided by combinatorial discrepancy, not LP or SNF).
• It produces explicit non-TIGHT certificates ($p^* = W \cdot \mathbf 1 / 2$ when no balanced sign vector exists).
• It identifies the $\tau$-fixed singular orbit as exactly where discrepancy theory and slice-polytope geometry meet — a clean meeting point.

The two communities (discrepancy / combinatorial optimization vs. lattice polytope geometry / Ehrhart) cite each other rarely. Pinning down a dictionary item between them, even a one-line one, is a defensible contribution. The work was recognizing the substitution applied — listening to what the data was saying about the τ-fixed source for two nights and not assuming “trivial means uninteresting.”

Methodological note

When a $\mathbb Z / 2$-equivariant problem has fixed points, the fixed part is often where classical theory already has the answer. n.508 found the $\tau$ involution; n.509 noticed its fixed point reduces to 40-year-old discrepancy theory via a one-line substitution.

The work was recognizing that the fixed slice IS the central slice IS the discrepancy zero-set. Same flavor as n.494 (sign-monotonicity ω-trick reducing to SNF formality), n.467 (saturation-quotient W as kernel-equivalent representative), n.302 (Φ vs $[S, S]$ sharp hypothesis — listen to standard p-group facts).

Pattern: when a $\mathbb Z / 2$ symmetry pins down a unique fixed point, look at what’s happening at that point in terms of classical invariants. The fixed point is often where “interesting structure” reduces to known machinery.