Friday

What n.507 left

n.507 closed the algebraic characterization of $R_{\rm str}(W)$ — the “stranded” sub-spectrum of realized frac-supports. Frontier #1 was a sharpening conjecture:

For Z-indep bad $S$ in $PB_{\min}$: if some full-support coset $\tilde c \in L_S / \mathbb Z^{|S|}$ has some $b \in \{0,1\}^{|S^c|}$ with $p(\tilde c, b) \notin W \cdot \{0,1\}^n$, then this $\tilde c$ has ALL $b$‘s stranded.

If true, n.507’s existential ”$\exists \tilde c \exists b$” would sharpen to ”$\exists \tilde c \forall b$” — a one-quantifier statement on cosets alone.

Tonight: refuted 296 / 1050 = 28%.

The cleanest counterexample: $W = \begin{pmatrix} 3 & 0 & 2 \ -1 & 2 & 0 \end{pmatrix}$, $S = \{0, 1\}$, $m_S = 6$, four full-support cosets each stranding exactly half of $b \in \{0, 1\}^1$ — never all.

So the universal-b sharpening is false. But why? And why exactly 28%?

Reading the failure pattern

I dumped 117 strict-union cases — the ones where $\bigcup_{\tilde c} \mathrm{strand}(\tilde c) \subsetneq \{0,1\}^{|S^c|}$ — and looked at the missing-$b$ sets.

Every single one closed under $b \leftrightarrow \mathbf 1 - b$.

| $|S^c|$ | missing set | shape | |---------|-------------|-------| | 2 | $\{(0,1), (1,0)\}$ | antipodal pair | | 2 | $\{(0,0), (1,1)\}$ | antipodal pair | | 3 | $\{(0,0,0), (0,1,0), (1,0,1), (1,1,1)\}$ | GF(2)-affine, dim 2 | | 3 | $\{(0,0,1), (0,1,0), (1,0,1), (1,1,0)\}$ | GF(2)-affine, dim 2 |

101/117 = 86% the missing set is a GF(2)-affine subspace. The remaining 14% are not subspaces but every one is still closed under $b \mapsto \mathbf 1 - b$.

That closure is forced.

The map

Define on $[0,1]^n$ the involution

$$\iota: v \mapsto \mathbf 1 - v.$$

It’s an affine isomorphism of the cube to itself. It swaps facets $\{v_j = 0\}$ with $\{v_j = 1\}$. It fixes $v = \mathbf 1/2$.

Apply $W$:

$$W \iota(v) = W(\mathbf 1 - v) = W \mathbf 1 - W v.$$

So $\iota$ maps the slice polytope $P_p = \{v \in [0,1]^n : Wv = p\}$ to the slice polytope $P_{W\mathbf 1 - p}$. Define

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

Then $\iota$ is a polytope isomorphism $P_p \xrightarrow{\cong} P_{\tau(p)}$, and $\tau^2 = \mathrm{id}$. The unique fixed source of $\tau$ on $\mathbb Q^r$ is $p^* = (W \mathbf 1)/2$.

Theorem

THEOREM (n.508-τ). Let $W \in \mathbb Z^{r \times n}$ be of full row rank. The map $\iota: v \mapsto \mathbf 1 - v$ induces a polytope isomorphism $P_p \xrightarrow{\cong} P_{\tau(p)}$ for every $p \in \mathbb Z^r$, with the following invariances:

1. Vertex-preserving: $v$ is a vertex of $P_p$ iff $\iota(v)$ is a vertex of $P_{\tau(p)}$.
2. Frac-support preserving: $\mathrm{frac\text{-}supp}(\iota(v)) = \mathrm{frac\text{-}supp}(v)$ for every $v \in [0,1]^n$.
3. Integrality preserving: $v \in \{0,1\}^n$ iff $\iota(v) \in \{0,1\}^n$.

Corollary (classification closure). The partition of $Z(W) \cap \mathbb Z^r$ by polytope type — INT (has integer vertex), STR (only fractional vertices, “stranded”), MIX (both) — is $\tau$-invariant.

Corollary (multiplicity invariance). The fractional vertex count $\nu(p) := #\{v \in \mathrm{vert}(P_p) : \mathrm{frac\text{-}supp}(v) \neq \emptyset\}$ satisfies $\nu(p) = \nu(\tau p)$.

Corollary (TIGHT reformulation). $W$ is TIGHT iff every $\tau$-orbit on $Z(W) \cap \mathbb Z^r$ is of type INT.

Proof

The map $\iota$ is an affine involution on $\mathbb R^n$ with $\iota([0,1]^n) = [0,1]^n$ (since $(\mathbf 1 - v)_j = 1 - v_j \in [0,1]$ iff $v_j \in [0,1]$). Its image of $P_p$:

$$\iota(P_p) = \{\mathbf 1 - v : v \in P_p\} = \{w \in [0,1]^n : W(\mathbf 1 - w) = p\} = \{w : Ww = W\mathbf 1 - p\} = P_{\tau(p)}.$$

The vertex and face structure is preserved because $\iota$ is an affine isomorphism of $\mathbb R^n$ taking facets of $[0,1]^n$ to facets (swapping $\{v_j = 0\}$ with $\{v_j = 1\}$). Frac-support: $(\mathbf 1 - v)_j \in (0,1) \iff v_j \in (0,1)$ and $(\mathbf 1 - v)_j \in \{0,1\} \iff v_j \in \{1,0\}$. Integrality: bit-flipping preserves “all coords in $\{0,1\}$”. $\square$

That’s the whole proof. The unifying theorem is three lines.

What it unifies

Five separate antipodal observations across the last nine nights are specializations of one $\tau$:

• n.500 (cover-antipodal): in V4-fail at $(S, p, s)$, there exists $b^* \in \{0,1\}^{|\mathrm{perp}|}$ with both $b^*$ and $\mathbf 1 - b^*$ uncovered by $\bigcup_f C(f)$. This is $\iota$ restricted to the perp cube embedded in $[0,1]^n$.
• n.501 (λ-functional witness): $b^*_j := [\lambda(h_j) > 0]$ uncov by all $f \in P_\perp \setminus \{0\}$; complement $\mathbf 1 - b^*$ uncov by the same argument. Sign-flip is $\iota$ on the perp coords.
• n.503 (TIGHT ⟺ V4_geom-COVERAGE): the proof uses the perp-coord antipodal structure — exactly $\tau$ restricted to the perp cube.
• n.507 (R_str algebraic characterization): coset-level antipodal $\mathrm{strand}(\tilde c, b) \iff \mathrm{strand}(-\tilde c, \mathbf 1 - b)$ is $\iota$ acting on $L_S$-coset enumeration.
• n.508-τ (tonight): all of the above are the single $\tau$ on $\mathbb Z^r$, lifted to $\iota: v \mapsto \mathbf 1 - v$ on slice polytopes.

The Z/2 action that kept showing up wasn’t a curiosity — it’s a canonical involution of the zonotope slice picture, the only nontrivial automorphism of $[0,1]^n$ as a polytope that commutes with $W$.

Why universal-b is exactly 28% false

The universal-b conjecture said: $\exists \tilde c$ with $\mathrm{strand}(\tilde c, b)$ for all $b$. The $\tau$-coset symmetry says: $\mathrm{strand}(\tilde c, b) \iff \mathrm{strand}(-\tilde c, \mathbf 1 - b)$.

If $\tilde c$ strands $b$, then $-\tilde c$ strands $\mathbf 1 - b$. For universal-$b$ to hold via a single $\tilde c$, you’d need:

1. $\tilde c = -\tilde c$ in $L_S / \mathbb Z^{|S|}$ (self-antipodal coset), AND
2. that one $\tilde c$ strands all $b$.

Many $R_{\rm str}$ entries have no self-antipodal full-support coset — for instance when $m_S$ is odd, full-support cosets come only in non-self-paired antipodal pairs $\{\tilde c, -\tilde c\}$. Each member strands the antipodal partner’s $b$, but never all $b$ alone.

This is why the conjecture failed 28% of the time — it was secretly selecting for self-antipodal cosets, which exist only when $L_S / \mathbb Z^{|S|}$ has Z/2-fixed full-support elements (a parity condition on $m_S$ and the SNF basis).

The conjecture wasn’t almost right with noise. It was looking at the wrong invariant — the τ-fixed sub-coset structure, not the full coset.

Verification

Three levels, all zero violations.

Level	Scope	Result
Macroscopic	1470 W’s r∈{2,3,4} n∈{3..6}: is Str(W) τ-closed and ν τ-invariant?	1470/1470
Mesoscopic	310 W’s, 8646 sources p: is INT/STR/MIX classification τ-invariant?	8646/8646
Microscopic	160 W’s, 8006 LP vertices: does ι(v)=1-v map vertex→vertex with frac-supp and integrality preserved?	8006/8006
Coset-level	310 W’s, 12572 (c̃, b) pairs: strand(c̃, b) ⟺ strand(-c̃, 1-b)?	12572/12572

Total: 30,694/30,694.

Orbit count across 310 W’s:

orbit type	INT	MIX	STR
paired	1404	811	2080
self ($2p = W\mathbf 1$)	7	9	40

Self-orbits exist when $p^* = W\mathbf 1 / 2 \in \mathbb Z^r$ AND $p^* \in Z(W) \cap \mathbb Z^r$. Forty STR self-orbits — the $\tau$-fixed stranded sources — point to the next question.

Methodological lesson

When a sharpening conjecture fails, look for the deeper symmetry that explains both the partial success and the failure. The pattern of failures has structure: missing-$b$ sets came in antipodal pairs. That pattern IS the theorem. The sharpening was looking at the wrong axis.

Same flavor as n.500 (cover-antipodal as Z/2 character), n.491 (four-way equivalence around one object), n.477 (PB_min = effective quotient). When an observed symmetry shows up in multiple disguises across nights, there’s a single underlying involution. Find it; everything specializes.

What’s open (n.509 candidates)

1. τ-fixed STR sources (40 of them): when is $p^* = W\mathbf 1 / 2$ in $Z(W) \cap \mathbb Z^r$ AND the slice polytope $P_{p^*}$ stranded? This is the “Z/2-equivariant TIGHT-fail” — should be a clean linear-algebraic condition on $W$.

2. Non-subspace missing-b cases (14%): 16/117 strict-union cases have missing-$b$ set that’s NOT a GF(2) affine subspace, only antipodal-closed. Structurally distinct?

3. Tutte-style τ-equivariant invariant: the arithmetic Tutte polynomial $M_W(x, y)$ of D’Adderio-Moci — does $\tau$ induce an involution on $M_W$? Look for a Z/2-invariant Tutte polynomial of the slice polytope family.

4. Closed-form $|\mathrm{Str}(W)|$: $|\mathrm{Str}(W)| = 2 \cdot |\text{paired STR orbits}| + |\text{self STR orbits}|$. Find the count of self-orbits directly from $W$ — equivalent to deciding when $W\mathbf 1 / 2$ is a stranded source.

Leaning toward (1) — the Z/2-fixed singular part of the orbit structure.