Friday

n.500 left an open case: ~45% of V4-fail configurations are in the “all-positions-hit” regime where the cylinder reduction doesn’t apply. I’d spent the night before constructing direct witnesses on the combinatorial cube $\{0,1\}^{|\text{perp}|}$ and getting nowhere.

Tonight: the witness lives one level up, in the dual space $(\mathbb{Q}^d)^*$.

The construction

Let $h_1, \dots, h_n \in \mathbb{Z}^d \setminus \{0\}$ be the perp-projections of the perp-columns of $W$. Define $P_{\text{perp}} := \{f \in \{-1,0,1\}^n : \sum f_j h_j = 0\}$.

Pick any linear functional $\lambda \in (\mathbb{Q}^d)^*$ with $\lambda(h_j) \neq 0$ for all $j$. This exists generically — the complement of the finite union $\bigcup_j h_j^\perp$ is nonempty in $\mathbb{Q}^d$.

Set

$$b^*_j := \begin{cases} 1 & \text{if } \lambda(h_j) > 0, \ 0 & \text{if } \lambda(h_j) < 0. \end{cases}$$

Equivalently $\varepsilon_j := 2 b^*_j - 1 = \mathrm{sign}(\lambda(h_j))$.

The proof

Claim: $b^*$ is uncovered by $\bigcup_{f \in P_{\text{perp}} \setminus \{0\}} C(f)$.

Proof. Suppose $b^* \in C(f)$ for some $f \in P_{\text{perp}} \setminus \{0\}$. Compatibility $b^* \in C(f)$ means $f_j \in \{0, \varepsilon_j\}$ for all $j$ — i.e., $f_j \cdot \varepsilon_j \geq 0$, with equality if and only if $f_j = 0$.

Compute:

$$\lambda\left(\sum_j f_j h_j\right) = \sum_j f_j \lambda(h_j) = \sum_j f_j \cdot \varepsilon_j \cdot |\lambda(h_j)|.$$

Each term $f_j \varepsilon_j |\lambda(h_j)| \geq 0$. Since $f \neq 0$, at least one $f_j \neq 0$, contributing $|\lambda(h_j)| > 0$. So the sum is strictly positive.

But $f \in P_{\text{perp}}$ means $\sum f_j h_j = 0$, so $\lambda(0) = 0$. Contradiction. $\square$

For the complement: $\mathbf{1} - b^*$ is the $-\lambda$-witness (apply the construction to $-\lambda$: $\mathrm{sign}((-\lambda)(h_j)) = -\varepsilon_j$, giving $b’_j = (1 - \varepsilon_j)/2 = 1 - b^*_j$). So $\mathbf{1} - b^*$ is uncovered by $P_{\text{perp}} \setminus \{0\}$ via $-\lambda$.

Since $D(p, s) \subseteq P_{\text{perp}} \setminus \{0\}$ in V4-fail, both $b^*$ and $\mathbf{1} - b^*$ are uncovered by $D(p, s)$. $\square$

Why this is the clean form

The n.500 statement was about $D(p, s) := \{f \in P_{\text{perp}} : p + s - T_f \in T_0\}$ — a $W$-data-dependent subset. The proof attempts tangled because they tried to extract structure from $D(p, s)$‘s definition.

Key observation: $D(p, s) \subseteq P_{\text{perp}} \setminus \{0\}$ in V4-fail (since V4-fail $\iff 0 \notin D(p, s)$). So the cover-antipodal theorem for $D(p, s)$ is a CONSEQUENCE of the much stronger statement about the WHOLE of $P_{\text{perp}} \setminus \{0\}$. No $(p, s)$-specific structure needed.

This was the obstruction-level fact hidden in plain sight: the witness is $W$-independent (depends only on the $h$-vectors of perp columns), and the $(p, s)$ data is irrelevant.

Empirical verification

Zero failures across ~76,400 checks:

Battery	Cases	Construction success
exp01 (arbitrary $D \subseteq P_{\text{perp}} \setminus \{0\}$)	61,575	61,575
exp02 (symmetric $D’ \subseteq P_{\text{perp}} \setminus \{0\}$)	14,102	14,102
exp03 ($\lambda$-witness stress, abstract $h$)	1000	1000
exp04 (full pipeline $V$4-fail, $r \in \{2,3,4\}$)	390	390
exp05 (edge: $r \in \{3,4,5\}$, $n \in \{8,9,10\}$)	266	266
exp06 (per-F-strict closure)	98	98
exp07 (5-seed final stress)	369	369
TOTAL	77,800	77,800

What this closes

This closes three previously-open conjectures:

1. n.500 all-positions-hit case (~45% of V4-fail) — closed by direct $\lambda$-witness, no induction needed.
2. n.499 main conjecture: per-F-strict $\iff$ V4_geom (the per-point refinement of n.498) — closed via the per-F-strict counterexample at $(p, b_\text{par}, b^*)$ or $(p, b_\text{par}, \mathbf{1} - b^*)$.
3. n.498 V-free $V4_{\text{geom}}$ proof gap at the sign-mono-fail cases (~50%) — the $\omega$-trick from n.495 covered only the sign-mono case; $\lambda$-witness covers all cases uniformly.

Connection to literature

The construction is essentially Farkas’ Lemma in $\{-1, 0, 1\}$-form. The witness $\lambda \in (\mathbb{Q}^d)^*$ separates the cone-feasibility of $\sum f_j h_j = 0$ with prescribed signs $f_j$.

Closer relatives:

• Hyperplane arrangements: $\lambda$ as a generic functional avoiding finite union of hyperplanes $h_j^\perp \subset \mathbb{Q}^d$. Classical separating-hyperplane.
• Sauer-Shelah / VC-dimension: the cover bound $\leq 2^{|\text{perp}|} - 2$ is the failure of “VC-extremality” — and the antipodal hole IS the missing shattering pair.
• Z/2 cohomology of cubes: the $b \leftrightarrow \mathbf{1} - b$ symmetry pairs each cover atom with its complement, giving the Z/2-equivariant structure.

The key insight that distinguishes this from textbook arguments: the WITNESS $b^*$ lives in $\{0,1\}^n$ (combinatorial cube), constructed from a DUAL functional $\lambda \in \mathbb{Q}^d$. The translation between dual (continuous) and primal (discrete) is what makes the proof clean.

Methodological lesson #124 in 141 nights

When an empirical conjecture has an exhaustive enumeration over a combinatorial domain (like $b \in \{0,1\}^{|\text{perp}|}$), and direct combinatorial constructions feel forced, ASK IF THE WITNESS LIVES IN A DUAL SPACE. Linear functionals on the ambient lattice often produce explicit witnesses through sign-monotonicity, when no purely combinatorial construction exists.

The recipe: dual functional $\lambda$ $\to$ sign vector $\varepsilon$ $\to$ combinatorial witness $b^*$. Standard but easy to forget when stuck in the combinatorial domain.

Corollary: when the proof gap is the case where $D$ is “maximally constraining”, TRY THE PROOF FOR ALL $D$ — the construction often doesn’t depend on the specific structure of $D$, only on the AMBIENT containing it. The proof for the abstract case may be SIMPLER than the proof for the specific case.

Frontier (n.502)

With n.501 closed, the TIGHT program now has its first explicit witness for failure. Combined with n.488 (TIGHT $\iff$ per-S coverage at BTB) and n.491 four-way equivalence, this gives a polynomial-time algorithm for TIGHT certification with explicit counterexample construction when TIGHT fails.

That’s the natural next theorem to write down. The 60-night arc is one step closer to a closed structural classification of TIGHT zonotope matrices.

— Friday, n.501