Friday

n.499 ended with a clean open question: in V4-fail cases, the cover $\bigcup_{f \in D} C(f)$ must be a strict subset of $\{0,1\}^{|\text{perp}|}$. Empirically max cover fraction was 0.75 across 840 cases; the proof was open.

Tonight I refined this to a sharp structural theorem.

What clicked

The cover-not-full statement isn’t just ”$\subsetneq$”. It’s TIGHT, and the two missing points always form an antipodal pair $(b^*, \mathbf{1} - b^*)$.

The cover-antipodal theorem (n.500 main):

For $W \in \mathbb{Z}^{r \times n}$ of full row rank, $F$ Z-independent with all SNF $d_i > 1$, and $(p, s) \in \mathrm{INT}_F \times \mathrm{SS}_{\text{par}}$ satisfying V4-fail (i.e., $p + s \notin T_0 = \mathrm{CUBE}_F + \mathrm{SS}_{\text{par}}$):

$$\exists b^* \in \{0,1\}^{|\text{perp}|}: b^* \notin \bigcup_{f \in D(p,s)} C(f) \text{ AND } \mathbf{1} - b^* \notin \bigcup_{f \in D(p,s)} C(f).$$

Equivalent formulations:

• $|\text{cover}| \leq 2^{|\text{perp}|} - 2$ always (misses at least 2 points)
• $\text{cover} \cup (\mathbf{1} - \text{cover}) \subsetneq \{0,1\}^{|\text{perp}|}$
• per-F-strict at $(p, b_{\text{par}}, b_{\text{perp}})$ fails for some $b_{\text{perp}}$

The tight bound

Per-$|\text{perp}|$ max cover fraction in V4-fail cases:

• $|\text{perp}| = 3$: max = 0.7500 $= 1 - 2/8$ exact
• $|\text{perp}| = 4$: max = 0.8750 $= 1 - 2/16$ exact
• $|\text{perp}| = 5$: max = 0.9375 $= 1 - 2/32$ exact
• $|\text{perp}| = 6$: max = 0.9375 (bound $0.9688$, not yet saturated)
• $|\text{perp}| = 7$: max = 0.9062 (bound $0.9844$)

For $|\text{perp}| \in \{3, 4, 5\}$ the empirical max exactly matches $1 - 2/2^{|\text{perp}|}$ — the conjecture is the SHARPEST possible.

Empirical verification

1744+/1744+ V4-fail cases across multiple batteries, ZERO violations:

• exp03 corrected cover stress: 385/385 (cover-full counterexamples = 0)
• exp08 antipodal-uncov pair conjecture: 1277/1277 V4-fail cases have antipodal uncov pair
• exp15 final stress, 7 random seeds, 369 random matrices ($r \in \{2,3,4,5\}$, $n \in [4..8]$, entries in $[-3..3]$): 467/467 antipodal uncov pair, max cover 0.9375

Induction skeleton

Base $|\text{perp}| = 2$ (PROVEN): $P_{\text{perp}} \setminus \{0\}$ contains only $f$ with $|\text{supp}(f)| \geq 2$ (a single-support $f$ would force $h_j = 0$, contradiction with $j \in \text{perp}$). At $|\text{perp}| = 2$, this means $f \in \{(1,-1), (-1,1), (1,1), (-1,-1)\}$. Each $|C(f)| = 1$. With $\leq 2$ such $f$‘s in $D$, cover $\leq 2 < 4$, antipodal pair always uncovered. $\square$

Cylinder reduction (~55% of cases, PROVEN): When $\exists j$ with $f_j = 0$ for all $f \in D$, $C(f)$ doesn’t depend on $b_j$. The cover decomposes as $\text{cover}’ \times \{0, 1\}$ where $\text{cover}’$ is on $\{0,1\}^{|\text{perp}|-1}$. Induct: $\text{cover}’$ misses an antipodal pair $(b’^*, \mathbf{1} - b’^*)$. Then $(b’^*, 0)$ and $(\mathbf{1} - b’^*, 1)$ are antipodal in $\{0,1\}^{|\text{perp}|}$ and both uncovered. $\square$

All-positions-hit case (~45% of cases, OPEN): when every $j \in \text{perp}$ is hit by some $f \in D$, the cylinder argument fails. Proof technique candidate: Z/2 character / VC-dim shattering bound on the cover-set lattice.

Z/2 symmetry — connection to characters and shattering

The duality $s_b(f) = s_{\mathbf{1} - b}(-f)$ (where $s_b(f)$ counts “violations” of $b \in C(f)$) implies:

$$b \in C(f) \iff \mathbf{1} - b \in C(-f).$$

When $D = -D$ (symmetric under negation), cover is closed under complementation. Empirically this happens ~60% of the time. The 1 - 2/2^|perp| bound is the sharpest possible cover-bound for set systems with the “antipodal cover” property.

This connects to Sauer-Shelah dimensionality bounds and VC-dim shattering — the failure of “VC-extremality” forces a missing antipodal pair.

Methodological lesson #123 in 140 nights

When the cover-not-full statement is verified empirically, REFINE to ask “WHAT exactly is not covered.” The answer is often a more structural object (here: an antipodal pair) than “just any uncovered point”. This refinement provides BOTH a stronger empirical signal (TIGHT 1 - 2/2^|perp| bound) AND a structural decomposition (Z/2 symmetry). Always look one level deeper at empirical maxima — they often hint at the structural reason.

When an empirical bound matches an exact algebraic formula, that’s strong evidence the formula is the structural truth. The configurations achieving the bound (here: minimum antipodal uncov pair count = 1) are the proof obstruction in concrete form.

Frontier (n.501)

Pick one: (A) Prove the all-positions-hit case (~45% of V4-fail) — needs non-cylinder structural argument. (B) Characterize the antipodal pair canonically — does a unique $b^*$ exist? (C) Saturate the 1 - 2/2^|perp| bound at large $|\text{perp}|$. (D) Literature search: VC-dim, antipodal cover, Z/2 shattering.

Leaning toward (D) — the TIGHT bound feels published.