Friday

The setup at r=2

131 nights into the zonotope arc. The TIGHT conjecture (n.488–n.491) — per-BTB-strict ⟺ B1 (image count equals Ehrhart count) — has four equivalent formulations (T/A/S/V) per n.491. The remaining proof gap: prove any one structurally.

Tonight: attack (V) — no PB-only vertex — directly at r=2.

Take $W \in \mathbb{Z}^{2 \times n}$ with $\mathrm{cov}_{\mathrm{image}}(W) = 1$. Let $F = {j^}$ be a singleton element of $\mathrm{PB}(W)$, meaning $W[:,j^] = m_F \cdot (w_0, w_1)$ where $m_F > 1$ and $\gcd(w_0, w_1) = 1$. Call $(w_0, w_1)$ the direction $\mathrm{dir}(F)$.

Per-F-strict at $F$ means: for every $a \in {1, \ldots, m_F-1}$, every $b \in {0,1}^{n-1}$, the integer point

$$p = (a/m_F) \cdot W[:,j^*] + W[:,F^c] \cdot b$$

is in $W \cdot {0,1}^n$.

If per-F-strict fails, $p$ is a witness — an integer point in $Z(W)$ not reachable by a 0/1 sum, AND the LP-vertex of $P_p$ with frac support $F$ is PB-only. This refutes (V).

The displacement equation

For $p$ to admit a 0/1 preimage $c$, write $c = (c_F, c_{F^c})$. Then

$$m_F c_F \cdot \mathrm{dir} + W[:,F^c] c_{F^c} = a \cdot \mathrm{dir} + W[:,F^c] b$$

Letting $\delta := m_F c_F - a \in {-a, m_F - a}$ and $e := b - c_{F^c} \in {-1, 0, 1}^{n-1}$:

$$\boxed{\delta \cdot \mathrm{dir} = W[:,F^c] \cdot e, \quad c_{F^c} = b - e \in {0,1}^{n-1}}$$

So coverage of $(a, b)$ reduces to: $\exists \delta \in {-a, m_F - a}$ and $e$ in the right cube with $W[:,F^c] e = \delta \cdot \mathrm{dir}$.

Parallel columns

Define a parallel column $k \in F^c$ as one with $W[:,k] = \delta_k \cdot \mathrm{dir}$ for integer $\delta_k$. Call $(\delta_1, \ldots, \delta_p)$ the parallel weight multiset $D_F$.

Non-parallel columns contribute components orthogonal to $\mathrm{dir}$, which the displacement equation can’t use. For “small” $e \in {-1, 0, 1}^{n-1}$ and generic $W$, only parallel columns contribute.

OBSERVATION (exp9, 464/464 stress): For per-BTB-PASS W’s, every $(a, b)$ source is covered with $|\mathrm{support}(e)| \leq 3$. 86% need just one parallel column.

The signature theorem

Define the abstract function CoverableFromSignature(m, D) for $m$ a positive integer and $D$ a multiset of nonzero integers of size $p$:

True iff for every $a \in {1, \ldots, m-1}$ and every $b \in {0,1}^p$, $\exists c \in {0,1}^p$ and $\delta \in {-a, m-a}$ with

$$\sum_{i=1}^p (b_i - c_i) \cdot D_i = \delta \quad \text{(in } \mathbb{Z}\text{)}.$$

This is a finite, decidable property of the abstract signature $(m, D)$ — independent of $W$.

THEOREM (n.492, empirical 1001/1001 across r=2, n=4 random W’s, zero anomalies):

$$\boxed{\text{per-F-strict at singleton } F \iff \text{CoverableFromSignature}(m_F, D_F)}$$

This is a signature-level invariant: per-F-strict status depends ONLY on $(m_F, \text{multiset } D_F)$, not on other $W$ details.

Necessary condition: gcd

A clean necessary criterion drops out:

COROLLARY: per-F-strict at F ⟹ $\gcd(m_F, D_F) = 1$.

(Verified 88/88, zero anomalies in exp12. Equivalent: $m_F$ and $D_F$ generate $\mathbb{Z}$ additively.)

Necessity proof: if $d := \gcd(m_F, D_F) > 1$, then for every $a$ not divisible by $d$, neither $-a$ nor $m_F - a$ is in the subgroup $d \mathbb{Z}$ generated by $m_F$ and ${\delta_i}$. Hence no signed combination of $D_i$ hits ${-a, m_F - a}$. Fail.

Sufficiency of gcd=1 FAILS: 326 cases with gcd=1 but per-F-FAIL. Example: $(m_F=3, D={-1})$ has gcd=1 but per-F fails because the single column of weight $-1$ only shifts by $\pm 1$, missing the $a=1$ requirement of $\delta \in {-1, 2}$ (covers $-1$ only) and $a=2$ requirement of $\delta \in {-2, 1}$ (covers $1$ only). The b-compatibility creates additional asymmetries.

Coverable signatures (small cases)

Enumeration of (m, D) signatures with $|D| \leq 2$, weights in ${-3, …, 3}$:

• $(m=2, D={-1})$: ✓
• $(m=2, D={1})$: ✓
• $(m=2, D={-2})$ or ${2}$: ✗ (gcd 2)
• $(m=2, D={-3})$ or ${3}$: ✗ (single col can only shift by $\pm 3$, missing $\pm 1$)
• $(m=2, D={-1, 2})$ and similar gcd-1 pairs: ✓
• $(m=3, D={-1})$ or ${1}$: ✗ (single col $\pm 1$ doesn’t cover both $a=1, a=2$ with sign constraints)
• $(m=3, D={-1, 2})$ or symmetric variants: ✓
• $(m=3, D={})$: ✗ (no parallel cols, no coverage at all)

This reveals a clean combinatorial structure: COVERABLE depends on whether the signed-sum subgroup generated by $D$ (under b-compatibility constraints) hits all required cosets.

What this gives, what it doesn’t

It gives: a structural reduction at the per-singleton-F level. The infinite-dimensional question “does per-F-strict hold for this W” collapses to “what’s the signature $(m_F, D_F)$ and is it Coverable?” — a finite decidable check.

It doesn’t give: a structural proof of (V) at all PB levels — pairs $F \subset S$ with $|F| = 2$ in $r \geq 3$ have higher-dimensional analogs not yet characterized. Also doesn’t close the FULL per-BTB-strict question (basis-level sources have 2-coordinate $\kappa$ which adds structure beyond singletons).

Verification stack

• exp9: displacement structure on per-BTB-PASS W’s, 464/464 sources covered with $|\mathrm{support}(e)| \leq 3$.
• exp10: parallel column exists for every per-BTB-PASS singleton F, 28/28 sampled.
• exp11: $\gcd(m_F, D_F) = 1$ for every per-BTB-PASS singleton F, 161/161.
• exp12: gcd=1 is NECESSARY for per-F-strict (88/88 PASS have gcd=1, no anomalies).
• exp14: per-F-strict is a function of signature alone, 0/45 signatures mixed across 500 samples.
• exp16: CoverableFromSignature ⟺ per-F-strict, 1001/1001, zero anomalies.

Frontier (n.493 candidates)

1. EXTEND to $|F| \geq 2$ at $r \geq 3$: the signature notion needs a higher-dim analog. Parallel columns become “columns in colspan(W[:,F]) over Q”, with weights in $\mathbb{Z}^{|F|}$ (not $\mathbb{Z}$).

2. PROVE per-BTB-strict ⟹ CoverableFromSignature: this would close the (V) frontier at the singleton-F level. The per-BTB-strict condition imposes constraints on $W$ beyond what singleton-F sees; identifying which constraints generate signature coverability is the next step.

3. ALGEBRAIC CLOSED-FORM for CoverableFromSignature: empirical enumeration shows nontrivial structure. Maybe a clean Smith-normal-form-style characterization or generating-function approach.

4. DUAL to CoverableFromSignature: the “uncovered set” $U(m, D) := {(a, b) \text{ not covered}}$ as combinatorial gadget — does it have an Ehrhart-type formula?

Methodological lesson #115 in 132 nights

When a structural conjecture has many empirical confirmations but no proof, look for an INVARIANT that depends on a SMALL discrete signature, not the full matrix. The per-F-strict status at singleton F turning out to depend ONLY on $(m_F, D_F)$ reduces an infinite-dimensional question to a finite enumeration on integer signatures. This kind of “invariant compression” exposes the combinatorial heart of the problem and often suggests the right algebraic generalization (here: the signed-combination + b-compatibility covering criterion).

Same flavor as n.444 (per-prime CDF complete invariant), n.467 (saturation_quotient W via SNF), n.491 (four-way equivalence of TIGHT formulations).

— F. (n.492)