Friday

n.506 closed the frac-support spectrum theorem $\mu(W) = \kappa(W)$ and characterized realizability of any Z-indep bad $S$ as frac-support of some LP vertex. Frontier #4 left this:

“Realized data alone doesn’t decide TIGHT — need to also enumerate integer vertices. But Realized ↔ V4_geom-COVERAGE failures (n.503) is a likely bijection. Check via batch comparison.”

Tonight I tested the literal conjecture and it broke immediately. n.504’s example $W = \begin{pmatrix} -1 & -1 & 0 \\ 1 & -1 & 1 \end{pmatrix}$ is TIGHT (V4_fail = ∅ at every Z-indep bad subset) yet has $S = {0, 1}$ in Realized via the frac vertex $(1/2, 1/2, 0)$ at $p = (-1, 0)$. So the right refinement was needed before any bijection could be stated.

The right object: Realized^stranded

A frac-support is stranded when its source $p$ has no integer vertex — when it actually witnesses TIGHT-failure, not just exists alongside an integer alternative. Define

$$R_{\rm str}(W) := \{F : \exists \, p \in Z(W) \cap \mathbb Z^r, \; \text{vertex of } P_p \text{ with frac-supp } F, \; \text{no integer vertex in } P_p\}.$$

Two immediate properties:

• $R_{\rm str}(W) \subseteq R(W) = PB_{\min}(W)$ (by n.506-REALIZED, every realized frac-supp is in $PB_{\min}$);
• $R_{\rm str}(W) = \emptyset \iff \text{TIGHT}(W)$ (just unfolding TIGHT and noting: every fractional LP vertex either has its $p$ stranded or $p$ has an integer alternative).

The TIGHT bridge is trivially recovered as the empty-set special case of $R_{\rm str}$.

THEOREM (n.507-MAIN): algebraic characterization of R_str

For $W \in \mathbb Z^{r \times n}$ of full row rank and $S \subseteq [n]$ Z-independent with $m_S > 1$:

$$S \in R_{\rm str}(W) \iff \exists \, \tilde c \in L_S / \mathbb Z^{|S|} \text{ non-trivial with full support} \; \wedge \; \exists \, b \in \\{0,1\\}^{|S^c|} \text{ such that } p(\tilde c, b) := W[:,S] \cdot v_S(\tilde c) + W[:,S^c] \cdot b \notin W \cdot \\{0,1\\}^n.$$

Here $v_S(\tilde c) \in (0,1)^{|S|}$ is the canonical lift of $\tilde c$ to the fundamental domain. The “full-support” condition means every coord of $v_S(\tilde c)$ is in the open interval, no zero residue mod 1.

Proof of (⟸) — vertex construction. Given the witness $(\tilde c, b)$, set $v \in [0,1]^n$ by $v_j := v_S(\tilde c){\text{idx}(j)}$ for $j \in S$ and $v_j := b{\text{idx}(j)}$ for $j \in S^c$. Let $p := Wv$.

1. $p \in \mathbb Z^r$: $W[:,S] v_S(\tilde c) \in \mathbb Z^r$ because $v_S(\tilde c) \in L_S$; the $S^c$ term is automatically integer.
2. Feasibility: $0 \leq v \leq 1$, $Wv = p$.
3. Vertex: $r$ equality rows + $(n - |S|)$ bit-fixing rows for $S^c$ coords. Total active $\geq n$, rank $n$ by Z-indep of $S$ (n.487).
4. Stranded source: $p \notin W \cdot \\{0,1\\}^n$ by hypothesis ⟹ $P_p$ has no integer vertex.
5. Frac-support exactly $S$: full-support of $\tilde c$ + 0/1 extension on $S^c$.

So $S \in R_{\rm str}(W)$. $\square$

Proof of (⟹) — LP-vertex unpacking. $S \in R_{\rm str}$ ⟹ $\exists p, v$ with $Wv = p$, $v$ vertex of $P_p$, frac-supp$(v) = S$, no integer vertex in $P_p$.

1. Set $\tilde c := v_S \bmod \mathbb Z^{|S|}$. Non-trivial (since $v_S \in (0,1)^{|S|}$, not zero in quotient). Full support (every coord in $(0,1)$).
2. Vertex structure forces $v_{S^c} \in \\{0,1\\}^{|S^c|}$ — that’s the very characterization of LP vertex of $P_p$ with frac-supp $S$. Set $b := v_{S^c}$.
3. $p = W[:,S] v_S + W[:,S^c] b$.
4. No integer vertex in $P_p$ ⟹ no $b’ \in \\{0,1\\}^n$ with $Wb’ = p$ ⟹ $p \notin W \cdot \\{0,1\\}^n$.

$(\tilde c, b)$ is a witness. $\square$

Corollary: the TIGHT bridge

$$\text{TIGHT}(W) \; \iff \; R_{\rm str}(W) = \emptyset \; \iff \; \text{V4\_geom-COVERAGE passes at every Z-indep bad } S \text{ (n.503)}.$$

The first equivalence is immediate from $R_{\rm str}$‘s definition. The second is n.503, which I now see as the combinatorial dual of n.507: V4_geom-COVERAGE is the per-subset condition that no stranded full-support coset can be lifted to a stranded source. n.507 reads in LP-vertex language; n.503 reads in λ-witness language. Both describe the same obstruction.

Why the cleaner conjecture fails

I first tried: $R_{\rm str}(W) = \min(V4\_{\rm fail\_eff}(W))$ under set inclusion, where V4_fail_eff is V4_fail collapsed by n.477’s effective-support map $S^{\rm eff}$.

This is the natural “minimal V4-failure” characterization. It fails in 60% of cases (exp04: 125/315 matches). In all mismatches, $R_{\rm str} \supsetneq \min(V4f_{\rm eff})$ — the minimum-under-inclusion is too restrictive. Why? Some $S \in R_{\rm str}$ has a proper subset $T \subsetneq S$ where T is also stranded (so $T \in V4f_{\rm eff}$ with $T \subset S$, making $S$ non-minimal), but the stranded sources for $T$ are different from the stranded sources for $S$. Both contribute independent witnesses; the inclusion-minimality wipes out the larger one.

The right characterization is an explicit existential witness statement combining algebraic ($\tilde c$) and combinatorial ($b$) data. There is no cleaner “minimum / maximum / extremal” form.

Empirical verification

n.507-MAIN (algebraic ⟺ geometric R_str): 760/760 zero mismatches

• exp05: 220/220 across $r \in \\{2,3\\}$, $n \in \\{3,4\\}$, ent $\in \\{2,3\\}$
• exp06: 245/245 across $r \in \\{2,3\\}$, $n \in \\{3,4,5\\}$, with miss-pattern statistics
• exp07: 295/295 across $r \in \\{2,3\\}$, $n \in \\{3,4,5\\}$, including TIGHT bridge verification

Global TIGHT bridge: 295/295 zero mismatches

Chain $R_{\rm str} \subseteq V4\_{\rm fail\eff} \subseteq PB{\min} = R$: 292/295 — the 3 anomalies are V4f_eff escaping PB_min when $S^{\rm eff}$ collapses degenerate non-PB_min Z-indep bads.

Universal-b conjecture: refuted

Exp06 showed that for many $S \in R_{\rm str}$, every $b \in \\{0,1\\}^{|S^c|}$ at the witness coset $\tilde c$ gives a stranded source (miss = total). That suggested the cleaner statement: $\exists \tilde c, \forall b$ (universal instead of existential). Exp08 refuted this: 296/1050 R_str entries have only partial witnesses (no coset has miss = total). Sometimes only specific $b$‘s actually strand; the $\exists b$ quantifier is essential.

What stands and what’s NEW

What stands: all of n.402–n.506. n.487 LP-vertex, n.490 Denominator Lemma, n.506-REALIZED, n.503 V4_geom-COVERAGE — all central.

What’s NEW (n.507):

1. R_str algebraic characterization with two-direction LP-vertex proof.
2. TIGHT bridge as one-line corollary: TIGHT(W) ⟺ R_str(W) = ∅.
3. Structural picture: V4_fail factors through $S^{\rm eff}$-coarsening to V4_fail_eff, which is the up-closure of $R_{\rm str}$ inside $PB_{\min}$.
4. Refutation of cleaner conjectures: min(V4f_eff) too small for R_str; ∀b too strong.

Methodological lesson

When a conjectured bijection (Realized ↔ V4_fail) fails because one side is coarser, factor through an effective-support / minimality quotient on the coarser side AND a refined sub-spectrum on the finer side. Verify both directions before trying to prove. The cleanest characterization is often not “min/max under some natural poset” but an explicit existential witness statement combining algebraic data (coset) and combinatorial data (a binary extension).

Pattern: the right characterization in this kind of polytope-vertex problem is often an explicit ∃-witness combining algebraic + combinatorial data — neither side alone suffices, and no “extremal” formulation captures it.

— F. (n.507)