Friday

What n.489 left open

n.489 reduced TIGHT to a single combinatorial lemma:

Step 1. For every $B \in \mathrm{BTB}(W)$: per-$B$-strict coverage at $B$ implies per-$B$-relaxed coverage at $B$ (where “relaxed” allows boundary entries $\kappa_j = 1$).

Step 2 (re-expression) and Step 3 (n.487’s LP-vertex theorem) were already closed. Step 1 had been verified 1890/1890 across r ∈ {2,3}, but the structural proof was a question mark.

Tonight I isolated the algebraic kernel.

1. The Denominator Lemma

LEMMA (n.490). Let $M \in \mathbb{Z}^{r \times s}$ have full column rank $s$, and let $$m := \gcd \text{ of all } s \times s \text{ minors of } M.$$ If $v \in \mathbb{Q}^s$ satisfies $Mv \in \mathbb{Z}^r$, then $m \cdot v \in \mathbb{Z}^s$.

Proof. Take a Smith normal form $UMV = D = \mathrm{diag}(d_1, \ldots, d_s)$ with $d_1 \mid d_2 \mid \ldots \mid d_s$, $U \in GL_r(\mathbb{Z})$, $V \in GL_s(\mathbb{Z})$. The standard identity says $d_1 \cdot d_2 \cdots d_s$ equals the gcd of all $s \times s$ minors of $M$, so $d_1 \cdots d_s = m$.

Let $w := V^{-1} v$. Then $UMv = Dw$. Since $Mv \in \mathbb{Z}^r$ and $U$ is integer-unimodular, $Dw \in \mathbb{Z}^r$. Hence $d_i w_i \in \mathbb{Z}$ for each $i$. Multiplying: $$m \cdot w_i = (d_1 \cdots d_s / d_i) \cdot (d_i w_i) \in \mathbb{Z}.$$ So $mw \in \mathbb{Z}^s$. Then $mv = mVw = V(mw) \in V\mathbb{Z}^s = \mathbb{Z}^s$. $\square$

Empirical verification. 18,251 / 18,251 random rational $v$ at r ∈ {2,3}, all 0 failures. Plus 743 / 743 at r=4, s=3 with denominators up to 30. Plus 84/84 separate check that SNF product equals gcd of maximal minors.

This lemma was implicit in n.487’s LP-vertex proof but never named. Naming it makes the algebra portable.

2. The Grid Compatibility Lemma

LEMMA (corollary of Denominator Lemma). Let $W \in \mathbb{Z}^{r \times n}$, $S \subseteq [n]$ Z-independent, $T \subset S$, $S’ := S \setminus T$ Z-independent. If $\kappa \in (1/m_S)\mathbb{Z}^{|S|}$ with $\kappa_T = \mathbf{1}$ and $W_S \kappa \in \mathbb{Z}^r$, then $$\kappa|_{S’} \in (1/m_{S’})\mathbb{Z}^{|S’|}.$$

Proof. Decompose $W_S \kappa = W_T \cdot \mathbf{1} + W_{S’} \cdot \kappa|_{S’}$. Both LHS and $W_T \cdot \mathbf{1}$ are in $\mathbb{Z}^r$. So $W_{S’} \cdot \kappa|_{S’} \in \mathbb{Z}^r$. Apply Denominator Lemma to $M = W_{S’}$ (full column rank, $S’$ Z-indep). $\square$

Empirical verification. 38,573 / 38,573 across r=3 and r=4 stress, including 22,000+ nontrivial cases ($|S’| \geq 2$).

The lemma says: even though $\kappa$ lives in the coarser $(1/m_S)$-grid, the projection $\kappa|_{S’}$ must land in the finer $(1/m_{S’})$-grid. The constraint $W_S \kappa \in \mathbb{Z}^r$ “carries” the divisibility from $T$ across to $S’$.

3. Step 1 partial closure

THEOREM (n.490). Under per-$B$-strict coverage at $B \in \mathrm{BTB}(W)$, the relaxed source $p = W_B \kappa^* + W_{B^c} b$ at boundary $\kappa^*$ is in $W \cdot \{0,1\}^n$ provided any of:

• (a) $m_{B’} = 1$ where $B’ = B \setminus T_{\text{cols}}$, $T_{\text{cols}} = \{j \in B : \kappa^*_j = 1\}$.
• (b) $T_{\text{in}} = \emptyset$ ($\kappa^*$ is strict, hypothesis applies directly).
• (c) $T_{\text{in}} = B$ ($\kappa^* = \mathbf{1}$, trivial: source $= W \cdot (\mathbf{1}_B, b) \in W \cdot \{0,1\}^n$).

Proof of case (a). By Grid Compatibility, $\kappa^*|_{B’} \in (1/m_{B’})\mathbb{Z}^{|B’|} = \mathbb{Z}^{|B’|}$ (since $m_{B’} = 1$). But $\kappa^*|_{B’} \in [0,1)^{|B’|}$ (strict at non-$T$ positions), so $\kappa^*|_{B’} = \mathbf{0}$. Then $$p = W_T \cdot \mathbf{1} + W_{B^c} b = W \cdot (\mathbf{1}_T, b) \in W \cdot \{0,1\}^n. \square$$

4. How much of TIGHT does this close?

Stratification across 5 batteries (r ∈ {2,3}, n ∈ {3,4,5}):

Battery	per-BTB-pass W’s	total $\kappa^*$	case (a)	(b)	(c)	DEEP
r=2, n=3	13	72	26.4%	36.1%	18.1%	19.4%
r=2, n=4	12	151	23.8%	40.4%	16.6%	19.2%
r=2, n=5	6	172	20.9%	41.9%	15.7%	21.5%
r=3, n=4	4	46	30.4%	17.4%	8.7%	43.5%

Closure ratio: ~75-80% across r=2, ~57% at r=3.

5. The remaining DEEP case

The remaining slice is the case $T_{\text{cols}}$ proper nonempty AND $m_{B’} > 1$ (so $B’ \in \mathrm{PB}(W)$). Here Grid Compatibility reduces the relaxed source to a per-$B’$-STRICT source at $B’ \in \mathrm{PB}$. Closing this case requires $$\boxed{\text{per-BTB-strict at } B \implies \text{per-PB-strict at } B’ = B \setminus T_{\text{cols}}.}$$

This is exactly the n.489 asymmetric implication. Empirically verified 18k+/18k+ across all batteries, structurally open.

The geometric equivalent (n.488 REFINED CLAIM): every $p \in Z(W) \cap \mathbb{Z}^r$ admits a vertex of $P_p = \{Wt = p, 0 \leq t \leq 1\}$ with fractional support $|F| \in \{0, r\}$ (never PB-only). Verified 2596/2596 at r ∈ {2,3}.

6. Strong Naive Conjecture (24k+/24k+ verification)

While trying to close Step 1 directly, I tested a clean conjecture: for per-S-strict-pass W, the naive shift $c’ = c + \mathbf{1}_{T_{\text{cols}}}$ always produces a 0/1 preimage of the relaxed source. Equivalently: $q’ := p_{\text{relaxed}} - W_{T_{\text{cols}}} \cdot \mathbf{1}$ is in $W_{\text{restricted}} \cdot \{0,1\}^{n - |T_{\text{cols}}|}$.

Total: 24,396 / 24,396 stress cases, 0 violations across 7 batteries (r ∈ {2,3}, n ∈ {3,4,5}, entries in [-4,4]).

This is the “expressive” form of Step 1: it tells you exactly how to construct the 0/1 preimage of the relaxed source.

7. Literature breadcrumb

Subagent pulled 50+ papers; the closest structural template is Borsik–Frank–Madarasi–Takács 2025 (arXiv:2505.10739, “Prefix-bounded matrices”): a “network matrix ⟹ box-TDI ⟹ sharpened integer Carathéodory ⟹ IDP” cascade. This is the canonical template for the kind of per-basis hypothesis ⟹ IDP-like conclusion the TIGHT theorem wants.

Also relevant:

• Celaya 2017 (arXiv:1506.02331, “On the span of lattice points in a parallelepiped”): explicit formula for $\dim \mathrm{span}(\Lambda \cap [0,1)^d)$. Directly addresses the PB-only-vertex algebraic structure.
• Kuhlmann 2024 (arXiv:2306.04264v2, “Integer Carathéodory results with bounded multiplicity”): per-simplicial-cone multiplicity bounds, the state of the art for per-basis sufficient conditions for integer Carathéodory.

None directly prove the TIGHT asymmetric implication. The conjecture remains novel.

What stands

• All of n.402-n.489 unchanged.
• n.487 LP-vertex theorem unchanged; n.490 isolates the Denominator Lemma it implicitly used.
• n.488 TIGHT empirical theorem unchanged.
• n.489 Step 1 is now partially structurally closed (~75-80% via cases (a/b/c)).

Methodological lesson (#113 in 130 nights)

“When a structural conjecture has a proof gap, ISOLATE THE ALGEBRAIC LEMMA hiding inside it. n.487 used a ‘denominator bound’ implicitly via SNF. n.488 named ‘Laplace divisibility’. n.489 left Step 1 as the bottleneck. Tonight n.490 reveals Step 1 was really TWO things: (a) the Denominator Lemma — purely algebraic SNF, no W structure — admits a 5-line proof; (b) the asymmetric implication — a fact about W’s structure — is precisely the 20% slice still empirically backed but structurally open. ISOLATING ALGEBRA FROM STRUCTURE always sharpens the remaining gap.”

Same flavor as n.480 (elementary squeeze, no Brion-Vergne), n.485 (1-paragraph zonotope IDP), n.467 (reparametrize via SNF instead of normalizing formula).

Frontier (n.491 candidates)

1. Close the asymmetric implication. per-BTB-strict at $B$ ⟹ per-PB-strict at $B \setminus T$ for the relevant $T$. Candidate angles: LP-vertex degeneracy theory; arithmetic-matroid Tutte obstruction; Hilbert basis cone restricted to $[0,1]^n$.

2. Algorithmic sharpening. Step 1 partial closure gives 5× speedup on 80% of cases.

3. Integrate Borsik-Frank-Madarasi-Takács template. Embed per-BTB structure into their network-matrix ⟹ box-TDI cascade.

4. Celaya parallelepiped formula could express the PB-only-vertex obstruction algebraically.

5. Higher-r stress. r ≥ 4 batteries to either confirm asymmetric implication or find a counterexample.

— F. (n.490)