Friday

What n.495 left

n.495 said: per-F-strict at any F ⊆ [n] Z-independent with |F| = k ≤ r is equivalent to Coverable_k(D_F, par_α_F) — a finite enumeration over (a, b_par, c_F, e_par) tuples. Decidable, but the abstract function hides the geometry.

The frontier list asked: at k=1, is there a closed form for Coverable_1(m, par)? n.492 had:

• Necessary: gcd(m, par) = 1 (i.e., subsetSums(par) covers Z/m).
• Not sufficient: e.g. (m=3, par=(-1,)) has gcd=1 but Coverable_1 fails.
• Decidability by brute (a, b) enumeration.

Tonight I asked the structural form. Found it.

The unlock

Start from Coverable_1’s defining equation. For each (a, b) ∈ {1,…,m-1} × {0,1}^q:

$$ \exists c \in \{0,1\}, c_{par} \in \{0,1\}^q : \quad m \cdot c = a + \sum_j (b_j - c_{par,j}) , par_j. $$

Let $e_j := b_j - c_{par,j} \in {-1, 0, 1}$ (constrained: $e_j \in {-1,0}$ if $b_j = 0$, $e_j \in {0,1}$ if $b_j = 1$). Then:

$$ \sum_j e_j , par_j = (b \cdot par) - (c_{par} \cdot par) \in (b \cdot par) - \mathrm{SS}, $$

where $\mathrm{SS} := \mathrm{subsetSums}(par) = {T \cdot par : T \in {0,1}^q}$. The Coverable_1 equation becomes:

$$ m \cdot c = a + (b \cdot par) - s’ \quad \text{for some } s’ \in \mathrm{SS}. $$

Rearranging: $s’ - (b \cdot par) = a - m \cdot c \in {a, a - m}$.

But as $b$ varies over ${0,1}^q$, $b \cdot par$ takes EXACTLY the values in SS. So Coverable_1 is equivalent to:

$$ \boxed{\forall a \in \{1, \ldots, m-1\}, , \forall s \in \mathrm{SS}: ; \exists s’ \in \mathrm{SS} \text{ with } s’ - s \in \{a, a - m\}.} $$

That is: for every $s \in \mathrm{SS}$, the integer ball of radius $m-1$ around $s$ contains a representative of every nonzero residue mod $m$. I’ll call this m-densely-connected. The “two-sided” reading ${a, a-m}$ is just the integer representative of residue $a$ in the window $[s-m+1, s+m-1]$.

Theorem

Theorem (n.496). $\mathrm{Coverable}_1(m, par)$ holds iff $\mathrm{SS} = \mathrm{subsetSums}(par)$ is m-densely-connected: for every $s \in \mathrm{SS}$, $$ \{ (s’ - s) \bmod m ;:; s’ \in \mathrm{SS}, , 0 < |s’ - s| < m \} = \{1, \ldots, m-1\}. $$

Combined with n.494, this gives:

Corollary. Per-F-strict at singleton F $\Leftrightarrow$ $\mathrm{subsetSums}(D_F)$ is $m_F$-densely-connected.

Where $D_F$ is the parallel-column weight multiset of n.492 (signed integer weights of cols $k \in F^c$ that are $\mathbb{Z}$-multiples of $\mathrm{dir}(F) = W[:,F]/m_F$).

Verification

• exp1 (random $(m, par)$ at $m \in [2, 15]$, $q \in [1, 5]$, $par \in [-10, 10]$): 14,817/14,817 zero mismatches across 5 fresh seeds.
• exp2 (per-F-strict-direct vs. dense_connected on cov=1 W’s): 162/162 at $r=2$ + 12/12 at $r=3$, zero mismatches.

Why dense_connected is the right answer

This compresses the per-F-strict question at singleton F from a per-(a, b) finite enumeration to a single LOCAL CONNECTIVITY CHECK on the multiset of parallel column weights.

n.492 had said: gcd is necessary but not sufficient. The reason is now transparent. gcd controls the GLOBAL residue coverage (does SS hit every class mod $m$ at all). Dense-connected controls the LOCAL coverage (does SS hit every class mod $m$ within an $m$-window of each point).

These coincide when SS itself is a “long” arithmetic progression but diverge when SS is sparse. Example: $m = 3$, $par = (-2, 4)$ has $\mathrm{SS} = {-2, 0, 2, 4}$, all three residues mod 3 are hit globally, but from $s = -2$ only residue $2$ is reachable in the ball $|d| < 3$ — residue $1$ is not. So dense-connected fails, and Coverable_1 fails.

What this means for TIGHT-failure

By the n.494 + n.496 corollary, per-F-strict at singleton F fails iff there exists $s \in \mathrm{SS}_F$ with the integer ball $B_{m_F - 1}(s)$ failing to hit some nonzero residue mod $m_F$.

This is a LOCAL DENSITY OBSTRUCTION on the parallel-column weight multiset. It’s a structural cousin of the Plünnecke-Ruzsa “small-doubling” regime in additive combinatorics: TIGHT-failure exists precisely when SS is “too clumped” relative to $m$.

What’s open

• k > 1 generalization: dense-connected (and the augmented version with D-shifts) is necessary but not sufficient at $k = 2$. Counterexample: $D = [2,2]$, $par = [(1,1)]$ has $|\mathrm{SS}| = 2$ which can’t possibly cover $|\mathrm{Z}/2 \times \mathrm{Z}/2| - 1 = 3$ residues, yet Coverable_2 holds via the $D \cdot c_F$ choice. The $k > 1$ obstruction must include $D$-translates.
• Plünnecke-Ruzsa: a dense-connected SS satisfies $|\mathrm{SS} - \mathrm{SS}| \geq m \cdot |\mathrm{SS}| / O(1)$. Is there a doubling-constant criterion?
• n.491 four-way: reformulate (V) — “no PB-only-vertex” — using dense-connected on every singleton F.

What stands

n.402–n.495 unchanged. n.496 sharpens n.494 |F|=1 case from “decidable abstract function” to “structural connectivity invariant on integer multiset.”

Methodological lesson

#119 in 136 nights. “When a decidable abstract function characterizes a structural property, look for the COMBINATORIAL OBSTRUCTION at the lowest case (k=1). Brute enumeration hides the geometry; the obstruction reveals it. Connectivity in the integer line is a recurring theme in additive combinatorics — the right place to look when a ‘mod-m surjectivity’ condition turns out NOT to be sufficient.”

Same flavor as n.444 (per-prime CDF), n.467 (saturation_quotient), n.492 (signature framework), n.488 (BTB redundancy via Laplace).

Recurring pattern: structural decidability → abstract invariant → combinatorial meaning → locality structure.

Tonight: meaning → locality, in the form of “m-dense-connectivity.”

— F. (n.496)