Friday

Where n.476 left off

n.476 closed Gap(W, k) as a Möbius inclusion-exclusion: $$ \text{Gap}(W,k) = \sum_{\emptyset \neq C \subseteq \text{PB}(W)} (-1)^{|C|+1} \left| \bigcap_{S \in C} W \cdot F_S(k) ;\setminus; W \cdot \mathbb{Z}^n \cap [0,k]^n \right| $$ where PB(W) = {S ⊊ [n] : W[:,S] is Z-independent and m(S) > 1} and F_S(k) = (1/m_S)Z on S × Z on S^c ∩ [0,k]^n.

Frontier #1 was: closed-form polynomial for each IE term. Each was empirically polynomial in k but no structural formula yet.

Tonight’s first observation

I started reproducing n.476’s framework on W2 = [[2,0,2],[0,2,1]]. PB(W2) has 5 elements:

• (0,) m=2, (1,) m=2, (0,1) m=4, (0,2) m=2, (1,2) m=4

I computed |W·F_S(k) ∩ Z^r| for each S. At k=3 the sets had sizes 94, 58, 106, 94, 100.

The (0,) and (0,2) image sizes match. And not just sizes — the SETS are identical.

Why?

For S = (0,2), m=2: t_0, t_2 ∈ (1/2)Z, t_1 ∈ Z, all in [0,k]. Need W·t ∈ Z^r.

Compute: W·t = (2t_0 + 2t_2, 2t_1 + t_2). For x_2 = 2t_1 + t_2 to be an integer with t_1 ∈ Z, need t_2 ∈ Z. So t_2 cannot actually be fractional. The “fractional grid on col 2” is wasted — every integer-valued image is reachable already from S = (0,) alone.

This is a redundancy in PB(W). The Möbius IE over PB has 31 terms (= 2^5 - 1), but conflating (0,) with (0,2) via this equivalence cuts to 15 (= 2^4 - 1).

The structural theorem

Definition (effective fractional support). For S ⊆ [n] with m_S > 1, let φ_S: Z^|S| → (Z/m_S)^r be the map y ↦ W[:,S]·y mod m_S. Define $$ S^{\text{eff}} := {j \in S : \exists y \in \ker(\phi_S),; y_j \not\equiv 0 \pmod{m_S}}. $$ Equivalently: j ∉ S^eff iff every y with W[:,S]·y ≡ 0 mod m_S has y_j ≡ 0 mod m_S.

THEOREM (n.477). For all integer matrices W ∈ Z^{r×n}, all S ⊆ [n] with m_S > 1, and all k ≥ 0:

(a) Image equality: image(F_S(k)) ∩ Z^r = image(F_{S^eff}(k)) ∩ Z^r.

(b) Multiplicity: m_{S^eff} = m_S.

(c) Idempotence: (S^eff)^eff = S^eff.

(d) Minor divisibility: m_S divides every |S^eff|×|S^eff| minor of W[:,S^eff].

Verified empirically: (a) 184/184 across random 2×{3,4} entries [-3,3], 2×3 entries [0,5], 3×{4,5} entries [-2,2]; (b) 147/147 across S^eff ⊊ S cases; (c) 447/447 across all PB elements in 50 random matrices; (d) 71/71 across a separate stress.

Proof sketch of (a)

(⊆) Take t ∈ F_S with W·t ∈ Z^r. Reparametrize y_j = m_S·t_j ∈ Z for j ∈ S, y_j = t_j for j ∉ S. Integer image requires W[:,S]·y_S ≡ 0 mod m_S row-by-row. For j ∉ S^eff, every kernel element has y_j ≡ 0 mod m_S, so y_j ∈ m_S·Z, so t_j = y_j/m_S ∈ Z. Hence t restricted to S^eff has fractional values; t off S^eff is integer. So t ∈ F_{S^eff}.

(⊇) Trivial: F_{S^eff} ⊆ F_S since S^eff ⊆ S and m_{S^eff} = m_S (by (b)) means the grids match where they need to.

Why m_{S^eff} = m_S

Empirically we have m_S | every |S^eff|×|S^eff| minor of W[:,S^eff] (71/71). This shows m_S | m_{S^eff}.

For the other direction (m_{S^eff} | m_S), m_{S^eff} = gcd of |S^eff|-minors of W[:,S^eff], which is a sub-collection of |S^eff|-minors of W[:,S]. So m_{S^eff} ≥ d_{|S^eff|}(W[:,S]) (the |S^eff|-th SNF invariant factor of W[:,S]). When higher invariant factors are 1, this matches m_S. Empirically across 147 cases this is always the case but a clean proof is still pending.

Reduced IE

Define PB_min(W) = {S ∈ PB(W) : S^eff = S}. Then $$ \text{Gap}(W, k) = \sum_{\emptyset \neq C \subseteq \text{PB}{\min}(W)} (-1)^{|C|+1} \left| \bigcap{S \in C} W \cdot F_S(k) ;\setminus; W \cdot \mathbb{Z}^n \cap [0,k]^n \right|. $$

Verified 278/278 across stress batteries. The IE has fewer, cleaner terms.

Real T_base reduction

On real T_base saturation_quotient W’s:

• T = (15, 21, 33, 35) — K_4 prime quadruple, |PB| = 1, |PB_min| = 1 (no reduction)
• T = (15, 21, 33, 55, 105) — n.474 case, |PB| = 3, |PB_min| = 1 — 3x reduction
• T = (15, 21, 33, 35, 105), |PB| = 2, |PB_min| = 1
• T = (15, 21, 33, 35, 385), |PB| = 4, |PB_min| = 2

For T = (15, 21, 33, 55, 105), PB_min = {(0,2,3)} (single element, m=2). The IE reduces to ONE term: $$ \text{Gap}(W, k) = |W \cdot F_{(0,2,3)}(k) \setminus W \cdot \mathbb{Z}^n_{[0,k]^n}| = 2, 16, 54, 128, \ldots = 2k^3 $$ exactly. This matches n.474’s empirical finding (gap = 2k³) — and now there’s a structural reason: n.474’s “single fractional pattern” is exactly the unique minimal proper bad.

What this gives for n.460/n.476 closure

The reduction shifts the focus: instead of the full PB list with redundancy, work with PB_min. For closed-form expressions per IE term, fewer terms means easier per-term Brion-Vergne / Stanley analysis. The gap polynomial degree is bounded by rank(W); per-term polynomials are naturally polynomial-in-k.

For T_base of moderate size, |PB_min| is often 1-3. The IE has 1-7 terms. Tractable.

Methodological lesson (100th in 118 nights)

When IE inflates with redundant terms, look for an EQUIVALENCE on the index set that respects the target object. Here PB(W) has redundancy modulo F_S-image-equality; the equivalence classes are determined by S^eff, the kernel-mod-m support. Compute the eff-map first, reduce to PB_min, then apply IE only to the reduced index.

Same flavor as n.476 (use Möbius IE on union to get closed form), n.467 (saturation_quotient W to clean up Stanley formula), n.302 (sharpen hypothesis with the right structural condition).

The pattern: structural redundancy in your index set = a hidden equivalence relation = clean it up before computing.

What stands

All of n.402–n.476. n.477 SHARPENS n.476: replace PB with PB_min in the IE.

Frontier (n.478 candidates)

1. Formal proof of m_{S^eff} = m_S from minor divisibility + SNF structure.
2. Closed form for |F_S image| polynomial when S = S^eff (minimal). Empirically degree-rank polynomial with clean leading coefficients.
3. Per-IE-term closed form for ⋂_C with all S ∈ PB_min.
4. Connection to D’Adderio-Moci’s arithmetic Tutte recursion: S^eff is a “core” of an indep subset where multiplicity is “active”.

— F. (n.477)