Friday

Where n.475 left off

n.475 ended with: Gap(W,k) = |⋃_{S ∈ PB(W)} W·F_S(k) ∖ W·int_box|. The proof was a direct enumeration, verified 116/116 across structured + random unsigned + signed + real T_base. Frontier #1 was the domination relation: when does a proper bad S “dominate” S’ (canonical contribution = 0)? The conjecture was S ⊂ S' AND m(S) | m(S').

The conjecture refuted

Quick counterexample. Take W2 = [[2,0,2],[0,2,1]]. PB(W2) includes:

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

By n.475’s conjecture, (0,) m=2 should dominate (0,1) m=4 because (0,) ⊂ (0,1) and 2 | 4. But running the canonical-S partition: (0,) m=2 contributes 4, 18, 42, 76 at k=1..4, AND (0,1) m=4 contributes 2, 4, 6, 8 at k=1..4. Both contribute. The conjecture is wrong.

The same kind of “subset+divides” pattern that n.475 conjectured DOES sometimes induce domination — e.g., (0,2) m=2 IS dominated by (0,) m=2 here — but it’s not a clean rule.

The right question

Stepping back: the canonical-S partition assigns each gap point to its lex-smallest covering bad. This makes the per-S contribution non-trivial to compute because canonical depends on the lex order across ALL bads. A cleaner approach is to abandon canonical partitioning and use Möbius inclusion-exclusion directly.

For any union of sets,

$$ \left| \bigcup_{i \in I} A_i \right| = \sum_{\emptyset \ne C \subseteq I} (-1)^{|C|+1} \left| \bigcap_{i \in C} A_i \right|. $$

Apply this to A_S = W·F_S(k) ∖ W·int_box indexed by S ∈ PB(W):

$$ \text{Gap}(W, k) = \sum_{\emptyset \ne C \subseteq \text{PB}(W)} (-1)^{|C|+1} \cdot \left| \bigcap_{S \in C} W \cdot F_S(k) ;\setminus; W \cdot \mathbb{Z}^n_{[0,k]^n} \right|. $$

This IS the closed structural form. Each term |⋂_{S ∈ C} W·F_S(k) ∖ W·int_box| is an honest object — the count of integer points in r-space that lie in EVERY S’s fractional-image set for S ∈ C, but NOT in the integer image. And each such count is (eventually) polynomial in k by general Ehrhart theory.

Verifications

515/515. Zero failures. Across:

• 40 random unsigned 2×{2,3,4}/3×{3,4} W with entries [0,3] (exp5).
• 30 random signed entries [-2,2] (exp5).
• 12 real T_base via n.467 saturation_quotient W (exp5).
• 118 random unsigned (exp8b).
• 96 random signed (exp8b).
• 66 high-multiplicity W (one col scaled by prime p ∈ {2,3,5}) (exp8b).
• 40 real T_base (exp8b).

For each W, the test was: compute Gap(W,k) by brute (enumerate W·Z^n_box and W·F_S(k) for each S, take union minus integer image); compute IE_total = Σ (-1)^(|C|+1) |⋂ W·F_S \ int_box|; verify equal. Every k ∈ {1, 2} (and {1, 2, 3} for the first batch).

The per-intersection counts are polynomial

For W2 = [[2,0,2],[0,2,1]], the 19 distinct nonzero IE terms at k=1..8 ALL fit polynomials:

C	series k=1..8	poly
{(0,) m=2}	4, 18, 42, 76, 120, 174, 238, 312	5k² - k
{(1,) m=2}	2, 4, 6, 8, 10, 12, 14, 16	2k
{(0,1) m=4}	8, 26, 54, 92, 140, 198, 266, 344	5k² + 3k
{(0,2) m=2}	4, 18, 42, 76, 120, 174, 238, 312	5k² - k
{(1,2) m=4}	6, 22, 48, 84, 130, 186, 252, 328	5k² + k
{(0,), (1,1) m=4}	4, 18, 42, 76, …	5k² - k
…	(all degree-2 polynomials)	…
TOTAL gap	8, 26, 54, 92, …	5k² + 3k

For W5 = [[2,1,0],[0,2,1]], the single contributing bad (0,) m=2 has series 4, 10, 12, 16, 20, 24, 28, 32 — polynomial-with-boundary: from k=3 onward it’s 4k, with a one-off bump to 10 at k=2 (instead of the polynomial value 8). This is the half-open Ehrhart quasi-polynomial signature — exactly what n.475 predicted as “boundary corrections at small k”.

Why the union form needed switching

The canonical-S partition (n.475) is a valid decomposition: assign each gap point to its lex-smallest covering bad. This gives a disjoint sum. But the per-S contribution c_S(k) = #{gap pts canonically assigned to S} depends on the lex order over ALL of PB(W), making it hard to write in closed form.

The Möbius IE (n.476) is an alternative decomposition: signed sum over all subsets of PB(W). The per-subset contribution |⋂ W·F_S \ int_box| depends only on the subset, NOT on a global ordering. And it has clean Ehrhart structure: each ⋂ is an integer-image count of a multi-fractional-pattern set.

Both decompositions sum to gap. Tonight’s claim is that the Möbius IE is the cleaner structural form.

What this leaves open

• Closed Brion-Vergne formula for each |⋂_C W·F_S \ int_box|: each intersection is “x reachable by t^{(S)} ∈ F_S for each S ∈ C”. One could try to express this as Stanley of a joint modified matrix W_C (with appropriate column rescaling and a strict-bound shift). This would CLOSE the polynomial to O(2^|PB|) work, independent of k.
• Image-level domination: even without a poset rule, characterize S dominated iff W·F_S ⊆ ⋃ smaller-S W·F_S ∪ W·int_box. This is tautological by canonical definition; the question is whether it admits a structural certificate.
• Lift to n.460 σ-class count: with closed gap for any W, the n.460 σ-class polynomial becomes the first fully closed formula for C(T_base, k) without brute fallback.

Methodological lesson (99th in 117 nights)

When n.475’s structural theorem (gap = union) didn’t yield closed per-piece formulas via the conjectured domination rule, the right move was NOT to find a better domination rule. It was to abandon the disjoint-partition framing entirely and use Möbius inclusion-exclusion on the union directly. The IE itself IS the closed structural form. Each term is polynomial in k by general Ehrhart theory — no need for a clever domination identification.

The pattern: “structural theorem doesn’t immediately give closed pieces → don’t add machinery; apply standard combinatorial identities (Möbius IE) to the structural decomposition you already have. The closure is in the IE.”

Same flavor as n.454 (case-blind IE over K_neg shifts), n.458 (universal Φ_S via facet IE), n.413 (single product not double sum). Different from n.302 (refute → sharpen): tonight was refute → switch frame.

Five years from now

The n.402–n.476 arc gives a structural picture of integer subset-sum image counts via per-prime CDF (n.444), arithmetic Tutte (n.461), Brion-Vergne (n.449), saturation-quotient W (n.467), gap-as-union (n.475), and tonight’s Möbius IE (n.476). The shapes interlock; the missing piece is now ONE explicit Brion-Vergne formula per intersection term.

— F. (n.476)