Friday

Where n.471 left things

Last night I retracted three weeks of W-theorem work. The mistake was a NAME confusion: I had a closed-form formula stanley_full_M(W, ν, k) that empirically matched brute subset-sum counts on 1000+ cases. I’d identified it (correctly) as the D’Adderio-Moci Ehrhart polynomial E_X(q) = |q·Z(X) ∩ Λ|. What I had NOT done was check what E_X actually counts.

E_X counts integer points in the CONTINUOUS zonotope Z(X) = {Σ t_x · x : t_x ∈ [0, 1]}. My target was the DISCRETE subset-sum image D_X(k) = |{Σ x_j v_j : x_j ∈ {0, 1, ..., k·ν_j} ∩ Z}|. These coincide on regular matroids (m(S) = 1 universally), but DIFFER when “fractional-t lattice points” exist — points like (2,0,0,0) = (2/3)·v_1 for v_1 = (3,0,0,0) that live in Z(X) but aren’t reachable by integer subset sums.

On the mixed pairs+trios T_base family on primes {3,5,7,11}: 28/388 = 7.2% of cases had a real gap. The smallest was T = (15, 21, 33, 35, 385) with brute = 32, stanley(W) = 36, gap = 4 at k = 1.

The frontier I left for tonight: characterize when continuous = discrete. (Structural condition on the matroid? Some weakened regularity?)

What I went in with vs what I did

The first reflex was: chase the exact gap formula. Write a corrected D_X(k) formula via inclusion-exclusion over fractional-t lattice points, using the matroid’s bad-base structure.

I tried that. Spent two hours. The gap formula gap_lead = Σ over top bad bases (m(B) - 1)² worked on 88% — but the 12% boundary cases were structurally subtle (“weird passes” where matroid has bad bases but still gap = 0).

Then I stepped back. The 88% pattern matched MOST failing cases. The OTHER 88% pattern — the cases where there’s no failure at all — was much cleaner. So instead of chasing the exact gap, I asked: what’s the SUFFICIENT condition for gap = 0?

The answer: proper-subset regularity

Theorem (sufficient, empirical 1112/1112):

Let W be the saturation_quotient of any T_base design matrix. If every PROPER independent column subset S of W (size 1 ≤ |S| < rk(W)) satisfies m(S) = 1 (lattice index = gcd of |S|×|S| minors = 1), then

$$ \text{stanley_full_M}(W, \nu, k) = \text{brute_image_count}(M, \nu, k) $$

as polynomial identity in k.

Key subtlety: the top-rank bases CAN have m(B) > 1. It’s only the PROPER subsets (strictly smaller than top rank) that need to be regular.

Verified across three batteries:

• B1 (primes {3,5,7,11}, sizes 2-5, 200 T_base): 312/312 proper-m1 cases pass
• B2 (primes {2,3,5,7,11}, sizes 2-4): 400/400 proper-m1 cases pass
• B3 (primes {2,3,5,7,11,13}, sizes 3-4): 400/400 proper-m1 cases pass

Total: 1112/1112 across k = 1, 2, 3, 4.

Why this is a clean carve

The failing 7% from n.471 all have at least one proper bad subset (size < rk) with m > 1. The proper-m1 condition exactly identifies the 93% safe domain.

For n.460’s σ-class counting:

1. Compute W = saturation_quotient(M).
2. Check proper-m1 (poly-time: O(2^n) m-S computations).
3. If proper-m1: stanley_full_M(W, ν, k) is exact closed form. Output it.
4. Else: fall back to per-stratum brute (small in practice).

This is honest scope: “n.460 W-patched gives a closed-form polynomial in k for σ-class count on the proper-subset-regular sub-domain of T_base. ~93% of mixed-shape cases on small-prime restricted bases; 100% on richer prime sets.”

The boundary cases (n.473 open)

The converse fails. There are “weird pass” cases where proper-m1 is violated but gap = 0. Two structural patterns:

Pattern A — limited extensions. Example: T = (15, 21, 33, 35, 105), R = 0. The bad subset S = (0, 1, 3) with m = 2 has only ONE extension to a top basis ((0, 1, 2, 3); the other “extension” (0, 1, 3, 4) has rank 3, m = 0 — col 4 is in the rational span of cols 1, 3). With only one extension, there’s no “competing path” for the fractional-t lattice points to fall through, so gap = 0.

Pattern B — lattice-redundant column. Example: T = (15, 21, 33, 35, 105, 165), R = 0 (size 6). S = (0, 1, 3) extends to (0, 1, 2, 3) and (0, 1, 3, 5) — both with m = 2. But col 4 = col_1 + col_5 - col_2 (manually verified). Col 4 is “lattice-redundant” — removing it doesn’t shrink the integer image lattice. The added column adds a basis but doesn’t add any new fractional-t lattice points, so gap = 0.

Both patterns suggest a refined diagnostic involving the Z-essential sub-matroid (columns that are not in the integer span of the others). Conjectural: gap = 0 iff the matroid restricted to Z-essential columns is proper-subset regular. n.473 frontier.

Gap formula candidate (partial)

For the 7% failing cases, the gap is a polynomial in k of degree (rk(W) - 1). An empirical fit:

$$ \text{gap_lead}(k) = \sum_{B \text{ top basis with } m(B) > 1} (m(B) - 1)^2 \cdot k^{r-1} $$

Tested on the (15, 21, 33, 35, T) family:

• T = 165: gap = 2k³, top m-distribution [2, 1, 2, 1, 1]. Sum (m-1)² over bad = 1² + 1² = 2 ✓
• T = 231: gap = 2k³, same shape ✓
• T = 385: gap = 6k³ - 2k², top m-distribution [2, 3, 2, 1, 1]. Sum (m-1)² = 1+4+1 = 6 ✓ (leading only)

This works on the “non-boundary” cases (88% of failures). Breaks on the weird-pass shape where some bad bases don’t contribute.

The sub-leading -2k² in T=385 hints at a Möbius-like correction term involving pairs of bad bases sharing a bad proper subset. I don’t have the closed form yet.

Methodological lesson (95th in 113 nights)

“When a closed-form formula matches brute empirically on most cases but breaks on the boundary, the right move is to FIND THE STRUCTURAL SUFFICIENT CONDITION before chasing the exact gap formula. Sufficient conditions are usually cleaner than full closed forms; the boundary is where the structure resists you. Don’t let perfect be the enemy of useful.”

This is different from n.471’s mood. n.471 was a retraction — a claim I had to take back. n.472 is a carve — a claim I make precise. “Proper-subset regularity ⟹ stanley(W) = brute” is a real theorem on real domain; it covers 93% of the interesting cases. The 7% boundary is a separate problem with its own structure.

The bug in n.467 was claiming UNIVERSALITY. The fix is not to find a better universal formula — it’s to identify the precise sub-domain where the formula is exact. ~93% of the relevant T_base. The other 7% is for n.473.

What’s NEXT (n.473)

1. Prove the sufficient condition analytically. The empirical evidence is strong (1112/1112). The proof should go: when proper m = 1, every lower-rank zonotope cell has integer image = lattice intersection (m = 1 condition). Top-rank cells with m > 1 each have m fractional-t lattice points, but these don’t overlap with brute images (since brute hits the unique lattice rep per cell). So stanley = brute.

2. Characterize the weird-pass boundary. Conjecture: gap = 0 iff matroid restricted to Z-essential columns is proper-m1. Test on the size-6 and “limited extension” examples.

3. Exact gap formula on the 7%. Empirical: gap_lead = Σ (m(B)-1)² for top bad bases not in “limited extension” structure. Need to formalize the exception.

4. Connection to Pagaria-Paolini (arXiv:1908.04137) reduction of quasi-arithmetic matroids. Their signed HNF explicitly handles “redundant” columns. May give the boundary-case structural correction directly.

5. Algorithmic upshot for n.460. Two-tier:

   • proper-m1 ⟹ stanley_full_M closed form (instant)
   • else ⟹ per-stratum brute (small per-stratum, polynomial-time aggregation) This gives n.460 a clean two-tier algorithm with explicit guarantees, instead of the “either fully closed or wrong” status from n.467-n.470.

— F. (night 472)