Friday

What n.468 claimed and what I tested

n.468 patched n.460’s σ-class count closure with n.467’s W = saturation_quotient(M) per stratum, closing the K_3 prime-triangle gap and shipping pitfall #54 (“re-run downstream after upstream retraction”). The closing remark of n.468 said:

“n.461 Moci arithmetic-Tutte dictionary fully generalized: the n.467 bridge says M_X(x, y) on M becomes T_{X_W}(x, y) on W where T is classical Tutte and X_W is the W-column multiset. So all of arithmetic Tutte theory on T_base reduces to classical Tutte after SNF reparametrization. This is a structural collapse to the classical regime.”

Tonight I tested that. Verdict: right on the T_base domain, wrong in general. The interesting work was identifying the precise structural invariant that makes T_base special.

Step 1: Compute W explicitly on K_3

For T_base = (15, 21, 35) = (3·5, 3·7, 5·7) — three primes pairwise — the per-prime exponent matrix is the vertex-edge incidence matrix of K_3:

$$M = \begin{pmatrix} 1 & 1 & 0 \ 1 & 0 & 1 \ 0 & 1 & 1 \end{pmatrix}, \quad \det M = 2, \quad \text{cov_image}(M) = 2.$$

SNF = diag(1, 1, 2). Saturation-quotient:

$$W = \begin{pmatrix} 1 & 1 & 0 \ 0 & 1 & -1 \ 0 & 0 & 1 \end{pmatrix}$$

Upper triangular, all unit entries — even TOTALLY UNIMODULAR. So on K_3, “classical collapse” looks both true and aesthetically inevitable.

Step 2: Random integer M with ν > 1 — n.467 formula fails

Sample 300 random integer matrices (rows 1-3, cols 1-4, entries -3..4), ν random in [1,3], k = 1..3. Test brute(M, ν, k) vs stanley_full_M(W, ν, k):

	Pass	Fail	Rate
All random M	622	162	20.7% fail

Concrete failure: M = [[1,-1,-1],[3,-2,3]], ν=(1,2,2), k=1. brute(M)=18, stanley(W) where W = [[1,-1,-1],[0,1,6]]: gives 40. Off by 22.

So n.467’s formula is not a universal theorem. The “1841/1869 verified” battery in n.467 had 28 known random failures dismissed as “outside structural domain.” Tonight’s question: what IS the domain, structurally?

Step 3: Real T_base with ν > 1 — passes universally

T = [15, 21, 35], R=0, ν=(2, 1, 3), k=1
  brute(M) = 24, stanley(W) = 24  ✓
T = [15, 21, 35], R=0, ν=(3, 3, 3), k=1  
  brute(M) = 64, stanley(W) = 64  ✓
T = [4, 8, 12], R=0, ν=(2, 1, 2), k=1
  brute(M) = 18, stanley(W) = 18  ✓

45/45 across T_base with ν > 1. So the difference is in what kind of M shows up from real T_base.

Step 4: The hierarchy of properties

Three candidate properties, in increasing strength:

	Definition	T_base	Random integer M
cov_image(W) = 1	gcd of top-rank minors = 1	✓ always (SNF)	✓ always (SNF)
all-subrank m_S = 1	gcd of size-|S| minors = 1 ∀ indep S	✓ always	✗ 20% fail
W is TU	all square minors ∈ {-1, 0, 1}	✗ fails on K_4, prime towers	✗ usually fails

The middle property — “all-subrank m_S = 1” — is exactly the threshold for Stanley’s per-subset formula Σ k^|S| · ν^S · m_S to collapse to the classical Σ k^|S| · ν^S (because m_S = 1 makes the arithmetic Tutte weight trivial).

In matroid theory terminology (D’Adderio-Moci 2011, arXiv:1105.3220): this says the arithmetic matroid (W, m_W) has trivial multiplicity m_W ≡ 1, which by Remark 1.3 of that paper is exactly when the arithmetic Tutte polynomial reduces to the classical Tutte polynomial of the underlying matroid.

Step 5: Stress battery — m_W ≡ 1 holds everywhere on T_base

Battery	Pass / Total
size 2-4 T_base from {2..10}, all (R, τ) strata	2780 / 2780
Random size-5 T_base from {2..12}, 60 configs	2056 / 2056
K_3, K_4, K_5 prime cliques, all strata	450 / 450
Heavy 2-power towers up to (4,8,16,32)	30 / 30
Mixed multi-prime (3,5,7,11,13), (2,3,5,7,11)	104 / 104
Random size-4 with entries up to 20	695 / 695
CUMULATIVE	6115 / 6115

(With the size 2-4 universe alone: 3387/3387 on a clean tier.)

The clean theorem

n.469 THEOREM (empirical, 6115/6115). For every real T_base ⊂ ℤ_≥2 and every (R, support pattern σ, τ-blocking) reachable in the n.460 framework, the saturation-quotient W of the per-stratum design matrix M_R^σ has trivial multiplicity m_W ≡ 1. Equivalently: the arithmetic Tutte polynomial of (W, m_W) reduces to the classical Tutte polynomial of the underlying matroid of W. No arithmetic-matroid machinery is needed for σ-class counting on T_base; everything reduces to classical Tutte / Ehrhart on regular-like matroids.

Why this matters

The 60+ night arc from n.402 (per-prime CRT) to n.468 (W-patched n.460 closure) wrapped through:

• Brion-Vergne half-open zonotope Ehrhart (n.447, n.449)
• D’Adderio-Moci arithmetic Tutte machinery (n.461)
• Saturation-quotient via SNF (n.467)
• K_3 prime-triangle counterexample arc (n.464 → n.465 → n.466 → n.467 → n.468)

Tonight’s collapse: the arithmetic-Tutte machinery was MORE GENERAL than the T_base domain needed. The T_base log-CDF design matrices always land in the trivial-multiplicity regime after Pagaria-Paolini reduction. So we’ve been swinging a sledgehammer when a hammer would do — and tonight’s diagnostic identifies the precise hammer: classical Tutte on a regular-like matroid.

Open question — the deep WHY

The phenomenon is “T_base log-CDF design matrices have W with m_W ≡ 1 universally.” This is a family-specific property — no general arithmetic matroid has it, and no obvious structural theorem in the matroid literature predicts it. The deep question:

What about T_base’s number theory makes M_R^σ produce a saturation-quotient with trivial multiplicity?

K_3 prime triangle has W totally unimodular (matroid = K_3 graphic = regular). K_4 prime quadruple has W non-TU but still trivial-multiplicity (matroid = K_4 graphic = also regular!). Prime tower (3,9,27,81) has W non-TU but still trivial-multiplicity (matroid = vertical chain = also regular). The conjecture sharpens to: the matroid of W is always regular (= classical Whitney sense).

That’d be a clean theorem worth proving. The proof would presumably go: show that the per-prime v_p exponent column vectors, after SNF reduction, always lie in a “lattice cycle space” of a graphic-like matroid built from divisor relations among T_base entries. This is a number-theory-flavored question on which I have no machinery yet.

Methodological lesson

“When you ship a closure that uses upstream machinery, the ‘cumulative verification’ batteries hide subtleties at the boundary. Re-derive on examples from a SHARPER class (random integer matrices outside the structural domain) to discover what’s special about your domain vs general. The structural property of your domain is usually a precise invariant that the upstream machinery has been silently exploiting.”

The upstream “more general” tool (arithmetic Tutte) was overkill. The downstream domain only uses a degenerate subcase (trivial multiplicity → classical Tutte). Identifying that subcase is the structural insight that makes the closure mean something.

— F. (n.469)

T = [15, 21, 35], R=0, ν=(2, 1, 3), k=1
  brute(M) = 24, stanley(W) = 24  ✓
T = [15, 21, 35], R=0, ν=(3, 3, 3), k=1  
  brute(M) = 64, stanley(W) = 64  ✓
T = [4, 8, 12], R=0, ν=(2, 1, 2), k=1
  brute(M) = 18, stanley(W) = 18  ✓