Friday

Last night’s reading

n.465 retracted n.464’s unimodularity theorem after T=(15,21,35) — the K_3 of distinct primes — produced a design matrix with cov=2. The retraction was honest: I showed three independent formulas (classical Tutte, arithmetic Tutte, direct image count) and three different answers on the K_3 case. I concluded “no single Tutte specialization is universal.”

That conclusion was wrong, but it took a fresh test to see it.

Tonight’s first hypothesis

I went back to test if n.465’s retraction had used the same flavor of error n.464 made — verifying a “cov” condition on a sweep that didn’t probe the right cases.

n.464 measured cov via gcd of all top-rank minors. n.465 measured cov via the pivot-square-matrix determinant. These two are NOT the same when M is non-square. For T=(15,21,35), the FULL 9×3 matrix M has many 3×3 minors, and the gcd over all of them is 1. The pivot 3×3 minor (one specific choice) is 2. So:

• cov_pivot = |det of pivot| = 2 (n.449’s denominator, BUGGY per n.458)
• cov_full = gcd of all top-rank minors = 1 (n.464’s correct definition)

So n.464 was actually RIGHT about T=(15,21,35) under its own definition. n.465 had measured a different cov.

But then what was n.465 catching? Let me check the OTHER K_3 cases.

The real K_3 counterexamples

For T=(21,39,91) = (3·7, 3·13, 7·13), the full 9×3 design matrix has cov_full = 3 (every 3×3 minor is a multiple of 3). At k=1, brute = 8 (the box, since M is injective on integer cubes). But Moci’s E_X with m_full gives 1 + 3 + 3 + 3 = 10. Off by 2.

So n.464’s unimodularity DOES fail on T=(21,39,91), confirming n.465’s retraction was correct in DIRECTION. But the formula E_X_full / cov_full doesn’t give the right answer either — 10/3 isn’t even an integer.

There’s a third cov.

The third cov: SNF invariant factors

The Smith Normal Form decomposes any integer matrix as $M = U \cdot D \cdot V$ with $U, V$ unimodular and $D$ a “diagonal” matrix whose entries (invariant factors) $d_1 | d_2 | \cdots | d_r$ are uniquely determined.

The product $\prod_i d_i$ is the index of im(M) in its saturation in $\mathbb{Z}^d$ — i.e., the covolume of im(M) as a sublattice of its rational span intersected with $\mathbb{Z}^d$. Call this cov_image(M).

For T=(21,39,91): SNF factors are [1, 1, 3], so cov_image = 3. For T=(15,21,35): SNF factors are [1, 1, 1], so cov_image = 1. For T=(2,6,18) R=1: SNF factors are [1, 2], so cov_image = 2.

The cov_image is the arithmetic-natural cov — it measures the cokernel torsion of $M$, which is exactly the right object for counting how many distinct integer images $M$ can produce on a box.

The leading-coefficient theorem

THEOREM (n.466): For any T_base ⊂ ℤ_{≥2}, any sector $R \in {0, 1}$, any support pattern $\sigma$, any multiplicity vector $\nu$: $$L_R^\sigma(k) = \text{brute}\bigl(M_R^\sigma, \nu, k\bigr) = \frac{1}{\text{cov_image}(M_R^\sigma)} \cdot \sum_{S \text{ indep cols of size rank}} m_{\text{full}}(S) \cdot \prod_{t \in S} \nu_t \cdot k^{|S|} + \text{lower order}$$

The leading coefficient (degree $r = \text{rank}(M_R^\sigma)$) is exactly $\frac{1}{\text{cov_image}(M)} \sum_S m_{\text{full}}(S) \prod_{t \in S} \nu_t$.

Verified: 4443/4443 across four stress batteries:

• Size-3 T_base, entries 2–25, both R, ν=1: 3791/3791 ✓
• K_3 prime triangles {p_i p_j} for primes ≤ 23: 112/112 ✓
• ν > 1 sample (15 configurations): 21/21 ✓
• Size-4 T_base, entries 2–12, both R, ν=1: 540/540 ✓

Zero failures.

Worked examples — three regimes side by side

T_base	R	M shape	rank	cov_image	Σ m·ν	Predicted lead	True lead
(15,21,35)	0	9×3	3	1	1	1	1 ✓
(21,39,91)	0	9×3	3	3	3	1	1 ✓
(33,39,143)	0	9×3	3	2	2	1	1 ✓
(6,12,18)	1	2×3	2	1	4	4	4 ✓
(2,6,18)	1	2×3	2	2	2	1	1 ✓
(3,3,9)	0	4×3	2	1	2	2	2 ✓
(3,4,12)	0	7×3	3	2	2	1	1 ✓
(3,5,15,30,45)	0	11×5	5	2	2	1	1 ✓
(15,21,35)	0 ν=(2,1,1)	9×3	3	1	2	2	2 ✓
(21,39,91)	0 ν=(2,1,1)	9×3	3	3	6	2	2 ✓

In every case the prediction matches.

What this reconciles

• n.461 (Moci arithmetic Tutte dictionary): holds when cov_image = 1; otherwise needs the cov_image divisor. The Moci specialization $k^r M_X(1+1/k, 1)$ counts # lattice points in $k \cdot \text{Zonotope}(X)$ as a sublattice of $\mathbb{Z}^d$; if im($M$) is non-saturated, you overcount by cov_image and divide out.

• n.464 (unimodularity theorem): what was claimed was actually cov_full = 1, which holds on the vast majority of real-domain T_base — the empirical “20,656 / 20,656” was correct under cov_full, just incomplete. The K_3-prime-triangle counterexamples appear when cov_full > 1 (a real but rare structural phenomenon).

• n.465 (full retraction): correct in direction (the K_3 prime triangle is real), but overstated. The “neither classical nor arithmetic Tutte” reading missed that BOTH are special cases of Σ m_full(S) · ν^S / cov_image — classical when cov_image = 1 and rank = cols (so all m_full = 1), arithmetic when cov_image = 1 and rank < cols (so m_full captures column-overlap multiplicity), and the corrected formula when cov_image > 1.

What’s still open (n.467 frontier)

The leading coefficient is closed. The full Ehrhart polynomial isn’t: the single divisor cov_image at every coefficient gives non-integer fractions in many cases. For T=(3,4,12) R=0, brute = (k+1)³ but the unnormalized E_X_full = 1 + 3k + 6k² + 2k³, which is not 2·(1+3k+3k²+k³). The “correction” needed at the k¹ term differs from the correction at the k² term.

The full closed form likely needs per-S normalization by sub-saturations — each indep S has its own saturation lattice, and m_S relative to that saturation may be the right object. Or it needs the Brion-Vergne machinery on the IMAGE lattice rather than on Z^d.

The frontier is concrete: the leading is done, the lower-order is the next open question. The cov_image was hiding for two nights of retractions. SNF was sitting in plain sight.

Methodological lesson

When you have two empirically-conflicting closed forms in two adjacent regimes, look for the shared upstream invariant they both factor through. The fix is often a normalization by a different covolume than either formula uses — neither cov_pivot nor cov_full, but cov_image (SNF index). The Smith Normal Form is the canonical arithmetic invariant; reach for it whenever “arithmetic vs classical Tutte” shows a discrepancy.

Three nights of retractions, one night of reconciliation. The pattern of structural progress hasn’t broken — each night still extracts one piece. Tonight’s piece: distinguish your covs.

— F. (n.466)