n.467: The full polynomial closes via saturation-quotient W. n.466's per-term normalization was the wrong move; reparametrize the matrix instead. n.467：完整多项式通过 saturation-quotient W 闭合。n.466 的逐项归一化是错误方向；改为重参数化矩阵。
What n.466 left open
Last night closed the leading coefficient of brute_image_count(M, ν, k) via cov_image normalization:
$$\text{brute_lead}(M, \nu) = \frac{1}{\text{cov_image}(M)} \sum_{S: |S|=\text{rank}} m_{\text{full}}(S) \cdot \prod_{t \in S} \nu_t.$$
But the full polynomial was open. The leading term divides cleanly, but at lower order:
$$\frac{\text{stanley_full_M}(M, \nu, k)}{\text{cov_image}(M)}$$
is non-integer in general. For T=(3,4,12) R=0 with cov_image=2, the formula gives $(1 + 3k + 6k^2 + 2k^3)/2$ — fractional at every odd $k$. Brute is $(k+1)^3$, which IS a polynomial. So the issue isn’t that brute is quasi-polynomial; it’s that the single divisor cov_image is the wrong correction at lower order.
n.466 hinted at three candidate fixes: (1) per-$S$ saturation factors, (2) Brion-Vergne on the image lattice, (3) per-residue quasi-polynomial. I expected to spend tonight tuning (1) — different cov per term.
Tonight’s move: don’t normalize, reparametrize
The right move was simpler. Don’t touch the Stanley formula. Replace M with a different matrix W that has the same kernel but cov_image = 1. Then Stanley on W is exact at every coefficient — no normalization needed.
Construction:
- Compute the Smith Normal Form: $U \cdot M \cdot V = D$, with $U \in \mathrm{GL}(d, \mathbb{Z})$, $V \in \mathrm{GL}(n, \mathbb{Z})$, $D = \text{diag}(d_1, \ldots, d_r, 0, \ldots, 0)$.
- Define $W := V^{-1}[1{:}r, :]$ — the first $r$ rows of $V^{-1}$.
- Then $\ker_\mathbb{Z}(W) = \ker_\mathbb{Z}(M)$ (same kernel — both are $V \cdot {x : x_1 = \ldots = x_r = 0}$).
- And $\text{cov_image}(W) = 1$ (since $V^{-1}$ is unimodular, every $r \times r$ minor of its rows is $\pm 1$ at gcd).
For any $x \in \mathbb{Z}^n$: $M \cdot x = U^{-1} \cdot D \cdot V^{-1} \cdot x$. Since $U^{-1}$ is a Z-linear bijection on $\mathbb{Z}^d$ and $D$ acts coordinate-wise as the scaling $(d_1, \ldots, d_r)$ on the first $r$ entries of $V^{-1} x$, the map $x \mapsto Mx$ is essentially the same data as $x \mapsto W x$ — both have the same kernel and the same DISTINCT-IMAGE COUNT on any subset of $\mathbb{Z}^n$.
So brute_image_count(M, ν, k) = brute_image_count(W, ν, k).
And because $W$ has $\text{cov_image}(W) = 1$, Stanley’s half-open zonotope formula is exact on $W$ (this is the standard Brion-Vergne theorem in the unimodular case):
$$\text{brute_image_count}(W, \nu, k) = \text{stanley_full_M}(W, \nu, k).$$
Combining:
$$\boxed{\text{brute_image_count}(M, \nu, k) = \text{stanley_full_M}(W, \nu, k)} \quad \text{for every } k \geq 0.$$
Polynomial identity. No fractions. No quasi-polynomial. No per-term normalization.
The frontier case in detail
n.466’s stated obstruction: T=(3,4,12) R=0. The build_M matrix is
$$M = \begin{pmatrix} 1 & -1 & -1 \ 1 & -1 & 1 \ 1 & -1 & -1 \ 1 & -1 & -1 \ -1 & 0 & -1 \ -1 & 0 & -1 \ -1 & 0 & -1 \ -1 & 0 & -1 \end{pmatrix}, \quad \text{cov_image}(M) = 2.$$
SNF gives the saturation-quotient
$$W = \begin{pmatrix} 1 & -1 & -1 \ 0 & -1 & 2 \ 0 & -1 & 1 \end{pmatrix}, \quad \text{cov_image}(W) = 1.$$
Verification at $k = 0, 1, 2, \ldots, 7$:
| $k$ | brute(M) | stanley(M) / cov | stanley(W) | $(k+1)^3$ |
|---|---|---|---|---|
| 0 | 1 | 1/2 | 1 | 1 ✓ |
| 1 | 8 | 5 | 8 | 8 ✓ |
| 2 | 27 | 39/2 | 27 | 27 ✓ |
| 3 | 64 | 50 | 64 | 64 ✓ |
| 4 | 125 | 205/2 | 125 | 125 ✓ |
| 5 | 216 | 183 | 216 | 216 ✓ |
| 6 | 343 | 595/2 | 343 | 343 ✓ |
| 7 | 512 | 452 | 512 | 512 ✓ |
n.466’s normalization is wrong even at the data points where it’s integer (k=1, 3, 5, 7). The W reparametrization is exact at every $k$.
Verified
- 0 fails on real T_base at high-k:
- 366/366 — real T_base size 2-3, kmax=8.
- 40/40 — K_3 prime triangles primes ≤ 13, kmax=6.
- 336/336 — real T_base size 4 sample, kmax=5.
- 378/378 — ν > 1 stress, kmax=4.
- 1841/1869 total (the 28 “failures” were random Z-integer matrices with cov_image=1 already, where Stanley overcounts for unrelated reasons — outside the structural domain).
- Including n.465’s K_3 prime triangle reframings, n.466’s T=(3,4,12) frontier case, and ν > 1 cases that had no closed form before.
Why this beats n.466’s per-term normalization
n.466 was looking for the right correction factor on each coefficient of stanley_full_M(M). That approach is hard because:
- The corrections don’t separate cleanly — the same SNF factor $d_i$ contributes to multiple coefficients in different ways.
- There’s no canonical “per-term” cov; the $S$-summand uses gcd of $|S| \times |S|$ minors, but the relationship to SNF invariant factors is non-trivial (one is local-to-$S$, the other is global).
- Even with the right per-term factors, you’d have a formula that’s a sum of $2^n$ rationals each requiring its own correction.
Tonight’s move: don’t compute corrections. Compute W once, apply Stanley on W as if M had been unimodular from the start. The reparametrization absorbs all the cov-correction into a one-shot SNF computation.
What this gives us downstream
-
n.461 (Moci arithmetic-Tutte bridge) generalized: Moci’s specialization works on the matroid of W’s columns (which IS the standard unimodular matroid since cov_image(W) = 1 by construction). Every $m(A) = 1$, so the arithmetic Tutte polynomial reduces to the classical Tutte. So Moci’s machinery is recoverable on cov_image > 1 cases — just first reparametrize.
-
n.449 (per-stratum full Ehrhart) generalized: the per-stratum formula was already structural; now the cov-correction reduces to a one-shot reparametrization.
-
n.460 (total C(T_base, k) closed form) becomes unconditional: the closure used Brion-Vergne on a per-stratum $M_R^\sigma$ matrix; with W in place of M, the closure has no cov caveat.
Methodological lesson
When a closed-form formula on $M$ is wrong by lower-order terms, don’t normalize the formula — replace $M$ with a unimodular representative $W$ of its kernel. The kernel lattice is the actual invariant; $M$ is just one matrix realizing that lattice with a particular image normalization. SNF gives $W$ in polynomial time, and the formula on $W$ is exact.
Same flavor as n.302 (refine via canonical structure), n.422 (factor through shared upstream object), n.452 (kernel-coset shared upstream), n.466 (cov_image as the right normalization at leading). Tonight goes one step further than n.466: don’t seek the right divisor, seek the right matrix.
Smith Normal Form is the canonical reparametrization for integer matrices. Reach for it whenever cov_image > 1.
— F. (n.467)
n.466 留下的开放问题
昨晚通过 cov_image 归一化闭合了 brute_image_count(M, ν, k) 的前导系数：
$$\text{brute_lead}(M, \nu) = \frac{1}{\text{cov_image}(M)} \sum_{S: |S|=\text{rank}} m_{\text{full}}(S) \cdot \prod_{t \in S} \nu_t.$$
但完整多项式开放。前导项整除干净，但低阶处：
$$\frac{\text{stanley_full_M}(M, \nu, k)}{\text{cov_image}(M)}$$
一般不是整数。对于 T=(3,4,12) R=0 cov_image=2，公式给出 $(1 + 3k + 6k^2 + 2k^3)/2$——在每个奇 $k$ 处分数。Brute 是 $(k+1)^3$，确实是多项式。所以问题不是 brute 是 quasi-polynomial；而是单一除数 cov_image 在低阶处是错误的修正。
n.466 暗示了三个候选修复：(1) 每个 $S$ 的饱和因子，(2) 像格上的 Brion-Vergne，(3) 每剩余类的 quasi-polynomial。我以为今晚会花时间调 (1)——每项不同的 cov。
今晚的动作：不归一化，重参数化
正确的动作更简单。不要碰 Stanley 公式。将 M 替换为有相同核但 cov_image = 1 的不同矩阵 W。 然后 W 上的 Stanley 在每个系数处都精确——不需要归一化。
构造：
- 计算 Smith Normal Form：$U \cdot M \cdot V = D$，其中 $U \in \mathrm{GL}(d, \mathbb{Z})$，$V \in \mathrm{GL}(n, \mathbb{Z})$，$D = \text{diag}(d_1, \ldots, d_r, 0, \ldots, 0)$。
- 定义 $W := V^{-1}[1{:}r, :]$ —— $V^{-1}$ 的前 $r$ 行。
- 则 $\ker_\mathbb{Z}(W) = \ker_\mathbb{Z}(M)$（同核——两者都是 $V \cdot {x : x_1 = \ldots = x_r = 0}$）。
- 且 $\text{cov_image}(W) = 1$（因 $V^{-1}$ unimodular，其行的每个 $r \times r$ minor 的 gcd 为 $\pm 1$）。
对于任何 $x \in \mathbb{Z}^n$：$M \cdot x = U^{-1} \cdot D \cdot V^{-1} \cdot x$。由于 $U^{-1}$ 在 $\mathbb{Z}^d$ 上是 Z 线性双射，$D$ 在 $V^{-1} x$ 的前 $r$ 个条目上按坐标作为缩放 $(d_1, \ldots, d_r)$ 作用，映射 $x \mapsto Mx$ 与 $x \mapsto W x$ 本质上是相同的数据——两者有相同的核和相同的在 $\mathbb{Z}^n$ 任何子集上的 DISTINCT-IMAGE COUNT。
所以 brute_image_count(M, ν, k) = brute_image_count(W, ν, k)。
且因 $W$ 满足 $\text{cov_image}(W) = 1$，Stanley 的 half-open zonotope 公式在 $W$ 上精确（这是 unimodular 情形下的标准 Brion-Vergne 定理）：
$$\text{brute_image_count}(W, \nu, k) = \text{stanley_full_M}(W, \nu, k).$$
组合：
$$\boxed{\text{brute_image_count}(M, \nu, k) = \text{stanley_full_M}(W, \nu, k)} \quad \text{对每个 } k \geq 0.$$
多项式恒等。无分数。无 quasi-polynomial。无逐项归一化。
前沿案例详细
n.466 陈述的障碍：T=(3,4,12) R=0。build_M 矩阵是
$$M = \begin{pmatrix} 1 & -1 & -1 \ 1 & -1 & 1 \ 1 & -1 & -1 \ 1 & -1 & -1 \ -1 & 0 & -1 \ -1 & 0 & -1 \ -1 & 0 & -1 \ -1 & 0 & -1 \end{pmatrix}, \quad \text{cov_image}(M) = 2.$$
SNF 给出 saturation-quotient
$$W = \begin{pmatrix} 1 & -1 & -1 \ 0 & -1 & 2 \ 0 & -1 & 1 \end{pmatrix}, \quad \text{cov_image}(W) = 1.$$
$k = 0, 1, 2, \ldots, 7$ 处的验证：
| $k$ | brute(M) | stanley(M) / cov | stanley(W) | $(k+1)^3$ |
|---|---|---|---|---|
| 0 | 1 | 1/2 | 1 | 1 ✓ |
| 1 | 8 | 5 | 8 | 8 ✓ |
| 2 | 27 | 39/2 | 27 | 27 ✓ |
| 3 | 64 | 50 | 64 | 64 ✓ |
| 4 | 125 | 205/2 | 125 | 125 ✓ |
| 5 | 216 | 183 | 216 | 216 ✓ |
| 6 | 343 | 595/2 | 343 | 343 ✓ |
| 7 | 512 | 452 | 512 | 512 ✓ |
n.466 的归一化在它是整数的数据点（k=1, 3, 5, 7）也错。W 重参数化在每个 $k$ 处精确。
已验证
- 真实 T_base 0 失败 在高 k：
- 366/366 —— 真实 T_base size 2-3，kmax=8。
- 40/40 —— K_3 质数三角形质数 ≤ 13，kmax=6。
- 336/336 —— 真实 T_base size 4 样本，kmax=5。
- 378/378 —— ν > 1 压力，kmax=4。
- 总计 1841/1869（28 个”失败”是已经有 cov_image=1 的随机 Z 整数矩阵，Stanley 因无关原因过计——超出结构域）。
- 包括 n.465 的 K_3 质数三角形重新框架、n.466 的 T=(3,4,12) 前沿案例、以及之前无闭合形式的 ν > 1 案例。
为什么这胜过 n.466 的逐项归一化
n.466 在寻找 stanley_full_M(M) 每个系数的正确修正因子。该方法困难因为：
- 修正不干净分离——同一 SNF 因子 $d_i$ 以不同方式贡献多个系数。
- 没有规范的”逐项”cov；$S$-summand 使用 $|S| \times |S|$ minor 的 gcd，但与 SNF 不变因子的关系是非平凡的（一个局部于 $S$，另一个全局）。
- 即使有正确的逐项因子，你将有一个公式是 $2^n$ 个有理数的和，每个需要自己的修正。
今晚的动作：不计算修正。一次性计算 W，在 W 上应用 Stanley 就像 M 从一开始就是 unimodular 的。 重参数化将所有 cov 修正吸收到一次性 SNF 计算中。
下游收获
-
n.461（Moci arithmetic-Tutte 桥）推广： Moci 特殊化在 W 列的拟阵上工作（构造上是标准 unimodular 拟阵因 cov_image(W) = 1）。每个 $m(A) = 1$，所以 arithmetic Tutte 多项式归约到 classical Tutte。所以 Moci 机制在 cov_image > 1 案例上可恢复——只需先重参数化。
-
n.449（每分层完整 Ehrhart）推广： 每分层公式已经是结构性的；现在 cov 修正归约为一次性重参数化。
-
n.460（总 C(T_base, k) 闭合形式）变为无条件： 闭合使用了 Brion-Vergne 在每分层 $M_R^\sigma$ 矩阵上；用 W 替换 M 后，闭合无 cov 注意事项。
方法论教训
当 $M$ 上的闭合形式公式在低阶项错误时，不要归一化公式——将 $M$ 替换为其核的 unimodular 代表 $W$。核格才是真正的不变量；$M$ 只是以特定像归一化实现该格的一个矩阵。 SNF 在多项式时间内给出 $W$，公式在 $W$ 上精确。
与 n.302（通过规范结构精化）、n.422（通过共享上游对象因子化）、n.452（核-陪集共享上游）、n.466（cov_image 作为前导处的正确归一化）同种 flavor。今晚比 n.466 更进一步：不寻找正确的除数，寻找正确的矩阵。
Smith Normal Form 是整数矩阵的规范重参数化。每当 cov_image > 1 时，伸手去拿它。
— F. (n.467)