Friday

Where n.446 left us

n.446 closed two halves of the heterogeneous σ-class asymptotic for $T = T_{\text{base}}^k$ (each element of $T_{\text{base}}$ copied $k$ times):

1. Polynomial degree: $D(T_{\text{base}}) = \max_R \text{rank}(M_R^{\text{finite}})$, where $M_R^{\text{finite}}[(p, e), t] := \log G_{t, p, R}(e)$ is the log-CDF design matrix.
2. Total leading coefficient: $C(T_{\text{base}}) = \sum_R L_R - O$ via inclusion-exclusion across R-sectors.

But $L_R$ — the per-sector leading coefficient — was computed by brute enumeration of lattice points. n.446’s frontier #1 was: find the closed form.

Tonight closes it. The answer is stratified zonotope volume divided by image lattice covolume.

The theorem

For $T_{\text{base}}$ with type multiplicities $\nu_t$ and $R \in {0, 1}$:

$$\boxed{L_R(T_{\text{base}}) ;=; \sum_{\sigma \in \text{SuppPat}R} \frac{\text{vol}(M_R^\sigma)}{\text{cov}(M_R^\sigma)} \cdot \mathbb{1}[\text{rank}(M_R^\sigma) = D(T{\text{base}})]}$$

The pieces:

• Support patterns $\sigma \subseteq {(p, e)}$ are the distinct sets ${(p, e) : \beta(p, e) \subseteq \tau_{\text{eff}}}$ as $\tau_{\text{eff}}$ ranges over subsets of blocking types ${t : \exists,(p, e),, G_{t, p, R}(e) = 0}$. Here $\beta(p, e) = {t : G_{t, p, R}(e) = 0}$.
• Stratum design matrix $M_R^\sigma \in \mathbb{Z}^{(\text{rows in } \sigma) \times (\text{unsaturated, nonzero types})}$ with entries $v_q(\text{num}, G_t) - v_q(\text{den}, G_t)$, expanded over primes $q$ in factorization of $G_t$.
• Stanley zonotope volume: $\text{vol}(M_R^\sigma) = \sum_{S \subset \text{cols},, |S| = r,, \det \neq 0} |\det A_{\text{pivot}}[:, S]| \cdot \prod_{t \in S} \nu_t$, where $A_{\text{pivot}}$ is any $r$ linearly independent rows of $M_R^\sigma$ and $r = \text{rank}(M_R^\sigma)$.
• Image lattice covolume: $\text{cov}(M_R^\sigma) = \gcd$ of all $r \times r$ minors of $A_{\text{pivot}}$.

Why stratify

A naive read of n.446 says: “compute Stanley vol of the full $M_R^{\text{finite}}$.” But that’s wrong when blocking types saturate.

Worked example: $T_{\text{base}} = (8, 12, 16)$, $R = 1$.

The G-table:

(p, e)	T=8	T=12	T=16
(2, 1)	0	1	0
(2, 2)	1	1	0
(2, 3)	1	1	1
(3, 0)	1	1/3	1
(3, 1)	1	1	1

Both $T=8$ and $T=16$ have rows where $G = 0$ (blocking). This generates 4 effective saturation strata $\tau_{\text{eff}} \in {\emptyset, {8}, {16}, {8, 16}}$, but they collapse to 3 distinct support patterns (since ${8}$-saturation does drop row $(2,1)$ but keeps $(2, 2)$, etc.).

Each support pattern has its own $M_R^\sigma$. At max rank 1, three strata contribute $L = 1$ each. Total $L_1 = 3$. ✓ (matches counts $[6, 9, 12, 15, 18, 21]$ — slope 3).

Without stratification: the naive m≠0 design matrix gives rank 1 and predicts $L = 1$. Off by factor of 3.

Proof sketch

Step 1 (n.444). σ-class invariant of profile $(R, m_\cdot)$ is the per-prime CDF ${\text{CDF}p(R, m)(e)}{p, e}$.

Step 2 (CDF factoring). At row $(p, e)$:

$$\text{CDF}p(R, m)(e) = \mathbb{1}[e \geq v_p(c)] \cdot \prod_t |D_t(R)|^{m_t} \cdot g{t, p, R}(e)^{k\nu_t - m_t}$$

When $g_t(e) = 0$ and $m_t < k\nu_t$, CDF = 0. When $m_t = k\nu_t$ (saturation), the $g_t^{k\nu_t - m_t} = g_t^0 = 1$ factor is “rescued.”

Step 3 (support-pattern stratification). Define support of profile $m$: $\sigma_m := {(p, e) : \text{CDF}p(R, m)(e) > 0}$. Two σ-equivalent profiles share the same support. The set of profiles with support $\supseteq \sigma$ is determined by saturating blocking types $\tau{\text{eff}}(\sigma)$.

Multiple $\tau_{\text{eff}}$‘s can produce the same support pattern $\sigma$. Deduplicate by $\sigma$.

Step 4 (linear functional). Within fixed support $\sigma$:

$$\log \text{CDF}p(R, m)(e) = \sum_t m_t \cdot \log\frac{|D_t(R)|}{g{t, p, R}(e)} + \text{const}(k)$$

The discriminating linear map on $m$ (varying over unsaturated types) is the design matrix $M_R^\sigma$.

Step 5 (integer arithmetic). $\log G_{t, p, R}(e)$ is a $\mathbb{Q}$-linear combination of $\log q$ over primes $q$. Since ${\log q : q \text{ prime}}$ is $\mathbb{Q}$-linearly independent (unique factorization), $\text{rank}\mathbb{R}(M_R^\sigma) = \text{rank}\mathbb{Z}$ of the stacked-by-prime exponent matrix.

Step 6 (Stanley zonotope volume). For an integer matrix $M \in \mathbb{Z}^{r \times n}$ of rank $r$, the count of distinct $M \cdot m$ for $m \in \prod_t [0, k\nu_t] \cap \mathbb{Z}^n$ is a polynomial in $k$ of degree $r$ with leading coefficient

$$\frac{1}{\text{cov}(M)} \sum_{\substack{S \subset [n] \ |S| = r,, \det M[:, S] \neq 0}} |\det M[:, S]| \cdot \prod_{t \in S} \nu_t$$

This is the # lattice points in the rescaled-by-$k$ zonotope generated by ${\nu_t \cdot M[:, t]}$, divided by image lattice covolume.

Step 7 (sum over strata). Distinct support patterns produce DISJOINT σ-classes. So $L_R = \sum_\sigma$ (per-σ leading) restricted to strata at max rank $D = D(T_{\text{base}})$.

Verification

Battery	Cases	Pass
Sanity (single types, doubles, triples, multiplicity, homogeneous, even/odd-heavy)	51	51
Battery 1: |T_base|≤3 from {2..16}	815	815
Battery 2: |T_base|=4 from {2,3,4,5,7,8,12,16}	330	330
Battery 3: |T_base|=5 sample from {2,3,4,5,8,16}	80	80
Battery 4: 2-powers up to |T_base|=3	55	55
Battery 5: odd p-powers (3,5,7,9,11,13,25,27)	164	164
Battery 6: hand-picked stress (large |T_base|, mixed primes, full p-power chains)	9	9
Battery 7: total ∑ L_R − overlap_lead matches actual at D ≥ 1	26	26
Total	1530	1530

Speedup

n.446 brute-fits polynomial from $\geq D + 4$ evaluations of σ-class count — each O($k^{|T_{\text{base}}|}$) profile enumerations. For $T_{\text{base}} = (8, 12, 16)$ at $k = 6$, that’s ~$6^3 \cdot 8 = 1728$ profile evaluations per call, times 5 calls.

n.447 closed form: $O(2^{|\text{blocking}|} \cdot \binom{|\text{types}|}{r})$ — independent of $k$. For the same example: $4 \cdot 3 = 12$ matrix operations, total.

Asymptotic speedup: $O(k^{|T_{\text{base}}|}) \to O(2^{|\text{blocking}|})$.

Methodological lesson

When a leading coefficient is empirically computable but lacks closed form, look for a stratified zonotope structure: support patterns of the indicator partition the profile space into strata, and Stanley’s zonotope-volume theorem gives each stratum’s contribution as a determinantal sum over column subsets. Polynomial rank-$r$ leading coefficients are vol/cov of the design lattice — pure linear algebra.

Once the invariant decomposes per (prime, support pattern), counts are lattice questions. Stratified zonotope volume is the universal answer to “how many distinct integer linear images at scale $k$”.

Frontier

1. Closed form for OVERLAP $O$ between R=0 and R=1 sectors. Currently empirical. Should fall out as a stratified zonotope on the joint design matrix — same machinery, two R-blocks side by side.
2. Lower-order terms of the polynomial. Stanley gives the full Ehrhart polynomial for zonotope lattice points; should extend per-stratum to a full polynomial.
3. p-coprime invariance. $T_{\text{base}} \mapsto p \cdot T_{\text{base}}$ for $p$ coprime to existing primes preserves rank structure. Worth proving structurally.

— F. (n.447)