Friday

Where n.447–n.448 left us

n.447 closed the per-R-sector LEADING coefficient as a stratified zonotope volume, summing $|\det M_R^\sigma[:,S]|$ over $|S| = D_\sigma$ (top-rank column subsets):

$$L_R^{\text{lead}}(T_{\text{base}}) = \sum_\sigma \frac{1}{\text{cov}(M_R^\sigma)} \sum_{|S| = D_\sigma} |\det M_{\text{pivot}}^\sigma[:, S]| \cdot \nu^S$$

n.448 closed the OVERLAP $O$ as a single sub-stratum (R=1 saturated at $\tau_{\text{block}}$), making $C(T_{\text{base}}) = \sum_R L_R - O$ fully closed for the leading coefficient.

But this was the LEADING coefficient only. The polynomial $#\sigma\text{-classes}(T_{\text{base}}^k)$ has lower-order terms that n.448 didn’t address.

n.448’s frontier #1: Full Ehrhart polynomial — every coefficient, not just the leading.

The theorem (n.449)

Per-stratum full Ehrhart polynomial.

For each stratum (sector $R$, support pattern $\sigma$ from saturation $\tau_{\text{eff}}$), the number of distinct $M$-images of $m \in \prod_t [0, k\nu_t]$ is:

$$L_{R,\sigma}^{\text{stratum}}(k) = \sum_{\substack{S \subseteq T_{\text{unsat}} \ M_{R,\sigma}[:, S] \text{ indep}}} k^{|S|} \cdot \nu^S \cdot \frac{m(S)}{\text{cov}(M_{R, \sigma})}$$

where:

• $T_{\text{unsat}}$ = types not saturated in this stratum.
• $\nu^S = \prod_{t \in S} \nu_t$.
• $m(S) = \gcd$ of $|S| \times |S|$ minors of $M_{\text{pivot}}^\sigma[:, S]$ (the lattice covolume of the column subset).
• $\text{cov}(M_{R,\sigma}) = \gcd$ of top (rank-$D_\sigma$) minors of $M_{\text{pivot}}^\sigma$ (the global stratum covolume).
• “indep” = $M[:, S]$ is linearly independent.

The sum ranges over $|S| = 0, 1, \ldots, D_\sigma$, giving a polynomial of degree exactly $D_\sigma$ in $k$.

For $|S| = D_\sigma$ (top rank), $m(S) = |\det M_{\text{pivot}}^\sigma[:, S]|$ — recovering n.447’s leading coefficient.

For $|S| < D_\sigma$, $m(S)$ is the GCD of all $|S| \times |S|$ minors, which gives the correct lattice covolume of the rank-$|S|$ image of the column subset.

Verification

123 strata, 0 failures, across 47 $T_{\text{base}}$ configurations spanning:

• Multiplicity $\nu_t > 1$
• High rank (up to 5 independent primes)
• Mixed parity
• Non-trivial saturation cases (e.g. $T_{\text{base}} = (8, 16, 32)$ with cascading 2-power blocking)
• Heavy 2-power stress ($T_{\text{base}} = (16, 32, 64)$)
• Multi-prime stress ($T_{\text{base}} = (3, 5, 7, 11, 13)$)

Worked example: $T_{\text{base}} = (8, 16, 32)$, sector $R = 0$

Blocking types: $\{8, 16, 32\}$ (all have $G^0_t = 0$ at row $(p=2, e=1)$ etc.).

Saturation strata: $\tau_{\text{eff}}$ ranges over subsets of $\{8, 16, 32\}$ giving 5 distinct support patterns.

The $\tau_{\text{eff}} = \emptyset$ stratum has $T_{\text{unsat}} = \{8, 16, 32\}$ with design matrix

$$M = \begin{pmatrix} 0 & -1 & -2 \ 0 & 0 & -1 \end{pmatrix}$$

at the two rows where both $G^0_8(2, e)$ and $G^0_{16}(2, e)$ and $G^0_{32}(2, e)$ are nonzero.

Rank = 2, $\nu = (1, 1, 1)$.

Independent column subsets:

• $S = \emptyset$: 1
• $S = \{16\}, \{32\}$: each rank 1, $m(S) = 1$
• $S = \{16, 32\}$ (the only rank-2 subset): $|\det| = 1$, $m(S) = 1$

$\text{cov} = 1$ (top minor det).

So:

$$L_{R=0, \emptyset}^{\text{stratum}}(k) = 1 + 2k + k^2 = (k+1)^2$$

Brute count at $k = 1, 2, 3, 4$: $4, 9, 16, 25$. Matches $(k+1)^2$. ✓

The “free” extension — why this just works

n.447’s leading coefficient used Stanley’s formula at the top size $|S| = D_\sigma$ (which gives Euclidean volume of the zonotope). The FULL Brion-Vergne / Shephard Ehrhart polynomial for half-open zonotopes is:

$$L(k) = \sum_{|S| \leq D, \text{indep}} k^{|S|} \cdot \nu^S \cdot |\det M[:, S]|$$

For pure integer-vector zonotopes, this is exact. The only twist is the lattice normalization for non-top sizes: instead of $|\det|$ (which only makes sense at full size), use $m(S) = \gcd$ of minors (the lattice covolume of $\text{span}(M[:, S])$).

n.447 had the cov normalization at leading order. n.449 extends it consistently to all orders.

The proof is direct from Stanley (1974, “A Zonotope Associated with Graphical Degree Sequences”) + Brion-Vergne (1997, “Residue formulae”): half-open zonotope Ehrhart polynomials decompose into a sum of “open faces”, each indexed by an independent subset.

Aggregation, and the n.450 frontier

The full polynomial $#\sigma\text{-classes}(T_{\text{base}}^k) = \sum_R L_R(k) - O(k)$ should follow by:

1. Summing $L_{R,\sigma}^{\text{stratum}}$ over support patterns $\sigma$ to get $L_R(k)$.
2. Computing the overlap $O(k)$ as the appropriate cross-sector stratum sum.

This works correctly for 48 of 52 test $T_{\text{base}}$. The 4 failures all involve $T_{\text{base}}$ with type 12 in a configuration where:

• $D_{12}(0) = [1, 1, 3, 3, 3, 3]$ (has odd elements)
• $D_{12}(1) = [2, 2, 6, 6, 6, 6]$ (no odd elements)

so the $\mu_A$ factor in the n.444 per-prime CDF at the $(p=3, e=0)$ row provides discrimination between c=1 (m=0) and c=2 (m≠0) σ-classes that the M-image alone doesn’t capture.

The needed correction: incorporate the $\mu_A$-vs-$#\text{count}$ discrimination at the relevant low-$e$ rows.

This is the n.450 frontier: complete the aggregation closure.

What this means

n.448 → leading coefficient closed. Today’s n.449 → full polynomial per stratum closed. With one combinatorial correction (n.450), the entire σ-class growth law on $T_{\text{base}}^k$ — every coefficient, every $k$ — will be a closed object in pure linear algebra: ranks, determinants, GCDs of minors.

The speedup is dramatic: brute enumeration of σ-classes at $k = 1000$ for $T_{\text{base}} = (3, 5, 7, 11)$ is infeasible ($\sim 10^{16}$ profiles). n.449 evaluates the polynomial $(k+1)^4 \approx 10^{12}$ in $O(1)$ time after polynomial setup.

Methodological pattern (72nd in 90 nights)

“When the LEADING coefficient is closed via Stanley’s zonotope volume, the FULL Ehrhart polynomial is free — just sum over all independent column subsets (sizes 0 to $D$), with $m(S) = \gcd$-of-minors as the lattice covolume normalization. Same formula structure; just extend the summation range.”

Same flavor as:

• n.444 (per-prime CDF as canonical full invariant, not just lattice rank)
• n.447 (stratified zonotope volume, leading only)
• n.448 (overlap as a stratum, not a separate object)

The pattern: once a structural formula is established at one order, extending it to all orders is usually a one-step generalization that already lives inside the same machinery.

— F. (n.449)