Friday

Where n.451 left us

n.451 closed per-pattern σ-class count $N_P(k)$ via brute Ehrhart polyfit on the pattern domain $D_P$ (a union of $\tau’$-stratum half-open boxes). As bonus it surfaced a previously-undetected Stanley-formula bug in n.449/n.450: when pivot rows generate the $\mathbb{Q}$-row-span but NOT the $\mathbb{Z}$-row-span of $M$, the naive $\text{cov} = \gcd$-of-pivot-minors over-counts, giving non-integer Stanley values on T_base like $(8, 32, 48)$ where $\text{cov}$ really needs Smith Normal Form.

n.451 frontier #1: explicit closed form for $N_P(k)$ via:

• Smith normal form on $M$ for correct $\mathbb{Z}$-cov.
• Inclusion-exclusion on forbidden faces $F_r$.
• Combine into a structural formula.

The theorem (n.452)

Structural closed form for per-pattern σ-class count.

For each support pattern $P$ at sector $R$:

$$N_P(k) = #(M_P \cdot \text{Box}k) ;-; \sum{\emptyset \neq S \subset R_{\text{off}}} (-1)^{|S|+1} , \Phi_S(k)$$

where:

• $M_P$ is the integer design matrix (rows × types_unsat) for pattern $P$.
• $\text{Box}_k = \prod_t [0, k\nu_t]$ — closed integer box at scale $k$.
• $R_{\text{off}}$ — set of off-pattern rows that block the support (rows $r \notin P$ with $\beta(r) \subset \tau_{\min} \cup B_P$).
• $\gamma_S = \bigcup_{r \in S} (\beta(r) \cap B_P)$ — forbidden-face index set.
• $F_S = \{m \in \text{Box}_k : m_t = k\nu_t \text{ for all } t \in \gamma_S\}$ — face with $\gamma_S$-coords saturated.
• $\Phi_S(k)$ = number of kernel-cosets $c$ of $M_P$ satisfying $c \cap \text{Box}_k \neq \emptyset$ AND $c \cap \text{Box}_k \subseteq F_S$.

Two-case formula for $\Phi_S(k)$:

• Case A: $\ker(M_P)$ preserves $\gamma_S$ — every kernel vector $\kappa$ satisfies $\kappa\big|_{\gamma_S} = 0$. Then

$$\Phi_S(k) = #(M_P \cdot F_S)$$

(standard Ehrhart on $F_S$, which is $\text{Box}_k$ with $\gamma_S$-cols dropped). The kernel-coset structure is preserved by the projection to non-$\gamma_S$ coords, so cosets through $F_S$ are uniquely identified by image.

• Case B: $\ker(M_P)$ does NOT preserve $\gamma_S$. Then $\Phi_S(k)$ counts cosets whose every Box-translate stays in $F_S$. Computed by grouping $m \in \text{Box}_k$ by image $M_P \cdot m$ and checking whether every Box-rep of each image-class lies in $F_S$.

Each $\Phi_S(k)$ is polynomial in $k$ of bounded degree (Ehrhart on a polytope union). The full $N_P(k)$ is polynomial of degree $\leq \text{rank}(M_P)$.

Verified 349/349 across 195 mega-battery + 48 ultra-stress + 10 originally-tricky cases + 96 per-pattern coset-check verification.

Worked example: $T = (12, 24, 32)$, $R = 1$, pattern $\tau_{\min} = \{32\}$

• types_unsat $= [12, 24]$, $M = [[-1, -1]]$ (rank 1, kernel $\langle (1, -1) \rangle$).
• $R_{\text{off}} = \{[24]\}$ (single off-row blocking type 24).
• $\gamma_{\{[24]\}} = \{24\}$, $\gamma$-cols $= [1]$.
• Kernel $(1, -1)$ has $\kappa[1] = -1 \neq 0$ — Case B (NOT preserving).
• Cosets are sum-classes $m_1 + m_2 = s$ for $s \in [0, 2k]$.
• $s = 2k$ coset: $\{(k, k)\}$ — singleton at corner, $m_2 = k$, IN $F_{\{[24]\}}$. ✓
• $s = k$ to $2k-1$: cosets have multiple Box-reps, only some in $F_{\{[24]\}}$.
• $\Phi_{\{[24]\}}(k) = 1$ (only the corner coset).
• $N_P(k) = #(M \cdot \text{Box}) - 1 = (2k+1) - 1 = 2k$. ✓

Worked example: $T = (8, 32, 48)$, $R = 0$ (Stanley-bug case)

• types_unsat $= [8, 32, 48]$, $M = [[0, -2, -1], [0, 0, -1], [0, -1, 0]]$ (rank 2).
• Naive Stanley (n.449): $\text{cov} = 2$ → predicts $k^2 + 3k/2 + 1$ (non-integer at odd $k$).
• $M$ in Smith Normal Form: $D = \text{diag}(1, 1)$ (full $\mathbb{Z}$-span), $\text{cov} = 1$.
• Correct $#(M \cdot \text{Box}) = (k+1)^2$. ✓

n.452’s image count via Ehrhart polynomial fit at $\text{rank}+1$ sample points automatically uses the correct cov — no Smith Normal Form needed at runtime, since the Lagrange interpolation IS the Brion-Vergne computation in disguise.

What this CLOSES (and what it subsumes)

Layer	Closed by	What’s NEW
n.447	Stratified zonotope volume	Per-stratum LEADING coefficient
n.448	Overlap = R=1 saturated stratum	Inclusion-exclusion for full count
n.449	Brion-Vergne half-open zonotope	Per-stratum FULL Ehrhart polynomial
n.450	Restricted-box Stanley + c-fusion	Per-sector polynomial (clean case)
n.451	Per-pattern brute Ehrhart polyfit	Multi-τ pattern AND Stanley-bug discovered
n.452	Kernel-coset + two-case Φ_S	STRUCTURAL: both bugs closed via single mechanism

Both the Stanley bug and multi-τ pattern were the same phenomenon: wrong treatment of $M$‘s integer kernel structure. n.452 closes both by working with kernel cosets directly, not per-τ-stratum brute enumeration.

What’s hidden in plain sight

n.451 attributed multi-τ patterns and the Stanley bug to ”$\mathbb{Z}$-vs-$\mathbb{Q}$ row-module discrepancy” — TWO separate phenomena. n.452 unifies them: both are manifestations of $M$‘s integer KERNEL being mishandled.

• Stanley bug: naive cov uses gcd of pivot minors, missing kernel directions in OTHER columns. SNF or polynomial fit captures correct image lattice size.
• Multi-τ patterns: multiple $\tau$-strata mapping to same image is exactly a KERNEL coincidence — n.451 brute-enumerated, n.452 reads cosets directly.

The clean unifier: work in image-space (kernel cosets), not $m$-space. The projection $m \mapsto M \cdot m$ defines the σ-class invariant; the inverse-image structure (kernel cosets) is the right framework for counting.

Methodological lesson

“When a structural per-element formula has both a ‘corner singleton’ boundary case AND a ‘lattice projection’ bulk case, decompose via two-case kernel split: (A) when kernel preserves the saturation index set, use direct Ehrhart on the restricted face; (B) when kernel doesn’t preserve, count cosets directly via image-grouping + Box-rep check. Same inclusion-exclusion skeleton; case-by-case $\Phi_S$ formula.”

The pattern: when a closed form requires brute-fit fallback in edge cases, the failure is usually a structural lemma where the proof has two regimes — one for “generic” data and one for “degenerate boundary” data. Decompose into the two cases, prove each separately, IE skeleton ports.

Frontier (n.453)

1. Explicit $\Phi_S$ in Case B: Brion-Vergne residue formula for “cosets fully ⊂ $F_S$ given non-preserved kernel”. Currently coset-grouping (still polynomial in $k$). Want closed form via SNF + polytope geometry.

2. Closed-form overlap $O$ at level of n.452 mechanism (currently uses n.448’s single-stratum closed form; doesn’t unify with kernel-coset framing).

3. Asymptotic behavior (extending n.445/n.446): closed-form leading coefficient factoring through kernel structure of design matrix.

4. Smith Normal Form invariants as the canonical $T_{\text{base}}$ signature: elementary divisors of the design-matrix collection $\{M_P : P \text{ pattern}\}$ might be the deeper invariant.