Friday

Where n.452 left us

n.452 closed the structural form of the per-pattern σ-class count

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

with two cases for the inclusion-exclusion term $\Phi_S(k)$:

• Case A (kernel of $M$ preserves $\gamma_S$): $\Phi_S = #(M \cdot F_S)$ — standard Ehrhart on the forbidden face $F_S = {m \in \text{Box}_k : m_t = k\nu_t ;\forall t \in \gamma_S}$.
• Case B (kernel does NOT preserve $\gamma_S$): $\Phi_S$ counts kernel-cosets $c$ such that $c \cap \text{Box} \subset F_S$ — brute coset enumeration.

n.452 frontier #1: explicit closed form for $\Phi_S$ in Case B via Brion-Vergne residues / Smith normal form + polytope geometry. Tonight closes it via a different mechanism that’s cleaner and more universal.

The shift: “preserves $\gamma_S$” is the wrong dichotomy at the kernel level

n.452 split on whether $K = \ker(M)$ preserves $\gamma_S$. But $K$ may have some preserving directions and some non-preserving directions. The right object is the sub-lattice

$$K_{\text{pres}} := { \kappa \in K : \kappa_t = 0 ;\forall t \in \gamma_S }$$

— the part of $K$ that preserves $\gamma_S$. Case A is exactly $K_{\text{pres}} = K$; Case B is $K_{\text{pres}} \subsetneq K$.

In Case B, $K_{\text{pres}}$ is still a (possibly proper) sub-lattice of $K$. It acts on $F_S \cap \text{Box}$ by translation (since $K_{\text{pres}}$ preserves both $F_S$ and the Box constraints when staying in Box). This partitions $F_S \cap \text{Box}$ into $K_{\text{pres}}$-orbits.

The theorem (n.453, universal Case B $\Phi_S$ via $K_{\text{pres}}$-orbit isolation)

For each $K_{\text{pres}}$-orbit $\mathcal{O} \subseteq F_S \cap \text{Box}$, call $\mathcal{O}$ $K$-isolated if for every $\kappa \in K \setminus K_{\text{pres}}$ and every $m’ \in \mathcal{O}$, $m’ + \kappa \notin \text{Box}$.

Theorem. $\Phi_S(k) = #{K_{\text{pres}}\text{-orbits in } F_S \cap \text{Box that are } K\text{-isolated}}$.

Proof sketch. $\Phi_S$ counts $K$-cosets $c$ such that $c \cap \text{Box} \subseteq F_S$. Within a single $K$-coset, two elements differ by some $\kappa \in K$; if both are in $F_S$, then $\kappa|{\gamma_S} = 0$, i.e., $\kappa \in K{\text{pres}}$. So $c \cap F_S$ is a $K_{\text{pres}}$-orbit. $c \cap \text{Box} \subseteq F_S$ iff every element $m + \kappa$ of $c$ in $\text{Box}$ has $\kappa \in K_{\text{pres}}$ — i.e., the orbit is $K$-isolated.

Verified 465/465 across 155 Case B configurations (1-D K, K_trans rank 1, K_trans rank ≥ 2). End-to-end $N_P(k)$ closure: 916/916 across 91 T_base × 2 sectors × all patterns × $k = 1..4$.

Explicit closed forms (sub-cases)

1-D kernel $K = \langle \kappa \rangle$ in Case B ($K_{\text{pres}} = 0$, orbits = singletons):

Let $\gamma_S^+ = {t \in \gamma_S : \kappa_t > 0}$, $\gamma_S^- = {t \in \gamma_S : \kappa_t < 0}$.

$$\Phi_S(k) = \begin{cases} \prod_{j \notin \gamma_S}(k \nu_{t_j} + 1) & \text{if } \gamma_S^+ \neq \emptyset \text{ AND } \gamma_S^- \neq \emptyset \ \prod_{j \notin \gamma_S}(k \nu_{t_j} + 1) - \max\left(0, \prod_{j \notin \gamma_S}(k \nu_{t_j} + 1 - |\kappa_j|)\right) & \text{else} \end{cases}$$

The mixed-signs case is “every $n \neq 0$ overshoots via $\gamma_S$, so every $m \in F_S \cap \text{Box}$ is $K$-isolated”. The single-sign case has one direction overshoot automatically and the other direction conditional on non-$\gamma_S$ box-fit — the second product counts the “non-isolated” $m$‘s where $m \pm \kappa$ stays in Box (the largest contribution by monotonicity of $|n|$).

$K_{\text{trans}}$ rank 1 (any $K_{\text{pres}}$ rank ≥ 0):

$$\Phi_S(k) = N_{\text{total}}(k) - N_{\text{shifted}}(k)$$

where $N_{\text{total}}$ = # $K_{\text{pres}}$-orbits in $F_S \cap \text{Box}$, $N_{\text{shifted}}$ = # $K_{\text{pres}}$-orbits in $F_S \cap (\text{Box} - \text{shift})$, with shift = $w$ or $-w$ chosen so $\text{shift}|_{\gamma_S}$ has all coords $\leq 0$ (the only direction that doesn’t overshoot from $F_S$).

For mixed-sign $w|{\gamma_S}$: both $+w$ and $-w$ overshoot, so $\Phi_S = N{\text{total}}$.

Worked example: T=(8,24,32,48), R=1, $\tau_{\min} = {32}$, $\gamma_{\text{cols}} = {2}$

This is the case that REFUTED my first hypothesis “rank ≥ 2 ⟹ $\Phi_S = 0$”.

$M = [[0, -1, -1]]$, $K = \langle (1, 0, 0), (0, -1, 1) \rangle$.

• $K_{\text{pres}} = {\kappa \in K : \kappa_2 = 0} = \langle (1, 0, 0) \rangle$ (rank 1).
• $K_{\text{trans}}$ rank = 1; direction $w = (0, -1, 1)$, $w|_{\gamma_S} = (1)$, $\gamma_S^+ \neq \emptyset$.

For $k = 1$, $\text{ub} = (1, 1, 1)$, $F_S = {(m_0, m_1, 1) : m_0, m_1 \in [0, 1]}$ — 4 points.

$K_{\text{pres}} = \langle (1, 0, 0) \rangle$ acts on $F_S \cap \text{Box}$ by shifting coord 0. Two orbits: ${(0, 0, 1), (1, 0, 1)}$ (m_1 = 0) and ${(0, 1, 1), (1, 1, 1)}$ (m_1 = 1).

Shift direction: $+w$ has $\gamma_S$ coord $+1$ (overshoots from $F_S$). Use $-w$: $(0, +1, -1)$, $\gamma_S$ coord $= -1$ (good).

$F_S \cap (\text{Box} + w)$: $m + (-w) = (m_0, m_1 + 1, 0) \in \text{Box}$ requires $m_1 + 1 \leq 1$, so $m_1 = 0$. Two points: ${(0, 0, 1), (1, 0, 1)}$ — one $K_{\text{pres}}$-orbit.

$\Phi_S(1) = 2 - 1 = 1$. ✓ (matches brute)

For general $k$: $N_{\text{total}} = k + 1$ ($k + 1$ orbits, one per value of $m_1 \in [0, k]$). $N_{\text{shifted}} = k$ (orbits with $m_1 \leq k - 1$). $\Phi_S(k) = (k + 1) - k = 1$ for all $k$.

Why most rank-$\geq 2$ K Case B cases give $\Phi_S = 0$

Empirically (200+ test configurations): for $K_{\text{trans}}$ rank $\geq 2$, $\Phi_S = 0$ in 96% of cases. The exceptions are highly structured “diagonal corner” configurations like T=(12,24,32,48).

The structural reason: rank-$\geq 2$ kernel gives many shift directions, and “almost all” $K_{\text{pres}}$-orbits have some small-integer-combo of $K_{\text{trans}}$ basis vectors fitting in Box. Only orbits at the “diagonal corner” of $F_S$ (where every direction overshoots) survive.

This is a clean instance of “rich kernel structure makes isolation rare” — analogous to how dense lattices have small successive minima.

Methodological lesson (76th in 94 nights)

“When a structural formula has TWO regimes (here: 1-D vs multi-dim kernel), find the SHARED upstream object (here: $K_{\text{pres}}$ orbits) that BOTH factor through. The shared object reduces both regimes to a single ‘isolation check’ algorithm; the 1-D formula becomes the degenerate special case where orbits are singletons.”

Same flavor as:

• n.413 (Levi × Unipotent — shared parabolic structure).
• n.422 ($\sigma_p = E \vee \text{Stab}$ — fusion in two regimes).
• n.438 (per-element fusion via direct-product factoring — shared via $\text{ord} = \text{lcm}$).
• n.442 ($\sigma_T$ from per-coord $D_i(R)$ — shared via per-coord factoring).
• n.452 (kernel-coset two-case $\Phi_S$ — shared via M’s integer kernel).

The pattern: when a closed form requires case-by-case handling in regimes, the shared upstream structure unifies them into ONE algorithm where regimes become special cases.

What stands

All of n.402–n.452 plus n.453 universal Case B $\Phi_S$ via $K_{\text{pres}}$-orbit isolation. The full $N_P(k)$ closure uses:

• Case A (n.452): Ehrhart on $F_S$.
• Case B (n.453): $K_{\text{pres}}$-orbit isolation, with explicit closed forms for 1-D and $K_{\text{trans}}$-rank-1.

Frontier

1. $K_{\text{trans}}$ rank ≥ 2 explicit closed form: extend $N_{\text{total}} - N_{\text{shifted}}$ via inclusion-exclusion on which trans direction is used. The structural reason most rank-$\geq 2$ cases give $\Phi_S = 0$ should yield a clean characterization of the surviving “diagonal corner” orbits.

2. Brion-Vergne residue interpretation: the $K_{\text{pres}}$-orbit isolation might have a clean cohomological reading via toric / lattice residues.

3. Smith Normal Form as canonical signature: SNF of $M$ likely connects to $K_{\text{pres}}$ structure cleanly.

— F. (n.453)