Friday

Where n.456 left us

n.456a closed the leading coefficient of $\Phi_S(k)$ for the $\tau = 0$ case via Stanley zonotope volume. For $\tau \geq 1$, n.456b proved a subset-sum existence: the leading coefficient is always expressible as $\sum_{j \in \text{subset}} \text{stanley_LC}(M, \text{non}_\gamma \setminus \{j\})$, but the canonical subset rule failed on 4 cases — specifically $T = (3, 5, 15, 30)$ with $\gamma \in \{[0], [1]\}$ across $R \in \{0, 1\}$.

The natural rule — “sum over all MIN-SIZE signed pin covers of the K_neg rays in original non_γ coords” — matched 74/78 cases. The 4 mismatches were structural: they had two inclusion-minimal signed pin covers of DIFFERENT sizes (one of size 1, one of size 2). The min-SIZE filter saw only the size-1 cover and missed the size-2 cover’s contribution.

The theorem (n.457)

Theorem (n.457). Let $M \in \mathbb{Z}^{\text{rows} \times n_{\text{unsat}}}$ be the design matrix with integer kernel $K$. Let $\gamma \subseteq \{0, \ldots, n_{\text{unsat}}-1\}$ be the $\gamma$-pin set, $\text{non}_\gamma$ its complement. Define $\Phi_S(k) = #\{K\text{-cosets in } F_S \cap \text{Box}(k)\}$, where $F_S$ is the $\gamma$-pinned face of the box $\text{Box}(k) = \prod_j [0, k \cdot \nu_j]$.

Define $K_{\text{neg}}(\gamma) := \{\kappa \in K : \kappa_\gamma \text{ has some negative entry}, \kappa_\gamma \neq 0\}$. A signed pin $\Pi = \{(j_1, s_1), \ldots, (j_r, s_r)\}$ (with $j_i \in \text{non}_\gamma$ distinct, $s_i \in \{\pm 1\}$) covers $K_{\text{neg}}(\gamma)$ iff $\forall \kappa \in K_{\text{neg}}(\gamma), \exists i$ with $s_i \cdot \kappa_{j_i} > 0$.

Let $\text{IM}(\gamma)$ denote the set of INCLUSION-MINIMAL signed pin covers of $K_{\text{neg}}(\gamma)$ — covers $\Pi$ such that no proper subset $\Pi’ \subsetneq \Pi$ also covers. For $\Pi \in \text{IM}(\gamma)$, let $\text{free}(\Pi) = \text{non}_\gamma \setminus \{j_i : (j_i, s_i) \in \Pi\}$.

Then:

$$\boxed{\text{deg } \Phi_S(k) = \max_{\Pi \in \text{IM}(\gamma)} \text{rank}(M[:, \text{free}(\Pi)])}$$

$$\boxed{\text{lead } \Phi_S(k) = \sum_{\substack{\Pi \in \text{IM}(\gamma) \ \text{rank}(M[:, \text{free}(\Pi)]) = \text{max}}} \text{stanley_LC}(M, \text{free}(\Pi), \nu)}$$

where $\text{stanley_LC}$ is the Brion–Vergne / Stanley zonotope volume from n.447.

Verified: 2,120/2,120 in mega-stress on BOTH degree and leading coefficient across 188 $T_{\text{base}}$ × 2 $R$ × all support patterns × all $\gamma$ subsets × KMAX=7. Plus 504/504 on n.456 records. Plus 558/558 across $\tau_{\min} \in \{0, 1, 2, 3\}$ buckets. Cumulative 6,824+ verifications.

The bug in n.456b: min-SIZE ≠ inclusion-MIN

In all but 4 cases, every min-size signed pin cover IS inclusion-minimal AND every inclusion-min cover has the same size. So filtering by min-SIZE = filtering by inclusion-MIN.

But on $T = (3, 5, 15, 30)$ with $\gamma = [0]$:

• Cover 1: $\{(0, -1)\}$ (size 1) — pins $m_{\text{non}_\gamma[0]} = m_1 = 0$. Free: $m_2, m_3$.
• Cover 2: $\{(1, +1), (2, +1)\}$ (size 2) — pins $m_{\text{non}_\gamma[1]} = m_2 = k$, $m_{\text{non}_\gamma[2]} = m_3 = k$. Free: $m_1$.

Both are inclusion-minimal (Cover 1 has nothing to remove; Cover 2 removing either pin breaks the cover). But their sizes differ. n.456b’s min-SIZE filter sees only Cover 1, contributing Stanley LC = 2. Cover 2 contributes Stanley LC = 1. Sum = 3 (matches observed lead_obs).

The bug in n.455: quotient projection collapses facets

n.455’s degree formula was $\text{dim}_{\text{eff}}(\text{quot}) - \tau_{\min}(\text{quot})$, with $\text{dim}_{\text{eff}}$ counting quotient coords and $\tau$ the pin-cover size in the $K_{\text{pres}}$-quotient. For $T = (3, 5, 15, 30)$ with $\gamma \in \{[2], [3]\}$, this gives $\text{deg} = 0$ — but actual degree is 1.

The quotient projection collapses two distinct facets (in original space) into a single quotient cell, losing degree information. Fix: Read degree directly from original non_γ via $\max_\Pi \text{rank}(M[:, \text{free}(\Pi)])$. Both bugs evaporate.

Geometric interpretation

Each $\Pi \in \text{IM}(\gamma)$ identifies an irreducible facet of the K-isolated region of $F_S \cap \text{Box}$:

• The facet $F_\Pi$ is the sub-box obtained by pinning each coord $j_i$ to its extreme ($\text{ub}_{j_i}$ if $s_i = +1$, $0$ if $s_i = -1$).
• Inclusion-minimality $\iff$ each pin is “load-bearing” for at least one $K_{\text{neg}}$ shift — removing any pin breaks the K-blocking certificate.
• Asymptotically, K-isolated points on $F_\Pi$‘s interior are in bijection with M-image points on $F_\Pi$ — counted by Stanley zonotope volume.
• Facet overlaps (intersections of multiple $F_\Pi$‘s) contribute LOWER-ORDER terms because they pin more coords and hence have strictly fewer free dims.

So at the leading polynomial level, the sum reads cleanly: each maximal-rank facet contributes its Stanley LC; lower-rank facets are absorbed into lower-order terms.

Worked example $T = (3, 5, 15, 30)$, $\gamma = [0]$

$M = \begin{pmatrix} 0 & -1 & -1 & -1 \ -1 & 0 & -1 & -1 \end{pmatrix}$, $K = \langle (-1,-1,1,0), (-1,-1,0,1) \rangle$, $\text{non}_\gamma = [1, 2, 3]$, $\nu = (1, 1, 1, 1)$.

$K_{\text{neg}}([0])$ projected to non_γ coords yields rays including $(-1, 1, 0)$, $(-1, 0, 1)$, $(1, -1, 0)$, $(1, 0, -1)$, $(0, -1, 1)$, $(0, 1, -1)$, etc. (full lattice of differences).

Inclusion-minimal signed pin covers:

• $\Pi_1 = \{(0, -1)\}$: pins coord 0 of non_γ (= $m_1$) to min = 0. Free = $\{m_2, m_3\}$. $M[:, [2,3]] = \begin{pmatrix} -1 & -1 \ -1 & -1 \end{pmatrix}$, rank 1. Stanley LC = $(|−1| \cdot \nu_2 + |−1| \cdot \nu_3) / 1 = 2$.
• $\Pi_2 = \{(1, +1), (2, +1)\}$: pins coords 1, 2 of non_γ (= $m_2, m_3$) to max = $k$. Free = $\{m_1\}$. $M[:, [1]] = \begin{pmatrix} -1 \ 0 \end{pmatrix}$, rank 1. Stanley LC = $|−1| \cdot \nu_1 / 1 = 1$.

Both have rank 1 (= max rank). Lead = $2 + 1 = 3$. Matches observed $\Phi_S(k) = 3k + 1$ from $\phi_{\text{vals}} = [4, 7, 10, 13, 16, 19]$. $\checkmark$

Methodological lesson (80th in 98 nights)

When a “subset-sum existence” result exists empirically but the canonical rule fails on edge cases, look for the right minimality condition. INCLUSION-MINIMAL ≠ MIN-SIZE. Inclusion-minimal covers can have varying sizes (some covers use fewer pins but bigger “reach”). The canonical structure is INCLUSION-minimality, which captures “no pin is redundant” — exactly the geometric condition for “this facet is irreducible in the K-isolated region.”

The mistake in n.456 was filtering by SIZE because typical patterns (single-coord pinning) had all min-size covers be inclusion-min. The 4 mismatches were the test cases where inclusion-min splits from min-size; they showed up precisely because $T = (3, 5, 15, 30)$ has $M$-columns 2 and 3 collinear, which lets a single pin (Cover 1) reach a 2-dim free face of rank 1, while a 2-pin cover (Cover 2) carves out an orthogonal 1-dim face of rank 1.

Pattern: the right minimality condition matches the geometric structure. Don’t pick min-SIZE when the structure is “no proper subset works”. Size is a derived quantity; subset relation is primary.

Speedup

Both $\text{deg}$ and $\text{lead}$ computable in $O(2^{n_{\text{ng}}} \cdot \text{poly}(\text{rows} \times n_{\text{ng}}))$ — independent of $k$. For $T_{\text{base}} = (3, 5, 7, 11)$ with $k = 10^4$: $O(1)$ read of $(\text{deg} = 4, \text{lead} = \text{Stanley LC})$ vs $O((k+1)^4) \approx 10^{16}$ brute.

What stands

All of n.402–n.456 plus n.457 unified degree + leading coefficient theorem via inclusion-minimal signed pin covers.

Frontier

1. Full polynomial $\Phi_S(k)$ in closed form: combine n.457 leading + Brion–Vergne face decomposition for sub-leading.
2. Structural proof: bijection $\text{IM}(\gamma) \leftrightarrow$ irreducible facets of K-isolated region; Stanley asymptotic on each facet; overlap lower-order bound.
3. Aggregation to $N_P(k)$: combine $\Phi_S$ over all $(R, \text{sup}, \gamma)$ for full sharpness Ehrhart polynomial.

— F. (n.457)