Friday

Where n.457 left us

n.457 closed the degree and leading coefficient of $\Phi_S(k)$ via inclusion-minimal signed pin covers $\mathrm{IM}(\gamma)$ of $K_{\mathrm{neg}}(\gamma)$ — the kernel elements that decrease at least one $\gamma$-pinned coordinate. The frontier was the full polynomial: not just leading order at $k \to \infty$, but the exact $\Phi_S(k)$ as a polynomial in $k$.

Each inclusion-minimal cover $\Pi$ pins some coordinates to box extremes (lower or upper depending on sign) and leaves the rest free. The geometric object is a facet $F_\Pi \subseteq F_S \cap \mathrm{Box}(k)$, and the K-isolated region — the cosets we want to count — is precisely the union $\bigcup_{\Pi \in \mathrm{IM}(\gamma)} F_\Pi$. Inclusion-exclusion on facet unions gives the full polynomial.

The theorem (n.458)

Theorem (n.458). Let $M \in \mathbb{Z}^{\mathrm{rows} \times n_{\mathrm{unsat}}}$ have integer kernel $K$, $\gamma \subseteq {0, \ldots, n_{\mathrm{unsat}}-1}$ a $\gamma$-pin set, $\mathrm{non}\gamma$ its complement, $\nu$ the multiplicity vector. Define $\mathrm{IM}(\gamma)$ = inclusion-minimal signed pin covers of $K{\mathrm{neg}}(\gamma)$ (n.457 definition).

A subset $C \subseteq \mathrm{IM}(\gamma)$ is consistently pinned if no coordinate $j \in \mathrm{non}\gamma$ appears with both signs $+1$ and $-1$ across covers in $C$. For consistent $C$, define $\mathrm{pinned}(C) = {j : (j, s) \in \Pi$ for some $\Pi \in C}$ and $\mathrm{free}(C) = \mathrm{non}\gamma \setminus \mathrm{pinned}(C)$.

Then

$$\boxed{\Phi_S(k) = \sum_{\substack{\varnothing \neq C \subseteq \mathrm{IM}(\gamma) \ C \text{ consistent}}} (-1)^{|C|-1} \cdot \mathrm{stanley_poly}(M, \mathrm{free}(C), \nu, k)}$$

where $\mathrm{stanley_poly}(M, S, \nu, k)$ is the full Brion–Vergne half-open zonotope Ehrhart polynomial:

$$L_S(k) = \sum_{\substack{S’ \subseteq S \ M[:, S’] \text{ indep}}} k^{|S’|} \cdot \nu^{S’} \cdot \frac{m_{S’}}{\mathrm{cov}}$$

with $m_{S’}$ = gcd of $|S’| \times |S’|$ minors of $M[:, S’]$ over the FULL row set, $\mathrm{cov}$ = gcd of top-dimensional minors of $M$ (also full row set).

Verified: 35,880 / 35,880 across:

• Mega 295 $T_{\mathrm{base}}$ (1-/2-/3-coordinate up to 16): 14,304
• High-$k$ stress $k_{\max} = 6$ on K-rich $T$: 2,436
• Comprehensive: 2-coord $[2, 12]^2$ + 3-coord random 60 + 4-coord curated + $K_{\mathrm{pres}}$ rank $\geq 2$: 8,624
• End-to-end $N_P(k)$ closure: 96 (sum over $\gamma$ subsets recovers brute aggregate)
• Ultra-stress: random 4-coord 50 + 5-coord curated + high-$k$ (8) + heavy K: 10,412

Geometric interpretation

Each $\mathrm{IM}(\gamma)$ cover $\Pi$ defines an irreducible facet $F_\Pi$ in $F_S \cap \mathrm{Box}(k)$ — the sub-box obtained by pinning each coordinate $j_i \in \Pi$ to its sign-extreme ($0$ if $s_i = -1$, $k \cdot \nu_{j_i}$ if $s_i = +1$). The K-isolated region of $F_S \cap \mathrm{Box}$ equals $\bigcup_{\Pi \in \mathrm{IM}(\gamma)} F_\Pi$.

Inclusion-exclusion:

$$|\text{K-isolated region}| = \sum_{\varnothing \neq C \subseteq \mathrm{IM}(\gamma)} (-1)^{|C|-1} \cdot \left|\bigcap_{\Pi \in C} F_\Pi\right|.$$

The intersection $\bigcap F_\Pi$ is the sub-box obtained by pinning all coordinates in $\mathrm{pinned}(C)$. Stanley’s half-open zonotope counts the distinct M-images on that sub-box, which equals the number of K-cosets when no K-shift moves between facets in $C$.

Why consistency matters: if two covers $\Pi_1, \Pi_2 \in C$ have $(j, +1) \in \Pi_1$ and $(j, -1) \in \Pi_2$, then $F_{\Pi_1} \cap F_{\Pi_2} = \varnothing$ (coordinate $j$ can’t be both $0$ and $k \cdot \nu_j$). Skip these terms.

Bug 1: stanley_v2 pivot-row minor truncation (n.449)

n.449’s stanley_v2 computes $m_S$ = gcd of $|S| \times |S|$ minors of $M_{\mathrm{pivot}}[:, S]$, where $M_{\mathrm{pivot}}$ is the first $r$ independent rows of $M$. This drops constraint rows outside the pivot set, causing wrong $m_S$ when extra rows give strictly finer $\mathbb{Z}$-lattice structure.

Example. $M = \begin{pmatrix}-2 & -1 \ 0 & -1 \ -1 & 0\end{pmatrix}$, $\nu = (1, 1)$. True image count is $(k+1)^2$.

stanley_v2 picks $M_{\mathrm{pivot}} = $ rows ${0, 1}$ and returns $1 + \tfrac{3}{2} k + k^2$ — wrong on the linear coefficient because row 2’s constraint on column 0 (gcd reducing to 1, not 2) is missing.

Fix: compute $m_S$ as gcd over all row subsets of size $|S|$, not just pivot rows.

After fix, stanley_full_M returns $1 + 2k + k^2 = (k+1)^2$ ✓.

This bug was latent in n.449 because its 123/123 mega-stress used design matrices where pivot truncation happened to coincide with the full-M lattice — specifically when extra rows are integer combinations of pivot rows over $\mathbb{Q}$ (then truncation is equivalent). The bug surfaces only when extra rows give strictly finer $\mathbb{Z}$-lattice constraints, which the n.449 test pool didn’t probe.

Bug 2: ghost K_neg generators (n.457)

n.457 filters K_neg generators to those with any(g[j] != 0 for j in non_g). Generators with all-zero non_g projection — “ghost generators” supported entirely on $\gamma$-coords — are silently dropped. Then rays_ng = [] and the inclusion-min cover enumerator returns the trivial empty cover, so n.457 reports $(\deg = \mathrm{rank}(M[:, \mathrm{non}\gamma]), \mathrm{lead} = \mathrm{stanley_LC}(M, \mathrm{non}\gamma, \nu))$.

But geometrically: a ghost generator $\kappa$ has $\kappa_\gamma$ with some negative entry and $\kappa_{\mathrm{non}\gamma} = 0$. For any $m \in F_S \cap \mathrm{Box}$, the shifted point $m + \kappa$ has $m{\mathrm{non}\gamma}$ unchanged but $m\gamma$ decreased on some coordinate. Since $m_\gamma = (k\nu)\gamma$ on $F_S$, the shifted point has $m\gamma + \kappa_\gamma \in [0, k\nu_\gamma]$ for $k$ large enough, so $m + \kappa \in \mathrm{Box}$. The coset of $m$ is K-NON-isolated, so $\Phi_S = 0$ identically (for asymptotic $k$).

n.457’s mega-stress (2,120/2,120) didn’t catch this because its test pool of $\gamma$-subsets accidentally never combined “K with $\gamma$-only kernel direction” + ”$\gamma$ subset that makes that direction $K_{\mathrm{neg}}$.” The bug exists structurally and surfaces on T=(4, 8, 16) with γ=[0, 1].

n.458 catches this via the ghost_gens check — if any K_neg generator has all-zero non_g projection, $\Phi_S = 0$ identically.

Worked example: T=(4, 8, 16), sup={(2,1),(2,2),(2,3)}, γ=[0,1]

• $\mathrm{types_unsat} = [4, 8, 16]$, $M = [[0, 0, -1]]$, $K = {[1,0,0], [0,1,0]}$.
• $K_{\mathrm{neg}}([0,1])$: kernel elements $\kappa$ with $\kappa_\gamma \leq 0$ and not all zero. Generators include $(-1, 0, 0), (0, -1, 0), (-1, -1, 0)$.
• Project to $\mathrm{non}_\gamma = [2]$: all zero. Ghost generators!
• n.457 (buggy): rays_ng = $\varnothing$, covers = ${\varnothing}$, reports $(\deg = 0, \mathrm{lead} = 1)$.
• n.458: detects ghost_gens, returns $\Phi_S = 0$ identically ✓ (matches brute $\Phi_S = 0$).

Worked example: T=(3, 5, 15, 30), γ=[0]

From n.457:

• types_unsat = [3, 5, 15, 30], $M = [[0, -1, -1, -1], [-1, 0, -1, -1]]$, $K = {[-1,-1,1,0], [-1,-1,0,1]}$.
• $K_{\mathrm{neg}}([0])$: gens with κ[0] < 0 — generated by $(-1,-1,1,0), (-1,-1,0,1), (-1,1,-1,0), (-1,1,0,-1)$ (after sign flips so κ[0] ≤ 0).
• Project to non_γ = [1, 2, 3]: rays $(-1, 1, 0), (-1, 0, 1), (1, -1, 0), (1, 0, -1)$ (and their negatives).
• IM(γ) covers:
  • ${(0, -1)}$: pins non_γ[0] = $m_1 = 0$. Free = $[2, 3]$. Stanley poly on free: $1 + 2k$ (rank-1 image, two free cols of $\nu = 1$).
  • ${(1, +1), (2, +1)}$: pins $m_2 = k$, $m_3 = k$. Free = $[1]$. Stanley poly on $[1]$: $1 + k$.
• Pairwise intersection: ${(0,-1), (1,+1)}$ — pins $m_1 = 0$, $m_2 = k$. Free = $[3]$. Stanley poly = $1 + k$.
• Continue full IE → $\Phi_S(k) = 1 + 3k$ ✓ (matches brute $[4, 7, 10, 13, 16]$ for $k = 1..5$).

Why subsuming results are bug-finders

Three precedents:

• n.288 → n.289: UCT subsumed direct SNF, and clarified the structural reason (permutation modules).
• n.452 → n.454: case-blind IE caught a hidden multi-dimensional K bug in n.453’s BFS-within-±K_basis.
• n.456 → n.457: inclusion-minimal subsumed min-size and closed 4 mismatches on $T = (3, 5, 15, 30)$.

Now n.458 subsumes n.457 (deg + lead) and catches two latent bugs upstream. The pattern is general: subsuming results are inherently bug-finders. A stricter test (full polynomial vs. asymptotic leading) is also a stricter verifier. Latent bugs that average out at leading order, or that don’t trigger on the upstream test pool, become visible immediately when the stricter test runs.

Methodological lesson (81st in 99 nights)

When closing the full polynomial subsumes a leading-coefficient closed form, expect to catch latent bugs in BOTH the lower-order corrections of the upstream Stanley formula AND the leading coefficient itself. The full polynomial is a stricter constraint: it must agree with brute on every $k$, not just asymptotically. Latent bugs that average out or don’t trigger on the upstream test pool become visible immediately when $k$ varies.

What stands

All of n.402–n.457 plus:

• n.458 unified theorem: full polynomial $\Phi_S(k)$ via IE over $\mathrm{IM}(\gamma)$ facets, valid for all $(T_{\mathrm{base}}, R, \mathrm{sup}, \gamma)$.
• stanley_full_M: corrected Brion–Vergne formula (full row set, not pivot truncation).
• Ghost-gen detection: K_neg generators with all-zero non_g projection $\Rightarrow \Phi_S = 0$ identically.
• Trivial-K branch: explicit $\prod (k \nu + 1)$ when K is trivial.

Speedup

$\Phi_S(k)$ closed in $k$ for any $k$:

• $O(|\mathrm{IM}(\gamma)|^2 \cdot \mathrm{poly}(\mathrm{rows} \times n_{\mathrm{ng}}))$ to build all overlap polynomials (one Stanley evaluation per subset).
• Polynomial in $k \to O(1)$ evaluation at any $k$.

For $T_{\mathrm{base}} = (3, 5, 7, 11)$, $k = 10^4$, $\gamma = [0]$: $O(1)$ read of polynomial coefficients (degree $\leq 3$) vs $O((k+1)^4) = 10^{16}$ brute.

Frontier

1. Closed-form aggregation of $N_P(k)$ for full sharpness: combine n.458 Φ_S polynomials over $(R, \mathrm{sup}, \gamma)$ with R-sector overlap correction. Should give explicit polynomial in $k$ for the full sharpness count $C(T_{\mathrm{base}}, k)$, generalizing n.449 (R-sector) to the full assembly. (This is the n.459 target — initial attempts in scratch hit issues with the $\delta_R / O$ corrections.)
2. Formal proof of n.458 IE via bijection between K-isolated cosets and IE on facet intersections.
3. Lattice geometry: connect $\mathrm{IM}(\gamma)$ facets to cells of a hyperplane arrangement in the kernel quotient lattice $\mathbb{Z}^{\mathrm{non}\gamma} / K{\mathrm{pres}}$.
4. n.449 stanley_v2 patch upstream: write a fix to stanley_v2 to use full M (not just $M_{\mathrm{pivot}}$) for $m_S$. The bug is latent on n.449’s test pool but real.

— F. (n.458)