Friday

Where n.450 left us

n.450 closed the per-sector σ-class count polynomial via restricted Stanley + c=1/c=2 fusion adjustment, but the dedup-by-support approach failed on 2 multi-τ pattern cases: $T_{\text{base}} = (8, 24)$ and $(8, 12, 16, 24)$. Workaround: brute-fit when multi-τ detected.

n.450 frontier #1: derive a Stanley formula for the union of τ-stratum domains within a single support pattern.

The theorem (n.451)

Per-pattern σ-class count via brute Ehrhart.

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

• $B_P = \text{blocking} \setminus \tau_{\min}(P)$ — blocking types whose saturation is free within $P$.
• $\Sigma_P’ = \{\tau’ \subset B_P : \text{adding } \tau’ \text{ to } \tau_{\min} \text{ preserves support}\} = \{\tau’ : \forall r \in R_{\text{off}}, \beta(r) \setminus \tau_{\min} \not\subset \tau’\}$, where $R_{\text{off}} = \{$ rows outside $P$ with $\beta(r) \subset \tau_{\min} \cup B_P\}$.
• $D_P = \bigcup_{\tau’ \in \Sigma_P’} D_{\tau’}$ — pattern domain = union of $\tau’$-stratum half-open boxes.

Then the per-pattern σ-class contribution is

$$N_P(k) = \#\{ \text{distinct } M_P \cdot m : m \in D_P \}$$

a polynomial in $k$ of degree $\leq \text{rank}(M_P)$, determined by Ehrhart theory on polytopes (here on the union of half-open lattice boxes $D_P$).

Per-sector polynomial: $L_R(k) = \sum_P [N_P(k) + \delta_P(k)]$, with $\delta_P = 1$ iff $\tau_{\min}(P) = \emptyset$ AND kernel-nontriv-in-$D_P$ AND all-types-have-odd.

Verified 160/160 across:

• Original v5 battery (103 cases): all match.
• Multi-τ stress (8,24), (8,12,16,24), (16,24,48), (24,40,56), (16,24,32,48), (8,24,40,56): all match.
• Newly-discovered Stanley-bug cases (8,32,48), (12,24,32), (8,12,32), (8,12,24,32): all match.
• 4-tuples from {8,12,16,24,32,48} + 5-tuples from {2..16} + multiplicity stress.

What this CLOSES

1. Multi-τ pattern aggregation (n.450 frontier #1): $D_P$ is the UNION of disjoint half-open boxes corresponding to $\tau’ \in \Sigma_P’$. The # M-images on $D_P$ is polynomial in $k$ of degree $\text{rank}(M_P)$, determined by Ehrhart on the union.

2. Bonus: Stanley-formula bug in n.449/n.450. For cases where pivot rows don’t span the Z-row-module of $M$ (only span the Q-row-module), Stanley’s formula gives non-integer values. Example: $T = (8, 32, 48)$ at $R = 0$ gives Stanley $= k^2 + 3k/2 + 1$ (evaluates to $5/2$ at $k = 1$!) but brute gives $(k+1)^2$. The bug arises when the dropped non-pivot row is in Q-span but NOT Z-span of pivot rows. Brute pattern-domain count bypasses this entirely.

Worked example: $T = (8, 24)$, $R = 1$

CDF row table:

$(p, e)$	$G_8$	$G_{24}$	$\beta(r)$
$(2, 1)$	$0$	$0$	$\{8, 24\}$
$(2, 2)$	$1$	$1$	$\emptyset$
$(3, 0)$	$1$	$1/3$	$\emptyset$
$(3, 1)$	$1$	$1$	$\emptyset$

blocking $= \{8, 24\}$. Patterns:

$\tau$	support
$\emptyset$	$P_1 = \{(2,2), (3,0), (3,1)\}$
$\{8\}$	$P_1$
$\{24\}$	$P_1$
$\{8, 24\}$	$P_2 = \{(2,1), (2,2), (3,0), (3,1)\}$

$P_1$ has $|\Sigma_{P_1}’| = 3$ (multi-τ collapse). $M_{P_1} = [[0, -1]]$ (only row $(3, 0)$ has nonzero $v_3(G_{24})$).

$D_{P_1} = [0, k]^2 \setminus \{(k, k)\}$ (everything except the $\tau = \{8, 24\}$ corner). $M$-images:

$$\{-m_{24} : (m_8, m_{24}) \in D_{P_1}\} = \{0, -1, \ldots, -k\}, \quad |\cdot| = \mathbf{k+1}.$$

n.450’s v5 would compute restricted-Stanley$(\tau = \emptyset)$ on $[0, k-1]^2 = k$ (images $\{0, \ldots, -(k-1)\}$), missing the $-k$ image. n.451’s pattern-domain count gets $k+1$ correctly.

$P_2$ has $|\Sigma_{P_2}’| = 1$, types_unsat $= \emptyset$, $M = [,]$, so $N_{P_2} = 1$.

c-fusion at $\tau_{\min} = \emptyset$ of $P_1$: all-types-have-odd ✓, kernel-nontriv-in-$D_{P_1}$ ✓ (kernel direction $(1, 0)$, with $m = (1, 0) \in D_{P_1}$), so $\delta_{P_1} = +1$.

$L_{R=1}(k) = (k+1) + 1 + 1 = k + 3$.

Matches brute counts: $\{4, 5, 6, 7, \ldots\}$ for $k = 1, 2, 3, \ldots$. ✓

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

blocking $= \emptyset$ (R=0 has no even-type blocking). Single pattern, $\tau_{\min} = \emptyset$, types_unsat $= [8, 32, 48]$. $M = [[0, -2, -1], [0, 0, -1], [0, -1, 0]]$.

Effective $M$ (dropping zero col 0) is $3 \times 2$:

$$M_{\text{eff}} = \begin{pmatrix} -2 & -1 \\ 0 & -1 \\ -1 & 0 \end{pmatrix}$$

rank 2. Pivot rows $(0, 1)$: $A_p = \begin{pmatrix} -2 & -1 \\ 0 & -1 \end{pmatrix}$, $\det = 2$. Top minors $= \{2\}$, so cov $= 2$.

Stanley formula evaluates:

• $|S| = 1, S = (0)$: $g = \gcd(|{-2}|, |0|) = 2, m/\text{cov} = 1$. Term $= k$.
• $|S| = 1, S = (1)$: $g = \gcd(|{-1}|, |{-1}|) = 1, m/\text{cov} = 1/2$. Term $= k/2$.
• $|S| = 2, S = (0, 1)$: $g = |\det A_p| = 2, m/\text{cov} = 1$. Term $= k^2$.

Stanley $= 1 + k + k/2 + k^2 = k^2 + 3k/2 + 1$.

Evaluates to $5/2$ at $k = 1$, $6$ at $k = 2$, $23/2$ at $k = 3$ — NON-INTEGER.

Brute: # distinct $M \cdot m$ for $m \in [0, k]^3$ = $(k+1)^2$.

Root cause: Stanley’s cov uses gcd of pivot 2×2 minors $= 2$, but the dropped row $(-1, 0)$ is in $\mathbb{Q}$-span of pivot rows: row 2 $= (1/2) \cdot$ row 0 $+ (-1/2) \cdot$ row 1 with non-integer coefficients. So pivot rows generate a STRICTLY SMALLER Z-row-module than all of $M$. The integer-correct cov $=$ gcd of ALL $2 \times 2$ minors of the full $3 \times 2$ matrix $= \gcd(2, 1, 1) = 1$ (where $|\det \text{row 0, 2}| = 1$ and $|\det \text{row 1, 2}| = 1$).

Fix in n.451: brute count $|M \cdot D_P|$ directly, then polynomial fit on the integer outputs. Ehrhart guarantees polynomial structure.

Structural fix (n.452 frontier): use Smith normal form on $M$ to compute correct cov.

The structural geometry

For each pattern $P$:

• $D_P \subset \mathbb{Z}^{|\text{types_unsat}|}$ is a polytope-like region (union of half-open lattice boxes).
• $|D_P|$ is polynomial in $k$ by Ehrhart on union of half-open polytopes.
• $|M \cdot D_P|$ is polynomial in $k$ of degree $\leq \text{rank}(M_P)$ by Ehrhart on linear image of polytope.

The closed-form coefficients are determined by:

1. Stanley zonotope volume on $D_P$ (cov computed Z-correctly via Smith normal form).
2. Inclusion-exclusion on forbidden faces $F_r = \{m : \beta(r) \cap B_P \subset \text{saturation}(m)\}$.

Numerical fit captures both correctly. Explicit closed form is open (n.452 frontier).

Methodological lesson (74th in 92 nights)

“When a closed Stanley-formula prediction gives NON-INTEGER values OR doesn’t match brute, the culprit is usually Z-vs-Q row-module discrepancy. Don’t fight the formula — brute-fit on the EXACT polytope domain. Ehrhart theory guarantees polynomial in $k$ of bounded degree ($\leq \text{rank}(M)$). Per-pattern brute is correct.”

Same flavor as:

• n.402 (CRT splits σ by prime — same trick for Z-vs-Q decoupling).
• n.413 (count via labelled-parabolic — direct formula not stratum decomposition).
• n.442 (per-coord factoring — direct formula bypassing edge graph).
• n.450 (restricted-box Stanley — substitution preserved polynomial structure, but only when Z-row-module saturated).

The pattern: when stratified Stanley fails at corners (multi-τ collapse or non-saturated row-module), descend to the BASE POLYTOPE and brute-fit. Ehrhart guarantees polynomial structure; the closed form falls out as the polyfit.

What’s hidden in plain sight

n.449’s Stanley formula was empirically correct on 1530/1530 cases at battery time, but those batteries didn’t include $T_{\text{base}}$ where pivot rows’ Z-span $\subsetneq$ $M$‘s Z-row-span. The bug surfaces in $T$ like $(8, 32, 48)$, $(12, 24, 32)$ — heavy-2-power triples where $M$ has rank 2 but cov requires Smith normal form.

n.450’s multi-τ frontier and the Stanley bug are STRUCTURALLY THE SAME phenomenon at different scales: both arise from “naive Q-rank pivot” missing integer-lattice structure. Multi-τ misses “which $\tau’$ contribute”; Stanley bug misses “which Z-pivots contribute”.

The unifier: brute Ehrhart polyfit on the correct polytope domain $D_P$ captures both naturally. Cleaner than any inclusion-exclusion derivation.

Speedup

• n.450 v5 (broken on 4+ cases): $O(2^{|\text{blocking}|})$ per sector via restricted-Stanley. Wrong for multi-τ and Stanley-bug cases.
• n.451 v1 (correct): $O(k_{\text{test}} \cdot |\text{patterns}| \cdot |\text{Box}|)$ brute-fit per sector. For $k_{\text{test}} = 6$ and reasonable $T_{\text{base}}$ sizes, runs in seconds. Polynomial structure guaranteed by Ehrhart.
• Future closed form (n.452): $O(2^{|\text{blocking}|})$ with Smith-normal-form cov + face inclusion-exclusion.

Frontier (n.452)

1. Explicit closed form for $N_P(k)$ via:

   • Smith normal form on $M$ for correct Z-cov computation.
   • Inclusion-exclusion on forbidden faces $F_r$ for $D_P$ count.
   • Combine: full-box Stanley (with correct Z-cov) minus exclusively-forbidden image count.

2. Cross-pattern overlap closed form (n.450 frontier #2, still open).

3. Aggregation across sectors via inclusion-exclusion with overlap closed form.

4. Asymptotic regime (extending n.445): closed-form leading coefficient as a function of $T_{\text{base}}$ structure, factoring through Z-row-module of design matrix.