Wilf's Conjecture from the First Kunz Layer

1. Introduction

Let $S\subseteq \mathbb{N}$ be a numerical semigroup. We write

$$m:=m\left(S\right)=min(S\setminus \{0\left\}\right),e:=e\left(S\right),c:=c\left(S\right)=F\left(S\right)+1,$$

for its multiplicity, embedding dimension, and conductor, respectively, and

$$L:=L\left(S\right):=S\cap [0,c)$$

for its set of left elements. Wilf asked in 1978 whether every numerical semigroup satisfies [19]

$$e\left|L\right|\ge c.$$

(1.1)

Equivalently, if $g\left(S\right):=|\mathbb{N}\setminus S|$ denotes the genus, then $\left|L\right|=c-g\left(S\right)$ and (1.1) may be rewritten as

$$e\left(c-g\left(S\right)\right)\ge c.$$

For general background on the conjecture, we refer to Delgado’s survey [5] and to the monograph [17].

Despite its deceptively simple form, Wilf’s conjecture remains open in full generality. Nevertheless, a large body of partial results is known. The embedding-dimension-3 case is classical; see Fröberg, Gottlieb, and Häggkvist [12], Kaplan [15], and the recent revisit by Fel [11]. Sammartano proved Wilf’s conjecture whenever $e\ge m/2$, and in the same paper also settled the cases $m\le 8$ and numerical semigroups generated by generalized arithmetic sequences [18]. Eliahou later proved the conjecture for all semigroups with $c\le 3m$ [8], and then extended the general range to $e\ge m/3$ by a graph-theoretic method [9]. A computational and polyhedral approach of Bruns, García-Sánchez, O’Neill, and Wilburne proves the conjecture for all semigroups of multiplicity $m\le 18$ [3]. Moscariello and Sammartano obtained a complementary number-theoretic result showing that, for each fixed value of $\lceil m/e\rceil$, Wilf’s conjecture holds for all sufficiently large multiplicities outside a finite exceptional set of prime divisors [16]. More recently, D’Anna and Moscariello established new bounds for semigroup invariants and, as a corollary, recovered Wilf’s conjecture for almost-symmetric numerical semigroups [4]. In the divisible-conductor case, Eliahou proved in 2025 that Wilf’s conjecture holds whenever $m\mid c$ and $e\ge m/4$ [10].

A complementary line of work has pushed exhaustive verification to steadily larger genera. Bras-Amorós reported verification up to genus 50 [1]; Fromentin and Hivert extended this to genus 60 [13]; Bras-Amorós and Marín Rodríguez reached genus 65 [2]; and Delgado, Eliahou, and Fromentin proved in 2025 that Wilf’s conjecture holds for all numerical semigroups of genus at most 100, and up to genus 120 in the subcase $m\mid c$ [6].

There is also an important asymptotic perspective. Zhai proved that the proportion of numerical semigroups of genus g satisfying $f<3m$ tends to 1 as $g\to \infty$[20]. Since $c=f+1$, this implies that asymptotically almost all numerical semigroups satisfy Wilf’s conjecture once combined with Eliahou’s theorem for $c\le 3m$. More recently, Delgado, Kumar, and Marion introduced a new counting framework based on the maximum primitive and proved, among other things, the first-layer criterion

$$|S\cap (m,2m)|\ge \sqrt{3m}⟹S\mathrm{satisfies} \mathrm{Wilf}\text{’}\mathrm{s} \mathrm{conjecture}$$

(1.2)

[7]. Their work also shows that almost all numerical semigroups with sufficiently large maximum primitive satisfy Wilf’s conjecture.

The purpose of the present paper is to extract substantially more information from the first Kunz layer, namely from the interval $(m,2m)$. Set

$$\eta :=\eta \left(S\right):=|S\cap (m,2m\left)\right|.$$

Every element of $S\cap (m,2m)$ is primitive, so $\eta$ counts a distinguished subfamily of the minimal generators of S. Writing the conductor in the form

$$c=qm-\rho ,q=⌈{\displaystyle \frac{c}{m}}⌉,0\le \rho <m,$$

(1.3)

we also define the trimmed first-layer count

$${\eta }_{\rho }:={\eta }_{\rho }\left(S\right):=|S\cap (m,2m-\rho )|.$$

Our first main result is the following.

Theorem 1.1. 

For every numerical semigroup S,

$$\left|L\right|\ge q+(q-2)\eta +{\eta }_{\rho },$$

(1.4)

and therefore

$$W\left(S\right)\ge q\left(e(\eta +1)-m\right)-2e\eta +e{\eta }_{\rho }+\rho .$$

(1.5)

Two immediate consequences deserve to be highlighted.

Theorem 1.2. 

Let S be a numerical semigroup.

(i)

If $e(\eta +2)\ge 2m$, then S satisfies Wilf’s conjecture.

(ii)

In particular, if $(\eta +1)(\eta +2)\ge 2m$, then S satisfies Wilf’s conjecture.

The second statement yields the explicit threshold

$$\eta \ge ⌈{\displaystyle \frac{\sqrt{8m+1}-3}{2}}⌉⟹S\mathrm{satisfies}\mathrm{Wilf}\text{’}\mathrm{s}\mathrm{conjecture},$$

(1.6)

which is asymptotically $\eta \gtrsim \sqrt{2m}$. In particular, it improves the 2025 criterion (1.2) from $\sqrt{3m}$ to $\sqrt{2m}$.

In the divisible-conductor case we obtain a stronger estimate.

Theorem 1.3. 

Assume $m\mid c$. Then:

(i)

if $e(3\eta +4)\ge 4m$, then S satisfies Wilf’s conjecture;

(ii)

in particular, if $3{\eta }^2+7\eta +4\ge 4m$, then S satisfies Wilf’s conjecture.

We also show that the staircase method extends naturally from the first Kunz layer to cumulative Kunz layers. Writing

$${\Theta }_t:=\left|\{i\in \{1,\dots ,m-1\}:x_i\le t\}\right|,$$

for the cumulative population of the first t Kunz layers, we derive an exact formula for $\left|L\right|$ in terms of the numbers ${\Theta }_t$, and from it a hierarchy of sufficient Wilf criteria indexed by two integers $(Q,t)$. Even the choice $t=2$ already strengthens the interval-generated application from $r\asymp \sqrt{2m}$ to $r\asymp \sqrt{m}$.

The proofs are elementary once a suitable Apéry/Kunz expression for $\left|L\right|$ has been written down. Conceptually, the first-layer method isolates precisely those residues whose Kunz coordinates are equal to 1 and then propagates them upward by translation by the multiplicity. This produces disjoint residue staircases lying entirely below the conductor. The multi-layer extension reorganizes the same exact residue count into cumulative populations of low Kunz layers. The resulting lower bounds already detect infinite families beyond the broad regimes

$$c\le 3m,e\ge m/3,m\mid c\mathrm{with}e\ge m/4,\eta \ge \sqrt{3m}.$$

Structure of the paper.Section 2 fixes notation and derives an exact residue-by-residue formula for $\left|L\right|$ in Kunz coordinates. Section 3 extracts from it the first-layer staircase estimate. Section 4 proves the new first-layer Wilf criteria. Section 5 develops the cumulative-layer extension of the staircase method. Section 6 studies interval-generated semigroups, first as a test family for the first-layer criteria and then as an illustration of the additional strength supplied by the second cumulative layer.

2. Preliminaries

We follow the standard notation from [17]. Throughout the paper, $\mathbb{N}=\{0,1,2,\dots \}$ and $S\subseteq \mathbb{N}$ is a numerical semigroup, assumed distinct from $\mathbb{N}$. Thus $c=c\left(S\right)\ge 1$. We write

$$m=m\left(S\right),e=e\left(S\right),L=S\cap [0,c),$$

and

$$c=qm-\rho ,q=⌈{\displaystyle \frac{c}{m}}⌉,0\le \rho <m.$$

Definition 2.1. 

An element $s\in S\setminus \left\{0\right\}$ is called primitive if it does not belong to $(S\setminus \{0\left\}\right)+(S\setminus \{0\left\}\right)$; equivalently, if it belongs to the unique minimal system of generators of S.

Definition 2.2. 

The Apéry set of S with respect to m is

$$Ap(S,m):=\{w\in S:w-m\notin S\}.$$

Enumerating its elements by residue classes modulo m, we write

$$Ap(S,m)=\{w_0,w_1,\dots ,w_{m-1}\},w_0=0,w_i\equiv i(modm).$$

Thus

$$w_i=i+m{x}_i(1\le i\le m-1)$$

for uniquely determined integers $x_i\ge 1$. The vector

$$(x_1,\dots ,x_{m-1})\in {\mathbb{Z}}_{\ge 1}^{m-1}$$

is the Kunz coordinate vector of S.

Recall that every element of S in residue class i modulo m is of the form $w_i+tm$ with $t\in \mathbb{N}$.

Lemma 2.3. 

Every element of $S\cap [m,2m)$ is primitive. In particular,

$$\eta =|S\cap (m,2m\left)\right|\le e-1.$$

(2.1)

Proof. 

Let $s\in S\cap [m,2m)$. If $s=a+b$ with $a,b\in S\setminus \left\{0\right\}$, then $a\ge m$ and $b\ge m$, hence $s\ge 2m$, a contradiction. Thus s is primitive. Since m itself is primitive, the set

$$\left\{m\right\}\cup \left(S\cap (m,2m)\right)$$

consists of $\eta +1$ distinct primitive elements, and therefore $e\ge \eta +1$. □

For $0\le i\le m-1$ let

$$L_i:=\{\ell \in L:\ell \equiv i(modm)\}.$$

Then

$$\left|L\right|=\sum _{i=0}^{m-1}\left|L_i\right|.$$

The next proposition gives an exact formula. Empty sums are understood to be equal to 0.

Proposition 2.4. 

With the above notation,

$$\left|L\right|=q+\sum _{i=1}^{m-\rho -1}max\{0,q-x_i\}+\sum _{i=m-\rho }^{m-1}max\{0,q-1-x_i\}.$$

(2.2)

Moreover,

$$\eta =\left|\{i\in \{1,\dots ,m-1\}:x_i=1\}\right|,{\eta }_{\rho }=\left|\{i\in \{1,\dots ,m-\rho -1\}:x_i=1\}\right|.$$

(2.3)

Proof. 

Since

$$L_0=\{0,m,2m,\dots ,(q-1)m\},$$

we have $\left|L_0\right|=q$.

Fix $1\le i\le m-1$. The elements of S in residue class i are precisely

$$w_i,w_i+m,w_i+2m,\dots .$$

Hence

$$L_i=\{w_i+tm:t\in \mathbb{N},w_i+tm<c\},$$

and therefore

$$\left|L_i\right|=max\left\{0,⌊{\displaystyle \frac{c-1-w_i}{m}}⌋+1\right\}.$$

Using $c=qm-\rho$ and $w_i=i+m{x}_i$, we obtain

$${\displaystyle \frac{c-1-w_i}{m}}=q-x_i-{\displaystyle \frac{\rho +1+i}{m}}.$$

Assume first that $1\le i\le m-\rho -1$. Then

$$1\le \rho +1+i\le m,$$

so

$$⌊{\displaystyle \frac{c-1-w_i}{m}}⌋=q-x_i-1$$

and hence

$$\left|L_i\right|=max\{0,q-x_i\}.$$

Now assume $m-\rho \le i\le m-1$. Then

$$m+1\le \rho +1+i\le 2m-1,$$

so

$$⌊{\displaystyle \frac{c-1-w_i}{m}}⌋=q-x_i-2$$

and therefore

$$\left|L_i\right|=max\{0,q-1-x_i\}.$$

Summing over the residue classes yields (2.2).

Finally, $x_i=1$ if and only if $w_i=i+m\in S$, that is, if and only if the residue class i contributes an element in $(m,2m)$. This proves the first identity in (2.3). The second follows by observing that

$$m+i<2m-\rho ⟺i\le m-\rho -1.$$

□

3. The First-Layer Staircase Estimate

The formula (2.2) immediately yields a lower bound that depends only on the first Kunz layer $\{i:x_i=1\}$.

Proposition 3.1 

(First-layer staircase bound). For every numerical semigroup S,

$$\left|L\right|\ge q+(q-2)\eta +{\eta }_{\rho }.$$

(3.1)

Proof. 

By (2.2), every residue i with $x_i=1$ contributes

$$q-1\mathrm{if}1\le i\le m-\rho -1,$$

and contributes

$$q-2\mathrm{if}m-\rho \le i\le m-1.$$

All residues with $x_i\ge 2$ contribute nonnegative quantities. Therefore

$$\left|L\right|\ge q+(q-1){\eta }_{\rho }+(q-2)(\eta -{\eta }_{\rho })=q+(q-2)\eta +{\eta }_{\rho },$$

as claimed. □

Remark 3.2 

(Geometric interpretation). For each residue class i with $x_i=1$, the element $m+i\in S$ generates a vertical staircase

$$m+i,2m+i,\dots$$

inside the same residue class. The staircase stops one step earlier when $i\ge m-\rho$, because then $(q-1)m+i\ge qm-\rho =c$. This is the source of the two coefficients $q-1$ and $q-2$ in (3.1). Figure 1 illustrates the two possible staircase lengths.

A direct Wilf-type consequence is already available.

Corollary 3.3. 

If

$$e\left(q+(q-2)\eta +{\eta }_{\rho }\right)\ge c,$$

(3.2)

then S satisfies Wilf’s conjecture.

Proof. 

By (3.1) we have

$$W\left(S\right)=e\left|L\right|-c\ge e\left(q+(q-2)\eta +{\eta }_{\rho }\right)-c.$$

Thus (3.2) implies $W\left(S\right)\ge 0$. □

4. Main Wilf Criteria

We now rewrite the staircase estimate as an explicit lower bound for Wilf’s number.

Theorem 4.1. 

For every numerical semigroup S,

$$W\left(S\right)\ge q\left(e(\eta +1)-m\right)-2e\eta +e{\eta }_{\rho }+\rho .$$

(4.1)

In particular, if $m\mid c$, so that $\rho =0$ and ${\eta }_{\rho }=\eta$, then

$$W\left(S\right)\ge q\left(e(\eta +1)-m\right)-e\eta .$$

(4.2)

Proof. 

From Proposition 3.1 and $c=qm-\rho$ we obtain

$$\begin{array}{cc}\hfill W\left(S\right)& =e\left|L\right|-c\hfill \\ \hfill & \ge e\left(q+(q-2)\eta +{\eta }_{\rho }\right)-(qm-\rho )\hfill \\ \hfill & =q\left(e(\eta +1)-m\right)-2e\eta +e{\eta }_{\rho }+\rho ,\hfill \end{array}$$

which is (4.1). If $m\mid c$, then $\rho =0$ and ${\eta }_{\rho }=\eta$, and (4.2) follows immediately. □

The first consequence is the mixed $(e,\eta )$-criterion announced in the introduction.

Corollary 4.2. 

If

$$e(\eta +2)\ge 2m,$$

(4.3)

then S satisfies Wilf’s conjecture.

Proof. 

If $q\le 3$, then $c\le 3m$, so Wilf’s conjecture holds by [8, Theorem 1.1]. Assume henceforth that $q\ge 4$ and set

$$A:=e(\eta +1)-m.$$

From (4.3) we get

$$A\ge e(\eta +1)-{\displaystyle \frac{e(\eta +2)}{2}}={\displaystyle \frac{e\eta }{2}}\ge 0.$$

Now Theorem 4.1 gives

$$W\left(S\right)\ge qA-2e\eta +e{\eta }_{\rho }+\rho .$$

Since $q\ge 4$, $A\ge 0$, and $e{\eta }_{\rho }+\rho \ge 0$, we conclude

$$W\left(S\right)\ge 4A-2e\eta =2\left(e(\eta +2)-2m\right)\ge 0.$$

Thus S satisfies Wilf’s conjecture. □

Corollary 4.3. 

If

$$(\eta +1)(\eta +2)\ge 2m,$$

(4.4)

then S satisfies Wilf’s conjecture.

Proof. 

By Lemma 2.3, $e\ge \eta +1$. Hence

$$e(\eta +2)\ge (\eta +1)(\eta +2)\ge 2m,$$

and Corollary 4.2 applies. □

Corollary 4.4. 

If

$$\eta \ge ⌈{\displaystyle \frac{\sqrt{8m+1}-3}{2}}⌉,$$

(4.5)

then S satisfies Wilf’s conjecture.

Proof. 

The inequality (4.4) is equivalent to

$${\eta }^2+3\eta +2-2m\ge 0,$$

whose positive root is $(\sqrt{8m+1}-3)/2$. Since $\eta$ is an integer, (4.5) is exactly the least integral threshold implying (4.4). The conclusion follows from Corollary 4.3. □

We next record the stronger divisible-conductor refinement.

Corollary 4.5. 

Assume $m\mid c$. If

$$e(3\eta +4)\ge 4m,$$

(4.6)

then S satisfies Wilf’s conjecture.

Proof. 

If $q\le 3$, then again $c\le 3m$, so there is nothing to prove. Assume therefore that $q\ge 4$ and put $A:=e(\eta +1)-m$. From (4.6) we obtain

$$A\ge e(\eta +1)-{\displaystyle \frac{e(3\eta +4)}{4}}={\displaystyle \frac{e\eta }{4}}\ge 0.$$

Since $m\mid c$, we have $\rho =0$ and ${\eta }_{\rho }=\eta$, so (4.2) yields

$$W\left(S\right)\ge qA-e\eta .$$

Using $q\ge 4$ and $A\ge 0$ we conclude

$$W\left(S\right)\ge 4A-e\eta =e(3\eta +4)-4m\ge 0.$$

Thus S satisfies Wilf’s conjecture. □

Corollary 4.6. 

Assume $m\mid c$. If

$$3{\eta }^2+7\eta +4\ge 4m,$$

(4.7)

then S satisfies Wilf’s conjecture.

Proof. 

By Lemma 2.3, $e\ge \eta +1$. Hence

$$e(3\eta +4)\ge (\eta +1)(3\eta +4)=3{\eta }^2+7\eta +4\ge 4m,$$

and Corollary 4.5 applies. □

Corollary 4.7. 

Assume $m\mid c$. If

$$\eta \ge ⌈{\displaystyle \frac{\sqrt{48m+1}-7}{6}}⌉,$$

(4.8)

then S satisfies Wilf’s conjecture.

Proof. 

The inequality (4.7) is equivalent to

$$3{\eta }^2+7\eta +4-4m\ge 0,$$

whose positive root is $(\sqrt{48m+1}-7)/6$. The claim follows from Corollary 4.6. □

Remark 4.8 

(Comparison with the 2025 threshold). The 2025 preprint [7] proves that $\eta \ge \sqrt{3m}$ implies Wilf. By Corollary 4.4, our unconditional threshold is

$$T\left(m\right):=⌈{\displaystyle \frac{\sqrt{8m+1}-3}{2}}⌉.$$

Since $\lceil x\rceil \le x+1$ for every real number x, we have

$$T\left(m\right)\le {\displaystyle \frac{\sqrt{8m+1}-1}{2}}.$$

The latter quantity is strictly smaller than $\sqrt{3m}$, because

$$\sqrt{8m+1}<2\sqrt{3m}+1$$

for every $m\ge 1$. Hence

$$T\left(m\right)<\sqrt{3m}(m\ge 1).$$

Thus the present criterion is strictly stronger at the level of the first-layer population. In the divisible-conductor case the threshold becomes even smaller, namely (4.8), which is asymptotic to $\sqrt{4m/3}$.

5. A Multi-Layer Extension of the Staircase Method

In this section we show that the argument of Section 3 extends naturally from the first Kunz layer to arbitrary cumulative Kunz layers. This yields a hierarchy of Wilf-type criteria that strictly contains Theorem 4.1 and its corollaries as the special case $t=1$.

Definition 5.1. 

For each integer $t\ge 0$, define

$${\Theta }_t={\Theta }_t\left(S\right):=\left|\{i\in \{1,\dots ,m-1\}:x_i\le t\}\right|,$$

and

$${\Theta }_{t,\rho }={\Theta }_{t,\rho }\left(S\right):=\left|\{i\in \{1,\dots ,m-\rho -1\}:x_i\le t\}\right|.$$

Thus

$${\Theta }_0={\Theta }_{0,\rho }=0,{\Theta }_1=\eta ,{\Theta }_{1,\rho }={\eta }_{\rho }.$$

The next statement rewrites Proposition 2.4 in terms of the cumulative Kunz layers.

Proposition 5.2 

(Exact cumulative-layer formula). For every numerical semigroup S,

$$\left|L\right|=q+\sum _{t=1}^{q-2}{\Theta }_t+{\Theta }_{q-1,\rho }.$$

(5.1)

Proof. 

Starting from Proposition 2.4, we have

$$\left|L\right|=q+\sum _{i=1}^{m-\rho -1}max\{0,q-x_i\}+\sum _{i=m-\rho }^{m-1}max\{0,q-1-x_i\}.$$

For integers $x_i\ge 1$,

$$max\{0,q-x_i\}=\sum _{t=1}^{q-1}{\mathbf{1}}_{x_i\le t},max\{0,q-1-x_i\}=\sum _{t=1}^{q-2}{\mathbf{1}}_{x_i\le t},$$

where ${\mathbf{1}}_{(·)}$ denotes the indicator function. Therefore

$$\begin{array}{cc}\hfill \left|L\right|& =q+\sum _{i=1}^{m-\rho -1}\sum _{t=1}^{q-1}{\mathbf{1}}_{x_i\le t}+\sum _{i=m-\rho }^{m-1}\sum _{t=1}^{q-2}{\mathbf{1}}_{x_i\le t}\hfill \\ \hfill & =q+\sum _{t=1}^{q-2}\sum _{i=1}^{m-1}{\mathbf{1}}_{x_i\le t}+\sum _{i=1}^{m-\rho -1}{\mathbf{1}}_{x_i\le q-1}\hfill \\ \hfill & =q+\sum _{t=1}^{q-2}{\Theta }_t+{\Theta }_{q-1,\rho },\hfill \end{array}$$

which proves (5.1). □

The monotonicity of the cumulative layers immediately yields a whole family of staircase lower bounds.

Theorem 5.3 

(Multi-layer staircase bound). For every integer t with $1\le t\le q-1$,

$$\left|L\right|\ge q+\sum _{s=1}^{t-1}{\Theta }_s+(q-1-t){\Theta }_t+{\Theta }_{t,\rho }.$$

(5.2)

Proof. 

By Proposition 5.2,

$$\left|L\right|=q+\sum _{s=1}^{q-2}{\Theta }_s+{\Theta }_{q-1,\rho }.$$

Since ${\Theta }_s$ is nondecreasing in s, we have

$${\Theta }_s\ge {\Theta }_t(s=t,t+1,\dots ,q-2),$$

and similarly

$${\Theta }_{q-1,\rho }\ge {\Theta }_{t,\rho }.$$

Hence

$$\begin{array}{cc}\hfill \left|L\right|& =q+\sum _{s=1}^{t-1}{\Theta }_s+\sum _{s=t}^{q-2}{\Theta }_s+{\Theta }_{q-1,\rho }\hfill \\ \hfill & \ge q+\sum _{s=1}^{t-1}{\Theta }_s+(q-1-t){\Theta }_t+{\Theta }_{t,\rho },\hfill \end{array}$$

as claimed. □

Remark 5.4. 

For $t=1$, Theorem 5.3 becomes

$$\left|L\right|\ge q+(q-2){\Theta }_1+{\Theta }_{1,\rho }=q+(q-2)\eta +{\eta }_{\rho },$$

which is exactly Proposition 3.1. Thus the first-layer estimate is the initial case of an infinite hierarchy.

Multiplying by e and subtracting $c=qm-\rho$, we obtain the corresponding Wilf lower bound.

Corollary 5.5 

(Multi-layer Wilf bound). For every integer t with $1\le t\le q-1$,

$$W\left(S\right)\ge q\left(e({\Theta }_t+1)-m\right)+e\left(\sum _{s=1}^{t-1}{\Theta }_s-(t+1){\Theta }_t+{\Theta }_{t,\rho }\right)+\rho .$$

(5.3)

Proof. 

By Theorem 5.3,

$$W\left(S\right)=e\left|L\right|-c\ge e\left(q+\sum _{s=1}^{t-1}{\Theta }_s+(q-1-t){\Theta }_t+{\Theta }_{t,\rho }\right)-(qm-\rho ).$$

Rearranging terms gives (5.3). □

We next record a convenient sufficient criterion obtained by keeping only the dominant terms.

Theorem 5.6 

(A $(Q,t)$-criterion). Let Q and t be integers such that

$$4\le Q\le q,1\le t\le Q-2.$$

If

$$e\left((Q-t-1){\Theta }_t+Q\right)\ge Qm,$$

(5.4)

then S satisfies Wilf’s conjecture.

Proof. 

Set

$$A_t:=e({\Theta }_t+1)-m.$$

From (5.4) we obtain

$$Q{A}_t=Qe({\Theta }_t+1)-Qm\ge (t+1)e{\Theta }_t,$$

that is,

$$A_t\ge {\displaystyle \frac{t+1}{Q}}e{\Theta }_t\ge 0.$$

By Corollary 5.5,

$$W\left(S\right)\ge q{A}_t+e\left(\sum _{s=1}^{t-1}{\Theta }_s-(t+1){\Theta }_t+{\Theta }_{t,\rho }\right)+\rho .$$

Since all omitted terms are nonnegative, we get

$$W\left(S\right)\ge q{A}_t-(t+1)e{\Theta }_t\ge Q{A}_t-(t+1)e{\Theta }_t\ge 0.$$

Thus S satisfies Wilf’s conjecture. □

Remark 5.7. 

Taking $t=1$ and $Q=4$ in Theorem 5.6 gives

$$e(\eta +2)\ge 2m,$$

which is precisely Corollary 4.2. Thus the mixed first-layer criterion is also just the initial member of the $(Q,t)$-family.

In the divisible-conductor case the criterion becomes cleaner.

Theorem 5.8 

(Divisible-conductor $(Q,t)$-criterion). Assume $m\mid c$. Let Q and t be integers such that

$$4\le Q\le q,1\le t\le Q-1.$$

If

$$e\left((Q-t){\Theta }_t+Q\right)\ge Qm,$$

(5.5)

then S satisfies Wilf’s conjecture.

Proof. 

When $m\mid c$, we have $\rho =0$ and ${\Theta }_{t,\rho }={\Theta }_t$. Hence Corollary 5.5 becomes

$$W\left(S\right)\ge q\left(e({\Theta }_t+1)-m\right)+e\left(\sum _{s=1}^{t-1}{\Theta }_s-t{\Theta }_t\right).$$

Let

$$A_t:=e({\Theta }_t+1)-m.$$

From (5.5) we obtain

$$Q{A}_t=Qe({\Theta }_t+1)-Qm\ge te{\Theta }_t,$$

so $A_t\ge {\displaystyle \frac{t}{Q}}e{\Theta }_t\ge 0$. Therefore

$$W\left(S\right)\ge q{A}_t-te{\Theta }_t\ge Q{A}_t-te{\Theta }_t\ge 0.$$

Thus S satisfies Wilf’s conjecture. □

Remark 5.9. 

Taking $t=1$ and $Q=4$ in Theorem 5.8 yields

$$e(3\eta +4)\ge 4m,$$

which is exactly Corollary 4.5.

6. Interval-Generated Semigroups Beyond the Standard Large Regimes

For integers $m\ge 2$ and $1\le r\le m-1$, let

$$I_{m,r}:=\langle m,m+1,\dots ,m+r\rangle .$$

Interval-generated numerical semigroups were studied in [14]. The family $I_{m,r}$ is also a special case of a generalized arithmetic sequence, so Wilf’s conjecture for $I_{m,r}$ is already known by Sammartano’s theorem [18]. The point of the present section is twofold. First, we show that the first-layer criteria proved above already detect an infinite interval-generated family outside the four broad parameter regimes

$$c\le 3m,e\ge m/3,m\mid c\mathrm{with}e\ge m/4,\eta \ge \sqrt{3m}.$$

Second, using the multi-layer machinery of Section 5, we show that the second cumulative layer yields a strictly stronger interval-generated family.

The next proposition collects the basic invariants of this family.

Proposition 6.1. 

Let $m\ge 2$ and $1\le r\le m-1$, and define

$$q:=⌈{\displaystyle \frac{m-1}{r}}⌉.$$

Then the numerical semigroup $I_{m,r}$ satisfies:

(i)

$\eta \left(I_{m,r}\right)=r$ and $e\left(I_{m,r}\right)=r+1$;

(ii)

$c\left(I_{m,r}\right)=qm$;

(iii)

the intervals

$$[hm,h(m+r\left)\right](1\le h\le q-1)$$

are pairwise disjoint, and

$$I_{m,r}=\left\{0\right\}\cup \bigcup _{h=1}^{q-1}[hm,h(m+r)]\cup [qm,\infty );$$

(6.1)

(iv)

therefore

$$\left|L\left(I_{m,r}\right)\right|=1+\sum _{h=1}^{q-1}(hr+1)=q+{\displaystyle \frac{rq(q-1)}{2}},$$

(6.2)

so that

$$W\left(I_{m,r}\right)=(r+1)\left(q+{\displaystyle \frac{rq(q-1)}{2}}\right)-qm.$$

(6.3)

Proof. 

Every generator of $I_{m,r}$ lies in $[m,2m)$, and no element of $[m,m+r]$ can be written as a sum of two positive semigroup elements, since any such sum is at least $2m>m+r$. Thus the displayed generating set is minimal. Hence $e\left(I_{m,r}\right)=r+1$, and since

$$I_{m,r}\cap (m,2m)=\{m+1,m+2,\dots ,m+r\},$$

we get $\eta \left(I_{m,r}\right)=r$. This proves (i).

For each $h\ge 0$, the set of sums of exactly h generators is the full interval

$$[hm,h(m+r\left)\right].$$

Indeed, every such sum is of the form $hm+t$ with $0\le t\le hr$. Conversely, if $t\in \{0,\dots ,hr\}$, write $t=ur+v$ with $0\le v<r$. Since $t\le hr$, one has $u\le h$, and if $v>0$ then in fact $u\le h-1$. Hence t can be realized as a sum of u copies of r, one copy of v when $v>0$, and the remaining copies equal to 0. Therefore

$$I_{m,r}=\bigcup _{h\ge 0}[hm,h(m+r)].$$

Since $qr\ge m-1$, for every $h\ge q$ we have

$$h(m+r)+1\ge hm+(m-1)+1=(h+1)m.$$

Thus consecutive intervals $[hm,h(m+r\left)\right]$ and $\left[\right(h+1)m,(h+1\left)\right(m+r\left)\right]$ meet or overlap for all $h\ge q$, and consequently

$$\bigcup _{h\ge q}[hm,h(m+r)]=[qm,\infty ).$$

On the other hand, $(q-1)r<m-1$, so

$$(q-1)(m+r)<qm-1.$$

Since every sum of at least q generators is at least $qm$, the integer $qm-1$ does not belong to $I_{m,r}$. Hence $c\left(I_{m,r}\right)=qm$, proving (ii).

For $1\le h\le q-2$ we have $hr\le (q-2)r<m-1$, whence

$$h(m+r)<(h+1)m.$$

Thus the intervals $[hm,h(m+r\left)\right]$ are pairwise disjoint for $1\le h\le q-1$, and (6.1) follows by combining this with the previous paragraph. Their lengths are $hr+1$, so

$$\left|L\left(I_{m,r}\right)\right|=1+\sum _{h=1}^{q-1}(hr+1)=q+{\displaystyle \frac{rq(q-1)}{2}},$$

which is (6.2). Multiplying by $e\left(I_{m,r}\right)=r+1$ and subtracting $c\left(I_{m,r}\right)=qm$ yields (6.3). This proves (iii) and (iv). □

We first record the family detected already by the pure first-layer threshold.

Theorem 6.2. 

For each integer $m\ge 40$, set

$$r_m:=⌈{\displaystyle \frac{\sqrt{8m+1}-3}{2}}⌉,S_m:=I_{m,r_m}=\langle m,m+1,\dots ,m+r_m\rangle .$$

Then:

(i)

$S_m$ satisfies Wilf’s conjecture;

(ii)

$\eta \left(S_m\right)=r_m<\sqrt{3m}$;

(iii)

$e\left(S_m\right)=r_m+1<m/4<m/3$;

(iv)

$c\left(S_m\right)=q_{m}m$ with

$$q_m=⌈{\displaystyle \frac{m-1}{r_m}}⌉\ge 4;$$

(v)

consequently, $S_m$ is not covered by any of the broad regimes

$$c\le 3m,e\ge m/3,m\mid c\mathrm{and}e\ge m/4,\eta \ge \sqrt{3m}.$$

Proof. 

By construction,

$$r_m=⌈{\displaystyle \frac{\sqrt{8m+1}-3}{2}}⌉,$$

so Corollary 4.4 yields (i).

By Proposition 6.1, $\eta \left(S_m\right)=r_m$ and $e\left(S_m\right)=r_m+1$. Since $\lceil x\rceil \le x+1$ for every real number x, we have

$$r_m\le {\displaystyle \frac{\sqrt{8m+1}-1}{2}}<\sqrt{3m},$$

which proves (ii).

Using again $\lceil x\rceil \le x+1$, we obtain

$$r_m+1\le {\displaystyle \frac{\sqrt{8m+1}+1}{2}}<\sqrt{2m}+1.$$

For $m\ge 40$ one has $\sqrt{2m}+1<m/4$, so

$$e\left(S_m\right)=r_m+1<m/4<m/3.$$

This proves (iii).

By Proposition 6.1,

$$c\left(S_m\right)=q_{m}m,q_m=⌈{\displaystyle \frac{m-1}{r_m}}⌉.$$

Moreover, (iii) implies

$$r_m<{\displaystyle \frac{m}{4}}-1<{\displaystyle \frac{m-1}{3}},$$

so $q_m\ge 4$. This proves (iv).

Finally, (ii)–(iv) show that $S_m$ lies in none of the four displayed parameter regimes. This proves (v). □

We now revisit the same family from the cumulative-layer viewpoint.

Proposition 6.3. 

Let $I_{m,r}=\langle m,m+1,\dots ,m+r\rangle$ with $1\le r\le m-1$. Then for every $1\le i\le m-1$,

$$x_i=⌈{\displaystyle \frac{i}{r}}⌉.$$

Consequently, for every integer $t\ge 0$,

$${\Theta }_t\left(I_{m,r}\right)=min\{tr,m-1\}.$$

(6.4)

Proof. 

Fix $1\le i\le m-1$. By Proposition 6.1, the elements of $I_{m,r}$ are exactly the integers belonging to the intervals

$$[hm,h(m+r\left)\right](h\ge 0).$$

Hence $i+hm\in I_{m,r}$ if and only if $i\le hr$, that is, if and only if $h\ge i/r$. It follows that the least such h is

$$x_i=⌈{\displaystyle \frac{i}{r}}⌉.$$

Therefore $x_i\le t$ if and only if $i\le tr$, and so

$${\Theta }_t\left(I_{m,r}\right)=\left|\{i\in \{1,\dots ,m-1\}:i\le tr\}\right|=min\{tr,m-1\}.$$

This proves (6.4). □

The second cumulative layer already yields a stronger family than the first-layer family from Theorem 6.2.

Theorem 6.4 

(A stronger interval-generated family). For each integer $m\ge 25$, set

$$r_m:=\lceil \sqrt{m}\rceil -1,T_m:=I_{m,r_m}=\langle m,m+1,\dots ,m+r_m\rangle .$$

Then:

(i)

$T_m$ satisfies Wilf’s conjecture;

(ii)

$\eta \left(T_m\right)=r_m<\sqrt{3m}$;

(iiii)

$e\left(T_m\right)=r_m+1=\lceil \sqrt{m}\rceil <m/4<m/3$;

(iv)

$c\left(T_m\right)=q_{m}m$ with

$$q_m=⌈{\displaystyle \frac{m-1}{r_m}}⌉\ge 4;$$

(v)

consequently, $T_m$ is not covered by any of the four broad regimes

$$c\le 3m,e\ge m/3,m\mid c\mathrm{and}e\ge m/4,\eta \ge \sqrt{3m}.$$

Proof. 

By Proposition 6.1,

$$c\left(T_m\right)=q_{m}m,q_m=⌈{\displaystyle \frac{m-1}{r_m}}⌉.$$

Since $r_m=\lceil \sqrt{m}\rceil -1\le \sqrt{m}$ and $m\ge 25$, we get

$${\displaystyle \frac{m-1}{r_m}}\ge {\displaystyle \frac{m-1}{\sqrt{m}}}>4,$$

hence $q_m\ge 4$.

Also,

$$e\left(T_m\right)=r_m+1=\lceil \sqrt{m}\rceil ,\eta \left(T_m\right)=r_m.$$

Since $2r_m<m$ for $m\ge 25$, Proposition 6.3 gives

$${\Theta }_2\left(T_m\right)=2r_m.$$

Therefore

$$e\left(T_m\right)\left({\Theta }_2\left(T_m\right)+2\right)=\lceil \sqrt{m}\rceil (2r_m+2)=2{\lceil \sqrt{m}\rceil }^2\ge 2m.$$

Equivalently,

$$e\left(T_m\right)\left(2{\Theta }_2\left(T_m\right)+4\right)\ge 4m.$$

Since $m\mid c\left(T_m\right)$ and $q_m\ge 4$, Theorem 5.8 with $Q=4$ and $t=2$ applies, proving (i).

Since $\eta \left(T_m\right)=r_m=\lceil \sqrt{m}\rceil -1$, we have

$$\eta \left(T_m\right)<\sqrt{m}<\sqrt{3m},$$

which proves (ii).

Moreover,

$$e\left(T_m\right)=\lceil \sqrt{m}\rceil <m/4<m/3(m\ge 25),$$

so (iii) follows.

Statement (iv) was already proved at the beginning of the argument. Finally, (ii)–(iv) show that $T_m$ is not covered by any of the four displayed parameter regimes. This proves (v). □

Remark 6.5. 

Theorem 6.4 improves Theorem 6.2 at the level of interval-generated semigroups: the detectable range drops from $r\asymp \sqrt{2m}$ to $r\asymp \sqrt{m}$. This reflects the fact that, in this family, the second cumulative Kunz layer already contains substantially more information than the first one alone.

Remark 6.6. 

Because $c\left(I_{m,r}\right)$ is always a multiple of m by Proposition 6.1(ii), the stronger divisible-conductor threshold from Corollary 4.7 yields the smaller explicit bound

$$\eta \left(I_{m,r}\right)=r\ge ⌈{\displaystyle \frac{\sqrt{48m+1}-7}{6}}⌉⟹I_{m,r}\mathrm{satisfies}\mathrm{Wilf}\text{’}\mathrm{s}\mathrm{conjecture}.$$

This observation is not needed for Theorem 6.2; it merely records that the interval-generated family lies in the divisible-conductor subcase.

Example 6.7. 

Take $m=40$. Then

$${\displaystyle \frac{\sqrt{8·40+1}-3}{2}}={\displaystyle \frac{\sqrt{321}-3}{2}}\approx 7.458,r_{40}=8.$$

Hence

$$S_{40}=\langle 40,41,42,43,44,45,46,47,48\rangle .$$

By Proposition 6.1,

$$\eta \left(S_{40}\right)=8,e\left(S_{40}\right)=9,q_{40}=⌈{\displaystyle \frac{39}{8}}⌉=5,c\left(S_{40}\right)=200.$$

Moreover,

$$\left|L\left(S_{40}\right)\right|=5+{\displaystyle \frac{8·5·4}{2}}=85,W\left(S_{40}\right)=9·85-200=565>0.$$

This example is not covered by $c\le 3m$, nor by $e\ge m/3$, nor by the 2025 divisible-conductor criterion $e\ge m/4$, nor by the 2025 first-layer threshold $\eta \ge \sqrt{3m}$, because

$$5>3,9<{\displaystyle \frac{40}{4}}=10,8<\sqrt{120}\approx 10.95.$$

7. Concluding Remarks

The argument of this paper isolates a simple and robust mechanism: every residue class whose Kunz coordinate equals 1 contributes a long vertical staircase of left elements. This information is invisible if one only keeps track of the total embedding dimension, yet it is strong enough to improve the 2025 first-layer threshold from $\sqrt{3m}$ to $\sqrt{2m}$, and to yield an even sharper $\sqrt{4m/3}$-type threshold when $m\mid c$.

The cumulative-layer extension developed in Section 5 shows that the same philosophy goes further: instead of recording only the residues with $x_i=1$, one can reorganize the exact count of left elements into the cumulative populations ${\Theta }_t=\left|\{i:x_i\le t\}\right|$. This already leads to a genuine improvement in the interval-generated family, where the second cumulative layer lowers the detectable range from $r\asymp \sqrt{2m}$ to $r\asymp \sqrt{m}$.

The exact formulas (2.2) and (5.1) suggest a broader program. The present paper develops the first-layer and cumulative-layer viewpoints, but a still more refined analysis should keep track of the exact low Kunz layers $\{i:x_i=2\}$, $\{i:x_i=3\}$, and of the interactions between different layers, rather than only their cumulative counts. Such refinements should lead to stronger lower bounds for $\left|L\right|$ and $W\left(S\right)$ and, one may hope, to a further enlargement of the known Wilf region.