The i-th gap is at most 2i − 1, and the tail δ ≥ 2

For a submonoid S = ⟨A⟩ ⊆ ℕ and a gap g ∉ S, the map s ↦ g − s injects the elements of S in [0, g] into the gaps in [0, g]. Hence if g is the i-th gap (i gaps in [0, g]), then g + 1 ≤ 2i, i.e. g_i ≤ 2i − 1; in particular every gap is at most 2δ − 1, δ the genus.

The second part is Corollary cor:delta2 of the paper, from the generating rule alone: if S_T = ⟨e, {j < e | M_j ≠ 0}⟩ with M_1 = 0 (the spectrum is centred) and e ≥ 4, then δ ≥ 2 ↔ M_2 M_3 = 0. The moments are abstracted to a predicate V j ("M_j vanishes"); the generating rule itself (Part I of the paper) enters as the definition of genSet.

Main results

• P3b.two_mul_card_gaps_ge: g ∉ ⟨A⟩ → g + 1 ≤ 2 · #{x ≤ g | x ∉ ⟨A⟩}.
• P3b.gap_le_two_mul_index_sub_one: g_i ≤ 2i − 1 with i = #{x ≤ g | x ∉ ⟨A⟩}.
• P3b.gap_le_two_mul_genus_sub_one: every gap is ≤ 2δ − 1 when the gaps are finite.
• P3b.two_le_genus_iff: cor:delta2, 2 ≤ δ ↔ V 2 ∨ V 3.

References

• C. Marín, The order generated by a column of the SU(n)_k S-matrix is locally a numerical semigroup ring (Part II), Zenodo (2026), doi:10.5281/zenodo.22864267, §6 (the bound on g_i) and Corollary cor:delta2.

The bound g_i ≤ 2i − 1

theorem P3b.card_mem_le_card_gaps (A : Set ℕ) {g : ℕ} (hg : g ∉ AddSubmonoid.closure A) : {x ∈ Finset.range (g + 1) | x ∈ AddSubmonoid.closure A}.card ≤ {x ∈ Finset.range (g + 1) | x ∉ AddSubmonoid.closure A}.card

s ↦ g − s injects the semigroup elements of [0, g] into the gaps of [0, g].

theorem P3b.two_mul_card_gaps_ge (A : Set ℕ) {g : ℕ} (hg : g ∉ AddSubmonoid.closure A) : g + 1 ≤ 2 * {x ∈ Finset.range (g + 1) | x ∉ AddSubmonoid.closure A}.card

If g is a gap, then g + 1 ≤ 2 · #{x ≤ g | x is a gap}.

theorem P3b.gap_le_two_mul_index_sub_one (A : Set ℕ) {g i : ℕ} (hg : g ∉ AddSubmonoid.closure A) (hi : {x ∈ Finset.range (g + 1) | x ∉ AddSubmonoid.closure A}.card = i) : g ≤ 2 * i - 1

g_i ≤ 2i − 1: if g is the i-th gap, i.e. there are exactly i gaps in [0, g], then g ≤ 2i − 1.

theorem P3b.gap_le_two_mul_genus_sub_one (A : Set ℕ) (hfin : (P3a.gaps A).Finite) {g : ℕ} (hg : g ∉ AddSubmonoid.closure A) : g ≤ 2 * P3a.genus A - 1

With finitely many gaps, every gap is at most 2δ − 1, δ the genus.

Corollary cor:delta2

def P3b.genSet (e : ℕ) (V : ℕ → Prop) : Set ℕ

The generating rule of Part I at h = 1: S_T is generated by e and the 0 < j < e with M_j ≠ 0; here V j stands for "M_j = 0".

Equations

• P3b.genSet e V = insert e {j : ℕ | 0 < j ∧ j < e ∧ ¬V j}

theorem P3b.closure_sub_of {A : Set ℕ} (P : ℕ → Prop) (h0 : P 0) (hadd : ∀ (x y : ℕ), P x → P y → P (x + y)) (hA : ∀ a ∈ A, P a) {x : ℕ} (hx : x ∈ AddSubmonoid.closure A) : P x

A closure is contained in any additively closed predicate holding at 0 and on A.

theorem P3b.mem_of_two_three {A : Set ℕ} (h2 : 2 ∈ AddSubmonoid.closure A) (h3 : 3 ∈ AddSubmonoid.closure A) (x : ℕ) : x ≠ 1 → x ∈ AddSubmonoid.closure A

If 2, 3 ∈ ⟨A⟩ then every x ≠ 1 lies in ⟨A⟩.

theorem P3b.two_le_genus_iff {e : ℕ} {V : ℕ → Prop} (he : 4 ≤ e) (hV1 : V 1) (hfin : (P3a.gaps (genSet e V)).Finite) : 2 ≤ P3a.genus (genSet e V) ↔ V 2 ∨ V 3

Corollary cor:delta2: for h = 1 (generating rule genSet e V), a centred spectrum (V 1, i.e. M_1 = 0) and e ≥ 4, with finitely many gaps, δ ≥ 2 ↔ M_2 M_3 = 0.