Expanded generators and factorization lengths

Abstract

Expanded generators refute Narayan’s delta-set formula by unit-length normalization.

Definition 1.1 (The expanded generating tuple).

$$\forall a\in \mathbb{N},\forall h\in \mathbb{N},\forall d\in \mathbb{N},\forall k\in \mathbb{N},\forall i\in \mathrm{Fin}\left(k+2\right),\mathrm{generators}\left(a,h,d,k,i\right)=\mathrm{ite}\left(\mathrm{val}\left(i\right)=0,a,a\cdot h+\left(\mathrm{val}\left(i\right)-1\right)\cdot d\right)$$

Formalization. D5/S3/Arith/Additive/NarayanExpandedGeneratorDeltaRefutation.generators (✓ std3).

Citation. Ashwin Narayan (2026). The Delta Set of a Numerical Monoid Generated by a Generalized Arithmetic Sequence. DOI: 10.48550/arXiv.2610.02495. URL: https://arxiv.org/abs/2610.02495v1.

Commentary.

Conjecture 18.3 (p. 22) states: “For Sₕ =⟨a,ah,ah+d,ah+2d,…ah+kd ⟩, Δ(Sₕ) = E(e,d)”. The index i : Fin (k + 2) retains a at zero and ah at one. All later coordinates are ah + (i.val − 1)d; no displayed generator is removed.

Definition 1.2 (Coordinate factorizations).

$$\forall p\in \mathbb{N},\forall g\in \mathrm{Fin}\left(p\right)\to \mathbb{N},\forall n\in \mathbb{N},Z\left(g,n\right)=\{z:\mathrm{Fin}\left(p\right)\to \mathbb{N}\mid \sum _{i:\mathrm{Fin}\left(p\right)}(z\left(i\right)\cdot g\left(i\right))=n\}$$

Formalization. D5/S3/Arith/Additive/NarayanExpandedGeneratorDeltaRefutation.Z (✓ std3).

Citation. Ashwin Narayan (2026). The Delta Set of a Numerical Monoid Generated by a Generalized Arithmetic Sequence. DOI: 10.48550/arXiv.2610.02495. URL: https://arxiv.org/abs/2610.02495v1.

Commentary.

Section 3 (p. 2) states: “Suppose the set of factorizations of n ∈ S is given by Z(n) = { (z₁, z₂, …, zₚ) ∈ ℕᵖ ∣ z₁ n₁ + z₂ n₂ + … + zₚ nₚ = n }”. The p coordinates are indexed from zero by Fin p and take nonnegative integer values, including zero as in the paper’s subsequent inequalities. The sum is over every coordinate.

Definition 1.3 (Length sets).

$$\forall p\in \mathbb{N},\forall g\in \mathrm{Fin}\left(p\right)\to \mathbb{N},\forall n\in \mathbb{N},L\left(g,n\right)=\mathrm{Set}.\mathrm{image}\left(\mathrm{fun}(z:\mathrm{Fin}\left(p\right)\to \mathbb{N})\mapsto \sum _{i:\mathrm{Fin}\left(p\right)}(z\left(i\right)),Z\left(g,n\right)\right)$$

Formalization. D5/S3/Arith/Additive/NarayanExpandedGeneratorDeltaRefutation.L (✓ std3).

Citation. Ashwin Narayan (2026). The Delta Set of a Numerical Monoid Generated by a Generalized Arithmetic Sequence. DOI: 10.48550/arXiv.2610.02495. URL: https://arxiv.org/abs/2610.02495v1.

Commentary.

Section 3 (p. 3) states: “If z = (z₁, z₂, z₃, …, zₚ) is a factorization of n, then the length of z is given by |z| = z₁ + z₂ + z₃ + … + zₚ”. It continues: “The set of lengths of n is defined by L(n) = {|z| | z ∈ Z(n)} ⊂ ℕ”. Set.image applies the coordinate sum to all factorizations of n.

Definition 1.4 (Positive consecutive length differences).

$$\forall p\in \mathbb{N},\forall g\in \mathrm{Fin}\left(p\right)\to \mathbb{N},\mathrm{delta}\left(g\right)=\{q:\mathbb{N}\mid \exists n\in \mathbb{N},\exists lone\in \mathbb{N},\exists ltwo\in \mathbb{N},(lone\in L\left(g,n\right))\wedge ((ltwo\in L\left(g,n\right))\wedge ((lone<ltwo)\wedge ((q=ltwo-lone)\wedge (\forall l\in \mathbb{N},(l\in L\left(g,n\right))\Rightarrow (\neg ((lone<l)\wedge (l<ltwo)))))))\}$$

Formalization. D5/S3/Arith/Additive/NarayanExpandedGeneratorDeltaRefutation.delta (✓ std3).

Citation. Ashwin Narayan (2026). The Delta Set of a Numerical Monoid Generated by a Generalized Arithmetic Sequence. DOI: 10.48550/arXiv.2610.02495. URL: https://arxiv.org/abs/2610.02495v1.

Commentary.

Definition 3.1 (p. 3) states: “We define the delta set as Δ(Sₕ) = { l₂ − l₁ : l₂,l₁ ∈ L(s), L(s) ∩ (l₁,l₂) = ∅ }”. Here s ranges over all represented natural numbers and l₁ < l₂, so differences are positive and the endpoints are distinct. The final quantified clause excludes every intermediate length. Subtraction is natural-number subtraction.

Definition 1.5 (Subtractive Euclidean step).

$$\forall s\in \mathbb{N}\times \mathbb{N},\mathrm{euclidStep}\left(s\right)=\mathrm{ite}\left(\mathrm{Prod}.\mathrm{snd}\left(s\right)<\mathrm{Prod}.\mathrm{fst}\left(s\right),(\mathrm{Prod}.\mathrm{fst}\left(s\right)-\mathrm{Prod}.\mathrm{snd}\left(s\right),\mathrm{Prod}.\mathrm{snd}\left(s\right)),\mathrm{ite}\left(\mathrm{Prod}.\mathrm{fst}\left(s\right)<\mathrm{Prod}.\mathrm{snd}\left(s\right),(\mathrm{Prod}.\mathrm{fst}\left(s\right),\mathrm{Prod}.\mathrm{snd}\left(s\right)-\mathrm{Prod}.\mathrm{fst}\left(s\right)),s\right)\right)$$

Formalization. D5/S3/Arith/Additive/NarayanExpandedGeneratorDeltaRefutation.euclidStep (✓ std3).

Citation. Ashwin Narayan (2026). The Delta Set of a Numerical Monoid Generated by a Generalized Arithmetic Sequence. DOI: 10.48550/arXiv.2610.02495. URL: https://arxiv.org/abs/2610.02495v1.

Commentary.

Definition 3.2 (p. 3) states: “We define the subtractive Euclidean Algorithm in this way: Let x₀ and y₀ = (e,d)” and “Recursively define” the next pair as (xᵢ−yᵢ,yᵢ) if xᵢ>yᵢ, (xᵢ,yᵢ−xᵢ) if yᵢ>xᵢ, and “terminate” if xᵢ=yᵢ. The function ite selects its second argument when its first argument holds and its third otherwise. The equal pair is stationary; Prod.fst and Prod.snd are its coordinates.

Definition 1.6 (Euclidean states).

$$\forall e\in \mathbb{N},\forall d\in \mathbb{N},(\mathrm{euclidState}\left(e,d,0\right)=(e,d))\wedge (\forall i\in \mathbb{N},\mathrm{euclidState}\left(e,d,i+1\right)=\mathrm{euclidStep}\left(\mathrm{euclidState}\left(e,d,i\right)\right))$$

Formalization. D5/S3/Arith/Additive/NarayanExpandedGeneratorDeltaRefutation.euclidState (✓ std3).

Citation. Ashwin Narayan (2026). The Delta Set of a Numerical Monoid Generated by a Generalized Arithmetic Sequence. DOI: 10.48550/arXiv.2610.02495. URL: https://arxiv.org/abs/2610.02495v1.

Commentary.

Definition 3.2 (p. 3) starts with (x₀,y₀)=(e,d) and applies the displayed subtractive step recursively. Retaining the final diagonal pair repeats a state already visited and adds no positive coordinate value.

Definition 1.7 (Values visited by the Euclidean algorithm).

$$\forall e\in \mathbb{N},\forall d\in \mathbb{N},E\left(e,d\right)=\{q:\mathbb{N}\mid (0<q)\wedge (\exists i\in \mathbb{N},(\mathrm{Prod}.\mathrm{fst}\left(\mathrm{euclidState}\left(e,d,i\right)\right)=q)\vee (\mathrm{Prod}.\mathrm{snd}\left(\mathrm{euclidState}\left(e,d,i\right)\right)=q))\}$$

Formalization. D5/S3/Arith/Additive/NarayanExpandedGeneratorDeltaRefutation.E (✓ std3).

Citation. Ashwin Narayan (2026). The Delta Set of a Numerical Monoid Generated by a Generalized Arithmetic Sequence. DOI: 10.48550/arXiv.2610.02495. URL: https://arxiv.org/abs/2610.02495v1.

Commentary.

The abstract (p. 1) states: “Writing e=h−1 and letting E(e,d) denote the set of values arising from the subtractive Euclidean algorithm applied to e and d”. The set contains the positive coordinates of all states, including the initial and terminal states.

Definition 1.8 (Conjecture 18.3).

$$\mathrm{claim}=(\forall a\in \mathbb{N},\forall h\in \mathbb{N},\forall d\in \mathbb{N},\forall k\in \mathbb{N},(0<a)\Rightarrow ((0<h)\Rightarrow ((0<d)\Rightarrow ((0<k)\Rightarrow ((\mathrm{Nat}.\mathrm{Coprime}\left(a,d\right))\Rightarrow ((1<k)\Rightarrow ((k<a)\Rightarrow ((1<h)\Rightarrow (\mathrm{delta}\left(\mathrm{generators}\left(a,h,d,k\right)\right)=E\left(h-1,d\right))))))))))$$

Formalization. D5/S3/Arith/Additive/NarayanExpandedGeneratorDeltaRefutation.claim (✓ std3).

Citation. Ashwin Narayan (2026). The Delta Set of a Numerical Monoid Generated by a Generalized Arithmetic Sequence. DOI: 10.48550/arXiv.2610.02495. URL: https://arxiv.org/abs/2610.02495v1.

Commentary.

Conjecture 18.3 (p. 22) states: “For Sₕ =⟨a,ah,ah+d,ah+2d,…ah+kd ⟩, Δ(Sₕ) = E(e,d)”. Section 3 (p. 2) assumes “gcd(a,d) =1, 1<k<a, h > 1” and sets “e =h−1.” All four parameters are positive natural numbers; Nat.Coprime expresses the stated greatest-common-divisor condition. The equality uses every coordinate of the expanded tuple.

Theorem 1.9 (The expanded-generator equality is false).

$$\neg (\forall a\in \mathbb{N},\forall h\in \mathbb{N},\forall d\in \mathbb{N},\forall k\in \mathbb{N},(0<a)\Rightarrow ((0<h)\Rightarrow ((0<d)\Rightarrow ((0<k)\Rightarrow ((\mathrm{Nat}.\mathrm{Coprime}\left(a,d\right))\Rightarrow ((1<k)\Rightarrow ((k<a)\Rightarrow ((1<h)\Rightarrow (\mathrm{delta}\left(\mathrm{generators}\left(a,h,d,k\right)\right)=E\left(h-1,d\right))))))))))$$

Proof. Machine-checked in Lean as D5/S3/Arith/Additive/NarayanExpandedGeneratorDeltaRefutation.result (✓ std3). ∎

Resolves. Problems/narayan-delta-set (refuted) by D5/S3/Arith/Additive/NarayanExpandedGeneratorDeltaRefutation.result.

Source. Repository-derived.

Acknowledgement. Ashwin Narayan (2026). The Delta Set of a Numerical Monoid Generated by a Generalized Arithmetic Sequence. DOI: 10.48550/arXiv.2610.02495. URL: https://arxiv.org/abs/2610.02495v1.

Commentary.

Take (a,h,d,k)=(3,2,2,2), whose expanded tuple is (3,6,8,10). For every positive a coprime to 2, the tuple (a,2a,2a+2,…,4a−2) has delta set {1}. Replacing 2a by two copies of a, and replacing two other occurrences according to their index sum, preserves value and increases length by one. Strong induction on value minus length reaches a terminal factorization. Coprimality makes that endpoint unique by its residue modulo a; induction on a rewrite trace supplies every intervening length. Thus no larger consecutive gap occurs. Lengths 1 and 2 both occur at value 2a. At a=3 this contradicts the initial Euclidean value 2 ∈ E(1,2). The equality for the tuple omitting ah and the catenary conjectures are separate assertions.

References

• Truth anchor: D5/S3/Arith/Additive/NarayanExpandedGeneratorDeltaRefutation.E
• Truth anchor: D5/S3/Arith/Additive/NarayanExpandedGeneratorDeltaRefutation.L
• Truth anchor: D5/S3/Arith/Additive/NarayanExpandedGeneratorDeltaRefutation.Z
• Truth anchor: D5/S3/Arith/Additive/NarayanExpandedGeneratorDeltaRefutation.claim
• Truth anchor: D5/S3/Arith/Additive/NarayanExpandedGeneratorDeltaRefutation.delta
• Truth anchor: D5/S3/Arith/Additive/NarayanExpandedGeneratorDeltaRefutation.euclidState
• Truth anchor: D5/S3/Arith/Additive/NarayanExpandedGeneratorDeltaRefutation.euclidStep
• Truth anchor: D5/S3/Arith/Additive/NarayanExpandedGeneratorDeltaRefutation.generators
• Truth anchor: D5/S3/Arith/Additive/NarayanExpandedGeneratorDeltaRefutation.result