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Extensions of integral orthoregular sets and icubes

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Pith's one-line read Every Gaussian-integer icube in dimension 4 extends to an equal-length orthogonal basis, and in dimension 3 exactly when its norm is a sum of two squares; over $\mathbb{Z}$, vectors in $\mathbb{Z}^{4k+2}$ need sum-of-two-squares norm…

desk verdict Solid new quaternion-order framework for Gaussian icube extension, but two printed proof steps are wrong and the load-bearing enough-divisors lemma has a false identity in its proof. read the letter →

arxiv 2508.01196 v1 pith:F4FDLFM4 submitted 2025-08-02 math.NT

Márton Erdélyi , Péter Maga , Gergely Zábrádi This is my paper · ORCID
classification math.NT MSC 11E2511R5252C07
keywords integralvectorsorthogonalbasesicubesGaussianintegersquaternionordersfactorizationinalgebrassup-normproblemofautomorphicformsSmithnormalform
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper studies when a set of mutually orthogonal integral vectors of equal length, an 'icube', can be completed to a full orthogonal basis of integral vectors of that same length, over both the ordinary integers $\mathbb{Z}$ and the Gaussian integers $\mathbb{Z}[i]$. Its main results are complete answers in low dimensions: over $\mathbb{Z}[i]$, every icube in dimension 3 whose common norm is a sum of two rational squares extends to a 3-icube (Theorem 7), and every icube in dimension 4 extends unconditionally (Theorem 8); in particular, every single vector in $\mathbb{Z}[i]^4$ extends (Theorem 4). Over $\mathbb{Z}$ it adds a necessary condition, a vector in $\mathbb{Z}^{4k+2}$ that lies in an $n$-icube must have norm a sum of two squares (Theorem 1), which combines with the Gaussian results to give an if-and-only-if characterization in dimension 6 (Corollary 1.6). The method is a dictionary between such extension problems and factorizations in orders of the Hamiltonian quaternions, and the same dictionary yields a counting lower bound demonstrating that the standard amplification method cannot succeed for the unitary groups $\mathrm{SU}_{2,1}$, $\mathrm{SU}_{3,2}$, and $\mathrm{SU}_{4,3}$.

What carries the argument

The engine is a bijection (Proposition 3.3) between $Q$-orthoregular bases of a binary hermitian form $Q$ and factorizations in the quaternion order $S = \{r + s\sqrt{\varepsilon}\, j \mid r, s \in R\} \subset \mathbb{H}$, where $K$ is $\mathbb{Q}$ or $\mathbb{Q}(i)$, $R$ is its ring of integers, and $\mathbb{H}$ is the algebra of Hamiltonian quaternions. For a form with Gram matrix entries $\alpha, \beta, \gamma$ and discriminant $\mu = \Delta\varepsilon$, a $Q$-orthoregular basis of norm $\lambda = \nu\Delta$ corresponds exactly to a factorization $uv = (\lambda/\Delta)(\beta + \delta\sqrt{\varepsilon}\, j)$ with $|u|^2 = \alpha\lambda/\Delta$, so the question 'can this icube be extended?' becomes 'does this quaternion element have a divisor of prescribed norm?'. The key arithmetic input is Lemma 3.6: for $R = \mathbb{Z}$ and $R = \mathbb{Z}[i]$ with $\varepsilon = 1$, the orders $\mathbb{Z}[j]$ and $A = \{r + sj \mid r, s \in \mathbb{Z}[i]\}$ have 'enough divisors', meaning that whenever $|y|^2$ divides $|t|^2$, the element $t$ has a left divisor $u$ with $|u|^2 = |y|^2$. Around this sits the lattice calculus of Section 4: the module $\Lambda$ of vectors orthogonal to a given $k$-icube has discriminant $\mathrm{disc}(Q) = \lambda^k/|d_k(A_0)|^2$ (Proposition 4.1), the cross-product identity of Corollary 4.3, and the Smith normal form pairing $\alpha_j\alpha_{n+1-j} = \lambda$ of Lemma 1.10, which together reduce extension in higher dimensions to the binary hermitian case. The 'enough divisors' property is delicate: Example 3.7 shows it fails for other choices of $\varepsilon$, which is why the theorems apply specifically to the two orders $\mathbb{Z}[j]$ and $A$.

What would settle it

A finite computer search in the order $A = \{r + sj : r, s \in \mathbb{Z}[i]\}$ for a rational prime $p$ and an element $t$ with $p \mid |t|^2$ but no left divisor of $t$ of norm $p$ would settle the load-bearing Lemma 3.6, since the proof splits into cases according to whether $|t|^2 < p^2$ and claims every such configuration factors. For the sup-norm application, one can compute the density of primitive vectors $a_1$ with $|a_1|^2 = |\ell_1\ell_2|^2$, with $a_2$ chosen as in display (16), for which all prime divisors of $d_2(A_0)$ have norm $4k+1$ and are pairwise non-conjugate; if that density fails to approach 1, the lower bound $\#S_4(\ell_1,\ell_2) \gg L^6$ collapses.

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Extended reading notes

Core claim

The paper's central claim is that, over the Gaussian integers, extension of icubes is governed only by the norm in dimension 3 and is unconditional in dimension 4. Theorem 7 states that if $\lambda$ is a sum of two rational squares, then every $k$-icube ($1 \le k \le 3$) in $\mathbb{Z}[i]^3$ of norm $\lambda$ extends to a 3-icube, and Theorem 8 states that every $k$-icube ($1 \le k \le 4$) in $\mathbb{Z}[i]^4$ extends to a 4-icube; these are the $\mathbb{Z}[i]$ analogues of the known extension theorems over $\mathbb{Z}$ in dimensions 3 and 4 ([GKMS12], [KK12]). Over $\mathbb{Z}$, Theorem 1 gives the necessary condition that a vector in $\mathbb{Z}^{4k+2}$ contained in an $n$-icube has norm a sum of two squares, and Corollary 1.6 turns this into an if-and-only-if for $n = 6$ by embedding $\mathbb{Z}[i]^3$ into $\mathbb{Z}^6$. Finally, Section 1.4 converts the extension theorems into the counting statement $\#S_n(\ell_1,\ell_2) \gg_n L^{2(n-1)}$ for $n = 2, 3, 4$, which the paper presents as a quantitative explanation of why the amplification method cannot, as currently understood, solve the sup-norm problem for $\mathrm{SU}_{n,n-1}$.

Load-bearing premise

The extension theorems all rest on Lemma 3.6, which asserts that the quaternion orders $\mathbb{Z}[j]$ and $\{r + sj : r, s \in \mathbb{Z}[i]\}$ have 'enough divisors', and whose large-norm case depends on the cited principal-right-ideal property of the Hurwitz order; if that property does not yield divisors of exactly the required norm, Theorems 7 and 8 fail, while the sup-norm conclusion separately leans on the unproved assertion that Proposition 1.13's hypothesis holds for almost all choices of the first column $a_1$.

Editorial extensions

If this is right

  • Every single Gaussian vector in $\mathbb{Z}[i]^4$ extends to a 4-icube, and every icube in $\mathbb{Z}[i]^4$ extends to a 4-icube (Theorems 4 and 8): in Gaussian 4-space the equal-length orthogonal basis completion problem has no exceptions.
  • In dimension 3 over $\mathbb{Z}[i]$, the norm condition is exact: an icube of norm $\lambda$ extends to a 3-icube if and only if $\lambda$ is a sum of two rational squares (Theorem 7 together with Proposition 1.3).
  • For the ordinary integers, a vector $v \in \mathbb{Z}^6$ lies in a 6-icube if and only if $|v|^2$ is a sum of two squares (Corollary 1.6).
  • The counting lower bound $\#S_n(\ell_1,\ell_2) \gg_n L^{2(n-1)}$ for $n = 2, 3, 4$ shows that the amplification method as currently understood cannot yield a sup-norm exponent improvement for $\mathrm{SU}_{2,1}$, $\mathrm{SU}_{3,2}$, and $\mathrm{SU}_{4,3}$ (Section 1.4).
  • If Conjecture 1.4 holds, that any icube in $\mathbb{Z}^8$ extends to an 8-icube, then for every $n < 8$ any icube in $\mathbb{Z}^n$ of norm a sum of $(4,n)$ squares extends to an $n$-icube (Proposition 1.5).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The enough-divisors property is the natural place to look for a higher-dimensional version: Conjecture 1.4 would follow from an 8-dimensional analogue of Lemma 3.6, and the proof pattern points toward searching for such a property inside the relevant quaternion or octonion orders rather than in the lattice geometry alone.
  • Corollary 1.6 is the only dimension in which the paper proves the converse of Theorem 1; testing $n = 10$, the next $4k+2$ dimension, would show whether 'norm is a sum of two squares' remains sufficient for single vectors there or whether new obstructions appear, because the proof of the necessity is uniform in $k$.
  • The Section 1.4 assertion that Proposition 1.13's condition $(d_2(A_0), d_2(A_0)) = 1$ holds for almost all choices of the first column $a_1$ is stated without proof; a density computation over the sphere $|a_1|^2 = |\ell_1\ell_2|^2$ would turn the $\#S_4 \gg L^6$ lower bound into a fully verified statement or reveal a hidden dependence on the choice of $\ell_1, \ell_2$.
  • The paper's non-extendable examples in $\mathbb{Z}^{10}$, $\mathbb{Z}^{18}$, and $\mathbb{Z}^{36}$ (Example 1.7) show that beyond dimension 6 the norm condition is not sufficient for collections of vectors, so the complete answers in dimensions 3, 4, and 6 do not propagate naively to higher dimensions.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper studies when an integral vector or a set of vectors over Z or Z[i] can be completed to an orthogonal basis of equal-length vectors (an n-icube). It proves that a vector in Z^n with n=4k+2 that lies in an n-icube must have norm expressible as a sum of two squares (Theorem 1), and that the analogous necessary condition for odd n over Z[i] holds (Proposition 1.3). The main results are Theorems 7 and 8: any icube in Z[i]^3 of norm that is a sum of two rational squares extends to a 3-icube, and any icube in Z[i]^4 extends to a 4-icube; Theorem 4 (every vector in Z[i]^4 extends) is a consequence. The proofs go through a correspondence, developed in Section 3, between Q-orthobalanced bases and factorizations in quaternion orders, and a key lemma asserting that the relevant orders have 'enough divisors' (Lemma 3.6). The final section gives an application to the sup-norm problem for SU(n,n−1), claiming lower bounds #S_n ≫_n L^{2(n−1)} for n=2,3,4.

Significance. The paper introduces a quaternion-order framework (orthobalanced bases, 'enough divisors') that gives a unified proof of previously known integer results and new Gaussian analogues in dimensions 3 and 4. The extension theorems for Gaussian icubes (Theorems 7 and 8) are natural and nontrivial, and the necessary condition for n=4k+2 (Theorem 1) with the resulting Corollary 1.6 is a clean characterization for n=6. The paper also gives a concrete counting obstruction relevant to the sup-norm amplification method. Several components are carefully proved and machine-checkable in principle, including the Smith normal form symmetry (Lemma 1.10), the bijection in Proposition 3.3, and the cross-product identities in Lemmas 4.6 and 4.7. If the load-bearing issues identified below are repaired, this would be a solid contribution to the arithmetic of integral orthogonal sets.

major comments (2)
  1. [Lemma 3.6, Case 2] The displayed identity in Case 2 is algebraically false. For ω ∈ S′\S with ω^2 ≠ 1, one has ((u′+ω)/2)·(2ω) − 1 = u′ω + ω^2 − 1, not u′ω. For example, ω = (1+i+j+k)/2 satisfies ω^2 = (−1+i+j+k)/2. Consequently the constructed element u is not shown to lie in S, and the proof does not establish the required left divisor of t in A of norm p. Since Lemma 3.6 is the hinge of Corollary 3.5 and hence of Proposition 1.9 and Theorems 3, 4, 7, 8 and Corollary 1.6, a corrected argument (or an independent proof of the 'enough divisors' property for A) is required.
  2. [Section 1.4] The assertion that 'for almost all choices of a1, the condition imposed in Proposition 1.13 below holds' is stated without proof. This condition — that d2(A0) is coprime to its conjugate, i.e. all prime divisors of d2(A0) have norm p ≡ 1 mod 4 and are pairwise non-conjugate — is exactly what allows Proposition 1.13 to be applied, and it is needed to pass from the ≫ L^6 choices of a1 to the lower bound #S4(ℓ1,ℓ2) ≫ L^6. The authors should either supply the missing density argument or downgrade the n=4 lower bound to a conditional statement.
minor comments (5)
  1. [Proof of Theorem 1] The line 'νp(αj) ≤ νp(n + 1 − αj)' should read 'νp(αj) ≤ νp(α_{n+1−j})'.
  2. [Proof of Proposition 1.13] The definition of δ appears to have a missing conjugate: δ = d2(A0)α2/α2 as printed equals d2(A0), making the subsequent expression ωλ^2α2/α2 = ωλ^2 independent of α2; presumably δ = d2(A0)α2/\bar{α2} is intended.
  3. [Section 1.4] The word 'irredubible' should be 'irreducible'.
  4. [Example 1.7] The word 'orhogonality' should be 'orthogonality'.
  5. [Corollary 1.2] The notation 'sum of (4,n) squares' should be clarified, e.g., as gcd(4,n).

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the icube extension theorems are derived from an explicit bijection and a directly proved divisor lemma; prior results are context and the sole self-citation is non-load-bearing.

full rationale

The central derivation chain is self-contained. Theorems 3, 4, 7, and 8 are proved through Proposition 3.3, an explicit bijection between Q-orthobalanced bases and explicit factorizations in the quaternion order S, and through Corollary 3.5, which reduces existence of such bases to the "enough divisors" property of S. Lemma 3.6 proves that property directly (using the external principal-right-ideal fact from [CS03, Section 5.1]), and the target extension statements are never used as inputs. Proposition 1.9 and the Smith-normal-form arguments then supply the extensions; this is a genuine reduction, not a renaming of the conclusion. External results such as [Sár61], [GKMS12], [KK12], and [LG20] are used as benchmarks, comparisons, or context rather than as premises feeding the new theorems. The only self-citation is [MZ24], co-authored by two of the present authors, and it appears in Section 1.4 merely as background for the sup-norm application; it does not support any of the icube extension theorems. The unproved assertion in Section 1.4 that "for almost all choices of a1, the condition imposed in Proposition 1.13 below holds" is a correctness gap in the counting application, not a circular step. Similarly, the alleged algebraic identity error in Lemma 3.6, Case 2, is a potential proof defect, not a circularity: even if the lemma needed a corrected proof, the lemma is independent of the theorems it supports. No fitted parameter is renamed as a prediction, and no load-bearing claim reduces to a self-citation chain. Hence the circularity score is low.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters: every constant and condition in the paper is derived, not fitted to data. The framework's own objects (the order S, the enough-divisors property, the bijection F of Proposition 3.3) are explicit constructions with proofs, so they do not belong in invented_entities. The external load is concentrated in Lemma 3.6, which invokes the principal-right-ideal structure of the Hurwitz order from [CS03, Section 5.1], and in Section 1.4, which imports the amplification-method reduction and asserts an unproved density claim for the n = 4 counting.

assumptions (5)
  • standard math Unique factorization holds in Z, Z[i], and in the rings of integers of Q and Q(i) (class number one).
    Used in Lemma 2.1 and Proposition 3.4(3) to compare prime valuations; standard for these rings.
  • domain assumption The Hurwitz order S' = S + (1+i+j+k)/2 admits a non-commutative Euclidean division algorithm, so every nonzero right ideal is principal.
    Invoked in Lemma 3.6, Case 2, to prove the enough-divisors property; the paper cites [CS03, Section 5.1] rather than proving it.
  • domain assumption The sup-norm problem for the groups SU(n, n-1) over Q reduces, via the amplification method, to proving the counting bound #S_n(l1,l2) << L^{2(n-1)-eta}.
    Imported in Section 1.4 after some simplifications, citing [IS95], [BM15], [MZ24], and [Mar]; the paper's obstruction conclusion inherits this reduction.
  • domain assumption For almost all primitive a1 with |a1|^2 = |l1*l2|^2, the pair (a1, a2) from (16) satisfies the coprimality hypothesis of Proposition 1.13.
    Section 1.4 asserts this without proof or reference; the #S_4 >> L^6 lower bound depends on it.
  • standard math Classical four-square representation asymptotics: a large odd number of size L^2 is a sum of four squares in more than L^2 ways, with asymptotically 100 percent of representations satisfying the required coprimality condition.
    Used in Section 1.4 for the n = 2 count; standard (Jacobi four-square theorem) but uncited.

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Pith. "Pith review of Extensions of integral orthoregular sets and icubes." pith.science (2026). https://pith.science/paper/F4FDLFM4

@misc{pith2026250801196,
  author       = {Pith},
  title        = {Pith review of: Extensions of integral orthoregular sets and icubes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F4FDLFM4}},
  note         = {Machine review of arXiv:2508.01196}
}
abstract

In this paper, we study the question when a (rational or Gaussian) integral vector can be extended to an integral orthogonal basis consisting of vectors of equal length. We also study when a set of integral vectors has such an extension. Some necessary conditions are given which are proven to be sufficient in dimensions $3$ and $4$.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

3 extracted references · 3 canonical work pages

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