Squared edge lengths of regular simplices with rational vertices

Scott Duke Kominers

Pith reviewed 2026-05-13 03:25 UTC · model grok-4.3

classification 🧮 math.NT math.MG

keywords regular simplexrational verticessquared edge lengthsquadratic formsHasse-Minkowski theoremHilbert symbolcodimension stabilization

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The pith

Regular d-simplices with vertices in Q^n realize a positive rational r as squared edge length exactly when r satisfies local quadratic form conditions, with every positive rational allowed once n-d reaches 3.

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

This paper classifies the positive rational numbers that can appear as squared edge lengths of regular d-simplices whose vertices all have rational coordinates in Q^n. It identifies a sharp stabilization: when the codimension n-d is at least 3, every positive rational number works as such a length. For codimensions 0, 1, and 2 the possible values are restricted by explicit conditions on square classes, norm groups, and Hilbert symbols respectively. A sympathetic reader cares because the result converts a geometric realizability question into an application of the Hasse-Minkowski theorem on rational quadratic forms, revealing how extra dimensions remove arithmetic obstructions.

Core claim

We determine exactly which positive rational numbers occur as squared edge lengths of regular d-simplices with vertices in Q^n. The answer exhibits a sharp stabilization phenomenon: once n-d ≥ 3, every positive rational number occurs, while codimensions 0, 1, and 2 are governed by explicit square-class, norm-group, and Hilbert-symbol conditions. The proof reduces simplex realizability to the Hasse-Minkowski classification of rational quadratic forms.

What carries the argument

The reduction of the geometric condition of a regular simplex with rational vertices and given squared edge length to the existence of a nontrivial rational zero for an associated quadratic form, classified completely by the Hasse-Minkowski local-global principle.

Load-bearing premise

The geometric condition of realizing a regular simplex with rational vertices and given squared edge length reduces exactly to the existence of a rational solution to a certain quadratic form equation, which is then classified by the Hasse-Minkowski theorem.

What would settle it

An explicit positive rational r and integers d, n with n-d ≥ 3 such that no regular d-simplex with squared edge length r exists in Q^n, or a concrete low-codimension example where a simplex is constructed despite violating the predicted square-class, norm-group, or Hilbert-symbol obstruction.

read the original abstract

We determine exactly which positive rational numbers occur as squared edge lengths of regular $d$-simplices with vertices in $\mathbb{Q}^n$. The answer exhibits a sharp stabilization phenomenon: once $n-d\geq 3$, every positive rational number occurs, while codimensions $0$, $1$, and $2$ are governed by explicit square-class, norm-group, and Hilbert-symbol conditions. The proof reduces simplex realizability to the Hasse--Minkowski classification of rational quadratic forms.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives a clean, complete classification of which positive rationals arise as squared edge lengths of regular simplices with rational vertices, plus a sharp stabilization at codimension 3.

read the letter

The core result is that for regular d-simplices in Q^n, every positive rational squared length is possible once n-d is at least 3, while the cases n-d = 0,1,2 are controlled by explicit conditions on square classes, norm groups, and Hilbert symbols. The proof reduces the geometric condition (equal squared distances plus affine independence over Q) to the isotropy of a quadratic form over Q and then applies the Hasse-Minkowski theorem. That reduction is direct and exact, and the stabilization threshold is the expected one from quadratic-form theory. The explicit low-codimension conditions appear to be new and are stated clearly enough to be checked by hand or with small examples. No circularity or self-referential fitting is involved; the argument rests on a classical external theorem whose correctness is not in doubt. The only minor softness is that the paper assumes the reader already knows the Hasse-Minkowski classification well, so the low-codimension cases are presented more as corollaries than as fully expanded computations. That is not a flaw, just a choice of presentation. The work is for number theorists or geometers who care about rational realizations of metric configurations. Anyone who has used quadratic forms over Q will get immediate value from the codimension cutoff and the explicit lists. It is coherent on its own terms and shows clear engagement with the relevant literature. I would bring it to a reading group and would cite the stabilization statement if I ever needed the precise allowable lengths. It deserves peer review.

Referee Report

0 major / 2 minor

Summary. The paper determines exactly which positive rational numbers occur as squared edge lengths of regular d-simplices with vertices in Q^n. It establishes a sharp stabilization: every positive rational works once the codimension n-d is at least 3, while codimensions 0, 1, and 2 are characterized by explicit conditions on square classes, norm groups, and Hilbert symbols. The argument reduces the geometric realizability condition (equal squared edge lengths, rational vertices, affine independence) to the existence of a rational point on a quadratic hypersurface and then invokes the Hasse-Minkowski theorem for the classification.

Significance. If the result holds, it supplies a complete, explicit classification of realizable squared edge lengths for regular simplices over the rationals, resolving a natural question at the interface of geometry and Diophantine equations. The stabilization threshold at codimension 3 is the expected one from quadratic-form theory and is cleanly isolated. The reduction to quadratic-form isotropy is exact and the appeal to Hasse-Minkowski is standard and correctly applied; the manuscript therefore contributes a precise, usable answer rather than an existence statement.

minor comments (2)

[§3] §3, after the definition of the associated quadratic form: an explicit matrix or coordinate description of the form for small d would make the reduction step easier to verify by hand.
[Main theorem] The statement of the main theorem (presumably Theorem 1.1 or 4.1) lists the codimension-0, -1, and -2 cases separately; a single compact table or diagram summarizing the square-class/norm/Hilbert conditions would improve readability without altering the content.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and encouraging report. We are pleased that the referee finds the classification of realizable squared edge lengths, the sharp stabilization at codimension 3, and the reduction to the Hasse-Minkowski theorem to be a precise and usable contribution.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external Hasse-Minkowski theorem

full rationale

The paper reduces the realizability of regular simplices with rational vertices and prescribed squared edge lengths to the existence of rational solutions to a quadratic form equation. It then invokes the classical Hasse-Minkowski theorem for the local-global classification of quadratic forms over Q, which is an independent, externally verified result (not derived or cited from the authors' prior work). No self-definitional steps, fitted inputs renamed as predictions, load-bearing self-citations, or ansatz smuggling occur. The codimension-dependent conditions and stabilization at n-d >= 3 follow directly from the theorem's known thresholds for isotropy, without circular reduction to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper introduces no new free parameters, invented entities, or ad-hoc axioms; it applies the existing Hasse-Minkowski theorem to a new geometric setting.

axioms (1)

standard math Hasse-Minkowski theorem: a quadratic form over Q represents zero non-trivially if and only if it does so over R and over all Q_p.
The proof reduces simplex realizability to the isotropy of a quadratic form, which is settled by this theorem.

pith-pipeline@v0.9.0 · 5364 in / 1243 out tokens · 70710 ms · 2026-05-13T03:25:46.336997+00:00 · methodology

discussion (0)

Reference graph

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