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For unit-sphere, centroid-zero complete frameworks with at least three distinct points, the second-largest stiffness eigenvalue is always exactly n/2.

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load-bearing objection Cleanly settles the eigenvalue half of the Lew et al. conjecture and supplies an explicit infinite family of multiplicity counterexamples; the math is elementary and fully written.

arxiv 2607.05472 v1 pith:RYD77NGZ submitted 2026-07-06 math.CO

The Second Largest Eigenvalue of Stiffness Matrices of Normalized Complete Frameworks

Tingting Wang , Lu Lu This is my paper
classification math.CO MSC 05C5052C2515A18
keywords rigidity matrixstiffness matrixsecond largest eigenvaluecomplete graphnormalized frameworksspectral gap
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper studies the stiffness matrix of a complete framework whose points lie on the unit sphere and sum to the origin. It proves that whenever the ambient dimension is at least 2 and the points take at least three distinct values, the second-largest eigenvalue of this matrix is exactly n/2. That settles the eigenvalue half of an earlier conjecture. At the same time the paper exhibits regular polygons lying in a plane for which the same eigenvalue has multiplicity 2n-4, strictly larger than the multiplicity the conjecture predicted. The value of the eigenvalue is therefore universal under the given normalization, while its multiplicity depends on the geometry of the point set.

Core claim

Under the normalization that every point has unit length and the centroid is at the origin, and provided at least three distinct points appear, the second-largest eigenvalue of the stiffness matrix of the complete framework is always n/2 for every ambient dimension d≥2. The multiplicity of that eigenvalue, however, is not fixed by the same hypotheses: regular n-gons embedded in a two-dimensional subspace realize multiplicity 2n-4.

What carries the argument

A quadratic-form upper bound (Lemma 9) showing that any vector orthogonal to the configuration vector itself satisfies xᵀLx ≤ (n/2)‖x‖^{2}; combined with a prior lower-bound multiplicity statement, this forces n/2 to be precisely the second-largest eigenvalue.

Load-bearing premise

The argument takes as given that n/2 is already known to be an eigenvalue of multiplicity at least n-1; only the matching upper bound is proved here.

What would settle it

Produce any unit-sphere, centroid-zero configuration with at least three distinct points whose stiffness matrix has an eigenvalue strictly larger than n/2 but strictly smaller than n, or compute the spectrum of a regular n-gon and obtain a multiplicity for n/2 other than 2n-4.

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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 4 minor

Summary. The paper studies the stiffness matrix L(K_n,p)=R(K_n,p)R(K_n,p)^T of a complete framework whose points lie on the unit sphere and have centroid at the origin. Theorem 1 asserts that, for d≥2 and whenever the image of p contains at least three distinct points, the second-largest eigenvalue of L is exactly n/2. The argument proceeds by deriving a quadratic-form upper bound (Lemma 9) that, for every vector orthogonal to the known Perron eigenvector (p(1),…,p(n))^T, yields x^T Lx ≤ (n/2)∥x∥^{2}; combined with the variational characterization (Lemma 7) and the already-published fact that n/2 is an eigenvalue of multiplicity at least n-1 (Lemma 2 of Lew et al.), this identifies n/2 as the second-largest eigenvalue. Theorem 2 then exhibits an infinite family of regular n-gons lying in a two-dimensional subspace for which the multiplicity of n/2 is 2n-4 rather than the conjectured n-1, and computes the full spectrum explicitly via a complex-linear representation of the planar component. The results therefore confirm the eigenvalue claim of Conjecture 1 of Lew et al. while showing that the multiplicity claim fails in general.

Significance. The work cleanly separates two parts of a published conjecture: the value of the second-largest eigenvalue is universal under the stated normalization, while its multiplicity is geometry-dependent. The upper-bound argument (Lemmas 8–9) is self-contained, relies only on elementary identities (centering, Cauchy–Schwarz, Frobenius products) and does not assume the target eigenvalue. The explicit spectral computation for regular polygons supplies a concrete infinite family of counter-examples to the multiplicity prediction and is fully rigorous. Together these contributions settle the eigenvalue half of the conjecture for all d≥2 and clarify the geometric sensitivity of the multiplicity, which is of independent interest for the spectral theory of stiffness matrices and higher-dimensional algebraic connectivity.

minor comments (4)
  1. In the statement of Lemma 8 the phrase “vectors that orthogonal with p_i” should be corrected to “vectors orthogonal to p_i”.
  2. Section 3, proof of Lemma 8: the reduction step that produces the centered family y' is clear, but a one-sentence reminder that D is translation-invariant under the particular correction t_i = u-⟨u,p_i⟩p_i would help the reader follow the subsequent vanishing of E(y,t).
  3. Section 4.2: the identification Φ:H o C is isometric, yet the real-linear operator L on C^n is introduced without an explicit remark that the spectrum is independent of the choice of orthonormal basis of H; a short sentence would remove any ambiguity.
  4. References: the arXiv identifier of the present paper appears as 2607.05472; if this is a placeholder, it should be updated before publication.

Circularity Check

0 steps flagged

No circularity: the upper bound x^T L x ≤ (n/2)‖x‖^{2} for x ⊥ p is derived from first-principles quadratic-form estimates; existence of eigenvalue n/2 is an external citation used only for the matching lower bound.

full rationale

The load-bearing argument for Theorem 1 is the variational upper bound of Lemma 9 (proved via centering reduction, Cauchy–Schwarz on the Frobenius tensors B_i, and the elementary identity of Lemma 5), which shows every eigenvalue other than the known largest eigenvalue n is at most n/2. Combined with the external fact (Lemma 2 of Lew et al.) that n/2 is already an eigenvalue, this identifies the second-largest eigenvalue. The upper-bound derivation never assumes the target value and does not rely on any self-citation or fitted parameter. Theorem 2 is an independent, fully explicit eigenspace computation for regular polygons (complex representation of the planar restriction, direct verification of the four invariant subspaces W1–W4). No step reduces by construction to its own input; the only external spectral fact is a published lower bound from distinct authors. This is ordinary mathematical practice, not circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

Pure-mathematics paper. No free parameters are fitted. All background facts are either standard linear-algebra identities or previously published lemmas on the stiffness matrix; no new physical or combinatorial entities are postulated.

axioms (3)
  • domain assumption Largest eigenvalue of L(K_n,p) is n with eigenvector (p(1),…,p(n))^T whenever p is non-constant (Lemma 1 of Jordán–Tanigawa).
    Invoked at the start of the proof of Theorem 1 to identify the Perron eigenvalue that must be excluded.
  • domain assumption Under the unit-sphere centroid-zero hypotheses, n/2 is an eigenvalue of multiplicity at least n-1 (Proposition 4.2 of Lew et al.).
    Used only to guarantee that n/2 actually appears in the spectrum so that the proved upper bound becomes sharp.
  • standard math Cauchy–Schwarz, the variational characterization of eigenvalues of symmetric matrices, and the Frobenius inner product on tensor products.
    Applied repeatedly in Lemmas 6–9 and in the complex-plane calculations of Section 4.

pith-pipeline@v1.1.0-grok45 · 17935 in / 2319 out tokens · 24361 ms · 2026-07-11T12:57:38.402574+00:00 · methodology

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read the original abstract

Let $R(G,p)$ be the normalized rigidity matrix of a framework $(G,p)$ in $\mathbb R^d$, and let \[ L(G,p)=R(G,p)R(G,p)^{T} \] be the associated stiffness matrix. We study the extremal eigenvalues of $L(K_n,p)$ for complete frameworks whose vertices lie on the unit sphere and have centroid at the origin. Our main result shows that, whenever $d\ge2$ and the image of $p$ contains at least three distinct points, the second largest eigenvalue of $L(K_n,p)$ is exactly $n/2$. This settles the eigenvalue part of a conjecture of Lew et al. [Israel J. Math. 256, 2023]. We further construct an infinite family of examples, given by regular polygons embedded in a two-dimensional subspace, for which the eigenvalue $n/2$ has multiplicity $2n-4$. Consequently, the multiplicity predicted in the conjecture is not correct in general. Our results reveal a dichotomy: the value of the second largest eigenvalue is universal, while its multiplicity is sensitive to the geometry of the underlying point configuration.

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Reference graph

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