Open questions and potential solutions

The contraction follows from invariance of the short arc, conservation of the lifted mean, the sine sector bound and the expander’s Laplacian gap. For $K_n$, the first Daido parameter obeys ${\dot{\rho }}_1=(n\mathrm{/}2)({\rho }_1-\overline{{\rho }_1}{\rho }_2)$, so $\mathrm{\mid }{\rho }_1(t)\mathrm{\mid }\le e^{nt}\mathrm{\mid }{\rho }_1(0)\mathrm{\mid }$. Entrance requires $\mathrm{\mid }{\rho }_1(T)\mathrm{\mid }\ge \sqrt{3}\mathrm{/}2$, whereas independent uniform initialization gives $\mathbb{E}\mathrm{\mid }{\rho }_1(0){\mathrm{\mid }}^2=1\mathrm{/}n$. Markov’s inequality yields the stated lower-time estimate. Steps below also prove a sufficient exponential-moment criterion for the requested upper tail.

Direct proof of the partial results: establish the Laplacian gap and short-arc invariance, derive post-entry contraction, and use exact complete-graph phasor dynamics to obtain a logarithmic lower bound. For the unresolved upper bound, work in normalized time $\tau =dt$ and seek a polynomial-in-$n$ exponential moment of the entrance time. Quantitative control of trajectories lingering near nonsynchronous equilibria is a possible research route, not a proved estimate.

1. 1.1 Given: Work in the usual ZFC foundations with the graph, dynamics and initial law in the question. The case $n=1$ has $T=0$. For $n\ge 2$, put $L=dI-A$ and ${\mathrm{\Delta }}_A=A-(d\mathrm{/}n)J$. If $v\perp \mathbf{1}$, then $v^{T}Lv=d\mathrm{\parallel }v{\mathrm{\parallel }}_2^2-v^T{\mathrm{\Delta }}_{A}v\ge (1-a)d\mathrm{\parallel }v{\mathrm{\parallel }}_2^2$. Thus ${\lambda }_2(L)\ge (1-a)d>0$; in particular the graph’s non-loop support is connected. Indeed, a disconnected support would supply a nonzero vector constant on each component, orthogonal to $\mathbf{1}$, and annihilated by $L$. [given, algebra] [dep-entrance-setting, dep-rayleigh-bound]
2. 1.2 The target arc set is compact: it is the image of $S^1\times [0,\gamma {]}^n$ under a common rotation followed by the coordinate offsets. Continuity of the flow therefore makes $T$ measurable: for every finite $s$, the event $\{T\le s\}$ is the projection of a closed subset of the compact product of phase space with $[0,s]$. At any finite $T$, choose real lifts in one interval of length $\gamma$. At a maximal coordinate the velocity is nonpositive, and at a minimal coordinate it is nonnegative, because all differences have magnitude below $\pi$. The finite maximum/minimum derivative rule and continuation imply that the lifted diameter remains at most $\gamma$. Symmetry cancels the terms in ${\sum }_x{\dot{\theta }}_x$, so the lifted mean $\bar{\theta }$ is conserved. [given, step 1.1, algebra] [dep-entrance-setting, def-expander-sync-kuramoto-system, dep-elementary-analysis, step:1]
3. 1.3 Put $u=\theta -\bar{\theta }\mathbf{1}$ after entrance. Pairing the symmetric interaction terms gives $\frac{1}{2}\frac{d}{dt}\mathrm{\parallel }u{\mathrm{\parallel }}_2^2=-\frac{1}{2}{\sum }_{x,y}{A}_{x,y}(u_x-u_y)\sin (u_x-u_y)$. Since $\mathrm{\mid }{u}_x-u_y\mathrm{\mid }\le \gamma$, this is at most $-\mathrm{sinc}(\gamma )u^{T}Lu\le -c_{a}d\mathrm{\parallel }u{\mathrm{\parallel }}_2^2$. Multiplication by $e^{2c_{a}dt}$ and integration prove the claimed norm contraction. This is a local derivation and does not require importing Dörfler–Bullo’s theorem. [step 1.1, step 1.2, algebra] ∎ [dep-elementary-analysis, dep-rayleigh-bound, step:1, step:2]
4. 2.1 Apply the existing positive-degree regular synchronization theorem under the question’s hypotheses. Its almost-everywhere convergence conclusion implies $T<\mathrm{\infty }$ almost surely: a trajectory converging to a synchronized state eventually lies in an arc of length strictly less than $\gamma$. Independent uniform coordinates give precisely normalized phase volume. This application supplies no rate of decay for $\mathbb{P}(T>t)$. [given, step 1.2] ∎ [thm-expander-sync-thm-1-4, def-expander-prereq-phase-volume, dep-entrance-setting, step:2]
5. 3.1 For the loop-free $K_n$, $A=J-I$ and $d=n-1$. The centered adjacency has eigenvalue zero on $\mathbf{1}$ and eigenvalue $-1$ on its orthogonal complement, so $K_n$ belongs to the question’s class whenever $n-1\ge 1\mathrm{/}a$. Retain the complex Daido parameters ${\rho }_k=n^{-1}{\sum }_x{e}^{ik{\theta }_x}$ without imposing a time-dependent phase rotation. Direct differentiation of the complete-graph equations yields ${\dot{\rho }}_1=(n\mathrm{/}2)({\rho }_1-\overline{{\rho }_1}{\rho }_2)$. Since $\mathrm{\mid }{\rho }_2\mathrm{\mid }\le 1$, $\frac{d}{dt}\mathrm{\mid }{\rho }_1{\mathrm{\mid }}^2\le 2n\mathrm{\mid }{\rho }_1{\mathrm{\mid }}^2$. Integrating after multiplication by $e^{-2nt}$ gives $\mathrm{\mid }{\rho }_1(t)\mathrm{\mid }\le e^{nt}\mathrm{\mid }{\rho }_1(0)\mathrm{\mid }$, including initial states with ${\rho }_1(0)=0$. [given, algebra] [dep-entrance-setting, def-expander-sync-kuramoto-system, dep-daido-calculus]
6. 3.2 If all phases lie in an arc of length $\gamma$, projection onto its midpoint direction gives $\mathrm{\mid }{\rho }_1\mathrm{\mid }\ge \cos (\gamma \mathrm{/}2)=\sqrt{3}\mathrm{/}2$. Consequently $nT\le s$ implies $\mathrm{\mid }{\rho }_1(0){\mathrm{\mid }}^2\ge (3\mathrm{/}4)e^{-2s}$. Uniformity gives $\mathbb{E}{e}^{i{\theta }_x(0)}=0$, and independence eliminates the off-diagonal terms, so $\mathbb{E}\mathrm{\mid }{\rho }_1(0){\mathrm{\mid }}^2=1\mathrm{/}n$. Markov’s inequality proves $\mathbb{P}(nT\le s)\le \min \{1,4e^{2s}\mathrm{/}(3n)\}$. In particular, $\mathbb{P}(dT\le [(n-1)\mathrm{/}(4n)]\log n)\le 4\mathrm{/}(3\sqrt{n})$. Along these admissible complete graphs, a uniform $o(\log n)$ upper quantile at fixed failure probability is therefore impossible. [step 3.1, algebra] ∎ [dep-daido-calculus, dep-elementary-probability, step:5]
7. 4.1 For any nonnegative random variable $X$, an estimate $\mathbb{E}{e}^{\eta X}\le n^K$ with $\eta >0$ and $K\ge 0$ implies $\mathbb{P}(X>(K\log n+\log (1\mathrm{/}\delta ))\mathrm{/}\eta )\le \delta$ by applying Markov’s inequality to $e^{\eta X}$. Hence uniform constants ${\eta }_a>0$ and $K_a\ge 0$ satisfying this moment estimate for $X=dT$ would establish the requested bound with $C_a=\max \{K_a,1\}\mathrm{/}{\eta }_a$. The implication is proved; existence of such constants for $dT$ is not asserted. [given, algebra] ∎ [dep-elementary-probability]

• dep-entrance-setting: Work in ZFC. Fix $0<a<0.0816$, an integer $n\ge 1$, and a symmetric binary matrix $A$ with row sum $d>0$ and $\mathrm{\parallel }A-(d\mathrm{/}n)J\mathrm{\parallel }\le ad$. Degrees count each loop once. Use the unnormalized rotating-frame equation in the question, independent uniform initial circle coordinates, $\gamma =\pi \mathrm{/}3$, and the first entrance time into a closed arc of length $\gamma$. The operator norm and vector norm are Euclidean, and logarithms are natural. Source: self-contained; Question formulation and proof step 1.1. Reading limits: Explicit hypotheses and definitions of this direction; no empirical or physical premise is used.
• def-expander-sync-kuramoto-system: The finite graph Kuramoto equation is well defined on the phase torus and has a unique global smooth trajectory through every initial state. The flow depends continuously on the initial state. A common intrinsic frequency is removed by a common rotating frame; diagonal interaction terms vanish. Source: def-expander-sync-kuramoto-system; Frozen context, Definition, Facts & Assumptions and Justification, steps 1.1–2.1. Reading limits: Read the frozen item body and its local justification. Its ODE and compact-manifold supplier proofs were not independently reconstructed here. The original article is attribution for the model, not proof of this well-definedness input.
• def-expander-prereq-phase-volume: Normalized phase volume is normalized Lebesgue measure on a fundamental angular cube pushed to the finite phase torus. Independent uniform circle coordinates have this law; almost-everywhere synchronization is an assertion with respect to that measure. Source: def-expander-prereq-phase-volume; Frozen context, Definition and Justification, especially steps 1.1 and 2.1. Reading limits: Read the complete frozen item body. Its underlying measure-theory supplier proofs were not independently reconstructed.
• thm-expander-sync-thm-1-4: Under countable choice, every positive-degree $d$-regular $(n,d,\alpha )$-expander with $\alpha \le 0.0816$ is globally synchronizing. Loops are permitted and counted once in degrees. Source: thm-expander-sync-thm-1-4; Frozen context, corrected Statement and Proof steps 1.1 and 2.1. Reading limits: Read the frozen statement, Facts & Assumptions and complete proof. Checked applicability to the present positive-degree hypotheses. The underlying PSD rigidity and analytic-flow arguments were not freshly reconstructed. This theorem is applied to deduce finite entrance, not cited as proof of itself.
• dep-rayleigh-bound: For a real symmetric matrix $B$, $\mathrm{\mid }{v}^{T}Bv\mathrm{\mid }\le \mathrm{\parallel }B\mathrm{\parallel }\mathrm{\parallel }v{\mathrm{\parallel }}_2^2$ by Cauchy–Schwarz. For symmetric nonnegative adjacency, $v^{T}Lv=\frac{1}{2}{\sum }_{x,y}{A}_{x,y}(v_x-v_y{)}^2$. The least Rayleigh quotient on ${\mathbf{1}}^{\perp }$ is ${\lambda }_2(L)$ when $L\mathbf{1}=0$ and this restriction is positive definite. Source: self-contained; Proof steps 1.1 and 1.3. Reading limits: The quadratic-form identity follows by expanding the finite sum; the Rayleigh characterization is elementary finite-dimensional spectral theory. No article theorem is needed for these calculations.
• dep-elementary-analysis: For finitely many differentiable coordinates, the upper right derivative of their maximum is the largest derivative among currently maximizing coordinates, with the corresponding minimum rule. On $\mathrm{\mid }s\mathrm{\mid }\le \gamma <\pi$, $s\sin s\ge \mathrm{sinc}(\gamma )s^2$: concavity of sine on $[0,\gamma ]$ gives $\sin s\ge s\sin \gamma \mathrm{/}\gamma$, and oddness handles negative $s$. If $q^{\mathrm{\prime }}\le -2cq$, differentiation of $e^{2ct}q$ gives $q(t)\le e^{-2c(t-t_0)}q(t_0)$. Source: self-contained; Proof steps 1.2–1.3. Reading limits: Elementary local arguments are specified here. The finite maximum rule follows by excluding coordinates strictly below the maximum for sufficiently short time; the remaining comparisons use differentiation and integration.
• dep-daido-calculus: Define ${\rho }_k=n^{-1}{\sum }_x{e}^{ik{\theta }_x}$, so $\mathrm{\mid }{\rho }_k\mathrm{\mid }\le 1$. The identities $e^{is}=\cos s+i\sin s$ and $\sin s=(e^{is}-e^{-is})\mathrm{/}(2i)$ permit direct differentiation of these finite sums. If the phases lie within $\gamma \mathrm{/}2$ of an arc midpoint, the real part of the correspondingly rotated ${\rho }_1$ is at least $\cos (\gamma \mathrm{/}2)$. Source: self-contained; Proof steps 3.1–3.2. Reading limits: Definitions and elementary identities are stated explicitly; the evolution equation and arc implication are locally derived. The notation agrees with the article’s equation (2.7), but that equation is not used as an independent theorem.
• dep-elementary-probability: For $Y\ge 0$ and $b>0$, $Y\ge b{\mathbf{1}}_{\{Y\ge b\}}$ implies $\mathbb{P}(Y\ge b)\le \mathbb{E}Y\mathrm{/}b$. A uniform circle coordinate satisfies $\mathbb{E}{e}^{i\theta }=(2\pi {)}^{-1}{\int }_0^{2\pi }{e}^{it}dt=0$. Independence factors expectations of bounded functions of distinct coordinates. Source: self-contained; Proof steps 3.2 and 4.1. Reading limits: Markov’s inequality and the uniform-coordinate integral are proved in the statement. Independence is part of the specified initial law. No concentration theorem, continuum limit or approximation is imported.

Decision: accept

The formulation has explicit graph, time-normalization and probability quantifiers. All seven numbered steps are valid at their stated scope: the spectral gap, entrance-time measurability, short-arc invariance, post-entry contraction, almost-sure finite entrance, complete-graph logarithmic lower bound and conditional exponential-moment reduction. The draft correctly distinguishes these partial results from the unproved upper bound. The attribution to the authors’ synchronization-speed direction agrees with Section 7, PDF p. 39 of the original article.

Dependency review: Checked all eight immediate dependency records against their uses. The explicit setting supplies symmetry, nonnegative binary adjacency, positive degree, uniform independent phases and ZFC, which includes the countable-choice assumptions of the frozen suppliers. The quadratic-form calculation gives lambda_2(L) $\ge$ (1-a)d for n$\ge$2, including the stated loop convention; $n=1$ is handled separately. Compactness and continuous dependence justify measurability and attainment at finite entrance. The maximum/minimum argument preserves the chosen short arc, symmetry conserves the lifted mean, and the sine-sector estimate gives exactly the claimed norm-decay constant. The positive-degree regular theorem applies with $alpha=a$ and yields finite entrance almost surely without supplying a rate. For $K_n$, independently checked the centered-adjacency spectrum, the exact unrotated Daido evolution, the 2n differential bound for its squared modulus, the entrance threshold 3/4 and the Markov calculation. The resulting lower bound uses nT and correctly translates to dT=(n-1)T. The exponential-moment implication and constant $C_a$ are correct. No substantive circular inference was found; the combined frozen declared dependency closure also contains no cycle. The two inspected physics:-namespaced suppliers explicitly have domain: mathematics. No physical observation or open question supplies a mathematical premise. Dörfler–Bullo, Theorem 4.8 concerns an already cohesive initial configuration and supports the draft’s qualification, while the contraction here is locally derived.

Review reading limits: Freshly read the complete assigned draft, its seven steps and eight immediate dependency records; read the frozen item bodies listed in items_read and inspected the stated interfaces of the two mathematics-domain physics: suppliers. Read original PDF text on pp. 1–3, 6–7, 10 and 39, and inspected original page images 6, 7 and 39. Read Dörfler–Bullo’s Section 4.3, Theorem 4.8, equation (4.23) and proof paragraph on PDF p. 22. Read the publisher-indexed Sinha–Ghosh abstract; direct publisher access failed and its full paper was not read. Reviewed the verification artifact’s recorded limits and relevant regular-theorem comparison, and confirmed its supplied SHA256. This direction review does not repeat the article-wide finding assessment, reconstruct the entire synchronization-theorem dependency chain, replay its numerical certificates or reaffirm inherited claims of all-page reading. No files were created or changed.

Write $G=X{X}^{\mathsf{T}}$ and $S(G)=\mathrm{diag}(\sqrt{(AG{A}^{\mathsf{T}}{)}_{ii}})$. Search for rank-$r$ unit-row Gram matrices satisfying $AG{A}^{\mathsf{T}}=SGS$. Recover $C=(X^{\mathsf{T}}X{)}^{-1}{X}^{\mathsf{T}}{S}^{-1}AX$, and require $\det C=1$ and $C\mathrm{\ne }{I}_r$. The derivative in steps 5–6 removes the infinitesimal common rotations and supplies a concrete stability calculation. Finding and rigorously certifying such a candidate, followed by a nonlinear stability argument, remains a proposed strategy.

Direct algebra proves the reductions and agent-count restriction without importing an attracting-rotation theorem. The proposed research strategy is to solve the Gram equations for $n\ge r+1$, check proper nontrivial rotation, and certify stability on feasible Gram variations. This strategy is not a solution of the existence question.

1. 1.1 For every unit-row configuration, the triangle inequality gives $\mathrm{\parallel }{\sum }_j{a}_{ij}{u}_j{\mathrm{\parallel }}_2\ge a_{ii}-{\sum }_{j\mathrm{\ne }i}{a}_{ij}>0$. Hence normalization is smooth everywhere. Also $A$ is invertible: if $Av=0$ and $\mathrm{\mid }{v}_i\mathrm{\mid }$ is a largest nonzero component, then $a_{ii}\mathrm{\mid }{v}_i\mathrm{\mid }\le {\sum }_{j\mathrm{\ne }i}{a}_{ij}\mathrm{\mid }{v}_j\mathrm{\mid }\le ({\sum }_{j\mathrm{\ne }i}{a}_{ij})\mathrm{\mid }{v}_i\mathrm{\mid }$, contradicting strict diagonal dominance. Row normalization therefore preserves the rank of $X$. [given, algebra] [def-directed-sphere-model, math:thm-cauchy-schwarz-and-the-euclidean-norm, math:cor-matrix-rank-nullity]
2. 1.2 Define $G=X{X}^{\mathsf{T}}$, $s_i=\sqrt{(AG{A}^{\mathsf{T}}{)}_{ii}}$ and $S=\mathrm{diag}(s_i)$. Direct multiplication gives $F_A(X)=S^{-1}AX$ and its Gram matrix ${\mathrm{\Phi }}_A(G)=S^{-1}AG{A}^{\mathsf{T}}{S}^{-1}$. Thus every relative equilibrium satisfies ${\mathrm{\Phi }}_A(G)=G$, equivalently $AG{A}^{\mathsf{T}}=SGS$. These formulas depend only on $G$, and step 1.1 ensures positive denominators and preservation of rank $r$. [step 1.1, algebra] [def-relative-equilibrium-stability, def-gram-variations, step:1]
3. 2.1 Conversely, suppose $X$ has rank $r$ and ${\mathrm{\Phi }}_A(X{X}^{\mathsf{T}})=X{X}^{\mathsf{T}}$. Set $Y=S^{-1}AX$. Equality $Y{Y}^{\mathsf{T}}=X{X}^{\mathsf{T}}$ implies $\ker Y^{\mathsf{T}}=\ker X^{\mathsf{T}}$, because the corresponding quadratic forms are $\mathrm{\parallel }{Y}^{\mathsf{T}}z{\mathrm{\parallel }}_2^2$ and $\mathrm{\parallel }{X}^{\mathsf{T}}z{\mathrm{\parallel }}_2^2$. Taking orthogonal complements gives equal column spaces. With $L=(X^{\mathsf{T}}X{)}^{-1}{X}^{\mathsf{T}}$ and $C=LY$, this yields $Y=XC$. Multiplying $XC{C}^{\mathsf{T}}{X}^{\mathsf{T}}=X{X}^{\mathsf{T}}$ by $L$ and $L^{\mathsf{T}}$ gives $C{C}^{\mathsf{T}}=I_r$. Thus $C$ is orthogonal. The additional checks $\det C=1$ and $C\mathrm{\ne }{I}_r$ give precisely a proper, nonstationary relative equilibrium with $R=C^{\mathsf{T}}$. A Gram fixed point alone does not distinguish a reflection from a proper rotation. [step 1.2, algebra] [def-relative-equilibrium-stability, def-finite-matrix-conventions, math:cor-double-orthogonal-complement-and-dimension, step:2]
4. 2.2 Every full-rank relative equilibrium under these hypotheses has $n\ge r+1$. Rank first gives $n\ge r$; suppose $n=r$. Put $B=S^{-1}A$ and $m=n-1$. Strong connectivity and positive diagonal imply $B^m$ is entrywise positive: each ordered vertex pair has a path of length at most $m$, extended to length $m$ by diagonal loops. From $BX=X{R}^{\mathsf{T}}$, every row of $B^{m}X=X(R^{\mathsf{T}}{)}^m$ has norm one. For positive weights $w_j$, expansion gives $({\sum }_j{w}_j{)}^2-\mathrm{\parallel }{\sum }_j{w}_j{u}_j{\mathrm{\parallel }}_2^2={\sum }_{j<\mathrm{\ell }}{w}_j{w}_{\mathrm{\ell }}\mathrm{\parallel }{u}_j-u_{\mathrm{\ell }}{\mathrm{\parallel }}_2^2$. Full rank with $r>1$ ensures distinct states, so every row sum of $B^m$ exceeds one. Let their minimum be $c>1$. Positivity gives $(B^m{)}^k\mathbf{1}\ge c^k\mathbf{1}$. But square invertibility of $X$ gives $B^{mk}=X(R^{\mathsf{T}}{)}^{mk}{X}^{-1}$, whose entries remain bounded because the middle factor is orthogonal. This contradiction excludes $n=r$. [step 1.2, algebra] [def-directed-sphere-model, def-relative-equilibrium-stability, def-finite-matrix-conventions, step:2]
5. 3.1 At a Gram fixed point, take a feasible variation $H=X{V}^{\mathsf{T}}+V{X}^{\mathsf{T}}$ with $u_i\cdot v_i=0$. Such variations are realized by the unit-row curves $(u_i+t{v}_i)\mathrm{/}\mathrm{\parallel }{u}_i+t{v}_i{\mathrm{\parallel }}_2$, which retain full rank for sufficiently small $t$. Put $N=AH{A}^{\mathsf{T}}$ and $E=\mathrm{diag}(N_{ii}\mathrm{/}(2s_i^2))$. Expanding the positive square roots gives $S(G+tH+o(t))=(I+tE)S+o(t)$, hence $D{\mathrm{\Phi }}_A(G)[H]=S^{-1}N{S}^{-1}-EG-GE$. This is the derivative along feasible rank-$r$ Gram variations; unrestricted symmetric-matrix directions need not be feasible. [step 1.1, step 1.2, algebra] [def-gram-variations, def-finite-matrix-conventions, step:1, step:2]
6. 3.2 The Gram differential removes exactly the infinitesimal common rotations. Indeed, if $X{V}^{\mathsf{T}}+V{X}^{\mathsf{T}}=0$, put $L=(X^{\mathsf{T}}X{)}^{-1}{X}^{\mathsf{T}}$ and $P=XL$. Multiplication on the left by $I-P$ and on the right by $X(X^{\mathsf{T}}X{)}^{-1}$ gives $(I-P)V=0$, so $V=XK$. Multiplication by $L$ and $L^{\mathsf{T}}$ then gives $K+K^{\mathsf{T}}=0$. Conversely every such $XK$ has zero Gram differential and satisfies the row-tangency conditions. The feasible Gram-variation space consequently has dimension $n(r-1)-r(r-1)\mathrm{/}2$. For a relative equilibrium, the co-rotating map $Z\mapsto F_A(Z)R$ fixes $X$ and has the same Gram update as $F_A$. Differentiating this identity identifies its tangent dynamics modulo common rotations with the operator in step 3.1. This proves the linear reduction, without asserting any contraction estimate. [step 1.2, step 3.1, algebra] ∎ [def-relative-equilibrium-stability, def-gram-variations, def-finite-matrix-conventions, math:cor-matrix-rank-nullity, step:2, step:5]

• def-directed-sphere-model: Finite $n$, ambient dimension $r>3$, unit vectors $u_i\in {\mathbb{R}}^r$, and $X$ with rows $u_i^{\mathsf{T}}$. Weights are nonnegative, directed support is strongly connected, and $a_{ii}>\sqrt{2}{\sum }_{j\mathrm{\ne }i}{a}_{ij}$. The update normalizes each row of $AX$. Source: self-contained; Question formulation; step 1 establishes well-definedness. Reading limits: Mathematical hypotheses adopted explicitly here, matching Sidorenko–Thunberg equations (1) and (3). No physical assumptions, observations or existence claim are used.
• def-relative-equilibrium-stability: $\mathrm{SO}(r)=\{R:R^{\mathsf{T}}R=I_r,\det R=1\}$. A relative equilibrium satisfies $F_A(X)=X{R}^{\mathsf{T}}$; full rank means rank $r$, and nonstationary means $R\mathrm{\ne }{I}_r$. Orbit stability means Lyapunov stability and attraction in the Frobenius distance to the common-rotation orbit. Source: self-contained; Question formulation and steps 2–6. Reading limits: Definitions are restated completely. Their source counterpart was checked in Sidorenko–Thunberg Section 4, equations (15)–(18), PDF p. 4. No stability theorem is imported into the local deductions.
• def-gram-variations: The rank-$r$ unit-row Gram set consists of $G=X{X}^{\mathsf{T}}$ for unit-row matrices $X$ of rank $r$. Its feasible first variations are $H=X{V}^{\mathsf{T}}+V{X}^{\mathsf{T}}$ with $u_i\cdot v_i=0$; realizability is proved in step 5. Source: self-contained; Steps 2, 5 and 6. Reading limits: Local definitions and direct calculations only; no global quotient-manifold theorem is assumed.
• def-finite-matrix-conventions: All matrices are finite and real; transpose, kernel, column space, rank and determinant have their usual linear-algebra meanings. Full column rank gives the left inverse $(X^{\mathsf{T}}X{)}^{-1}{X}^{\mathsf{T}}$, since $z^{\mathsf{T}}{X}^{\mathsf{T}}Xz=\mathrm{\parallel }Xz{\mathrm{\parallel }}_2^2>0$ for nonzero $z$. Scalar square-root expansion at $s>0$ is $\sqrt{s^2+t\eta +o(t)}=s+t\eta \mathrm{/}(2s)+o(t)$, obtained by rationalizing the difference. Source: self-contained; Elementary matrix operations in steps 1–6. Reading limits: The particular left-inverse and differentiation facts needed are stated with their elementary justifications. No Perron–Frobenius or nonlinear stability theorem is invoked.
• math:thm-cauchy-schwarz-and-the-euclidean-norm: On finite-dimensional real Euclidean space, the Euclidean norm is positive definite, homogeneous and satisfies the triangle inequality; Cauchy–Schwarz bounds the inner product. Source: math:thm-cauchy-schwarz-and-the-euclidean-norm; Statement clauses 1–2; Proof. Reading limits: Read the complete current supplier file, including metadata and proof. Used its finite-dimensional norm conclusions; did not recursively inspect its suppliers or external bibliography.
• math:cor-matrix-rank-nullity: For a finite matrix $M\in {\mathbb{R}}^{a\times b}$, $\mathrm{rank}M+\dim \ker M=b$. Source: math:cor-matrix-rank-nullity; Statement and Proof. Reading limits: Read the complete current supplier file. Applied over the real field for square invertibility and the dimension of the Gram differential's image; upstream suppliers were not recursively inspected.
• math:cor-double-orthogonal-complement-and-dimension: For a subspace $W$ of a finite-dimensional real inner-product space, $W^{\perp \perp }=W$. Together with the defining transpose identity, this gives $\mathrm{range}M=(\ker M^{\mathsf{T}}{)}^{\perp }$. Source: math:cor-double-orthogonal-complement-and-dimension; Statement and Proof; transpose application in step 3. Reading limits: Read the complete current supplier file. The range-kernel application uses the elementary identity $z\perp \mathrm{range}M$ exactly when $M^{\mathsf{T}}z=0$. No cited external book was freshly read.

Decision: accept

Accept unchanged in category question. Independently checked all six argument steps: well-definedness and rank preservation, the Gram equation, recovery of the orthogonal factor with orientation checks, the restriction n≥r+1, the feasible Gram derivative, and removal of infinitesimal common rotations. These deductions are valid under the stated hypotheses. They establish reductions, not existence or attraction.

Dependency review: Checked the four self-contained definition/convention dependencies and the complete current files for math:thm-cauchy-schwarz-and-the-euclidean-norm, math:cor-matrix-rank-nullity and math:cor-double-orthogonal-complement-and-dimension. Their finite-dimensional real hypotheses apply. The local deductions use neither physical claims nor the open question as premises; no substantive dependency cycle was identified. In step 4, strong connectivity and positive diagonal justify positivity of B^(n−1), and the strict row-sum growth contradicts bounded powers when $n=r$. In steps 5–6, the derivative is correctly restricted to feasible variations; its kernel consists exactly of XK with K skew-symmetric. The cited external results supply context rather than premises for these deductions. The model assumptions, Theorem 3, embedding caveat and concluding question match the cited Sidorenko–Thunberg paper.

Review reading limits: Read the frozen assigned direction, all six steps and all seven immediate dependency records. Read all three cited library supplier files, without recursively auditing their dependencies or bibliography. Independently read original-article PDF p. 39 and inspected its retained page image; this supports the broad higher-sphere attribution only. Read the cached Sidorenko–Thunberg PDF text on pp. 1–2, 4–5, 7 and 9, inspected the retained p. 9 image, and checked relevant primary online HTML passages. Read the current arXiv version record, which displays v1, and performed targeted status searches. Parsed the supplied verification artifact, confirmed its specified SHA256, and inspected its relevant higher-sphere attribution records; this was not an independent review of its entire finding inventory. The inherited all-pages reading declaration is not this review's reading coverage. No serving files, proposal files or assessment artifacts were written.