Open questions and potential solutions

At extreme lifted phases, the signs of the sine interactions make the maximum nonincreasing and the minimum nondecreasing while the diameter is below $\pi$; continuity makes this invariant for all later times. Symmetry conserves the arithmetic mean. For $y=\theta -\bar{\theta }\mathbf{1}$, pairwise cancellation gives $\frac{1}{2}\frac{d}{dt}\mathrm{\parallel }y{\mathrm{\parallel }}_2^2=-{\sum }_{x<z}{A}_{x,z}(y_x-y_z)\sin (y_x-y_z)$. Monotonicity of sinc on $[0,\pi )$ bounds this by $-\mathrm{sinc}(\gamma )y^{T}Ly$, and the spectral theorem bounds it further by $-r\mathrm{\parallel }y{\mathrm{\parallel }}_2^2$. Integration proves contraction and the tolerance bound. Restricting the centered Laplacian inequality to ${\mathbf{1}}^{\perp }$ gives the expander specialization. Moreover, the diameter is at most $\sqrt{2}\mathrm{\parallel }y{\mathrm{\parallel }}_2$, so restarting after it falls below any prescribed $\eta \in (0,\gamma )$ improves the certified rate to ${\lambda }_2(L)\mathrm{sinc}(\eta )$. A two-vertex calculation verifies sharpness of the initial universal factor and failure at diameter $\pi$.

proof_strategy: direct. Given: [F1] hyp-symmetric-dynamics; [F2] hyp-common-arc; [F3] math:def-finite-weighted-graph-laplacian; [F4] math:lem-weighted-graph-laplacian-quadratic-form-and-kernel; [F5] math:cor-real-spectral-theorem-for-self-adjoint-endomorphisms; [F6] hyp-centered-expansion, only for the expander specialization; [F7] fact-finite-extrema; [F8] def-sinc. Prove interval invariance and mean conservation, establish the Laplacian and scalar inequalities, and integrate the resulting energy inequality. Derive quantitative consequences and test sharpness explicitly.

1. 1.1 Self-loops contribute zero to the differential equations. Set $W=A-\mathrm{diag}(A_{x,x})$. Its zero-diagonal Laplacian equals $L=\mathrm{diag}(A\mathbf{1})-A$, so the library conventions apply without restricting the allowed self-loops. Also ${\sum }_x{\dot{\theta }}_x=-{\sum }_{x<z}{A}_{x,z}[\sin ({\theta }_x-{\theta }_z)+\sin ({\theta }_z-{\theta }_x)]=0$. Thus the arithmetic mean is constant and $y=\theta -\bar{\theta }\mathbf{1}$ belongs to ${\mathbf{1}}^{\perp }$. [F1, F2, F3, algebra] [hyp-symmetric-dynamics, hyp-common-arc, math:def-finite-weighted-graph-laplacian]
2. 1.2 Write $M(t)={\max }_x{\theta }_x(t)$ and $m(t)={\min }_x{\theta }_x(t)$. Whenever $M-m<\pi$, every active maximum has ${\theta }_x-{\theta }_z\in [0,\pi )$ and hence ${\dot{\theta }}_x\le 0$; every active minimum has derivative at least zero. The finite-extrema derivative formula therefore makes $M$ nonincreasing and $m$ nondecreasing on that interval. Starting from diameter $\gamma <\pi$, continuity prevents a first exit from the region of diameter below $\pi$: on any putative pre-exit interval the diameter is at most $\gamma$. Consequently all coordinates stay in $[m(T),M(T)]$ for every $t\ge T$. This also covers $\gamma =0$. [F1, F2, F7] [hyp-symmetric-dynamics, hyp-common-arc, fact-finite-extrema]
3. 2.1 The Laplacian identity gives $v^{T}Lv={\sum }_{x<z}{A}_{x,z}(v_x-v_z{)}^2\ge 0$. Its kernel consists of vectors constant on support components; connectedness therefore gives $\ker L=\mathrm{span}\{\mathbf{1}\}$. The real spectral theorem supplies an orthonormal eigenbasis, with $0={\lambda }_1(L)<{\lambda }_2(L)\le \cdots \le {\lambda }_n(L)$. Expanding $v\perp \mathbf{1}$ in that basis proves $v^{T}Lv\ge {\lambda }_2(L)\mathrm{\parallel }v{\mathrm{\parallel }}_2^2$. [F1, F4, F5, step 1.1] [hyp-symmetric-dynamics, math:lem-weighted-graph-laplacian-quadratic-form-and-kernel, math:cor-real-spectral-theorem-for-self-adjoint-endomorphisms, step:1]
4. 2.2 For $0<u<\pi$, put $h(u)=\sin u-u\cos u$. Since $h(0)=0$ and $h^{\mathrm{\prime }}(u)=u\sin u>0$, one has ${\mathrm{sinc}}^{\mathrm{\prime }}(u)=-h(u)\mathrm{/}{u}^2<0$. Continuity at zero and evenness imply $u\sin u\ge \mathrm{sinc}(\gamma )u^2$ whenever $\mathrm{\mid }u\mathrm{\mid }\le \gamma <\pi$, including $u=0$. In particular $\mathrm{sinc}(\gamma )>0$. The invariant interval makes this inequality applicable to every phase difference. [F8, step 1.2, algebra] [def-sinc, step:2]
5. 3.1 Pairing the symmetric interactions and using the conserved mean yields $\frac{1}{2}\frac{d}{dt}\mathrm{\parallel }y{\mathrm{\parallel }}_2^2=-{\sum }_{x<z}{A}_{x,z}(y_x-y_z)\sin (y_x-y_z)\le -\mathrm{sinc}(\gamma )y^{T}Ly\le -r\mathrm{\parallel }y{\mathrm{\parallel }}_2^2$, where $r={\lambda }_2(L)\mathrm{sinc}(\gamma )>0$. Thus the derivative of $e^{2r(t-T)}\mathrm{\parallel }y(t){\mathrm{\parallel }}_2^2$ is nonpositive. Evaluation at $T$ and taking square roots give $\mathrm{\parallel }y(t){\mathrm{\parallel }}_2\le E_T{e}^{-r(t-T)}$, including the case $E_T=0$. [F1, F4, step 1.1, step 1.2, step 2.1, step 2.2] [hyp-symmetric-dynamics, math:lem-weighted-graph-laplacian-quadratic-form-and-kernel, step:1, step:2, step:3, step:4]
6. 3.2 Under the optional centered-expansion hypothesis, $Jv=0$ for $v\perp \mathbf{1}$, so $v^{T}Lv=d^{\ast }\mathrm{\parallel }v{\mathrm{\parallel }}_2^2+v^T{\mathrm{\Delta }}_{L}v\ge (1+c^{-})d^{\ast }\mathrm{\parallel }v{\mathrm{\parallel }}_2^2$. Testing on an eigenvector for ${\lambda }_2(L)$ gives ${\lambda }_2(L)\ge (1+c^{-})d^{\ast }>0$. Substitution in step 3.1 proves the claimed expander rate. Neither the upper centered-Laplacian bound nor an adjacency-norm bound enters this deduction. [F6, step 2.1, step 3.1, algebra] [hyp-centered-expansion, step:3, step:5]
7. 4.1 For $E_T>0$, solving $E_T{e}^{-r(t-T)}\le \epsilon$ gives the stated tolerance time. Also $(y_x-y_z{)}^2\le 2(y_x^2+y_z^2)\le 2\mathrm{\parallel }y{\mathrm{\parallel }}_2^2$, hence the phase diameter is at most $\sqrt{2}{E}_T{e}^{-r(t-T)}$. If $0<\eta <\gamma$, define $T_{\eta }=T+r^{-1}\max \{0,\log (\sqrt{2}{E}_T\mathrm{/}\eta )\}$. At $T_{\eta }$ the diameter is at most $\eta$; restarting the same proof gives $\mathrm{\parallel }y(t){\mathrm{\parallel }}_2\le \mathrm{\parallel }y(T_{\eta }){\mathrm{\parallel }}_2{e}^{-{\lambda }_2(L)\mathrm{sinc}(\eta )(t-T_{\eta })}$ for $t\ge T_{\eta }$. Thus certified rates can approach ${\lambda }_2(L)$ after a controlled additional delay. [step 1.2, step 3.1, algebra] [step:2, step:5]
8. 4.2 Take two vertices with $A_{1,2}=A_{2,1}=a>0$, zero diagonal, and initial lifts ${\theta }_1(T)=\gamma \mathrm{/}2$, ${\theta }_2(T)=-\gamma \mathrm{/}2$, where $0<\gamma <\pi$. All graph hypotheses hold and ${\lambda }_2(L)=2a$. Their difference $\delta$ satisfies $\dot{\delta }=-2a\sin \delta$, while $\mathrm{\parallel }y{\mathrm{\parallel }}_2=\delta \mathrm{/}\sqrt{2}$. Therefore $-{\frac{d}{dt}\log \mathrm{\parallel }y(t){\mathrm{\parallel }}_2\mid }_{t=T}=2a\mathrm{sinc}(\gamma )$. Any larger universal exponential rate with prefactor one would fail immediately for this example. At initial difference exactly $\pi$, both derivatives vanish, so the strict common-arc condition cannot be extended to that endpoint. These examples concern the conditional estimate, not a defect in the article or a refutation of almost-everywhere synchronization. [F1, F2, F8, step 3.1, algebra] ∎ [hyp-symmetric-dynamics, hyp-common-arc, def-sinc, step:5]

• hyp-symmetric-dynamics: A finite vertex set has $n\ge 2$ elements; $A$ is a fixed symmetric nonnegative matrix with connected positive off-diagonal support. A $C^1$ real-valued solution on $[T,\mathrm{\infty })$ satisfies ${\dot{\theta }}_x=-{\sum }_z{A}_{x,z}\sin ({\theta }_x-{\theta }_z)$. This is the homogeneous system in a rotating frame, with unit time constants. Source: self-contained; Direction statement; proof steps 1, 2, 5 and 8. Reading limits: Explicit mathematical hypotheses, rather than a physical assumption or an imported synchronization conclusion. A solution on the stated interval is assumed; an existence theorem is not used as a proof premise.
• hyp-common-arc: At time $T$, choose consistent real lifts with $\gamma ={\max }_x{\theta }_x(T)-{\min }_x{\theta }_x(T)\in [0,\pi )$. Define $\bar{\theta }=n^{-1}{\sum }_x{\theta }_x(T)$, $y=\theta -\bar{\theta }\mathbf{1}$ and $E_T=\mathrm{\parallel }y(T){\mathrm{\parallel }}_2$. Source: self-contained; Direction statement; proof steps 1 and 2. Reading limits: Assumed entry condition and explicit definitions. No assertion about reaching this condition from arbitrary or random initial data is imported.
• math:def-finite-weighted-graph-laplacian: For a finite symmetric nonnegative zero-diagonal matrix $W$, its degree matrix is $D_W=\mathrm{diag}(W\mathbf{1})$, its Laplacian is $D_W-W$, and its support edges are the pairs with positive weight. Here apply this definition to $W=A-\mathrm{diag}(A_{x,x})$. Source: math:def-finite-weighted-graph-laplacian; ## Definition; loop removal and identification in proof step 1. Reading limits: Read the complete library item and checked its zero-diagonal convention. Its normalized-Laplacian definitions are unused. Bibliographic sources were not independently read.
• math:lem-weighted-graph-laplacian-quadratic-form-and-kernel: For a finite symmetric nonnegative zero-diagonal weight matrix, $v^{T}Lv=\frac{1}{2}{\sum }_{x,z}{W}_{x,z}(v_x-v_z{)}^2$, and the kernel comprises vectors constant on each support component. Source: math:lem-weighted-graph-laplacian-quadratic-form-and-kernel; ## Statement and ## Proof, steps 1.1–3.1. Reading limits: Read the complete item, including its proof, and checked application to the loop-deleted weights. No independent reading of its cited papers or audit of its supplier closure is claimed.
• math:cor-real-spectral-theorem-for-self-adjoint-endomorphisms: Every self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis. Applied to the symmetric positive semidefinite $L$ with kernel $\mathrm{span}\{\mathbf{1}\}$, this yields $v^{T}Lv\ge {\lambda }_2(L)\mathrm{\parallel }v{\mathrm{\parallel }}_2^2$ on ${\mathbf{1}}^{\perp }$. Source: math:cor-real-spectral-theorem-for-self-adjoint-endomorphisms; ## Statement and ## Proof, steps 1.1–3.1; application derived in direction proof step 3. Reading limits: Read the complete library item and checked the finite-dimensional self-adjoint hypotheses. Its underlying classification theorem and cited textbook were not independently reread.
• hyp-centered-expansion: For the optional specialization, $d^{\ast }>0$, $c^{-}>-1$, $J={\mathbf{11}}^T$ and ${\mathrm{\Delta }}_L=L-d^{\ast }I+(d^{\ast }\mathrm{/}n)J$ satisfy ${\mathrm{\Delta }}_L⪰c^{-}{d}^{\ast }I$. The Loewner inequality means $v^T{\mathrm{\Delta }}_{L}v\ge c^{-}{d}^{\ast }\mathrm{\parallel }v{\mathrm{\parallel }}_2^2$ for every real vector $v$. Source: self-contained; Direction statement; proof step 6. Reading limits: An explicitly imposed algebraic hypothesis. The article’s p. 6 notation and p. 8 Definition 1.9 were checked in text and images for correspondence; no original-article theorem is assumed to prove this consequence.
• fact-finite-extrema: For finitely many $C^1$ functions $f_x$, the right derivative of $M(t)={\max }_x{f}_x(t)$ is ${\max }_{x:f_x(t)=M(t)}{f}_x^{\mathrm{\prime }}(t)$, with the analogous minimum formula. Indeed, inactive indices have a positive gap at a fixed time, and first-order expansions reduce the right derivative to the active indices. These extrema are locally Lipschitz, so derivative signs imply monotonicity by integration. Source: self-contained; Proof step 2 and this finite-family derivative argument. Reading limits: Elementary local argument supplied explicitly; no nonsmooth-analysis theorem or article assertion is imported.
• def-sinc: Define $\mathrm{sinc}(u)=\sin (u)\mathrm{/}u$ for $u\mathrm{\ne }0$ and $\mathrm{sinc}(0)=1$. It is even, positive and nonincreasing on $[0,\pi )$, yielding $u\sin u\ge \mathrm{sinc}(\gamma )u^2$ for $\mathrm{\mid }u\mathrm{\mid }\le \gamma <\pi$. Source: self-contained; Definition in direction statement; elementary derivative proof in step 4. Reading limits: Definition and scalar inequality proved locally using elementary differentiation and the continuous limit at zero; no imported contraction theorem is needed.

Decision: accept

The stated hypotheses establish interval invariance, mean conservation and exponential contraction with rate λ₂(L)sinc(γ). The energy calculation has the correct factors. The tolerance time, improved rate after restarting, two-vertex sharpness example and obstruction at diameter π are valid. Restricting the centered Laplacian inequality to the mean-zero subspace proves the expander specialization. This establishes conditional decay after cohesive entry; it supplies no entrance-time estimate from random initial data.

Dependency review: Checked all eight immediate dependencies. The three library supplier files match their frozen SHA256 values; their definitions, statements and supplied proofs were read. Removing diagonal weights preserves both the dynamics and Laplacian, satisfying the suppliers’ zero-diagonal convention. Connected positive support gives a one-dimensional kernel, and the spectral theorem applies to the real symmetric Laplacian. The finite-extrema argument and sinc inequality are supplied locally and are valid. The optional expansion hypothesis matches the article’s notation. No physical premise, open problem or circular use of an article conclusion supports the proof. Dörfler–Bullo, Theorem 4.8 and equation (4.23) corroborate the stated rate specialization with unit time constants and zero phase shifts.

Review reading limits: Read the complete assigned draft, all eight proof steps and dependency records. Inspected frozen original-PDF text and rendered images of pp. 2, 6, 8 and 39, and searched extracted article text. These checks verify the dynamics, expansion conventions and discussion of synchronization speed in the original article. Read Dörfler–Bullo’s equation (2.8), Theorem 4.8, equation (4.23) and accompanying proof in PDF text. Did not reread either entire paper, the library suppliers’ bibliography or their full prerequisite closure. The draft’s broader historical source-reading records are not independent reading claims by this reviewer. No exhaustive novelty search or full original-article defect assessment was performed.

For each vertex $x$, consider the event that every phase lies in the clockwise $D$-arc starting at ${\theta }_x$. Any covering arc can be shifted until its starting point is an occupied phase. Away from coincidences, two different occupied starting points cannot both work: their clockwise separations would each be at most $D$, although those separations sum to $2\pi$. Each anchored event has probability $(D\mathrm{/}(2\pi ){)}^{n-1}$ by product integration. Their probabilities therefore add. Increasing $D$ to $\pi$ gives the probability of fitting in any arc strictly shorter than a semicircle. This bounds the applicability of any criterion requiring such initial containment, while leaving subsequent entry and convergence unrestricted.

Use a direct decomposition into occupied-starting-point events. Prove their equivalence to arc containment and their disjointness outside a null set, compute each event by finite product integration, and pass to the increasing union of all strictly subsemicircular arc events.

1. Work on $[0,2\pi {)}^V$ with the product of normalized angular-length probability measures, identifying coordinates modulo $2\pi$. Every singleton has measure zero. For distinct vertices $x,y$, the section of $\{{\theta }_x={\theta }_y\}$ at a fixed ${\theta }_x$ is a singleton in the $y$ coordinate, so product integration gives probability zero. A finite union shows that all phases are distinct almost surely. [hyp-uniform-phase-law, math:thm-tonelli-theorem-for-sigma-finite-product-spaces, math:def-measure]
2. For each $x\in V$, define $A_x(D)$ by requiring $({\theta }_y-{\theta }_x)<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>2\pi \in [0,D]$ for every $y\mathrm{\ne }x$. These are Borel events because modular subtraction is piecewise continuous. Clearly $A_x(D)\subseteq C_D$. Conversely, lift any covering clockwise $D$-arc to a real interval $[a,a+D]$, choose the smallest occupied lift $b$, and shift the interval to $[b,b+D]$. All occupied lifts remain inside, so $C_D={\bigcup }_{x\in V}{A}_x(D)$. This also proves measurability of the initially existential event $C_D$. [def-clockwise-covering-arc]
3. On the distinct-phase event, suppose $A_x(D)$ and $A_y(D)$ both hold for $x\mathrm{\ne }y$. Write $a=({\theta }_y-{\theta }_x)<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>2\pi \in (0,2\pi )$. The two anchored conditions imply $a\le D$ and $2\pi -a\le D$, hence $2\pi \le 2D<2\pi$, a contradiction. Thus the anchored events are pairwise disjoint outside the null set established in step 1. [def-clockwise-covering-arc, step:1, step:2]
4. Fix $x$ and its coordinate value ${\theta }_x=t$. Each other coordinate is restricted to the clockwise arc from $t$ of length $D$. In $[0,2\pi )$ this is one interval or two intervals with total length $D$, with endpoints immaterial to measure. The product section therefore has probability $(D\mathrm{/}(2\pi ){)}^{n-1}$. Integrating this constant section probability over $t$ gives $\mathbb{P}(A_x(D))=(D\mathrm{/}(2\pi ){)}^{n-1}$. Tonelli applies to the Borel indicator and finite, hence sigma-finite, factor measures. [hyp-uniform-phase-law, math:def-product-measure-on-sigma-finite-spaces, math:thm-tonelli-theorem-for-sigma-finite-product-spaces, step:2]
5. Finite additivity after removal of the null coincidence set gives $\mathbb{P}(C_D)={\sum }_{x\in V}\mathbb{P}(A_x(D))=n(D\mathrm{/}(2\pi ){)}^{n-1}$. For fixed $D<\pi$, putting $q=D\mathrm{/}(2\pi )<1\mathrm{/}2$ yields $\log \mathbb{P}(C_D)=\log n+(n-1)\log q$, which proves exponential decay up to the polynomial prefactor $n$. [math:def-measure, step:1, step:2, step:3, step:4]
6. Take $D_m=\pi (1-1\mathrm{/}m)$ for integers $m\ge 2$. The measurable events $C_{D_m}$ increase, and their union is exactly the event of containment in some arc of length strictly less than $\pi$: every such length is bounded above by some $D_m$. Continuity from below and step 5 give $\mathbb{P}({\bigcup }_{0<D<\pi }{C}_D)={\lim }_{m\to \mathrm{\infty }}n(D_m\mathrm{/}(2\pi ){)}^{n-1}=n\mathrm{/}{2}^{n-1}$. This also handles $n=2$, for which the probability is one. [math:thm-continuity-from-below-for-measures, step:5]
7. An initial applicability event requiring a covering arc shorter than a semicircle is a subset of the event in step 6, regardless of how its arc or allowed length is selected. Its probability is therefore at most $n\mathrm{/}{2}^{n-1}$. If a random graph on the fixed vertex set is independent of the phases, conditioning on any graph realization of positive probability leaves their product uniform law unchanged, so the same bound holds conditionally. This is only an applicability bound at time zero; it gives no bound on subsequent entry into the arc region. [hyp-arc-criterion-and-graph-independence, hyp-uniform-phase-law, math:def-measure, step:5, step:6]

• hyp-uniform-phase-law: The vertex set $V$ is finite with $\mathrm{\mid }V\mathrm{\mid }=n\ge 2$. Each phase lies in $S^1=\mathbb{R}\mathrm{/}(2\pi \mathbb{Z})$ and has normalized angular-length probability measure; the joint law is the finite product of these measures. Thus an arc of angular length $D$ has marginal probability $D\mathrm{/}(2\pi )$, and singleton phases have probability zero. Source: self-contained; Sampling hypothesis of this direction; proof steps 1 and 4.. Reading limits: An explicit mathematical sampling assumption, not an empirical observation or a conclusion imported from the article.
• def-clockwise-covering-arc: For $0<D<\pi$, a closed clockwise arc starting at $a\in S^1$ is $\{a+s<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>2\pi :0\le s\le D\}$. The event $C_D$ means that some such arc contains every phase. For distinct phases, the two directed separations between them sum to $2\pi$. Source: self-contained; Definitions and elementary modular geometry used in proof steps 2 and 3.. Reading limits: Defined here; the occupied-endpoint reduction and uniqueness argument are proved locally rather than imported.
• math:def-measure: A measure is nonnegative, assigns zero to the empty set, and is countably additive on disjoint measurable sequences. Finite additivity, monotonicity and preservation of probabilities after removing null sets follow directly. Source: math:def-measure; Definition, countable-additivity clause.. Reading limits: Read the complete local item, including metadata and Definition. Used its measure definition; did not freshly inspect its bibliographic sources or dependency closure.
• math:def-product-measure-on-sigma-finite-spaces: For sigma-finite measure spaces, the product measure of a product-measurable set equals either iterated integral of its section measures. Repetition gives the finite-product construction used for the phase law. Source: math:def-product-measure-on-sigma-finite-spaces; Definition, displayed section-integral formula.. Reading limits: Read the complete local item and its metadata. Applied the stated interface to finite probability measures; did not audit its suppliers or claim new library acceptance.
• math:thm-tonelli-theorem-for-sigma-finite-product-spaces: For two sigma-finite measure spaces and a nonnegative product-measurable function, the product integral equals both iterated integrals, with measurable inner-integral functions. Here the functions are Borel event indicators and every factor measure is finite. Source: math:thm-tonelli-theorem-for-sigma-finite-product-spaces; Statement; Proof steps 1.1 and 2.1.. Reading limits: Read the complete local theorem, including its Statement, Facts & Assumptions and two-step Proof. Checked applicability to the event indicators; did not freshly audit its dependency closure or external references.
• math:thm-continuity-from-below-for-measures: If measurable sets $E_m$ increase, then the measure of their union equals the supremum of their measures; no finiteness hypothesis is required. Source: math:thm-continuity-from-below-for-measures; Statement; Proof steps 1.1–3.1.. Reading limits: Read the complete local theorem and its proof. Applied it to the explicitly increasing sequence $C_{D_m}$; did not freshly inspect its external references or dependency closure.
• hyp-arc-criterion-and-graph-independence: The criterion being benchmarked requires initial containment of all phases in an arc of angular length strictly below $\pi$. For the optional conditional statement, the random graph has the fixed vertex set $V$ and is independent of the phase sample. No actual contraction theorem or convergence conclusion is assumed. Source: self-contained; Conditional interpretation of the probability calculation; proof step 7.. Reading limits: An explicit applicability condition. No graph-specific contraction result is imported or asserted.

Decision: accept

The occupied-endpoint reduction, almost-sure disjointness, product-measure calculation and increasing-union argument establish both probability identities for every stated n and D. The applicability bound permits choosing the arc and its length after observing the phases. Its exponential decay is valid. The direction correctly distinguishes this initialization benchmark from an entry-time estimate or a solution to the authors’ synchronization-speed question.

Dependency review: Read the complete four cited library records, including metadata, definitions, statements and available proofs. Their interfaces support countable additivity, null-set removal, finite-product section integration, Tonelli and continuity from below. All factor measures are finite, and the event indicators are Borel measurable, so the stated applications satisfy the hypotheses. The remaining three dependency entries explicitly supply the sampling assumption, arc definition and conditional applicability assumptions. No physical claim, open problem, contraction theorem or original-article theorem supplies a mathematical premise. The seven steps cite only declared dependencies or earlier steps; no substantive circularity was found. The $n=2$ case, closed endpoints, coincidences and the uncountable union over D are handled correctly.

Review reading limits: Read the frozen direction, all seven argument steps and all seven dependency entries. Read original-article PDF text pp. 2–3 and 39, inspected the retained rendered image of p. 39, and searched the full extracted text for relevant arc and probability terminology. Definition 1.1 and the Section 7 speed discussion support the contextual interpretation. Independently checked the indexed opening-page extraction of Wendel’s paper, including the planar formula and Darling attribution, and its publisher metadata. Direct retrieval of the older PDF URL failed. Did not inspect Wendel’s scan, reconstruct his proof, audit supplier dependency closures or perform the separate article-wide defects assessment. The inherited source inventory’s broader reading claims are not adopted as this reviewer’s reading.