Alphabeta MathArticle items · 12 — Asymptotically global synchrony via basin mass
Presents 28 items from the article and its immediate prerequisites, covering dependency levels 7–8.
1 · Prerequisites
- Equilibria, stable states, synchronized states, and spurious minima
- Globally synchronizing graph — Definition 1.1
- Point basins, equilibrium-set basins and aggregate volume
- Asymptotic quantifiers and uniform parameter ranges
- The Axiom of Countable Choice (AC_ω)
- Finite phase tori, local lifts and the flat metric
- Homogeneous Kuramoto energy
- Phasors, trigonometric identities and periodic derivatives
- Gradient trajectories and complete compact phase dynamics
- Rotating-frame gradient formulation
- Local minima and nonnegative second variation
- Adjacency, degrees, graph Laplacian, and spectral notation
- Normalized phase volume and almost-everywhere assertions
- A simple nonsynchronizing family approaching eleven sixteenths
- Dedekind completeness: the least-upper-bound property
- Degree large-deviation rate function and roots
- Entropy rates, inverse branches and shifted root contracts
- Graph Kuramoto differential equations
- Analytic potentials on a finite phase torus
- Differentiation of symmetric periodic pair potentials
- Point convergence for compact analytic periodic gradient flows
- A null basin for all strict saddles of a smooth compact gradient flow
- Twisted cycle equilibria and transverse positivity
- A positive transverse Hessian gives an open nonsynchronous basin
- Trees, forests, leaves and isolated vertices
- Deleting any edge of a tree separates it into exactly two tree components
- Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis
- The zero sets of sine and cosine and the least positive common period 2 pi
- Quarter-turn values and shifts by pi/2 and pi
- A rational entropy test above the critical degree ratio two
- Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm
- The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t
- On an interval I, for f continuous on I and differentiable at every interior point: f' ≥ 0 throughout gives f nondecreasing, f' > 0 gives f increasing, f' ≤ 0 and f' < 0 give the two decreasing forms; conversely a nondecreasing f has f' ≥ 0 and a nonincreasing f has f' ≤ 0 wherever it is differentiable, and no strict converse is claimed
- Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on [a,b] takes every value between f(a) and f(b)
- Logarithmic tangent bound and a quadratic entropy envelope
- Divergence to +∞ and to -∞
- Limits at +∞ and -∞, and infinite limits at a point
- Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function
- A composite of continuous functions is continuous, with no side hypothesis of the kind the composition of limits needs
- Nonnegative real square roots exist uniquely
- A negative adjacency eigenvalue survives all-ones centering
- Real powers for positive bases, with the zero-base positive-exponent convention
- The natural logarithm as the inverse of the exponential function
- The exponential function is strictly increasing
- Basic identities for a probability measure
- Probability measures and probability spaces
- Finite vertex Hilbert space in the article convention
- The operator norm is zero on the zero domain and otherwise is max_‖ v‖=1‖ Tv‖
- Transfer of the analytic gradient inequality to a smooth metric
- A critical cluster point for a compact gradient trajectory
- The local gradient inequality bounds trajectory length
- Trapping a gradient tail near a critical cluster point
- Riemannian distance on a connected manifold
- Riemannian distance is a metric
- The riemannian distance topology is the manifold topology
- Riemannian speed and length
- Two-sided spectral expander — Definition 1.9
- Oriented edge count between sets
- Internal-edge mixing bound — Lemma 3.3
- Mixing for two nearby nested sets — Lemma 3.7
- Centered graph matrices and reference degree
- Loewner order, Hermitian inner product, and operator norm
- Independent-copy symmetrization matrix
- Gaussian comparison matrix in the moment argument
- Symmetrization and moments of the adjacency difference
- Trace-moment comparison with Gaussian matrices
- Gaussian trace/norm estimate attributed to Slepian comparison
- The Axiom of Choice
- Mean value theorem on a closed real interval
- Sums, scalar multiples, products and quotients: (f+g)'(c) = f'(c) + g'(c), (α f)'(c) = α f'(c), (fg)'(c) = f'(c)g(c) + f(c)g'(c), and (f/g)'(c) = (f'(c)g(c) - f(c)g'(c))/g(c)² when g(c) ≠ 0
- Angular order and elementary inverse-trigonometric bounds
- Sphere configurations and tangent variations
- Galina Sidorenko and Johan Thunberg, Attracting Rotations Obstruct Almost-Global Consensus on Spheres over Strongly Connected Graphs (arXiv:2610.28341v1, 20 October 2026) · PDF p. 1, Section 1, equation (1); p. 2, Section 2.1, equation (3) and volume convention; p. 9, Section 6, final paragraph
- Johan Markdahl, Johan Thunberg and Jorge Gonçalves, Almost Global Consensus on the n-Sphere (arXiv:1609.04885v4) · PDF p. 2, System 1; p. 3, Algorithm 3, Definitions 4–5, Algorithm 8 and Remark 9; p. 5, Theorem 13
- Finite probability spaces, outcome weights, events, and event probabilities
- Adjacency spectral expander — Definition 1.3
- Entropy endpoint limits and the two monotone branches
- Differentiable entropy inverse branches and shifted contracts
- Sum, product, quotient, and chain rules from difference quotients
- Cᵏ Euclidean maps and diffeomorphisms
- Cᵏ maps and multi-index derivative notation in Euclidean space
- Gradient and Hessian formulas
- Real and complex equilibrium identities
- Uniform random regular graph convention
- Matthew J. Colbrook, Random cubic graphs admit nonsynchronized stable Kuramoto equilibria (11 September 2026) · Theorem 1, p. 2; Lemma 2 and Sections 3–4, pp. 2–6; packaging note p. 1
- MD-06 repository status and submission record · Resolution dated 11 September 2026; linked submission record
- Carvallo, Groisman and Mitsche, Random 3-regular graphs are not globally synchronizing, v1 · Definition 1.1 and Theorem 1.2, pp. 1–2
- arXiv:2610.09135v2 withdrawal record · Submission history and Comments, 10 October 2026
- Bordenave, A new proof of Friedman’s second eigenvalue theorem and its extension to random lifts · v4, p. 2, uniform simple graph model and Theorem 1
- Johnson, Exchangeable pairs, switchings, and random regular graphs · v5, p. 3, Section 2 and Proposition 1(a); p. 12, Theorem 11
- Colbrook MD-06 submission record: attribution, review boundaries and missing files · Results and boundaries; Original package and missing files
- Erdős–Rényi random graph and high probability
- Earlier thresholds with explicit source quantifiers and a fixed-scale deduction
- Extremal degrees at a fixed logarithmic mean
- Random graph process and connectivity hitting time
- Uniform edge orderings, graph-process times and independent labels
- Simultaneous global synchronization from connectivity
- The random graph process is globally synchronizing · Abstract
- Ramanujan graph convention used in the article
- Daido order parameters
- Diagonal phase rotations and relative-phase coordinates
- Point stability, attraction and stability of a set
- Łojasiewicz inequality and analytic point stability
- Sine and cosine defined by their real power series
- The sum of a real power series is real analytic throughout the open interval determined by its radius
- Real analytic germs in several variables
- Loop-free Bernoulli adjacency and its two centerings
- Johannes Alt, Raphael Ducatez and Antti Knowles, Extremal eigenvalues of critical Erdos-Renyi graphs · Section 2, PDF pp. 4-6: model, Theorem 2.1, definition of beta_l(d) preceding Corollary 2.4, and Corollary 2.4(i)
- Critical PSD phases: quadratic tests and small-alpha mass bound
- Critical PSD phases: kernel inequality and half-circle rigidity
- Kernel phases on two-sided expanders: the two tail-growth estimates
- Strict numerical budgets for the two-stage PSD amplification
- Geometric growth, truncation and total decrement budgets
- Principal inverse sine and inverse cosine
- Signs, monotonicity intervals, and ranges of sine and cosine
- Parity and the Pythagorean identity for sine and cosine
- A rare logarithmic degree occurs with probability of its first-moment order
- The unshifted lower Laplacian bound excludes a boundary degree
- The logarithm grows more slowly than every positive real power
- Regular critical PSD phases: a direct mass bootstrap at 0.0816
- Regular critical PSD phases: kernel and half-circle
- Certified regular amplification excludes a PSD half-circle tail
- Bounds, saturation, and rotation of order parameters
- Truncated sine-square function
- Counts of vertices of each degree
- Elementary estimates used for degree expectations
- Elementary binomial probability envelopes
- Indicator expectations and exceptional-vertex probabilities
- The Binomial count has its usual probability mass function
- The exponential addition formula exp(x+y)=exp(x)exp(y)
- Kernel for stability amplification
- The addition formulas for sine and cosine
- Spectral estimates for phase Fourier sums
- Finite symmetric networks and loop removal
- Connected graphs and connected components defined by the existence of vertex paths
- Walks, closed walks, trails, paths and cycles, with length equal to the number of traversed edges
2 · Logical flowchart
Arrows show item dependencies. A grouped prerequisite node points to items that use one or more of its members; their exact links are listed above.
3 · Items
DefinitionDependency level 7
Asymptotically global synchrony via basin mass
DefinitionDependency level 7
Critical minimum-degree connectivity parameter
DefinitionDependency level 7
Shifted rate roots for quantitative expansion
ExampleDependency level 7
Trees synchronize whereas cycles longer than five do not
LemmaDependency level 7
The deterministic critical degree root stays above nine quarters
LemmaDependency level 7
A fixed outlier gap dominates the critical-source error
LemmaDependency level 7
Finite tail length and point convergence on a compact analytic manifold
LemmaDependency level 7
Relative mixing for a small nested set — Lemma 3.8
LemmaDependency level 7
Expected-norm estimate from the corrected moment comparison
Open problemDependency level 7
Extensions to higher-dimensional phase spheres
Open problemDependency level 7
Synchronization for nonuniform and geometric random graph models
Open problemDependency level 7
Quantitative synchronization speed from random initial conditions
PropositionDependency level 7
Monotonicity, endpoints, and asymmetry of the rate roots
PropositionDependency level 7
Second-order necessary conditions for stable states
RemarkDependency level 7
Historical regular-graph synchronization question and subsequent negative-resolution claims
RemarkDependency level 7
Previous Erdős–Rényi synchronization bounds
RemarkDependency level 7
Extremal-degree asymptotics near connectivity
RemarkDependency level 7
Synchronization at the connectivity hitting time
RemarkDependency level 7
Numerical and structural limits of the regular-graph method
RemarkDependency level 7
Spurious minima obstruct global synchrony
CounterexampleDependency level 8
A near-threshold counterexample to the reported constant-two spectral bound
LemmaDependency level 8
Positive-reference expanders have only synchronized critical PSD phases
LemmaDependency level 8
The exact unshifted expansion event cannot have polynomial failure decay
LemmaDependency level 8
All critical PSD equilibria synchronize on positive regular expanders
LemmaDependency level 8
Centered-adjacency pointwise phase bounds
LemmaDependency level 8
Expected degree-count bound and logarithmic factorial estimate
LemmaDependency level 8
Pointwise comparisons used in kernel stability
LemmaDependency level 8