Alphabeta MathArticle items · 13 — Families with vanishing total spurious-basin measure
Presents 28 items from the article and its immediate prerequisites, covering dependency levels 8–11.
1 · Prerequisites
- Asymptotically global synchrony via basin mass
- Globally synchronizing graph — Definition 1.1
- Equilibria, stable states, synchronized states, and spurious minima
- Graph Kuramoto differential equations
- Point basins, equilibrium-set basins and aggregate volume
- Normalized phase volume and almost-everywhere assertions
- Asymptotic quantifiers and uniform parameter ranges
- The Axiom of Countable Choice (AC_ω)
- The size of the sync basin · p. 1, opening paragraph
- Adjacency, degrees, graph Laplacian, and spectral notation
- Erdős–Rényi random graph and high probability
- Centered graph matrices and reference degree
- Loewner order, Hermitian inner product, and operator norm
- Expected-norm estimate from the corrected moment comparison
- The Axiom of Choice
- The finite product of finite probability spaces
- The set [A]ᵏ of k-element subsets and the binomial coefficient C(n, k) := | [n]ᵏ|
- The factorial n! and the falling factorial n^k̲, defined by recursion in ℕ
- Cauchy-Schwarz |⟨ x,y⟩| ≤ ‖ x‖₂‖ y‖₂ with its equality case, the triangle inequality for ‖·‖₂, the parallelogram law and polarisation
- Real random variables on finite probability spaces and their finite distributions
- Expectation of a real random variable on a finite probability space
- Homogeneous Kuramoto energy
- Rotating-frame gradient formulation
- Phase-locked long-term behavior
- Point convergence for compact analytic periodic gradient flows
- Analytic potentials on a finite phase torus
- Diagonal phase rotations and relative-phase coordinates
- Transfer of the analytic gradient inequality to a smooth metric
- Finite tail length and point convergence on a compact analytic manifold
- Extended riemannian distance on a disconnected manifold
- Connected components, quasicomponents, and totally disconnected spaces
- The components of a space are its maximal connected subsets, they partition it, and each of them is closed
- A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact
- Every smooth vector field on a compact manifold is complete
- The Riemannian gradient is the metric dual of the differential
- Connected graphs and connected components defined by the existence of vertex paths
- Every affine hyperplane of ℝⁿ, and hence every proper linear subspace, is Lebesgue null
- Phasors, trigonometric identities and periodic derivatives
- Trees synchronize whereas cycles longer than five do not
- Classical Erdős–Rényi connectivity threshold
- Critical minimum-degree connectivity parameter
- Independent inputs for the critical-density bounds
- A simple nonsynchronizing family approaching eleven sixteenths
- Degree large-deviation rate function and roots
- Monotonicity, endpoints, and asymmetry of the rate roots
- Extremal-degree asymptotics near connectivity
- Self-adjoint order, norm bounds and diagonal tests
- The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t
- Two-sided spectral expander — Definition 1.9
- Oriented edge count between sets
- Phase tail sets for amplification
- Nested-cut polarization identity
- Internal-edge mixing bound — Lemma 3.3
- Cut-edge mixing bound — Lemma 3.4
- Mixing for two nearby nested sets — Lemma 3.7
- Relative mixing for a small nested set — Lemma 3.8
- Shifted rate roots for quantitative expansion
- Entropy rates, inverse branches and shifted root contracts
- Adjacency spectral expander — Definition 1.3
- Daido order parameters
- Bounds, saturation, and rotation of order parameters
- First and second order parameter inequality — Lemma 2.1
- A near-threshold counterexample to the reported constant-two spectral bound
- The gamma-two graph has a vertex of degree at most log n with high probability
- The original shifted roots impose a minimum degree above log n
- The logarithm grows more slowly than every positive real power
- The Erdős-Rényi finite random graph G(n,p)
- Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree
- Integer part: for every real x there is exactly one integer m with m ≤ x < m + 1
- Normalization, nonnegativity, monotonicity, complements, and differences in a finite probability space
- Truncated sine-square function
- Mean-zero projection of the regular phase vector
- Centered-adjacency pointwise phase bounds
- Real and imaginary parts, complex conjugation, and modulus
- Elementary properties of the function H
- Finite phase tori, local lifts and the flat metric
- Kernel for stability amplification
- Pointwise comparisons used in kernel stability
- Real and complex equilibrium identities
- Local minima and nonnegative second variation
- Cauchy–Schwarz: |⟨ u,v⟩|≤‖ u‖‖ v‖, with equality exactly for linearly dependent vectors
- Counts of vertices of each degree
- Expected degree-count bound and logarithmic factorial estimate
- Sum, product, quotient, and chain rules from difference quotients
- The mean value inequality: if f : [a,b] → ℝᵐ is continuous and differentiable on (a,b) with ‖ f'‖₂ ≤ M, then ‖ f(b)-f(a)‖₂ ≤ M(b-a)
- Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm
- The integral logarithm is unbounded above and below
- The exponential function is strictly increasing
- Strict saddle critical point
- Measure-zero convergence to strict saddles
- Clairaut--Schwarz theorem for continuous second partial derivatives
- Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely
- A fixed-law Wigner spectral-edge input
- Norm symmetrization at the actual matrix mean
- Range-reduced exponential, logarithm and arctangent series have certified tails
- Finite real interval operations enclose their exact values
- Bernoulli random variables and binomial random variables as sums of independent Bernoulli trials
- Pairwise and mutual independence of finite-valued random variables
- Variance, standard deviation, and covariance on a finite probability space
- A complete-graph second-order landscape application
- Ordered edge counts and centered cut vectors
- Lower degree-tail count estimate — Lemma 6.7
- Upper degree-tail count estimate — Lemma 6.8
- Indicator expectations and exceptional-vertex probabilities
- The restricted spectral estimate needed above the connectivity threshold
- The operator norm is zero on the zero domain and otherwise is max_‖ v‖=1‖ Tv‖
- Fermat's theorem: an interior differentiable local extremum has zero gradient
- Pi is the first positive zero of sine
- Signs, monotonicity intervals, and ranges of sine and cosine
- Quarter-turn values and shifts by pi/2 and pi
- Positive-reference expanders have only synchronized critical PSD phases
- The real exponential function and the number e by a power series
- The natural logarithm as the inverse of the exponential function
- Assuming countable choice, Lebesgue outer measure is an outer measure that restricts to elementary volume
- Assuming countable choice, L(ℝⁿ) is a sigma-algebra containing every elementary set and λₙ is a complete measure extending elementary volume
- For every ε > 0 in a complete ordered field there is a natural n ≥ 1 with 1/n < ε
- All critical PSD equilibria synchronize on positive regular expanders
- Ramanujan graph convention used in the article
- Uniform random regular graph convention
- Typical regular graphs are almost Ramanujan
- Uniform constant-two adjacency control above a large logarithmic density
- Uniform degree slack and the vanishing synchrony condition
- Degree control implies two-sided expansion — Proposition 3.2
- The finite union bound
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
Open problemDependency level 8
Families with vanishing total spurious-basin measure
PropositionDependency level 8
Expected centered-adjacency norm — Proposition 6.3
RemarkDependency level 8
Single-point convergence for analytic gradient flows
RemarkDependency level 8
Connectivity is necessary but not sufficient for synchrony
RemarkDependency level 8
Bounds on the critical connectivity threshold
RemarkDependency level 8
Failure of symmetric Laplacian control near threshold
RemarkDependency level 8
Strict forms of the nested mixing estimates — Remark 3.9
RemarkDependency level 8
Shifted-root comparison — Remark 6.10
CorollaryDependency level 9
Positive first order parameter — Corollary 2.2
CorollaryDependency level 9
Failure of the stated constant-two expansion bound — Corollary 6.1
LemmaDependency level 9
Two fixed-parameter expander events have vanishing success probabilities
LemmaDependency level 9
Refined H mass bound in a regular graph
LemmaDependency level 9
Kernel stability condition — Lemma 2.8
LemmaDependency level 9
Order parameter control of total H mass — Lemma 2.3
LemmaDependency level 9
Rounded degree thresholds and expectation bounds
PropositionDependency level 9
Positive-semidefinite-equilibrium criterion for global synchrony
RemarkDependency level 9
The reported centered-adjacency estimate and its sparse-law limit
RemarkDependency level 9
Numerical coefficients and thresholds — Remark 6.4
CorollaryDependency level 10
Lower bound for the second order parameter — Corollary 2.4
ExampleDependency level 10
Two cliques joined by one edge synchronize despite poor expansion
LemmaDependency level 10
Valid restricted spectral tails and a counterexample to the shifted degree-failure conclusion
LemmaDependency level 10
Half-circle lemma — Lemma 2.9
LemmaDependency level 10
Kernel bounds on three phase regions
PropositionDependency level 10
Quantitative two-sided expansion — Proposition 6.9
TheoremDependency level 10
General two-sided spectral criterion — Theorem 1.10
TheoremDependency level 10
Regular spectral expansion implies synchronization — Theorem 1.4
CorollaryDependency level 11
Ramanujan and typical regular graphs synchronize for degree at least 600
CorollaryDependency level 11