Alphabeta MathArticle items · 11 — Entropy rates, inverse branches and shifted root contracts
Presents 28 items from the article and its immediate prerequisites, covering dependency levels 5–6.
1 · Prerequisites
- The natural logarithm as the inverse of the exponential function
- Logarithmic tangent bound and a quadratic entropy envelope
- Entropy endpoint limits and the two monotone branches
- Differentiable entropy inverse branches and shifted contracts
- Parameter bounds giving the signs of shifted entropy roots
- Inward comparison and small-shift limits of entropy roots
- Two-sided spectral expander — Definition 1.9
- Centered graph matrices and reference degree
- Adjacency, degrees, graph Laplacian, and spectral notation
- Loewner order, Hermitian inner product, and operator norm
- Self-adjoint order, norm bounds and diagonal tests
- Adjacency spectral expander — Definition 1.3
- Ramanujan graph convention used in the article
- The fixed-degree Alon–Boppana lower bound
- Regular adjacency spectra and normalized expansion
- Bipartite Ramanujan existence and the centered spectrum
- Support connectivity and the unnormalized Laplacian kernel
- The constant projection and centering decomposition
- Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely
- Oriented edge count between sets
- Cauchy–Schwarz: |⟨ u,v⟩|≤‖ u‖‖ v‖, with equality exactly for linearly dependent vectors
- The operator norm is zero on the zero domain and otherwise is max_‖ v‖=1‖ Tv‖
- Uniform random regular graph convention
- Asymptotic quantifiers and uniform parameter ranges
- Friedman spectral expansion in the uniform simple model
- Ordered eigenvalues and the centered regular norm
- Finite symmetric networks and loop removal
- Loop removal in the weighted Kuramoto calculus
- Covariance under a common positive rescaling of weights
- Weighted spectral bounds control the degree without loops
- Loop-dependent weighted degree envelope
- A complete weighted network distinguishes a loop cap from spectral control
- Hamiltonicity of a uniform fixed-degree regular graph
- Homogeneous Kuramoto energy
- Rotating-frame gradient formulation
- Strict saddle critical point
- A null basin for all strict saddles of a smooth compact gradient flow
- Normalized phase volume and almost-everywhere assertions
- Finite phase tori, local lifts and the flat metric
- Phasors, trigonometric identities and periodic derivatives
- Gradient trajectories and complete compact phase dynamics
- The Axiom of Countable Choice (AC_ω)
- Signed vector-spin objectives and low-rank Gram factors
- A scoped interacting-particle transformer connection
- Triangular Gaussian coordinates: metric and moment control
- Finite vertex Hilbert space in the article convention
- The operator norm as the least bound and as the unit-sphere or unit-ball supremum
- Gaussian entry moments and triangular independence
- The Axiom of Choice
- Expected extreme eigenvalues from finite Gaussian comparison
- Extreme-value Lipschitz bounds and centered norm reduction
- Minkowski's inequality for integrals, including p = ∞
- Monotonicity and nonnegative homogeneity of the nonnegative integral
- Even trace moments transfer from the operator norm
- Random elements and real random variables
- Independent random elements
- Expectation of a nonnegative or integrable random variable
- Probability measures and probability spaces
- Afonso S. Bandeira and Ramon van Handel, Sharp nonasymptotic bounds on the norm of random matrices with independent entries · Lemma 2.2, printed/PDF pp. 6–7; Gaussian concentration display at top of p. 7; triangular matrix convention in Proposition 2.1 and Lemma 2.2 on p. 6
- Uniform edge orderings, graph-process times and independent labels
- Normalized graph Laplacian and its nontrivial spectrum
- Exponential clocks and the uniform edge-order process
- Uniform spectral control transfers past connectivity
- Christopher Hoffman, Matthew Kahle and Elliot Paquette, Spectral gaps of random graphs and applications · Section 6, Theorem 6.1 specialized to k=1 and delta=1/4; process and t0 definitions immediately preceding it (PDF pp. 22–23); compare Theorem 1.3 and following paragraph (PDF p. 3)
- Andrew D. McRae, Benign landscapes for synchronization on spheres via normalized Laplacian matrices · Theorem 3.4, positive implication for normalized Laplacian (PDF p. 10); Theorem 2.1 and complete proof Section 6.1 (PDF pp. 19–22); dynamics interface Section 1.2 (PDF p. 3); process application Section 3.4.2 (PDF p. 11)
- Vishesh Jain, Clayton Mizgerd and Mehtaab Sawhney, The random graph process is globally synchronizing · Theorem 1.2 and preceding process definition (PDF p. 2); Section 1.3 (PDF p. 4); Theorem 1.6 (PDF p. 3) and Sections 2–3 (PDF pp. 4–11)
- Graph Kuramoto differential equations
- Diagonal phase rotations and relative-phase coordinates
- Equilibria, stable states, synchronized states, and spurious minima
- The real exponential function and the number e by a power series
- 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
- Nonnegative real square roots exist uniquely
- For a natural n ≥ 1 the function x ↦ xⁿ is differentiable everywhere with derivative ι(n) x^ n-1; for n = 0 it is the constant 1, with derivative 0; for a natural n ≥ 1 the function x ↦ x⁻ⁿ is differentiable at every x ≠ 0 with derivative -ι(n) x⁻ⁿ⁻¹; consequently every polynomial function is differentiable at every real, with the derivative computed term by term
- 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
- A function differentiable at c is continuous at c
- Internal-edge mixing bound — Lemma 3.3
- Cut-edge mixing bound — Lemma 3.4
- The sum rule: a finite disjoint union is finite with | A ∪ B| = | A| + | B| and |⋃_i ∈ I Aᵢ| = ∑_i ∈ I| Aᵢ|, and a sum over a finite index set splits along a partition
- Nested-cut polarization identity
- Geometric growth, truncation and total decrement budgets
- Empirical phase measures and Fourier order statistics
- Terence Tao, Moser’s entropy compression argument (5 August 2009) · Opening discussion of finite iteration and the second bullet, “The density increment argument”, before the energy increment bullet.
- Single-vertex degree laws and the shared-edge dependence
- Elementary binomial probability envelopes
- The Erdős-Rényi finite random graph G(n,p)
- The Binomial count has its usual probability mass function
- Stirling's formula for factorials
- Chebyshev's inequality on a finite probability space
- Variance and covariance identities for random variables
- The finite union bound
- Gradient and Hessian formulas
- A simple nonsynchronizing family approaching eleven sixteenths
- Dedekind completeness: the least-upper-bound property
- Kassabov, Strogatz and Townsend, Sufficiently dense Kuramoto networks are globally synchronizing, arXiv:2105.11406v1 · Section I, p. 1, Eq. (1) and definition of global synchronization; Section II, p. 2, Eq. (2); Section V, p. 5, Theorem 5 and the paragraph immediately following its proof
- Eduardo A. Canale, From weighted to unweighted graphs in Synchronizing Graph Theory, arXiv:2209.06362v3 · Section II.A, printed p. 4 (strong density); Section V, printed pp. 14–15 (regular degree, parameters, and Theorem 2)
- Erdős–Rényi random graph and high probability
- Chernoff bound for independent bernoulli trials
- The finite product of finite probability spaces
- Product weights normalize, and coordinate events are mutually independent
- Gaussian comparison matrix in the moment argument
- Additivity of the nonnegative Lebesgue integral
- A finite stack of independent Gaussian vectors is jointly Gaussian
- Multivariate normal law, including singular covariance
- Change of variables for expectation
- Continuous functions on Euclidean spaces are Borel measurable
- Michel Ledoux, The Gaussian Poincaré inequality, The Gaussian Blog (v1 February 2024) · Section 1, Theorem 1, equation (3), printed/PDF p. 3
- Afonso S. Bandeira and Ramon van Handel, Sharp nonasymptotic bounds on the norm of random matrices with independent entries · Lemma 2.2, printed/PDF p. 6; proof continues on p. 7; matching triangular convention in Proposition 2.1 on p. 6
- Łojasiewicz inequality and analytic point stability
- Point stability, attraction and stability of a set
- Analytic potentials on a finite phase torus
- Fermat's theorem: an interior differentiable local extremum has zero gradient
- Edge isoperimetric constant convention
- Connected graphs and connected components defined by the existence of vertex paths
- One-vertex composition of phase-difference landscapes
- Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis
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
PropositionDependency level 5
Entropy rates, inverse branches and shifted root contracts
PropositionDependency level 5
Degree control implies two-sided expansion — Proposition 3.2
PropositionDependency level 5
Regular adjacency expansion supplies two-sided expansion
RemarkDependency level 5
Asymptotic optimality and existence background for Ramanujan graphs
RemarkDependency level 5
Connectivity and the lower Laplacian parameter
RemarkDependency level 5
Disjoint-set lower mixing needs additional hypotheses
RemarkDependency level 5
Typical regular graphs are almost Ramanujan
RemarkDependency level 5
Claimed extension to nonnegative weighted graphs
RemarkDependency level 5
Typical regular graphs are Hamiltonian
RemarkDependency level 5
Measure-zero convergence to strict saddles
RemarkDependency level 5
Connections to optimization and transformer dynamics
TheoremDependency level 5
An independent symmetric-Gaussian norm moment input
TheoremDependency level 5
Simultaneous global synchronization from connectivity
DefinitionDependency level 6
Globally synchronizing graph — Definition 1.1
DefinitionDependency level 6
Degree large-deviation rate function and roots
LemmaDependency level 6
A rational entropy test above the critical degree ratio two
LemmaDependency level 6
Total incident edge bounds — Lemma 3.6
LemmaDependency level 6
Mixing for two nearby nested sets — Lemma 3.7
Open problemDependency level 6
Adapting amplification beyond synchronization
PropositionDependency level 6
Extremal degrees at a fixed logarithmic mean
PropositionDependency level 6
Nonnegativity and synchronized global minima
PropositionDependency level 6
Real and complex equilibrium identities
RemarkDependency level 6
Independent inputs for the critical-density bounds
RemarkDependency level 6
Chernoff degree concentration used in Section 6
RemarkDependency level 6
Gaussian trace/norm estimate attributed to Slepian comparison
RemarkDependency level 6
Analytic local minima and Lyapunov stability
RemarkDependency level 6
Cheeger lower bound — Remark 3.5
RemarkDependency level 6