Alphabeta MathArticle items · 14 — Angle and mass sequences in the amplification argument
Presents 29 items from the article and its immediate prerequisites, covering dependency levels 11–15.
1 · Prerequisites
- Phase tail sets for amplification
- Half-circle lemma — Lemma 2.9
- Adjacency, degrees, graph Laplacian, and spectral notation
- Equilibria, stable states, synchronized states, and spurious minima
- Daido order parameters
- The Axiom of Countable Choice (AC_ω)
- Bounds, saturation, and rotation of order parameters
- Erdős–Rényi random graph and high probability
- Entropy rates, inverse branches and shifted root contracts
- Exact logarithmic binomial masses at bounded distance from an entropy root
- Single-vertex degree laws and the shared-edge dependence
- The Binomial count has its usual probability mass function
- Expectation of a real random variable on a finite probability space
- The restricted spectral estimate needed above the connectivity threshold
- Degree control implies two-sided expansion — Proposition 3.2
- Two-sided spectral expander — Definition 1.9
- General two-sided spectral criterion — Theorem 1.10
- Globally synchronizing graph — Definition 1.1
- The logarithm grows more slowly than every positive real power
- Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm
- Adjacency spectral expander — Definition 1.3
- Truncated sine-square function
- Centered graph matrices and reference degree
- Loewner order, Hermitian inner product, and operator norm
- Phasors, trigonometric identities and periodic derivatives
- Rational bootstrap through the regular endpoint
- Lower bound for the second order parameter — Corollary 2.4
- First and second order parameter inequality — Lemma 2.1
- Oriented edge count between sets
- Kernel bounds on three phase regions
- Strict forms of the nested mixing estimates — Remark 3.9
- Kernel for stability amplification
- Kernel stability condition — Lemma 2.8
- Total incident edge bounds — Lemma 3.6
- Signs, monotonicity intervals, and ranges of sine and cosine
- Parity and the Pythagorean identity for sine and cosine
- Quarter-turn values and shifts by pi/2 and pi
- Finite phase tori, local lifts and the flat metric
- Homogeneous Kuramoto energy
- Positive first order parameter — Corollary 2.2
- Root and sign certificates for parameter-dependent polynomials
- 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)
- 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
- 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
- Signed vector-spin objectives and low-rank Gram factors
- Nonconvex landscapes for Z₂ synchronization and graph clustering are benign near exact recovery thresholds · Abstract
- Geometric growth, truncation and total decrement budgets
- A bilinear centered-adjacency estimate without regularity
- Second-order necessary conditions for stable states
- Regular adjacency expansion supplies two-sided expansion
- Ramanujan graph convention used in the article
- The real exponential function and the number e by a power series
- The natural logarithm as the inverse of the exponential function
- The exponential function is strictly increasing
- Uniform two-sided expansion sufficient for global synchrony
- Vanishing numerical condition for fixed positive Laplacian margins
- The integral logarithm is unbounded above and below
- The integral logarithm is continuous and strictly increasing on (0,∞)
- The integral logarithm L is the published natural logarithm
- Ramanujan and typical regular graphs synchronize for degree at least 600
- Typical regular graphs are Hamiltonian
- Spurious minima obstruct global synchrony
- Trees synchronize whereas cycles longer than five do not
- Uniform random regular graph convention
- The finite union bound
- Graph Kuramoto differential equations
- Normalized phase volume and almost-everywhere assertions
- Refined H mass bound in a regular graph
- Angular order and elementary inverse-trigonometric bounds
- Pi is the first positive zero of sine
- Elementary properties of the function H
- Connectivity in the logarithmic random-graph window
- Asymptotic quantifiers and uniform parameter ranges
- Connected graphs and connected components defined by the existence of vertex paths
- Scoped independent low-rank landscape reports
- Degree large-deviation rate function and roots
- The exact unshifted expansion event cannot have polynomial failure decay
- Integer part: for every real x there is exactly one integer m with m ≤ x < m + 1
- Principal inverse sine and inverse cosine
- The zero sets of sine and cosine and the least positive common period 2 pi
- Double-angle and quadratic power-reduction identities
- The principle of mathematical induction
- Order parameter control of total H mass — Lemma 2.3
- Exact rational replay of the regular amplification constants
- Elementary angle and logarithm estimates used in amplification
- The addition formulas for sine and cosine
- Global rotation equivalence of phase configurations
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 11
Angle and mass sequences in the amplification argument
LemmaDependency level 11
Slack entropy thresholds preserve the quantitative synchronization bound
LemmaDependency level 11
Both tractable inequalities at the regular endpoint
LemmaDependency level 11
Small-set amplification — Lemma 4.2
LemmaDependency level 11
Almost antisymmetry and summed kernel bounds
LemmaDependency level 11
Cubic estimates used for the order parameter bootstrap
Open problemDependency level 11
General amplification for signed optimization landscapes
RemarkDependency level 11
Mass amplification outline and methodological claims
RemarkDependency level 11
Expansion and the distribution of phases in stable states
RemarkDependency level 11
Analytic versus numerical regular-expander bounds — Remark 1.5
TheoremDependency level 11
Erdős–Rényi global synchronization — Theorem 1.2
CorollaryDependency level 12
Destroying cycle spurious minima by bounded-degree edge additions
LemmaDependency level 12
Small-alpha first parameter and regular H mass
LemmaDependency level 12
Large-set amplification — Lemma 4.3
LemmaDependency level 12
Tractable order parameter inequalities — Lemma 2.5
LemmaDependency level 12
Mass-angle lower bound — Lemma 4.1
RemarkDependency level 12
Critical-window gap and vanishing epsilon in the authors’ method
RemarkDependency level 12
Reported progress on benign landscapes and an alternative synchronization proof
TheoremDependency level 12
Quantitative Erdős–Rényi synchrony and the false exact-expansion rate — Theorem 6.2
AlgorithmDependency level 13
Two-stage angle and mass construction
CorollaryDependency level 13
Uniform bound on average H mass — Corollary 2.6
ExampleDependency level 13
Published amplification parameters and bounds — Table 1
Open problemDependency level 13
Useful finite-size synchronization estimates near threshold
RemarkDependency level 13
Refined order estimates for regular graphs — Remark 2.7
CorollaryDependency level 14
Contradiction threshold for amplification
LemmaDependency level 14
Parameter and critical-step estimates in general amplification
LemmaDependency level 14
Mass growth, stopping, and final-angle claims
LemmaDependency level 14
Regular-graph numerical contradiction threshold
LemmaDependency level 15