Alphabeta MathSupporting prerequisites · 02 — A centered adjacency column retains a high-degree lower bound
Presents 19 supporting prerequisites, covering dependency levels 3–10.
1 · Prerequisites
- Adjacency, degrees, graph Laplacian, and spectral notation
- The operator norm is zero on the zero domain and otherwise is max_‖ v‖=1‖ Tv‖
- Centered graph matrices and reference degree
- Erdős–Rényi random graph and high probability
- The logarithm grows more slowly than every positive real power
- The integral logarithm is unbounded above and below
- The integral logarithm L is the published natural logarithm
- The integral logarithm is continuous and strictly increasing on (0,∞)
- The Cauchy-sequence reals are Archimedean
- The Cartesian product A × B := { z ∈ P(P(A ∪ B)) : ∃ a ∈ A ∃ b ∈ B z = (a,b) }
- The standard list e : n → Fⁿ with eᵢ(i) = 1_F and eᵢ(j) = 0_F for j ≠ i is an ordered basis of Fⁿ; hence dim_F Fⁿ = n, and F⁰ is the zero space with basis ∅ and dimension 0
- Eigenvalues, eigenvectors, eigenspaces E_λ(T)=ker(T-λ I), and the spectrum σ_F(T) of an endomorphism
- The complex numbers are algebraically closed
- Every endomorphism of a nonzero finite-dimensional vector space over an algebraically closed field has an eigenvalue
- An endomorphism of an n-dimensional space has at most n distinct eigenvalues
- Real and imaginary parts, complex conjugation, and modulus
- Every nonempty finite set of reals has a maximum and a minimum
- Expectation of a real random variable on a finite probability space
- Variance, standard deviation, and covariance on a finite probability space
- Normalize the logarithmic random graph for nonbacktracking estimates
- Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm
- The natural logarithm as the inverse of the exponential function
- Benaych-Georges, Bordenave and Knowles, Spectral radii of sparse random matrices (19 September 2019) · PDF p. 3, Definition 2.1; p. 4, Assumption 2.4 and Theorem 2.5
- Homogeneous Kuramoto energy
- Finite phase tori, local lifts and the flat metric
- Phasors, trigonometric identities and periodic derivatives
- Angular order and elementary inverse-trigonometric bounds
- The Hessian matrix and critical points of a scalar field
- Positive definite, negative definite, semidefinite, and indefinite quadratic forms
- Finite vertex Hilbert space in the article convention
- Cauchy–Schwarz: |⟨ u,v⟩|≤‖ u‖‖ v‖, with equality exactly for linearly dependent vectors
- Real symmetric quadratic forms tested on phasors
- Gradient and Hessian formulas
- The integral logarithm satisfies L'(x)=1/x for x>0 and L(1)=0
- Florent Benaych-Georges, Charles Bordenave and Antti Knowles, Spectral radii of sparse random matrices (19 September 2019) · Section 2, PDF p. 3 Definition 2.1; PDF p. 4 Assumption 2.4, Theorem 2.5 and Remark 2.6
- Chernoff bounds with the exact degree mean
- Two-sided spectral expander — Definition 1.9
- Quarter-turn values and shifts by pi/2 and pi
- Finite symmetric networks and loop removal
- Connected graphs and connected components defined by the existence of vertex paths
- The addition formulas for sine and cosine
- Parity and the Pythagorean identity for sine and cosine
- The zero sets of sine and cosine and the least positive common period 2 pi
- Transfer of the analytic gradient inequality to a smooth metric
- Energy dissipation along a downward gradient curve
- Every smooth vector field on a compact manifold is complete
- The Axiom of Countable Choice (AC_ω)
- A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value
- In any metric space compactness implies countable compactness and limit point compactness, and each of countable compactness and limit point compactness implies sequential compactness; every implication here is proved without a choice principle
- Riemannian distance on a connected manifold
- Riemannian distance is a metric
- The riemannian distance topology is the manifold topology
- Riemannian speed and length
- Continuous functions have an oriented Riemann integral
- If f is continuous on [a,b] and g is integrable with g ≥ 0, there is ξ ∈ [a,b] with ∫ₐᵇ fg = f(ξ)∫ₐᵇ g
- Continuity and derivatives of positive-base real powers
- The fundamental theorem of calculus and integration by parts
- The chain rule, in one line from Carathéodory: if g is differentiable at c and f is differentiable at g(c), then f ∘ g is differentiable at c with (f ∘ g)'(c) = f'(g(c)) g'(c)
- If f ≤ g on [a,b] and both are integrable then ∫ₐᵇ f ≤ ∫ₐᵇ g; and m(b-a) ≤ ∫ₐᵇ f ≤ M(b-a)
- Real powers for positive bases, with the zero-base positive-exponent convention
- The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents
- The exponential is positive and satisfies exp(-x)=1/exp(x)
- 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
- Every nonempty set bounded below has an infimum
- A composite of continuous functions is continuous, with no side hypothesis of the kind the composition of limits needs
- 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
- Signs, monotonicity intervals, and ranges of sine and cosine
- Adjacency spectral expander — Definition 1.3
- Equilibria, stable states, synchronized states, and spurious minima
- Daido order parameters
- Bounds, saturation, and rotation of order parameters
- Truncated sine-square function
- Loewner order, Hermitian inner product, and operator norm
- Regular adjacency spectra and normalized expansion
- Open ball, closed ball and sphere in a metric space
- Degree large-deviation rate function and roots
- Exact rational replay of the regular amplification constants
- Shifted rate roots for quantitative expansion
- Degree bounds from two-sided expansion — Lemma 3.1
- Refined H mass bound in a regular graph
- First and second order parameter inequality — Lemma 2.1
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
Lemma
A centered adjacency column retains a high-degree lower bound
Lemma
Uniform centered normalization across the full Bernoulli density range
Lemma
A nonbacktracking tail smaller than the required polynomial
Lemma
Regular critical PSD phases: a direct mass bootstrap at 0.0816
Lemma
A parameter-uniform nonbacktracking spectral tail
Lemma
Uniform squared-row control including dense graphs
Lemma
Critical PSD phases: quadratic tests and small-alpha mass bound
Lemma
Regular critical PSD phases: kernel and half-circle
Lemma
A critical cluster point for a compact gradient trajectory
Lemma
The local gradient inequality bounds trajectory length
Lemma
Critical PSD phases: kernel inequality and half-circle rigidity
Lemma
Mean-zero projection of the regular phase vector
Lemma
Critical PSD kernel phases: quantitative tail growth
Lemma
Trapping a gradient tail near a critical cluster point
Lemma
Kernel phases on two-sided expanders: the two tail-growth estimates
Lemma
The unshifted lower Laplacian bound excludes a boundary degree
Lemma
Certified regular amplification excludes a PSD half-circle tail
Lemma
The original shifted roots impose a minimum degree above log n
Lemma