Alphabeta MathSpectral estimates and phase landscapes
Supply graph and matrix conventions, centered cut estimates, angular and numerical controls, and graph-specific and vector-spin landscape applications.
1 · Prerequisites
- Range-reduced exponential, logarithm and arctangent series have certified tails
- Machin's formula π/4=4 arctan(1/5)-arctan(1/239)
- 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
- Principal arctangent: derivative, integral, power series, and the Gregory–Leibniz series
- Dedekind completeness implies the Archimedean property
- Finite vertex Hilbert space in the article convention
- Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely
- The operator norm is zero on the zero domain and otherwise is max_‖ v‖=1‖ Tv‖
- The orthogonal projection P_Wv is the W-component in V=W⊕ W^⊥
- Finite symmetric networks and loop removal
- Quadratic form and kernel of a weighted graph Laplacian
- Connected graphs and connected components defined by the existence of vertex paths
- Eigenvalues, eigenvectors, eigenspaces E_λ(T)=ker(T-λ I), and the spectrum σ_F(T) of an endomorphism
- Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space
- Cauchy–Schwarz: |⟨ u,v⟩|≤‖ u‖‖ v‖, with equality exactly for linearly dependent vectors
- The difference a ∖ b, the symmetric difference a △ b, and the complement X ∖ a relative to a set X
- A finite sum in a commutative monoid indexed by an arbitrary finite set
- Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis
- A symmetric matrix operator norm is its largest absolute Rayleigh quotient
- Regular multigraph and normalized adjacency
- Spectral edge and vertex expansion
- Constant vector is a top eigenvector
- Expander mixing lemma
- Conjugation is an involutive real-field automorphism, zz̄=|z|², and modulus is definite, multiplicative, and subadditive
- Sphere configurations and tangent variations
- A mathematical perspective on Transformers · Abstract; Sections 1, 2.2, 7.1 and 7.2, especially equations (SA), (2.6), (7.1) and (7.4)
- Signed vector-spin objectives and low-rank Gram factors
- The intrinsic Hessian of a smooth function at a critical point
- Nonconvex landscapes for Z2 synchronization and graph clustering are benign near exact recovery thresholds · Abstract
- Benign landscape for Burer-Monteiro factorizations of MaxCut-type semidefinite programs · Abstract
- Benign landscapes for synchronization on spheres via normalized Laplacian matrices · Abstract
- Rational sine enclosures from a finite factorial sum
- Principal inverse sine and inverse cosine
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
Rational logarithm and pi enclosures after range reduction
Proposition
Self-adjoint order, norm bounds and diagonal tests
Proposition
The constant projection and centering decomposition
Proposition
Support connectivity and the unnormalized Laplacian kernel
Proposition
A bilinear centered-adjacency estimate without regularity
Proposition
Ordered edge counts and centered cut vectors
Proposition
Regular adjacency spectra and normalized expansion
Proposition
Real symmetric quadratic forms tested on phasors
Remark
A scoped interacting-particle transformer connection
Remark
Scoped independent low-rank landscape reports
Lemma