Mathematical references

Every page on this list was opened and checked to say what it is credited with here. What the page explains about landas comes from the Kosmacrux book; these references back the standard mathematics it uses.

Mathematics pages

• Klein four-group — Wikipedia. Supports: V₄ is isomorphic to ℤ₂ × ℤ₂ (the direct product of two cyclic groups of order 2), every non-identity element has order 2, it is abelian and it is the smallest non-cyclic group.
• Vierergruppe — MathWorld (Wolfram). Supports: The abstract abelian group of four elements, isomorphic to C₂ × C₂ and to the dihedral D₂, with its multiplication table.
• Klein Four-Group / Cayley Table — ProofWiki. Supports: The Cayley table of V₄: it is symmetric (the group is abelian) and every element is its own inverse. It is the ⊕ table of the operation.
• Elementary abelian group — Wikipedia. Supports: An abelian group in which every non-identity element has the same prime order; the finite ones are (ℤ/pℤ)ⁿ, of order pⁿ. For p = 2 it is called a Boolean group and is a vector space over the field with two elements.
• Elementary abelian 2-group — Groupprops. Supports: It is equivalent to a vector space over the field with two elements, and all its non-identity elements are involutions.
• Cayley table — Wikipedia. Supports: Every row and every column of the table of a finite group is a permutation of its elements; a group is abelian if and only if its table is symmetric about the diagonal.
• Latin square — Wikipedia. Supports: A Latin square is an n × n array with n symbols, each exactly once per row and per column. The tables of finite groups are Latin squares, but not every Latin square comes from a group.
• Exclusive or — Wikipedia. Supports: ({T, F}, ⊕) is an abelian group; the bitwise exclusive disjunction of two n-bit strings is the vector sum in (ℤ/2ℤ)ⁿ; A ⊕ A is zero.
• Nim — Wikipedia. Supports: The nim sum is binary addition without carry (bitwise XOR); the winning strategy is to end every move with a nim sum of zero.
• Principal homogeneous space — Wikipedia. Supports: A torsor is a set on which a group acts freely and transitively: the group with its identity forgotten. An affine space is a vector space with its origin forgotten, and the difference of two points is a vector.
• Hadamard transform — Wikipedia. Supports: The Walsh–Hadamard transform is a generalized Fourier transform, equivalent to a multidimensional DFT of size 2 × 2 × … × 2; its matrix has entries (−1)^(k·n) with the bitwise dot product; it is an involution up to normalization.
• Fourier transform on finite groups — Wikipedia. Supports: Defines the transform of a function on a finite group; in a finite abelian group the irreducible representations have degree 1 and are the characters; over the Boolean group (ℤ/2ℤ)ⁿ it is the Hadamard transform.
• Hypercube graph — Wikipedia. Supports: The vertices are binary strings of n digits, adjacent when they differ in a single digit; there are 2ⁿ vertices and n! · 2ⁿ automorphisms; Q₄ is the graph of the tesseract.
• Hypercube Graph — MathWorld (Wolfram). Supports: Q_n has as vertices the n-bit strings, adjacent if they differ in exactly one coordinate; Q₄ is the graph of the tesseract; it is distance-transitive.

Books

• An Introduction to the Theory of Groups (4th ed.) — Joseph J. Rotman, Springer, Graduate Texts in Mathematics 148, 1995. Supports: The group theory text the Kosmacrux book itself cites (normal subgroups of Sₙ and the Klein group).
• Fourier Analysis on Finite Groups and Applications — Audrey Terras, Cambridge University Press, LMS Student Texts 43, 1999. Supports: Characters and the Fourier transform on finite groups, including Walsh–Hadamard; the Kosmacrux book cites it in its Fourier chapter.
• Nim, a game with a complete mathematical theory — Charles L. Bouton, Annals of Mathematics 3 (14), 1901–1902, pp. 35–39. Supports: The complete theory of the game of Nim, the basis of the nim sum. The citation is taken from the Wikipedia page on Nim.
• Introducción a la teoría de grupos — Fernando Barrera Mora, 2003. Supports: Groups, direct products of groups, permutation groups and Cayley's theorem, p-groups. It is in the project's bibliography; it is not cited in the text.
• Categories for the Working Mathematician — Saunders Mac Lane, Springer, Graduate Texts in Mathematics 5. Supports: It is in the project's bibliography; it is not cited in the text.

The book

Kosmacrux v1 (2026): chapter 2 (The landas, structure theorem), chapter 4 (The operation) and chapter 6 (Fourier). The proofs and the exhaustive check are there.