tropical semiring arithmetic
an encyclopedia of the mathematics behind trop's (min, +) algebra — from first principles to applications. every concept is grounded in the semiring we implement: tropical addition is min, tropical multiplication is ordinary addition, and shortest paths are matrix products.
foundations
tropical-semiring — field axioms fail, semiring axioms hold, idempotency changes everything
matrix-algebra — tropical matrices turn shortest paths into linear algebra
algorithms
kleene-star — the Kleene star is to (min, +) what matrix inversion is to (+, x), and Floyd-Warshall computes it
verification — LP duality provides optimality certificates, verified over F_p
context
applications — shortest paths, scheduling, phylogenetics, auction theory, string matching
f2-fp-trop — three arithmetics (F₂, F_p, (min, +)), when to use which, how they compose in a proof system
Dimensions
explanation
rs/docs/explanation
explanation
zheng/docs/explanation
zheng explanations zheng is a polynomial proof system. these documents explain the concepts, design decisions, and historical context. for formal definitions, see reference/. for the hash primitive, see hemera. for the VM whose traces we prove, see nox. reading path pages vision | page | topic |…
kuro/docs/explanation
binary field arithmetic an encyclopedia of the mathematics behind kuro's F₂ tower — from first principles to applications. every concept is grounded in the tower we implement: F₂ → F₂² → F₂⁴ → F₂⁸ → F₂¹⁶ → F₂³² → F₂⁶⁴ → F₂¹²⁸. foundations binary-fields — field axioms in characteristic 2, GF(2) as…
trident/docs/explanation
💡 Trident Explanation [← Documentation Index](/trident/docs/readme) Understanding-oriented. Deep dives into why Trident works the way it does, for readers who want the full picture. 🏗️ Core Architecture | Document | Description | |----------|-------------| |…
genies/docs/explanation
isogeny group action arithmetic an encyclopedia of the mathematics behind genies — supersingular isogenies, class group actions, and the CSIDH construction. every concept connects back to the central operation: the commutative group action of cl(O) on supersingular elliptic curves over F_q.…
nox/docs/explanation
nox explanations conceptual documentation — why nox works the way it does, the design choices behind it, and the consequences that emerge. these pages illuminate the architecture. for formal definitions, see reference/. for task-oriented instructions, see docs/guides/ (when available). pages | page…
bbg/docs/explanation
explanation
hemera/docs/explanation
why Hemera works the way it does design decisions behind the Hemera hash primitive. philosophy why-hemera — eight design principles: permanence, the tree, endofunction, self-reference, identity, unity, beauty, the name the-name — etymology: Hemera in the Protogenoi, genealogy of hash names…
jali/docs/explanation
polynomial ring arithmetic an encyclopedia of the mathematics behind jali's polynomial ring R_q = F_p[x]/(x^n+1) — from first principles to applications. every concept is grounded in the ring we implement: structured vectors of Goldilocks field elements with negacyclic convolution. foundations…
nebu/docs/explanation
finite field arithmetic an encyclopedia of the mathematics behind the Goldilocks field — from first principles to applications. every concept is grounded in the field we implement: p = 2⁶⁴ − 2³² + 1. foundations finite-fields — field axioms, existence and uniqueness, GF(p), characteristic, the…