dedekind.morphologies Module Reference

Concepts
concept	dedekind::morphologies::IsCyclic
	\emph{Operational} duck-typed check that T carries the member API of a cyclic structure: T::Domain alias, T::generator() returning a Domain element, and T::successor(a) walking the chain.
concept	dedekind::morphologies::IsSimplyInfinite
concept	dedekind::morphologies::IsCyclicRing
concept	dedekind::morphologies::IsOrderedField
concept	dedekind::morphologies::IsArchimedeanField
concept	dedekind::morphologies::IsDedekindCompleteField
concept	dedekind::morphologies::IsMinkowskiSummable
concept	dedekind::morphologies::IsInteger
concept	dedekind::morphologies::IsSaturatingInteger
	Saturating-ℤ-shape integer: integer surface + translation-invariant order (sentinels propagate cleanly; no cyclic wrap). The variant ℤ proxy SignedCardinality is the canonical witness.
concept	dedekind::morphologies::IsCyclicInteger
	Cyclic-ℤ-shape integer: integer surface + abelian-group-cyclic under +. SignedExtensionalCardinal<> (sign-magnitude ℤ/2^{N*64}ℤ) is the canonical witness via IsCyclicGroup<SignedExtensionalCardinal<>, std::plus<>> pinned in :sets:cardinality.
concept	dedekind::morphologies::HasEuclideanGcd

Classes
struct	dedekind::morphologies::CauchyPath< T >
class	dedekind::morphologies::CyclicRing< T, N >
	Minimal int-backed \(\mathbb{Z}/N\mathbb{Z}\) carrier from the experimental reintegration phase. Retained for parity with the existing test; Modular<N> below is the broader (auto-typed N) replacement. More...
struct	dedekind::morphologies::Modular< N >
	\(\mathbb{Z}/N\mathbb{Z}\) as a total finite cyclic ring. More...
struct	dedekind::morphologies::Congruence< N, R >
	The residue-class predicate \(x \equiv R \pmod N\): the characteristic map of the fiber of the reduction \(\mathbb{Z} \twoheadrightarrow \mathbb{Z}/N\mathbb{Z} =\) Modular<N> over the residue R. More...
struct	dedekind::morphologies::ModularCongruence< N >
	The binary congruence relation \(x \equiv y \pmod N\) on the integer carrier: the kernel of the reduction \(\mathbb{Z} \twoheadrightarrow \mathbb{Z}/N\mathbb{Z} =\) Modular<N>. More...
struct	dedekind::category::SpeciesTraits< dedekind::morphologies::Modular< N > >
struct	dedekind::morphologies::first_iso_mod_2::mod_2_arrow
	The parity homomorphism mod_2: int → bool. More...
struct	dedekind::morphologies::first_iso_mod_2::connector
	Hand-rolled iso connector Modular<2> → bool. More...
struct	dedekind::morphologies::first_iso_mod_2::connector_inv
	Inverse direction: bool → Modular<2>. More...
class	dedekind::morphologies::safe_float< F >
	Finite IEEE-754 subset: the finiteness invariant makes the order total, so (unlike raw F) this carrier reflexively compares. More...

Typedefs
using	dedekind::morphologies::𝔽ₚ = Modular<18446744073709551557ull>
	The largest prime field of a machine word: \(\mathbb{F}_p\) with \(p = 2^{64} - 59\), the largest prime below \(2^{64}\). The field beside the ring std::uint64_t \(= \mathbb{Z}/2^{64}\).
using	dedekind::morphologies::extensional_integer = int
	Current extensional machine integer carrier.
template<std::floating_point F>
using	dedekind::morphologies::𝕃 = safe_float<F>
	\(\mathbb{L}\langle F\rangle\) — the algebraic handle for the finite-float lattice. Names the (min, max) lattice reduct only; the carrier's (+, x) arithmetic is deliberately not claimed (it is non-associative under rounding). safe_float<F> is the ASCII spelling.

Functions
constexpr bool	dedekind::morphologies::is_prime (std::uint64_t n)
	Whether n is prime: Miller–Rabin with the first twelve primes as witnesses, which decides every 64-bit n (J. Sorenson and J. Webster, "Strong pseudoprimes to twelve prime bases", Math. Comp. 86 (2017), 985–1003). constexpr, so a modulus in a type is certified by the compiler.
template<auto N>
constexpr Modular< N >	dedekind::morphologies::nth_value (std::type_identity< Modular< N > >, std::uint64_t i)
	The i-th element of Modular<N> for a bounded search (sets:spans' search_value): the symmetric representatives in order of size, 0, 1, −1, 2, −2, …
constexpr connector_inv	dedekind::morphologies::first_iso_mod_2::inverse (connector) noexcept
	Free-function inverse hook satisfying :morphism::IsIsomorphism.
constexpr connector	dedekind::morphologies::first_iso_mod_2::inverse (connector_inv) noexcept
template<typename T, typename K, auto N, decltype(N) R, typename L, typename C> requires std::same_as<std::remove_cvref_t<T>, decltype(N)> && std::unsigned_integral<decltype(N)> && (std::has_single_bit( static_cast<std::make_unsigned_t<decltype(N)>>(N)))
constexpr auto	dedekind::morphologies::preimage (const Comprehension< 𝔸< std::pair< T, T >, L, C >, Translation< K > > &g, Congruence< N, R >)
	preimage of a residue class through the translation \(x\mapsto x+K\): again a residue class, shifted. \(\{x \mid x+K \equiv R \pmod N\} = \{x \mid x \equiv R-K \pmod N\}\), the grammar's residue atom π % N == (R−K) mod N over the carrier.
template<std::unsigned_integral U>
constexpr dedekind::sets::Cardinality	dedekind::morphologies::embed_uint_ℕ (U v)
	Universal machine-to-variant lift: std::unsigned_integral → Cardinality. Closes part of #417.
template<typename S> requires ( requires { typename std::remove_cvref_t<S>::Domain; requires std::same_as<typename std::remove_cvref_t<S>::Domain, unsigned>; } || requires { typename std::remove_cvref_t<S>::value_type; requires std::same_as< typename std::remove_cvref_t<S>::value_type, unsigned>; }) && requires(S&& s) { dedekind::sets::image(embed_uint_ℕ_, std::forward<S>(s)); }
constexpr auto	dedekind::morphologies::embed_uint_ℕ (S &&s)
	Set-level lift of embed_uint_ℕ_: image of an unsigned-set S under the canonical mono unsigned ↪ ℕ.
template<IsInteger Z, std::signed_integral S>
constexpr Z	dedekind::morphologies::embed_signed_integral (S v)
	Canonical injection from std::signed_integral into any IsInteger domain Z.
template<IsInteger Z>
constexpr Z	dedekind::morphologies::integer_abs (Z value)
	Absolute value in the integer spine.
template<IsInteger Z>
constexpr Z	dedekind::morphologies::euclidean_gcd (Z lhs, Z rhs)
	Euclidean GCD over an IsInteger carrier.
template<std::signed_integral S>
constexpr dedekind::sets::SignedCardinality	dedekind::morphologies::embed_sint_ℤ (S v)
	Universal machine-to-variant lift: std::signed_integral → SignedCardinality. Closes part of #418.
template<typename S> requires ( requires { typename std::remove_cvref_t<S>::Domain; requires std::same_as<typename std::remove_cvref_t<S>::Domain, int>; } || requires { typename std::remove_cvref_t<S>::value_type; requires std::same_as< typename std::remove_cvref_t<S>::value_type, int>; }) && requires(S&& s) { dedekind::sets::image(embed_sint_ℤ_, std::forward<S>(s)); }
constexpr auto	dedekind::morphologies::embed_sint_ℤ (S &&s)
	Set-level lift of embed_sint_ℤ_: image of an int-set S under the canonical mono int ↪ ℤ.
constexpr unsigned	dedekind::morphologies::abs_int_unsigned (int x) noexcept
	Type-honest absolute value at the machine layer: int → unsigned.
template<std::floating_point F>
constexpr std::optional< safe_float< F > >	dedekind::morphologies::try_safe_float (F x) noexcept
	Free-function spelling of safe_float<F>::lift.

Variables
constexpr auto	dedekind::morphologies::GF_p
	GF_p — the universe over the field, 𝔸 <𝔽ₚ>{}: the twin of Python's GF_p (A(𝔽ₚ)), named for the Galois field GF(p).
constexpr auto	dedekind::morphologies::embed_𝔹_uint_
	Canonical embedding 𝔹 ↪ ℕ: bool → unsigned.
constexpr auto	dedekind::morphologies::embed_uint_ℕ_
	Concrete monic arrow: unsigned ↪ Cardinality (variant ℕ-proxy).
constexpr auto	dedekind::morphologies::embed_sint_ℤ_
	Concrete monic arrow: int ↪ SignedCardinality (variant ℤ-proxy).

Files
file	src/main/modules/dedekind/morphologies/morphologies.cppm
	Cyclic and Archimedean morphology carriers.
file	src/main/modules/dedekind/morphologies/archimedean.cppm
	Archimedean / ordered-field convergence stubs.
file	src/main/modules/dedekind/morphologies/cyclic.cppm
	Cyclic-morphology carriers and their concept witnesses.
file	src/main/modules/dedekind/morphologies/floating_point.cppm
file	src/main/modules/dedekind/morphologies/integral.cppm
file	src/main/modules/dedekind/morphologies/sint.cppm
file	src/main/modules/dedekind/morphologies/uint.cppm