Exported Modules
module	dedekind.topology
module	dedekind.sets
module	dedekind.sets
module	dedekind.order
module	dedekind.order
module	dedekind.order

Concepts
concept	dedekind::numbers::IsNatural
	Structural concept for a commutative semiring with total order (the intensional ℕ).
concept	dedekind::numbers::IsIntegralScalar
	Numbers-layer alias for the algebra scalar bridge concept.
concept	dedekind::numbers::IsUnsignedIntegralScalar
	Numbers-layer alias for the algebra scalar bridge concept.
concept	dedekind::numbers::IsFloatingScalar
	Numbers-layer alias for the algebra scalar bridge concept.
concept	dedekind::numbers::IsScalar
concept	dedekind::numbers::IsQuaternionScalar
	Scalar type suitable as quaternion components.
concept	dedekind::numbers::RejectsNonFinite
	The rejection contract a saturating backing must offer ℚ's tripwire: `is_finite(i)`, found by ADL, says whether `i` is a point of ℤ or a sentinel (`±ℵ_0` / `NaZ`). `Rational` rejects a non-finite value wherever one can arise (construction, comparison); a saturating carrier without this hook would clamp silently and is not admitted as a backing (see `FractionBacking`).
concept	dedekind::numbers::IsRational
	`Q` is the rational field `Rational<Z>` — the canonical exact, dense, strongly-ordered coefficient carrier. Used to gate the real carriers (`Cut`, `QuadraticReal`): over $\mathbb{Q}$ a nonsquare integer `D` has $\sqrt D\notin\mathbb{Q}$ (a theorem), so those carriers are genuine (canonical representation, sound conjugate/norm) — which fails for a coefficient field that already contains $\sqrt D$ (e.g. the degenerate `QuadraticReal<2, QuadraticReal<2>>`).
concept	dedekind::numbers::IsComplexScalar

Classes
class	dedekind::numbers::DivergencePath< R >
	The escape signal of a Mandelbrot orbit, tagged as a sequence-shape witness: a monotone `{Unknown`,True}-valued `Path<Ternary>` that is structurally absorptive for every parameter `c`. More...
class	dedekind::numbers::EmbedRationalToReal< I >
	The field embedding $\mathbb{Q}\hookrightarrow\mathbb{R} =\mathbb{Q}(\sqrt2)$, $q\mapsto q+0\sqrt2$. More...
struct	dedekind::numbers::LatticeFactory< AmbientSet, N >
	Primary template for lattice factory values. More...
struct	dedekind::sets::ResidueClass< L >
	The class $\{x \in \mathbb{N} \mid x \equiv r \pmod n\}$, read analytically: the materialisation of `ℕ` \| (π `%` `n` `==` `r`). More...
class	dedekind::numbers::Quaternion< R >
	Hamilton's hypercomplex numbers q = a + bi + cj + dk. More...
class	dedekind::numbers::QuadraticReal< D, Q >
	A real of $\mathbb{Q}(\sqrt D)$: $a+b\sqrt D$. More...
class	dedekind::numbers::Rational< Z >
	The Field of Fractions over an Integral Domain Z. More...
struct	dedekind::numbers::PartialAddRational< I >
	Partial addition transform for Rational. More...
struct	dedekind::numbers::PartialMulRational< I >
	Partial multiplication transform for Rational. More...
struct	dedekind::numbers::HonestDivRational< I >
	Division transform for Rational. More...
struct	dedekind::numbers::PartialEmbedIntegerToRational< I >
	Identity and Associativity traits for Rational arithmetic. More...
struct	dedekind::numbers::ReachPath
	The reach indicator's type: a `Path<Ternary>` monotone in `{Unknown`, True} (`True` absorbs), constructible only through `collatz_reach_path`, which guarantees that shape; registered absorptive in `sequences` below, as :mandelbrot's `DivergencePath` is. More...
class	dedekind::numbers::Cut< Q >
	A decidable Dedekind real: a principal rational cut or a rational radical $\pm\sqrt{c}$. More...
struct	dedekind::numbers::LowerCut< Q >
	The datum of a real's lower set: the real itself, χ(q) = `real.contains(q)`. More...
class	dedekind::numbers::Complex< R >
struct	dedekind::numbers::PartialAddComplex< R >
	Partial addition transform for Complex<R>. More...
struct	dedekind::numbers::PartialMulComplex< R >
	Partial multiplication transform for Complex<R>. More...
struct	dedekind::numbers::UnitRoot< N >
	A SYMBOLIC N-th root of unity $\zeta_N^{\,p}$, carried by its exponent `p` ∈ ℤ/N (`Modular<N>`). The unit harmonics $\mu_N=\{z:z^N=1\}$: the torsion subgroup of the unit circle `U(1)`, cut out of ℂ, cyclic of order `N` and $\cong\mathbb{Z}/N$ via de Moivre. More...
struct	dedekind::numbers::Character< N >
	The de Moivre CHARACTER $\chi_m : \mathbb{Z}/N \to \mu_N$, $k \mapsto \zeta_N^{\,mk}$ — the plane wave "as such", carried SYMBOLICALLY in `UnitRoot<N>` (consumes the μ_N layer; `resolve()` collapses to exact ℂ at `N=8`). More...
struct	dedekind::numbers::DiscreteCos< N >
	The exact discrete COSINE $\cos_N(k) = \cos(2\pi k/N) = \Re\,\zeta_N^k$ over the `Modular<N>` angle — the EVEN part of the de Moivre character (`=` the $\tfrac12(I+U)$ projection's real value). More...
struct	dedekind::numbers::DiscreteSin< N >
	The exact discrete SINE $\sin_N(k) = \sin(2\pi k/N) = \Im\,\zeta_N^k$ — the ODD part of the character (`Im` of `character<N>(1)` resolved). An `IsArrow` `Modular<N>` → ℝ, exact at N=8. More...

Enumerations
enum class	dedekind::numbers::KleenePolicy { Inclusive , Exclusive }

Typedefs
template<IsComplexScalar R>
using	dedekind::numbers::OrbitPath = Path<Complex<R>>
using	dedekind::numbers::machine_real_scalar = double
using	dedekind::numbers::default_integer = dedekind::sets::SignedCardinality
	Default signed-integer carrier used by downstream numeric layers (Rational, embeddings). Post-#670-sibling (ℚ retarget): anchored on `sets::SignedCardinality` (saturating ℤ proxy with ±ℵ_0 / NaZ escalation), mirroring how `ℤ` = `𝔸<SignedCardinality>` in :integer. This is the discipline-consistent canonical ℤ carrier — `ℚ` = `𝔸<Rational<default_integer>>` now uses the saturating variant uniformly.
using	dedekind::numbers::C1 = dedekind::sets::ExtensionalCardinal<>
using	dedekind::numbers::Z1 = dedekind::sets::SignedExtensionalCardinal<>
using	dedekind::numbers::SC = dedekind::sets::SignedCardinality
using	dedekind::numbers::Card = dedekind::sets::Cardinality
using	dedekind::numbers::machine_integer = int
template<IsInteger I = default_integer>
using	dedekind::numbers::RationalPolynomial = dedekind::algebra::Polynomial<Rational<I>>
	The polynomial ring over the rationals in one variable, ℚ[x]: `algebra::Polynomial` over the quotient field of the integer carrier `I`.
template<IsRational Q = Rational<default_integer>>
using	dedekind::numbers::LowerSet = Comprehension<𝔸<Q, Boole>, LowerCut<Q>>
	The lower set of a real, $\{q\in\mathbb{Q}\mid q<r\}$, as a set over $\mathbb{Q}$.
template<IsRational Q = Rational<default_integer>>
using	dedekind::numbers::PrincipalRay = Interval<Q>
	The ray $\{q<p\}$ over $\mathbb{Q}$: the lower set of a principal cut, $\mathbb{Q}\hookrightarrow\mathbb{R}$ as the order datum.

Functions
template<IsComplexScalar R>
constexpr auto	dedekind::numbers::euclidean_escape_radius_squared (R escape_radius_squared=R{4})
template<IsComplexScalar R>
constexpr auto	dedekind::numbers::mandelbrot_step (const Complex< R > &c)
template<IsComplexScalar R>
constexpr auto	dedekind::numbers::mandelbrot_orbit (const Complex< R > &c)
template<IsComplexScalar R, typename EscapeCriterion> requires IsEscapeCriterion<EscapeCriterion, Complex<R>>
constexpr auto	dedekind::numbers::orbit_divergence_path (const OrbitPath< R > &orbit, EscapeCriterion criterion) -> DivergencePath< R >
template<IsComplexScalar R, typename EscapeCriterion> requires IsEscapeCriterion<EscapeCriterion, Complex<R>>
constexpr std::optional< std::size_t >	dedekind::numbers::orbit_escape_time (const OrbitPath< R > &orbit, std::size_t max_iter, const EscapeCriterion &criterion)
template<IsComplexScalar R>
constexpr std::optional< std::size_t >	dedekind::numbers::orbit_escape_time (const OrbitPath< R > &orbit, std::size_t max_iter, R escape_radius_squared=R{4})
template<IsComplexScalar R, typename EscapeCriterion> requires IsEscapeCriterion<EscapeCriterion, Complex<R>>
constexpr auto	dedekind::numbers::M_kleene_N (std::size_t max_iter, EscapeCriterion criterion)
template<IsComplexScalar R, typename EscapeCriterion> requires IsEscapeCriterion<EscapeCriterion, Complex<R>>
constexpr auto	dedekind::numbers::M_N (std::size_t max_iter, EscapeCriterion criterion, KleenePolicy policy=KleenePolicy::Inclusive)
template<IsComplexScalar R, typename EscapeCriterion> requires IsEscapeCriterion<EscapeCriterion, Complex<R>>
constexpr auto	dedekind::numbers::M_tower (EscapeCriterion criterion, KleenePolicy policy=KleenePolicy::Inclusive)
template<typename S> requires requires(S&& s) { dedekind::sets::image(embed_𝔹_ℕ_, std::forward<S>(s)); }
constexpr auto	dedekind::numbers::embed_𝔹_ℕ (S &&s)
	Set-level lift of `embed_𝔹_ℕ_:` image of a Boolean set `S` under the canonical mono 𝔹 ↪ ℕ.
template<IsNatural N, std::unsigned_integral U>
constexpr N	dedekind::numbers::embed_unsigned_integral (U v)
	Canonical injection from std::unsigned_integral into any IsNatural domain N via its single-argument constructor.
template<typename L, typename C, std::integral K1, std::integral K2>
constexpr ResidueClass< L >	dedekind::sets::operator& (const 𝔸< Cardinality, L, C > &, const Cmp< Rel::Eq, BinOp< std::modulus<>, Projection< 0 >, Const< K1 > >, Const< K2 > > &c)
	`ℕ` `&` (π `%` `n` `==` `r`): the residue class, decided analytically. A residue outside $[0, n)$ is never met, so `π` `%` `3` `==` `3` is empty. ADL finds it through the `𝔸<Cardinality>` operand.
template<typename Q>
constexpr auto	dedekind::numbers::Sqrt2_Symbolic ()
template<typename R> requires std::regular<R>
constexpr auto	dedekind::numbers::TranscendentalSet ()
	The Transcendental Set (𝕋).
constexpr std::size_t	dedekind::numbers::collatz_rule (std::size_t n)
	The recurrence, spelled out explicitly as a named function (not a lambda): `n` even `↦` `n/2`, `n` odd `↦` `3n+1`. Computed on the native `size_t` shadow, which saturates where `3n+1` would leave the word, the posture of ℕ's proxy (the point-free relation on `Cardinality` saturates to ℵ₀ the same way). The word's top is odd, so it is a fixpoint of the rule: an orbit that reaches it never reaches 1, and the verdict below stays `Unknown` rather than wrapping into a false `True`.
constexpr auto	dedekind::numbers::collatz_orbit (std::size_t n)
	The orbit $\{n\} ; T^{\le N}$ presented as the arrow's iterate — a lazy `Path<std::size_t>`.
constexpr ReachPath	dedekind::numbers::collatz_reach_path (std::size_t n)
	`True` once 1 is present, `Unknown` before — a monotone `{Unknown`,True} absorptive path (the ℕ analogue of Mandelbrot's escape indicator).
constexpr std::optional< std::size_t >	dedekind::numbers::collatz_reach_time (std::size_t n, std::size_t budget)
	The first index $k \le$ `budget` at which the orbit hits 1, else `nullopt` — the efficient materialization (one forward pass).
constexpr Ternary	dedekind::numbers::reaches_1_within (std::size_t n, std::size_t budget)
	The Rosolini-dominance verdict: reaches 1 within `budget` `⟹` `True` (IN); not yet `⟹` `Unknown` (U). Never `False`.
template<std::size_t W, std::size_t B>
constexpr bool	dedekind::numbers::all_reach_1_within ()
	The collapse: the open $\forall$-conjecture, restricted to a finite window $[1, W)$ and a budget `B`, is a decidable compile-time fact.
template<IsRational Q>
constexpr LowerSet< Q >	dedekind::numbers::lower_set (const Cut< Q > &r)
	$r \mapsto \{q<r\}$: the real read as its lower set.
template<IsRational Q>
constexpr PrincipalRay< Q >	dedekind::numbers::principal_ray (const Q &p)
	$p \mapsto \{q<p\}$: the rational embedded as the principal ray, extensionally `lower_set`(Cut<Q>{p}).
template<IsComplexScalar R>
constexpr Complex< R >	dedekind::numbers::conj (const Complex< R > &z)
	Complex conjugate: conj(a + bi) = a − bi.
template<IsComplexScalar R>
constexpr R	dedekind::numbers::dot (const Complex< R > &z, const Complex< R > &w)
	The real inner product on ℂ ≅ ℝ²: dot(z, w) = Re(z · conj w) = re_z·re_w + im_z·im_w.
template<IsComplexScalar R>
constexpr R	dedekind::numbers::abs2 (const Complex< R > &z)
	Squared norm abs2(z) = <z, z> = \|z\|² = re² + im²: exact, √-free.
template<IsComplexScalar R>
constexpr R	dedekind::numbers::euclidean_norm_squared (const Complex< R > &z)
	Squared Euclidean norm on Complex<R>: a synonym for `abs2`, kept so it stays reachable by ADL in `dedekind::numbers` for callers that do not import `dedekind.geometry` (e.g. the Mandelbrot escape test and the set-algebra predicates). Kept squared to avoid a square root in hot paths.
template<unsigned N>
constexpr UnitRoot< N >	dedekind::numbers::unit_root ()
	The primitive N-th root $\zeta_N=\zeta_N^{1}$ (the μ_N generator).
template<unsigned N>
constexpr Character< N >	dedekind::numbers::character (dedekind::morphologies::Modular< N > m)
	The character $\chi_m$ at frequency `m`.
template<unsigned N> requires (N == 8u)
constexpr DiscreteCos< N >	dedekind::numbers::cos_n ()
	$\cos_N = \Re\,\zeta_N^{(\cdot)}$ as an arrow.
template<unsigned N> requires (N == 8u)
constexpr DiscreteSin< N >	dedekind::numbers::sin_n ()
	$\sin_N = \Im\,\zeta_N^{(\cdot)}$ as an arrow.

Variables
template<dedekind::numbers::IsComplexScalar R>
constexpr bool	dedekind::sequences::is_absorptive_sequence_v< dedekind::numbers::DivergencePath< R > > = true
	Opt-in: every `DivergencePath<R>` is absorptive (eventually constant — `True` after escape, or the degenerate constant `Unknown` when bounded). The eventual-constancy is structural, holding for all `c`, so it is a sound type-level registration. See `dedekind::numbers::DivergencePath`.
constexpr EmbedRationalToReal	dedekind::numbers::embed_ℚ_ℝ {}
	The canonical S-leg instance $\mathbb{Q}\hookrightarrow\mathbb{R}$.
constexpr auto	dedekind::numbers::ℝ
	The canonical real-number universe `ℝ` `=` `𝔸<QuadraticReal<2>`, Boole, ℶ_1> — the coat-hanger.
constexpr auto	dedekind::numbers::ℝ_d
	The machine-real ambient: the finite doubles `𝕃<double>`.
template<auto AmbientSet, std::size_t N = 1>
constexpr LatticeFactory< AmbientSet, N >	dedekind::numbers::lattice {}
	First-class lattice value: lattice<ℝ_d>.
constexpr auto	dedekind::numbers::embed_𝔹_ℕ_
	Canonical embedding 𝔹 ↪ ℕ: bool → Cardinality.
constexpr Cardinality	dedekind::numbers::aleph_0 = Cardinality{ℵ_0{}}
	The countably-infinite cardinal $\aleph_0$ as a `Cardinality` value: the saturation point, and the unique fixpoint of `Successor<Cardinality>` (`succ`(ℵ₀) `=` `ℵ₀`). A readable spelling of `Cardinality{ℵ_0{}}`, sibling to `finite_cardinality(n)`.
template<typename T>
constexpr bool	dedekind::numbers::is_transcendental_v = false
	Trait to mark a species as Transcendental (π, e).
constexpr auto	dedekind::numbers::ℤ
constexpr 𝔸< Rational< default_integer >, Boole, ℵ_0 >	dedekind::numbers::ℚ
	The canonical rational-number universe ℚ = 𝔸<Rational<default_integer>> (post-#559).
constexpr auto	dedekind::numbers::embed_ℤ_ℚ_
	Restored variant-layer subalgebra embedding ℤ ↪ ℚ: `SignedCardinality` → `Rational<default_integer>`, $n\mapsto n/1$.
template<IsInteger I = default_integer>
constexpr auto	dedekind::numbers::embed_𝕃_ℚ
	Exact dyadic embedding of the finite doubles into ℚ: `𝕃<double>` → `Rational`.
constexpr auto	dedekind::numbers::collatz
	The Trsk relation $T \subseteq \mathbb{N}\times\mathbb{N}$, spelled point-free as the McCarthy conditional $T = (\text{even} \cap \text{halve}) \cup (\text{odd} \cap \text{triple})$ — no lambda, no graph, the recurrence is the relation, spelled in uniform set-grammar: `&` (meet), `~` (complement), \| (join, the structural `OrPredicate` union of #365). Mirrors the divides relation of §4; the parity test appears exactly once.
constexpr auto	dedekind::numbers::converges_in_1
	`converges_in_1` $= \mathrm{collatz}^{\circ}(1) = \{n \mid \mathrm{collatz}(n) = 1\}$ — the pre-image of the fixed-point target $\{1\}$: the naturals that reach 1 in exactly one step. Point-free, via the converse fibre.
constexpr auto	dedekind::numbers::collatz2 = collatzM >> collatzM
	`collatz2` $= \mathrm{collatz};\mathrm{collatz}$ over the finite $[0,64)$ middle — two steps, the bare relative product, middle inferred from the bound (no argument, no shadow, no graph, no array).
constexpr auto	dedekind::numbers::converged_2
constexpr auto	dedekind::numbers::pending_2 = ~converged_2
	`pending_2` $= \sim$`converged_2` — the naturals not yet captured within two steps: the honest "Unknown", complemented on the domain (unary), not on the pair universe.
template<>
constexpr bool	dedekind::sequences::is_absorptive_sequence_v< dedekind::numbers::ReachPath >
	Opt-in: every `ReachPath` is absorptive (eventually constant: `True` once 1 is seen, or the constant `Unknown`). The shape is guaranteed by its only constructor, so the registration is sound.
constexpr auto	dedekind::numbers::embed_ℝ_ℂ
	The Birkhoff S leg $\mathbb{R}\hookrightarrow\mathbb{C}$ over the coat-hanger: `QuadraticReal<2>` → `Complex<QuadraticReal<2>>`, $r\mapsto (r,0)$.
constexpr auto	dedekind::numbers::root8
	The exact de Moivre exponential $\zeta_8 : \mathbb{Z}/8 \to \mathbb{C}^\times$ over the coat-hanger ℝ = ℚ(√2): $k \mapsto \zeta_8^{\,k} = (\cos\tfrac{2\pi k}{8},\ \sin\tfrac{2\pi k}{8})$.
constexpr Complex< QuadraticReal< 2 > >	dedekind::numbers::ζ8
	The primitive 8th root of unity ζ8 = e^{2πi/8} = ½√2 + ½√2·i, exact in ℂ(ℚ(√2)) — the generator of the de Moivre map `root8`.
constexpr auto	dedekind::numbers::ℂ
	The canonical complex-number universe ℂ = 𝔸<Complex<QuadraticReal<2>>, Boole, ℶ_1> — the coat-hanger ℂ = Cplx(ℝ) over the genuine ℝ = ℚ(√2) (mirroring ℝ and 𝔻). Equality on ℚ(√2)[i] is decidable, so ℂ answers in `Boole`; `ℶ_1` is the cardinality of the ℂ it models, not a logic.

Files
file	src/main/modules/dedekind/numbers/numbers.cppm
	Level -1 to 8: The Numerical Tower (ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ).
file	src/main/modules/dedekind/numbers/boolean.cppm
	Formal definition of the Boolean system 𝔹 within the numeric ontology.
file	src/main/modules/dedekind/numbers/cardinality.cppm
	Re-export of dedekind.sets:cardinality with numbers-level proofs.
file	src/main/modules/dedekind/numbers/collatz.cppm
	Bounded Collatz reachability — a §4 relational exhibit on ℕ.
file	src/main/modules/dedekind/numbers/complex.cppm
	Minimal complex wrapper for experimental reintegration.
file	src/main/modules/dedekind/numbers/constants.cppm
	Reserved for Dedekind-cut realisations of named reals (empty).
file	src/main/modules/dedekind/numbers/cut.cppm
	The genuine real: a decidable Dedekind cut on $\mathbb{Q}$.
file	src/main/modules/dedekind/numbers/integer.cppm
	Minimal number taxonomy concepts for reintegration.
file	src/main/modules/dedekind/numbers/lattice.cppm
	Integer lattices in ℝ_d and ℝ_dⁿ.
file	src/main/modules/dedekind/numbers/mandelbrot.cppm
	Core Mandelbrot recurrence helpers over complex carriers.
file	src/main/modules/dedekind/numbers/natural.cppm
	The Dictionary of Species (The Registry).
file	src/main/modules/dedekind/numbers/quadratic.cppm
	$\mathbb{Q}(\sqrt D)$ — a genuine, decidable quadratic real field.
file	src/main/modules/dedekind/numbers/quaternion.cppm
	The Quaternion Division Ring (ℍ).
file	src/main/modules/dedekind/numbers/rational.cppm
	ℚ — rationals as numerator/denominator pairs.
file	src/main/modules/dedekind/numbers/real.cppm
	The real line: `ℝ` (the coat-hanger — a set-indexed field over the order-complete carrier `QuadraticReal<2>`) and the machine ambient `ℝ_d` over the finite doubles `𝕃<double>`.
file	src/main/modules/dedekind/numbers/real_bridge.cppm
	The Birkhoff S leg $\mathbb{Q}\hookrightarrow\mathbb{R}$: a genuine monic homomorphism into the $\mathbb{Q}(\sqrt2)$ coat-hanger, with computed witnesses, plus its graph $\Gamma=\operatorname{graph}(\text{embed})\subseteq\mathbb{Q}\times \mathbb{R}$ reifying the arrow relationally.
file	src/main/modules/dedekind/numbers/scalars.cppm
	Semirings as the formal algebraic basis for scalar number systems.
file	src/main/modules/dedekind/numbers/symbolic.cppm
	Symbolic constants ($\sqrt{2}$, …) as Dedekind cuts.

Exported Modules

Concepts

Classes

Enumerations

Typedefs

Functions

Variables

Files