dedekind.analysis Module Reference

Concepts
concept	dedekind::analysis::IsKernel
concept	dedekind::analysis::IsDifferentialForm
	A mapping from a Tangent Space to a Scalar.
concept	dedekind::analysis::IsHamiltonian
	Formal verification of the energy mapping.
concept	dedekind::analysis::IsPoissonAlgebra
	A Ring equipped with a bracket satisfying the Jacobi Identity.

Classes
class	dedekind::analysis::Dual< F >
	Represents f(x) + f'(x)ε. More...
class	dedekind::analysis::TwoForm< F, N >
	An antisymmetric bilinear form ω(u, v) = -ω(v, u). More...
struct	dedekind::analysis::GaussianKernel< T >
	Gaussian radial basis function (RBF) kernel. More...
struct	dedekind::analysis::ReproducingKernel< DomainSet, T >
class	dedekind::analysis::OneForm< F, N >
	The cotangent vector (The 'Gradient' components). More...
struct	dedekind::analysis::HarmonicOscillator< R >
	Concrete Hamiltonian carrier for the 1-D harmonic oscillator. More...

Typedefs
template<std::floating_point R>
using	dedekind::analysis::PhasePoint = Vector<R, 2>
	Canonical 1D phase-space state (q, p).

Functions
template<typename F>
constexpr F	dedekind::analysis::component (const Dual< F > &d, dedekind::category::At< 1 >)
	Canonical product projections for Dual (the ADL component hook of dedekind::category::π_1 / π_2): 𝔻's storage is val / der, not the pair-like .first / .second the default projection reads, so we provide the overload here. This is what makes Dual satisfy IsProduct uniformly with Complex (P-leg), while value() / derivative() stay as the domain-specific aliases that agree with π_1 / π_2.
template<typename F>
constexpr F	dedekind::analysis::component (const Dual< F > &d, dedekind::category::At< 2 >)
template<detail::IsNumericalBridgeScalar R, typename F> requires std::invocable<F, R>
constexpr R	dedekind::analysis::derivative_at (F &&f, R x, R h=static_cast< R >(1e-5))
	Numerical derivative witness via central difference.
template<detail::IsNumericalBridgeScalar R, typename F> requires std::invocable<F, R>
constexpr R	dedekind::analysis::integral_over (F &&f, R a, R b, std::size_t slices=4096)
	Numerical integral witness via trapezoidal rule (Riemann sum).
template<detail::IsNumericalBridgeScalar R, typename F> requires std::invocable<F, R>
constexpr bool	dedekind::analysis::ftc_part_i_hypotheses (F &&f, R a, R b, std::size_t samples=32)
	FTC Part I bridge: d/dx integral_a^x f(t) dt ~= f(x).
template<detail::IsNumericalBridgeScalar R, typename F, typename Antiderivative> requires std::invocable<F, R> && std::invocable<Antiderivative, R>
constexpr bool	dedekind::analysis::ftc_part_ii_hypotheses (F &&f, Antiderivative &&antiderivative, R a, R b, std::size_t samples=32, R tolerance=static_cast< R >(1e-4))
	Numerical bridge for the sampled Part II hypothesis F' ~= f.
template<detail::IsNumericalBridgeScalar R, typename F> requires std::invocable<F, R>
constexpr bool	dedekind::analysis::ftc_part_i_bridge (F &&f, R a, R x, R tolerance=static_cast< R >(1e-4))
	FTC Part I bridge: d/dx integral_a^x f(t) dt ~= f(x).
template<detail::IsNumericalBridgeScalar R, typename F, typename Antiderivative> requires std::invocable<F, R> && std::invocable<Antiderivative, R>
constexpr bool	dedekind::analysis::ftc_part_ii_bridge (F &&f, Antiderivative &&antiderivative, R a, R b, R tolerance=static_cast< R >(1e-4))
	FTC Part II bridge: integral_a^b f(x) dx ~= F(b)-F(a).
template<detail::IsNumericalBridgeScalar R, typename F, typename Antiderivative> requires std::invocable<F, R> && std::invocable<Antiderivative, R>
constexpr bool	dedekind::analysis::ftc_worked_theorem_chain (F &&f, Antiderivative &&antiderivative, R a, R x, R b, R tolerance=static_cast< R >(1e-4))
	Worked theorem chain combining both FTC directions.
template<detail::IsNumericalBridgeScalar R, typename F> requires std::invocable<F, R>
constexpr std::pair< R, R >	dedekind::analysis::suggest_derivative_step_size (F &&f, R x, R tolerance=static_cast< R >(1e-6))
	Diagnostic: suggests optimal derivative step size via convergence testing.
template<detail::IsNumericalBridgeScalar R, typename F> requires std::invocable<F, R>
constexpr std::pair< std::size_t, R >	dedekind::analysis::diagnose_integral_convergence (F &&f, R a, R b, R convergence_threshold=static_cast< R >(1e-5))
	Diagnostic: validates integral convergence via Richardson extrapolation-style sampling.
template<std::floating_point F, std::size_t N>
constexpr auto	dedekind::analysis::wedge (const OneForm< F, N > &a, const OneForm< F, N > &b)
	The Wedge Product (∧): α ∧ β
template<IsMatrixScalar F, std::size_t N>
constexpr Covector< F, N >	dedekind::analysis::to_covector (const OneForm< F, N > &w)
	Convert a OneForm to its Covector (row-vector) representation.
template<IsMatrixScalar F, std::size_t N>
constexpr OneForm< F, N >	dedekind::analysis::from_covector (const Covector< F, N > &cov)
	Convert a Covector to its OneForm representation.
template<std::floating_point R, std::size_t N>
constexpr R	dedekind::analysis::poisson_bracket (auto &&f, auto &&g, const Vector< R, N > &state)
	The Poisson Bracket: The fundamental operation of Classical Mechanics.
template<std::floating_point R>
constexpr PhasePoint< R >	dedekind::analysis::harmonic_oscillator_closed_form (R q0, R p0, R t, R ω=R{1})
	Closed-form harmonic-oscillator flow at time t.
template<std::floating_point R>
constexpr auto	dedekind::analysis::harmonic_oscillator_curve (R q0, R p0, R ω=R{1})
	Continuum-indexed Hamiltonian trajectory as a Curve.
template<std::floating_point R>
constexpr auto	dedekind::analysis::harmonic_oscillator_leapfrog_path (R q0, R p0, R dt, R ω=R{1})
	Discrete Hamiltonian trajectory via leapfrog/Verlet (infinite path).
template<std::floating_point R>
constexpr auto	dedekind::analysis::harmonic_oscillator_leapfrog_finite_path (R q0, R p0, R dt, std::size_t steps, R ω=R{1})

Variables
constexpr auto	dedekind::analysis::𝔻
	The canonical dual-number universe 𝔻 = 𝔸<Dual<QuadraticReal<2>>, Boole, ℶ_1> — the coat-hanger 𝔻 = Dual(ℝ) over the genuine ℝ = ℚ(√2) (mirroring ℝ and ℂ).
constexpr auto	dedekind::analysis::embed_ℝ_𝔻
	The Birkhoff S leg \(\mathbb{R}\hookrightarrow\mathbb{D}\) over the coat-hanger: QuadraticReal<2> → Dual<QuadraticReal<2>>, \(r\mapsto r+0\varepsilon\) (primal r, tangent 0).

Files
file	src/main/modules/dedekind/analysis/analysis.cppm
	kernels, forms, exterior calculus, Hamiltonian flow.
file	src/main/modules/dedekind/analysis/dual.cppm
	Dual numbers 𝔻 = a + bε with ε² = 0 — forward-mode AD carrier.
file	src/main/modules/dedekind/analysis/exterior.cppm
	The Wedge Product and Symplectic Geometry.
file	src/main/modules/dedekind/analysis/forms.cppm
	The Exterior Algebra (Differential Forms).
file	src/main/modules/dedekind/analysis/ftc.cppm
	Fundamental Theorem of Calculus bridge hooks.
file	src/main/modules/dedekind/analysis/hamilton.cppm
	The Principle of Least Action (Hamiltonian Dynamics).
file	src/main/modules/dedekind/analysis/kernels.cppm
	Kernel primitives – Gaussian and reproducing kernels.