🌳 Oakfield Operator Calculus

Name: Oakfield Operator Calculus
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Author: Oakfield

Mathematics

1. Ambient Spaces and Notation

Let M be a smooth, σ-compact d-dimensional manifold (e.g. M = ℝ^d or the d-torus 𝕋^d).
Fix n ∈ ℕ. A field is a map u : M → ℂⁿ.
Choose a Banach (or Hilbert) function space X ⊆ (ℂⁿ)^M with the following properties:
- X is continuously embedded in L^∞(M; ℂⁿ) and is a Banach algebra under pointwise multiplication (e.g. Sobolev H^s(M; ℂⁿ) with s > d/2, or C^k with k ≥ 1).
- Composition with smooth diffeomorphisms preserves X.

We write ℱ = X for the field space. (This corresponds to OOC's "field space ℱ" and its role as the state manifold.)

For time-dependent problems we consider u : [0, T] → X. Unless otherwise stated, norms are |·|_X, and D denotes Fréchet derivatives on X.

2. Operators and Their Algebra

2.1 The Operator Category

For fixed X, let Lip_loc(X, X) be the class of (possibly nonlinear) mappings 𝒪 : X → X that are locally Lipschitz on bounded sets. Composition ∘ makes this a noncommutative monoid (associative with identity).

Notation. For linear operators we use the standard End(X) = 𝓛(X). For nonlinear maps we write Lip_loc or Hom(X, Y) when the domain/codomain differ. Pointwise addition and scalar scaling require linearity.

Composition (always). (𝒪₂ ∘ 𝒪₁)(u) = 𝒪₂(𝒪₁(u))
Linear combinations (linear case). For 𝒪₁, 𝒪₂ ∈ End(X) linear, (a𝒪₁ + b𝒪₂)(u) = a 𝒪₁(u) + b 𝒪₂(u).

In the linear setting, End(X) is an algebra under addition and composition; the commutator is

[𝒪₁, 𝒪₂] = 𝒪₁ ∘ 𝒪₂ − 𝒪₂ ∘ 𝒪₁

For nonlinear operators the commutator expression captures order-sensitivity but subtraction may fall outside the class; OOC's "operator algebra 𝒪" should be read as Lip_loc(X, X).

Remark (categorical view). One can form a category O whose single object is X and whose morphisms are Lip_loc(X, X); for multiple field spaces X_i the objects are {X_i} and the morphisms X_i → X_j are admissible maps X_i → X_j. (Matches OOC's "objects = field spaces, morphisms = operators".)

2.2 Canonical Operator Families

All operators below are maps in Lip_loc(X, X) under standard analytic hypotheses; when an operator is unbounded, we point to its generated semigroup/action which lies in that class.

Stimulus (Source) Operators

Fix u₀ ∈ X. Define S_u₀ : X → X by S_u₀(u) = u₀ (source) or S_u₀(u) = u + u₀ (injection). (OOC "stimulus".)

Phase/Feature Operators

A local functional G of finite order:

(G[u])(x) = g(x, u(x), (Du)(x), …, (D^mu)(x))

with g smooth and compatible with X. (OOC "phase feature".)

Analytic (Nemytskii) Transforms

For an analytic f : Ω ⊂ ℂⁿ → ℂⁿ with u(M) ⊂ Ω,

(N_f[u])(x) = f(u(x))

(OOC "analytic warp" includes such pointwise nonlinearities.)

Geometric Warps (Diffeomorphic Action)

For Φ ∈ Diff(M) define the pullback

(P_Φu)(x) = u(Φ(x))

(This realizes OOC's "diffeomorphic warps".)

Combined Analytic Warp

W_f,Φ = N_f ∘ P_Φ, i.e.

(W_f,Φ[u])(x) = f(u(Φ(x)))

(OOC's analytic warp as geometry + nonlinearity.)

Mixers (Bilinear Couplings)

For a, b ∈ ℂ and c ∈ X,

M_a,b,c(u₁, u₂) = a u₁ + b u₂ + c

As a two-input bilinear map X × X → X, one can fix one argument to view a unary operator in Lip_loc(X, X). (OOC's "mixer".)

Linear Dissipative Operators

Closed, densely defined linear A : D(A) ⊂ X → X that is (quasi)-m-dissipative, generating a contraction semigroup (e^tA)_t≥0. The semigroup maps e^tA belong to Lip_loc(X, X). (OOC "linear dissipative".)

Fractional-Memory (Volterra) Operators

For α ∈ (0,1), the Caputo derivative D^α_tu defined via

D^α_t u(t) = (1/Γ(1−α)) ∫₀^t (t−τ)^−α ∂_τu(τ) dτ

or equivalently via convolution with a kernel K_α that is locally integrable and yields bounded Volterra maps on X; with these kernel regularity hypotheses, the associated solution operator sits in Lip_loc(X, X). (OOC "fractional operator".)

Stochastic Perturbations

Additive or multiplicative noise in a (separable) Hilbert setting X = H:

du = ··· dt + Σ(u) dW_t

with W_t a Q-Wiener process and Σ Lipschitz as an H-valued Hilbert–Schmidt map. These hypotheses ensure the noise-induced flow is well-defined; the stochastic term is not a bare X → X endomorphism without this Hilbert structure. (OOC "stochastic operator".)

Neural Operators

A parameterized, measurable map 𝒩_θ : X → X (e.g. continuous on bounded sets), treated abstractly as an element of Lip_loc(X, X). (OOC "neural operator".)

Quantum (Heisenberg) Lifts

On a C*-algebra 𝔄 of observables, derivations δ_H(A) = i[H, A] generate *-automorphism groups; these are the "superoperators" in OOC's quantum extension.

3. Dynamics and the Integrator Functor

Continuity Modes, Limiters, and Diagnostics

OOC instruments every operator with a continuity policy (SimContinuityMode) plus clamp/tolerance metadata. The runtime now exposes:

none / strict (dirty): raw evaluation, only domain checks
clamped (hard limiter): values clipped to [clamp_min, clamp_max]
limited (soft limiter): blends analytic form with asymptotic caps inside a tolerance window

Limiter catalogue. Each continuity mode is implemented by composing the following limiter strategies:

hard clip: deterministic clamp to [clamp_min, clamp_max]
soft clip: convex blend between raw output and clamped range inside the tolerance window
sigmoid compression: smoothstep/sigmoid remap that suppresses runaway growth before clamps engage
adaptive bias: nudges the operator bias based on recent tolerance violations
stat-aware throttling: shrinks/expands clamps using SimFieldStats (RMS / max tracking)
per-field overrides: honours per-field CLI/API overrides so critical fields stay stabilized

Continuity Mode	Strategies Used
`none`	none
`strict`	per-field overrides only (trust the operator; diagnostics still record dirtied fields)
`clamped`	hard clip · stat-aware throttling · per-field overrides
`limited`	soft clip · sigmoid compression · adaptive bias · stat-aware throttling · per-field overrides

3.1 Deterministic Evolution

A (nonautonomous) vector field on X is a map F : [0, T] × X → X that is measurable in t and locally Lipschitz in u. The evolution equation

u̇(t) = F(t, u(t)), u(0) = u₀ ∈ X

has a unique (maximal) solution u ∈ C([0, T*); X) by the Picard–Lindelöf theorem on Banach spaces (local theory).

Define the integrator functor

ℐ : Vect(X) → Flow(X), F ↦ Φ_F

where Φ_F(t) ∈ Lip_loc(X, X) is the solution operator Φ_F(t)u₀ = u(t). Composition in time corresponds to composition in that monoid, Φ_F(t + s) = Φ_F(t) ∘ Φ_F(s) when F is autonomous. (This formalizes OOC's "integrators as functors that map operators into flows".)

3.2 Stochastic Evolution

On a separable Hilbert space H, consider the Itô SDE

du(t) = F(t, u(t)) dt + Σ(t, u(t)) dW_t

Under standard Lipschitz/linear-growth hypotheses with Σ Hilbert–Schmidt (and with A dissipative when present), there exists a unique H-valued adapted process with continuous paths in H (mild/strong solution). This extends ℐ to a functor into a category of Markov transition kernels / stochastic flows; the stochastic forcing is not a bare X → X endomorphism outside this Hilbert setting. (OOC's Euler–Maruyama integrator is a numerical realization of this case.)

3.3 Fractional Evolution

For α ∈ (0,1), the Caputo equation

D^α_t u(t) = F(u(t)), u(0) = u₀

is equivalent to the Volterra integral equation

u(t) = u₀ + (1/Γ(α)) ∫₀^t (t−τ)^α−1 F(u(τ)) dτ

which is well-posed under analogous Lipschitz conditions; this yields a non-Markovian flow Φ_F,α(t). (OOC "fractional integrators".)

3.4 Operator Splitting

Let F = A + B, with A and B (possibly unbounded) generators that satisfy the Trotter–Kato hypotheses. Then

(Lie–Trotter) lim_n→∞ (e^tA/n e^tB/n)ⁿ = e^t(A+B) (strongly)

and the Strang scheme has local error O(t³):

(e^tA/(2n) e^tB/n e^tA/(2n))ⁿ = e^t(A+B) + O(t³/n²)

(This matches OOC's operator-splitting layer and integrator toolbox.)

4. Remainders and Sieves (Measurement/Analysis Layer)

4.1 Remainder (Geometric Residue)

For an analytic f and diffeomorphism Φ, define the remainder operator

ℛ_f,Φ[u] := f(u ∘ Φ) − f(u) = N_f(P_Φ u) − N_f(u)

Basic properties:

ℛ_f,id[u] = 0 for all u
If f is affine linear, then ℛ_f,Φ[u] = f(u ∘ Φ − u)
Linearization: let Φ_ε = exp(εv) be the time-ε flow of a smooth vector field v. For g = ∇u · v,
ℛ_{f,Φ_ε}[u] = ε Df(u) g + (ε²/2)(D²f[u](g,g) + Df(u) ∇²u[v,v]) + O(ε³)
The first variation is the Lie derivative of f ∘ u along v; higher-order terms encode geometric curvature signatures of the warp.

4.2 Sieves (Filters/Projections)

A sieve is a bounded linear operator 𝒮_σ : X → X, indexed by a scale/resolution parameter σ > 0. Two standard classes:

Convolution sieves (approximate identities): on M = ℝ^d,

𝒮_σ u = K_σ * u, K_σ(x) = σ^−d K(x/σ), K ∈ L¹, ∫ K = 1

Spectral/projective sieves: 𝒮_σ is an orthogonal projection onto a subspace (e.g. low-pass Fourier modes |ξ| ≤ σ⁻¹).

Given a remainder, one defines complexity metrics

C_f,Φ,σ(u) := |𝒮_σ[ℛ_f,Φ[u]]|_X

quantifying structure at resolution σ.

5. Unified Equation of Motion

OOC's dynamics can be expressed as a single evolution law on X:

∂_t u = (𝒪_analytic + 𝒪_lin/diss + 𝒪_fractional + 𝒪_neural)[u] + 𝒪_stoch[u] Ẇ_t

with appropriate interpretations of each term as elements of Lip_loc(X, X) (deterministic parts locally Lipschitz; stochastic part satisfying Hilbert–space SDE hypotheses).

An ensemble view is obtained by equipping X with a probability measure μ (e.g. a Gibbs-type law) and studying expectations 𝔼_μ[Ψ(u)] of observables Ψ ∈ X*.

6. Numerical Realization

Mathematical statements that justify the integration and splitting strategies OOC employs:

Deterministic well-posedness. If F(t, ·) is (locally) Lipschitz on X uniformly in t, the IVP admits a unique maximal solution; if F = A + B with A a generator of a C₀-semigroup and B globally Lipschitz, mild solutions exist globally.
Stochastic well-posedness. On separable Hilbert H, if F and Σ are Lipschitz with linear growth (and standard coercivity for A), the SDE admits a unique mild solution with finite moments.
Fractional well-posedness. For α ∈ (0,1), if F is Lipschitz, the Caputo/Volterra formulation yields a unique solution u ∈ C([0, T]; X).
Operator splitting. Under Trotter–Kato conditions, Lie–Trotter and Strang splittings converge strongly to the exact flow; Strang is second-order accurate.

7. Categorical Semantics

Let Fld be the category of admissible field spaces and Op the category whose objects are those spaces and whose morphisms are the admissible operators between them.

Let VF be the category of vector fields on objects of Fld, and Flow the category of deterministic or stochastic flows. Then:

The integrator is a functor ℐ : VF → Flow sending F to its flow Φ_F.
The remainder can be viewed as a natural transformation between functors built from N_f and P_Φ:
ℛ_f,Φ := N_f ∘ P_Φ − N_f
i.e., a canonical 2-cell measuring how far N_f fails to commute with the diffeomorphism action P_Φ.

8. How OOC's Operator Families Fit the Formalism

Stimulus, feature, analytic warp, mixer, linear dissipative (via semigroups), stochastic noise (Hilbert setting), fractional memory, neural, quantum/superoperator are all instances of admissible operators in Lip_loc(X, X) or its quantum lift.
Composition in Lip_loc(X, X) mirrors chaining modules; in the linear subalgebra, the commutator quantifies non-commutativity.
Integrator functors implement time evolution (RKF45, Euler–Maruyama, fractional convolution schemes correspond to flows Φ_F).
Remainder–Sieve loop is the pair (ℛ_f,Φ, 𝒮_σ) with complexity observable C_f,Φ,σ(u).

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