A unified framework in which the operator derivative D = δ*,
defined by duality with the stochastic integral, separates representation from
calculus and exposes representability obstructions across Gaussian,
Banach/stable-Lévy, financial, and AI-facing applications.
PAPER I
The Operator Derivative in Continuous Stochastic Calculus: A Hilbert Energy Space Framework
Defines D = δ* via adjoint identity. Clark-Ocone, variance identity,
and chain rule follow from duality alone. The representer rigidity theorem —
deterministic representers force deterministic volatility — is the genuinely
non-Malliavin headline result.
PAPER II-A
The One-Parameter Banach Factorization for Stable Lévy Processes: Representability Obstructions and Leibniz Defects
Extends the operator-adjoint framework to stable Lévy integrals in
Banach spaces. The one-parameter integral δL(u) = ∫ u dL
only sees jump kernels of the form h(t,z) = utz,
producing a representability obstruction for nonlinear jump-size
functionals. The Lean formalization verifies the abstract Banach-adjoint,
factorization, obstruction, and Leibniz-defect structure.
PAPER II-B
ν-Regular Representations and Singular Obstructions in Stable Lévy Lp Calculus
Continues the stable Lévy analysis at the two-parameter
compensated-Poisson level. The paper separates ν-regular
compensated-Poisson integrands from ν-singular stable-integral terms,
showing why a single Lp(ν)-regular representation space cannot
contain the stable identity integrand h(t,z) = z. The Itô
formula naturally splits into a singular stable-integral term, a regular
jump-defect martingale, and a compensator fluctuation.
PAPER III
The Boundary of Hedgeability: Pricing and Hedging in Volterra-Lévy Markets
First application: mathematical finance. The kernel-weighted Leibniz defect
equals the structurally unhedgeable risk in markets with memory and jumps.
The VRNC (Volterra Risk-Neutral Constraint) governs pricing.
Formally verified in Lean 4 (zero sorry, zero axioms).
PAPER IV
Temporal Memory for Language Models
First AI-facing application: temporal memory for frozen language
models. The model weights remain fixed; adaptation occurs through an
external memory state. The paper studies a normalize–route–rerank
architecture for persistent memory, comparing exponential, power-law,
finite-mixture, and fractional/Mittag-Leffler temporal selectors
across synthetic, MiniLM, matched-scale, contrastive TempLAMA-style,
and structured decoder tests.
PAPER V
Scores and Markov Auxiliary Reverse Sampling for Type II Fractional Brownian Noising: An Operator-Derivative Framework
First operator-derivative score formula for non-Markov noising.
The Type II fBM-driven forward has an explicit Tweedie-type score
with the intrinsic bracket ‖Π D Xt‖2
as denominator, and a Markov auxiliary reverse sampler whose
one-time marginals match. Identifies the K → ∞
GFDM limit at the level of kernels, variances, and pointwise
scores — without asserting path-space reversal of fBM.