A unified framework in which the operator derivative D = δ* —
defined by duality with the stochastic integral — yields representation theorems,
Leibniz defects, and information-theoretic structure across stochastic calculus,
mathematical finance, and AI memory architecture.
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
Operator Factorization Beyond Hilbert Spaces: Representability Obstructions and Leibniz Defects for Stable Lévy Processes
Extends the framework to Banach spaces and Lévy processes.
The Leibniz defect Γ(F,G) appears as a structural invariant measuring
the non-Gaussian content — the part of the interaction that has no
stochastic integral representation.
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
Fractional Memory for Attention-Based Systems: Theory, Information Limits, and Implementation
Second application: AI memory. The fractional memory process provides
persistent state with power-law retention, O(N) query cost, and
provable information limits. The Leibniz defect becomes the memory
gap — the irreducible information loss from discontinuous updates.