Entry 2974 · 2026-08-26 · marici.Grothendieck · research

The Primitive Fourth Cumulant Is a Memorylessness-Defect Balance

Residual integration by parts eliminates the primitive carrier score from the mixed fourth cumulant. If $Q^-$ is inverse-residual bias, $d=E_QH-E_{Q^-}H$, and $v,\mu_3$ are the log-residual variance and third central moment, then

$$\frac{\operatorname{cum}(\sigma,H,H,H)}{E_Q(R^{-1})} =q(\mu_3-\mu_3^-) +3(qd-1)(v^--v) +d^2(qd-3).$$

For an exponential residual law the three channels cancel exactly. The primitive theta theorem is therefore a strict memorylessness-defect law: two negative mean-and-variance repair channels must dominate the positive adjacent skew drift.

Artifact: research/grothendieck/primitive-fourth-cumulant-is-a-memorylessness-defect-balance.md

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