Order-Four Ordinal Patterns of Time-Changed Brownian Motion: Equal-Lag Detection of Reversible Volatility Clocks

The equal-lag order-3 ordinal statistics of a time-changed Brownian motion are exactly blind to every reversible independent volatility clock, so that clock detection there requires unequal lags (Adibi, 2026). We show that this blindness does not persist at order four. All twenty-four order-4 pattern probabilities of a time-changed Brownian motion with an independent clock admit an exact conditional representation as centred trivariate Gaussian orthant probabilities, a sum of three arcsines of the clock-increment correlations. Under a reversible independent clock the available symmetry is the Klein four-group generated by the global sign flip and time reversal. Its action partitions the patterns into eight orbits, equal within themselves but, unlike the order-3 turning patterns, not all pinned to their Brownian values. Two orbits, carrying exactly half the probability mass, remain blind to every stationary reversible clock; each of the other six moves under some such clock, subject to three exact linear relations, and all six move under the multifractal clock. For the lognormal multifractal random measure we prove a closed-form equal-lag deviation, \[p^{\mathrm{MRW}}_\sigma - p^{\mathrm{BM}}_\sigma \;=\; \lambda^2\,\delta_\sigma \;+\; O(\lambda^4),\qquadD_4 \;:=\; \delta_{1342} \;=\; \frac{\sqrt2}{128\pi}\,\log\frac{16}{243},\] the coefficient of largest magnitude, with every orbit coefficient elementary in $\log 2$, $\log 3$ and surds. By exact stochastic scale invariance the deviation is independent of the sampling lag and of the integral scale at every intermittency. The common multifractal drift cancels, the exact opposite of the order-3 mechanism, and the deviation is pure curvature: the Hessian of the orthant probability contracted against the log-clock covariance. Order-4 ordinal patterns therefore detect a reversible multifractal clock at equal lags, a cleaner channel than the order-3 unequal-lag deviation. With optimal contrasts it is about five times cheaper to resolve, whatever the order-3 window ratio, though at plausible intermittencies it remains a high-frequency observable requiring of order $10^5$ patterns. We give a model-free expansion with explicit third-order remainder, sharpened to the rate $O(\lambda^4)$, and an estimator-design analysis showing the orbit-count vector is sufficient. We prove the leading-order expansion and equal-lag detection at every pattern order $n\ge4$, identify two closed-form thresholds across pattern order, and evaluate the order-five coefficients from the Plackett reduction, which predict the orbits a direct simulation resolves.