Ordinal Patterns of Time-Changed Brownian Motion: Exact Blindness and a Detection Formula

We study the ordinal pattern (permutation) statistics of time-changed Brownian motion, the canonical structure of stochastic-volatility and multifractal asset-price models. Our first results are negative and exact: for any independent volatility clock the sign-pattern probabilities equal their Brownian values identically; and if consecutive equal-length clock increments are exchangeable (as they are for every time-reversible clock, including the Multifractal Random Walk (MRW) at arbitrary intermittency and stationary rough-volatility models without leverage), then all order-3 ordinal pattern probabilities at equal lags take exactly their Brownian values $(\tfrac14,\tfrac18,\tfrac18,\tfrac18,\tfrac18,\tfrac14)$. Gaussianity is not required: the classical universality theorem for random walks with i.i.d. symmetric increments extends verbatim to conditionally independent increments with exchangeable volatilities. These results imply that equal-lag permutation entropy provably cannot detect multifractality or stochastic volatility of any kind, and they explain, from first principles, the "order self-similarity" that Bandt (2020) reported in financial data as a property the time-change structure forces for every reversible clock. Our second contribution is positive: the blindness is broken exactly where the symmetry is, at unequal lags. For the MRW with intermittency $\lambda^2$ and window-length ratio $\kappa$ we prove the closed-form deviation \[p_{132}^{\mathrm{MRW}} - p_{132}^{\mathrm{BM}} \;=\; \lambda^{2}\,\frac{\sqrt{\kappa}}{16\pi} \Bigl[\tfrac{\kappa-1}{\kappa}\log(1+\kappa)-\log\kappa\Bigr] \;+\;O(\lambda^{3}),\] independent of the sampling lag and of the integral scale. The deviation is carried jointly by the fluctuation of the log clock ratio and by the Jensen gap between typical and mean scaling, the defining feature of multifractality, and the latter dominates. All proofs are self-contained and largely elementary, given the exact moment formulas and standard moment bounds for the multifractal measure; in particular, no small-deviation theory for multiplicative chaos is needed. Together, the results give an identifiability theorem: two ordinal statistics separate model classes that moment-scaling estimators provably conflate, with the equal-lag coordinate freely measurable at any sample size and the unequal-lag coordinate a high-frequency phenomenon whose cost is itself diagnostic of the regime it probes.