Abstract
<p>Hierarchies form even among near-equals. Their inequality is usually read as meritocracy sorting on ability, as a coordination device a group needs, or as cumulative advantage in which leads compound. We give a falsifiable normal form for one further route, a symmetry-breaking amplifier of position. This is a multiplicative cascade that turns tiny, often capability-irrelevant differences into a large positional dispersion Var(ln rho_eff) &gt;&gt; Var(ln k), where k is capability and rho_eff the density of favourable possibilities the environment makes accessible. On top of the normal form we draw the line the title names, between what such a hierarchy's trajectory can and cannot recover about its own origin. The dynamics (question A, whether the process amplifies or merely diffuses) are identifiable from a trajectory. We fix them with three results: a necessity theorem, a non-monotone dispersion-maximising gain that plain cumulative advantage does not produce, and a sqrt(tau) early-lead law that separates amplification from diffusion on a homogeneous population, and, where entities differ, does so through a contrast rather than the magnitude of the excess. Cumulative advantage is told apart not by the magnitude of early-lead persistence, which it can match, but by the non-monotone gain and by the collapse of that persistence under revocation. The seed (question B, whether what got inflated was capability or position, real skill or amplified noise) is not identifiable. We prove a non-identifiability theorem: the map depends on capability and position only through their sum, and it carries an early fluctuation and an inherited head start through one identical term, so no statistic of the trajectory recovers the split. The informal statement that the amplifier keeps no signature of what it amplified is therefore a theorem rather than a metaphor, and it cannot be refuted by data. Identifying B requires an exogenous handle on capability that does not pass through the amplifier, such as quality fixed by construction, randomised, or a convergent later estimate. That handle returns not a verdict but a manufactured share R = 1 - corr^2. We demonstrate the boundary on one large public dataset. An online-chess cohort of near-equal entry ratings shows an order that looks manufactured: it is decoupled from entry and locks in faster than diffusion, though neither signature establishes as much once the cohort's underlying heterogeneity is acknowledged. Once a convergent skill estimate is admitted, that order is found to be about 60% revealed skill (R ~ 0.4). The remaining behaviours of the model follow. Development requires the feedback, since at zero gain the process is a martingale, and the motion is non-ergodic, with the ensemble advancing while the typical trajectory peaks and declines. The contribution is not that hierarchy is noise. It is a precise account of what a hierarchy's own trajectory can, and cannot, tell us about where it came from.</p>