First-passage race

In plain terms

Suppose a measurement outcome is decided by something physical rather than declared by a rule. That something then has to be specified in detail, and every detail a theory specifies is a detail a critic can call arbitrary, so the first-passage race is CSD's way of settling an outcome physically while leaving as little as possible to choice.

The picture is a race. Each possible outcome carries its own clock, and the clocks run at speeds set by the quantum state, so an outcome the state favours has a fast clock and an outcome it disfavours a slow one, and the outcome that happens is the one whose clock fires first. Set up this way, the race reproduces the quantum probabilities exactly, but it seems to buy that result by choosing how the clocks behave. The clocks have exponential waiting times, which means the chance of a clock firing in the next instant never depends on how long it has already been running, much as a radioactive atom is no more likely to decay for having survived a long time, so you can ask why that law and not some other.

The answer is that no other law works. If the rule that the first clock to fire wins is to reproduce the right proportions for every possible number of outcomes, the waiting times have to be exponential, so it isn't a natural choice or a convenient one but the only one. The argument runs both ways, as the exponential law makes the race work, and any law that makes the race work is exponential.

In CSD

The race is the record layer's order-free way of deciding an outcome. An earlier construction cut an interval into consecutive pieces stacked in a fixed order, which quietly privileged whichever outcome came first in the list, and in the race no outcome is privileged, as each simply has a clock. The clock rates are read off the moment map, the map that extracts the Born weights directly from the geometry of the state space, so the squared amplitudes that set the speeds are geometry rather than an extra input, and the clock law itself is a theorem in the corpus rather than a citation to the probability literature.

Two qualifications travel with the result and should never be dropped. First, the characterisation quantifies over every number of outcomes, as at any one fixed number of outcomes the race constrains only finitely many summary statistics of the waiting-time law and determines nothing on its own. What's forced is the exponential law given that a single clock law serves every case, and that's a commitment the programme already makes, as the behaviour of the clocks doesn't depend on which measurement is being performed.

Second, the result removes a posit and doesn't supply a mechanism. No dynamics is shown to carve the cells of the race, and the earlier attempt to derive them from a mixing flow was retired as mis-specified rather than merely unfinished.

Mathematically

Take independent, identically distributed clock readings xi_j drawn from a common probability law on the reals, and let clock j fire at time xi_j / b_j for a rate vector b with positive entries. The law has the race property when, for every number of clocks and every such rate vector, the probability that clock i fires first is b_i / sum_j b_j. The theorem hasRaceProperty_iff_exists_expMeasure (CsdLean4/Mathlib/Probability/IidClockRace.lean) states that a probability law has the race property if and only if it's an exponential law with some positive parameter.

The forward direction instantiates the race at k clocks for every k, which turns the race condition into a family of moment identities for the unknown law, and atomlessness and support in the positive reals are derived from the race property along the way rather than assumed. The quantification over the number of clocks is load-bearing, as at a fixed number of clocks only finitely many moments are constrained, so what's characterised is the law that serves every number of clocks.

Related entries

• De-isolationNothing collapses, as a measured system stops being isolated, and measurement is what de-isolation dynamics looks like from inside.
• FibreThe arena of CSD is larger than the space of quantum states by exactly one compact factor, and from three dimensions upward the natural ways of doing without that factor stop working.
• Record layerA measurement is over when the world itself carries which outcome happened, and the record layer is where that fact lives, on the arena and not in anyone's notebook.
• Measurement flowA deterministic, volume-preserving flow carries out a measurement, and the volumes of its pointer blocks are the Born weights, for every preparation.
• Single trajectoryThere's one world and it follows one path, so probability lives in how experiments are repeated and not in how the world evolves, and theorems rather than wording keep the two apart.