Typicality volume

In plain terms

A deterministic world has no dice, so you can ask where quantum probabilities come from. The classical answer, ignorance of initial conditions, always carried a hidden debt, as which initial conditions count as typical depends on how you measure sets of them, that is, on the rule you use to say how large a set of starting points is, and choosing that rule to get the answer you wanted is circular.

Typicality volume is the debt paid. Fix the one measure the symmetry of the state space allows, then ask what fraction of all the microstates, the complete states of the world, compatible with how the experiment was prepared sits in the region belonging to a given outcome. That fraction is the outcome's weight, and if you run the experiment many times, drawing a fresh microstate each run, the observed frequency converges to it.

No randomness enters the dynamics. The probability lies entirely in the ignorance, and the ignorance is priced by a ruler nobody chose.

In CSD

This is the ontic half of the Born rule, the quantum rule for outcome probabilities. The outcome regions are epistemic, partitions of the sector fixed by the measurement context, but the weight of each region is the Liouville volume of its preimage on the constraint surface, the arena's own volume applied to the set of arena points that project into the region, and conditioned on the preparation region, meaning measured as a fraction of the states compatible with the preparation. Typicality here is repeated-preparation typicality, as each trial draws a fresh microstate from the conditional measure, and the strong law of large numbers, the theorem that averages of independent draws settle to their expected value, does the rest.

The volume reading is proved at full generality, as frequencies converge jointly to the complete Born vector, the full list of outcome probabilities, for every unit preparation and without genericity hypotheses. What remains open is concentration, which would upgrade "the mean is the Born weight" to "a single long run concentrates around it" at an explicit polynomial rate.

Mathematically

The chain runs LF1 to LF4, with i.i.d. preparation sampling, indicator variables per outcome region, the strong law of large numbers, and the identification of the limiting weight with a Fubini-Study volume ratio via the moment map and the Duistermaat-Heckman law.

The general-N statement is unconditional and foundational-triple-only, with no Gleason-type input anywhere on the volume route.

Related entries

• Born ruleQuantum probabilities aren't a law of nature in CSD, as they're the sizes of regions in the space of states, measured with a ruler nobody was free to choose.
• TypicalityTypicality is probability without randomness, as the world is deterministic, you don't know which state you have, and the odds are the sizes of the possibilities.
• Born volume ratioIn every finite dimension the Born weight of an outcome equals a ratio of volumes in the space of quantum states, for every outcome and every state, with no genericity condition attached.
• Outcome regionAn outcome is a region, and the apparatus alone draws its boundary, so the preparation decides where the trajectory lands and doesn't get to draw the map.
• Fubini-Study measureThe space of quantum states allows exactly one way of measuring size that respects its symmetry, and that one measure is what turns regions of the space into Born-rule probabilities.