Why PC Is the Internality Axiom on the Weight Axis, and Why PC and RC Were Never the Same Kind of Debt
Why PC Is the Internality Axiom on the Weight Axis, and Why PC and RC Were Never the Same Kind of Debt
This paper builds directly on the sequence of recent VERSF papers exploring why quantum mechanics uses the square of an amplitude when calculating probabilities. Earlier work gradually narrowed the problem. The Operational Distinguishability Geometry papers showed that the Born rule is tied to a deeper geometric structure. The finite packing papers then pushed the question further down by asking where that geometry comes from. More recently, Bath or Ledger and No Pre-Individuation and the Seal-Trichotomy reduced the problem to a remarkably small set of possibilities. They showed that many apparently different questions—about distinguishability, refinement, probability, and possibility—were actually connected.
This new paper tackles what had become the last unexplored assumption in that chain: the decomposition-independence principle, or PC. Throughout the previous papers, PC had been treated as the one genuinely new bookkeeping rule required to complete the argument. It was carried forward because it seemed obvious, but it had never been examined directly. The natural question was whether PC was truly a new assumption, or whether it was already hidden inside principles the programme had previously established.
The paper’s central result is that most of PC is not new at all. Earlier work showed that outcomes and refinements must be generated from within the distinguishability capacity of a region rather than being supplied from outside. This was formalised as the Internality Axiom. The new paper observes that if refinements themselves must be internally generated, then the weights assigned to those refinements should also be internally generated. A weight should depend on the thing it weighs, not on the wider context in which it happens to appear. In this sense, decomposition-independence turns out to be the Internality Axiom wearing different clothes. The same principle that prevents externally supplied outcomes also prevents externally supplied weighting rules.
The paper also revisits how probability weights combine when larger structures are split into smaller ones. If a region’s distinguishability capacity is divided into subregions, the total weight should still be recoverable by adding the pieces back together. Rather than treating this as a separate assumption, the paper shows that it follows naturally from the idea that the weight is faithfully measuring the capacity it represents. In other words, the bookkeeping behaves the way it does because that is what a faithful measure is supposed to do.
Perhaps the most important conclusion is conceptual rather than mathematical. The paper argues that PC and Reversible Connectedness (RC) were never the same kind of problem. PC belongs to the “data” side of the theory—it concerns how information, capacity, and weight are assigned. RC belongs to the “dynamics” side—it concerns how refinements move and connect under reversible evolution. Once that distinction is made, the landscape becomes much clearer. The data side is now largely settled: no pre-given outcomes, no externally supplied refinement menus, and no context-dependent weighting rules. What remains is a single question about dynamics.
As a result, the programme has now reduced the Born-rule problem further than before. The remaining obstacle is no longer a collection of assumptions scattered across different parts of the framework. It is one focused question: are admissible refinements connected by reversible motion, or can they remain permanently sealed from one another? That question is what the programme calls Reversible Connectedness. The present paper does not answer it, but it shows that almost every alternative route around it has now been eliminated.
Taken together, the recent papers form a coherent progression. First the programme asked where quantum probability comes from. Then it asked where the underlying geometry comes from. Then it asked whether classical alternatives could survive. Finally, it examined the last unexplored bookkeeping assumption and found that it was largely a consequence of principles already established. The result is that the mystery has become much smaller, sharper, and more clearly defined. The question is no longer “Why does the Born rule exist?” but increasingly “What principle governs the continuity of reversible motion in refinement space?” That is now the key frontier for the programme.