What: Four candidate weighting routes. The stationary-measure route yields Born-like weighting Pi = |ci|² in the equilibrium regime where |Ψ|² = Pst. The Born rule emerges as the sector measure of the underlying closure process, not as an independent axiom.
The Gap
Admissibility (Paper XXIX) determines which outcomes are possible. It doesn’t determine their relative frequencies. The Born rule gap: given admissible outcomes, what determines Pi?
The standard formulation simply postulates Pi = |ci|². This is empirically impeccable but explanatorily silent. Why squared modulus? Why not some other function of the coefficients? The question is not whether the Born rule works — it does — but whether it can be understood as arising from something deeper.
Probability is not a primitive stochastic ingredient. It is the induced measure over observer-compatible admissible sectors.
The Programme Chain
The crystallisation programme has followed a specific logical sequence, each step building on the previous:
- Structure (Papers I–V) — geometric substrate and constraint architecture
- Dynamics (Papers XII–XIV) — stochastic closure evolution and equilibrium
- Quantum recovery (Papers XV–XXI) — interference, unitarity, Born-adjacent structure
- Observer (Paper XXIX) — observer as constraint substructure
- Measurement (Paper XXIX) — measurement as constraint-induced stabilisation
- Probability (this paper) — Born rule from closure geometry
Step 6 is the natural continuation. With the observer placed inside the dynamics and measurement defined as constraint stabilisation, the remaining question is: what determines the relative weights of the stabilised outcomes?
Outcome Sectors
The observer-admissible set (Paper XXIX) partitions into outcome sectors — connected components or basin neighbourhoods of the crystallisation flow. Each sector corresponds to a distinguishable measurement outcome.
Admissibility is binary: a configuration either satisfies the observer constraint or it does not. But the sectors may have different “sizes” under various natural measures. The question of probability reduces to the question of weighting.
Four Routes to Born Weighting
The paper examines four candidate mechanisms by which outcome sectors acquire weights:
Basin-Volume Measure
Weight = volume of attractor basin under the crystallisation flow. Each outcome sector inherits a weight proportional to the configuration-space volume that flows into it.
Structural but not proven to produce quadratic weighting.
Stability-Depth Weighting
Weight from closure functional depth at basin minimum. Deeper basins (lower F) attract more strongly and receive higher weight.
Related to quadratic form but indirect.
Stationary-Measure Route
In the equilibrium regime (Paper XIV), the stationary distribution is Pst = e−2F/σ². Nelson density |Ψ|² = ρ = Pst. Sector weights inherit the stationary measure.
Yields Pi = |ci|² directly.
Gleason-Type Argument
Once admissible-sector geometry has Hilbert-space structure, quadratic weighting is the unique compatible measure (Gleason’s theorem).
Structural uniqueness complement.
In the equilibrium closure regime, the Born rule need not be introduced as an independent axiom. The Born weighting coincides with the stationary sector measure of the underlying closure process.
The Toy Model
Consider a two-basin system: a configuration space with two stable minima under the closure functional F. The observer admits both outcomes. The stochastic closure dynamics have a stationary distribution Pst that assigns occupation probability to each basin.
In the equilibrium regime, the stationary measure assigns weights proportional to basin occupation under Pst. The deeper basin — the one with lower F at its minimum — accumulates more stationary measure and therefore receives higher probability.
Result: the Born rule is demonstrated in miniature. The probability of each outcome is not imposed externally but inherited from the equilibrium geometry of the closure dynamics. For a superposition ψ = c1φ1 + c2φ2, the sector measures are P1 = |c1|² and P2 = |c2|².
Routes C and D Together
The two strongest routes are complementary, not competing:
- Route C provides the dynamical origin — why the weighting is quadratic. The stationary measure of the stochastic closure dynamics concentrates on basins in proportion to |ci|².
- Route D provides structural uniqueness — quadratic weighting is the only option compatible with Hilbert-space sector geometry (Gleason’s theorem, dimension ≥ 3).
Together they say: the closure dynamics produce a weighting, and the only weighting consistent with the Hilbert-space structure of the admissible sectors is the Born rule. The dynamical route explains why it happens; the structural route explains why nothing else could.
The Equilibrium Assumption
The stationary-measure derivation depends on an equilibrium assumption: the system has reached (or is sufficiently close to) the stationary distribution Pst of the stochastic closure dynamics.
This is analogous to the quantum-equilibrium hypothesis in Bohmian mechanics. The justification: Pst is the unique stationary distribution of the stochastic closure dynamics; any initial distribution converges to it under ergodic evolution. The Born rule is the equilibrium prediction.
What remains open: whether convergence occurs on physically relevant timescales. If the mixing time is short relative to observation, the equilibrium assumption is effectively exact. If not, there could in principle be transient deviations from Born statistics — though none have ever been observed.
What Changes
The standard quantum predictions are not modified. No new experimental signatures are proposed. The Born rule remains Pi = |ci|².
Stated Limitations
- Not a complete, assumption-free derivation of the Born rule
- Depends on the equilibrium assumption and well-separated basins
- No extension to continuous spectra
- No concrete construction from the 600-cell
- No proof that equilibrium is reached in finite time
- No decoherence treatment
- No corrections to the Born rule computed
Probability is not added to measurement. It is the geometry of constraint satisfaction, inherited by the observer.