Coupling-Induced Rank Transitions in Statistical Manifolds
What's this about?
When systems interact, genuinely new properties become measurable — properties that weren't accessible in either system alone. This paper provides a mathematical definition of that phenomenon.
The key concept is Fisher rank: the number of parameters you can actually distinguish from observations. This is different from:
- Entropy (how many microstates exist)
- Dimensionality (how many degrees of freedom move together)
- Parameter count (how many numbers describe the system)
Fisher rank asks: how many of those parameters actually change what you can see?
When two systems couple, parameters that previously had no observable effect can suddenly become detectable. We call this manifold expansion. The correlation between two oscillators, for instance, is unmeasurable when they're independent (Fisher rank = 0 for that coordinate) but becomes measurable the moment they interact (Fisher rank = 1).
Two checkable conditions trigger this: (1) the constraint-release criterion — coupling moves the system off a constraint surface; (2) the symmetry-breaking criterion — coupling breaks a group invariance. Both are proven to increase Fisher rank by at least one.
Key findings
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Fisher rank = Jacobian rank of the dynamics-to-distribution map (via pullback identity)
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Emergence defined: coordinates with zero Fisher info in isolation, nonzero under coupling
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Constraint-release and symmetry-breaking as checkable conditions for rank increase
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Eigenvalue emergence as operational signature — detectable from time-series data
Citation
Todd, I. (2026). Coupling-Induced Rank Transitions in Statistical Manifolds. Annals of the Institute of Statistical Mathematics (in preparation).
Workflow: Claude Code with Opus 4.5 (Anthropic) for drafting and figures. Author reviewed all content and takes full responsibility.