Alphabeta MathPhase dynamics and oscillator models
Build phase geometry, analytic gradient dynamics and basin measure; place oscillator assumptions, historical observations and continuum context in their separate stated scopes.
1 · Prerequisites
- Finite phase tori, local lifts and the flat metric
- Łojasiewicz inequality and analytic point stability
- Energy dissipation along a downward gradient curve
- The Riemannian gradient is the metric dual of the differential
- The Axiom of Countable Choice (AC_ω)
- In any metric space compactness implies countable compactness and limit point compactness, and each of countable compactness and limit point compactness implies sequential compactness; every implication here is proved without a choice principle
- A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value
- Continuity and derivatives of positive-base real powers
- The fundamental theorem of calculus and integration by parts
- Riemannian speed and length
- Real powers for positive bases, with the zero-base positive-exponent convention
- The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents
- Mean value theorem on a closed real interval
- Real analytic germs in several variables
- Holomorphic functions of several variables are smooth and their complex derivatives are holomorphic
- Sum, product, quotient, and chain rules from difference quotients
- Normalized phase volume and almost-everywhere assertions
- Gradient trajectories and complete compact phase dynamics
- Local center-stable trapping disc for a time map
- C¹ Euclidean maximal flows, variational dependence and the finite C² upgrade
- Finite matrix exponential series and its identities
- Assuming countable choice, every second countable space is Lindelöf
- The image of a lower-dimensional C¹ manifold is null
- An equidimensional C¹ map sends null sets to null sets
- Countable unions and subsets of manifold null sets are null
- A subset of ℝᵐ has Lebesgue outer measure zero if and only if it is null in the sense of countable closed-cube covers
- Null subsets of a smooth manifold
2 · Logical flowchart
Arrows show item dependencies. A grouped prerequisite node points to items that use one or more of its members; their exact links are listed above.
3 · Items
Proposition
Point convergence for compact analytic periodic gradient flows
Proposition