A Spectral–Observable–Cone Equivalence Theorem for Nonlinear Cocycles under Monotone Output

This preprint establishes an equivalence theorem linking four properties
of nonlinear dynamical systems observed through a smooth output map:

(P1) a strict Lyapunov spectral gap λ1 > λ2,
(P2) exponential projective contraction in output space,
(P3) absence of strictly invariant cone fields on the dominant Oseledets plane,
(P4) impossibility of orthant confinement under finite-family monotone directional criteria.

Under standard structural assumptions (strong irreducibility and proximality),
these four properties are equivalent.

The key technical result shows that projective contraction in state space
is preserved up to bounded distortion through the output map under a minimal
non-degeneracy condition on Dh, strictly weaker than full observability.

The framework is extended to the degenerate case λ1 = λ2 > λ3,
where a rotation-frequency estimator provides a complete observable
dichotomy between colinearization (ω = 0) and persistent angular exploration (ω ≠ 0),
with equivalence to rank-2 Persistent Excitation.

This work closes the logical cycle initiated in:

Cone certificate — doi:10.5281/zenodo.18789449
FMDC criterion — doi:10.5281/zenodo.18803213