The claim
Channels. Linearize the dynamics about a trajectory: δẋ = A(t)δx + B_s(t)u + B_a(t)v + Σ(t)ξ, with u the agent’s own input, v an outside party’s and ξ noise. By variation of constants every input reaches the final state through the same Φ(T, t). The Lyapunov spectrum is a property of Φ. What each input can reach is a property of (Φ, B_w), its Gramian W_w(T) = ∫₀ᵀ Φ(T, t)B_wB_wᵀΦ(T, t)ᵀ dt, and its least energy for a displacement Δx in range is Δxᵀ W_w^† Δx (CLAIM-345). So two systems equally chaotic can differ arbitrarily in how control is shared among the three. Over long horizons any channel with a generic component along the leading unstable direction grows at the same rate λ, so the channels differ in their projections onto the unstable directions and in what they reach at finite horizons, not in exponential rate.
One unstable direction. dx = (λx + bu)dt + σ dW, x(0) = 0, constant u.
- Least effort to reach a displacement: W_T = b²(e^{2λT} − 1)/(2λ), which grows with λ, so the cost Δx²/W_T falls.
- Open loop against noise: the terminal mean is bu(e^{λT} − 1)/λ and the variance σ²(e^{2λT} − 1)/(2λ), so the signal-to-noise ratio tends to (bu/σ)√(2/λ) for λT large. It falls as λ grows. Noise is fed in all along, while only the early input is amplified.
Simulated (20,000 paths, T = 8, b = u = σ = 1): λ = 0.5, 1, 2 give ratios 1.98, 1.40, 1.00, against the exact 1.96, 1.41, 1.00 and the limit 2.00, 1.41, 1.00.
So sensitivity lowers the cost of control but does not raise the controller’s share of the outcome against disturbances in the same direction. Feedback that observes the state changes that, which is what Ott, Grebogi and Yorke’s control of chaos does (LIT-1997).
Mode by mode (A28 §4 of the exchange). For ż = −κLz + B_s u + B_e v with L symmetric and eigenvectors w_k, a constant outside input along mode k displaces it by (w_kᵀB_e v)/(κλ_k), so the slowest modes are the most displaced. If w_kᵀB_s = 0 the agent’s own channel cannot reach that mode at all, by the PBH eigenvector test. And a channel of rank one cannot control a system with a repeated eigenvalue, so a single channel cannot control a sheaf with dim H⁰ ≥ 2. Stabilizability and detectability with H⁰ = 0 are NOTE-1222’s Thms 6.1–6.2.
What it does not say
- Anything beyond the linearization. Nonlinear basins, separatrices and slow modes are where an effect can outlast its cause; the claim is local.
- That the agent is the author of what its channel does. A channel’s reach is not whose input it carries (CLAIM-022, QUESTION-071).
- That comparisons of energy across channels are free. As CLAIM-345 says, they need inputs in common units and budgets.
Note of 2026-11-03: the stable side
On a stable direction, dx = (−|λ|x + bu)dt + σ dW, the stationary signal-to-noise ratio is (bu/σ)√(2/|λ|), which rises as a fold is approached (the reader of Scheffer et al., NOTE-1536). So the open-loop ratio goes as 1/√|λ| on both sides of marginal stability: near the fold a constant input is most decisive against noise and cheapest to sustain, and slowing down is where it is. Linear regime only.
Note of 2026-11-03 (later): Kane’s relabelling
Kane’s self-forming action (NOTE-1518, p. 40) makes the noise that hinders each choice the agent’s own competing effort. In this claim’s terms that moves Σ into B_self: two of the agent’s channels, each the other’s disturbance. The share of the outcome against outside noise is then undefined, because there is none. The claim’s results hold wherever some of the noise is exogenous, which neural noise generally is.