The claim
A9 §2: “Two systems may both be formally controllable but have radically different minimum energies for the same change.” A12’s Proposition 1: “For x_{t+1}=x_t+bu_t, any b≠0 gives one-step controllability. But achieving a fixed displacement Δ with quadratic effort costs E_min=Δ²/b². The required effort is unbounded as b→0, despite unchanged controllability rank.” A15 kept it, adding “proportional to b⁻²”.
Rank is a yes-or-no summary: the system is controllable for every b ≠ 0 and not at b = 0. The cost varies continuously and without bound inside the controllable range. That is the sense in which controllability “isn’t binary” (the owner, U9), and it is why CLAIM-354 counts control cost as a dimension of its own.
Proof
Scalar case. Take x_{t+1} = x_t + b·u_t with b ≠ 0, one step, cost u². Reaching x₁ = x₀ + Δ forces b·u₀ = Δ, so u₀ = Δ/b is the only admissible input, and its cost is Δ²/b². For any two nonzero gains b₁, b₂ the costs of the same Δ stand in the ratio (b₂/b₁)², which is any positive number. As b → 0 the cost diverges. Over T steps the least cost is Δ²/(T b²): the constraint is b·Σu_k = Δ, and by Cauchy–Schwarz Σu_k² ≥ (Σu_k)²/T, with equality at equal inputs. It still diverges as b → 0.
General case. Take x_{t+1} = A x_t + B u_t over T steps, cost Σ‖u_k‖². Then
x_T = A^T x₀ + 𝒞U, 𝒞 = [A^{T−1}B, …, AB, B], U = (u₀, …, u_{T−1}),
so the constraint is 𝒞U = Δx with Δx = x_T − A^T x₀. Write W_T = 𝒞𝒞ᵀ = Σ_{k=0}^{T−1} A^k B Bᵀ (Aᵀ)^k, the finite-horizon controllability Gramian. Its range is the range of 𝒞.
- Δx is reachable exactly when it lies in the range of 𝒞, that is, of W_T.
- Every solution splits as U = U* + N with U* in the range of 𝒞ᵀ and N in the kernel of 𝒞. The two are orthogonal, so ‖U‖² = ‖U*‖² + ‖N‖², and the least cost is at N = 0.
- U* = 𝒞ᵀ W_T^† Δx solves the constraint: 𝒞U* = W_T W_T^† Δx = Δx, since W_T W_T^† projects onto the range of W_T, which contains Δx. It lies in the range of 𝒞ᵀ.
- Its cost is UᵀU = Δxᵀ W_T^† 𝒞𝒞ᵀ W_T^† Δx = Δxᵀ W_T^† W_T W_T^† Δx = Δxᵀ W_T^† Δx.
So E_min = Δxᵀ W_T^† Δx, which is A9’s formula. The scalar case is A = 1, B = b, T = 1, W_1 = b². The rank of W_T says which directions are reachable. Its eigenvalues say what they cost: the cost along a unit eigenvector of W_T is the reciprocal of its eigenvalue. Two systems with Gramians of the same rank can have any eigenvalues, and so any costs.
What it does not say
- Not a fact about units. The cost is for a fixed input and a fixed norm. Put v = b·u and every scalar system costs Δ². A comparison of b₁ with b₂ is meaningful only when u means the same in both, a common unit of effort. That is the modelling step A9 names: “We would need to establish a defensible system model, interventions, units, and cost function before interpreting its values.”
- Not a psychological measure. A9: “Importantly, a Gramian is not automatically a meaningful psychological measure.” The claim is linear algebra. That a habit is costly to change in this sense is CLAIM-354’s interpretive step, not this claim’s.
- Not new. A9: “This is already a standard result from linear control theory.” The record holds it as a proof because it holds no source for it.
- Not constrained or nonlinear control. With bounded inputs a costly displacement may be unreachable in T steps, and nonlinear systems have no Gramian in this form.