Same path, different period array

Two inputs with the same mean μ = (1, 0) — same macroscopic path, same diagonal. But input A oscillates while drifting, and H₂[T]/T oscillates permanently — the sub-diagonal Λ₂(1) exists only along subsequences.
Scale T
—
H₂[T] / T² (A)
—
H₂[T] / T² (B)
—
H₂[T] / T (A)
—
H₂[T] / T (B)
—
Λ₂(1) target (A)
—
α = 1.0
Speed: 3.0
Input A: drift + oscillation

Blow-down ℓT(u) — wiggles shrink, path → straight

KA(t) = (1, α sin 2πt). The oscillation sweeps area on every period, but at Carnot scale the wiggles vanish.
Input B: pure drift

Blow-down ℓT(u) — always straight

KB(t) = (1, 0). No oscillation, no area, straight at every scale.

Diagonal: H₂[T] / T² → 0 for both

Same macroscopic path ⟹ same diagonal Λ₂(2) = 0. Both traces collapse together.
Input A Input B

Sub-diagonal: H₂[T] / T — persistent oscillation

Peel off the leading T² term (it's zero). The remainder H₂[T]/T oscillates permanently with period 1 between 0 and α/(2π) — the full limit Λ₂(1) does not exist. Only along subsequences where cos 2πTj has a limit does Λ₂(1) converge.
Input A: oscillates between 0 and α/(2π) Input B: identically 0

Period arrays (through step 2)

The diagonal (grey) is path-determined and identical for both inputs. The sub-diagonal entry Λ₂(1) (bold) is free — for input A it oscillates permanently and only converges along subsequences. This is the "persistent oscillations" obstruction of line 2323.
Input A (drift + oscillation)
Λ₁(1) = μ = (1, 0)
Λ₂(2) = 0 Λ₂(1) = ?
Input B (pure drift)
Λ₁(1) = μ = (1, 0)
Λ₂(2) = 0 Λ₂(1) = 0