Critical Flow Theory

Geometry of trajectories approaching extreme fluid states

Pack
τ 0.6200 · ΔR 0.20 · resolved

Left: physical collapse — the core shrinks, elongates, spiral accelerates. Right: similarity holds still if the scaling is right.

ℓ_r 0.787 · ℓ_z 0.834 · Γ∼ 1.06 · exploratory h

gold RMWmagenta evencyan oddlime inflow

run clocks

class

D_M

late distance

φ_K

Q axis

Δĥ_clock

R axis

λ̂

floor

D_⊥

2-plane

Λ

Cesàro λ̂

The field is a snapshot until you run the clocks. Evolve is physical time. Run is a log–log fit.

Exclusion space

φ_K × Δĥ_clock. Point scale is 1 − λ̂. S sits at the origin and grows. T sits on φ_K with scale ~ 0.

φ_KΔĥ
φ_K
Δĥ_clock
1 − λ̂
OfforiginQφ_KRΔĥ_clockS1 − λ̂Tλ̂ spoofUCesàro Λ

Floor axis

λ̂ = 1 − 2(ĥ_geom − ĥ_vel)

Recovers the rebound clock of S. Invariants cannot. The Q–R plane cannot. dλ/ds = −2λ(1−λ) on the cartoon.

λ=1s = −log τ
λ̂ late
ODE residual
ε̂
lead (decades)

Run S after Off. The algebra stays quiet. λ̂ falls.

self-check passed · max |λ̂−λ| 1.1e-16

2-plane · I^⊥

a = λ a0 + μ v

λ = 1 is the affine line: β_r = −1/2, |u_r| blows. S walks toward the origin. The shaded half is regularizing. T leaves this plane.

λ=1M00λμ
λ late
μ late
D_⊥
β_r late

Run Off, then S, then T. Off sits at (1, h). S walks the dashed ray. T leaves the page.

self-check passed · Gram det 3.5 · P5–P7 algebraic

Mechanism space

Each point is a sliding-window fit. M0 is the diagonal ĥ_geom = ĥ_vel. Physical collapse should map to a path that approaches that line.

ĥ_geomĥ_vel
ĥ_geom
ĥ_vel
ĥ_in

Hole plot · two channels

K = |u_θ| ℓ_r²/ℓ_z should sit still on the variety (φ_K ≈ 0). ℓ_r|u_r| should sit still if inflow is the heat kernel (B2). Q tilts K. R tilts inflow. Δh sees neither of those tilts by itself.

Run trajectory. Two lines: K (swirl) and ℓ_r|u_r| (inflow / B2).

Layer 8 · intervention catalog

Designed attacks are the only nudges this envelope has. Inverse field perturbation is dark. Not an actuator.

Run a trajectory, then pick a specimen. Off is the origin.

Layer 6 · exclusion

Q lights φ_K. R lights Δĥ_clock. S lights the floor. T spoofs the floor. U oscillates λ̂; Cesàro still blows. Idle is Unresolved.

ĥ_ℓ

ĥ_in

φ_K

Δĥ_clock

Slice (r–z)