PYTHON · NUMPY

When does a fast/slow split stop paying?

A cascade's break-even hit rate depends on which of these?

Derive the break-even hit rate from the cost ratio alone, then sweep the gate cost and check the formula against measured energy.

01Challenge

A cascade runs a cheap path plus a gate, and escalates to an expensive path when the gate fires. The gate is not free. Before you derive it: what fraction of steps must the cheap path handle for the whole arrangement to beat simply always deliberating?

Predict firstA cascade's break-even hit rate depends on which of these?

02Model

Per step a flat deliberative arm costs C_slow. A cascade costs C_fast plus C_gate always, plus C_slow on the fraction it escalates. Set the second below the first and everything cancels except a ratio: the cascade pays when the escalation rate is below one minus (C_fast plus C_gate) over C_slow. It does not depend on the task, the body or the disturbance. It is arithmetic you can do before building anything.

03Guided practice

Derive the break-even by hand. Then sweep the gate cost and compare the formula's prediction against measured compute energy at every setting.

04Feedback

PASS WHEN You predicted five crossings from two constants and the simulation agreed with all five.

If your run fails, check:

The break-even is (C_fast + C_gate) / C_slow and nothing else enters it. At a gate costing 2% of the slow path it is 2.1%; at 20% it is 20.1%.
The measured hit rate here is 19.8%, so the cascade pays while the gate costs 10% of the slow path and stops paying at 20%. The crossing sits between two adjacent rows, and the formula calls it correctly on both sides.
This is the same shape as speculative decoding, which lowers energy by up to 29% at small batch and RAISES total energy at large batch. An always-on gate is not free, and below break-even you are paying for two systems and using one.

05Retrieve & space

Write the break-even hit rate in terms of the three costs.
Which of task, body and disturbance appears in that formula?
A gate costs 30% of the expensive path and the cheap path handles 20% of steps. Does the cascade pay?

06Mastery & project

Decide, before building, whether a proposed two-tier stack pays on a given body, using only its two path costs and an estimated hit rate.

Take a dual-system robot policy and estimate its two path costs in operations. Compute the hit rate it would need to break even.