The Null Result
This is the hardest kind of post to write. The kind where the answer is no.
The claim
A pulsed offset gyro: two counter-rotating assemblies of masses on circular race tracks, where the track centre is offset from the spin axis. Spin speed varies within each revolution — faster through the wide side, slower through the narrow side.
The idea: break time symmetry. Centripetal force is proportional to ω². If the masses move faster when they're furthest from the spin axis, the outward force during the power stroke is larger than the inward reaction on the return. Integrated over a full cycle, the housing feels a net bias in one direction.
That was the claim. It's a reasonable claim. It doesn't obviously violate anything. Worth building a sim to find out.
What we built
A real-time 3D GPU simulation in Taichi — two counter-rotating assemblies, three balls each, offset race geometry computed analytically, pulsed speed profile ω = ω_base × (1 + pulse × cos(3θ)). The visualiser shows instantaneous force vectors, cumulative impulse, and a force history trail so you can watch the oscillation pattern over time.
Then an energy ledger to verify the sim itself wasn't lying. The residual question: does Σ m·a·v (rate of KE change from computed accelerations) match the actual change in kinetic energy? If the sim has numerical errors, the ledger catches them. If the ledger is clean, the sim is telling the truth.
It took several iterations to get the ledger right. The key fix: integrate work in theta, not time. KE is an exact function of theta, so trapezoid-in-theta isolates derivation and quadrature errors cleanly. Final residuals:
- RPM 96 / Pulse 0.51: −0.238 J residual (0.052% of KE)
- RPM 40 / Pulse 0.30: −0.001 J residual (0.0013% of KE)
Clean. The sim is honest about what it's computing.
The audit
Before drawing conclusions, an independent audit. Full code submitted to Gemini with no shared context — fresh eyes on the derivatives, the force formula, the quadrature, the counter-rotating symmetry, the rebase logic.
Result:
The ledger code itself is remarkably solid. The small residuals for offset < R are strong indicators that the simulator is numerically sound under those conditions.
No math errors. No sign errors. No bookkeeping tricks. The near-zero residuals are real.
And then this:
The small |p| cyc avg values for the valid cases indicate that the momentum components are indeed largely canceling out over a cycle as intended by the design.
What that means
The counter-rotation was designed to cancel angular momentum so the device doesn't just spin its housing. It works — the total momentum of the ball assemblies stays near zero.
But here's the thing: if the balls' collective momentum is always near zero, the integral of force on the housing per cycle is also near zero. The reaction that would push the housing is the rate of change of the balls' momentum. If momentum is conserved and near-zero throughout, there's no sustained net push.
The instantaneous forces are real and asymmetric. Watch the visualiser — the force vector oscillates, peaks are higher on one side. But integrated over a full cycle, they cancel. Classical mechanics, closed system, momentum conservation. The counter-rotation that prevents housing spin also prevents the propulsive bias.
The sim isn't broken. It's telling the truth.
Where it goes
Three paths, all running in parallel:
1. The loophole search. A frictionless, perfectly closed system can't produce net momentum. But real systems aren't that. Does adding a non-conservative interaction — drag, gravity coupling, external field — break the cycle's symmetry in a usable way? Worth a structured research pass before closing the book.
2. The actuator reframe. The mechanism produces real asymmetric instantaneous forces, even if they cancel over a cycle. That's potentially useful for precision inertial actuators, vibration isolation, positioning systems — applications that care about the instantaneous profile, not the cycle average. Different market, honest application.
3. This post. A documented null result is worth more than a speculative positive. If anyone else builds this mechanism and gets confused by the same instantaneous-vs-cycle-average question, this is here now.
The prototype design is still in CAD. The phase-flip test would still be the discriminator — if the carriage walks with pulsing and reverses with phase flip, something is happening. If it doesn't walk at all, the sim called it correctly.
That test will happen. For now: the sim says no, the math is clean, and this is what we found.
The most useful thing you can publish is the truth of what the measurement said.