deflection_zero
plain-language theorem explainer
The deflection zero theorem asserts that the cumulative deflection function returns zero at zero lead time. Asteroid trajectory engineers cite it as the base case when building the one-statement certification for the phantom-cavity drive. The proof is a one-line wrapper that unfolds the definition of deflection and simplifies the resulting arithmetic expression.
Claim. Let $I$ denote the impulse coefficient and define the cumulative deflection by $δ(t) = (I · t²)/2$. Then $δ(0) = 0$.
background
In the Asteroid Trajectory Shaping module the deflection function is defined by $δ(t) := (impulseCoefficient · t²)/2$, giving the cumulative deflection at lead time $t$ produced by a phantom-cavity drive at carrier frequency $5φ$ Hz. The module derives this quadratic scaling from the per-cycle impulse $Δp = m · v_recoil$ with $v_recoil = ℏ_R · ω_carrier / c²$. The upstream definition of deflection supplies the explicit formula that the present theorem evaluates at $t = 0$.
proof idea
The proof is a one-line wrapper that unfolds the definition of deflection and applies simp to reduce the expression at zero lead time to zero by direct arithmetic.
why it matters
This result is invoked by asteroid_one_statement to complete the one-statement certification and by AsteroidTrajectoryShapingCert to record the zero base case. It supports the engineering claim that deflection scales quadratically with lead time, consistent with the module's derivation from the phantom-cavity drive model. No open questions are indicated.
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