deflection_double
plain-language theorem explainer
Doubling lead time quadruples cumulative deflection under the quadratic impulse model. Engineers planning phantom-cavity trajectory corrections cite this scaling when sizing lead times. The proof is a direct algebraic reduction after unfolding the deflection definition.
Claim. For every real number $t$, the deflection at lead time $2t$ equals four times the deflection at lead time $t$, where deflection is defined by $δ(t) = (κ · t²)/2$ and $κ$ is the impulse coefficient.
background
The Asteroid Trajectory Shaping module models a phantom-cavity drive that produces per-cycle impulse $Δp = m · v_recoil$ at carrier frequency $5φ$ Hz. Cumulative deflection is introduced by the upstream definition $δ(t) = (impulseCoefficient · t²)/2$. The module establishes positivity, non-negativity, zero at $t=0$, and the doubling relation to support a certification structure for trajectory shaping.
proof idea
The proof unfolds the definition of deflection and applies the ring tactic to verify the algebraic identity.
why it matters
This theorem supplies the doubling relation required by the AsteroidTrajectoryShapingCert structure and the asteroid_one_statement summary. It confirms the $t²$ scaling of deflection with lead time, consistent with the engineering derivation in the Recognition Science framework. The result closes part of the certification for trajectory shaping applications.
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