deflection
plain-language theorem explainer
Cumulative deflection at lead time t equals half the impulse coefficient times t squared in dimensionless units. Asteroid trajectory modelers cite this when deriving t-squared growth from per-cycle impulses at carrier frequency near 5 phi Hz. The declaration is supplied as a direct definition that enables the algebraic derivations of nonnegativity, zero at t equals zero, and the quadrupling property upon doubling t.
Claim. $δ(t) = (k t^2)/2$, where $k$ is the per-cycle impulse coefficient equal to the carrier frequency.
background
The Asteroid Trajectory Shaping module derives deflection from a phantom-cavity drive that produces per-cycle impulse Δp = m v_recoil, with v_recoil tied to carrier frequency at 5φ Hz. Cumulative deflection at lead time t is δ(t) = (Δp/m) t²/2 in dimensionless units. The upstream definition states that the per-cycle impulse coefficient is the dimensionless analogue that scales with carrier frequency and is set equal to carrier_frequency.
proof idea
The declaration is a direct definition that writes cumulative deflection as half the product of the impulse coefficient and the square of lead time. No lemmas are applied inside the definition. Downstream results unfold the expression and apply ring or positivity tactics to obtain the doubling law and nonnegativity.
why it matters
This definition supplies the explicit quadratic form required by the AsteroidTrajectoryShapingCert structure and the asteroid_one_statement theorem, which asserts carrier frequency bounds together with deflection zero at t=0 and the doubling property. It completes the engineering derivation for asteroid trajectory shaping, linking carrier frequency to observable t² scaling. The module falsifier is a tracked NEO showing deflection inconsistent with t² growth to 3σ over a 12-month window.
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