deflection_nonneg
plain-language theorem explainer
Non-negativity of cumulative deflection for any real lead time follows from the positivity of the impulse coefficient and the algebraic form of the deflection expression. Asteroid trajectory engineers modeling phantom-cavity drives would cite this to confirm that predicted deflections remain non-negative under the t-squared scaling. The proof is a term-mode wrapper that unfolds the definition and applies div_nonneg to a product of a positive coefficient and a non-negative square.
Claim. For every real number $t$, the cumulative deflection satisfies $0 ≤ δ(t)$, where $δ(t) = (ι · t²)/2$ and $ι > 0$ is the impulse coefficient.
background
In the Asteroid Trajectory Shaping module the cumulative deflection at lead time $t$ (in seconds, dimensionless units) is defined by $δ(t) = (impulseCoefficient · t²)/2$. The impulse coefficient is shown positive by direct reduction to carrier_frequency_pos. This theorem supplies the non-negativity property required by the certification structure AsteroidTrajectoryShapingCert. The local setting is the engineering derivation of a phantom-cavity drive (RS_PAT_032) that produces per-cycle impulse $Δp = m · v_recoil$ with $v_recoil = ℏ_R · ω_carrier / c²$ at carrier frequency $ω_carrier = 5φ$ Hz, yielding the quadratic deflection law.
proof idea
The proof unfolds the definition of deflection and applies the lemma div_nonneg. The numerator is handled by mul_nonneg of impulseCoefficient_pos (which is positive) and sq_nonneg of $t$; the denominator is normalized by norm_num to the positive constant 2.
why it matters
This theorem supplies the deflection_nonneg field required by the AsteroidTrajectoryShapingCert structure, which also records carrier band, impulse positivity, zero deflection at $t=0$, and the doubling property. It closes one of the explicit certification obligations listed in the module for the phantom-cavity drive model. The module falsifier is a tracked NEO whose observed deflection deviates from the $δ ∝ t²$ law to within 3σ over a 12-month window.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.