asteroid_one_statement
Engineers modeling phantom-cavity asteroid deflection cite this theorem as the single consolidated statement that the carrier frequency 5φ lies in (8.05, 8.10) Hz, that cumulative deflection scales quadratically with lead time, and that doubling lead time quadruples deflection. The result packages the frequency band membership with the algebraic scaling properties of the deflection function. The proof is a direct term construction that invokes the band theorem and the two scaling lemmas without further reasoning.
claim$8.05 < 5φ < 8.10$ and, writing $δ(t)$ for cumulative deflection at lead time $t$, one has $δ(2t) = 4δ(t)$ for all real $t$ together with $δ(0) = 0$.
background
The Asteroid Trajectory Shaping module treats a phantom-cavity drive coupled to a small body. Carrier frequency is defined as five times phi in hertz. Cumulative deflection at lead time t is defined as (impulseCoefficient times t squared) over two. The band theorem establishes the numerical interval by unfolding the carrier frequency definition and applying the known bounds phi greater than 1.61 and less than 1.62. The double theorem shows that deflection of twice the argument equals four times the original by direct ring simplification after unfolding. The zero theorem shows deflection at zero equals zero by unfolding and simplification.
proof idea
The proof is a one-line term wrapper that assembles the conjunction directly from the two projections of the carrier_frequency_band theorem together with the deflection_double and deflection_zero theorems.
why it matters in Recognition Science
This theorem supplies the single reference statement for the J1 engineering track in Plan v5. It consolidates the frequency interval and the t-squared scaling that appear in the module falsifier criterion of deflection inconsistent with quadratic growth to within 3 sigma over twelve months. The result sits downstream of the carrier frequency definition and the deflection scaling lemmas, providing a compact citation point for trajectory shaping calculations.
scope and limits
- Does not establish physical realizability of the phantom-cavity drive.
- Does not compute explicit deflection magnitudes for named asteroids.
- Does not incorporate relativistic or multi-body gravitational corrections.
- Does not prove uniqueness of the chosen carrier frequency.
formal statement (Lean)
108theorem asteroid_one_statement :
109 (8.05 : ℝ) < carrier_frequency ∧ carrier_frequency < 8.10 ∧
110 (∀ t, deflection (2 * t) = 4 * deflection t) ∧
111 deflection 0 = 0 :=
proof body
Term-mode proof.
112 ⟨carrier_frequency_band.1, carrier_frequency_band.2,
113 deflection_double, deflection_zero⟩
114
115end
116
117end AsteroidTrajectoryShaping
118end Engineering
119end IndisputableMonolith