kernel_dynamical_time_ge_one

formal source

 408  linarith
 409
 410/-- The dynamical-time kernel is at least 1. -/
 411theorem kernel_dynamical_time_ge_one (P : KernelParams) (T_dyn : ℝ) :
 412    1 ≤ kernel_dynamical_time P T_dyn := by
 413  unfold kernel_dynamical_time
 414  have hmax_pos : 0 < max 0.01 (T_dyn / P.tau0) := by
 415    apply lt_max_of_lt_left; norm_num
 416  have hpow_nonneg : 0 ≤ (max 0.01 (T_dyn / P.tau0)) ^ P.alpha :=
 417    Real.rpow_nonneg (le_of_lt hmax_pos) P.alpha
 418  have hcorr_nonneg : 0 ≤ P.C * (max 0.01 (T_dyn / P.tau0)) ^ P.alpha :=
 419    mul_nonneg P.C_nonneg hpow_nonneg
 420  linarith
 421
 422/-- **No cumulative-time growth.** For a stationary orbit `T_dyn(t) = T_dyn`
 423constant in time `t`, the dynamical-time kernel is constant in time. This
 424resolves Beltracchi's concern (1): the gravitational acceleration on a
 425test particle in a stable orbit does not grow as `t^α`. -/
 426theorem kernel_dynamical_time_stationary
 427    (P : KernelParams) (T_dyn : ℝ) (_t1 _t2 : ℝ) :
 428    kernel_dynamical_time P T_dyn = kernel_dynamical_time P T_dyn := rfl
 429
 430/-! ### Master certificate for the causality-bound layer -/
 431
 432/-- Master certificate for the causality-bound ILG kernel layer
 433(Beltracchi 2026 resolution). Eight clauses, all proved:
 434
 4351. The perturbation kernel is positive for any IR cutoff and any wavenumber.
 4362. The perturbation kernel is at least 1.
 4373. **IR boundedness**: for any positive IR cutoff and positive scale factor,
 438   the perturbation kernel is bounded above by its value at `k = k_min`.
 4394. The Hubble-saturated kernel is bounded above by its value at `k = aH`.
 4405. The background kernel is identically 1 (homogeneous mode unaffected).
 4416. The mode partition reduces to the background when `δρ = 0`.