The side conditions hold for all large `n` (round six, `par6`)
ProvedIntMul.Kappa.R6.P18.par6The side conditions hold for all large n.
The following holds (Lean statement, in the scope of module Solutions.R6P18Par):
theorem IntMul.Kappa.R6.P18.par6 (P : Prm) (hP : P.Ok) {Ew : ℕ → ℕ} (hEw : EwOK6 P Ew) (K₀ : ℕ) :
∃ n₁ : ℕ, ∀ n, n₁ ≤ n → Par6 P Ew K₀ n
Context. Round six, Proposition 18: the numeric side conditions at the round-six parameters
Par6 P Ew K₀ n collects every numeric hypothesis of the Proposition 18 program (the frozen
Par18Need list with D = d - 1, K = ℓ - 1, p = pP6) together with the guard bound
Ew D ≤ p + 2 and the three size hypotheses of IL3.P14NeedH' (constant K₀, at D = d - 1).
par6: they all hold for large n (for P.Ok, a guard with EwOK6 P Ew, any K₀).
Every condition is "polynomial in b below 2^K" (big_K: b ≤ (2ℓ)^m), except the guard
(ε C₁ < 1 + x) and P14NeedH' (3) (IL3.p14_hyps_r6, i.e. ε (2 - lamp) < 1).
Formalization Note. Part of the Lean formalization of the round-six κ-improvement (integer-multiplication-kappa campaign, target IntMul.Kappa.jain_round6_framework). The statement is copied verbatim from the Lean development; the platform node carries the same declaration name. Definitions it mentions live in the IntMul_Kappa_R6_Defs* bundles (on top of the published κ bundles IntMul_Kappa_K_Defs*).
-- Round-six κ development (integer-multiplication-kappa), generated by skeleton subtraction. import Definitions.Def_IntMul_Kappa_R6_Defs70 set_option autoImplicit false set_option relaxedAutoImplicit false set_option maxHeartbeats 1000000 open IntMul IntMul.Kappa IntMul.Kappa.R6 IntMul.Kappa.R6.P18 open IntMul NLogN Real IntMul.Kappa.R6.IL3
theorem IntMul.Kappa.R6.P18.par6 (P : Prm) (hP : P.Ok) {Ew : ℕ → ℕ} (hEw : EwOK6 P Ew) (K₀ : ℕ) :
∃ n₁ : ℕ, ∀ n, n₁ ≤ n → Par6 P Ew K₀ n := by sorry