Λ(T,H) Kernel Equivalences

N (Eta Terms): 500

Max Height T: 50

H (Kernel Width): 1.50

Kernel A ■

Kernel B ■

Λ(T,H) = ∫ |ζ(½+i(T+u))|² · K(u,H) du / ∫ K(u,H) du

lim_H→0⁺ Λ(T,H) = |ζ(½+iT)|² = 0 ⇔ ζ(½+iT) = 0

Traces Displayed:

|ζ(½ + iT)| — Raw zeta magnitude on the critical line (via Dirichlet Eta). Zeros appear as dips to 0.
√Λ_A(T,H) — Kernel-smoothed zeta mirror using Kernel A. Tracks |ζ| and drops toward 0 at the same zeros.
√Λ_B(T,H) — Kernel-smoothed zeta mirror using Kernel B. All 6 kernel forms are equivalent; switch between them to verify.

The 6 Equivalent Kernel Forms (all ≡ sech²):

K₁	sech²(u/H) = 1/cosh²(u/H)	Primary form
K₂	4 / (e^u/H + e^−u/H)²	Exponential expansion
K₃	d/du[tanh(u/H)] · H	Derivative form
K₄	4e^2u/H / (e^2u/H + 1)²	Exponential ratio
K₅	1 − tanh²(u/H)	Pythagorean identity
K₆	4σ(2u/H)(1 − σ(2u/H))	Logistic / sigmoid form

Key property: All 6 kernel forms yield identical Λ values — select different forms for Kernel A and B to verify equivalence visually. Poles lie at ±iπH/2 ≈ ±i·2.356 (H=1.5), safely outside the Weil strip |Im(t)| < 0.5.

Λ(T,H) Kernel Equivalences — Zeta Mirror

Traces Displayed:

The 6 Equivalent Kernel Forms (all ≡ sech²):