Trilinear Hilbert transform: three_core_noref_side #
Blueprint lines 2540 and 2562-2567 on a leaf cell of the side states (three cores, side role
l, no reference: the tuple is discarded). When no reference lies in the full box (threeRefSet
empty), every anchored expansion frequency T_l + (P₂ - l P₁)·ξ of the transported core inputs has
r |·| > U₀ (its integer part lies in the full box, expP_fullBox), so the cell form with the side
X in slot l is at most twice the torus-form and gauge errors, plus
4 r⁻¹ C_{J,A} U₀^{-J} 2^{A+4} W Π_{j ≠ l} ((1 + τ_c) Σ_{F_j} |α|) (r ∫_I |S_phys|²)^{1/2},
plus the far tails
multNorm cutNorm Π_{j ≠ l} (cutNorm Σ_{F_j} |α|) r⁻¹ ∫_I |S_phys| 12 C_J (4/U₀)^{J-2}
(three_core_noref_of).
The dyadic window weights sum to at most two: Σ_{i ≤ I} 2^{-i(A+1)} ≤ 2.
The absolute Fourier costs of a three-core expansion: with full Fourier norms at most D_*,
multNorm (cutNorm Π_{j ∈ s} (cutNorm S_j)) ≤ D_*^{|s| + 2} Π_{j ∈ s} S_j.
The window split of a three-core tuple (blueprint eq:core-numerical): if the core factor
W Π (a'_j + τ_c S_j) is at most P_a + 31 u P and every window energy root is at most R, the
window sum is at most the principal window sum with the true torus sizes P_a, plus
31 u P R Σ_i c_i.
The coset replacement and tail errors of a three-core tuple (blueprint eq:core-numerical):
with full Fourier norms at most D_*, replacement precision ε ≤ D_*^{-999} (near error
N ≤ C √ε), far separation H ≥ D_*^{999}, J ≥ 3, and normalized L¹ side size at most σ,
multNorm (cutNorm Π_{j ∈ s} (cutNorm S_j)) L¹ (N + 12 C_J (4/H)^{J-2}) ≤ (C + 48 C_J) D_*^{-100} Π_{j ∈ s} S_j σ for three roles.
The absolute constant of the numerical errors of a three-core tuple (blueprint
eq:core-numerical, with the local side size as an extra factor): torus-form and gauge errors,
truncation parts of the core sizes, coset replacement and tail errors, shells above U₀, and
discarded tuples.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The principal quantity of a three-core tuple with side role l (blueprint eq:side-windows):
2 Σ_{Λ = 2^i ≤ 2^{I_w}} r⁻¹ C_{J,A} μ ν^{-J} 2^{A+4} Λ^{-(A+1)} (Π_{j ≠ l} a'_j) b_Λ with the true
torus sizes a'_j and the offset-averaged window energies b_Λ of the side X in the
mask-independent windows threeWin(Λ).
Equations
- One or more equations did not get rendered due to their size.
Instances For
The window energies of a side on a cell are at most r ∫_I |S_phys|² (Bessel for each offset,
averaged over the offsets).