Trilinear Hilbert transform: three_core_principal #
Blueprint lines 2463-2582 for the three-core pattern with side role l on a leaf cell of the
side states (eq:principal-core before binning, with eq:core-numerical): with anchors of the
final classes in the core roles (AnchorOKAt), the pattern form is at most
2⁴ Z / (1 - 4 (2δ_*)^{1/4}) plus the
absolute tuple errors C_{J,A} D_*^{-100} σ Π_{j ≠ l} Σ_{m ∈ r_j} |α_m| (σ the local side size),
where Z = Σ_{h ∈ H} |H|⁻¹ Σ_p c_p Π_{j ≠ l} 6 σ_{p_j} with the principal window coefficients
c_p (threeCoef, 0 for discarded tuples) of the side S (pattern_leaf_bound,
masked_group_side, three_core_tuple_bound, role_size_moments_cls, principal_word_average;
assignments of a failure to the side slot leave no masked tuples).
The value 2 C_{J,A} 2^{A+4} (2μ) ν^{-J} Λ^{-(A+1)} (2 b) Π_{j ≠ l} 6 (2σ_j) of a three-core
dyadic bin (μ, ν, i, b, σ) with Λ = 2^i (blueprint eq:principal-core; the side role l bins
the window size b_Λ).
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The window size b_Λ of the side X at the shell Λ = 2^i (blueprint eq:side-windows): the
offset-averaged ℓ² norm of the normalized Fourier coefficients of the side in the window.
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The dyadic bin (μ, ν, i, b_Λ, (σ_j)_j) of a three-core tuple at the shell Λ = 2^i.
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The tested three-core tuples and shells of a family: the pairs (p, i) passing the three-core
test with positive window size b_Λ and positive core sizes.
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- D.threeTestedOf G I xs h' l T a X σv Iw = {q ∈ T ×ˢ Finset.range (Iw + 1) | D.threePasses G I xs h' l (a q.1) ∧ 0 < D.winB G I xs h' l (a q.1) X q.2 ∧ ∀ (j : Fin 4), j ≠ l → 0 < σv q.1 j}
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Blueprint eq:principal-core (lines 2577-2582) for three cores (side role l), at one final
height: the principal quantity Σ_p c_p Π_{j ≠ l} 6 σ_{p,j} (c_p the window coefficient of a
passing tuple, 0 for discarded ones) is at most the sum over the occupied dyadic bins
(μ, ν, Λ, b_Λ, σ) of C_{J,A} μ ν^{-J} Λ^{-(A+1)} b_Λ Π_{j ≠ l} σ_j (with the factors 2 of the
bands) times the number N of tested tuples and shells in the bin (zero factors contribute
nothing; threeBands_spec, sum_by_bins).
The tested three-core tuples and shells at a final height h (side role l, side X):
tuples of final classes of good core components in the core roles passing the three-core test
with the slopes of their anchors, together with the shells Λ = 2^i ≤ 2^{I_w} of positive window
size, all core class sizes positive.
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The dyadic bin of a three-core tuple of final classes and a shell at the final height h.
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Blueprint eq:principal-core (lines 2577-2582) for the three-core pattern with side role l
on a leaf cell of the side states: the pattern form is at most
2⁴/(1 - 4(2δ_*)^{1/4}) Σ_{h ∈ H} |H|⁻¹ Σ_{bins} C_{J,A} μ ν^{-J} Λ^{-(A+1)} b_Λ Π_{j ≠ l} σ_j N
(with the factors 2 of the bands), plus the absolute tuple errors of eq:core-numerical. Here
N = N_{h,bin} is the number of tested tuples of final classes and shells Λ = 2^i in the bin
(at most once per tuple and shell). The side bins its window size b_Λ, the core roles bin their
component sizes.