Trilinear Hilbert transform: three_core_test #
Blueprint eq:side-windows (lines 2536-2560) on a leaf cell I of the side states, for a
three-core tuple with side role l of the group G (all roles in the chart of role 0): let
F_j, j ≠ l, be the selected modes of the core roles with their torus forms
eq:core-torus-form at the final height h ≤ J_* (representatives a_j, shifts of size at most
D_* E_{h-Δ}, error D_*^{-150}), let a role j₀ be good at h (a successful component, whose
gap gives the tiny/huge dichotomy), and let the tuple have a reference for the three-core test
(m = threeRef, β = threeBeta; the threshold |β| ≥ U₀⁻¹ is not used). Then for every side
column X (S_phys = augPhys X) the cell form with the three selected core subtotals and the side
X in slot l is at most twice
- the torus-form and gauge errors
2‖w‖_∞ (4 D_*^{-150} + E_g) Π_{j ≠ l} Σ_{F_j} |α|timesr⁻¹ ∫_I |S_phys|, plus - the shells above
2^{I_w}:C_{J,A} μ ν^{-J} 2^{-(I_w+1)(A+1)} W Π_{j ≠ l} a''_jtimes(r ∫_I |S_phys|²)^{1/2}, withW = (1 + τ_m)(1 + τ_c),a''_j = a'_j + τ_c Σ_{F_j} |α|anda'_jthe torus size of the true periodic factor of rolej(eq:torus-size), plus - the coset errors: tiny replacement errors (
ε₁ = T⁻¹/L_{h-Δ}) and far-coset tails (H = T/L_{h-Δ} - U₀ ≥ 4) times the absolute coefficient summultNorm cutNorm Π_{j ≠ l} (cutNorm Σ_{F_j} |α|)andr⁻¹ ∫_I |S_phys|, plus - the window sum
Σ_{Λ = 2^i ≤ 2^{I_w}} C_{J,A} μ ν^{-J} Λ^{-(A+1)} W Π_{j ≠ l} a''_j b_Λofeq:side-windows, whereb_Λ² = E_{ρ_l} Σ_{k ∈ threeWin(Λ)} |SFourier'_ρ(k)|²is the offset average of the side energies in the mask-independent windows|k - r(Q + n·ξ)| ≤ 2Λ,|n|_∞ ≤ 16 B'. HereC_{J,A} μ ν^{-J} = entryC κ J A βwithμ = min(1, |β|),ν = max(1, |β|).
The absolute numerical errors (blueprint eq:core-numerical, lines 2562-2567) #
Π_{j ∈ s} (a_j + τ S_j) ≤ Π_{j ∈ s} a_j + ((1 + τ)^{|s|} - 1) Π_{j ∈ s} S_j for
0 ≤ a_j ≤ S_j and τ ≥ 0 (the truncation parts of the core sizes are replacement errors).
The anchored expansion frequencies P₂ - l P₁ of the core product (anchor l, including
l = 0, where it is P₂): with indices at most B_x and shifts at most N_b, their entries are
at most 4 L B_x + 6 (QL B_x + N_b) (Σ_j |j - l| ≤ 6).
Blueprint lines 2512-2514 and 2562-2567 (a four-core tuple without reference is discarded) with
the Fourier truncations made explicit: if every expansion frequency has
ν = max(1, r|q'|, r|t'|) ≥ U₀, every kernel sample is at most C₁ U₀⁻¹ (fourK_discard), and the
local form is at most the numerical errors plus C₁ U₀⁻¹ times the absolute expansion coefficient
sum.
The torus size of the true periodic factor is at most its absolute coefficient sum: the true
periodic cutoff is at most one, the characters are unimodular, and 𝓗 is a probability space.
Blueprint eq:frequency-box for the combined expansion frequencies: with shifts at most
D_* E_{h'}, truncated cutoff indices Q₀ L_{h'} v and multiplier indices L_{h'} v
(|v|_∞ ≤ 2^{K_t+1}, Q₀ 2^{K_t+1} ≤ B_*), the horizontal frequency P₁ and every anchored
frequency P₂ - l P₁ of the core product lie in the full box B'_{h'} = D_*^5 E_{h'}.
The torus size a'_j = ‖g_j‖_{L²(𝓗)} (blueprint eq:torus-size) of the true periodic factor of
a core role j with selected modes F and shifts n at a leaf cell I and torus height h'
(𝓗 the annihilator torus of the tiny-frequency space of the chart of role j).
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The absolute coefficient sum Σ_{m ∈ F} |α_m| of selected modes of a role j.
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The torus size of a core role is at most its absolute coefficient sum
(torusGT_eLpNorm_le_sum).
At the chosen schedule L_{h'} Ξ ≤ T for h' ≤ J_* (T > H Ξ, H > L_{J_*}).
Blueprint lines 2512-2514 and 2562-2567 on a leaf cell of the side states (four cores, no
reference: the tuple is discarded). When no reference lies in the full box (fourRefSet empty),
every expansion frequency of the four transported core inputs has ν ≥ U₀ (its horizontal and
vertical frequencies lie in the full box, expP_fullBox), so the cell form of the four selected
subtotals is at most the torus-form and gauge errors plus
C₁ U₀⁻¹ multNorm Π_j (cutNorm Σ_{F_j} |α|) (four_core_noref_of).
The torus-form and gauge errors at the enlarged gauge cutoff δ/2 (zero gauge error,
gaugeErr_half) with truncation tails at most D_*^{-300} (blueprint eq:core-numerical):
2K (4 D_*^{-150} + E_g) P ≤ 104 K D_*^{-100} P.
The truncation parts of the core sizes (blueprint eq:core-numerical): for 0 ≤ a_j ≤ S_j and
tails τ_c, τ_m ≤ u ≤ 1, (1 + τ_m) Π_j (a_j + τ_c S_j) ≤ Π_j a_j + 31 u Π_j S_j.
The absolute Fourier costs of the expansion: multNorm Π_j (cutNorm S_j) ≤ D_*^5 Π_j S_j when
both full Fourier norms are at most D_* (blueprint line 1944).
The coset replacement and tail errors (blueprint eq:core-numerical): with replacement
precision ε ≤ D_*^{-999}, far separation H ≥ D_*^{999}, J ≥ 1 and full Fourier norms at
most D_*, max(C ε, C_J H^{-J}) multNorm Π_j (cutNorm S_j) ≤ (C + C_J) D_*^{-100} Π_j S_j.
The arithmetic of a tuple with a reference (blueprint eq:core-numerical): from
Λ ≤ E₁ + (F T + E₃) with E₁ ≤ A₁ u P, T ≤ P_a + 31 u P, E₃ ≤ A₃ u P and 0 ≤ F ≤ C₀, the
form is at most the principal quantity F P_a plus (A₁ + 31 C₀ + A₃) u P.
The absolute constant of the numerical errors of a four-core tuple (blueprint
eq:core-numerical): torus-form and gauge errors, truncation parts of the core sizes, coset
replacement and tail errors, and discarded tuples.
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The schedule facts used for the numerical errors: T⁻¹/L_{h'} ≤ D_*^{-999} and
T/L_{h'} - U₀ ≥ D_*^{999} for h' ≤ J_* (L_{h'} Ξ ≤ T, Ξ = D_*^{1000}, U₀ = D_*^{250}).
Blueprint eq:core-numerical (lines 2562-2567) for a four-core tuple on a leaf cell of the
side states (common chart of role 0, torus forms with error D_*^{-150}, anchor role good at the
final height h; enlarged gauge cutoff δ/2, Fourier truncation tails at most D_*^{-300} and
full Fourier norms at most D_*, as chosen in blueprint line 1944): the cell form of the four
selected subtotals is at most the principal quantity |WFourier_s(r q', r t')| Π_j a'_j if the
tuple
passes the four-core test, and 0 if it is discarded, plus the absolute error
C_J D_*^{-100} Π_j Σ_{F_j} |α|. The error collects the torus-form and gauge errors, the truncation
parts of the core sizes (a'_j ≤ Σ_{F_j} |α|, coreSize_le), the coset replacement and tail errors
(T⁻¹/L ≤ D_*^{-999}, T/L - U₀ ≥ D_*^{999}), discarded tuples with a reference
(|WFourier_s| ≤ C₁ U₀⁻¹, fourK_discard) and tuples without reference
(four_core_noref_side).
The anchor entry constant C_{J,A} μ ν^{-J} is at most C_{J,A} min(1, |β|) (a discarded
three-core tuple with |β| < U₀⁻¹).
The anchor entry constant C_{J,A} μ ν^{-J} is at most C_{J,A} U^{-J} when |β| ≥ U ≥ 1
(all anchored frequencies far: a three-core tuple without reference).
Blueprint lines 2540 and 2562-2567 (a three-core tuple without reference is discarded) with the
Fourier truncations made explicit: if every anchored expansion frequency is far,
r |T_l + (P₂ - l P₁)·ξ| ≥ U ≥ 4, the averaged three-core bound (three_core_averaged_of, with no
near coset) gives the form at most twice the torus-form and gauge errors, plus
4 r⁻¹ C_{J,A} U^{-J} 2^{A+4} W Π_{j ≠ l} ‖g_j‖₂ (r ∫_I |f_l|²)^{1/2}, plus the far-coset tails
multNorm Π_j (cutNorm Σ|c|) r⁻¹ ∫_I |f_l| 12 C_J (4/U)^{J-2}.
The constant of the near-coset replacement error: nearB κ ε ≤ nearBC κ √ε for ε ≤ 1.
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The truncation parts of three core sizes with the multiplier and side-cutoff factor
W = (1 + τ_m)(1 + τ_c) (blueprint eq:core-numerical): for 0 ≤ a_j ≤ S_j, three roles and tails
τ_c, τ_m ≤ u ≤ 1, W Π_{j ∈ s} (a_j + τ_c S_j) ≤ Π_{j ∈ s} a_j + 31 u Π_{j ∈ s} S_j.
Cauchy-Schwarz on a cell: the normalized L¹ norm of a square-integrable side on
[a, a + r] (or [a, a + r)) is at most its normalized L² size (r⁻¹ ∫ |g|²)^{1/2}.
The L²(𝓗) norm of a character polynomial Σ_{k ∈ F} c_k χ_{n_k} is at most its absolute
coefficient sum (𝓗 a probability space).