Trilinear Hilbert transform: core_reduction #
Blueprint line 2461, the reduction of the core terms on the side states. Take the cut family
C of the side estimate (two-side bound B₂, side cap T_c, three-side power bound B₃(B_c) on
retained cells with cores of normalized size B_c) and a second cut C₂ of root fraction at most
ρ₂ for the integrated budgets of the rolling sides (eq:variation) and the linear sides
(eq:linear-energy). On the cells retained by both, the cores, errors and sides have the P^C
retained sizes coreBd, errBd and √T_c (the lift is pointwise bounded), and the weighted
active minimum of the whole leaf outputs is at most B₂, 82 times the error patterns (bounded by
eq:physical-error, the active multiplicity M and Cauchy-Schwarz), 82 B₃ and the weighted
active minimum of the patterns with at least three cores. A basic U role remains a side.
Blueprint line 2461 with the side estimate (lines 2446-2456): on one localization leaf of the
guarded search, after the cut family of the side estimate (root fraction 2ρ) and the second cut
for the rolling and linear sides (root fraction ρ₂), the cores, errors and sides have the
retained sizes coreBd, errBd, √T_c, the side cap of the side estimate holds along the
retained paths, and the weighted active minimum of the whole leaf
outputs is at most the two-side bound of the side estimate, 82 times the error patterns
(eq:physical-error, the active multiplicity and Cauchy-Schwarz), 82 times the three-side power
bound with cores of size coreBd, and the weighted active minimum of the patterns with at least
three cores (core_reduction, side_estimate_leaf).
The exact common-height selection on the side states (blueprint lines 2464-2466) #
A column function vanishing off a set restricts to zero on a disjoint set, also after taking its physical part.
The pre-cell J = anc_{b - 2K}(I) of the block of a cell (blueprint line 1975).
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The core components of the slot l and group G at the cell I: those of the retained
pre-cell of its block (none in the basic U slot or when the pre-cell is not retained).
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The physical subtotal on the cell I of a core component on its pre-cell.
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Blueprint lines 2463-2464: on a cell the core is the sum of the subtotals of the core components of its retained pre-cell (the other retained pre-cells of the block are disjoint from the cell).
The goodness at a height of the entry of a component tuple in a slot: the goodness of the
component on the pre-cell of the block of I in a core slot, always good otherwise.
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Blueprint lines 2464-2466 ("each core mode has bad proportion at most 2δ_*, so success
probability is at least 1 - 2 m₀ δ_*"): every tuple of core components of a pattern on a cell
has an upper height at which all its slots are good.
Blueprint lines 2463-2466, the exact common-height selection on one actual cell and group of
the side states: the form of a core/side/error pattern is the sum over the tuples of core
components of the forms of their subtotals (the finite multilinear expansion), and it is the
convergent sum over the trial counts n, the assignments a of the failed trials to their first
failing roles, the words w of failed upper heights and the final heights h, with the weights
|H|^{-(n+1)} of the uniformly sampled heights, of the sums over the tuples in the product
masks; the first-success rule has total weight one for every tuple.
Failure masks and representatives on the side states (blueprint lines 2466-2467) #
good_same_predecessor with the stabilization only at the heights h ≤ J.
core_good_iff with the stabilization only at the heights h ≤ J.
mem_pred_of_good_class with the stabilization only at the heights h ≤ J.
compGood_iff_of_same_class with the stabilization only at the heights h ≤ J_*.
Blueprint lem:stabilize on the pre-cells of the side states (blockComps_stab): two alive
modes of one key good at a height h ≤ J_* lying in one CURRENT component at h are PRE-joined
at h - Δ.
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At the chosen schedule (S.Valid, side parameters S.sideParams δ_*) the stabilization holds
on the pre-cells of the side states (blockComps_stab): the alive modes number at most D_*,
the dimensions are at most d_max and D_* covers the heights of the mode differences.
The final class at height h of a component index of a core slot (blueprint line 2467):
the union of the CURRENT components at h of its modes.
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Blueprint lines 2466-2467 ("for a fixed assignment, the conditions in a role form a product
mask ... the upper-height coherence proved in Section sec:modes means these masks select unions
of entire core subtotals in minimal CURRENT components"): two core components of a slot in one
minimal CURRENT component have the same role mask for every assignment and every failure history
with heights in the upper heights, and the same final class at every height.
Blueprint line 2467 for any representative of the predecessor (in particular the hereditary
representative of lines 2172-2183): every mode of a core component good at the final height h
lies in the PRE predecessor E_{h-Δ}(a) of any mode a of the predecessor E_{h-Δ}(m) of a good
mode m of its final class (the unique good predecessor of the final component).
The core torus form on the side states (blueprint eq:core-torus-form, lines 2467-2471) #
A cell at least j depths below s has length at most 2^{-j} times that of depth s.
Points of one dyadic cell are at distance at most its length.
The common torus frequency ξ = ϑ / L_h (ϑ = θ / Q₀) of a selected group at height h
(blueprint line 2467).
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The periodic factor g_j of eq:core-torus-form: the true periodic cutoff (re-centered at
x*) times the finite trigonometric polynomial of the selected modes with their integer shifts
(blueprint lines 2468-2470).
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The periodic factor is periodic on the ξ torus (L_h is an integer).
Blueprint eq:core-torus-form (lines 2467-2471) for a group of selected modes F in the PRE
component E_h(a) of a representative a, on an actual cell I of the block b (I.1 ≥ b)
inside the pre-cell J (J.1 = b - 2K), at any x* ∈ I: the wave sum is
e(c p_ρ(y)) e(γ*(y - x*)) g(ξ (y - x*)) with γ* = ω_a(x*), ξ = ϑ / L_h and the periodic
factor g = torusG (the true cutoff times a trigonometric polynomial with integer shifts of size
at most |alive| E_h), up to D_*^{-150} times the absolute coefficient sum: the PRE path gives
slope error D_* Ξ / r_J and the pre-lag 2^K ≥ D_*^{600} makes it and the quadratic Taylor
error (curvature at most Λ ≤ D_* at the admission scale) negligible.
Blueprint lines 2463-2470 (the core subtotals as physical functions): the physical function of
the subtotal on a cell of a core component is a.e. 1_I Σ_{m ∈ K} c_m wave_m.
The frequency box (blueprint eq:frequency-box, lines 2471-2476) #
The common cutoff B_* is at most A₀ (blueprint line 1955: A₀ is a power of
C_* 6^{R_*} B_* D_*).
Blueprint lines 1944 and 2471 ("truncate the Fourier series to tails at most D_*^{-300}"):
with the Fourier cutoff K_f of the schedule (2 gaugeTail / 2^{K_f} ≤ D_*^{-300}), the
truncation tail of every true periodic cutoff of width δ in dimension d is at most
D_*^{-300}.
Blueprint eq:frequency-box for the truncated cutoffs: in the torus coordinates ξ = ϑ / L_h
the frequencies Q₀ L_h v of the truncated cutoffs (|v|_∞ ≤ 2^{K_f + 1}, Q₀ 2^{K_f + 1} ≤ B_*)
lie in the box B'_h = D_*^5 E_h (the cutoff indices B_* L_h ≤ E_h).
Blueprint eq:frequency-box for the truncated common progression multiplier: its frequencies
L_h v (|v|_∞ ≤ 2^{K_f + 1} ≤ B_*) lie in the box B'_h = D_*^5 E_h.
The full Fourier norm bound of the true periodic cutoff.
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The full Fourier norm of the cutoff is at most that of the squared gauge (which the schedule
covers by D_*, blueprint line 1944).
Blueprint lines 2471-2476 ("absolute Fourier costs ... are fixed powers of D_*; dilation of
indices does not change coefficient sums"), for the true periodic cutoff: its Fourier
coefficients have modulus at most one, every finite absolute coefficient sum is at most cutNorm
(at most gaugeSqNorm ≤ D_*), and the truncation to the box |v|_∞ ≤ 2^{K+1} is uniformly
within cutTail (at most D_*^{-300} at the Fourier cutoff of the schedule).
The truncated multiplier R_z (blueprint line 2478) #
Blueprint line 2478 ("in coordinates u = ξ(y_{l'} - x*), v = ξ t, denote its shifted
truncated version by R_z(u, v); it has uniformly bounded supremum, including for different
offsets and anchors"): the truncated common progression multiplier of any anchor l', re-centered
at x* and lifted to the torus by L, is bounded by 1 + τ_m at every pair of torus points,
where τ_m is its truncation tail.