Trilinear Hilbert transform: retained_numeric_exp #
The exponential standing bounds of the retained-attempt estimate at accuracy k (blueprint
lines 2870-2888: P = ⌈η₀^{-α}⌉ dominates every fixed power of k and of the unit
G = e^{(1 + log 2k)²}, and the closure, mode and catalog errors are below η₀^{c_*}), with
Y = L₄ + 2 k^b, F = 4 (6000^{30} Y^{30} + 1), k_D = (2 C_l + 2) k^{cc+1},
τ₃ = e^{-k^{c₃}} and η₀ = e^{-2 k^{θ₀}}.
The final absorption on one root (blueprint lines 2851-2888, from eq:before-absorption to
eq:active-final) for the explicit leaf bound summed over N_lv leaves plus a localization error
E_r |R|. With ε = 1/1200, C₄ = C_l + 1, α = ε/(4(C₄+1)), P = ⌈η₀^{-α}⌉ and η = 4η₀:
if P dominates the leaf count, the powers of k_D and the unit G = e^{(1 + log 2k)²}
(h1-h3), and the remaining errors are at most η₀^{c_*} (h4), the total is at most
((3 + C_q)(2^{C₄} + 1) + 1) η₀^{c_*} (1 + C_N) |R| with c_* = min(ε/2, α/4).
Blueprint lines 2839-2841 on one root ("choose their thresholds so the union of exceptional
roots has length at most |R|/4"), assembled with the localization to the search leaves and the
leaf bounds: the failure roots (length at most |R|/8) and the six cut families of each of the
N_l leaves (lengths at most 2ρ|R|, 12 N_l ρ ≤ 1/8) have maximal union cells X of length
at most |R|/4, and the active minima of the pattern below no cell of X are at most
N_l times the leaf bound plus the localization error M_p |R| W.
A basic projection without columns vanishes almost everywhere: its current projection lies in the span of no columns.
Without vertices of the basic families (no basic columns) all pattern forms vanish on the cells of the setup: every pattern has a basic slot, and the basic arguments vanish there.