High-contrast homogenization: Provider.Entry.ScaleAccount #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The burn and scale account of the entry argument #
The proof of t.polynomial.entry closes by turning the
selector's additive estimate m_ent ≤ j_† + (C_sel + C_gap) Λ_Π
into the printed ceiling m_ent ≤ ⌈C(d,g) log_3(2 + Π K_{Ψ_S})⌉ and its
exponential form 3^{m_ent} ≤ 3(2 + Π K_{Ψ_S})^{C(d,g)}. Three accounts feed
that step.
The coupled burn. e.global.selection.lower.scale is the
larger of the source burn j_S of e.source.lower.scale and the ceiling
that makes the pair a coupled window; with d/(4d+3) ≤ 1 and
16(d+1)^2/(4d+3) ≤ 8(d+1) the second is below C_exec Λ_Π + 2 + 8(d+1) log_3 K_bar_S.
The source burn. Its three entries are the dimensional floor k_0(d), the
ceiling of 2 log_3 K_bar_S, and the ceiling of
2 + 4(d+1) log_3 K_bar_S + log_3 𝔐_Q(K_bar_S). The moment multiplier of the source
gauge satisfies log_3 𝔐_Q(K_bar) ≤ C_mom(d,g)(1 + log_3 K_bar), because 1 + T ≤ 2T
on T ≥ 1 and log(1 + log K_bar) ≤ log K_bar; hence j_S ≤ C_burn log_3 K_bar_S.
The gauge comparison. Since K_{Ψ_S} > 1 one has K_bar_S ≤ 2K_{Ψ_S} and
Λ_Π ≤ Λ_S := log_3(2 + Π K_{Ψ_S}), while Λ_S ≥ 1; therefore
log_3 K_bar_S ≤ log_3 2 + Λ_S ≤ (1 + log_3 2)Λ_S, and every additive constant of
the account is itself below its own multiple of Λ_S. The exponential form is
then immediate from 3^{Λ_S} = 2 + Π K_{Ψ_S}.
The moment multiplier and the source burn #
The coupled execution burn #
The account in abstract form #
The two printed readings of the entry generation, from the single additive
bound: the ceiling form and the polynomial form, the latter through
3^{log_3 b} = b.