PrimeNumberTheoremAnd

Generalized Buthe floor assembler: like FloorButhe.floor_buthe_of_curve but with an arbitrary split point xlo ≥ 5 and slab list slabsFrom slabLo n, so the slab interval can start above √5 for rows whose curve the Buthe bound only clears higher up (e.g. row 3 at √6). xlo ≥ 5 keeps Epi_le_evalLhsE applicable.

admissible_bound A (1/4) C R x in terms of s = √(log x): = (A/R^{1/4})·√s·exp(−(C/√R)·s). (√s = (log x)^{1/4}.)

Buthe Eπ-upper-bound reconciliation over the WIDE range [e^5, e^40] (vs FloorButhe.Epi_le_evalLhsE's [e^5,e^10]): identical proof, only the x ≤ 10^19 derivation uses x ≤ e^40 (e^40 ≈ 2.4e17 < 10^19). Needed for row 1's near-threshold boundary [e^22.955, e^40].

Quarter (B=1/4) Buthe floor assembler on [e^xlo, e^10]: the SQUARED slab check lhsE² ≤ rhsE2 plus the squared-curve domination, sqrt taken outside the kernel. Companion of floor_buthe_of_curve_gen for the half-power rows.

Wide quarter Buthe floor assembler over [e^xlo, e^xhi] (5 ≤ xlo, xhi ≤ 40), using Epi_le_evalLhsE_wide (for row 1's near-threshold boundary, depth 8).

FKS2 Corollary 23 (complete). Every Table-6 row gives an admissible asymptotic bound for Eπ with R = 5.5666305.

Row-8/9 tail source: theorem_3 applied to corollary_14 (the un-normalized Eθ B=3/2, C=2, R=5.5666305 bound) at x₁ = e^20000. Same rate as rows 8/9 (C = 2); transported constant 121.0961·(1+μ_asymp) ≤ 121.107 by mu_asymp_num_le_cor14. This IS row 8's curve, and DOMINATES row 9's B=2 curve.

Row-1 tail [e^20000, ∞): Cor 22 domination, B=1/4 with tiny A; √s ≥ 1 drops the √s, reducing to 9.2211·s³·e^{-0.8476s} ≤ 7.811e-5·e^{-0.4239s}.

Row-1 mid-range [e^40, e^20000] via the quarter transport on the RESTRICTED cell set (b' ≥ 40).

Row-2 tail [e^20000, ∞): Cor 22 domination, B=1/4. The row-2 √s on the RHS is bounded below by √s ≥ 1 (s ≥ 141), reducing to 9.2211·s³·e^{-0.8476 s} ≤ 0.5377·e^{-0.4239 s} (cube trick s³·e^{-0.15s}=(s·e^{-0.05s})³).

Row-2 mid-range [e^10, e^20000] via the B=1/4 quarter transport.

Row-3 tail [e^20000, ∞): Cor 22 domination, B=1/2. Surplus s² over the Cor 22 s³ absorbed via s²·e^{-0.1 s} = (s·e^{-0.05 s})² ≤ (20 e^{-1})².

admissible_bound A 1 C R x in terms of s = √(log x): = (A / R) · s² · exp(−(C/√R)·s). The B = 1 analogue of admissible_three_halves_eq, used for the row-4 tail domination.

Row-4 tail [e^20000, ∞): Eπ ≤ the row-4 curve, by domination of the (faster-decaying) Corollary 22 curve. Unlike row 5, the Cor 22 curve carries an s³ while the row-4 curve carries s²; the surplus s is absorbed into the exponential gap 0.8476 − 1.5/√R ≥ 0.21183 via s·e^{-0.1 s} ≤ 10·e^{-1}.