Helfgott–Platt block ladders, group 1 of 10

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WeakGoldbach.helfgott_platt_block_ladders_grp01

by Ethan Cheung Wang · Oct 21, 2026 · Mathlib 0df444a (Lean v4.33.1)

finite-certificatesgoldbachnumber-theoryprimes

Put $H = 2\cdot 10^{18}$ and $W = 2^{54}\cdot 10^9 = 18014398509481984000000000$ . For every integer $b$ in the range

$0 \le b < 49270$,

there is a finite increasing sequence of primes $p_0,\ldots,p_k$ satisfying

bW < p_0 \le bW + H,\qquad p_i < p_{i+1} < p_i + 2H \quad (0 \le i < k),

p_k \le (b+1)W < p_k + H.

Role: this is group $1$ of $10$ consecutive groups of Helfgott–Platt computation blocks covering $0 \le b < 492700$ . Each group has width $49270$ in the block index $b$ . The groups meet end to end: group $j$ 's upper endpoint is group $j+1$ 's lower endpoint, the first lower endpoint is $0$ , and the last upper endpoint is $492700$ .

This isolates the local computational certificate for $49270$ of the $492700$ intervals of width $W$ reported in Section 4 of the source. The endpoint conditions allow consecutive blocks (and consecutive groups) to be joined without a gap of size $2H$ .

The hypothesis is $b < 49270$ . No certificate data are supplied here; the statement is an open certificate obligation for this index range.

Formalization Note The conclusion matches the parent WeakGoldbach.helfgott_platt_block_ladders exactly; only the range of $b$ is restricted.

Preamble

import Mathlib.Data.Nat.Prime.Basic

Formal statement

theorem WeakGoldbach.helfgott_platt_block_ladders_grp01 (b : ℕ) (hb : b < 49270) :
    ∃ (k : ℕ) (p : ℕ → ℕ),
      (∀ i, i ≤ k → Nat.Prime (p i)) ∧
      (b*18014398509481984000000000 < p 0 ∧
        p 0 ≤ b*18014398509481984000000000 + 2*10^18) ∧
      (∀ i, i < k → p i < p (i+1) ∧ p (i+1) < p i + 4*10^18) ∧
      (p k ≤ (b+1)*18014398509481984000000000 ∧
        (b+1)*18014398509481984000000000 < p k + 2*10^18) := by sorry

Source

H. A. Helfgott and D. J. Platt, Numerical Verification of the Ternary Goldbach Conjecture up to 8.875e30, arXiv:1305.3062v2, Section 3 (prime ladder algorithm) and Section 4 (width 2^54*10^9, endpoint distances 2*10^18, and 492700 intervals). https://arxiv.org/pdf/1305.3062. Group 1 of 10 consecutive partitions of the parent WeakGoldbach.helfgott_platt_block_ladders covering block indices [0, 49270).

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