Theorem 10.3 — Residual tail exactification given central anchor

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Erdos390.eventual_residual_assembly_of_central_anchor

by doctosil · Sep 6, 2026 · Mathlib c5ea003 (Lean v4.30.0)

asymptoticscombinatoricserdos-problemsnumber-theory

Theorem 10.3 (Residual Tail Exactification given Central Anchor)

Fix a constant $c > C_0 = \frac{4029639598}{25970038185}$ , and put $h = \left\lceil c \frac{n}{\log n} \right\rceil$ .

For all sufficiently large $n \in \mathbb{N}$ , given any integer divisor $D \in \mathbb{N}$ and central factor subset $\mathrm{central} \subseteq (n, 2n]$ such that $\prod_{\mathrm{central}} a = \binom{2n}{n} D$ and $D \mid \prod_{(2n, 2n+h]} a$ , there exists a residual factor subset $\mathrm{residual} \subseteq (n, 2n + h]$ satisfying:

Disjointness from the central anchor set: $\mathrm{central} \cap \mathrm{residual} = \emptyset$ .
Exact product realization:

\left(\prod_{a \in \mathrm{residual}} a\right) \cdot D = \prod_{a \in (2n, 2n + h]} a.

This isolates the discrete exactification and floating rounding of the smooth tail (Shouqiao Wang's BankPaperGuardedUpperProductAssembly.lean).

Preamble

import Definitions.Def_erdos390_problem
open Filter

Formal statement

namespace Erdos390

open Filter

/-- Theorem 10.3 (Residual tail exactification):
For every constant `c > C0`, for sufficiently large `n`, given any divisor `D` and central
subset of `(n, 2n]` satisfying `central.prod id = binom(2n, n) * D` and `D ∣ tailProduct`,
there exists a disjoint residual subset of `(n, 2n + ⌈c n / log n⌉]` whose product times `D`
equals the full upper tail product. -/
theorem eventual_residual_assembly_of_central_anchor :
    ∀ c : ℝ, C0 < c →
      ∀ᶠ n : ℕ in atTop,
        ∀ (D : ℕ) (central : Finset ℕ),
          central ⊆ factorInterval n (2 * n) →
          central.prod id = Nat.choose (2 * n) n * D →
          D ∣ (factorInterval (2 * n) (2 * n + Nat.ceil (c * secondOrderScale n))).prod id →
          ∃ residual : Finset ℕ,
            residual ⊆ factorInterval n (2 * n + Nat.ceil (c * secondOrderScale n)) ∧
            Disjoint central residual ∧
            residual.prod id * D = (factorInterval (2 * n) (2 * n + Nat.ceil (c * secondOrderScale n))).prod id := by sorry

end Erdos390

Source

Shouqiao Wang, A Proposed Solution to Erdős Problem 390, Section 10, BankPaperGuardedUpperProductAssembly.lean (GitHub 61325b1)

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