unsorry

Upstream packet: nicomachus-sum-cubes

Status: packet-ready · generated mechanically (ADR-020 / SPEC-020-A) · sponsor: Chris Barlow

The statement (as proved here)

import Mathlib

theorem nicomachus_sum_cubes (n : ℕ) :
    (∑ k ∈ Finset.range n, k ^ 3) = (∑ k ∈ Finset.range n, k) ^ 2 := by
  sorry

Kernel-verified on main: library/Unsorry/NicomachusSumCubes.lean (theorem nicomachus_sum_cubes), through Gate A (build --wfail, axiom audit against the standard whitelist, leanchecker kernel replay, regenerated ADR-011 binding obligation).

Proposed contribution

The git apply-able new-file diff is at nicomachus-sum-cubes.patch. The target path Mathlib/Unsorry/NicomachusSumCubes.lean is a placeholder — file placement and the final name are Zulip questions, not ours to decide. Content:

/-
Copyright (c) 2026 Chris Barlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Barlow
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith

theorem nicomachus_sum_cubes (n : ℕ) :
    (∑ k ∈ Finset.range n, k ^ 3) = (∑ k ∈ Finset.range n, k) ^ 2 := by
  induction n with
  | zero => simp
  | succ n ih =>
    rw [Finset.sum_range_succ, Finset.sum_range_succ, ih]
    cases n with
    | zero => simp
    | succ m =>
      set S := ∑ k ∈ Finset.range (m + 1), k
      have h : S * 2 = (m + 1) * m := by
        have h' := Finset.sum_range_id_mul_two (m + 1)
        rw [show (m + 1) - 1 = m from by omega] at h'
        exact h'
      have expand : (S + (m + 1)) ^ 2 = S ^ 2 + (m + 1) ^ 3 :=
        calc (S + (m + 1)) ^ 2
            = S ^ 2 + S * 2 * (m + 1) + (m + 1) ^ 2 := by ring
          _ = S ^ 2 + (m + 1) * m * (m + 1) + (m + 1) ^ 2 := by rw [h]
          _ = S ^ 2 + (m + 1) ^ 3 := by ring
      linarith

Dedup at mathlib HEAD

A name-grep is a pre-filter, not a proof of absence; the kernel build at HEAD (tools/upstream/verify_head.sh) is the strong evidence and its result belongs in the PR conversation.

Provenance dossier

Field Value
source Phase-2 seeded target (pre-ADR-012)
reference Nicomachus of Gerasa, Introduction to Arithmetic II.20; left as a reader exercise in Mathematics in Lean §5
absence machine-checked no-local-match (verified against pinned mathlib v4.30.0, 2026-06-10: only the general Bernoulli sum_range_pow exists — a different statement); normalized to the ADR-012 field format 2026-06-12, evidence unchanged
difficulty 3
title Nicomachus’s theorem: the sum of the first n cubes equals the square of the sum

Proof produced by an autonomous Claude agent swarm (model policy ADR-013/ADR-015: fable, progressive effort), merged with no human review through two CI gates (ADR-006 soundness, Gate B hygiene). Full machine history: the goal’s PR trail in this repository.

AI disclosure (paste-ready facts)

The Lean proof in this PR was produced by an autonomous LLM agent (Anthropic Claude, model fable) operating in the unsorry proof swarm (github.com/agenticsnz/unsorry), and was machine-verified there by kernel replay, an axiom audit against the standard whitelist (propext, Classical.choice, Quot.sound), and a CI-regenerated statement-binding obligation. I have read and understood the proof in full and can justify each step without AI assistance. Label: LLM-generated.

For the sponsor

  1. Read the proof until you can justify every step without AI assistance — mathlib reviewers will expect exactly that.
  2. Zulip first, in your own words: is the lemma wanted, where does it live, what should it be called? The PR-description narrative and every review reply likewise must be rewritten in your own words — mathlib policy forbids LLM-written conversation; only the lemma itself (disclosed) and the factual disclosure block above may be pasted.
  3. Raise the draft PR with one command once you’ve done 1–2 — from the unsorry repo root: 
    python3 -m tools.upstream.raise_pr --goal nicomachus-sum-cubes --fork <your-github-user> --understood

    It clones mathlib master, applies the patch to a fresh branch, pushes to your fork, and opens a draft PR pre-filled with the factual disclosure and a placeholder where your narrative goes. (--understood is your attestation that you’ve read the proof; --dry-run shows the plan first.) The machine never marks it ready and never writes a review reply.

  4. Write your narrative in the draft, apply the LLM-generated label, then you flip draft → ready. Expect the linter to want golfing (binder names, line length) — that editing is yours. See docs/upstreaming.md.
  5. Record the outcome on the targets board (in-discussion → pr-open → merged | declined). Declined is a valid, recorded result.

HEAD verification: PASS at mathlib 68c609a0f0fdc49ba2e09efa25146c80e28bc895 (2026-06-12T03:27Z)

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