unsorry

Upstream packet: nicomachus-sum-cubes-triangular

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

The statement (as proved here)

import Mathlib

theorem sum_range_cube_eq_triangular_sq (n : ℕ) : ∑ i ∈ Finset.range (n + 1), i ^ 3 = (n * (n + 1) / 2) ^ 2 := by
  sorry

Kernel-verified on main: library/Unsorry/NicomachusSumCubesTriangular.lean (theorem sum_range_cube_eq_triangular_sq), 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-triangular.patch. The target path Mathlib/Unsorry/NicomachusSumCubesTriangular.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

theorem sum_range_cube_eq_triangular_sq (n : ℕ) : ∑ i ∈ Finset.range (n + 1), i ^ 3 = (n * (n + 1) / 2) ^ 2 := by
  rw [nicomachus_sum_cubes (n + 1), Finset.sum_range_id, Nat.add_sub_cancel,
    Nat.mul_comm (n + 1) n]

Dependencies on sibling lemmas

The proof imports unsorry library modules that mathlib does not have — the sponsor must bundle or inline them (or upstream the dependency first):

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 Freek 100 / classic
reference Nicomachus of Gerasa, Introduction to Arithmetic, Book II, in explicit closed form; Graham, Knuth & Patashnik, Concrete Mathematics, §2.5 (∑k^3 = (n(n+1)/2)^2).
absence machine-checked no-local-match (grep of pinned mathlib rev c5ea00351c28, 2026-06-10); related lemmas exist but are different identities
difficulty 3
decomposition sketch Path A: prove the Nicomachus form ∑ i^3 = (∑ i)^2 (mirror the active goal), then rewrite ∑ i over range(n+1) as n(n+1)/2 via the Finset.sum_range_id closed form. Path B: direct induction, but the ℕ division means you must discharge 2 ∣ n*(n+1) via Nat.even_mul_succ_self so squaring commutes with /2.
title For every natural n, the sum over i in 0..n of i^3 equals (n(n+1)/2)^2 (the explicit triangular-number form of Nicomachus’ theorem).

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-triangular --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.
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