Upstream packet: sum-range-pow-four-closed-form

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

The statement (as proved here)

import Mathlib

theorem sum_range_pow_four_closed (n : ℕ) : 30 * (∑ k ∈ Finset.range (n + 1), (k : ℤ) ^ 4) = n * (n + 1) * (2 * n + 1) * (3 * n ^ 2 + 3 * n - 1) := by
  sorry

Kernel-verified on main: library/Unsorry/SumRangePowFourClosedForm.lean (theorem sum_range_pow_four_closed), 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 sum-range-pow-four-closed-form.patch. The target path Mathlib/Unsorry/SumRangePowFourClosedForm.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

theorem sum_range_pow_four_closed (n : ℕ) : 30 * (∑ k ∈ Finset.range (n + 1), (k : ℤ) ^ 4) = n * (n + 1) * (2 * n + 1) * (3 * n ^ 2 + 3 * n - 1) := by
  induction n with
  | zero => simp
  | succ n ih =>
    rw [Finset.sum_range_succ, mul_add, ih]
    push_cast
    ring

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	classic identities
reference	Faulhaber’s formula, case p=4. Conway & Guy, The Book of Numbers, Ch. 2; Knuth, ‘Johann Faulhaber and sums of powers’, Math. Comp. 61 (1993).
absence	machine-checked no-local-match (grep of pinned mathlib rev c5ea00351c28, 2026-06-10); related lemmas exist but are different identities
difficulty	2
decomposition sketch	Induction on n over ℤ. Base n=0 (both sides 0). Step via Finset.sum_range_succ then ring closes the polynomial identity. PRE-FLIGHT: before admission, confirm Finset.sum_range_pow specialization doesn’t trivialize it; if it does, downgrade to a corollary-application rather than a fresh induction.
title	For every natural n, 30 * (sum of k^4 for k in 0..n) = n(n+1)(2n+1)(3n^2+3n-1), the integer (ℤ) form of the Faulhaber p=4 identity.

Field

Value

source

classic identities

reference

Faulhaber’s formula, case p=4. Conway & Guy, The Book of Numbers, Ch. 2; Knuth, ‘Johann Faulhaber and sums of powers’, Math. Comp. 61 (1993).

absence

machine-checked no-local-match (grep of pinned mathlib rev c5ea00351c28, 2026-06-10); related lemmas exist but are different identities

difficulty

decomposition sketch

Induction on n over ℤ. Base n=0 (both sides 0). Step via Finset.sum_range_succ then ring closes the polynomial identity. PRE-FLIGHT: before admission, confirm Finset.sum_range_pow specialization doesn’t trivialize it; if it does, downgrade to a corollary-application rather than a fresh induction.

title

For every natural n, 30 * (sum of k^4 for k in 0..n) = n(n+1)(2n+1)(3n^2+3n-1), the integer (ℤ) form of the Faulhaber p=4 identity.

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

Read the proof until you can justify every step without AI assistance — mathlib reviewers will expect exactly that.

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.

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 sum-range-pow-four-closed-form --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.

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.

Record the outcome on the targets board (

in-discussion → pr-open →
merged | declined

). Declined is a valid, recorded result.