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Upstream packet: sum-two-squares-zmod-four-ne-three

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

The statement (as proved here)

import Mathlib

theorem sum_two_squares_zmod_four_ne_three (a b : ℤ) : (((a ^ 2 + b ^ 2 : ℤ)) : ZMod 4) ≠ 3 := by
  sorry

Kernel-verified on main: library/Unsorry/SumTwoSquaresZmodFourNeThree.lean (theorem sum_two_squares_zmod_four_ne_three), 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-two-squares-zmod-four-ne-three.patch. The target path Mathlib/Unsorry/SumTwoSquaresZmodFourNeThree.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.Data.ZMod.Basic
import Mathlib.Tactic.Ring

theorem sum_two_squares_zmod_four_ne_three (a b : ℤ) : (((a ^ 2 + b ^ 2 : ℤ)) : ZMod 4) ≠ 3 := by
  have key : ∀ x y : ZMod 4, x ^ 2 + y ^ 2 ≠ 3 := by decide
  have hcast : ((a ^ 2 + b ^ 2 : ℤ) : ZMod 4) = (a : ZMod 4) ^ 2 + (b : ZMod 4) ^ 2 := by
    push_cast; ring
  rw [hcast]; exact key _ _

Dedup at mathlib HEAD

• mathlib revision scanned: 571b8a8e54219b4d393f75f4b8653fac08197fcc
• patterns: \bsum_two_squares_zmod_four_ne_three\b
• verdict: no-local-match
• matches:
• none

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

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 sum-two-squares-zmod-four-ne-three --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.