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Upstream packet: fourth-power-mod-fortyone-mem

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

The statement (as proved here)

import Mathlib

theorem fourth_power_mod_fortyone_mem (r : ℕ) (hr : r < 41) : (∃ n : ℕ, n ^ 4 % 41 = r) ↔ r ∈ ({0, 1, 4, 10, 16, 18, 23, 25, 31, 37, 40} : Finset ℕ) := by
  sorry

Kernel-verified on main: library/Unsorry/FourthPowerModFortyoneMem.lean (theorem fourth_power_mod_fortyone_mem), 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 fourth-power-mod-fortyone-mem.patch. The target path Mathlib/Unsorry/FourthPowerModFortyoneMem.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

theorem fourth_power_mod_fortyone_mem (r : ℕ) (hr : r < 41) : (∃ n : ℕ, n ^ 4 % 41 = r) ↔ r ∈ ({0, 1, 4, 10, 16, 18, 23, 25, 31, 37, 40} : Finset ℕ) := by
  constructor
  · rintro ⟨n, rfl⟩
    have h : n ^ 4 % 41 = (n % 41) ^ 4 % 41 := by
      rw [Nat.pow_mod]
    rw [h]
    have hlt : n % 41 < 41 := Nat.mod_lt _ (by norm_num)
    interval_cases (n % 41) <;> decide
  · intro hr2
    fin_cases hr2
    · exact ⟨0, by decide⟩
    · exact ⟨1, by decide⟩
    · exact ⟨11, by decide⟩
    · exact ⟨4, by decide⟩
    · exact ⟨2, by decide⟩
    · exact ⟨16, by decide⟩
    · exact ⟨7, by decide⟩
    · exact ⟨6, by decide⟩
    · exact ⟨12, by decide⟩
    · exact ⟨8, by decide⟩
    · exact ⟨3, by decide⟩

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 #400 Identity Engine (ADR-043) — power-residue family; promoted from candidate backlog.  
reference The fourth-power residues modulo the prime 41 are exactly {0,1,4,10,16,18,23,25,31,37,40}. Not a named mathlib lemma in this form.  
absence no-local-match (grep of pinned mathlib rev c5ea00351c, 2026-06-15); generic-lemma instances dropped via direct mathlib grep (sub_dvd_pow_sub_pow / Odd.add_dvd_pow_add_pow / add_pow / centralBinom / Vandermonde / fib_dvd).  
triviality machine-checked non-trivial (battery v1, rev c5ea00351c, 2026-06-15).  
difficulty 2  
decomposition sketch 4 40 gives only 11 quartic residues; Nat.pow_mod and decide over n % 41 with raised maxRecDepth. Verified to build (lake env lean) at sourcing.
title The fourth-power residues modulo the prime 41 are exactly {0,1,4,10,16,18,23,25,31,37,40}.  

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 fourth-power-mod-fortyone-mem --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.