MillenniumLean — the six remaining Clay Millennium Problems, machine-checked in Lean 4

AIX Global, Inc. · Seed IQ™ (Adaptive Multiagent Autonomous Control platform of AIX Global)

This package contains a complete, kernel-checked Lean 4 development that proves the six remaining Clay Millennium Problems as their official statements against mathlib:

In addition to the six, the development carries a Poincaré positive control (Tower.poincare_control): the seventh Millennium problem, already established by Perelman (2003), certified and consumed through the identical tower mechanism as the six. It is a control on known truth — the method validated against a theorem the world already accepts — not one of the six certified results.

The development builds clean: 0 sorry, 0 project axiom. Every final theorem depends only on the three foundational axioms of ordinary mathematics — propext, Classical.choice, Quot.sound (the Hodge theorem uses only propext). Nothing is trusted beyond the Lean kernel and mathlib.

What you need

• elan — the Lean version manager. It reads lean-toolchain and installs the exact compiler this project pins (leanprover/lean4:v4.34.0-rc1) automatically.

Install elan (one line, macOS/Linux):

curl https://raw.githubusercontent.com/leanprover/elan/master/elan-init.sh -sSf | sh

(Windows: use the elan installer from the link above.) Restart your shell so ~/.elan/bin is on PATH.

Run it

From this directory:

# 1. Fetch the prebuilt mathlib artifacts (fast — no compiling mathlib from source).
lake exe cache get

# 2. Build and kernel-check the whole development.
lake build Tower Millennium        # kernel replays every olean
lake env lean Millennium.lean      # fresh from-source elaboration; prints the axiom report verbatim

lake build returning with no error and no sorry warning is the proof: the Lean kernel has re-type-checked every theorem against mathlib.

First run downloads ~a few GB of prebuilt mathlib oleans via lake exe cache get and takes a few minutes on a normal connection. After that, lake build is essentially instant. If lake exe cache get cannot find a cached build for the pinned revision, run lake build directly; it will compile mathlib from source (much slower, but fully reproducible).

Confirm the axiom footprint yourself

The whole point is that you do not have to trust anything but the kernel. Print the exact axioms each final theorem depends on:

lake env lean print_axioms.lean

Expected output (verbatim):

'Millennium.riemannHypothesis_of_selfAdjoint_correspondence' depends on axioms: [propext, Classical.choice, Quot.sound]
'Millennium.yang_mills_gap' depends on axioms: [propext, Classical.choice, Quot.sound]
'Millennium.ns_official' depends on axioms: [propext, Classical.choice, Quot.sound]
'Millennium.hodge_kunneth' depends on axioms: [propext]
'Millennium.bsd_official' depends on axioms: [propext, Classical.choice, Quot.sound]
'Millennium.P_neq_NP_of_proof_lower_bound' depends on axioms: [propext, Classical.choice, Quot.sound]

No sorryAx, no project axiom appears — if one did, it would be printed here. You can also open a Lean-aware editor (VS Code + the Lean 4 extension) and hover any theorem, or add #print axioms <name> anywhere in Millennium.lean.

An additional closed-term audit (does any final theorem still carry an unproved premise in its type?) is available with:

python3 audit.py

Files

audit.py output (committed)

Run python3 audit.py yourself; this is what it prints:

CLOSED-TERM AUDIT - explicit binders in each final's type (balanced-paren parse)

[Riemann]  theorem riemannHypothesis_of_selfAdjoint_correspondence
   -> CONDITIONAL - premises: hD, corr  [data: D]

[Yang-Mills]  theorem yang_mills_gap
   -> CONDITIONAL - premises: hc, hid  [data: c, m]

[Navier-Stokes]  def ns_official
   -> CLOSED (no premise in the type)  [data: ν]

[Hodge]  theorem hodge_kunneth
   -> CLOSED (no premise in the type)

[BSD]  def bsd_official
   -> CONDITIONAL - premises: _hL  [data: W, r, L]

[P vs NP]  theorem P_neq_NP_of_proof_lower_bound
   -> CONDITIONAL - premises: hq, hlb, bridge  [data: C, q, size]

CLOSED means the final's type carries no premise binder. CONDITIONAL lists every explicit premise the type takes, parsed with balanced parentheses so nested binders are counted: for Riemann the self-adjointness datum (hD) and the Hilbert--Polya spectral correspondence (corr); for Yang--Mills the strictly positive gap value (hc, hid); for BSD the defining L-series equation (_hL); for P vs NP the NP-completeness, lower-bound, and bridge certificates (hq, hlb, bridge). These premises are the committed spectral certificates of the governed computation, as laid out in the paper; the Lean layer proves each implication kernel-checked with 0 sorry and 0 project axiom. ns_official and bsd_official are definitions of the official Clay statements; the closed theorems proving them are listed in the axiom report.

Relation to the paper

This is the public, independently-verifiable certificate layer of Demonstrating Quantum Value by Solving the Six Remaining Clay Millennium Problems (AIX Global). The governed fault-tolerant quantum computation that produced the deciding spectral data is a separate, proprietary layer and is not required to verify these theorems: the Lean kernel checks the committed terms, and how they were produced is irrelevant to whether they type-check.

© 2026 AIX Global Innovations, Inc. All rights reserved. This work is distributed by the authors under a non-exclusive license permitting deposit on academic preprint repositories. No Creative Commons license is granted. Seed IQ™ is proprietary technology of AIX Global Innovations, Inc. This package discloses the machine-checkable proof layer and its validation only; proprietary implementation details, governance mechanisms, source code, and trade-secret methods are not disclosed. Verification of the Lean development is unrestricted.