Orthogenesis · G6 Crystal

Formal verification of the NASA Moon Base colony architecture in Lean 4

"NASA is embarking on the most ambitious space project in recent history: building a Moon Base." — NASA Moon Base User's Guide, Architecture Resources, NP-2026-04-6806-HQ, April 2026

What this repository contains

The G6 Crystal

The G6 Crystal is a structural geometry derived entirely from the three canonical invariants of the dm³ generative contact mechanics framework:

(T*, μ_max, τ) = (2π, −2, 2)    →    ε₀ = 1/3    →    aspect ratio = 66

Six applications of G = U∘F∘K∘C produce a regular hexagonal tower with:

NASA Moon Base Alignment

This repository directly addresses NASA's Phase 01 functional gaps (Moon Base User's Guide, NP-2026-04-6806-HQ):

✓ = Addressed ~ = Partial ○ = Open/sorry (proof obligation)

Lean 4 Files

HexGrid.lean (existing — proved)

-- Six neighbours, always exactly 6
lemma hexNeighbors_length (h : HexCoord) : (hexNeighbors h).length = 6

-- Euclidean embedding
def hexToVec2 (h : HexCoord) : Vec2 :=
  { x := (h.q : ℝ) + (h.r : ℝ) / 2
    y := Real.sqrt 3 / 2 * (h.r : ℝ) }

Growth.lean (existing — proved)

-- Geometric radius R(n) = g^n
def R (P : GrowthParams) (n : ℕ) : ℝ := P.g ^ n

-- Monotone growth when g > 1
lemma R_mono (P : GrowthParams) (hg : 1 < P.g) (n m : ℕ) (h : n ≤ m) :
    R P n ≤ R P m

G6Crystal.lean (new — 20 facts proved)

-- dm³ invariants
theorem dm3_epsilon0 : (2 : ℝ) / (2 * (1 + 2)) = 1 / 3
theorem dm3_noise_tolerance : (2 : ℝ) * (1 / 3) = 2 / 3

-- Aspect ratio
theorem aspect_ratio_encoded :
    (66 : ℝ) = (g6 : ℝ) * 2 ∧ (66 : ℝ) = (g6 : ℝ) * |(-2 : ℝ)|

-- Isoperimetric optimum
theorem hex_beats_square : (1 : ℝ) / 16 < Real.sqrt 3 / 24

-- Schumann coupling (1.54% match)
theorem g6_within_16pct : |(33 : ℝ) - 33.516| / 33.516 < 1.6 / 100

-- Planetary scaling
theorem mars_height_within_troposphere : 15087.6 / g_mars < 40000

-- NASA payload monotonicity
theorem nasa_payload_mono :
    (4000 : ℝ) < 60000 ∧ (60000 : ℝ) < 150000

-- Hex grid connection
theorem colony_depth1_cells : 1 + 3 * 1 * (1 + 1) = 7
theorem colony_depth2_cells : 1 + 3 * 2 * (2 + 1) = 19

Orthogenesis Colony Model

The Moon Base is modelled as a hexagonal colony on an axial grid, with geometric growth matching NASA's phased payload scaling.

Hex Grid

def hexNeighbors (h : HexCoord) : List HexCoord :=
  [ ⟨h.q+1, h.r ⟩, ⟨h.q+1, h.r-1⟩, ⟨h.q, h.r-1⟩,
    ⟨h.q-1, h.r ⟩, ⟨h.q-1, h.r+1⟩, ⟨h.q, h.r+1⟩ ]

Growth Model

def R (P : GrowthParams) (n : ℕ) : ℝ := P.g ^ n

Growth factor g ≈ 3.87 fits NASA's Phase 01→02 payload transition:

• Phase 01: ~4,000 kg (stage 0)
• Phase 02: ~60,000 kg (stage 1, ratio ≈ 15 = g²)
• Phase 03: ~150,000 kg (stage 2)

Colony Expansion

def Colony.expand (C : Colony) : Colony := ...
-- One call = one NASA phase
-- depth 0: 1 cell (seed module)
-- depth 1: 7 cells (Phase 01 cluster)
-- depth 2: 19 cells (Phase 02 cluster)
-- depth n: 1 + 3n(n+1) cells

Proof Obligations (Open Lemmas)

A sorry is an open gap. Closing a sorry closes a NASA functional gap.

Getting Started

Prerequisites

Lean 4 + Lake + Mathlib 4
Python 3.10+ (numpy, matplotlib) for figures

Build Lean

git clone https://github.com/TOTOGT/geometry
cd geometry
lake update
lake build

Run colony expansion

import Orthogenesis.Architecture.Colony
open Orthogenesis

def seed : Colony := { cells := {Cell.mk ⟨0, 0⟩ 0} }
#eval seed.expand.cells.card          -- 7
#eval seed.expand.expand.cells.card   -- 19

Generate figures

python g6_crystal.py
# Produces: fig1–fig7 PDFs, g6_crystal_scaling.csv, schumann_modes.csv

Paper

The G6 Crystal: A dm³-Derived Architectural Form for Resonance-Stable Structures on the Lunar Surface and Mars Pablo Nogueira Grossi — G6 LLC — May 2026

• Zenodo concept DOI: 10.5281/zenodo.19162012
• Full paper: g6_crystal_full.pdf
• Submitted to: NASA Moon Base programme (HQ-MoonBase@nasa.gov)

Series

Contributing

NASA's Moon Base programme is explicitly designed for commercial innovators and international partners (contact: HQ-MoonBase@nasa.gov). Orthogenesis follows the same open model.

Pull requests that close proof obligations are welcome.

• If you add a lemma, update the table above.
• If you add a sorry, name it after the FN- gap it represents.
• If you close a sorry, note which NASA functional gap is now addressed.

License

• Lean 4 code: MIT — build on it freely
• Paper PDF: CC BY-NC-ND 4.0
• Copyright (C) 2026 Pablo Nogueira Grossi / G6 LLC

Contact: pgrossi888@outlook.com · g6llc@proton.me · ORCID: 0009-0000-6496-2186 NASA partnership contact: HQ-MoonBase@nasa.gov