The G6 Crystal: A dm³-Derived Architectural Form for Resonance-Stable Tall Structures

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 File Description HexGrid. lean Axial hex coordinates, Euclidean embedding, hexNeighbors (proved) Growth. lean Geometric radius model R (n) = gⁿ, Rₘono (proved) G6Crystal. lean NEW — G6 Crystal formal verification (20 facts, 0 sorry) g6crystal. py Python figure generator (7 figures, 2 CSV data tables) g6crystalfull. pdf Full paper — NASA Moon Base contribution Orthogenesis/Geometry/ HexGrid + Growth module tree README. md This file 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*, μₘax, τ) = (2π, −2, 2) → ε₀ = 1/3 → aspect ratio = 66 Six applications of G = U∘F∘K∘C produce a regular hexagonal tower with: Parameter Value Derivation Total height 15, 087. 6 m (33, 000 cubits) 1, 000·g⁶·τ Layer height 2, 514. 6 m (5, 500 cubits) Total / 6 Base side 114. 30 m (250 cubits) 500/2 Aspect ratio 66 = 33·τ g⁶·τ = g⁶·|μₘax| Stability radius ε₀ = 1/3 Theorem D 2 Noise tolerance τ·ε₀ = 2/3 Locked constants NASA Moon Base Alignment This repository directly addresses NASA's Phase 01 functional gaps (Moon Base User's Guide, NP-2026-04-6806-HQ): NASA Gap Description G6 Crystal Response Status FN-H-101L / FN-H-102L Pressurised habitable environment G¹ hex module → G⁶ full structure ✓ FN-L-101L Pressurised mating Standardised hex interface geometry ✓ FN-U-103L ISRU operations Material self-sufficiency (6 layers → 6 seeds) ~ FN-T-201L / FN-T-202L Cargo transport to surface Phased hex payload scaling (g ≈ 3. 87) ✓ FN-P-101L / FN-P-402L Power, year+ duration Arnold tongue A₄: ₁ passive stability ~ FN-A-104L Robotic manipulation Colony. expand (Orthogenesis) ○ ✓ = Addressed ~ = Partial ○ = Open/sorry (proof obligation) Lean 4 Files HexGrid. lean (existing — proved) -- Six neighbours, always exactly 6 lemma hexNeighborsₗength (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) = gⁿ def R (P: GrowthParams) (n: ℕ): ℝ: = P. g ^ n -- Monotone growth when g > 1 lemma Rₘono (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 dm3ₑpsilon0: (2: ℝ) / (2 * (1 + 2) ) = 1 / 3 theorem dm3ₙoiseₜolerance: (2: ℝ) * (1 / 3) = 2 / 3 -- Aspect ratio theorem aspectᵣatioₑncoded: (66: ℝ) = (g6: ℝ) * 2 ∧ (66: ℝ) = (g6: ℝ) * | (-2: ℝ) | -- Isoperimetric optimum theorem hexbeatsₛquare: (1: ℝ) / 16 < Real. sqrt 3 / 24 -- Schumann coupling (1. 54% match) theorem g6within₁6pct: | (33: ℝ) - 33. 516| / 33. 516 < 1. 6 / 100 -- Planetary scaling theorem marsₕeightwithinₜroposphere: 15087. 6 / gₘars < 40000 -- NASA payload monotonicity theorem nasaₚayloadₘono: (4000: ℝ) < 60000 ∧ (60000: ℝ) < 150000 -- Hex grid connection theorem colonydepth1cells: 1 + 3 * 1 * (1 + 1) = 7 theorem colonydepth2cells: 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) Lemma Statement Status hexNeighborsₗength (hexNeighbors h). length = 6 ✓ proved Rₘono 1 < g → n ≤ m → R P n ≤ R P m ✓ proved hexbeatsₛquare (1: ℝ) /16 < √3/24 ✓ proved nasaₚayloadₘono 4000 < 60000 ∧ 60000 < 150000 ✓ proved expandₘono C. cells ⊆ (C. expand). cells ◑ in progress stagebound Every cell in expandⁿ has stage ≤ n ○ open coordcoverage Ring at distance k has 6k cells ○ open nocoordcollision Well-formed colonies have unique coords ○ open S1ₐrnoldₜongue Arnold tongue A₄: ₁ coupling ○ open (ODE flow) S2ₕexagridcollapse Hexagrid collapse superiority ○ open (FEM data) 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 g6crystal. py # Produces: fig1–fig7 PDFs, g6crystalₛcaling. csv, schumannₘodes. 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: g6crystalfull. pdf Submitted to: NASA Moon Base programme (HQ-MoonBase@nasa. gov) Series Role Record Series root (Principia Orthogona) 10. 5281/zenodo. 19117399 Volume One (mathematics) HAL hal-05555216v1 Volume Two (contact geometry) HAL hal-05559997v1 Biological transitions V2 10. 5281/zenodo. 20230612 Drosophila connectome V2 10. 5281/zenodo. 20128568 k-nacci spine (polylaminin) 10. 5281/zenodo. 20230633 G6 Crystal (this repo) 10. 5281/zenodo. 19162012 AXLE (formal verification hub) github. com/TOTOGT/AXLE 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