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#Geometry
Orthogenesis · Geometry · Lean 4 · leanprover/lean4:v4.32.0 · page 2026-09-14

Cell Blueprint

The geometry layer the architecture stands on. Six Lean files — a hex grid, a cell, a colony, its growth, the hexagonal form and a discrete Gauss–Bonnet — carrying fifteen declarations with no sorry. Everything in the NASA gap ledger is built from these.
Files6 in Orthogenesis/Geometry/
all imported by Orthogenesis.lean
Status15 declarations, no sorry
compiles; no axiom report yet
Depends on nothingno material, no gravity, no atmosphere
which is why the results travel
This page used to say “Lean 4 · HexGrid · Cell · Colony · Growth” and nothing else. The files it named are the ones every architecture claim in this corpus is built on, so they are worth more than a list.

1 · The six files

FileDeclsWhat it establishes
HexGrid.lean2Axial coordinates, and that every cell has exactly six neighbours, all distinct
Cell.lean0The unit: a coordinate and a stage. Definitions only
Colony.lean4A set of cells and its expand; reachability is monotone — the basis of the NASA autonomy row
Growth.lean1Stage-gating: a cell at stage n cannot appear before stage n−1 exists
HexForm.lean5The hexagon itself — the isoperimetric optimum the habitation row rests on
GaussBonnet.lean3Discrete Gauss–Bonnet on the tiling: curvature is bookkeeping, and it closes

2 · Why this layer is the one that travels

Not one of these files mentions a material, a gravity, an atmosphere or a load. That is not modesty — it is the reason the results hold equally in vacuum, in Newark soil and at Chichén Itzá. Everything above this layer carries a hypothesis; everything in it carries none.

Which theorem may be cited where, and which may not, is set out in WP-119 · Which Theorem Travels. The short version: the tiling and the graphene honeycomb are duals, both carry D₆ symmetry, and only the tiling carries the load-sharing result. Sharing a symmetry group is the weakest possible reason to think two results are one result.

Compiles is not audited

All six are imported by Orthogenesis.lean and build with the target, so they are known to elaborate. None currently carries a #print axioms report, so nothing has checked what their theorems rest on. That is a real gap, it is listed as declared, no gate in the ledger, and it is stated here rather than left for a reader to discover.

3 · What is built on it

NASA ArchitectureThe Moon Base gap ledger — thirteen NASA functional gap codes, twelve theorems
Ch 18 · SeismicLoad share, crack tortuosity, detuning
Ch 19 · AcousticThe staircase chirp
G6 Earth HouseThe same geometry as compressed-earth-block housing