Cosmology Finite Cell Boundary Bounded Voxel

The bounded voxel

A bounded voxel is a finite, three-dimensional grid of cells with no wrapping edges. Think of a box of fixed size, divided into smaller cubes, where each cube has a definite position and the box has a definite boundary. The declaration BoundedVoxel in the Recognition Science library establishes exactly this: given three positive natural numbers nx, ny, and nz, it defines a structure that requires each dimension to have at least one cell. It is a scaffold, not a physical claim, a formal container that later work can build upon.

The declaration belongs to a family of finite cell boundary structures in the framework's machine-checked library of formal theorems. Its siblings include the periodic 1D ring, which wraps around on itself like a circle of sites, and the open 2D patch, a flat grid without wrap. The bounded voxel is the 3D member of this family: open, finite, and bounded in all three directions. The only theorem attached to this structure is a simple one: if you have a periodic ring with n sites, then n is greater than zero. The voxel itself carries no theorems yet, only the positivity requirements that make it a valid finite object.

In Recognition Science, this structure is part of an engineering scaffold for cosmology. The library's documentation states that Python exact cells in scripts mirror these boundary tags, with no new axioms, and certificate targets for periodic ring and open patch serialization. In plain terms: the framework is setting up the formal vocabulary for describing finite regions of space in its cosmological models. The bounded voxel is the 3D building block that says, here is a finite box with a boundary, nothing more.

What the declaration does not claim is equally important. It does not say that physical space is made of voxels, that the universe is finite, or that any particular cosmology follows from this structure. It does not assert that recognition events happen inside voxels, nor that the eight-tick cycle or three-dimensional forcing theorems apply to this specific container. The bounded voxel is a definitional choice, a tool, not a result. It establishes a finite 3D grid with positive dimensions and no wrap, and it leaves all physical interpretation to later work.

/-- Bounded 3D voxel without wrap. -/
structure BoundedVoxel (nx ny nz : Nat) where
  nx_pos : 0 < nx
  ny_pos : 0 < ny
  nz_pos : 0 < nz

theorem periodic_ring_n_pos {n : Nat} (c : PeriodicRing n) : 0 < n := c.n_pos

/-- Bounded 3D voxel without wrap. -/
structure BoundedVoxel (nx ny nz : Nat) where
  nx_pos : 0 < nx
  ny_pos : 0 < ny
  nz_pos : 0 < nz

What this page does not claim

Physical space is made of voxels. The universe is finite. Recognition events happen inside voxels.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Cosmology/FiniteCellBoundary.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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