Cosmology Finite Cell Boundary Open Patch

The open patch

In cosmology, a finite patch of space often needs a boundary condition. The open patch is one such condition: a rectangular grid of cells with a finite width and height, where the edges simply stop. No cell wraps around to meet another edge, unlike a ring or torus. The framework's machine-checked library of formal theorems defines this as a structure requiring only that both dimensions be positive, so a 1 by 1 patch is allowed but a 0 by anything is not.

The open patch belongs to a family of boundary tags for finite recognition cell commitments. Its neighbors include a periodic 1D ring, where sites wrap so the last cell posts to the first, and a bounded 3D voxel, a finite block without wrap. These are engineering scaffolds, not physical claims. The library docstring says Python exact cells mirror these boundary tags, and the certificate targets cover periodic ring and open patch serialization. Nothing here asserts that the cosmos actually uses an open patch; the declaration only fixes what the word means inside the framework.

In Recognition Science, the open patch models a discrete spatial region whose boundary is not identified with itself. The framework treats conventional physics as itself RS-derived, so this boundary condition is a candidate for how a finite recognition ledger, a discrete record of events, could give rise to a patch of space. But the declaration itself proves nothing about cosmology. It is a definitional choice, a structure with two positive natural numbers, and the only theorem attached to the whole file is a trivial one about the periodic ring's positivity.

What the open patch does not claim is worth stating plainly. It does not say that space is discrete, that the cosmos is finite, or that any particular boundary condition is physical. It does not derive the three spatial dimensions from linking, nor does it assert that the physical recognition-to-linking bridge is closed. The open patch is a tool for serialization and bookkeeping, a named shape that later work can use. A reader should take it as a precise vocabulary item, not as a cosmological discovery.

/-- Open 2D patch: finite grid without wrap. -/
structure OpenPatch (nx ny : Nat) where
  nx_pos : 0 < nx
  ny_pos : 0 < ny

/-- Open 2D patch: finite grid without wrap. -/
structure OpenPatch (nx ny : Nat) where
  nx_pos : 0 < nx
  ny_pos : 0 < ny

What this page does not claim

No claim that space is discrete or that the cosmos is finite. No derivation of three spatial dimensions from this boundary condition. No assertion that the physical recognition-to-linking bridge is closed.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: