IndisputableMonolith.Cosmology.DomainCoarsening3D
Three-dimensional locked-domain coarsening on a charge grid indexed by (x,y,z). Each fixed-(x,y) column is a 1D z-fiber that the engine collapses into maximal equal-charge runs; fiber costs are depth-independent and at most the fiber volume. Cosmologists tracking foam interface cost versus bulk volume would cite the fiber lemmas and the foam-cost-tracks-interface bound. The module lifts the 2D perimeter localization by flattening x-y and reusing the 1D run engine on each fiber.
claimOn a 3D charge grid $Q[x][y][z]$, the $z$-fibers are the columns at fixed $(x,y)$. Each fiber is coarsened into its maximal equal-charge runs. The recognition cost of a fiber equals the 1D run cost, is independent of fiber depth, and is at most the fiber volume; the total foam cost tracks the active interface rather than the bulk volume.
background
Recognition Science treats locked domains as maximal connected equal-charge regions; the engine pays only on the recognition-active interface between domains. In one dimension that interface is a set of run boundaries. The upstream 2D module lifts this to a plane: locked domains become maximal 4-connected components, the interface is a 1D curve, and cost localizes to a perimeter (sub-extensive in area).
This module steps to three spatial dimensions. A 3D grid is sliced into 1D $z$-fibers by fixing $(x,y)$ and flattening the $x$ and $y$ levels. Each fiber is a charge column; the same 1D coarsening engine collapses it into maximal equal-charge runs. The sibling definitions package the fiber list (zFibers), emptiness and non-emptiness facts, equality of fiber cost with the 1D run cost, the volume upper bound, depth-independence of that cost, and the global statement that foam cost tracks interface rather than bulk.
proof idea
Definition-heavy module with short supporting lemmas, not a single monolithic theorem. Fibers are obtained by flattening the $x$ and $y$ indices so each column is an ordinary 1D charge list; nil and nonempty facts are list bookkeeping. Fiber cost is identified with the 1D run cost by construction, hence inherits the volume bound and becomes independent of how deep the column is. The foam-cost-tracks-interface claim assembles the per-fiber bounds: total cost sums interface contributions across fibers and therefore cannot scale with bulk volume. The argument reuses the locked 2D coarsening module only as the dimensional stepping stone; the actual algebra stays on 1D runs.
why it matters in Recognition Science
In the RS forcing chain, $D=3$ is forced at T8; cosmology therefore needs a 3D domain picture whose recognition cost stays on interfaces (foam walls) rather than filling volume. This module supplies that picture: $z$-fiber coarsening makes the 3D cost a sum of 1D run costs, so foam cost tracks interface. It sits directly above DomainCoarsening2D (perimeter localization in the plane) and feeds any downstream cosmogenesis or foam-energy accounting that needs a Lean-backed 3D cost bound. No further used-by edges are recorded yet; the natural parents are global foam-energy or structure-formation statements that quote depth-independent fiber cost and the interface-tracking inequality.
scope and limits
- Does not derive $D=3$; that is T8 upstream.
- Does not model time evolution or dynamics of domain walls.
- Does not bound continuum PDE free-boundary problems; only discrete grids.
- Does not claim optimality of the fiber slicing versus other 3D partitions.
- Does not connect fiber cost to observed cosmological parameters.