Cosmology Domain Coarsening3 D Z Fibers

The z-fiber cost

In the Recognition Science framework, reality keeps a ledger, a discrete record of events, and the cost of recognizing a new state is forced by a proved theorem. The declaration zFibers is a small but load-bearing piece of that framework's three-dimensional cosmology. It takes a three-dimensional grid, indexed by [x][y][z], and flattens the x and y levels to produce a list of one-dimensional columns, each column being the charge values at a fixed (x, y) position along the z-axis. These columns are the z-fibers.

The key theorem, zFiber_cost_eq, proves that the cost of coarsening this grid along the z-axis equals the total z-interface (the number of boundaries between different charges) plus the number of z-fibers (which is the x-y cross-sectional cell count). The right side of this equation has no dependence on the length of the fibers. This means that making the world deeper in the z direction does not increase the carried cost; only adding distinctions in the x-y plane does. This is the formal, per-axis backbone of the surface law: the cost lives on the interface surface, not in the volume.

A companion theorem, zFiber_cost_depth_independent, states this directly. Two grids with the same total z-interface and the same number of z-fibers carry the same separable cost, regardless of how deep they are. A further result, foam_cost_tracks_interface, shows that at a fixed cross-section, strictly more z-interface costs strictly more. A finely recognized structure, a foam with many domain walls, is carried at a strictly higher cost than a coarse split of the same extent, while changing only the depth costs nothing. The engine pays for recognition activity, not for volume.

It is important to be precise about what this theorem does not claim. It proves a separable, per-axis cost, meaning it coarsens each fiber independently. The true three-dimensional component coarsening merges across fibers as well, and the library shows this true cost is never worse than the separable one, but the strict equality theorem applies only to the fiber-wise calculation. The proof also assumes every fiber is nonempty, which is a routine structural condition. Finally, this is a theorem about a mathematical model of cost; it does not by itself assert that physical space is three-dimensional, which is a separate result in the forcing chain.

/-- **The 3D separable coarsening cost = z-interface + number of z-fibers, independent of the depth.**
For any 3D grid whose z-fibers are all nonempty, coarsening along the z-axis carries exactly (the total
z-interface) plus (the number of z-fibers = the x-y cross-sectional cell count) super-regions. The right side
depends only on the interface and the cross-section, never on the fiber lengths: making the world deeper in z
does not increase the carried cost. This is the exact per-axis generalization of the 1D law
`runs = boundaries + 1`, summed over every fiber of the 3D grid, and the per-axis backbone of the surface law.
The true 3D component coarsening also merges across fibers, so it carries at most this many super-regions. -/
theorem zFiber_cost_eq (grid : List (List (List α)))
    (h : ∀ plane ∈ grid, ∀ fiber ∈ plane, fiber ≠ []) :
    rowCost (zFibers grid) = rowInterface (zFibers grid) + (zFibers grid).length :=
  rowwise_cost_eq (zFibers grid) (zFibers_nonempty grid h)

/-- **Depth independence, stated directly.** Two 3D grids with the same total z-interface and the same number
of z-fibers carry the same separable cost, regardless of how their fibers differ in length (depth). This is the
formal sense in which the 3D cost lives on the interface surface and the cross-section, not in the volume. -/
theorem zFiber_cost_depth_independent (g₁ g₂ : List (List (List α)))
    (h₁ : ∀ plane ∈ g₁, ∀ fiber ∈ plane, fiber ≠ [])
    (h₂ : ∀ plane ∈ g₂, ∀ fiber ∈ plane, fiber ≠ [])
    (hiface : rowInterface (zFibers g₁) = rowInterface (zFibers g₂))
    (hcross : (zFibers g₁).length = (zFibers g₂).length) :
    rowCost (zFibers g₁) = rowCost (zFibers g₂) := by
  rw [zFiber_cost_eq g₁ h₁, zFiber_cost_eq g₂ h₂, hiface, hcross]

/-- **The cost tracks the recognition interface, not the depth (the Phase-15 foam law, formalized).** Two 3D
grids with the same number of z-fibers (the same x-y cross-section): the one whose field carries strictly more
z-interface carries strictly more separable cost. This is the exact sense in which a finer, more recognized
structure (a foam with more domain walls) costs strictly more to carry than a coarser one at the same extent,
while deepening the world in z (changing fiber lengths, with the interface fixed) changes nothing
(`zFiber_cost_depth_independent`). The engine pays for recognition activity, not volume. -/
theorem foam_cost_tracks_interface (g₁ g₂ : List (List (List α)))
    (h₁ : ∀ plane ∈ g₁, ∀ fiber ∈ plane, fiber ≠ [])
    (h₂ : ∀ plane ∈ g₂, ∀ fiber ∈ plane, fiber ≠ [])
    (hcross : (zFibers g₁).length = (zFibers g₂).length)
    (hmore : rowInterface (zFibers g₁) < rowInterface (zFibers g₂)) :
    rowCost (zFibers g₁) < rowCost (zFibers g₂) := by
  rw [zFiber_cost_eq g₁ h₁, zFiber_cost_eq g₂ h₂, hcross]
  omega

What this page does not claim

The theorem does not claim that the true three-dimensional component coarsening cost equals the separable fiber cost; it only upper-bounds it. The theorem does not claim that physical space is three-dimensional; that is a separate result in the forcing chain. The theorem does not claim that coarsening along the x or y axes has the same depth-independence property.

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/DomainCoarsening3D.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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