Cosmology Domain Coarsening3 D Z Fibers Nonempty

The nonempty fibers lemma

In the Recognition Science framework, a 3D field is modeled as a list of planes, each plane a list of 1D fibers, and each fiber a list of charge values. The declaration zFibers_nonempty states a simple condition: if every fiber in every plane is nonempty, then every z-fiber (the vertical column at a fixed x, y position) is also nonempty. This is a theorem in the framework's machine-checked library of formal theorems, proved by unfolding the definition of zFibers and applying the hypothesis to each plane and fiber. It is a basic structural guarantee, not a physical law.

The lemma matters because the framework's coarsening cost along the z-axis, expressed by the theorem zFiber_cost_eq, requires that every z-fiber be nonempty. That cost equals the total z-interface plus the number of z-fibers, which is the x-y cross-sectional cell count. The nonempty condition ensures the cost formula is well-defined for every column. Without it, a fiber could be empty, and the cost accounting would have no cells to carry. The lemma is the hygiene that makes the depth-independence result, zFiber_cost_depth_independent, applicable: two grids with the same interface and cross-section carry the same cost regardless of depth.

The lemma does not claim that the 3D component coarsening itself is separable, nor that the true 6-connected component count equals the separable cost. The framework states that the true component coarsening merges across fibers, so it carries at most the separable cost, bracketed by the cross-section below and the volume above. The nonempty lemma is a stepping stone, not a standalone physical statement. It does not assert anything about the physical world directly; it is a formal property of the list representation used in the framework's cost model.

/-- Every z-fiber is nonempty when every fiber in every plane is nonempty. -/
theorem zFibers_nonempty (grid : List (List (List α)))
    (h : ∀ plane ∈ grid, ∀ fiber ∈ plane, fiber ≠ []) :
    ∀ f ∈ zFibers grid, f ≠ [] := by
  intro f hf
  rw [zFibers, List.mem_flatten] at hf
  obtain ⟨plane, hplane, hfplane⟩ := hf
  exact h plane hplane f hfplane

/-- **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)

What this page does not claim

The lemma does not prove that the true 3D component coarsening is separable or equals the separable cost. The lemma does not make any claim about the physical universe; it is a formal property of the list representation. The lemma does not establish that the framework's cost model is the only possible one.

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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