Cosmology Domain Coarsening3 D Foam Cost Tracks Interface

The surface cost law

Coarsening is a way to simplify a field of data by merging neighboring cells that share the same value. In one dimension, a line of cells splits into runs of equal values, and the number of runs equals the number of boundaries plus one. In three dimensions, the same idea applies to a stack of planes, each a grid of cells. The framework's ledger, a discrete record of events, carries each run as one super-region, and the cost of carrying the whole field is the number of super-regions it must hold.

The machine-checked theorem foam_cost_tracks_interface formalizes a specific comparison. Take two three-dimensional grids with the same cross-section, meaning the same number of vertical columns. If one grid has strictly more vertical boundaries between differing cells, then it carries strictly more super-regions, and therefore strictly higher cost. The theorem proves this for the separable coarsening cost, which treats each vertical column independently before merging across columns. A companion result, zFiber_cost_depth_independent, shows that changing only the depth of the columns, leaving the interface and cross-section fixed, does not change the carried cost at all.

These two results together express a surface law. The cost of carrying a three-dimensional structure is pinned to its internal interface, the surfaces where recognized regions meet, and to its cross-section, not to the volume it fills. A finely divided foam, with many domain walls, costs more to carry than a coarse split of the same extent, while a deeper world with the same boundaries costs nothing extra. The engine pays for recognition activity, not for bulk.

The theorem is proved in the framework's machine-checked library of formal theorems. It relies on a dimension-free graph fact: in any connected finite graph, the number of monochromatic components is at most the number of bichromatic edges plus one. The three-dimensional octahedral lattice is one instance of that connected-ambient hypothesis. The proof is complete for the separable cost, which upper-bounds the true component coarsening that merges across fibers, so the surface law holds as an upper bound for the full three-dimensional case.

What the theorem does not claim is a full proof for the true six-connected component coarsening in three dimensions. The separable cost is the proved backbone; the step that wires the specific lattice graph and identifies flood-fill components with graph components remains routine and unfinished. The theorem also does not claim that the physical recognition-to-linking bridge, which would connect this cost law to actual spatial dimensions, is closed. That bridge remains open. The result is a formal statement about the cost of carrying a grid, not a claim about the physics of space itself.

/-- **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

/-- **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 carried separable cost never exceeds the volume: the number of coarse super-regions along z is at most
the total number of cells (each fiber coarsens into at most as many runs as it has cells, `runs_le_length`).
Together with `zFiber_cost_eq`, the carried cost is bracketed `(#z-fibers) <= rowCost <= (volume)` and pinned
to the z-interface, so when the interface grows as a surface while the volume grows as `t^3`, the carried cost
is sub-extensive in the volume. -/
theorem zFiber_cost_le_volume (grid : List (List (List α))) :
    rowCost (zFibers grid) ≤ ((zFibers grid).map List.length).sum := by
  show ((zFibers grid).map runs).sum ≤ ((zFibers grid).map List.length).sum
  exact List.sum_le_sum (fun fiber _ => runs_le_length fiber)

What this page does not claim

The theorem does not prove the full three-dimensional component coarsening cost, only the separable per-axis cost that upper-bounds it. The theorem does not claim that the physical recognition-to-linking bridge for three dimensions is closed. The theorem does not state that a deeper world always costs the same, only that depth changes nothing when the interface and cross-section are fixed.

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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