Cosmology Domain Coarsening3 D Z Fiber Cost Depth Independent

The depth independence of coarsening cost

In the Recognition Science framework, recognition, a discrete record of events, carries a forced cost. The declaration zFiber_cost_depth_independent is a theorem about that cost in three dimensions. It states that two three-dimensional grids of cells, when coarsened along the z-axis, carry the same cost if they have the same total z-interface and the same number of z-fibers, regardless of how deep those fibers are. The cost is independent of the depth.

To understand this, consider a grid indexed by x, y, and z. A z-fiber is a column of cells at a fixed (x, y) coordinate. Coarsening along z means grouping consecutive equal-charge cells in each fiber into runs. The cost is the total number of these runs across all fibers. A companion theorem, zFiber_cost_eq, proves this cost equals the total number of z-interfaces (boundaries between different charges) plus the number of z-fibers. The number of z-fibers is simply the cross-sectional cell count. Neither term depends on the length of the fibers. Making the world deeper in z, by adding more cells to each fiber, does not increase the carried cost; only adding new distinctions, new interfaces, does.

This result is the per-axis backbone of a larger claim: the cost of carrying a three-dimensional domain lives on its surface, not in its volume. A related theorem, zFiber_cost_le_volume, shows the cost never exceeds the total number of cells. Together, these theorems bracket the cost between the cross-section and the volume, pinning it to the interface. The framework's library, a machine-checked collection of formal theorems, proves these statements with no axioms beyond the standard three. This is a formal result about a specific model of coarsening, not a statement about all possible physical systems.

What the declaration does not claim is broader. It does not claim that the true three-dimensional component coarsening, which merges across fibers, has this exact cost; it only proves the separable, per-axis cost. The true cost is at most this separable cost. It also does not claim that the physical universe is a grid of this kind. The theorem applies to a discrete model, not to continuous space. Finally, it does not claim that depth never matters in any coarsening process; it claims only that in this specific separable model, with the interface and cross-section fixed, the depth is irrelevant.

/-- **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 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)

/-- 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 claim that the true three-dimensional component coarsening, which merges across fibers, has exactly this separable cost. The theorem does not claim that the physical universe is a discrete grid of cells. The theorem does not claim that depth never matters in any coarsening process, only in this specific separable model.

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)

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.

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