Cosmology Polarized Birth Domains Mem Fmono

The birth field's three regions

A recognition event is a discrete record of a comparison: two cells of a lattice are found to agree or differ. The declaration mem_Fmono is a small but load-bearing piece of that ledger. It says that if two neighboring cells carry the same charge, then the edge between them is included in a list called Fmono, the list of edges that stay inside a single charged region. The theorem is proved for two lattice shapes, the diamond in two dimensions and the octahedron in three, and it holds for every radius t from 1 upward.

The field in question is the forced conjugate-birth field, the configuration a shell creation installs at the start of each cadence cycle. It assigns +1 to the fine half of the lattice, -1 to the coarse half, and 0 to the spine between them. The theorem polarized_components_eq_three then proves the sharp result: the number of connected monochromatic regions in this field is exactly 3, independent of the radius. A world of Θ(t^d) cells, where d is the dimension, is carried as only three super-regions, so the carried-state fraction components divided by volume tends to 0. This is the strongest possible sub-extensivity: the cost of carrying the birth field does not grow with the world.

The proof uses a descent argument. A height function (|x| minus an indicator of x being nonzero, plus |y|, plus |z| in three dimensions) decreases along monochromatic edges until it reaches one of three roots: (1,0), (-1,0), or (0,0) in two dimensions, with the analogous triple in three. Since every cell descends to one of these roots, every cell is connected to a root, and the three roots cover all cells, so the component count is at most three. A separate argument shows three distinct charges exist, so the count is exactly three.

In Recognition Science, this result makes the North-Star claim "carry each region at the coarsest φ-rung its recognition allows" exact for the birth configuration. The framework's machine-checked library of formal theorems proves it as a theorem, not as a numeric estimate. The consequence is that the birth field is carried at O(1) cost, which means the ledger's memory footprint for the initial state does not scale with the universe's volume.

What the theorem does not claim is equally important. The live engine that follows the birth diffuses the spine by forced resolution, creating additional small interface components; those are bounded above by a separate Phase-48 interface bound, not counted here. The O(1) statement is exact only for the forced conjugate birth, the configuration the shell creation installs each cadence cycle. It is not a statement about the steady state of the engine, nor about arbitrary charge fields on the lattice.

/-- **The polarized birth field has exactly 3 locked domains, every radius (2-D).** The upper bound
`≤ 3` (descent toward the three roots) meets the lower bound `≥ 3` (the three roots carry the three
distinct charges `+1`, `-1`, `0`, and charge is a closure invariant). So the carried state is exactly
3 for all `t ≥ 1`: the fine half, the coarse half, and the spine. -/
theorem polarized_components_eq_three (t : ℕ) (ht : 1 ≤ t) : comp (Fmono t) = 3 := by
  refine le_antisymm (polarized_components_le_three t ht) ?_
  exact three_le_comp_of_three_charges (edges t) (polarized t)
    ⟨(1, 0), by rw [mem_ball_iff]; omega⟩
    ⟨(-1, 0), by rw [mem_ball_iff]; omega⟩
    ⟨(0, 0), by rw [mem_ball_iff]; omega⟩
    (by simp only [polarized]; dsimp only; decide)
    (by simp only [polarized]; dsimp only; decide)
    (by simp only [polarized]; dsimp only; decide)

/-- **Carried-state sub-extensivity for the birth field (2-D).** The polarized diamond fills
`2t² + 2t + 1` cells (Phase 49) but is carried as exactly 3 super-regions, so the carried state times
the radius is at most the area for every `t ≥ 1`: `3 t ≤ card (ball t)`, the carried cost is `O(1)`
while the world is `Θ(t²)`. -/
theorem polarized_carried_subextensive (t : ℕ) (ht : 1 ≤ t) :
    comp (Fmono t) = 3 ∧ 3 * t ≤ (InterfaceComponentBound.Diamond.ball t).card := by
  refine ⟨polarized_components_eq_three t ht, ?_⟩
  rw [LatticeBallVolume.Diamond.card_ball]
  nlinarith [ht]

/-- A monochromatic edge between two diamond cells: adjacency plus equal polarized charge. -/
theorem mem_Fmono (t : ℕ) (a b : Vtx t)
    (hadj : adj a.val b.val) (hc : polarized t a = polarized t b) : (a, b) ∈ Fmono t := by
  rw [Fmono, List.mem_filter]
  exact ⟨(mem_edges t a b).2 hadj, by simpa using hc⟩

What this page does not claim

The O(1) carried-state statement does not apply to the live engine's steady state after spine diffusion. The theorem does not bound components for arbitrary charge fields on the lattice, only for the polarized birth field. The component count of exactly three is not claimed for any field other than the forced conjugate-birth configuration.

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

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