Cosmology Polarized Birth Domains Clos Mono Charge

The charge-closure lemma

In graph theory, a connected component is a set of vertices linked by edges. The lemma clos_mono_charge concerns a graph whose edges have been filtered to keep only those joining vertices with the same charge value. It states that if two vertices are connected in this filtered graph, then they must have the same charge. The proof is a direct induction on the path: each step along an edge preserves the charge, so the whole path does.

This is a foundational lemma in the framework's cosmology module. It guarantees that the monochromatic regions, the areas of uniform charge, are exactly the connected components of the filtered graph. The lemma itself is a theorem in the machine-checked library of formal theorems, proved with no axioms beyond the standard three. It does not by itself say how many such regions exist, only that they are well-defined.

The payoff comes in the companion theorems. For the specific "polarized" birth field, which assigns +1 to one half of a diamond or octahedron lattice, -1 to the other half, and 0 to the dividing spine, the framework proves that the number of these monochromatic regions is exactly three, independent of the lattice radius. A world with Θ(t^d) cells is thus carried as just three super-regions, the sharpest possible compression. This is the sense in which the framework models the carried state of a birth configuration as sub-extensive.

What the lemma does not claim is also precise. It does not describe the live engine's behavior after birth, when the spine diffuses and creates additional small interface components. Those are bounded by a separate interface bound, not by this lemma. The "carried at O(1)" statement is exact only for the forced conjugate birth configuration, not for the subsequent dynamics.

/-- **Charge is a monochromatic-closure invariant.** Two cells connected through monochromatic edges
carry the same charge. Induction on the equivalence-closure derivation: the generating (monochromatic)
edges are equal-charge by construction; reflexivity, symmetry, and transitivity preserve equality. -/
theorem clos_mono_charge {β : Type*} [DecidableEq β] (E : List (V × V)) (c : V → β) {u v : V}
    (h : clos (E.filter (fun p => decide (c p.1 = c p.2))) u v) : c u = c v := by
  induction h with
  | rel x y hxy =>
      simp only [gen, List.mem_filter, decide_eq_true_eq] at hxy
      exact hxy.2
  | refl x => rfl
  | symm x y _ ih => exact ih.symm
  | trans x y z _ _ ih1 ih2 => exact ih1.trans ih2

/-- **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]

What this page does not claim

The lemma does not describe the number of regions for any charge field other than the polarized birth field. The O(1) carried-state claim does not apply to the live engine's post-birth dynamics. The lemma does not establish the existence of the birth field itself, only its properties.

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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