Cosmology Polarized Birth Domains Comp Le Of Roots

The root bound

A discrete structure like a lattice or a graph can be divided into connected regions, often called components. The declaration comp_le_of_roots states a simple counting fact: if every vertex of a finite graph can reach at least one vertex from a chosen list of roots, then the number of connected components is at most the length of that list. The proof is direct: each component must contain at least one root, and no root can lie in two distinct components, so the components cannot outnumber the roots.

In the Recognition Science framework, this lemma is applied to the birth field, the initial configuration installed when a world is created. The field assigns a charge of +1 to one half of the space, -1 to the other half, and 0 to the dividing spine. The framework's machine-checked library of formal theorems proves that for this field, on both the diamond and octahedron lattices, the number of connected monochromatic regions is exactly three: the positive half, the negative half, and the spine. These three regions are the roots, and the lemma comp_le_of_roots guarantees that no finer subdivision can appear.

The consequence is a statement about information cost. A world of radius t contains on the order of t² or t³ cells, but the birth field can be described by just three regions, independent of t. The framework calls this sub-extensivity: the carried state does not grow with the world's size. The theorem polarized_carried_subextensive states this exactly, showing that the component count is three while the volume grows without bound.

What the lemma does not claim is equally important. It does not say anything about the later evolution of the world. After birth, the framework's live engine diffuses the spine, creating additional small interface components; those are bounded separately by an interface bound, not by this lemma. The O(1) carried-state statement is exact only for the forced conjugate-birth configuration, not for any later state.

/-- **Few roots bound the component count.** If every cell's component is represented by some root in
`roots`, the number of components is at most `roots.length`. (The quotient map, restricted to the
roots, hits every class.) -/
theorem comp_le_of_roots [Finite V] (E : List (V × V)) (roots : List V)
    (hcov : ∀ v : V, ∃ r ∈ roots, clos E v r) : comp E ≤ roots.length := by
  classical
  haveI : Fintype (Quotient (cs E)) := Fintype.ofFinite _
  have himg : (Finset.univ : Finset (Quotient (cs E)))
      ⊆ roots.toFinset.image (Quotient.mk (cs E)) := by
    intro q _
    obtain ⟨v, rfl⟩ := Quotient.exists_rep q
    obtain ⟨r, hr, hvr⟩ := hcov v
    rw [Finset.mem_image]
    exact ⟨r, List.mem_toFinset.2 hr, Quotient.sound ((clos_equiv E).symm hvr)⟩
  calc comp E = Fintype.card (Quotient (cs E)) := by
            rw [comp, Nat.card_eq_fintype_card]
    _ = (Finset.univ : Finset (Quotient (cs E))).card := by rw [Finset.card_univ]
    _ ≤ (roots.toFinset.image (Quotient.mk (cs E))).card := Finset.card_le_card himg
    _ ≤ roots.toFinset.card := Finset.card_image_le
    _ ≤ roots.length := List.toFinset_card_le roots

/-- **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 bound components for any field other than the specific polarized birth field. The O(1) carried state does not persist after the live engine begins diffusing the spine. The result does not imply anything about the total information content of the world, only the 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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