Cosmology Polarized Birth Domains Polarized Components Eq Three

The three-region birth field

In the Recognition Science framework, a ledger is a discrete record of events, and a recognition is the act of matching a new event to a stored one. The framework models a growing world as a lattice of cells, each carrying a charge. The declaration polarized_components_eq_three is a proved theorem about a specific birth configuration: a field that assigns +1 to cells on one side of a central spine, -1 to cells on the other side, and 0 to the spine itself. The theorem states that this field always splits into exactly three connected monochromatic regions: the positive half, the negative half, and the zero spine. This holds for both the 2D diamond lattice and the 3D octahedron lattice, for any radius t ≥ 1.

The proof is a descent argument. The framework's machine-checked library of formal theorems defines a height function on each cell, and shows that from any cell not at a root, you can step along a same-charge edge to a cell of strictly smaller height. Since heights are natural numbers, this descent must terminate. The only cells where descent stops are the three roots: (1,0), (-1,0), and (0,0) in 2D, or (1,0,0), (-1,0,0), and (0,0,0) in 3D. Every cell therefore connects to one of these three roots, so the number of connected components is at most three. A separate argument shows that the three charges are genuinely distinct, so the count is exactly three, not fewer.

The importance is that the number of regions does not grow with the world's size. A lattice of radius t has on the order of t² cells in 2D and t³ in 3D, yet the birth field is carried by only three super-regions. This is the sharpest possible sub-extensivity: the carried-state fraction, components divided by volume, tends to zero as t grows. The framework's North-Star claim, that each region is carried at the coarsest rung its recognition allows, is here made exact for the birth configuration, as a theorem rather than a numerical observation.

In Recognition Science, this theorem is a step in the forcing chain that derives physical structure from the cost of recognition. It shows that a simple polarized field, the kind a shell creation installs each cadence cycle, can be described by a constant number of regions. The theorem is proved for the birth field only. The live engine then diffuses the spine by forced resolution, creating additional small interface components; those are bounded by a separate interface bound, not counted here. The O(1) carried-state statement is exact for the forced conjugate birth, which is the configuration the shell creation installs.

/-- **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)

/-- **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

This theorem does not claim that the live engine's diffused state has only three components. This theorem does not claim that the O(1) carried-state property holds for any field other than the specific polarized birth field. This theorem does not prove that the recognition cost function itself is the unique one; that is a separate theorem in the framework.

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)

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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