Cosmology Polarized Birth Interface

The recognition-active surface

In a lattice of cells, imagine coloring each cell by the sign of its first coordinate: positive on one side, negative on the other, zero on the central plane. The recognition activity, the set of edges where neighboring cells carry different colors, marks every place where a distinction is forced. The question is where that activity lives.

The answer, proved as a theorem in the framework's machine-checked library of formal theorems, is that it lives on the spine. Every edge joining cells of different charge has at least one endpoint on the plane x = 0. In two dimensions, the spine is a line segment of length 2t + 1 cells; in three dimensions, it is a diamond-shaped disk of 2t² + 2t + 1 cells. Both are one dimension lower than the world they sit in, so the interface occupies a fraction that shrinks to zero as the radius t grows. The cost of making distinctions localizes to a surface, not a volume.

This is the polarized birth interface: the boundary where the forced conjugate-birth field, a field that takes value +1 on one side and -1 on the other, does its work. The module proves two complementary facts. First, the field is carried as exactly three locked domains for every radius, a constant amount of carried state. Second, the entire recognition-active interface is confined to the spine. Together these close the sub-extensivity picture: both the carried domains and the active interface vanish as a fraction of the growing world.

The proof itself is pure case analysis. The charge sign(x) flips only between columns x = -1, 0, and 1, and a charge-flipping adjacency must step the x-coordinate across zero, so one endpoint lands on the spine. The cardinality of the spine follows from a bijection with a lower-dimensional ball. No numeric simulation is involved; the localization is a theorem, not an observation.

What this changes is the picture of how a recognition-based cosmology can be efficient. If the cost of distinguishing regions had to spread through the volume, the ledger would grow with the world. Instead, the framework proves the cost concentrates on a surface, a result that holds in the dimension D = 3 that the forcing chain selects. The interface is where the universe's bookkeeping happens, and it is thin.

/-- **Every active edge touches the spine.** For the polarized diamond field, any two adjacent cells
of different charge have an endpoint on the spine `x = 0`. The charge `sign(x)` flips only between
columns `x = -1, 0, 1`, and a charge-flipping adjacency must step the `x`-coordinate across `0`, so
one endpoint sits on the spine. Pure case analysis: split the `sign` on each endpoint, then the
unit-distance adjacency forces the spine. -/
theorem bichromatic_endpoint_on_spine (t : ℕ) (a b : Vtx t)
    (hadj : adj a.val b.val) (hc : polarized t a ≠ polarized t b) :
    a.val.1 = 0 ∨ b.val.1 = 0 := by
  unfold adj at hadj
  simp only [polarized] at hc
  split_ifs at hc <;> omega

/-- **The 2-D spine is `2t + 1` cells.** The interface of the birth field is confined to this
codimension-1 set (a 1-D ball), which is `Θ(t)` while the world is `Θ(t²)`. -/
theorem spine_card (t : ℕ) : (spine t).card = 2 * t + 1 := by
  rw [spine_eq_image, Finset.card_image_of_injective _ (by
    intro u v h; simpa using h)]
  rw [Int.card_Icc]
  omega

/-- **The 2-D spine is `2t + 1` cells.** The interface of the birth field is confined to this
codimension-1 set (a 1-D ball), which is `Θ(t)` while the world is `Θ(t²)`. -/
theorem spine_card (t : ℕ) : (spine t).card = 2 * t + 1 := by
  rw [spine_eq_image, Finset.card_image_of_injective _ (by
    intro u v h; simpa using h)]
  rw [Int.card_Icc]
  omega

/-- **Birth-field sub-extensivity, both halves (2-D).** The single capstone tying Phase 50 to Phase 51
for the forced conjugate-birth field: (1) it is carried as exactly 3 locked domains for every radius
(`O(1)` carried state, Phase 50), (2) its entire recognition-active interface is spine-incident, and
(3) the spine times the radius fits in the area, so the interface is confined to a codimension-1 set.
Both the carried-domain fraction and the interface support vanish as a fraction of the `Θ(t²)` world:
the North-Star "carry each region at the coarsest φ-rung its recognition allows" made exact for the
birth field, as a THEOREM. -/
theorem birth_field_subextensive (t : ℕ) (ht : 1 ≤ t) :
    comp (PolarizedBirthDomains.Diamond.Fmono t) = 3
    ∧ (∀ p ∈ (edges t).filter (fun q => decide (polarized t q.1 ≠ polarized t q.2)),
         p.1.val.1 = 0 ∨ p.2.val.1 = 0)
    ∧ (spine t).card * t ≤ (InterfaceComponentBound.Diamond.ball t).card :=
  ⟨PolarizedBirthDomains.Diamond.polarized_components_eq_three t ht,
   interface_on_spine t, (interface_subextensive t ht).2⟩

/-- **The whole recognition-active interface is on the spine.** Every bichromatic edge in the
interface list (the forced distinctions the engine posts on the birth field) has an endpoint on the
spine `x = 0`. This is the exact list `InterfaceComponentBound.Diamond.mono_le_interface_succ` bounds,
now shown to be spine-confined. -/
theorem interface_on_spine (t : ℕ) :
    ∀ p ∈ (edges t).filter (fun q => decide (polarized t q.1 ≠ polarized t q.2)),
      p.1.val.1 = 0 ∨ p.2.val.1 = 0 := by
  intro p hp
  rw [List.mem_filter] at hp
  obtain ⟨hpe, hpc⟩ := hp
  rw [mem_edges] at hpe
  rw [decide_eq_true_eq] at hpc
  exact bichromatic_endpoint_on_spine t p.1 p.2 hpe hpc

What this page does not claim

This does not claim the exact number of interface edges is proved; the docstring notes that count would need a separate enumeration. This does not claim the physical recognition-to-linking bridge is established; that remains open. This does not claim the framework derives the fine-structure constant or any other specific coupling.

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

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