Cosmology Polarized Birth Interface Birth Field Subextensive

The sub-extensive birth field

A birth field is a discrete, grid-based model of a physical quantity that takes one of three values: +1, -1, or 0. The zero values sit on a central line or plane called the spine, and the +1 and -1 values occupy the regions on either side. This field is a toy model of how a fundamental distinction might arise in a discrete spacetime, and the question is where the action, the changes between +1 and -1, actually happens.

The answer, now a proved theorem in the framework's machine-checked library of formal theorems, is that all the action is confined to the spine. In two dimensions, the spine is a line of length 2t+1; in three dimensions, it is a disk of area 2t²+2t+1. The theorem birth_field_subextensive states that every edge connecting a +1 cell to a -1 cell has an endpoint on the spine. The cost of recognition, the framework's term for the work of maintaining these distinctions, therefore lives on a surface of dimension one less than the world it describes.

This is a statement about how the cost scales. The world is a diamond of area Θ(t²) in 2D or a volume Θ(t³) in 3D, but the interface where the field changes is only Θ(t) or Θ(t²). The fraction of the world that is active, the interface divided by the volume, tends to zero as t grows. The framework calls this property sub-extensivity: the cost of maintaining the field is not spread through the whole volume but concentrated on a lower-dimensional surface.

In Recognition Science, this is a structural result about the birth field, the simplest forced field in the framework's cosmology. It is not a numerical approximation but an exact statement about a discrete model. The theorem also confirms that the field is carried as exactly three locked domains, the +1 region, the -1 region, and the spine, for every radius. This is the framework's North-Star principle, that each region is carried at the coarsest scale its recognition allows, made precise for this field.

The theorem does not count the exact number of interface edges. The docstring notes that the number of ordered edges in 2D is numerically 8t-4, but proving this would require a separate enumeration of the edge set. The sub-extensivity claim, that the cost lives on a lower-dimensional surface, is fully established by the spine-confinement and cardinality theorems. The result is a sharp statement about where the action is, not a count of every individual edge.

/-- **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

/-- **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

/-- **Interface sub-extensivity (2-D).** The recognition-active interface of the birth field is
confined to the spine, a set of `2t + 1` cells, so `spine · t ≤ area`: the interface fraction falls
as `~ 1/t`. The cost lives on a 1-D curve while the world is the 2-D area `2t² + 2t + 1`. -/
theorem interface_subextensive (t : ℕ) (ht : 1 ≤ t) :
    (spine t).card = 2 * t + 1 ∧
      (spine t).card * t ≤ (InterfaceComponentBound.Diamond.ball t).card := by
  refine ⟨spine_card t, ?_⟩
  rw [spine_card t, LatticeBallVolume.Diamond.card_ball]
  nlinarith [ht]

/-- **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⟩

What this page does not claim

This theorem does not count the exact number of interface edges. This theorem does not describe a physical spacetime in the conventional sense; it is a discrete combinatorial model. This theorem does not claim the birth field is the only field with sub-extensive 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/PolarizedBirthInterface.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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