Cosmology Polarized Birth Interface Spine Eq Image

The spine as an image

In the Recognition Science framework, a ledger (a discrete record of events) can carry a field that assigns a charge to each cell of a lattice. For the two-dimensional diamond-shaped world of radius t, the declaration spine_eq_image proves that the set of cells where the charge changes, the spine, is exactly the image of the integer interval from -t to t under the map y ↦ (0, y). This is a formal theorem in the framework's machine-checked library of formal theorems, with no unproved assumptions beyond the ambient type theory's standard three axioms.

What this means in plain terms: the recognition-active interface, the collection of edges where the charge flips between adjacent cells, is confined to a single column of cells at x = 0. The theorem states that this column is a one-dimensional ball, a line segment of length 2t + 1. The library also proves that the entire interface, every edge connecting cells of different charge, has an endpoint on this spine. So the cost of recognition, the forced distinctions the engine posts, localizes to a lower-dimensional surface rather than spreading through the area.

The result is a structural identity, not a numerical coincidence. It shows that the spine, which carries the interface, is a codimension-1 set with size Θ(t) while the world is Θ(t²). The fraction of the world occupied by the interface falls as roughly 1/t. This is the sub-extensivity picture: the carried state is O(1) and the active interface is confined to a surface, both vanishing as a fraction of the volume. The same structure holds in three dimensions, where the spine is a two-dimensional diamond of size 2t² + 2t + 1, one dimension below the octahedral world.

The theorem does not claim to count the exact number of interface edges. That number, 8t - 4 in two dimensions, would require a separate enumeration of the edge set. The sub-extensivity content, that the cost lives on a lower-dimensional surface, is exactly the spine-confinement plus the spine cardinality, and both are proved as theorems here. The declaration establishes a precise geometric description of where recognition activity sits, not a count of how many edges are active.

/-- The spine is the image of the segment `[-t, t]` under `y ↦ (0, y)`: it is a 1-D ball. -/
theorem spine_eq_image (t : ℕ) :
    spine t = (Finset.Icc (-(t : ℤ)) t).image (fun y => ((0 : ℤ), y)) := by
  apply Finset.ext
  rintro ⟨x, y⟩
  simp only [spine, Finset.mem_filter, Finset.mem_image, Finset.mem_Icc,
    InterfaceComponentBound.Diamond.mem_ball_iff, Prod.mk.injEq]
  constructor
  · rintro ⟨hb, rfl⟩
    exact ⟨y, by omega, rfl, rfl⟩
  · rintro ⟨z, hz, rfl, rfl⟩
    exact ⟨by omega, rfl⟩

/-- **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

/-- **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

What this page does not claim

The declaration does not count the exact number of interface edges, only their confinement to the spine. The theorem does not derive the fine-structure constant or any coupling constant. The framework's geometric results do not imply a physical model of spacetime without additional bridging assumptions.

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)

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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