Cosmology Polarized Birth Interface Count Idx Card

The exact interface count

In the Recognition Science account, a growing world is a discrete lattice of points, and the engine that drives it posts forced distinctions along the boundary between two polarities. The declaration idx_card (a lemma in the framework's machine-checked library of formal theorems) states the exact size of that boundary set. In a two-dimensional diamond world of radius t, the count is 8t - 4. In the three-dimensional octahedron world, the count is 8t² - 8t + 4.

The count matters because it measures the cost of recognition: the framework models cost as scaling with the number of forced distinctions, not with the volume of the world. In two dimensions, the boundary grows by exactly 8 edges per cycle, a constant independent of world size. In three dimensions, the boundary is a growing disk, so the per-cycle increment is 16t, linear in the radius. This is the honest dimensional contrast: the activity is sub-extensive against the volume in both cases, but only in two dimensions is it constant per cycle.

The theorem is proved by a bijection: every bichromatic edge has exactly one endpoint on the central spine and one neighbor beside it, so the data of the edge is precisely a spine coordinate, a side, and an orientation. The count follows by summing over the spine. The same structure lifts to the octahedron, where the spine is itself a two-dimensional diamond.

What idx_card does not claim: it does not say the framework's three-dimensional world is the physical universe, nor that the linear growth is observed in nature. It is a theorem about a defined combinatorial object. The bridge from this combinatorial count to physical cosmology remains open.

/-- The index set has `8t - 4` elements: `(2t - 1)` interior spine cells times `4` (side, orient). -/
theorem idx_card (t : ℕ) (ht : 1 ≤ t) : (idx t).card = 8 * t - 4 := by
  rw [idx, Finset.card_product, Int.card_Icc]
  have h4 : (Finset.univ : Finset (Bool × Bool)).card = 4 := by decide
  rw [h4]
  omega

/-- The index set has `8t - 4` elements: `(2t - 1)` interior spine cells times `4` (side, orient). -/
theorem idx_card (t : ℕ) (ht : 1 ≤ t) : (idx t).card = 8 * t - 4 := by
  rw [idx, Finset.card_product, Int.card_Icc]
  have h4 : (Finset.univ : Finset (Bool × Bool)).card = 4 := by decide
  rw [h4]
  omega

/-- **Constant per-cycle recognition activity (the compute-watch law, in Lean).** Advancing the
diamond birth field by one cadence cycle (`t → t + 1`) adds exactly `8` ordered interface edges,
*independent of `t`* and hence independent of the world volume (which grows as `Θ(t²)`). The forced
distinctions the engine must post per cycle are `O(1)`, so the simulation's cost scales with
recognition activity, never with volume. -/
theorem interface_increment_const (t : ℕ) (ht : 1 ≤ t) :
    (B (t + 1)).card - (B t).card = 8 := by
  rw [interface_card_eq (t + 1) (by omega), interface_card_eq t ht]
  omega

/-- **Linear per-cycle recognition activity (3-D).** Advancing the octahedron birth field by one
cadence cycle (`t → t + 1`) adds exactly `16t` ordered interface edges. Unlike the 2-D case (where the
increment is the constant `8`), in three dimensions the forced recognition activity per cycle grows
`Θ(t)`: the interface is a 2-D surface whose area grows linearly per shell. This is the honest 3-D
form of the compute-watch law: cost per cycle still tracks recognition activity, but in D = 3 that
activity is `Θ(t)`, not `O(1)`, because the recognition-active interface is a growing codim-1 disk. -/
theorem interface_increment_linear (t : ℕ) (ht : 1 ≤ t) :
    (B (t + 1)).card - (B t).card = 16 * t := by
  rw [interface_card_eq (t + 1) (by omega), interface_card_eq t ht]
  have e : (t + 1) ^ 2 = t ^ 2 + 2 * t + 1 := by ring
  have hsq : t ≤ t ^ 2 := by nlinarith [ht]
  rw [e]
  omega

What this page does not claim

The theorem does not identify the three-dimensional octahedron lattice with the physical universe. The theorem does not state that the linear growth rate is observed in cosmological data. The bridge from the combinatorial boundary count to physical cosmology is not established by this declaration.

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/PolarizedBirthInterfaceCount.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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