Cosmology Polarized Birth Interface Cost Edge Cost Carried Zero

The carried edge is free

In the Recognition Science framework, a ledger (a discrete record of events) assigns a forced cost to every adjacency between cells in a polarized birth field. Each cell carries a charge of +1, -1, or 0. The cost function J(x) = (x + x⁻¹)/2 - 1, uniquely forced by five plain conditions, evaluates the cost of an edge based on the phi-rung gap between the two cells' charges. The declaration edgeCost_carried_zero proves that when two adjacent cells have equal charges, the cost is exactly zero.

The proof is direct. A carried edge connects cells of the same charge, so the charge difference is zero phi-rungs. The cost function at zero is J(1) = 0, which the theorem establishes exactly. This is not an approximation or a limit; it is a precise equality. The bulk of the field, the carried interior, is recognized for literally zero cost.

The consequence is that the total recognition cost of the field is determined entirely by its interface, the boundary between regions of different charge. An interface edge spans exactly one phi-rung, and its cost is J(φ) = (√5 - 2)/2, a positive number. In a two-dimensional diamond lattice, the total cost at time t is (8t - 4) times this positive constant. In a three-dimensional octahedral lattice, it is (8t² - 8t + 4) times the same constant. In both cases, the carried cost is zero, and the total cost equals the interface cost.

This is the cost-unit statement of a compute-watch law: the cost of recognition scales with the activity at the boundary, not with the volume of the bulk that the engine carries for free. The theorem holds over the real and natural numbers, with no unproved assumptions beyond the three standard axioms of the ambient type theory.

/-- A carried (monochromatic) edge costs exactly `0`: equal charges span no phi-rung. -/
theorem edgeCost_carried_zero (t : ℕ) (p : Vtx t × Vtx t) (hp : p ∈ carried t) :
    edgeCost t p = 0 := by
  rw [carried, Finset.mem_filter] at hp
  have hd : polarized t p.1 - polarized t p.2 = 0 := sub_eq_zero.mpr hp.2
  rw [edgeCost, hd, Jpow_zero]

/-- A zero-rung gap (a carried, monochromatic edge) costs nothing: `J(φ^0) = J(1) = 0`. -/
lemma Jpow_zero : Jpow 0 = 0 := by
  rw [Jpow, zpow_zero, Cost.Jcost_unit0]

/-- The total cost in real-multiplication form, `(8t - 4) * J(φ)` (`t ≥ 1`). -/
theorem totalCost_mul (t : ℕ) (ht : 1 ≤ t) :
    totalCost t = ((8 * t - 4 : ℕ) : ℝ) * Cost.Jcost Constants.phi := by
  rw [totalCost_card t ht, nsmul_eq_mul]

/-- The total cost in real-multiplication form, `(8t - 4) * J(φ)` (`t ≥ 1`). -/
theorem totalCost_mul (t : ℕ) (ht : 1 ≤ t) :
    totalCost t = ((8 * t - 4 : ℕ) : ℝ) * Cost.Jcost Constants.phi := by
  rw [totalCost_card t ht, nsmul_eq_mul]

/-- **2D headline (Phase 55).** For a polarized diamond of radius `t ≥ 1`: the carried bulk costs
exactly zero recognition, the total cost is `(8t - 4) • J(φ)` (the interface count times the
one-rung cost), and `J(φ) > 0` is a genuine positive cost. "Carry the bulk free, pay only for the
interface," in exact cost units. -/
theorem t55_cost_ledger (t : ℕ) (ht : 1 ≤ t) :
    carriedCost t = 0
    ∧ totalCost t = (8 * t - 4) • Cost.Jcost Constants.phi
    ∧ totalCost t = interfaceCost t
    ∧ 0 < Cost.Jcost Constants.phi :=
  ⟨carriedCost_eq_zero t, totalCost_card t ht, totalCost_eq_interfaceCost t, Jcost_phi_pos⟩

What this page does not claim

This theorem does not claim that the cost function J is the only possible cost function; it is unique under the five stated conditions. This theorem does not claim that the polarized birth field itself is physically realized; it is a mathematical model within 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/PolarizedBirthInterfaceCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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