Cosmology Polarized Birth Interface Cost T55 Cost Ledger

The cost ledger

The declaration t55_cost_ledger is a theorem in the Recognition Science framework's machine-checked library of formal theorems. It concerns a ledger, a discrete record of recognition events, that tracks the cost of a polarized birth field, a mathematical structure where each cell carries a charge of +1, 0, or -1. The theorem proves, for two specific lattice shapes, that the total recognition cost is exactly equal to the cost of the interface edges alone, and that the cost of the carried, monochromatic edges is precisely zero.

The core result is that the cost of a single adjacency between two cells is given by the forced cost function J(x) = (x + x⁻¹)/2 - 1, evaluated at the phi-rung gap between their charges. Here, phi is the golden ratio. A carried edge connects cells with equal charges, so the gap is zero, and J(1) = 0 exactly. An interface edge connects cells with charges differing by exactly one, so the gap is one phi-rung, and its cost is J(phi) = (√5 - 2)/2, a positive number. This is the literal, cost-unit statement of a compute-watch law: cost scales with the recognition activity at the interface, not with the bulk volume.

For the 2D diamond lattice at time t, the theorem proves the total cost is (8t - 4) · J(phi), and for the 3D octahedron it is (8t² - 8t + 4) · J(phi). In both cases, the carried cost is zero. The theorem is proved over the real and natural numbers with zero 'sorry' and only the three standard axioms of the ambient type theory: propext, Classical.choice, and Quot.sound.

What the declaration does not claim is broader physical significance. It does not claim that this specific lattice model describes the actual universe, nor that the compute-watch law is a physical law. It establishes a formal result about a specific mathematical structure. The identification of this structure with physical cosmology remains an open question, a target for future work, not a proven fact.

/-- **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⟩

/-- 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]

/-- `J(φ) > 0`: a forced interface distinction has a genuine, strictly positive recognition cost.
`J(φ) = (φ - 1)²/(2φ) = (√5 - 2)/2`. -/
lemma Jcost_phi_pos : 0 < Cost.Jcost Constants.phi := by
  rw [Cost.Jcost_eq_sq Constants.phi_ne_zero]
  apply div_pos
  · have hne : Constants.phi - 1 ≠ 0 := sub_ne_zero.mpr Constants.phi_ne_one
    positivity
  · have := Constants.phi_pos; linarith

/-- A forced interface (bichromatic) edge costs exactly `J(φ)`: its charges differ by one phi-rung. -/
theorem edgeCost_interface (t : ℕ) (p : Vtx t × Vtx t) (hp : p ∈ B t) :
    edgeCost t p = Cost.Jcost Constants.phi := by
  simp only [B, Finset.mem_filter, Finset.mem_univ, true_and] at hp
  obtain ⟨hadj, hpol⟩ := hp
  simp only [edgeCost]
  exact Jpow_of_abs_one (level_diff t p.1 p.2 hadj hpol)

/-- 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]

What this page does not claim

The theorem does not claim that this lattice model describes the actual physical universe. The theorem does not claim that the compute-watch law is a physical law. The theorem does not claim that the identification of this structure with physical cosmology is proven.

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)

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.

Derived articles

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