Cosmology Graded Rung Cost Polarized Total Cost

The graded-rung cost ledger

The declaration polarized_totalCost is a theorem in the Recognition Science framework's machine-checked library of formal theorems. It states that for a specific two-dimensional lattice shape, the Diamond, the total recognition cost of a polarized field equals a simple expression: (8t - 4) times the constant J(φ), where t is a natural number counting time steps and φ is the golden ratio. The same library proves an analogous formula for the Octahedron shape: (8t² - 8t + 4) times J(φ). These are not fitted parameters; the theorems are proved from the framework's axioms with zero gaps.

The deeper content is the graded-rung cost ledger, a discrete record of how much recognition costs when regions of a field sit at different levels, or rungs. The theorem proves that for any finite set of edges and any integer-valued rung field satisfying the unit-step property, meaning adjacent regions differ by at most one rung, the total cost equals the number of interface edges times J(φ). Carried edges, where the rung is the same, cost exactly zero. The polarized case is the special case where the rung field takes only the values +1, 0, or -1, recovering the earlier binary birth field result.

The unit-step property is not an assumption pulled from thin air. The library proves it holds for the polarized birth field, and the engine maintains it because the T-3 refiner descends one rung at a time. So the theorem covers any rung profile, not just the binary case. The cost per forced distinction is always J(φ), and J(φ) is proved positive. This is the general law: the engine pays exactly one fixed price per unit-rung interface and carries the entire same-rung bulk for free.

What the declaration does not claim is equally important. It does not claim that the polarized field is the only field, or that the specific shapes Diamond and Octahedron exhaust the possible lattices. It does not claim that the rung field can take arbitrary values; it requires the unit-step property. It does not claim that J(φ) is derived here; that is a separate theorem. The declaration establishes a cost formula under a precise hypothesis, and the hypothesis is proved for the birth field, not assumed as a free parameter.

/-- The general graded-rung `totalCost` at `k = polarized t` is definitionally the Phase-55
`PolarizedBirthInterfaceCost.Diamond.totalCost t`. -/
theorem polarized_totalCost (t : ℕ) :
    totalCost (polarized t) (E t) = PolarizedBirthInterfaceCost.Diamond.totalCost t := rfl

/-- **The general law recovers the Phase-55 2D closed form.** Instantiating the graded-rung ledger at
the polarized birth field gives `totalCost = (8t - 4) * J(phi)` (`t >= 1`): Phase 55 is the binary
special case of the graded-rung cost law. -/
theorem polarized_totalCost_card (t : ℕ) (ht : 1 ≤ t) :
    totalCost (polarized t) (E t) = (8 * t - 4) • Cost.Jcost Constants.phi := by
  rw [polarized_totalCost, PolarizedBirthInterfaceCost.Diamond.totalCost_card t ht]

/-- **The general law recovers the Phase-55 2D closed form.** Instantiating the graded-rung ledger at
the polarized birth field gives `totalCost = (8t - 4) * J(phi)` (`t >= 1`): Phase 55 is the binary
special case of the graded-rung cost law. -/
theorem polarized_totalCost_card (t : ℕ) (ht : 1 ≤ t) :
    totalCost (polarized t) (E t) = (8 * t - 4) • Cost.Jcost Constants.phi := by
  rw [polarized_totalCost, PolarizedBirthInterfaceCost.Diamond.totalCost_card t ht]

/-- **The total recognition cost of any unit-step rung field is `(interface edge count) * J(phi)`.**
Carried bulk is free; the whole cost sits on the forced unit-rung distinctions. -/
theorem totalCost_eq_card (k : V → ℤ) (E : Finset (V × V)) (hk : UnitStep k E) :
    totalCost k E = (E.filter (fun p => k p.1 ≠ k p.2)).card • Cost.Jcost Constants.phi := by
  rw [totalCost_eq_interfaceCost, interfaceCost_eq_card k E hk]

/-- **The polarized birth field satisfies the forced minimal-distinction property.** Every adjacency
of the diamond either keeps the charge (carried, gap `0`) or flips it across the spine, and the
Phase-55 `level_diff` shows a flip is exactly `±1` rung. So `UnitStep (polarized t) (E t)`. -/
theorem polarized_unitStep (t : ℕ) : UnitStep (polarized t) (E t) := by
  intro p hp
  rw [E, Finset.mem_filter] at hp
  by_cases h : polarized t p.1 = polarized t p.2
  · exact Or.inl (sub_eq_zero.mpr h)
  · rcases PolarizedBirthInterfaceCost.Diamond.level_diff t p.1 p.2 hp.2 h with h1 | hm1
    · exact Or.inr (Or.inl h1)
    · exact Or.inr (Or.inr hm1)

What this page does not claim

The declaration does not claim that the polarized field is the only possible field. The declaration does not claim that the unit-step property holds for all fields; it is proved for the birth field and maintained by the engine. The declaration does not derive the value of J(φ); that is a separate theorem in 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/GradedRungCost.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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