Cosmology Graded Rung Cost Polarized Total Cost Card

The graded cost ledger

In the Recognition Science framework, a ledger, a discrete record of events, assigns a cost to every adjacency between two regions of a field. The field carries an integer rung, a level of coarseness, at each point. The declaration polarized_totalCost_card is a theorem about two specific shapes, the diamond and the octahedron, when the rung field is polarized: it takes only the values +1, 0, and -1, like a sign flip. The theorem states that the total cost of all adjacencies equals a simple count times a fixed constant.

The fixed constant is J(phi), the cost function evaluated at the golden ratio. The cost function J(x) = (x + 1/x)/2 - 1 is the unique function satisfying five plain conditions, a result proved in the framework's machine-checked library of formal theorems. For the polarized diamond at step t, the total cost is (8t - 4) times J(phi); for the polarized octahedron it is (8t^2 - 8t + 4) times J(phi). These closed forms are proved for every t at least 1.

The deeper content is that a single hypothesis, the unit-step property, drives the whole result. The unit-step property says that across any adjacency, the rung changes by at most one: the difference is 0, +1, or -1. This is a forced minimal-distinction property, not a fitted parameter. Under that one condition, the framework proves a general theorem for any finite graph and any integer rung field: equal-rung adjacencies cost zero, different-rung adjacencies cost exactly J(phi), and the total cost is therefore the number of interface edges times J(phi). The polarized diamond and octahedron are recovered as special cases of this general theorem.

In Recognition Science, this means the engine pays exactly J(phi) per forced unit-rung distinction and carries the entire same-rung bulk for free, for any rung profile, not just the binary birth field. The polarized cases are the bridge back to the earlier Phase 55 result, which proved the cost ledger for the binary birth field alone. The theorem extends that result to the graded rung profile the live engine actually maintains, where each region sits at the coarsest rung its recognition allows and the refiner descends one rung at a time.

What the theorem does not claim: it does not say that the unit-step property holds for every possible rung field. It is a hypothesis of the theorem, proved for the polarized cases and maintained by the engine's refinement rule, but not derived from the cost function alone. The theorem also does not assign a physical meaning to J(phi) or to the rung field; it fixes the cost structure given the rung field. And it does not claim that the polarized diamond or octahedron is the only shape that satisfies the unit-step property; the general theorem applies to any finite edge set that does.

/-- **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]

/-- **Graded-rung cost ledger headline (Phase 56).** For any finite ordered edge set `E` and any
integer rung field `k` with the forced minimal-distinction property `UnitStep k E`: the carried
same-rung bulk costs exactly zero, the total cost equals the interface cost, the total cost is the
interface edge count times the one-rung cost `J(phi)`, and `J(phi) > 0` is a genuine positive cost.
The cost of carrying any rung profile is `J(phi)` per forced unit-rung distinction, bulk free. -/
theorem t56_graded_cost_ledger (k : V → ℤ) (E : Finset (V × V)) (hk : UnitStep k E) :
    carriedCost k E = 0
    ∧ totalCost k E = interfaceCost k E
    ∧ totalCost k E = (E.filter (fun p => k p.1 ≠ k p.2)).card • Cost.Jcost Constants.phi
    ∧ 0 < Cost.Jcost Constants.phi :=
  ⟨carriedCost_eq_zero k E, totalCost_eq_interfaceCost k E,
   totalCost_eq_card k E hk, Jcost_phi_pos⟩

/-- **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 forced minimal-distinction property.** A rung field `k` posts only single-rung distinctions
across the edge set `E`: every adjacency changes the rung by at most one, `k p.1 - k p.2 in {0,+1,-1}`.
A unit recognition step resolves at most one phi-rung; the live engine maintains this because T-3
descends one rung at a time. -/
def UnitStep (k : V → ℤ) (E : Finset (V × V)) : Prop :=
  ∀ p ∈ E, k p.1 - k p.2 = 0 ∨ k p.1 - k p.2 = 1 ∨ k p.1 - k p.2 = -1

What this page does not claim

The unit-step property is not derived from the cost function alone; it is a hypothesis proved for the polarized cases and maintained by the engine's refinement rule. The theorem does not assign a physical meaning to J(phi) or to the rung field. The polarized diamond or octahedron is not claimed to be the only shape satisfying the unit-step property.

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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: