Cosmology Graded Rung Cost T56 Graded Cost Ledger

The graded cost ledger

A ledger is a discrete record of events, and in Recognition Science the record is kept in units of cost. The declaration t56_graded_cost_ledger proves a general rule about such ledgers when the field being recorded is graded: each region carries an integer level, and adjacent regions may differ by at most one level. The theorem states that under this unit-step condition, the total cost of the whole ledger is exactly the number of boundaries between different levels, multiplied by a single fixed constant J(phi).

The result is a formal theorem in the framework's machine-checked library of formal theorems. It applies to any finite set of edges and any integer-valued field on the vertices, provided the field satisfies the unit-step property. The proof uses only the three standard axioms of the ambient type theory, with zero gaps. The theorem combines four separate facts: carried adjacencies, where levels are equal, cost zero; interface adjacencies, where levels differ, cost exactly J(phi); the total cost equals the interface cost; and the constant J(phi) is positive.

The unit-step condition is not a fitted parameter. It is the forced minimal-distinction property: a single recognition step resolves at most one phi-rung, so across every adjacency the rung changes by at most one. The theorem proves this property for the polarized birth field, and the engine maintains it by refining one rung at a time. The polarized birth field of an earlier phase is recovered as a special case, with the same total cost formula.

What the theorem does not claim is equally important. It does not say that any particular physical field satisfies the unit-step condition; that is a separate hypothesis. It does not assign a numerical value to J(phi), only proving that it is positive. And it does not assert that the cost ledger describes the actual universe, only that if the unit-step condition holds, the cost formula follows.

/-- **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⟩

/-- **Carried cost is exactly zero.** The whole same-rung bulk the engine carries coarse is free. -/
theorem carriedCost_eq_zero (k : V → ℤ) (E : Finset (V × V)) :
    carriedCost k E = 0 := by
  simp only [carriedCost]
  apply Finset.sum_eq_zero
  intro p hp
  rw [Finset.mem_filter] at hp
  exact edgeCost_carried k hp.2

/-- Under the unit-step law, an interface (different-rung) adjacency costs exactly `J(phi)`: the gap
is forced to `±1` rung, and `J(phi^(±1)) = J(phi)` by reciprocal symmetry. -/
theorem edgeCost_interface (k : V → ℤ) {E : Finset (V × V)} (hk : UnitStep k E)
    {p : V × V} (hp : p ∈ E) (hne : k p.1 ≠ k p.2) :
    edgeCost k p = Cost.Jcost Constants.phi := by
  have hd : k p.1 - k p.2 = 1 ∨ k p.1 - k p.2 = -1 := by
    rcases hk p hp with h0 | h1 | hm1
    · exact absurd (sub_eq_zero.mp h0) hne
    · exact Or.inl h1
    · exact Or.inr hm1
  rw [edgeCost]
  exact Jpow_of_abs_one hd

/-- **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 theorem does not claim that any particular physical field satisfies the unit-step condition. The theorem does not assign a numerical value to J(phi), only proving that it is positive. The theorem does not assert that the cost ledger describes the actual universe.

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.

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