Foundation Quantum Ledger Entry Cost Zero Iff Unity

The zero-cost condition

A ledger is a discrete record of recognition events, each with a positive ratio and a cost. The theorem entry_cost_zero_iff_unity states that an entry has cost zero if and only if its ratio equals one. This is a biconditional: zero cost implies unity, and unity implies zero cost.

The proof is a direct rewrite. Each entry's cost is defined as Jcost of its ratio. The theorem Jcost_eq_zero_iff, already in the library, says Jcost(r) = 0 exactly when r = 1. The declaration chains these two facts together, so the result is immediate and axiom-clean.

In the framework's quantum ledger, this theorem does real work. A quantum state is a superposition over ledger configurations. The Born rule, the probability of a configuration, is derived from J-cost minimization. The zero-cost condition identifies the unique configuration a measurement collapses to: the one where every ratio is one, meaning no recognition cost is paid.

The theorem does not claim that a ratio of one is the only possible state, nor that zero cost is achievable in every ledger. It only establishes the equivalence for a single entry. The framework's own ledger conservation theorem shows that updates preserve total balance, but that is a separate result.

/-- The J-cost is zero iff the ratio is 1. -/
theorem entry_cost_zero_iff_unity (e : LedgerEntry) : e.cost = 0 ↔ e.ratio = 1 := by
  rw [e.cost_eq]
  exact Jcost_eq_zero_iff e.ratio e.ratio_pos

/-- Create a ledger entry from a ratio and phase. -/
noncomputable def mkEntry (id : ℕ) (r : ℝ) (hr : 0 < r) (p : Fin 8) : LedgerEntry := {
  id := id
  ratio := r
  ratio_pos := hr
  cost := Jcost r
  phase := p
  cost_eq := rfl
}

What this page does not claim

The theorem does not prove that a ratio of one is the only possible ledger state. It does not claim that zero cost is achievable for every ledger entry. It does not establish that measurement always collapses to a zero-cost configuration.

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/Foundation/QuantumLedger.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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