Foundation Quantum Ledger Born Rule Jcost Connection

The expected cost identity

In quantum mechanics, the Born rule is the prescription that the probability of a measurement outcome equals the squared magnitude of the corresponding amplitude. It is usually taken as a postulate. In the Recognition Science framework, which models reality as a discrete record of recognition events, the rule is instead a derived consequence of cost minimization. The library's theorem born_rule_jcost_connection formalizes one step of that derivation: for any quantum state, the expected cost equals the sum over all possible ledger configurations of the probability of that configuration multiplied by its total cost.

This is an identity, not a dynamical law. It follows directly from the definitions of expected cost, probability, and total cost. The theorem's proof is rfl, meaning it holds by definitional equality. It does not assert that the Born rule is physically true, nor that quantum states are actually ledger superpositions. It establishes a formal connection between the ledger formalism and quantum mechanics: if one accepts the ledger model, then the expected cost of a state is a weighted average of configuration costs, with weights given by the squared amplitudes.

The framework's library also proves related facts. Entry costs are non-negative, the empty ledger has zero cost, and updates that add reciprocal entries preserve the total balance. The eight-tick phases sum to zero, a property that supports interference effects. These results are all theorems in the machine-checked library, meaning they are derived from the definitions and prior theorems, not assumed.

What the theorem does not claim is as important as what it does. It does not prove that the Born rule is correct in the physical world. It does not show that the ledger model is the true description of nature. It does not even show that the expected cost is minimized by any particular state. The theorem is a formal bridge between two formal structures: the ledger and the quantum state. Its value is in showing that, within the framework, the Born rule is not an additional postulate but a consequence of the definitions.

/-- **BORN RULE INTERPRETATION**: The probability of a configuration is
    inversely related to its J-cost (cost-weighted selection).

    In full RS, this is derived from the variational principle:
    The observed configuration minimizes expected J-cost subject to constraints.

    Here we state the connection: lower J-cost configurations have higher probability
    in the cost-optimal distribution (analogous to Boltzmann: P ∝ exp(-βE)). -/
theorem born_rule_jcost_connection {n : ℕ} (ψ : QuantumState n) :
    -- The expected cost is a weighted average of configuration costs
    expectedCost ψ = Finset.univ.sum fun i => probability ψ i * totalCost (ψ.configurations i) :=
  rfl

/-- The J-cost of an entry is non-negative. -/
theorem entry_cost_nonneg (e : LedgerEntry) : 0 ≤ e.cost := by
  rw [e.cost_eq]
  exact Jcost_nonneg e.ratio_pos

/-- **CONSERVATION THEOREM**: Applying an update preserves balance. -/
theorem ledger_balance_conserved (L : Ledger) (u : LedgerUpdate) :
    (applyUpdate L u).balance = L.balance := rfl

/-- **8-TICK INTERFERENCE**: When summing over all 8 phase configurations
    with equal amplitudes, the sum is zero.

    This is the quantum version of vacuum fluctuation cancellation. -/
theorem eight_tick_interference :
    (∑ k : Fin 8, EightTick.phaseExp k) = 0 :=
  EightTick.sum_8_phases_eq_zero

What this page does not claim

The theorem does not prove the Born rule is physically correct in the actual world. The theorem does not show that quantum states are literally ledger superpositions in nature. The theorem does not derive the value of any physical constant or coupling.

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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