Foundation Primitive Recognition Calculus Valid Comparison Examples Hilbert Disp

What the Hilbert display theorem proves

In quantum mechanics, the Born rule turns a wavefunction into a probability: the chance of finding a particle at a given spot is the squared magnitude of the amplitude there. The theorem hilbert_display_valid_iff takes this classical idea and states it as a precise equivalence. For a system with N+1 possible states, two wavefunctions are considered validly comparable in the framework exactly when the sum of their Born weights over all states is equal. In plain terms, the recognition protocol sees two quantum states as equivalent precisely when they predict the same total probability distribution.

The theorem is proved in a machine-checked library of formal theorems, and it is a direct consequence of a more general principle: a comparison is valid exactly when the native observation and the displayed value agree. Here, the native observation is the Born weight sum, and the display is the real number that results. The proof is a simple unfolding of definitions, which means the statement is not a new physical law but a formal bridge connecting the framework's notion of valid comparison to the standard quantum mechanical Born rule.

This matters because it shows the framework's core concept of a valid comparison, a ledger of recognition events, can be instantiated for quantum states without altering the usual probability calculus. The theorem does not say anything about measurement collapse, entanglement, or the physical interpretation of the wavefunction. It only establishes that the comparison protocol is consistent with the Born rule for finite-dimensional systems. It also does not claim that the Born rule itself is derived from recognition principles; the rule is assumed as part of the definition of the bridge.

The practical upshot is a clean interface: any reasoning about recognition in quantum settings can use the Born rule as the standard of comparison, knowing the framework's validity condition reduces to it. This is a foundation stone, not a physical prediction.

theorem hilbert_display_valid_iff (N : ℕ)
    (ψ φ : FRSComplexAmplitude.FRSIAmp N) :
    ValidComparison.IsValidComparison (hilbertNormBridge N) ψ φ
      ↔ Finset.univ.sum (fun i : Fin (N + 1) => FRSComplexAmplitude.bornWeight ψ i)
        = Finset.univ.sum (fun i : Fin (N + 1) => FRSComplexAmplitude.bornWeight φ i) :=
  ValidComparison.validComparison_iff_native (hilbertNormBridge N) ψ φ

What this page does not claim

The Born rule is not derived from recognition principles; it is assumed by the bridge definition. The theorem does not address measurement collapse or entanglement. The theorem only covers finite-dimensional systems with N+1 states.

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

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