Foundation Pair Kernel Native Action Quantum Native Action Quantum Eq Inv Phi Po

The action quantum

The golden ratio φ, the number satisfying φ² = φ + 1, has a long history in art and geometry. In the Recognition Science framework, it appears in a more surprising role: as the base of the smallest possible unit of action, the framework's analogue of a quantum of change. The declaration nativeActionQuantum_eq_inv_phi_pow_five proves that this unit equals φ⁻⁵, the reciprocal of the fifth power of the golden ratio. This is a theorem in the framework's machine-checked library of formal theorems, meaning its proof has been verified step by step by a computer, with no gaps and no additional assumptions.

The path to this result begins with counting. The framework models a recognition event, a discrete record of a change, as having a configuration dimension: the number of independent degrees of freedom it possesses. At the spatial dimension of three, which the framework derives elsewhere, this count is five: three spatial degrees, one temporal tick, and one degree for the ledger balance, the framework's term for the record of past events. The theorem then evaluates the ledger expression for the action quantum, showing that its value is exactly the inverse of φ raised to the fifth power. A companion theorem confirms this value is positive and that it is pi-free, meaning it contains no factor of π.

What the theorem does not claim is equally important. It does not, by itself, select the magnitude of the pair-kernel source, a separate quantity in the framework. The theorem establishes the value of the action quantum as a standalone result, but it does not link that value to any other physical constant or measurement. It also does not claim that this value is the measured Planck constant; the framework's constants are derived from its axioms, and their comparison to measured values is a separate empirical check, not part of this theorem. The theorem is a statement about the framework's internal structure, not a prediction about the external world.

theorem nativeActionQuantum_eq_inv_phi_pow_five :
    nativeActionQuantum = (goldenRatio ^ (5 : ℕ))⁻¹ := by
  unfold nativeActionQuantum nativeActionQuantumExpr LedgerExpr.eval
  rw [phiPow5Expr_eval]

theorem configDim_at_spatial3 : configDim spatialDim = 5 := by
  decide

theorem nativeActionQuantum_pos : 0 < nativeActionQuantum := by
  rw [nativeActionQuantum_eq_inv_phi_pow_five]
  exact inv_pos.mpr (pow_pos goldenRatio_pos 5)

/-- The native action quantum is pi-free by construction. -/
theorem nativeActionQuantum_piFree : PiFree nativeActionQuantum :=
  piFree_eval nativeActionQuantumExpr

What this page does not claim

The theorem does not select the pair-kernel source magnitude. The theorem does not claim the value matches the measured Planck constant. The theorem does not establish any link to other physical constants.

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

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