Foundation Pair Kernel Native Action Quantum Native Action Quantum Pos

A positive action quantum

In physics, action is a quantity that describes how a system evolves, with units of energy multiplied by time. The Recognition Science framework defines a native action quantum, its own smallest unit of action, and proves that this unit is a positive real number. The declaration nativeActionQuantum_pos establishes that this value is greater than zero, a foundational fact for any quantity meant to represent a physical magnitude.

The value itself is not arbitrary. The framework derives it from a chain of theorems starting from a cost function, which is a measure of the effort involved in a recognition event, a discrete record of something happening. The native action quantum is defined as the reciprocal of the golden ratio raised to the fifth power, written as φ⁻⁵. The golden ratio is the number approximately equal to 1.618, famous for its appearance in geometry and art, and here it emerges from the framework's internal logic rather than being chosen as a free parameter.

The proof of positivity is direct once the value is known. Since the golden ratio is positive, its fifth power is positive, and the reciprocal of a positive number is also positive. The declaration nativeActionQuantum_pos records this reasoning in a machine-checked library of formal theorems, meaning the proof has been verified step by step by a computer. This is a theorem, not a hypothesis: it follows from the framework's definitions and prior results without any unproven assumptions.

What the declaration does not claim is equally important. It does not by itself determine the overall scale of action in the framework, nor does it connect this quantum to any measured physical constant. The certificate that establishes this positivity is built to be free of the number pi, but that is a property of the construction, not a claim about the physical universe. The declaration is a small, precise building block: it confirms that the framework's basic unit of action is a valid positive number, ready for use in further derivations.

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

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

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

What this page does not claim

The declaration does not determine the numerical scale of action in physical units. The positivity proof does not connect the action quantum to any measured physical constant. The pi-free property of the construction is not a claim about the physical 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/Foundation/PairKernelNativeActionQuantum.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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