Foundation Pair Kernel Native Action Quantum Native Action Quantum Inv Mul

The reciprocal pair

In physics, an action quantum is a fixed amount of the quantity called action, which has units of energy multiplied by time. Planck's constant is the familiar example. The Recognition Science framework defines its own native action quantum as the number φ⁻⁵, where φ is the golden ratio, approximately 1.618. Its reciprocal, φ⁵, is the inverse action quantum. The declaration nativeActionQuantumInv_mul is a machine-checked theorem stating that these two numbers multiply to exactly 1: φ⁻⁵ × φ⁵ = 1.

This is not a discovery about nature but a definitional consistency check. The framework chooses the ledger expression for the action quantum and its inverse; the theorem proves that the chosen expressions behave as inverses. The proof is a formal calculation: it expands the definitions and applies the algebraic fact that a nonzero number times its reciprocal is one. The golden ratio is positive, so its fifth power is nonzero, and the cancellation is valid.

The theorem also connects to the framework's counting of dimensions. A recognition event has a configuration dimension of D + 2, where D is the spatial dimension. At D = 3, this gives 5, and the exponent in φ⁵ matches that count. The theorem thus ties the reciprocal pair to the framework's spatial dimension, though it does not by itself derive that dimension.

In Recognition Science, the framework's library proves this as a formal theorem with no unproved assumptions. The theorem is a small piece of a larger structure: it certifies that the native action quantum and its inverse form a consistent pair, free of π by construction. The practical consequence is that the framework's unit conventions are internally coherent: the action quantum and its reciprocal are exact inverses, not approximations.

theorem nativeActionQuantumInv_mul :
    nativeActionQuantum * nativeActionQuantumInv = 1 := by
  rw [nativeActionQuantum_eq_inv_phi_pow_five,
    nativeActionQuantumInv_eq_phi_pow_five]
  exact inv_mul_cancel₀ (pow_ne_zero 5 (ne_of_gt goldenRatio_pos))

/-- Configuration dimension of one recognition event:
spatial degrees + one temporal tick + one ledger-balance degree. -/
def configDim (d : ℕ) : ℕ := d + 2

theorem configDim_at_spatial3 : configDim spatialDim = 5 := by
  decide

/-- Configuration-dimension bridge: the Nat exponent equals `configDim 3`. -/
theorem nativeActionQuantum_exponent_eq_configDim :
    nativeActionQuantum = (goldenRatio ^ configDim spatialDim)⁻¹ := by
  rw [nativeActionQuantum_eq_inv_phi_pow_five, configDim_at_spatial3]

What this page does not claim

The theorem does not claim that the native action quantum is measured in any experiment. The theorem does not claim that the spatial dimension D = 3 is derived in this module. The theorem does not claim that the native action quantum equals Planck's constant.

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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