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

The action quantum

The golden ratio, usually written φ and equal to (1 + √5)/2, is the number whose square equals itself plus one. Its fifth power, φ⁵, is approximately 11.09. The Recognition Science declaration nativeActionQuantumInv_eq_phi_pow_five states that a quantity called the native action quantum, when inverted, equals φ⁵. The declaration is a theorem in the framework's machine-checked library of formal theorems, meaning it is proved from earlier definitions rather than assumed.

In Recognition Science, an action quantum is the smallest discrete unit of action, analogous to Planck's constant in conventional physics. The framework models action as a ledger, a discrete record of recognition events. Each recognition event has a configuration dimension: it adds one temporal tick and one ledger-balance degree to the spatial dimensions. At the framework's forced spatial dimension of three, that count is five. The theorem connects this count to the ledger expression: the native action quantum is the reciprocal of φ raised to the fifth power, and its inverse is φ⁵.

The declaration also establishes that this value is pi-free by construction. It is built only from the golden ratio and the number five, with no π appearing in its ledger expression. The proof that the inverse equals φ⁵ proceeds by evaluating the ledger expression and identifying the golden ratio atom. A companion theorem shows the action quantum and its inverse multiply to one, confirming they are genuine reciprocals.

What the declaration does not claim is equally important. It does not, by itself, select the pair-kernel source magnitude; that is a separate identification. It does not assert that the native action quantum equals Planck's constant in conventional units, nor does it derive the fine-structure constant. The theorem is a certificate that a particular ledger expression evaluates to φ⁵, not a claim about measured physics.

The consequence is that within the framework, the golden ratio's fifth power is not a coincidence but a forced result of counting recognition events. A reader can now see why φ⁵ appears in the framework's constants: it is the reciprocal of the native action quantum, derived from the configuration dimension of a recognition event at spatial dimension three.

theorem nativeActionQuantumInv_eq_phi_pow_five :
    nativeActionQuantumInv = goldenRatio ^ (5 : ℕ) :=
  phiPow5Expr_eval

theorem configDim_at_spatial3 : configDim spatialDim = 5 := by
  decide

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))

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

What this page does not claim

This declaration does not select the pair-kernel source magnitude. This declaration does not equate the native action quantum with Planck's constant in conventional units. This declaration does not derive the fine-structure 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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