Foundation Pair Kernel Native Action Quantum Native Action Quantum Pi Free

A quantum without pi

In physics, an action quantum is the smallest meaningful unit of change in a system, a discrete step that reality takes. The most famous one is Planck's constant, which sets the scale of quantum mechanics. In Recognition Science, the framework's native action quantum is a specific number: the golden ratio raised to the fifth power, then inverted, written φ⁻⁵. The golden ratio φ is the number whose square equals itself plus one, about 1.618. The framework's action quantum is therefore about 0.09017, a small, positive, and fixed value.

The declaration nativeActionQuantum_piFree establishes a precise fact about this number: it is pi-free. This means the value can be written using only the golden ratio and arithmetic operations, with no appearance of the circle constant π anywhere in its construction. The proof is a theorem in the framework's machine-checked library of formal theorems, meaning a computer program has verified every step of the reasoning. The declaration also shows the action quantum is positive, and that multiplying it by its reciprocal, φ⁵, gives exactly 1.

The framework builds this value from a ledger, a discrete record of recognition events. A recognition event is a unit of reality tracking itself. The framework counts the configuration dimension of one such event: at spatial dimension 3, the count is 5, which is the exponent in φ⁵. The action quantum is the inverse of this, φ⁻⁵. The declaration proves this exponent matches the configuration dimension, tying the value to the framework's spatial structure.

In Recognition Science, the framework models the native action quantum as a ledger expression built from the golden ratio alone. The declaration does not, by itself, select the pair-kernel source magnitude, which is a separate quantity. It does not claim the value is the same as Planck's constant, nor does it say anything about how this action quantum interacts with other parts of the framework. It is a certificate, a standalone proof that this particular number can be expressed without pi.

The consequence is that one of the framework's fundamental constants is independent of the circle constant. This is a structural fact, not a numerical coincidence. It means the framework's smallest action step is determined by the golden ratio, a number that appears throughout geometry and nature, rather than by π. This distinction matters because it shows the framework's constants are not arbitrary; they follow from the framework's own rules.

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

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

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]

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

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))

What this page does not claim

The declaration does not claim that the native action quantum equals Planck's constant. The declaration does not claim that the framework's action quantum is the only possible one. The declaration does not claim that the pi-free property has a direct physical consequence beyond the framework's internal structure.

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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