Constants Codata Hbar Ne Zero

The reduced Planck constant

The reduced Planck constant, written hbar and pronounced h-bar, is the unit of angular momentum in quantum mechanics. Its value is 1.054571817 times 10 to the minus 34 joule-seconds, a number fixed by measurement rather than by theory. In the framework's machine-checked library of formal theorems, the declaration hbar_ne_zero states that this number is not zero. That is the whole content: the constant is positive, hence nonzero, because it is defined as a positive decimal.

The proof is a direct computation from the definition. The library defines hbar as the decimal 1.054571817e-34, then proves it is greater than zero by evaluating the number, and finally derives that it is not equal to zero from that positivity. The chain is three short lemmas: hbar_pos, then hbar_ne_zero. Nothing about quantum theory, measurement error, or the origin of the constant is involved.

In Recognition Science, the constant plays a different role from its empirical one. The framework derives its own hbar as phi to the minus 5, where phi is the golden ratio, from the forcing chain that starts with the cost function. That derived value is a theorem in the library, separate from this empirical constant. The two are not the same object; the empirical hbar is quarantined in a nested namespace so that the certified surface does not depend on measured numbers.

The declaration does not claim that the reduced Planck constant is derived from first principles, nor that its numerical value is exact. It only says the empirical value is nonzero, which is true by construction. The framework's own derived hbar is a different claim, tagged as a theorem, and is not what this page establishes.

lemma hbar_ne_zero : hbar ≠ 0 := ne_of_gt hbar_pos

/-- Reduced Planck constant (CODATA 2018). -/
@[simp] noncomputable def hbar : ℝ := 1.054571817e-34

lemma hbar_pos : 0 < hbar := by unfold hbar; norm_num

What this page does not claim

This does not claim that the reduced Planck constant is derived from theory. This does not claim the numerical value is exact beyond its decimal definition. This does not claim the empirical hbar is the same as the framework's derived hbar.

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

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Derived articles

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