Constants Codata Hbar Pos

The positivity lemma

The reduced Planck constant, written ℏ (h-bar), is the quantum of angular momentum: the natural unit in which spin and energy-frequency relations are counted. Its SI value is 1.054571817 × 10⁻³⁴ joule-seconds, as set by the 2018 CODATA adjustment. The declaration hbar_pos is a machine-checked lemma stating that this number is greater than zero. That is all it states: it does not derive the value from first principles, nor does it connect ℏ to the framework's own constants.

The lemma lives in a module explicitly labeled quarantined. The framework's certified surface, the chain of theorems that forces constants like φ and G, deliberately avoids depending on these empirical SI numbers. The quarantine exists because the derived constants and the empirical constants would otherwise share names in the same namespace, a collision that broke two importing modules. The positivity lemma exists so that any code comparing or dividing by ℏ can rely on it being nonzero and positive, a routine but necessary guard.

In Recognition Science, the framework models physical structure from a ledger, a discrete record of recognition events whose cost is forced. From that ledger the framework derives its own ℏ in native units as φ⁻⁵, where φ is the golden ratio. That derived value is a theorem of the forcing chain; the empirical CODATA value is a measurement, not a theorem. The two live in separate namespaces and are not identified with each other. The positivity lemma concerns only the empirical number.

What the lemma does not claim is worth stating plainly. It does not prove that the reduced Planck constant is positive in any physical sense; it proves that the decimal 1.054571817e-34, as defined in the module, is positive. It does not assert that the empirical value equals the framework's derived φ⁻⁵. It does not establish that ℏ is fundamental, quantized, or even physically meaningful. It is a bookkeeping fact about a number, not a statement about nature.

The practical consequence is small but real: any formal argument that needs to divide by ℏ, or to know its sign, can do so without adding an unproved assumption. The lemma closes a gap that would otherwise be open in every proof that touches the empirical constant. That is the entire payoff, and it is enough.

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

/-- 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

The lemma does not prove the reduced Planck constant is positive in any physical sense beyond the defined decimal. The lemma does not assert that the empirical CODATA value equals the framework's derived φ⁻⁵. The lemma does not establish that ℏ is fundamental, quantized, or physically meaningful.

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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