half_eq
plain-language theorem explainer
The equality states that the half-ladder embedding applied to any integer k returns exactly the rational k/2. Researchers refining quark or neutrino masses via fractional phi-ladder placements would cite it to fix the half-rung convention. The proof is a one-line reflexivity that unfolds the defining equation of the half function.
Claim. For every integer $k$, the half-ladder embedding satisfies half$(k) = k/2$ in the rationals.
background
The module supplies a minimal representation for fractional rungs on the phi-ladder. Core mass models use integer rungs for rigidity while selected physics modules invoke fractions such as quarters to match observed ratios; this file records the half convention explicitly as a reporting seam. The sibling definition half (k : ℤ) : Rung := (k : ℚ)/2 supplies the embedding itself. Upstream lemmas in SpinStatistics and RecogSpec.Core reuse the same half label for spin-1/2 and for PhiClosed at 1/2, respectively.
proof idea
The proof is a one-line term that applies reflexivity directly to the definition of half.
why it matters
The result anchors the half-rung representation inside the fractional phi-ladder support layer. It supplies the mechanical link required by any mass formula that places particles on half-integer rungs of the phi-ladder, consistent with the Recognition Science mass formula yardstick * phi^(rung-8+gap(Z)). No downstream theorems currently depend on it, leaving the seam open for later refinement.
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