U1_gapless
plain-language theorem explainer
The abelian U(1) sector has a vanishing mass gap in the Recognition Science model, corresponding to the massless photon. This separates it from the gapped non-abelian sectors. The result is used in the certificate for the Yang-Mills mass gap problem. The proof reduces immediately to the definition of the gauge mass gaps.
Claim. The U(1) hypercharge sector in the Recognition Science gauge mass gap structure has a mass gap of zero, $Δ_{U(1)}=0$, as required for the massless photon.
background
Recognition Science derives the mass gap from the J-cost functional on the golden ratio lattice. The gauge sector mass gap structure assigns the same positive gap to the color and weak sectors while setting the hypercharge gap to zero. This setup follows from the forcing chain that produces three spatial dimensions and the eight-tick octave. The module doc emphasizes that the gap emerges exactly from the J-cost without free parameters.
proof idea
This is a one-line wrapper that applies reflexivity to the explicit definition of the gauge sector mass gap structure, where the hypercharge component is set to zero by construction.
why it matters
It completes the abelian case in the yang_mills_gap_cert theorem, which inhabits the Yang-Mills mass gap certificate (QG-005). The declaration distinguishes the gapless photon from the gapped gluons and weak bosons, consistent with the phi-ladder spectral gap for non-abelian excitations. It touches the central theorem that the minimum excitation cost is positive only for non-trivial phi-ladder steps.
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