IsContractible
plain-language theorem explainer
A gauge bond on the phi-ladder is contractible precisely when its rung index n equals zero. This definition is cited in proofs that separate the massless U(1) photon from the gapped SU(2) and SU(3) sectors in the Recognition Science treatment of the Yang-Mills mass gap. The declaration is a direct equality on the integer rung with no auxiliary lemmas.
Claim. A gauge bond with rung index $n$ on the golden-ratio lattice is contractible if and only if $n=0$.
background
Recognition Science equips the golden-ratio lattice with the J-cost functional $J(x)=½(x+x^{-1})-1$. The phi-ladder maps each integer rung to a state whose cost is obtained from the multiplicative recognizer or observer-forcing cost maps. Contractibility marks the vacuum rung where this cost vanishes. The module derives the mass gap from the J-functional alone on the lattice forced by the T5-T6 chain, with native units fixing $c=1$. Upstream results supply the RS-native gauge $U$ and the derived cost functions used to evaluate excitations.
proof idea
The declaration is a direct definition that sets contractibility equal to the predicate $n=0$. No lemmas or tactics are invoked; the equality is introduced by abbreviation.
why it matters
The definition supplies the hypothesis for contractible_bond_zero_cost, which shows that contractible bonds carry zero J-cost, and for gap_separates_sectors, which proves the logical equivalence between contractibility and vanishing cost. It thereby implements the topological separation between the massless photon and gapped non-abelian sectors required by the central theorem of the Yang-Mills mass gap module. The construction rests on the J-uniqueness property (T5) and the self-similar fixed point (T6) that force the spectral gap above the vacuum.
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