gap_separates_sectors
plain-language theorem explainer
The equivalence states that a gauge bond on the phi-ladder is contractible precisely when its J-cost vanishes. Researchers addressing the Yang-Mills mass gap in discrete recognition models would reference this to separate the vacuum from excitations. The term proof combines the zero-cost property of the zero rung with a contradiction from the strict positivity of costs for nonzero rungs.
Claim. A gauge bond with rung $n$ is contractible if and only if the J-cost of $phi^n$ equals zero.
background
In the Recognition Science treatment of the Yang-Mills mass gap, the phi-ladder consists of elements $phi^n$ for integer $n$, with the J-cost functional defined by $J(x) = 1/2(x + x^{-1}) - 1$. Contractible bonds are those with rung exactly zero, corresponding to the massless U(1) photon. The module establishes a strict spectral gap separating zero cost from all positive costs on the ladder. This builds on the cost definitions from multiplicative recognizers and observer forcing, where cost is always non-negative, and on the phi-ladder positivity.
proof idea
The proof proceeds in two directions. One direction invokes the lemma that contractible bonds have zero cost. The converse assumes a zero-cost non-contractible bond and derives a contradiction by applying the strict spectral gap inequality for nonzero rungs.
why it matters
This result places the mass gap at the boundary between contractible and non-contractible sectors on the phi-ladder. It supports the topological protection claim in the module's central theorem, confirming that no non-trivial excitation can reach zero cost. The equivalence closes the separation of sectors required for the RS resolution of the Millennium problem, consistent with the J-uniqueness from the forcing chain.
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