vacuum_cost_zero
plain-language theorem explainer
The vacuum gauge configuration carries zero total J-cost under the Recognition Science functional. Researchers on the Yang-Mills mass gap would cite it as the zero baseline from which the strict spectral gap Δ = J(φ) follows. The proof is a one-line term simplification that unfolds the total cost definition and applies the zero-cost property of the phi-ladder at rung zero.
Claim. The summed recognition cost over all gauge bonds in the vacuum configuration equals zero: $J(1) = 0$ for every bond variable at the identity element of the phi-lattice.
background
Recognition Science defines the cost functional $J(x) = ½(x + x^{-1}) - 1$ on the golden-ratio lattice generated by powers of φ. The vacuum state is the configuration in which every bond sits at rung zero, so that each local factor equals the identity element with $J = 0$. The module places this result inside the derivation of the Yang-Mills mass gap from the J-functional alone, with no free parameters, on the discrete substrate forced by the T5–T8 chain.
proof idea
One-line term proof that applies simp to the definitions of total gauge cost and the vacuum state, then reduces via the lemma that J-cost vanishes on the phi-ladder at rung zero.
why it matters
This supplies the zero-energy baseline required for the spectral-gap theorem stated in the module's central claim. It closes the vacuum case before the positivity and monotonicity results that establish Δ = J(φ) > 0 for every non-trivial excitation. The result sits inside the Recognition framework's derivation of the mass gap from T5 J-uniqueness and the RCL identity.
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