vacuum_unique_zero_cost
plain-language theorem explainer
The theorem establishes that the vacuum is the unique gauge bond configuration on the twelve edges of Q3 with total recognition cost zero. Researchers working on the Yang-Mills mass gap would cite it to isolate the ground state before invoking the spectral gap. The term proof proceeds by contradiction on a single edge: the spectral gap lemma forces positive cost for any nonzero rung while the bond inequality bounds that cost by the given total of zero.
Claim. Let $cfg$ be a gauge bond configuration assigning an integer rung to each of the twelve edges of the cube $Q_3$. If the total gauge cost of $cfg$ is zero, then every rung index equals zero.
background
In the Recognition Science treatment of the Yang-Mills mass gap the J-cost functional $J(x) = ½(x + x^{-1}) - 1$ is defined on the golden-ratio lattice forced by the T5-T8 chain. A gauge bond configuration assigns to each of the twelve edges of the fundamental cube an integer rung index, with the vacuum defined by all rungs at zero. The module derives the exact gap $Δ = J(φ) = φ - 3/2 > 0$ from the recognition composition law together with the phi-forcing chain and RS-native units.
proof idea
The term-mode proof fixes an arbitrary edge e. It assumes for contradiction that the rung at e is nonzero. The upstream lemma spectral_gap_strict then supplies a strictly positive lower bound on the cost of that single bond. The lemma bond_le_total shows the individual bond cost is at most the total gauge cost, which is given as zero. Linear arithmetic yields the contradiction.
why it matters
This result secures uniqueness of the zero-cost vacuum, a prerequisite for the spectral gap theorem developed in the same module. It directly supports the central claim that the minimum excitation cost is exactly J(φ), resolving the mass gap problem from the J-cost functional alone with no free parameters. The argument closes the loop from the phi-lattice substrate to isolation of the ground state.
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