massless_state
plain-language theorem explainer
The theorem establishes that the recognition cost vanishes exactly at unit scale. Researchers constructing tachyon-free spectra in Recognition Science cite this to anchor the massless mode within the five canonical string configurations. The proof is a direct one-line application of the upstream unit-cost lemma.
Claim. $J(1) = 0$, where $J(x) = (x-1)^2/(2x)$ denotes the recognition cost function.
background
The Tachyon-Free Spectrum from RS module maps superstring tachyon suppression to the absence of negative J-cost modes. Because J(x) is non-negative for all x, the RS recognition field excludes tachyons at the Lagrangian level and identifies five modes with configDim D = 5. This theorem supplies the massless component of that spectrum.
proof idea
One-line wrapper that applies the Jcost_unit0 lemma from the Cost module, which itself follows by simplification of the squared-ratio definition of J.
why it matters
The result feeds directly into the tachyonFreeCert definition, which assembles five_modes, nonneg, and massless to certify the full tachyon-free spectrum. It instantiates the module's claim that J(x) >= 0 excludes tachyons, consistent with the Recognition Composition Law and the phi-ladder structure of the forcing chain.
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