jcost_energy_zero_iff_ground
plain-language theorem explainer
The theorem states that the energy cost vanishes precisely when the scaling factor equals unity, marking the ground state as the unique minimum. Engineers deriving storage hierarchies from the phi-ladder would cite this to anchor the zero-energy reference. The proof is a biconditional that unfolds the product, invokes coherence positivity, and reduces via the squared J-cost identity to force the numerator to vanish only at x=1.
Claim. For every real number $x > 0$, $J(x) E = 0$ if and only if $x = 1$, where $J(x) = (x + x^{-1})/2 - 1$ and $E = phi^{-5}$ is the fixed coherence quantum.
background
Recognition Science expresses stored energy as the product of the J-cost function and a coherence quantum. The J-cost is $J(x) = (x + x^{-1})/2 - 1$, nonnegative for $x > 0$ and zero only at unity. The coherence energy is the constant $E = phi^{-5}$ in native units. The module derives storage-density limits from the phi-ladder and this cost structure. Upstream lemmas supply the algebraic reduction $J(x) = (x-1)^2/(2x)$ for $x neq 0$ and the normalization $J(1) = 0$.
proof idea
The forward direction assumes the product is zero, uses positivity of the coherence quantum to obtain $J(x) = 0$, substitutes the squared identity, and shows the numerator $(x-1)^2$ must vanish. The reverse substitutes $x = 1$ and applies the normalization lemma. The argument relies on Jcost_eq_sq, Jcost_unit0, and E_coh_storage_pos.
why it matters
The result anchors the energy-storage hierarchy by confirming the ground state is the sole zero of the cost function. It feeds the EN-004 certificate that assembles all derived storage properties. The theorem realizes the T5 J-uniqueness step of the forcing chain before the phi-ladder is applied to chemical and nuclear rungs.
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