J_one_zero
plain-language theorem explainer
J-cost vanishes exactly at the identity state x=1. Researchers deriving the phi-ladder mass hierarchy or the zero-defect consciousness ground state cite this base case. The result follows from the Recognition Science cost function and anchors all higher-rung calculations on the Q3 structure. The proof is a direct term application of the unit lemma already established in the Cost module.
Claim. $J(1)=0$, where the J-cost is given by $J(x)=((x-1)^2)/(2x)$.
background
The J-cost function is introduced in the Cost module as the squared-ratio expression J(x)=(x-1)^2/(2x), which is zero only at the fixed point x=1. This theorem sits inside the Spectral Emergence module, which derives the full Standard Model gauge content and three-generation structure from the forced D=3 via the binary cube Q3 and its automorphism group of order 48. Upstream, the identity event is defined as the canonical RecognitionEvent with state exactly 1, and the Jcost_unit0 lemma states that Jcost 1=0 by direct simplification of the cost definition.
proof idea
The proof is a one-line term wrapper that directly invokes the Jcost_unit0 lemma from the Cost module.
why it matters
This base case supplies the zero-cost identity required for the unique consciousness ground state listed in the module's self-consistency loop. It feeds the phi-ladder mass formula (yardstick times phi to the power rung-8+gap) and the Recognition Composition Law applications that produce the SU(3)xSU(2)xU(1) sector dimensions. The result closes the T5 J-uniqueness step for all subsequent spectral emergence theorems.
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