any_deviation_costs
plain-language theorem explainer
Any non-unit ratio assigned to a vertex of the Q3 lattice produces strictly positive J-cost. Researchers deriving the Standard Model gauge group and three generations from the Recognition Science forcing chain cite this to confirm the ground state is the sole zero-cost configuration. The argument rewrites Jcost via its squared-ratio identity and invokes positivity of squares and denominators.
Claim. Let $s$ be a Q3State, i.e., an assignment of positive real ratios to the eight vertices of the cube $Q_3$. For any index $i$ such that the entry $s_i$ differs from 1, the J-cost satisfies $J(s_i) > 0$.
background
Q3State is the structure that assigns a positive real ratio to each of the eight vertices of the binary cube $Q_3$. Its total_cost sums J-cost over all entries, while is_zero_defect holds exactly when every entry equals one. The J-cost function is defined from the Recognition Composition Law as $J(x) = (x + x^{-1})/2 - 1$, and the upstream lemma Jcost_eq_sq rewrites it as $(x-1)^2/(2x)$ for $x > 0$ (or $x < 0$).
proof idea
The proof is a term-mode reduction. It applies the lemma Jcost_eq_sq to replace Jcost with the squared form, using entry positivity to discharge the non-zero hypothesis. Then div_pos splits the goal: sub_ne_zero plus positivity shows the numerator is positive, while linarith on entries_pos shows the denominator is positive.
why it matters
This result is invoked by the master theorem spectral_emergence to certify that the zero-defect state on $Q_3$ is the unique ground state under cost minimization. It fills the step establishing that deviations from the identity assignment incur positive cost, supporting the claim that only $D=3$ yields the full spectral structure with $|Aut(Q_3)|=48$ matching the chiral fermion count. It touches the question of whether other cost functions could stabilize different dimensions.
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