any_deviation_costs
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.
claimLet $s$ 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 in Recognition Science
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.
scope and limits
- Does not establish positivity of the summed total cost over all eight entries.
- Does not apply when any entry is non-positive.
- Does not compute the numerical value of the cost, only its sign.
- Does not address states outside the Q3 lattice.
formal statement (Lean)
378theorem any_deviation_costs (s : Q3State) (i : Fin 8) (hi : s.entries i ≠ 1) :
379 0 < Jcost (s.entries i) := by
proof body
Term-mode proof.
380 rw [Jcost_eq_sq (ne_of_gt (s.entries_pos i))]
381 apply div_pos
382 · have : s.entries i - 1 ≠ 0 := sub_ne_zero.mpr hi
383 positivity
384 · linarith [s.entries_pos i]
385
386/-! ## Part 7: Dimensional Uniqueness — Only D = 3 Works
387
388No other dimension supports the full spectral emergence structure.
389This proves that the Standard Model is the UNIQUE physics compatible
390with the cost-minimization principle. -/
391
392/-- The essential requirements for spectral emergence at dimension D:
393 1. Non-trivial linking (forces D = 3 by Alexander duality)
394 2. At least 3 face pairs (for 3 generations)
395 3. Gap-45 synchronization: lcm(2^D, 45) = 360 -/