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 -/
used by (1)
From the project-wide theorem graph. These declarations reference this one in their body.