Explanation of schumannRS_strictMono
(1) In plain English, the declaration asserts that the RS-predicted Schumann resonance frequency function is strictly increasing: whenever the harmonic index m is a smaller natural number than n, the computed frequency at m is strictly less than the frequency at n.
(2) In Recognition Science this matters because the zero-parameter formula f(n) = (4n−1)·φ + 3 (forced by T6 self-similarity and T8 dimension D=3) must produce an ordered sequence of frequencies to be consistent with the observed Schumann harmonics, which themselves increase. The strict monotonicity is a structural consequence of the positive spacing 4φ.
(3) The formal statement reads: ∀ m n : ℕ, m < n → schumannRS m < schumannRS n. In standard mathematical notation this is “for all natural numbers m, n, if m < n then schumannRS(m) < schumannRS(n)”. The supplied proof proceeds by exhibiting the difference schumannRS n − schumannRS m = 4·φ·(n−m), showing the right-hand side is positive when m < n (via the auxiliary fact that 4·φ > 0), and concluding the inequality by linear arithmetic.
(4) Visible dependencies inside the supplied source are the definition schumannRS, the spacing identity spacing_eq, and the positivity lemma spacing_pos. The proof also relies on the concrete bounds phi_gt_1618 and phi_lt_1619 only indirectly through the positivity of φ. No external axioms or sorrys appear in this declaration.
(5) The declaration does not prove numerical agreement with measured Schumann values (those appear in separate theorems such as harmonic1_matches through harmonic5_matches), nor does it establish the EEG-band classification or the master certificate earthBrainResonance_forced. It likewise does not address the contingent physical radius of Earth or any falsification criteria beyond the mathematical ordering.