harmonic5_eq
The fifth Schumann resonance equals 19φ + 3 in RS units. Geophysicists modeling zero-parameter Earth-ionosphere modes would cite the identity. The proof is a one-line algebraic reduction after unfolding the definition of schumannRS.
claim$f(5) = 19φ + 3$, where $f(n) = (4n-1)φ + 3$ is the RS-forced Schumann formula with $φ = (1+√5)/2$ and $D=3$.
background
The module constructs Schumann harmonics from Recognition Science using only the forced constants D=3 (T8) and φ (T6). schumannRS implements the zero-parameter formula f(n)=(4n−1)·φ + 3, which decomposes as fundamental 3φ² and spacing 4φ = half the eight-tick period. Upstream results supply the primitive distinction axioms and collision-free empirical program that license the structural identities.
proof idea
The proof is a one-line wrapper that unfolds schumannRS, pushes the cast on the natural number 5, and applies the ring tactic to confirm the arithmetic identity.
why it matters in Recognition Science
This supplies the exact value used by harmonic5_bounds and harmonic5_matches, which place f(5) inside (33.742, 33.761) and within 0.06 Hz of the measured 33.8 Hz. It instantiates the general formula forced by D=3 and the eight-tick octave in the Recognition framework. The result closes the structural decomposition for the fifth harmonic in the Earth-brain resonance model.
scope and limits
- Does not derive the Schumann formula from Maxwell equations.
- Does not predict measured values without the supplied phi bounds.
- Does not address harmonics beyond n=5.
- Does not incorporate atmospheric conductivity corrections.
Lean usage
rw [harmonic5_eq]formal statement (Lean)
122theorem harmonic5_eq : schumannRS 5 = 19 * phi + 3 := by
proof body
Term-mode proof.
123 unfold schumannRS; push_cast; ring
124
125/-! ## Part IV: Structural Identities
126
127The formula's structure is forced by D = 3 and φ. -/
128
129/-- The RS-forced spatial dimension (from T8). -/