EarthBrainResonanceCert
plain-language theorem explainer
The EarthBrainResonanceCert structure records that Schumann frequencies obey the zero-parameter formula f(n)=(4n-1)φ+3 forced by the Recognition Composition Law, with the fundamental at 3φ²≈7.854 Hz, constant spacing 4φ, and all five harmonics matching measured values to within 0.06 Hz while landing in theta-beta-gamma EEG bands. Researchers deriving natural frequencies from foundational axioms would cite this when connecting Earth's electromagnetic cavity to brain rhythms. The declaration is a structure definition that assembles the forced RS-
Claim. A certificate asserts that the Schumann resonance function satisfies $f(n)=(4n-1)φ+3$ for all natural $n$, the identities $f(1)=3φ^2=φ^4+1$, constant spacing $Δf=4φ$, and strict increase, together with the bounds $|f(1)-7.83|<0.06$, $|f(2)-14.3|<0.06$, $|f(3)-20.8|<0.06$, $|f(4)-27.3|<0.06$, $|f(5)-33.8|<0.06$, and membership of $f(1)$ in the 4-8 Hz band, $f(2)$--$f(4)$ in 13-30 Hz, and $f(5)$ in 30-100 Hz, where $φ$ is the golden ratio and provenance labels separate the algebraically forced components from the empirical checks.
background
The module works in the setting where the Recognition Composition Law (RCL) is the sole starting axiom: for positive $x,y$, $F(xy)+F(x/y)=2F(x)F(y)+2F(x)+2F(y)$. This forces the golden ratio $φ$ via the fixed-point equation $φ^2=φ+1$ (T6) and the spatial dimension $D=3$ via linking and synchronization conditions (T8). The structure uses the sibling abbreviation $D:=3$ and the inductive type Provenance with constructors forced and empirical to tag each field. Upstream results include the Composition class that encodes the RCL and the PrimitiveDistinction theorem that reduces seven axioms to four structural conditions.
proof idea
This is a structure definition with an empty proof body. Its fields are populated downstream in earthBrainResonance_forced by direct application of the sibling lemmas fundamental_eq_D_phi_sq (for $f(1)=3φ^2$), fundamental_eq_phi4_plus_1 (for $f(1)=φ^4+1$), spacing_eq (for the constant difference $4φ$), and schumannRS_strictMono (for monotonicity), together with rfl for the zero-parameter formula.
why it matters
The certificate supplies the concrete content for the master theorem earthBrainResonance_forced, which states that Earth-Brain Resonance follows from the RCL alone with zero free parameters and reproduces the five observed Schumann harmonics to <0.4% error while spanning EEG bands. It directly realizes the T6 and T8 landmarks by exhibiting the $φ$-ladder in the Earth's cavity spectrum. No open scaffolding remains in this declaration.
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