J_phi_Hz
plain-language theorem explainer
J_phi_Hz defines the real number phi minus 3/2 as the canonical J-cost ceiling in Hz units for bounding Schumann resonance offsets. Researchers deriving astrophysical frequencies from the Recognition Science BIT carrier would cite this constant when measuring the 0.118 Hz tolerance between the predicted 5 phi band and the empirical 7.83 Hz value. The declaration is a direct one-line definition with no proof steps or additional hypotheses.
Claim. Define $J(phi) := phi - 3/2$ in Hz units, where $phi$ is the golden-ratio fixed point satisfying the self-similar equation of the Recognition framework.
background
The Recognition Science J-cost satisfies the functional equation J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y) with uniqueness J(x) = (x + x^{-1})/2 - 1; its value at the forced fixed point phi is the ceiling used here. The module sets the Schumann fundamental equal to the BIT carrier 5 phi Hz, which lies numerically in (8.05, 8.10) Hz, and treats J(phi) as the structural tolerance that places the observed 7.83 Hz inside the predicted window after K-gate suppression. Local conventions follow the eight-tick octave of T7 and the gauge-boson scaffolding of Foundation.RecognitionSpectrum, with phi > 1.618 taken from upstream arithmetic lemmas.
proof idea
The declaration is a one-line definition that directly assigns phi - 3/2 to J_phi_Hz.
why it matters
This definition supplies the numerical tolerance required by SchumannResonanceCert and schumann_one_statement, closing the gap between the RS-predicted 5 phi band and the empirical Schumann value. It instantiates the J-uniqueness of T5 and the eight-tick octave of T7 inside the astrophysics module, with the same J(phi) factor appearing in the muon g-2 derivation via the 1/45 K-gate. The construction supports the structural claim that the Earth-ionosphere cavity mode is identified with the BIT carrier.
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