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Explain the Lean theorem `schumannRS_strictMono` in module `IndisputableMonolith.Physics.EarthBrainResonance`. Write for an educated reader who knows science and programming but may not know Lean. Cover: (1) what the declaration says in plain English, (2) why it matters in Recognition Science, (3) how to read the formal statement, (4) visible dependencies or certificates in the supplied source, and (5) what this declaration does not prove. Cite only declarations present in the supplied Recognition source.

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The declaration schumannRS_strictMono appears in module IndisputableMonolith.Physics.EarthBrainResonance.

(1) In plain English it asserts that the RS-predicted Schumann harmonic frequencies form a strictly increasing sequence: whenever the harmonic index m is smaller than n, the predicted frequency for m is smaller than the predicted frequency for n.

(2) In Recognition Science this matters because the underlying formula schumannRS n = (4n-1)·phi + 3 is built only from the RS-forced constants D=3 (T8) and phi (T6). Strict monotonicity is the minimal structural property needed for the sequence to be capable of matching the observed increasing Schumann resonances; it is invoked directly inside the master certificate earthBrainResonance_forced.

(3) The formal statement reads: ∀ (m n : ℕ), m < n → schumannRS m < schumannRS n. The proof first rewrites the difference schumannRS n - schumannRS m as 4·phi·(n-m) using the definition of schumannRS and the ring tactic, then shows the difference is positive by multiplying the already-proved spacing_pos by the positive quantity (n-m), and finally concludes the inequality by linarith.

(4) Visible dependencies inside the supplied source are the definition of schumannRS, the theorem spacing_pos (0 < 4·phi), the positivity of phi, and the tactics ring, mul_pos, sub_pos, Nat.cast_lt and linarith. The theorem is packaged into the strictly_increasing field of earthBrainResonance_forced.

(5) The declaration proves only the ordering property of the mathematical function schumannRS. It does not prove numerical agreement with measured Schumann values (those appear in harmonic1_matches through harmonic5_matches), EEG band membership, or any physical claim that the formula describes actual Earth-cavity resonances.

cited recognition theorems

outside recognition

Aspects Recognition does not yet address:

  • Any claim that the formula physically generates Schumann resonances (Earth radius is treated as contingent)
  • Derivation of EEG band boundaries from RS
  • Falsification criteria beyond the mathematical statement

recognition modules consulted

The Recognition library is at github.com/jonwashburn/shape-of-logic. The model is restricted to the supplied Lean source and instructed not to invent theorem names. Treat output as a starting point, not a verified proof.