atmospheric_layering_one_statement
plain-language theorem explainer
The theorem states that tropopause, stratopause, and thermosphere boundaries align with rungs 0, 3, and 7 on the phi-altitude ladder, with mesopause between 4 and 5, strict ordering preserved, the stratopause-to-tropopause ratio equal to phi cubed inside (4.22, 4.24), and that ratio below the thermosphere-to-tropopause ratio. Climate physicists working in Recognition Science would cite it as the compact closed-form statement for Track P4. The proof is a direct term that assembles reflexivity for the rung values with the ordering theorem, the pi
Claim. Let $z(k) = z_0 phi^k$ denote altitude on the phi-ladder. The tropopause lies at rung 0, the stratopause at rung 3, and the thermosphere base at rung 7. These satisfy the strict ordering $0 < 3 < 4$ (mesopause lower) and $5$ (mesopause upper) $< 7$. The ratio of stratopause to tropopause altitude equals $phi^3$ and lies in the open interval $(4.22, 4.24)$. Moreover $phi^3 < phi^7$.
background
The module treats atmospheric layer boundaries as minima of the J-cost on the radiative-convective recognition lattice. Each boundary corresponds to a rung k on the phi-ladder with altitude ratio $phi^k$ relative to the tropopause base. Upstream definitions fix tropopause_rung = 0, stratopause_rung = 3, thermosphere_rung = 7, mesopause_rung_lower = 4, mesopause_rung_upper = 5, and stratopause_tropopause_ratio = phi^3. The rung_strict_ordering theorem asserts the chain of inequalities among these natural-number rungs. The stratopause_tropopause_ratio_band theorem supplies the numerical interval (4.22, 4.24) inside the empirical band (3.5, 4.5).
proof idea
The proof is a term-mode wrapper. It supplies the triple of reflexivity proofs for the three rung equalities, then directly invokes rung_strict_ordering for the ordering conjunct, stratopause_tropopause_ratio_band for the ratio band, and thermosphere_above_stratopause_ratio for the final comparison.
why it matters
This declaration supplies the single-statement summary of atmospheric layering for Track P4 in Plan v7. It embeds the phi-ladder (T6 self-similar fixed point) into empirical climate boundaries and supplies closed-form ratios without additional axioms. No downstream theorems yet depend on it, leaving open its integration into larger Recognition Science climate models.
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