thermosphere_tropopause_ratio_pos
plain-language theorem explainer
The declaration proves that the thermosphere-to-tropopause altitude ratio, defined as phi to the seventh power, is strictly positive. Climate researchers using the Recognition Science phi-ladder for layer boundaries would cite this to confirm the geometric progression remains positive at every rung. The proof is a one-line term-mode reduction that unfolds the definition and invokes the standard positivity of powers for a positive base.
Claim. $0 < phi^7$, where $phi$ denotes the self-similar fixed point of the Recognition Science framework and the exponent 7 counts the cumulative rung separation between the tropopause and thermosphere boundaries on the altitude ladder.
background
In the Recognition Science treatment of Earth's atmosphere, layer boundaries arise from J-cost minima on the radiative-convective recognition lattice. Altitude at rung k is given by $z_0 phi^k$ with $z_0$ the base recognition altitude; the thermosphere-tropopause ratio is the special case $phi^7$ that accumulates the rung steps from the tropopause (near rung 0) to the thermosphere (rung 7). The module AtmosphericLayeringFromPhiLadder derives closed-form ratios for the five canonical layers (troposphere through exosphere) as structural theorems with zero axioms and zero sorrys. Upstream results include the direct definition thermosphere_tropopause_ratio := phi^7 together with the imported positivity infrastructure for real exponentiation.
proof idea
The proof is a one-line term-mode wrapper. It unfolds thermosphere_tropopause_ratio to the expression phi^7 and then applies the lemma pow_pos to the known positivity of phi.
why it matters
This positivity statement supplies an elementary but required certificate inside the master atmospheric layering certificate atmosphericLayeringFromPhiLadderCert. It closes the structural theorem for the phi-ladder description of the five atmospheric layers. Within the broader framework it instantiates the self-similar fixed point phi (T6) and the discrete rung steps that follow from the eight-tick octave. No open questions or scaffolding remain attached to this elementary result.
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