recoveryTime_strict_mono
plain-language theorem explainer
Recovery time after a cascade of depth k is strictly monotone increasing in k. Researchers modeling extinction cascades in Recognition Science ecosystems cite this to bound post-extinction recovery durations that scale as phi^k. The proof is a direct term-mode wrapper that unfolds the definition, rewrites the successor power, and closes with nlinarith using positivity and one_lt_phi.
Claim. For every natural number $k$, the recovery time after a cascade of depth $k$ is strictly less than after depth $k+1$, where recovery time equals the golden ratio to the power $k$.
background
The module models ecosystems as finite recognition graphs in which species rungs drop below the ignition threshold Z_life = phi^19 upon ledger bankruptcy, triggering monotone cascade closure. Recovery time after a cascade of depth k is defined as phi^k in units of the natural scale tau_0. Upstream, one_lt_phi supplies 1 < phi and the recoveryTime definition supplies the explicit power expression.
proof idea
Unfold recoveryTime to replace both sides by powers of phi. Rewrite the right-hand side via pow_succ. Introduce 0 < phi^k from pow_pos phi_pos k and 1 < phi from one_lt_phi. Apply nlinarith to obtain the strict inequality.
why it matters
The result is invoked inside ExtinctionCascadeCert and the one-statement theorem extinction_cascade_one_statement, which asserts that post-cascade recovery time scales as phi^k and remains unbounded above. It supplies the monotonicity needed for the module's claim that recovery after deep cascades (k=17) exceeds phi^16, matching 10^4-10^5 year K-Pg mammal radiation under canonical tau_0 calibration. It reinforces phi-ladder scaling within the Recognition Science framework.
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