# AMRAL × Lebesgue Universal Covering — Round 32
## Bulk Publication-Candidate Migration and Global Residual Closure Wave

**Document ID:** AMRAL-LUC-FC-R32  
**Version:** v0.1  
**Date:** 2026-09-20  
**Research Status:** Bulk shard migration wave 1 / Geometry residual continuation

---

# 0. Round Summary

Round 32 is the first time proof progress has been split into genuine stage coverage, rather than a single completion percentage.

Currently:

$$
\boxed{66/77}
$$

cells are strict geometry-close at some finite tested depth; remaining:

$$
\boxed{11}
$$

geometry residuals.

The latest complete common-budget is still:

$$
\boxed{\Gamma_{B_7}(36)=51/77.}
$$

Higher budgets, since not all residuals are synchronized, can only be reported as lower bounds:

$$
\boxed{
\Gamma(38)\ge57,\ 
\Gamma(40)\ge63,\
\Gamma(42)\ge64,\
\Gamma(44)\ge65,\
\Gamma(46)\ge66.
}
$$

The arithmetic team obtained the following this round:

- marker70: already `PUBLICATION-CANDIDATE-SHARD`;
- cell69: newly promoted to `PUBLICATION-CANDIDATE-SHARD`;
- cell47: promoted to `ARITHMETICALLY-CLOSED-PROTOTYPE`.

Therefore:

$$
\boxed{
\text{Publication-Candidate}=2/77
}
$$

And:

$$
\boxed{
\text{Arithmetic-Closed-or-Better}=3/77.
}
$$

The global theorem remains:

$$
\boxed{
a_{\mathrm{Leb}}\ge0.835
\text{ NOT CERTIFIED.}
}
$$

---

# 1. Geometry Progress

New strict closures this round:

- cell31: depth 42;
- cell27: depth 44;
- cell7: depth 46.

cell7's unresolved volume:

$$
9.43\times10^{-10}
\to
1.34\times10^{-10}
\to
1.93\times10^{-12}
\to
0.
$$

So the currently known strict geometry-complete count is:

$$
\boxed{66/77\approx85.71\%.}
$$

This is not a common-depth-46 statement; only the depth-36 figure of 51/77 is a complete common-budget result.

---

# 2. Cell69 Topology

cell69:

$$
\boxed{15711\text{ nodes}}
$$

$$
\boxed{7856\text{ terminal leaves}}
$$

closure depth:

$$
\boxed{30.}
$$

The explicit topology / split-axis stream is consistent with the original geometry ledger.

---

# 3. Cell69 mpmath/libmp Migration

The first arithmetic implementation:

`mpmath.iv/libmp + support/contact candidate generation`.

Result:

$$
\boxed{7856/7856\text{ PASS.}}
$$

minimum exact rational margin:

$$
\boxed{1.6525\times10^{-9}>0.}
$$

---

# 4. Cell69 MPFR Cross-Backend

The second implementation/backend:

`libMPFR/GMP + defining-disk support geometry`.

The first complete wave leaves only one leaf:

```text
011000100000100101100
```

At 32768 directions:

$$
A-167/200\approx-3.07\times10^{-9}.
$$

Under the fail-closed rule it is kept as `INCONCLUSIVE`.

Only for that leaf, the arithmetic density is increased:

$$
32768\to65536.
$$

This gives:

$$
\boxed{
A-167/200\approx2.5329\times10^{-9}>0.
}
$$

So no geometry resplit is needed.

Finally:

$$
\boxed{7856/7856\text{ MPFR PASS.}}
$$

---

# 5. Cell69 Marker Upper Bound

marker69's exact-rational outer bound:

$$
\boxed{
0.8349723974082344<0.835.
}
$$

margin:

$$
\boxed{
2.76026\times10^{-5}.
}
$$

Independent exact polar replay PASS;

MPFR support replay PASS.

cell69 therefore satisfies the local publication-candidate gate:

$$
\boxed{
\texttt{PUBLICATION-CANDIDATE-SHARD.}
}
$$

---

# 6. Cell47 Arithmetic Prototype

cell47:

$$
18455\text{ nodes},
\qquad
9228\text{ leaves},
$$

depth 30.

mpmath/libmp rational lower bound:

$$
\boxed{9228/9228\text{ PASS.}}
$$

minimum rational margin:

$$
\boxed{6.7502\times10^{-9}>0.}
$$

marker47 already had, from Round 29:

- rational upper PASS;
- independent exact polar replay PASS.

But this round has not yet done the full MPFR lower cross-backend, so it strictly remains at:

$$
\boxed{
\texttt{ARITHMETICALLY-CLOSED-PROTOTYPE.}
}
$$

---

# 7. Stage Coverage Dashboard

Currently:

$$
\boxed{
\text{Geometry Complete}=66
}
$$

$$
\boxed{
\text{Arithmetic Closed or Better}=3
}
$$

$$
\boxed{
\text{Publication Candidate}=2
}
$$

$$
\boxed{
\text{Geometry Complete but Arithmetic Pending}=63
}
$$

$$
\boxed{
\text{Geometry Residual}=11.
}
$$

Relative to the global 77 cells:

$$
\boxed{
\text{Publication-Candidate Coverage}
=
2/77
\approx2.60\%.
}
$$

Relative to the currently geometry-complete 66 cells:

$$
\boxed{
2/66
\approx3.03\%.
}
$$

---

# 8. Bulk Migration Lesson

Round 32 is the first time selective arithmetic refinement is actually triggered:

$$
\boxed{
\text{one leaf density escalation}
}
$$

rather than:

$$
\text{whole-tree geometry resplit}.
$$

This empirically confirms the correct fallback order for arithmetic migration:

$$
\boxed{
\text{increase finite certificate density}
\to
\text{increase arithmetic precision}
\to
\text{only then geometry resplit.}
}
$$

---

# 9. Separate Queues

Production must now maintain two independent queues:

## Geometry Queue

11 residuals, doing only $B_7$ closure continuation.

## Arithmetic Queue

63 cells that are geometry-complete but not yet arithmetic-complete.

These two queues should not block each other.

---

# 10. Round 33

The next round prioritizes completing the depth-30 batch:

$$
47,48,68,75,76.
$$

Arithmetic:

- cell47: fill in MPFR;
- 48/68/75/76: build topology and bulk-migrate;
- continue to increase the publication-candidate count.

Geometry:

- continue the 11 residuals;
- attempt to restore a new exact common-budget checkpoint.

---

# 11. Shortest Handoff

Round 32's core change is:

$$
\boxed{
\text{proof progress}
\to
\text{stage coverage dashboard.}
}
$$

It is no longer just a question of "how many cell closures," but tracking simultaneously:

$$
\boxed{
\text{Geometry}
\to
\text{Arithmetic}
\to
\text{A1}
\to
\text{Cross-Backend}
\to
\text{Publication Candidate}.
}
$$

Currently:

$$
\boxed{66/77\text{ geometry complete}}
$$

$$
\boxed{3/77\text{ arithmetic-closed or better}}
$$

$$
\boxed{2/77\text{ publication-candidate}.}
$$