# AMRAL × Lebesgue Universal Covering — Round 36
## Depth-30 Publication Batch Completion and Durable Frontier Recovery

**Document ID:** AMRAL-LUC-FC-R36  
**Version:** v0.1  
**Date:** 2026-09-20  

---

# 0. Round Summary

Round 36 completes two closures.

First, the Depth-30 arithmetic batch:

$$
\boxed{
\{47,48,68,69,70,75,76\}
}
$$

Now:

$$
\boxed{
7/7\text{ are publication-candidate shards.}
}
$$

Second, `DFRONT-v0.1` moves from specification into real geometry recovery.

cell33's old depth-36 runtime state is deterministically rebuilt from the witness root, exactly reproducing the Round 26 durable ledger:

$$
\boxed{129401\text{ nodes}}
$$

$$
\boxed{14027\text{ pending boxes}}
$$

$$
\boxed{
V_{36}
=
2.9748317031636626\times10^{-8}
}
$$

All three values match.

The new explicit pending frontier is only:

$$
\boxed{126327\text{ bytes}}.
$$

So:

$$
\boxed{
\text{runtime-local proof state}
\to
\text{durable DFRONT proof object}
}
$$

The first real closure.

Globally, still:

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

---

# 1. cell75 promotion

cell75:

$$
10948\text{ leaves},
\qquad
2{,}627{,}357\text{ rational vertices}.
$$

RHCERT raw bytes:

$$
44{,}382{,}455.
$$

bytes-only replay:

$$
\boxed{10948/10948\text{ PASS}}.
$$

minimum exact margin:

$$
\boxed{
6.2517761412\times10^{-8}>0.
}
$$

MPFR dominance:

$$
\boxed{
1690\text{ unique states},\quad0\text{ failures}.
}
$$

Therefore:

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

---

# 2. cell76 promotion

cell76:

$$
10932\text{ leaves},
\qquad
2{,}617{,}252\text{ rational vertices}.
$$

raw certificate:

$$
44{,}229{,}325\text{ bytes}.
$$

bytes-only replay:

$$
\boxed{10932/10932\text{ PASS}}.
$$

minimum margin:

$$
\boxed{
2.0477267270\times10^{-8}>0.
}
$$

MPFR dominance:

$$
\boxed{
1688\text{ unique states},\quad0\text{ failures}.
}
$$

Therefore:

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

---

# 3. Depth-30 batch completion

Depth-30 batch:

$$
\boxed{
\{47,48,68,69,70,75,76\}
}
$$

All of them now have:

- complete finite topology;
- explicit split-axis stream;
- rational marker upper;
- independent marker replay;
- MPFR marker replay;
- concrete rational lower proof object;
- bytes-only replay;
- MPFR enclosure audit.

So:

$$
\boxed{
\text{Depth-30 Publication Batch}=7/7.
}
$$

---

# 4. Publication-candidate coverage

global necessity atlas:

$$
\boxed{
7/77
\approx9.09\%.
}
$$

relative to the current 66 known geometry-complete cells:

$$
\boxed{
7/66
\approx10.61\%.
}
$$

There remain:

$$
\boxed{
59
}
$$

geometry-complete cells at an earlier arithmetic stage.

---

# 5. DFRONT-v0.1

Each unresolved witness box stores:

$$
\boxed{
(k_\phi,m_\phi),
(k_x,m_x),
(k_y,m_y)
}
$$

where:

$$
\left[
\frac{m}{2^k},
\frac{m+1}{2^k}
\right]
$$

is the dyadic sub-interval relative to the witness-root coordinate interval.

split semantics:

$$
(k,m)\xrightarrow{L}(k+1,2m),
$$

$$
(k,m)\xrightarrow{R}(k+1,2m+1).
$$

So the final box code is independent of the historical axis split order.

---

# 6. DFRONT verifier

The Round 36 implementation checks:

1. canonical sorted / unique;
2. $0\le m<2^k$;
3. exact interior-disjoint;
4. exact relative-volume sum;
5. dependency hash;
6. parent checkpoint hash.

interior-disjoint uses an exact dyadic sweep-line:

- selected-axis endpoints are lifted to a common integer grid;
- for equal coordinates, ends are processed before starts;
- the active set only checks the other two axes for exact overlap.

This replaces the original, unscalable $O(n^2)$ brute pair scan.

---

# 7. Cell33 exact recovery

Round 26 durable values:

$$
N=129401,
$$

$$
L_{\rm pending}=14027,
$$

$$
V_{36}
=
2.9748317031636626\times10^{-8}.
$$

Round 36 re-executes the exact/reference certificate semantics from the root, simultaneously generating dyadic codes.

Result:

$$
\boxed{
N_{\rm rebuild}=129401
}
$$

$$
\boxed{
L_{\rm pending,rebuild}=14027
}
$$

$$
\boxed{
V_{\rm rebuild}
=
2.9748317031636626\times10^{-8}.
}
$$

absolute volume error:

$$
\boxed{0}.
$$

So:

$$
\boxed{
\texttt{CELL33-DFRONT-RECOVERY: PASS}.
}
$$

frontier size:

$$
\boxed{126327\text{ bytes}}.
$$

---

# 8. Depth-38 continuation scope

Round 36 attempted to advance from the recovered DFRONT to depth 38.

The full exact-kernel job exceeded this round's 120-second single-execution cap before a new checkpoint could be produced.

So:

$$
\boxed{
\text{proof-state effect}=0.
}
$$

No new d38 volume or closure is claimed.

---

# 9. Geometry status

Known strict geometry-complete:

$$
\boxed{66/77}.
$$

Residual:

$$
\boxed{11}.
$$

latest exact common budget:

$$
\boxed{51/77\text{ at }d=36}.
$$

Round 36's geometry progress is durable recovery, not an increase in closure count.

---

# 10. Durable-source audit

Round 36 checked:

- mounted runtime;
- Round 26 package;
- Round 27 package;
- AMRAL Google Drive subtree filenames.

No old residual pending-frontier bytes were found.

Therefore cell33 is a deterministic rebuild, not a retrieval of an old pickle.

Going forward, new geometry checkpoints must persist DFRONT.

---

# 11. Depth-32 queue

The next arithmetic batch has:

$$
\boxed{12\text{ closure-depth-32 cells}}.
$$

Priority order starts at:

$$
\boxed{
49,\ 64,\ 66,\ 12,\ 59,\ 57,\ldots
}
$$

cell49 has already proven that geometry topology can be reconstructed without depending on the old runtime result JSON.

Using the durable marker axis/side trace + closure ledger:

$$
\boxed{
19971\text{ nodes},\quad9986\text{ leaves}.
}
$$

The node count exactly matches the durable ledger.

---

# 12. Round 37

Arithmetic:

1. cell49 bulk rational lower migration;
2. shardize 64 / 66;
3. marker upper + A1 + MPFR;
4. begin the depth-32 publication-candidate wave.

Geometry:

1. switch cell33 DFRONT continuation to sharded jobs;
2. persist child checkpoints segment by segment;
3. recover the remaining 10 residual frontiers;
4. update geometry progress only after a durable checkpoint is produced.

---

# 13. Shortest Handoff

Round 36 turns both kinds of proof state into genuine durable artifacts:

$$
\boxed{
\text{RHCERT = certified leaves}
}
$$

$$
\boxed{
\text{DFRONT = unresolved leaves}.
}
$$

Depth-30 batch:

$$
\boxed{7/7\text{ publication-candidate}}.
$$

cell33:

$$
\boxed{
\text{runtime state recovery}\to\text{DFRONT PASS}.
}
$$

Globally, still:

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