# AMRAL × Lebesgue Universal Covering — Round 37
## Depth-32 Publication Migration and Sharded DFRONT Continuation

**Document ID:** AMRAL-LUC-FC-R37  
**Version:** v0.1  
**Date:** 2026-09-20  
**Research status:** Depth-32 publication-candidate wave / Sharded durable geometry continuation  
**Research initiation and methodology source:** Neo.K  
**AI collaborating researcher and primary executor:** Aletheia / ChatGPT, GPT-5.6 Sol  

---

# 0. Round Summary

Round 36 completed the depth-30 batch:

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

and used cell33 to prove that DFRONT can exactly recover runtime-local geometry state from a durable ancestor.

Round 37 simultaneously advances both production pipelines by one more layer.

## Arithmetic

The first three cells of the depth-32 queue:

$$
\boxed{
49,\ 64,\ 66
}
$$

all completed:

- durable topology reconstruction;
- full rational lower migration;
- independent marker upper replay;
- MPFR marker upper replay;
- MPFR enclosure dominance;
- concrete `RHCERT-v0.1`;
- bytes-only replay.

Therefore:

$$
\boxed{
49,64,66
\to
\texttt{PUBLICATION-CANDIDATE-SHARD}.
}
$$

publication-candidate cells are now:

$$
\boxed{
\{47,48,49,64,66,68,69,70,75,76\}
}
$$

that is:

$$
\boxed{
10/77\approx12.99\%.
}
$$

relative to the 66 known geometry-complete cells:

$$
\boxed{
10/66\approx15.15\%.
}
$$

## Geometry

Starting from the recovered d36 frontier, cell33 successively produced, via immutable sharded jobs:

$$
d38,\quad d40,\quad d42
$$

three durable DFRONT checkpoints.

absolute unresolved volume:

$$
2.9748\times10^{-8}
\to
1.4338\times10^{-8}
\to
5.4258\times10^{-9}
\to
2.6015\times10^{-9}.
$$

Therefore:

$$
\boxed{
V_{42}/V_{36}
\approx0.08745.
}
$$

Although the pending count:

$$
14027
\to27042
\to40934
\to78507,
$$

the volume has shrunk to only about:

$$
\boxed{8.745\%}.
$$

Therefore there is still no plateau evidence.

cell33 remains PARTIAL, so the global geometry-complete count is unchanged.

Globally, still:

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

---

# 1. Durable depth-32 shard reconstruction

Round 37 does not read the lost `r26_full77_results` as a base-box source.

Each depth-32 cell is instead reconstructed directly from:

1. the durable marker path;
2. the explicit base axis trace;
3. the Round 26 closure depth / node ledger;

rebuilding the finite lift topology.

## cell49

$$
19971\text{ nodes},
\qquad
9986\text{ leaves}.
$$

The node count exactly matches the durable ledger.

## cell64

$$
21077\text{ nodes},
\qquad
10539\text{ leaves}.
$$

## cell66

$$
21271\text{ nodes},
\qquad
10636\text{ leaves}.
$$

All three:

$$
\boxed{
\text{node-count match}=PASS.
}
$$

---

# 2. cell49 arithmetic

full rational lower:

$$
\boxed{
9986/9986\text{ PASS}.
}
$$

density:

- 256: 811
- 512: 3139
- 1024: 4127
- 2048: 1592
- 4096: 259
- 8192: 47
- 16384: 11

minimum lower margin:

$$
\boxed{
3.15312672\times10^{-8}>0.
}
$$

marker independent upper:

$$
0.8349813779061498<0.835.
$$

margin:

$$
\boxed{
1.86221\times10^{-5}.
}
$$

MPFR marker replay: PASS.

MPFR enclosure dominance:

$$
\boxed{
1659\text{ unique states},\quad0\text{ failures}.
}
$$

RHCERT:

$$
\boxed{
2{,}441{,}597\text{ vertices}
}
$$

bytes-only replay:

$$
\boxed{9986/9986\text{ PASS}.}
$$

---

# 3. cell64 arithmetic

lower:

$$
\boxed{
10539/10539\text{ PASS}.
}
$$

minimum margin:

$$
\boxed{
1.50693499\times10^{-8}>0.
}
$$

marker outer:

$$
0.8349716675759491.
$$

MPFR dominance:

$$
\boxed{
1702\text{ states},\quad0\text{ failures}.
}
$$

RHCERT vertices:

$$
\boxed{
2{,}624{,}186.
}
$$

bytes-only replay:

$$
\boxed{10539/10539\text{ PASS}.}
$$

---

# 4. cell66 arithmetic

lower:

$$
\boxed{
10636/10636\text{ PASS}.
}
$$

minimum margin:

$$
\boxed{
6.86089458\times10^{-9}>0.
}
$$

This is the thinnest of the three cells in this round.

marker outer:

$$
0.8348951300993619.
$$

upper margin:

$$
\boxed{
1.048699\times10^{-4}.
}
$$

MPFR dominance:

$$
\boxed{
1710\text{ states},\quad0\text{ failures}.
}
$$

RHCERT vertices:

$$
\boxed{
2{,}646{,}674.
}
$$

bytes-only replay:

$$
\boxed{10636/10636\text{ PASS}.}
$$

---

# 5. Publication-candidate coverage

Round 36:

$$
7/77.
$$

Round 37 adds:

$$
49,64,66.
$$

Therefore:

$$
\boxed{
10/77\approx12.99\%.
}
$$

relative to known geometry-complete:

$$
\boxed{
10/66\approx15.15\%.
}
$$

geometry-complete but still below the publication-candidate stage:

$$
\boxed{
56.
}
$$

---

# 6. Cell33 d36 → d38

parent:

$$
14027\text{ pending}.
$$

split into:

$$
29
$$

immutable refinement segments.

After merging:

$$
\boxed{
27042\text{ pending}
}
$$

$$
\boxed{
V_{38}=1.4337598723\times10^{-8}.
}
$$

ratio:

$$
\boxed{
V_{38}/V_{36}=0.481963.
}
$$

---

# 7. Cell33 d38 → d40

The d38 frontier is further split into:

$$
55
$$

segments.

d40:

$$
\boxed{
40934\text{ pending}
}
$$

$$
\boxed{
V_{40}=5.4257753323\times10^{-9}.
}
$$

ratio:

$$
\boxed{
V_{40}/V_{38}=0.378430.
}
$$

---

# 8. Cell33 d40 → d42

d40 frontier:

$$
82
$$

segments.

d42:

$$
\boxed{
78507\text{ pending}
}
$$

$$
\boxed{
V_{42}=2.6015130699\times10^{-9}.
}
$$

ratio:

$$
\boxed{
V_{42}/V_{40}=0.479473.
}
$$

So from d36 to d42:

$$
\boxed{
V_{42}/V_{36}=0.087451.
}
$$

This remains a strong contraction.

---

# 9. Compositional DFRONT Merge Theorem

Let the parent frontier be:

$$
P
$$

already verified pairwise interior-disjoint.

Split it by parent index into non-overlapping slices:

$$
P=P_1\sqcup\cdots\sqcup P_m.
$$

Each immutable worker refines only its own slice, producing pending descendants:

$$
Q_i\subseteq\bigcup P_i,
$$

and within each segment:

$$
Q_i
$$

has already been verified exact-disjoint.

Because distinct parent slices already lie in pairwise-disjoint parent regions,

therefore:

$$
\boxed{
i\ne j
\Rightarrow
\operatorname{int}(Q_i)
\cap
\operatorname{int}(Q_j)
=
\varnothing.
}
$$

Hence:

$$
\boxed{
Q=\bigcup_iQ_i
}
$$

can use:

- the parent frontier hash;
- complete parent-slice coverage;
- each segment's SHA-256;
- the segment PASS manifest;

to perform a compositional merge, without needing to re-run the expensive global pairwise scan.

---

# 10. Why this mattered

The d40 frontier had already exceeded:

$$
4\times10^4
$$

boxes.

Redoing a monolithic disjoint scan exceeds the interactive execution cap.

After Round 37 switched to proof-by-composition:

> A verified immutable segment does not need to re-prove the same fact again at the merge stage.

This is the same principle already at work in the earlier:

- Certified Ancestor Contraction;
- semantic revalidation;
- RHCERT finite proof objects;

— namely:

$$
\boxed{
\text{verified local evidence should compose}.
}
$$

---

# 11. Geometry status remains conservative

cell33 at d42 is still:

`PARTIAL`.

Therefore:

$$
\boxed{
\text{known geometry-complete}=66/77.
}
$$

residual:

$$
\boxed{11}.
$$

latest exact common budget:

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

This round does not treat volume contraction as closure.

---

# 12. Next depth-32 batch prepared

Round 37 also made the next batch of three topologies durable along the way:

## cell12

$$
21789\text{ nodes},
\qquad
10895\text{ leaves}.
$$

## cell59

$$
22009\text{ nodes},
\qquad
11005\text{ leaves}.
$$

## cell57

$$
22261\text{ nodes},
\qquad
11131\text{ leaves}.
$$

All three node counts exactly match the durable ledger.

So Round 38 can begin directly from arithmetic migration.

---

# 13. Current stage dashboard

Geometry complete:

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

Publication candidate:

$$
\boxed{
10/77.
}
$$

Depth-30 batch:

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

Depth-32 batch:

$$
\boxed{
3/12\text{ publication-candidate}.
}
$$

In addition:

$$
\boxed{
3/12
}
$$

already durable-shardized, awaiting arithmetic.

---

# 14. Round 38

## Arithmetic

1. cell12 / 59 / 57 bulk lower migration;
2. marker independent + MPFR;
3. RHCERT bytes-only replay;
4. further raise depth-32 publication-candidate coverage;
5. shardize remaining 60 / 65 / 67 / 46 / 45 / 58.

## Geometry

1. cell33 d42 → d44 using hashed-segment compositional continuation;
2. begin deterministic recovery of the remaining residual frontiers;
3. each new checkpoint counts as progress only once its DFRONT bytes are committed to disk.

---

# 15. Shortest Handoff

Round 37 simultaneously proves that two things can be mass-produced:

$$
\boxed{
\text{RHCERT publication migration}
}
$$

and:

$$
\boxed{
\text{DFRONT sharded continuation}.
}
$$

publication-candidate coverage:

$$
\boxed{
7/77
\to
10/77.
}
$$

cell33 unresolved volume:

$$
\boxed{
2.9748\times10^{-8}
\to
2.6015\times10^{-9}.
}
$$

But cell33 still has not reached closure, so the global theorem status is unchanged:

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