# AMRAL × Lebesgue Universal Covering — Round 33
## Cross-Backend Enclosure Dominance and Depth-30 Batch Shardization

**Document ID:** AMRAL-LUC-FC-R33  
**Version:** v0.1  
**Date:** 2026-09-20  
**Research status:** Cross-backend scalability / Depth-30 batch arithmetic wave  
**Research initiation and methodology source:** Neo.K  
**AI collaborating researcher and primary executor:** Aletheia / ChatGPT, GPT-5.6 Sol  

---

# 0. Round Summary

Round 32's pipeline:

$$
66/77
$$

geometry-complete,

$$
3/77
$$

arithmetic-closed-or-better,

$$
2/77
$$

publication-candidate.

Round 33 did not relax the gate in order to inflate the publication-candidate number.

This round's most important new result is the **Cross-Backend Enclosure Dominance Theorem**, which compresses the second backend's audit from "regenerating every leaf's polygon" into "verifying the containment relation of the directed common-core enclosure."

cell47:

$$
\boxed{
9228\text{ leaves}
\to
1564\text{ unique witness-core states}
}
$$

cross-backend dominance:

$$
\boxed{\text{PASS},\quad 0\text{ failures}.}
$$

cell48:

$$
\boxed{
9285\text{ leaves}
\to
1574\text{ unique witness-core states}
}
$$

likewise:

$$
\boxed{\text{PASS},\quad 0\text{ failures}.}
$$

cell48's mpmath/libmp rational lower migration is also complete:

$$
\boxed{
9285/9285\text{ PASS}
}
$$

no inconclusive.

Its marker48:

- independent exact polar upper replay: PASS;
- MPFR upper replay: PASS.

So arithmetic-closed-or-better coverage rises from:

$$
3/77
$$

to:

$$
\boxed{4/77}.
$$

publication-candidate is still strictly held at:

$$
\boxed{2/77}
$$

that is:

$$
\{69,70\}.
$$

In addition, the depth-30 batch's cell68, 75, and 76 have all had explicit topology and witness-axis streams generated, and have entered the arithmetic migration queue.

Globally, still:

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

---

# 1. Cross-Backend Enclosure Dominance

For every defining disk of the common core, Backend A preserves:

$$
c_j\in R^A_j,
\qquad
\rho\ge\rho_A.
$$

Backend B independently reconstructs:

$$
c_j\in R^B_j,
\qquad
\rho\ge\rho_B.
$$

If:

$$
\boxed{
R^B_j\subseteq R^A_j
}
$$

and:

$$
\boxed{
\rho_B\ge\rho_A,
}
$$

then any exact rational point $p$ that passes A's worst-case membership:

$$
\sup_{c\in R^A_j}\|p-c\|<\rho_A,
$$

automatically has:

$$
\sup_{c\in R^B_j}\|p-c\|
\le
\sup_{c\in R^A_j}\|p-c\|
<
\rho_A
\le
\rho_B.
$$

So:

$$
\boxed{
p\in G_A^{\rm cert}
\Rightarrow
p\in G_B^{\rm cert}.
}
$$

---

# 2. Polygon Transfer Corollary

If an exact rational polygon:

$$
P=\operatorname{conv}\{p_1,\ldots,p_m\}
$$

has every vertex verified by Backend A's membership certificate,

and Backend B satisfies the dominance premises for every defining disk,

then that same set of exact rational vertices transfers without regenerating the candidate geometry:

$$
\boxed{
P\subseteq G_B.
}
$$

The exact rational shoelace area is backend-independent.

So the second backend only needs to audit enclosure semantics.

---

# 3. Why this is a large scalability gain

The cost of brute-force cross-backend tree replay scales with the number of leaves:

$$
O(L).
$$

But a large number of leaves in the B7 tree share the same:

- rotation interval;
- halfwidth tuple;
- directed core state.

cell47:

$$
L=9228,
$$

but the unique core states number only:

$$
\boxed{1564}.
$$

cell48:

$$
L=9285,
$$

unique:

$$
\boxed{1574}.
$$

roughly a:

$$
6\times
$$

reduction in expensive directed transcendental state construction.

---

# 4. Cell47 dominance audit

Root containment:

$$
t_3,t_5,t_7,\pi,\sqrt3:
\boxed{\text{PASS}}.
$$

Base directed box containment:

`PASS`.

Base $B_3$ core:

`PASS`.

Base $B_5$ core:

`PASS`.

Witness translation/halfwidth containment:

`PASS`.

B7 unique core states:

$$
1564.
$$

failures:

$$
\boxed{0}.
$$

Minimum gain of MPFR radius-lower minus mpmath radius-lower:

$$
\boxed{
2.15\times10^{-64}>0.
}
$$

---

# 5. Cell48 arithmetic migration

cell48 topology:

$$
18569\text{ nodes},
$$

$$
9285\text{ leaves},
$$

closure depth:

$$
30.
$$

mpmath/libmp rational lower:

$$
\boxed{
9285/9285\text{ PASS}.
}
$$

density:

- 256: 824
- 512: 3438
- 1024: 3725
- 2048: 1065
- 4096: 183
- 8192: 43
- 16384: 7

minimum rational margin:

$$
\boxed{
1.03164\times10^{-7}>0.
}
$$

---

# 6. Cell48 marker upper

independent exact polar replay:

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

outer:

$$
0.8349325735415322.
$$

margin:

$$
\boxed{
6.74265\times10^{-5}.
}
$$

MPFR replay reaches the same displayed outer area:

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

---

# 7. Cell48 dominance audit

Root containment:

`PASS`.

Base cores:

`PASS`.

Witness translation boxes:

`PASS`.

unique B7 core states:

$$
\boxed{1574}.
$$

failures:

$$
\boxed{0}.
$$

---

# 8. Certificate-Entity Gap

This round uncovered a production ABI gap:

> The computation ledger records `PASS / density / margin` — that is not the same as having preserved an independently replayable finite certificate entity.

The Round 32 emitter did not persist, for each leaf:

- the exact rational hull vertices;
- body source tags;
- the exact area fraction;

or an equivalent deterministic compact transcript.

So although Round 33 proves that:

> if these exact rational polygons were given explicitly, MPFR envelope dominance would automatically verify them,

the current summary ledger itself cannot substitute for the polygon certificate bytes.

So cell47/48 **are not promoted** to:

`PUBLICATION-CANDIDATE`.

---

# 9. New certificate persistence rule

Future bulk emitters must preserve at least one of the following:

## Option A — Full rational hull

For each leaf:

```text
vertex_count
(x_i,y_i) integer-grid coordinates
source tags
exact area numerator/denominator or hash
```

## Option B — Deterministic transcript

Sufficient to uniquely reconstruct the same rational hull:

```text
candidate-density
grid denominator
candidate algorithm id/hash
source selection transcript
directed-box dependency hash
```

An independent verifier must be able to recover the exact finite certificate without trusting the emitter's PASS bit.

---

# 10. Raw certificate-size implication

cell47's total final hull vertex count is approximately:

$$
2.16\times10^6.
$$

cell48:

$$
2.11\times10^6.
$$

If each vertex is first stored as two signed 64-bit grid integers,

the raw coordinate payload is approximately:

$$
16\text{ bytes/vertex}.
$$

A single shard's raw size is on the order of a few dozen MB.

So production needs:

- delta coding;
- zigzag varint;
- packed source tags;
- sibling/parent structural reuse;
- or a deterministic compact transcript.

This becomes a new certificate-engineering problem, not a mathematical-correctness problem.

---

# 11. Depth-30 batch shardization

depth-30 cells:

$$
47,48,68,69,70,75,76.
$$

Currently:

## Publication candidate

$$
\boxed{
69,70
}
$$

## Cross-backend enclosure audited prototypes

$$
\boxed{
47,48
}
$$

## Shardized / arithmetic pending

$$
\boxed{
68,75,76
}
$$

Three new topologies:

### cell68

$$
18509\text{ nodes},
\qquad
9255\text{ leaves}.
$$

### cell75

$$
21895\text{ nodes},
\qquad
10948\text{ leaves}.
$$

### cell76

$$
21863\text{ nodes},
\qquad
10932\text{ leaves}.
$$

All explicit topology / axis streams have been generated.

---

# 12. Stage coverage after Round 33

Geometry:

$$
\boxed{
66/77
}
$$

unchanged.

This round did not complete a new common-depth geometry wave, so no geometry progress is claimed that was not actually made.

Arithmetic-closed-or-better:

$$
\boxed{
\{47,48,69,70\}
}
$$

so:

$$
\boxed{
4/77\approx5.19\%.
}
$$

Publication candidate:

$$
\boxed{
\{69,70\}
}
$$

so:

$$
\boxed{
2/77\approx2.60\%.
}
$$

---

# 13. Why publication count did not increase

It is not a cross-backend failure.

On the contrary:

$$
\boxed{
47/48\text{ backend dominance fully PASS}.
}
$$

The real blocker is:

$$
\boxed{
\text{finite certificate entity persistence}.
}
$$

This is a production problem more worth fixing first than "running MPFR one more time."

---

# 14. Round 34

## Replayable Rational-Hull Certificate Format and Depth-30 Batch Migration

Next round's priorities:

1. design a compact replayable binary format for rational inner polygons;
2. re-emit the finite hull transcript for cell47/48;
3. use the MPFR dominance verifier to read the transcript directly for:
   - membership;
   - exact area;
4. if all PASS:
   - promote 47/48 to publication-candidate;
5. in parallel, launch 68/75/76 single-backend lower migration;
6. resume the common-depth wave for the geometry residual queue.

---

# 15. Shortest Handoff

Round 33's biggest advance is not "running yet one more cell," but compressing cross-backend verification from:

$$
\boxed{
\text{regenerating the entire polygon tree}
}
$$

down to:

$$
\boxed{
\text{finite core-enclosure dominance audit}.
}
$$

cell47/48:

$$
\boxed{
0\text{ dominance failures}.
}
$$

What now genuinely blocks publication promotion is:

> the certificate entity must be persisted — it is not enough to retain only a PASS summary.

Globally, still:

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