# AMRAL × Lebesgue Universal Covering — Round 28
## Exact Root Reconstruction and Bulk Rational-Interval Migration

**Document ID:** AMRAL-LUC-FC-R28  
**Version:** v0.1  
**Date:** 2026-09-20  
**Research status:** Round 28 / Directed master-root reconstruction / Bulk leaf migration  
**Research mode:** Human-Directed + Semi-Autonomous AI Mathematical Research  
**Research initiator and methodology source:** Neo.K  
**AI collaborating researcher and primary executor:** Aletheia / ChatGPT, GPT-5.6 Sol  
**Prerequisite documents:** AMRAL-LUC-FC-R00 v0.2; R01–R27 v0.1  

---

# 0. Round Summary

Round 27 was the first to pass cell0's 5 thinnest COMPLETE leaves, using:

$$
\text{directed interval membership}
+
\text{exact rational polygon area}.
$$

But one publication-structure gap remained:

> Round 27's interval prototype built its enclosure around the stored binary64 leaf state, rather than reconstructing everything from the master root / exact split semantics.

Round 28 fills this gap, and performs bulk migration on the entirety of cell0's:

$$
\boxed{9278}
$$

terminal leaves.

This round's main results:

1. the master target is expressed as the exact rational:
   $$
   T_M=\frac{1673}{2000}
   $$
   ;
2. $d_\star$ is obtained via directed interval bisection using the monotone a-priori equation;
3. $t_3,t_5,t_7$ are reconstructed using directed interval arithmetic;
4. the D3 master root is built directly from the directed endpoints;
5. cell0's 40-layer base path is changed to store:
   - an explicit split axis;
   - an explicit side bit;

   and no longer depends on recomputing via the scheduler;
6. along this cell0 base path, the actual count is:
   $$
   \boxed{0\text{ CPG overrides}}
   $$
   ;
7. for the B7 lift tree's 18,555 nodes, each internal split axis additionally outputs an explicit 2-bit stream;
8. the reconstructed directed leaf boxes no longer use the stored float as theorem semantics;
9. margin-adaptive rational inner-polygon migration is performed on all 9,278 leaves;
10. result:
    $$
    \boxed{9278/9278\text{ PASS}}
    $$
11. selective resplit:
    $$
    \boxed{0}
    $$
12. interval membership failures:
    $$
    \boxed{0}
    $$
13. thinnest exact rational margin:
    $$
    \boxed{
    3.1948922738\times10^{-8}>0.
    }
    $$

So in Round 28, cell0 can be upgraded to:

$$
\boxed{
\texttt{CELL0-ARITHMETIC-MIGRATED-PROTOTYPE}
}
$$

But it still cannot be upgraded to a global publication theorem, because:

- the production interval backend is not yet pinned / independently audited;
- the global base atlas has not yet completed arithmetic migration in full;
- the necessity markers have not yet all undergone directed upper replay;
- the final global shard merge has not yet undergone interval A1 replay.

Therefore:

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

---

# 1. Exact master target

Round 28 no longer uses the binary64 value:

`0.8365`

as the root semantics.

Instead it uses the exact:

$$
\boxed{
T_M=\frac{1673}{2000}.
}
$$

proof target:

$$
\boxed{
T=\frac{167}{200}.
}
$$

---

# 2. Directed $d_\star$

a-priori equation:

$$
F(d)
=
\frac14
\left(
\pi-\arccos\frac1{2d}
\right)
+
\frac12\sqrt{d^2-\frac14}.
$$

defined by:

$$
F(d_\star)=T_M.
$$

Using the monotonicity of $F$ on the relevant domain,

a directed bisection is performed:

$$
\boxed{
d_\star\in[d_-,d_+].
}
$$

final interval width approximately:

$$
O(10^{-63}).
$$

So the root-reconstruction uncertainty is far smaller than the stored binary64 rounding scale.

---

# 3. Directed translation roots

For the regular odd Reuleaux shape $B_n$:

$$
R_n
=
\frac1{2\sin((n-1)\pi/(2n))}.
$$

translation root:

$$
t_n
=
-R_n\cos\frac\pi n
+
\sqrt{
R_n^2\cos^2\frac\pi n
-R_n^2
+d_\star^2
}.
$$

Round 28 obtains, using directed interval arithmetic:

$$
[t_n^-,t_n^+]
$$

for:

$$
n=3,5,7.
$$

interval widths all far smaller than:

$$
10^{-50}.
$$

---

# 4. Directed D3 root

canonical base root:

$$
0\le x_3\le t_3,
$$

$$
0\le y_3\le\frac{\sqrt3}{2}t_3,
$$

$$
0\le\phi_5\le\frac{2\pi}{5},
$$

$$
|x_5|,|y_5|\le t_5.
$$

Every irrational endpoint is now a directed interval.

---

# 5. Scheduler decisions leave theorem semantics

The old reference path was:

> re-run the scheduler starting from the root, then infer each cut's split axis.

Round 28 changes this to:

$$
\boxed{
\text{certificate stores actual split axis + side}.
}
$$

The verifier only needs to check:

$$
C=C_0\cup C_1.
$$

It does not need to trust:

- the official heuristic;
- CPG;
- a learned scheduler;
- a cost model.

Therefore:

$$
\boxed{
\text{scheduler optimality is no longer a proof dependency}.
}
$$

---

# 6. Cell0 base path

cell0 base path depth:

$$
\boxed{40}.
$$

Round 28 recovers the reference axis trace.

actual override count:

$$
\boxed{0}.
$$

So this path itself happens to follow the official axis exactly,

but the certificate still stores explicit axis semantics.

Even if the scheduler is revised in the future,

cell0's box will not go stale.

---

# 7. Stored float vs intended box

After the directed root/path reconstruction,

the maximum deviation between the stored reference center and the intended center is on the order of:

$$
10^{-17}
$$

,

the witness tree's stored centers deviate by at most about:

$$
2.1\times10^{-16}.
$$

So the binary64 reference is very close to the intended geometry,

but:

$$
\boxed{
\text{close}
\neq
\text{identical}.
}
$$

From Round 28 onward, proof membership is always decided against the intended directed box,

and the stored float is not treated as the source of truth.

---

# 8. B7 witness root

directed witness root:

$$
0\le\phi_7\le\frac{2\pi}{7},
$$

$$
|x_7|,|y_7|\le t_7.
$$

Round 24 topology:

$$
18555\text{ nodes},
$$

$$
9278\text{ leaves}.
$$

Round 28 additionally outputs, for:

$$
\boxed{
9277
}
$$

internal split axes, a 2-bit stream.

---

# 9. Explicit-axis partition theorem

If the parent coordinate interval:

$$
[a,b]
$$

selects axis $k$,

with midpoint:

$$
m=\frac{a+b}{2},
$$

children:

$$
[a,m],
\qquad
[m,b].
$$

Then:

$$
\boxed{
[a,b]
=
[a,m]\cup[m,b].
}
$$

So the split axis requires no optimality theorem whatsoever.

It only requires:

- a valid axis;
- directed/exact midpoint arithmetic;
- a correct side bit.

---

# 10. Directed common-core inputs

For each reconstructed leaf:

1. an exact/directed base box;
2. an exact/directed B7 placement box;
3. a motion upper bound:
   $$
   \delta_{\rm hi};
   $$
4. a core radius lower bound:
   $$
   \rho_{\rm lo}=1-\delta_{\rm hi};
   $$
5. regular Reuleaux vertices via interval trigonometric enclosure.

So each defining disk:

$$
D(V_j,\rho)
$$

becomes:

- a center rectangle:
  $$
  V_j\in X_j\times Y_j;
  $$
- a radius lower bound:
  $$
  \rho\ge\rho_{\rm lo}.
  $$

---

# 11. Exact point-membership test

For an exact rational point:

$$
p=(p_x,p_y),
$$

and center rectangle:

$$
[c_x^-,c_x^+]
\times
[c_y^-,c_y^+],
$$

the maximum possible squared distance is:

$$
\boxed{
D_{\max}^2
=
\max\{
(p_x-c_x^-)^2,
(p_x-c_x^+)^2
\}
+
\max\{
(p_y-c_y^-)^2,
(p_y-c_y^+)^2
\}.
}
$$

If:

$$
\boxed{
D_{\max}^2<\rho_{\rm lo}^2,
}
$$

then no matter where the true center actually falls within the rectangle:

$$
p\in D(V_j,\rho).
$$

If this holds for all defining disks, it proves:

$$
p\in G_{\rm core}.
$$

This step uses exact rational comparison throughout.

---

# 12. Candidate generation remains performance-only

Candidate points can still be generated from the nominal geometry:

- dense boundary contacts;
- a slight shrink toward the interior;
- round-trip decimal rationalization.

But the proof does not trust:

> the nominal body boundary.

The proof only trusts:

$$
\boxed{
\text{final rational point passed directed membership}.
}
$$

So the candidate generator can be completely replaced in the future.

---

# 13. Rational polygon lower theorem

If every rational vertex of:

$$
P
$$

is certified inside the common cores, then:

$$
\operatorname{conv}(P)
\subseteq
\operatorname{conv}
(
G_D\cup G_3\cup G_5\cup G_7
).
$$

Therefore:

$$
\boxed{
A(P)
\le
A(
\operatorname{conv}
(
G_D\cup G_3\cup G_5\cup G_7
)
).
}
$$

and $A(P)$ uses the exact rational shoelace formula.

If:

$$
A(P)>\frac{167}{200},
$$

the leaf is certified.

---

# 14. Margin-adaptive density

Round 27's thinnest leaf needed 16,384 directions.

But most leaves have a far larger margin.

Round 28 uses the reference slack only to decide:

> which candidate density to start from.

The reference slack plays no part in final acceptance.

density levels:

$$
256,
512,
1024,
2048,
4096,
8192,
16384.
$$

---

# 15. Bulk migration result

cell0:

$$
\boxed{
9278
}
$$

terminal leaves.

Round 28:

$$
\boxed{
9278/9278
}
$$

all:

`RATIONAL-INTERVAL-PASS`.

So:

$$
\boxed{
\text{direct migration fraction}=1.
}
$$

and:

$$
\boxed{
\text{selective resplit fraction}=0.
}
$$

---

# 16. Density distribution

## 256

$$
821
$$

leaves.

## 512

$$
3425.
$$

## 1024

$$
3709.
$$

## 2048

$$
1115.
$$

## 4096

$$
163.
$$

## 8192

$$
39.
$$

## 16384

only:

$$
\boxed{6}.
$$

So:

$$
\boxed{
\text{high-density arithmetic certificates are a tiny tail}.
}
$$

---

# 17. Membership result

For all selected rational hull vertices, directed core membership:

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

and no leaf was forced into a resplit due to membership uncertainty.

---

# 18. Minimum migrated margin

Within the bulk migration, the thinnest exact rational certificate is still:

```text
1101011010010011111001110
```

exact rational margin:

$$
\boxed{
3.1948922738\times10^{-8}.
}
$$

consistent with Round 27's targeted replay.

---

# 19. Cell0 arithmetic state

Old:

$$
\texttt{REFERENCE-COMPLETE}.
$$

After Round 28:

$$
\boxed{
\texttt{CELL0-ARITHMETIC-MIGRATED-PROTOTYPE}.
}
$$

It now has:

- a directed master-root reconstruction;
- an explicit base split trace;
- an explicit witness axis stream;
- exact topology;
- 9278 finite rational lower certificates;
- 0 arithmetic-inconclusive leaves.

---

# 20. Why still not publication-grade

Still missing:

## Backend trust closure

The directed transcendental/root construction currently relies on the:

`mpmath.iv / libmp`

prototype.

publication should pin:

- implementation;
- version;
- backend hash;
- independent arithmetic audit.

## Global proof coverage

cell0 is only one cell of the necessity atlas.

## Marker arithmetic

The upper certificates for the 77 necessity markers have not all been migrated yet.

## Global base atlas

The base-side APR/CORE/SYM/prefix proof still needs the same arithmetic bundle.

## Independent A1 replay

The final migrated shard needs an independent verifier.

So:

$$
\boxed{
\text{local migration success}
\neq
\text{global theorem}.
}
$$

---

# 21. Arithmetic migration theorem strengthened

Round 26:

> interval-pass leaves are upgraded in place, inconclusive leaves are locally resplit.

Round 28's actual measurement on cell0:

$$
\boxed{
\text{all leaves pass without geometry resplit}.
}
$$

This proves, for at least one real complete witness tree reaching depth 30:

> arithmetic migration does not necessarily cause certificate explosion.

---

# 22. Certificate size economics

The leaf proof does not need to store all of the dense candidate points.

Production can keep:

- final rational hull vertices;
- body-source tags;
- directed box dependency hash;
- exact area numerator/denominator;
- density metadata, kept only for reproducibility.

And there are only 6 leaves at the high-density 16k tier.

Therefore arithmetic bytes can use:

$$
\boxed{
\text{margin-adaptive storage}.
}
$$

---

# 23. Root-reconstruction ABI

Round 28 proposes fixing the final root certificate to:

```text
exact target
exact master target
directed d_star interval
directed t3/t5/t7 intervals
D3 root endpoint intervals
base explicit axis/side path
witness topology
witness explicit axis stream
arithmetic backend id/hash
```

scheduler name does not enter the proof-critical bundle.

---

# 24. Status ledger

## Exact rational master target

`CLOSED`

## Directed $d_\star$

`IMPLEMENTED`

## Directed $t_3,t_5,t_7$

`IMPLEMENTED`

## D3 root reconstruction

`IMPLEMENTED`

## cell0 base path explicitization

`PASS`

## witness split-axis stream

`PASS`

## cell0 bulk arithmetic migration

$$
\boxed{9278/9278\text{ PASS}}
$$

## selective resplit

$$
\boxed{0}
$$

## publication backend pinning

`OPEN`

## global arithmetic replay

`OPEN`

## global $a_{\mathrm{Leb}}\ge0.835$

`NOT CERTIFIED`

---

# 25. Round 29

## Necessity-Marker Upper Migration and Whole-Shard Arithmetic ABI

Priorities for the next round:

1. use directed support half-plane outer polygon migration on the 77 necessity markers;
2. run the thinnest-margin markers first;
3. upgrade the base cell's exact path semantics into a reusable prefix certificate the same way;
4. compress cell0's 9278 rational leaf certificates into a production shard format;
5. perform an independent A1 arithmetic replay;
6. the geometry line continues only on the not-yet-closed residual;
7. if cell0's A1 arithmetic replay succeeds, formally establish the first:
   $$
   \boxed{\text{PUBLICATION-CANDIDATE LIFT SHARD}}
   $$
   but still without claiming the global theorem.

---

# 26. Shortest Handoff

Round 28 fills the largest arithmetic structural gap left by Round 27:

$$
\boxed{
\text{stored float semantics}
\to
\text{directed master-root/path semantics}.
}
$$

Then cell0:

$$
\boxed{
9278/9278
}
$$

leaves all complete exact rational lower migration,

with no selective resplit whatsoever.

So cell0's bottleneck is no longer leaf geometry,

but rather:

$$
\boxed{
\text{global arithmetic ABI + independent replay + global coverage}.
}
$$

Therefore:

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