# AMRAL × Lebesgue Universal Covering — Round 27
## Residual Tail Prioritization and First Directed-Interval Leaf Certificates

**Document ID:** AMRAL-LUC-FC-R27  
**Version:** v0.1  
**Date:** 2026-09-20  
**Research status:** Round 27 / Residual continuation / First interval-contained rational leaf replay  
**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–R26 v0.1  

---

# 0. Round Summary

At Round 26, on the full 77-cell necessity atlas, at the common budget:

$$
d=36
$$

we obtained:

$$
\boxed{
51/77
}
$$

strict reference $B_7$ closures,

and:

$$
26
$$

residuals still contracting.

Round 27 simultaneously advances two lines:

1. only extend the residual tail, without recomputing already-closed subtrees;
2. begin true interval-contained arithmetic replay on the thinnest reference leaves.

This round's most important new result:

## Geometry progress

The canonical common-depth result still holds:

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

Because Round 27 has not yet pushed all the broad residuals synchronously to the same 38/40 budget,

the higher-budget figures report only a **known strict closure lower bound**:

$$
\boxed{
\ge57/77
\text{ have known closure depth }\le38,
}
$$

$$
\boxed{
\ge63/77
\text{ have known closure depth }\le40.
}
$$

These two numbers are not a common-depth capacity curve.

## Arithmetic progress

For the 5 thinnest leaves in the cell0 COMPLETE tree,

we establish:

$$
\boxed{
\text{interval-verified rational inner polygon}
}
$$

and re-verify using exact rational convex hull / shoelace.

Result:

$$
\boxed{
5/5\text{ PASS}.
}
$$

The thinnest reference leaf's original reference slack is approximately:

$$
4.2074\times10^{-8}.
$$

The new exact rational inner-polygon area:

$$
\boxed{
0.8350000319489227
}
$$

so the exact rational margin:

$$
\boxed{
3.1949\times10^{-8}>0.
}
$$

and moreover:

- candidate hull vertices:
  $$
  3170;
  $$
- interval membership failures:
  $$
  \boxed{0}.
  $$

This round therefore obtains the first leaf-level lower certificate prototype that does not depend on floating hull area.

But the scope must be strictly bounded:

> The interval prototype currently in use encloses the stored reference binary64 leaf state — it is not a complete exact/directed split-semantics reconstruction starting from the master root.

So:

$$
\boxed{
\text{PUBLICATION THEOREM: NOT YET}.
}
$$

---

# 1. Residual continuation semantics

Round 26 already established a resumable wave state.

Round 27's continuation obeys:

$$
\boxed{
\text{only extend unresolved boxes}.
}
$$

Already-closed subtrees are never recomputed.

Compute cost therefore begins to scale with:

$$
\text{current residual volume}
$$

rather than with historical total tree size.

---

# 2. Common-budget discipline

If not all residual cells are pushed to the same depth:

$$
d,
$$

then one cannot pass off:

> known closure depth $\le d$

as:

$$
\Gamma_{B_7}(d).
$$

So Round 27 retains Round 26's:

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

as the latest complete common-budget value.

---

# 3. Known higher-budget closures

In this round and in Round 26's partial continuation,

it is known that:

$$
\boxed{
57
}
$$

cells have strict closure depth:

$$
\le38.
$$

and at least:

$$
\boxed{
63
}
$$

cells already have strict closure depth:

$$
\le40.
$$

These are all legitimate strict edges,

but because the remaining broad residuals have not all been run to the same budget,

we can only record:

$$
\boxed{
\Gamma_{B_7}(38)\ge57,
\qquad
\Gamma_{B_7}(40)\ge63.
}
$$

Equality cannot be written.

---

# 4. Residual scheduling

The current priority order for remaining cells:

1. terminal dust;
2. thin;
3. broad contracting.

The priority metric remains:

$$
V_d
$$

and:

$$
\rho_{d,\Delta}
=
V_{d+\Delta}/V_d.
$$

not pending leaf count.

---

# 5. Fast negative prefilter

Round 27 also tests a performance-only acceleration:

1. APR runs first;
2. a cheap approximate hull is used only as a negative prefilter;
3. if the approximate value is far below target:
   - the exact hull is not called;
   - split directly;
4. the exact hull is called only when the candidate is near target;
5. any `CERT` must still be confirmed by the original strict exact/reference lower rule.

So the fast prefilter:

$$
\boxed{
\text{can create false negatives, but cannot create false positives}.
}
$$

It only affects tree cost,

it does not change theorem semantics.

On the broadest residuals there is currently still an expensive exact-hull tail,

so this round did not promote it to the production scheduler default.

---

# 6. Arithmetic objective

Round 26 already pointed out:

> a larger uniform global safety pad would break many thin-margin leaves.

Round 27's objective is therefore not to:

$$
\text{guess a safer pad}.
$$

but rather to directly convert each individual leaf into a:

$$
\boxed{
\text{finite exact rational one-sided certificate}.
}
$$

---

# 7. Rational Inner-Polygon Theorem

## Theorem 7.1

Let:

$$
G_1,\ldots,G_m
$$

be certified common cores.

If a finite point set:

$$
P_i\subset G_i
$$

is entirely verified by exact / directed arithmetic,

let:

$$
P=\bigcup_iP_i.
$$

Then:

$$
\boxed{
\operatorname{conv}(P)
\subseteq
\operatorname{conv}
\left(
\bigcup_iG_i
\right).
}
$$

Hence:

$$
\boxed{
\operatorname{Area}(\operatorname{conv}(P))
\le
\operatorname{Area}
\operatorname{conv}
\left(
\bigcup_iG_i
\right).
}
$$

Therefore the exact rational polygon area is a legitimate lower bound.

Q.E.D.

---

# 8. Why this helps arithmetic

The original reference lower area depended on:

- support envelope;
- crossing detection;
- circular-segment area;
- floating transcendental operations.

The new leaf certificate needs only:

1. point membership;
2. exact rational convex hull;
3. exact rational shoelace.

The proof surface shrinks substantially.

---

# 9. Candidate generation is not proof-critical

How the points are found does not matter.

The Round 27 emitter uses:

- nominal common-core contacts;
- each point shrunk slightly toward the interior center;
- 17-digit round-trip decimal rationalization.

But the final verifier need not trust at all that:

> these points come from the nominal boundary.

It only needs to check, point by point:

$$
p\in G_i.
$$

So candidate generation belongs to the performance layer.

---

# 10. Disk membership

disk:

$$
D=B_{1/2}(0).
$$

rational point:

$$
p=(x,y).
$$

direct exact check:

$$
\boxed{
x^2+y^2<1/4.
}
$$

no interval transcendental arithmetic is needed.

---

# 11. Eroded Reuleaux membership

common core:

$$
G
=
\bigcap_jD(V_j,\rho).
$$

for a rational point:

$$
p,
$$

production-style membership only needs:

$$
\boxed{
\|p-V_j\|^2
<
\rho^2
\qquad
\forall j.
}
$$

Round 27 prototype:

- $V_j$ uses interval enclosure;
- $\rho$ uses interval enclosure;
- squared distance uses interval arithmetic;
- only when:
  $$
  d^2_{\rm hi}<(\rho^2)_{\rm lo}
  $$
  does it PASS.

This is fail-closed.

---

# 12. Prototype interval engine

This round's environment has no MPFR/Arb Python binding.

Available:

`mpmath.iv`

whose underlying `libmp` interval endpoints use downward/upward rounding.

Round 27 only defines it as a:

$$
\boxed{
\text{reference directed-interval prototype}.
}
$$

The publication backend should still:

- pin the implementation;
- pin the version/hash;
- preferably have an MPFR/Arb-class independent audit.

---

# 13. Stored-state limitation

The current common-core parameters —

- base center;
- base halfwidth;
- witness center;
- witness halfwidth;

come from the already-stored binary64 reference tree.

Round 27 uses:

$$
[\operatorname{nextdown}(x),\operatorname{nextup}(x)]
$$

to enclose each stored float.

So the prototype rigorously handles:

$$
\boxed{
\text{a local enclosure of the stored reference state}.
}
$$

but has not yet proven that:

> the stored reference box itself is the complete intended box obtained from the exact master root / exact split semantics.

This remains the next layer of publication migration.

---

# 14. Cell0 worst leaf

reference path:

```text
1101011010010011111001110
```

depth:

$$
25.
$$

reference lower:

$$
0.835000042074202.
$$

reference slack:

$$
\boxed{
4.20742\times10^{-8}.
}
$$

---

# 15. 16k rational replay

For each common-core body, take:

$$
16384
$$

candidate directions.

candidate-point inward factor:

$$
10^{-12}.
$$

float hull candidate vertices:

$$
3170.
$$

directed interval membership:

$$
\boxed{
3170/3170\text{ PASS}.
}
$$

failures:

$$
\boxed{0}.
$$

---

# 16. Exact polygon area

The verified points are re-run through an exact rational convex hull.

vertices:

$$
3170.
$$

exact rational area converted to decimal:

$$
\boxed{
0.8350000319489227.
}
$$

target:

$$
0.835
=
167/200.
$$

so, exactly in rationals:

$$
\boxed{
A_{\rm poly}-167/200
>0.
}
$$

decimal margin:

$$
\boxed{
3.19489\times10^{-8}.
}
$$

---

# 17. Tightest five leaf audit

The cell0 COMPLETE tree's 5 thinnest leaves, ranked by reference slack.

Round 27 runs the same rational/interval replay on all of them.

Result:

$$
\boxed{
5/5\text{ PASS}.
}
$$

exact rational margins, approximately:

1. $3.19\times10^{-8}$;
2. $6.42\times10^{-8}$;
3. $1.16\times10^{-7}$;
4. $1.57\times10^{-7}$;
5. $2.03\times10^{-7}$.

interval membership failures:

$$
\boxed{0}.
$$

---

# 18. What this actually proves

Under the prototype arithmetic semantics:

> for a small interval enclosure of the stored reference binary64 state, all five leaves have a finite rational inner polygon whose exact area is strictly $>0.835$.

So they no longer depend on:

- floating hull shoelace;
- floating arc caps;
- floating support crossing area;

to obtain the lower inequality.

---

# 19. What it does not prove

This round still does not have:

1. interval reconstruction of the master root;
2. interval reconstruction of the D3 base path;
3. interval verification of all split box endpoints;
4. interval replay of all active leaves;
5. directed upper replay of all necessity markers;
6. an independent second interval backend.

So:

$$
\boxed{
\text{theorem-ready remains false}.
}
$$

---

# 20. Arithmetic migration architecture

New production leaf flow:

```text
REFERENCE LEAF
      |
      v
reconstruct exact/directed box semantics
      |
      v
generate rational inner points
      |
      v
interval membership
      |
      v
exact rational hull area
      |
      +--> area > T : INTERVAL/RATIONAL-PASS
      |
      +--> area <= T : add points / precision / selective resplit
```

---

# 21. Selective-density advantage

Tight leaves can use:

$$
16384,\ 32768,\ldots
$$

directions.

High-margin leaves may need only:

$$
512,\ 1024,\ldots
$$

So the arithmetic certificate can also be margin-adaptive.

Not all leaves need to use the same number of points.

---

# 22. Relationship to Round 14

Round 14:

> stale evidence can be semantically revalidated.

Round 27 now obtains a true arithmetic version of that:

$$
\boxed{
\text{reference floating leaf}
\rightarrow
\text{new rational/interval leaf certificate}.
}
$$

Certificate bytes can change,

the geometry subtree does not need to change.

Only leaves whose new lower certificate cannot reach target need to be resplit.

---

# 23. Joint-witness status

Round 27's new geometric continuation still does not supply joint-witness necessity.

Known strict single-$B_7$ closures continue to increase.

So:

$$
\boxed{
\text{joint queue remains deferred}.
}
$$

---

# 24. Round 28

## Exact Root Reconstruction and Bulk Rational-Interval Migration

Priorities for the next round:

1. use directed arithmetic to reconstruct:
   - the master target;
   - $t_3,t_5,t_7$;
   - the D3 wedge root;
2. convert the stored path into exact/directed child boxes;
3. compare whether the reference boxes are contained in the new interval boxes;
4. for the cell0 COMPLETE tree:
   - run margin-adaptive rational polygon replay on all 9278 leaves;
5. count:
   - direct migration fraction;
   - only-resplit fraction;
6. begin directed upper polygon replay of necessity markers;
7. the geometry line continues only on the remaining broad residual, without recomputing the already-known 63+ strict edges.

---

# 25. Shortest Handoff

Round 27 completes two key transitions.

First:

$$
\boxed{
\text{known strict B7 closures continue rising beyond the common d36 checkpoint}.
}
$$

Currently known, at least:

$$
57/77\text{ by tested depth }\le38,
$$

$$
63/77\text{ by tested depth }\le40.
$$

But inconsistent budgets are not passed off as a common curve.

Second:

$$
\boxed{
\text{first rational/interval lower leaf certificates PASS}.
}
$$

cell0's five thinnest leaves:

$$
\boxed{5/5}
$$

can all be converted into exact rational area $>0.835$.

So arithmetic migration is no longer just a specification.

It has already begun producing genuine one-sided finite certificates.

But the global theorem gate still holds:

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