# AMRAL × Lebesgue Universal Covering — Round 25
## Deep Single-Witness Closure Wave and Residual Collapse

**Document ID:** AMRAL-LUC-FC-R25  
**Version:** v0.1  
**Date:** 2026-09-20  
**Research status:** Round 25 / Deep single-witness closure wave / Residual extraction  
**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  
**Prior documents:** AMRAL-LUC-FC-R00 v0.2; R01–R24 v0.1  

---

# 0. Round Summary

Round 23, under 12 necessity-marked reference cells, 5 portfolio witnesses, and lift depth 16, obtained:

$$
\boxed{0\text{ COMPLETE edges}.}
$$

After Round 24 deepened the $B_7$ tree for the first four hardest cells, 3 had already reached COMPLETE, with the fourth still contracting rapidly.

Round 25 pushes this strategy to the complete 12-cell reference subset.

Result:

$$
\boxed{
12/12
}
$$

reference necessity cells are, in the end, all closed by a single:

$$
\boxed{B_7}
$$

at finite lift depth.

Closure depth:

$$
\boxed{
30\le D_j\le36.
}
$$

Therefore, under the current reference certificate semantics:

$$
\boxed{
\mathcal W_{\rm ref}
=
\{B_7\}
}
$$

already suffices to cover this 12-cell necessity subset.

Joint-admissible residual:

$$
\boxed{
\varnothing.
}
$$

Important limitation:

> This is still reference / pilot arithmetic, and covers only Round 23's 12-cell subset. It cannot be extrapolated to the 77-cell atlas, still less to the global Lebesgue theorem.

This round completes:

1. the 12-cell deep $B_7$ closure wave;
2. the budget-to-coverage curve;
3. Unresolved-Volume Monotonicity;
4. the closure-depth ledger;
5. the singleton-portfolio closure result for the reference subset;
6. the joint residual collapse;
7. the 77-cell adaptive B7-first long-run policy.

This round's verdict:

$$
\boxed{
\text{12-CELL REFERENCE SINGLETON }B_7\text{ COVER: COMPLETE}
}
$$

$$
\boxed{
\text{REFERENCE JOINT RESIDUAL: EMPTY}
}
$$

$$
\boxed{
\text{77-CELL / GLOBAL RESULT: NOT YET COMPUTED}
}
$$

$$
\boxed{
a_{\mathrm{Leb}}\ge0.8350:
\text{STILL COMPUTE-DEFERRED}
}
$$

---

# 1. Strict closure depth

For a necessity cell:

$$
C_j,
$$

with a fixed witness:

$$
B_7.
$$

define the reference strict closure depth:

$$
\boxed{
D_j
=
\min
\{
d:
I_{j,B_7}^{(d)}=1
\}.
}
$$

where:

$$
I_{j,B_7}^{(d)}=1
$$

means that at budget depth $d$, the entire relevant $B_7$ placement root has no PENDING leaf whatsoever.

---

# 2. Measured closure depths

This round's 12 cells:

| Cell | Closure depth | Nodes | CERT leaves |
|---:|---:|---:|---:|
| 0 | 30 | 18,555 | 9,278 |
| 1 | 32 | 22,009 | 11,005 |
| 2 | 32 | 26,189 | 13,095 |
| 3 | 34 | 36,193 | 18,097 |
| 4 | 32 | 21,271 | 10,636 |
| 5 | 30 | 18,509 | 9,255 |
| 6 | 34 | 31,209 | 15,605 |
| 7 | 36 | 45,401 | 22,701 |
| 8 | 32 | 26,439 | 13,220 |
| 9 | 34 | 41,077 | 20,539 |
| 10 | 34 | 31,793 | 15,897 |
| 11 | 30 | 21,895 | 10,948 |

All complete trees satisfy:

$$
\boxed{
N=2L-1.
}
$$

---

# 3. Closure depth statistics

Maximum:

$$
\boxed{
D_{\max}=36.
}
$$

Mean:

$$
\boxed{
\bar D=32.5.
}
$$

Median:

$$
\boxed{
32.
}
$$

---

# 4. Budget-Coverage Curve

Define:

$$
\boxed{
F(d)
=
\frac1m
\left|
\{
j:D_j\le d
\}
\right|.
}
$$

For this reference subset:

## depth 30

$$
3/12
=
25\%.
$$

## depth 32

$$
7/12
\approx58.33\%.
$$

## depth 34

$$
11/12
\approx91.67\%.
$$

## depth 36

$$
\boxed{
12/12=100\%.
}
$$

So the residual:

$$
|\mathcal R_{30}|=9,
$$

$$
|\mathcal R_{32}|=5,
$$

$$
|\mathcal R_{34}|=1,
$$

$$
\boxed{
|\mathcal R_{36}|=0.
}
$$

---

# 5. Singleton Portfolio Closure

Round 23 portfolio:

$$
\{
B_7,H_{11},H_{13},H_{17},H_{19}
\}.
$$

Round 25's reference result shows:

$$
\boxed{
I_{j,B_7}=1
\qquad
\forall j=0,\ldots,11.
}
$$

Hence, on this finite subset:

$$
\boxed{
\{B_7\}
}
$$

is, by itself, a strict witness cover.

Not needed:

- $H_{11}$;
- $H_{13}$;
- $H_{17}$;
- $H_{19}$;
- a joint witness;

to complete these 12 reference cells.

They may still have performance / other-cell value.

---

# 6. Important scope

This result is:

$$
\boxed{
\text{reference subset closure}
}
$$

It is not:

$$
\boxed{
\text{global family closure}.
}
$$

Reasons:

1. only 12 necessity cells were tested;
2. the full atlas is 77 cells;
3. the global base atlas is not yet all theorem-ready;
4. the arithmetic is still the current pilot policy;
5. a publication-grade interval / formal verifier is not yet complete.

So it cannot be claimed that:

$$
\Lambda(D,B_3,B_5,B_7)\ge0.835.
$$

---

# 7. Pending leaf count is misleading

In branch-and-bound:

> the pending-leaf count can increase even while the unresolved region is shrinking rapidly.

For example, cell 7:

depth 26:

$$
1793\text{ pending},
$$

depth 30:

$$
1918\text{ pending}.
$$

The pending count actually increases.

But the absolute unresolved placement volume:

$$
3.89\times10^{-6}
\to
2.60\times10^{-7}.
$$

actually shrinks by about:

$$
15\times.
$$

So leaf count is not a plateau metric.

---

# 8. Unresolved Placement Volume

For budget depth:

$$
d,
$$

let the active unresolved boxes be:

$$
\mathcal P_d.
$$

Define:

$$
\boxed{
V_d
=
\sum_{C\in\mathcal P_d}
\operatorname{Vol}(C).
}
$$

This is the true size of the residual placement domain.

---

# 9. Unresolved-Volume Monotonicity Theorem

## Theorem 9.1

Suppose that on every split:

$$
C=C_0\cup C_1,
$$

the children's interiors are disjoint, and:

$$
\operatorname{Vol}(C)
=
\operatorname{Vol}(C_0)
+
\operatorname{Vol}(C_1).
$$

Certified children are removed from the unresolved set.

Then:

$$
\boxed{
V_{d+1}\le V_d.
}
$$

### Proof

Every unresolved parent falls into one of two cases:

1. both children remain unresolved:
   the volume total equals the parent's;
2. at least one child is certified:
   the unresolved children's total volume is strictly less than, or equal to, the parent's.

Summing over all unresolved parents.

Q.E.D.

---

# 10. Plateau metric

Performance contraction should therefore be read from:

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

and not from:

$$
\frac{P_{d+\Delta}}{P_d}.
$$

where $P_d$ is the pending leaf count.

---

# 11. Cell 7 example

cell 7:

$$
V_{22}
\approx
3.33\times10^{-5},
$$

$$
V_{24}
\approx
1.05\times10^{-5},
$$

$$
V_{26}
\approx
3.89\times10^{-6},
$$

$$
V_{28}
\approx
9.90\times10^{-7},
$$

$$
V_{30}
\approx
2.60\times10^{-7},
$$

$$
V_{32}
\approx
4.22\times10^{-8},
$$

$$
V_{34}
\approx
1.61\times10^{-10},
$$

$$
V_{36}=0.
$$

This is a typical:

$$
\boxed{
\text{deep tail collapse}.
}
$$

---

# 12. Round 24 joint admission revisited

Round 24 states:

- `JOINT-REDUNDANT`
- `JOINT-DEFERRED-CONTRACTING`
- `JOINT-CANDIDATE`
- `JOINT-NECESSARY-RELATIVE-TO-PORTFOLIO`

Round 25, on the 12-cell reference subset:

$$
\boxed{
\text{all cells end at JOINT-REDUNDANT}.
}
$$

because every cell ends up with a:

$$
B_7
$$

strict edge.

---

# 13. Why Round 23 depth-16 result was still useful

Round 23:

$$
0/60
$$

COMPLETE edges at depth 16.

This was not wrong.

It correctly described:

> the shallow proof budget was insufficient.

Round 25 shows:

> shallow infeasibility is not structural infeasibility.

So the budget must enter the incidence notation:

$$
I_{jK}^{(d)}.
$$

---

# 14. Closure Capacity Curve

For a fixed witness:

$$
K,
$$

and a finite cell set:

$$
\mathcal C,
$$

define:

$$
\boxed{
\Gamma_K(d)
=
\left|
\{
C_j:
I_{jK}^{(d)}=1
\}
\right|.
}
$$

For this reference $B_7$:

$$
\Gamma_{B_7}(30)=3,
$$

$$
\Gamma_{B_7}(32)=7,
$$

$$
\Gamma_{B_7}(34)=11,
$$

$$
\boxed{
\Gamma_{B_7}(36)=12.
}
$$

This is the witness's budget-dependent proof capacity.

---

# 15. Reference Maximum Closure Depth Theorem

For a finite reference cell set, if every:

$$
D_j<\infty,
$$

then:

$$
D_\star=\max_jD_j
$$

is a common finite budget such that:

$$
I_{jK}^{(D_\star)}=1
$$

for all cells.

This time:

$$
\boxed{
D_\star=36.
}
$$

This is only a finite reference statement.

---

# 16. Production scheduler update

Next-version priority for the full necessity atlas:

## Stage A

All cells first run the:

$$
B_7
$$

deep closure wave.

Checkpoints:

$$
30,\ 32,\ 34,\ 36.
$$

## Stage B

Once COMPLETE:

remove immediately from the residual / joint queue.

## Stage C

If still unresolved at depth 36:

look at the:

$$
V_d
$$

contraction.

If:

$$
\rho\ll1,
$$

continue B7 under the cap.

If it is a plateau:

only then move into portfolio / joint admission.

---

# 17. Why not start H19 first

At Round 23's shallow depth 16:

$$
H_{19}
$$

won on unresolved volume almost across the board.

But Round 24–25 show that:

$$
B_7
$$

's deep contraction is enough to truly COMPLETE all tested cells.

So if the current long-run objective is:

> obtaining a strict edge,

rather than:

> minimizing the residual over a short horizon,

then:

$$
\boxed{
B_7\text{-first}
}
$$

is more reasonable.

---

# 18. Portfolio is still useful

This does not mean:

$$
H_{11},H_{13},H_{17},H_{19}.
$$

can be dropped.

Because the full 77-cell atlas may still turn up:

- cells where B7 is slower;
- B7 counterexamples;
- plateaus;
- different active-contact regimes.

Round 25 only changes the priority:

$$
\boxed{
B_7\text{ first, portfolio on true residual}.
}
$$

---

# 19. Compute ledger

Total nodes across the 12 independent reference trees:

$$
\boxed{
340540.
}
$$

Average:

$$
\approx28378.
$$

Maximum:

$$
\boxed{
45401
}
$$

occurring at cell 7.

This statistic describes only these 12 independent lift trees.

It is not the global proof node count.

---

# 20. Validation status

All 12 cells:

- marker outer $<0.835$;
- final pending leaves = 0;
- forest identity:
  $$
  N=2L-1.
  $$

cell 0 additionally carries Round 24's actual topology stream + a deterministic replay PASS.

For the rest, this round preserves the complete adaptive run ledger and a reproducible runner.

---

# 21. What is still missing

Even with 12/12 complete,

still missing are:

1. the full 77-cell closure wave;
2. full base atlas coverage;
3. production arithmetic;
4. independent complete-edge replay for every final shard;
5. root / shard merge;
6. a global theorem-ready checkpoint.

So:

$$
\boxed{
a_{\mathrm{Leb}}\ge0.835
}
$$

has still not been obtained.

---

# 22. Round 26

## Full Necessity-Atlas $B_7$ Closure Wave and Production Arithmetic Upgrade

The next round's priority no longer opens a joint witness.

Instead, it will:

1. hand the B7-first adaptive runner over to the local runtime;
2. run the complete 77 necessity cells;
3. obtain the full:
   $$
   \Gamma_{B_7}(d)
   $$
   curve;
4. extract the true residual;
5. simultaneously begin replacing the current pilot $10^{-10}$ / floating-point arithmetic with a publication-grade interval / directed-rounding policy;
6. issue new A1 independent certificates for the closure cells.

Only if the full 77-cell atlas truly shows a plateau / disqualification residual

will joint-witness research be restarted.

---

# 23. Shortest Handoff

Round 25's most important result:

$$
\boxed{
\text{Round23's shallow portfolio infeasibility}
\neq
\text{single-witness infeasibility}.
}
$$

On the 12-cell reference subset:

$$
\boxed{
B_7\text{ alone closes }12/12.
}
$$

Maximum reference depth:

$$
\boxed{36}.
$$

So the current production order should be:

$$
\boxed{
\text{deep }B_7
\to
\text{true residual}
\to
\text{portfolio}
\to
\text{joint only if admitted}.
}
$$