# AMRAL × Lebesgue Universal Covering — Round 16
## Capacity-Aware Global Master-Root Pilot and Bottleneck Profiler

**Document ID:** AMRAL-LUC-FC-R16  
**Version:** v0.1  
**Date:** 2026-09-19  
**Research status:** Round 16 / Global-domain finite-depth pilot  
**Research mode:** Human-Directed + Semi-Autonomous AI Mathematical Research  
**Research initiation and methodology source:** Neo.K  
**AI collaborating researcher and primary executor:** Aletheia / ChatGPT, GPT-5.6 Sol  
**Predecessor documents:** AMRAL-LUC-FC-R00 v0.2; R01–R15 v0.1

---

# 0. The most important change this round

Round 10 was a local reference dry run.

Round 16 is the first time the pilot root is switched to the full five dimensions:

$$
\boxed{
D+B_3+B_5
}
$$

phase master target:

$$
\boxed{
T_M=0.8365.
}
$$

The next theorem target actually being profiled is still:

$$
\boxed{
T=0.8350.
}
$$

In other words:

> This time we genuinely start from the full global placement root, but only run to finite depth — we do not claim a complete certificate.

The purpose of this round is not to obtain a new lower bound, but to answer:

1. What does the prune structure look like on the full global root?
2. Which split scheduler actually saves the whole tree?
3. How large is the extra domain cost of the master root?
4. Of the raw `HARD` frontier, how much is a genuine near-minimizer, and how much is simply from bounds that are too loose?
5. When should B7 be activated?

---

# 1. Master root

Round 15's phase master target:

$$
T_M=0.8365.
$$

From the a-priori domain formula:

$$
\boxed{
t_3=0.20102352900741122,
}
$$

$$
\boxed{
t_5=0.20261828623043526,
}
$$

$$
\boxed{
t_7=0.20147344578628750.
}
$$

base root:

$$
x_3,y_3\in[-t_3,t_3],
$$

$$
\phi_5\in[0,2\pi/5),
$$

$$
x_5,y_5\in[-t_5,t_5].
$$

This is the complete five-dimensional enclosing root; square corners can still be excluded by the analytic a-priori rules.

---

# 2. Pilot proof modes

This round permits only one-sided lower routes.

## `APR`

Uses the analytic lower bound from a disk plus a body-contained center:

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

## `REP`

The cell-center inner hull:

$$
A^-_{\rm center}
$$

minus the Round 09 transfer error:

$$
\boxed{
4T\tau+\pi\tau^2.
}
$$

## `CORE`

Build Mishra-style common erosion cores for the entire placement cell:

$$
C_3,
\qquad
C_5,
$$

then use the convex-hull area of the finite boundary inner points as the lower witness.

## `HARD`

Cells that still cannot be closed by the three routes above before the pilot depth is reached.

Important:

this round's boundary sampling is one-sided geometry, but the floating policy is still just:

$$
N_{\rm sample}=40,
\qquad
e_{\rm pilot}=10^{-9}.
$$

A publication-grade floating theorem has not yet been established.

So:

$$
\boxed{
\text{Round 16's metrics are profiling / architecture evidence, not a }0.835\text{ final certificate}.
}
$$

---

# 3. Split scheduler A/B test

Comparing two schedulers.

## A — Mishra / official first-order weight

$$
\max
\left
\{
h_{x3},
h_{y3},
R_5h_{\phi5},
h_{x5},
h_{y5}
\right\
\}.
$$

## B — Round 09 exact-$\tau$ one-step scheduler

After each bisection, choose the axis with the smallest:

$$
\tau_{\rm child}
$$

Round 09 has already proved that B is one-step optimal for the certified Hausdorff radius.

What Round 16 tests is:

> is one-step $\tau$ optimality the same as global tree-cost optimality?

The answer:

$$
\boxed{
\text{No.}
}
$$

---

# 4. Scheduler pilot table

| Pilot depth | Scheduler | Nodes | CORE-pruned | HARD leaves |
|---:|---|---:|---:|---:|
| 10 | official | 2,047 | 370 | 654 |
| 10 | exact-$\tau$ | 2,047 | 0 | 1,024 |
| 12 | official | 4,919 | 1,126 | 1,334 |
| 12 | exact-$\tau$ | 6,519 | 1,936 | 1,324 |
| 14 | official | 11,639 | 3,218 | 2,602 |
| 14 | exact-$\tau$ | 13,319 | 3,146 | 3,514 |
| 16 | official | 23,455 | 7,616 | 4,112 |
| 16 | exact-$\tau$ | 28,771 | 9,134 | 5,252 |

At depth 16:

$$
\boxed{
\text{official has about }18.48\%\text{ fewer nodes}
}
$$

and:

$$
\boxed{
\text{official has about }21.71\%\text{ fewer HARD leaves}.
}
$$

---

# 5. This is not an error in Round 09

Round 09's theorem is:

> within a single coordinate bisection, the exact-$\tau$ split is one-step optimal for the certified Hausdorff-radius bound.

It never proved:

> this axis will minimize the whole branch-and-bound tree.

Round 16 shows:

$$
\boxed{
\text{uncertainty reduction objective}
\neq
\text{certificate prune objective}.
}
$$

So the status is:

`R09-STRENGTHENING-EMPIRICAL-001`

not a correction.

Production keeps the official split as the baseline for now.

Future work should study:

$$
\boxed{
\text{Expected Certified Prune Gain scheduler}.
}
$$

---

# 6. Master root vs immediate root

Same depth 16, official scheduler:

## Immediate root $T_M=0.835$

$$
N=24691,
$$

$$
CORE=8004,
$$

$$
HARD=4342.
$$

## Phase master root $T_M=0.8365$

$$
N=23455,
$$

$$
CORE=7616,
$$

$$
HARD=4112.
$$

The master root has a larger geometric volume, yet this finite-depth pilot actually has slightly fewer nodes.

This cannot be used to conclude that the master root is generally faster.

The real conclusion is:

$$
\boxed{
\text{domain volume overhead cannot be converted directly and linearly into certificate cost}.
}
$$

Split order, core geometry, and prune timing jointly determine tree cost.

So Round 15's roughly $11.66\%$ base translation-volume overhead is still a correct geometric description, but it is not a runtime multiplier.

---

# 7. Detailed global pilot: depth 18

The complete $T_M=0.8365$ root, official split:

$$
\boxed{
N=45651.
}
$$

Breakdown:

$$
APR=0,
$$

$$
REP=0,
$$

$$
CORE=12378,
$$

$$
\boxed{
HARD=10448.
}
$$

At the current resolution, what is actually doing the work is `CORE`.

`APR/REP=0` does not mean they are never useful — it only means that before depth 18:

- the APR contained-center bound is still not close enough to $0.835$;
- the cell's Hausdorff radius is still too large, so the REP transfer penalty is too high.

---

# 8. Raw HARD is not the same as the active frontier

The single most important profiling result:

the cell-center inner hull of the 10448 HARD cells:

$$
\boxed{
\min A^-_{\rm center}
=
0.8356828349502634
>
0.835.
}
$$

and:

$$
\boxed{
\#\{A^-_{\rm center}<0.835\}=0.
}
$$

So at depth 18, `HARD` mainly represents:

> the common-core / box uncertainty is still too loose — it is not yet possible to close the certificate.

It does not represent:

> we have already observed ten thousand actual configurations below the target.

These are two very different meanings.

---

# 9. Near-frontier compression

Using the distance of the cell-center inner area from the target as a profiling proxy.

Of the 10448 HARD cells:

| Center-area range | Cell count |
|---|---:|
| $\le T+0.00075$ | 2 |
| $\le T+0.001$ | 2 |
| $\le T+0.002$ | 8 |
| $\le T+0.005$ | 70 |
| $\le T+0.01$ | 196 |
| $\le T+0.02$ | 714 |

So:

$$
\boxed{
10448\text{ raw HARD}
\to
8\text{ very-near-frontier center cells at }+0.002.
}
$$

Note this is still a proxy; a cell may contain configurations lower than its center.

But it is already enough to drive scheduler priority.

---

# 10. HARD center-area distribution

Quantiles:

$$
Q_{0}=0.835682835,
$$

$$
Q_{0.01}=0.841831848,
$$

$$
Q_{0.10}=0.858184427,
$$

$$
Q_{0.50}=0.884118492,
$$

$$
Q_{0.99}=0.914281848.
$$

So the vast majority of HARD seed centers are actually far above:

$$
0.835.
$$

They are a certification-resolution problem, not an obvious-minimizer problem.

---

# 11. Current common-core looseness

HARD lower-bound quantiles:

$$
Q_{0.50}
\approx
0.822136323,
$$

$$
Q_{0.90}
\approx
0.832177159,
$$

$$
Q_{0.99}
\approx
0.834731887,
$$

max:

$$
0.834989575.
$$

So there is already a batch of seeds whose core bound is only this far from the target:

$$
\sim10^{-5}.
$$

while others still fall this short, due to erosion loss:

$$
\sim10^{-2}.
$$

This shows that the hard frontier itself is highly heterogeneous.

---

# 12. Margin-to-resolution profiler

Following Round 09:

$$
\tau_\star(m,T)
=
\frac{
\sqrt{4T^2+\pi m}-2T
}{\pi},
$$

where:

$$
m=A^-_{\rm center}-T.
$$

Converting the current:

$$
\tau=0.03855318146790454
$$

into a conservative proxy for "full parameter refinement cycles."

Result quantiles:

$$
Q_{0.10}=1,
$$

$$
Q_{0.50}=2,
$$

$$
Q_{0.90}=3,
$$

$$
Q_{0.99}=5,
$$

$$
\max=8.
$$

---

# 13. Tail counts

At least this many cycles are needed:

$$
\ge3\text{ cycles}:2186,
$$

$$
\ge4:432,
$$

$$
\ge5:142,
$$

$$
\ge6:48,
$$

$$
\ge7:8,
$$

$$
\boxed{
\ge8:2.
}
$$

In other words:

$$
\boxed{
\text{about }90\%\text{ HARD cells do not exceed 3 cycles under this proxy}.
}
$$

The real scheduler tail is quite small.

---

# 14. The two base cells at the very tail

The two cells with the lowest center inner area:

$$
A^-_{\rm center}
\approx
0.83568283495.
$$

Their current core lower bound is only:

$$
\approx0.79907297849.
$$

That is, an erosion / box loss of about:

$$
\boxed{
3.66\times10^{-2}.
}
$$

This is far larger than the milliscale improvement we currently want to obtain from B7.

So activating B7 immediately at this resolution is not justified.

---

# 15. B7 shallow-lift probe

For the two base cells above that are closest to the target, launch a complete phase-master B7 root for each:

$$
\phi_7\in[0,2\pi/7),
$$

$$
x_7,y_7\in[-t_7(0.8365),t_7(0.8365)].
$$

The B7 local tree runs to depth 12.

Both results are exactly identical:

$$
\boxed{
2403\text{ nodes}
}
$$

of which:

$$
568\text{ closed leaves},
$$

but still:

$$
\boxed{
634\text{ unresolved leaves}.
}
$$

Therefore:

$$
\boxed{
\text{B7-ACTIVATE-NOW: NO (at this base resolution)}.
}
$$

This does not mean B7 is ineffective.

Rather:

> the B7 tree is currently paying the cost of over-eroded $B_3/B_5$ base cores.

It makes more sense to refine the base first.

---

# 16. Base-First / Lift-Later policy

Round 16 recommends the following production policy:

## Phase A — Base refinement

Continue:

$$
D+B_3+B_5
$$

atlas refinement.

Priority tail:

- $\ge5$-cycle: 142 seeds;
- $\ge7$-cycle: 8 seeds;
- $\ge8$-cycle: 2 seeds.

## Phase B — Near-frontier re-profile

When:

- common-core loss drops noticeably;
- REP starts to be able to prune;
- the cell-center / core gap reaches milliscale;

then activate B7.

## Phase C — Witness lifting

Only open a:

$$
3\text{-D }B_7
$$

or a fallback witness batch, for truly persistent frontier.

This is a compute policy, not a theorem assumption.

---

# 17. Scheduler production decision

For now:

$$
\boxed{
\text{official first-order split = production baseline}.
}
$$

exact-$\tau$ is still retained, as:

- an uncertainty metric;
- a margin-to-resolution forecast;
- a tie-break feature.

But it no longer solely decides the split axis.

---

# 18. New scheduler research target

Round 16 exposes the true scheduler objective:

$$
\boxed{
\text{Expected Certified Prune Gain}.
}
$$

The ideal score should estimate:

$$
\frac{
\Pr(\text{child closes after split})
\times
\text{expected subtree saved}
}{
\text{split / bound evaluation cost}
}.
$$

It can use:

- $\tau$ reduction;
- current CORE gap;
- center margin;
- historical sibling prune rate;
- proof mode;
- path symmetry class.

This is a future scheduler; it does not enter the current verifier.

---

# 19. Symmetry signal

The tail-end seeds show a clear pattern of sign / orientation pairing.

For example, the two lowest center configurations have mirrored translation signs, with center areas that are almost identical.

Status:

`NUMERICAL-SYMMETRY-SIGNAL`

not a formal quotient theorem.

Worth establishing in the future:

$$
\boxed{
\text{global atlas symmetry canonicalization}.
}
$$

If a safe quotient can be found, it could directly cut the distributed workload further.

---

# 20. Master-root decision

Although the $T_M=0.8365$ root is slightly larger than the immediate $0.835$ root,

this round's finite-depth pilot observed no node penalty.

So we retain:

$$
\boxed{
T_M=0.8365
}
$$

as the B7-phase master domain.

Reasons:

1. the root overhead is geometrically moderate;
2. it can support Round 15's threshold ladder;
3. it avoids rebuilding seed identities every time the target is nudged up slightly;
4. the pilot showed no clear tree-cost degradation.

But:

`MASTER-ROOT-PERFORMANCE-NOT-A-THEOREM`.

---

# 21. Production shard tiers

Using the depth-18 pilot as the initial difficulty prior.

## Tail-A

$$
\ge5\text{ predicted cycles}
$$

total:

$$
\boxed{142}.
$$

Priority for long-running tasks / duplicate audit.

## Tail-B

3–4 cycles:

$$
2186-142
=
\boxed{2044}.
$$

## Bulk

1–2 cycles:

$$
10448-2186
=
\boxed{8262}.
$$

This can be fed directly into Round 13's tail-first scheduler.

---

# 22. Global long-run architecture after Round 16

The production flow now recommended:

```text
MASTER ROOT T_M=0.8365
        |
        v
base F3 refinement
        |
        +-- easy -> CORE/REP closure
        |
        +-- tail -> deeper base refinement
                      |
                      v
               near-frontier profiler
                      |
                      +-- base can close -> continue
                      |
                      +-- persistent -> activate B7
                                          |
                                          v
                                  conditional lift / batch
```

rather than:

```text
all HARD -> B7 immediately
```

---

# 23. New actual artifact

The Round 16 package preserves:

- the complete pilot methodology;
- the actual depth-18 raw JSON;
- the actual multi-depth scheduler metrics;
- the near-frontier top cells;
- the B7 shallow probes;
- reproduction code;
- the production long-run spec.

---

# 24. Status ledger

## Full global five-dimensional root used

`YES`

## Finite-depth global pilot

`VERIFIED-COMPUTATIONAL / PILOT`

## Official vs exact-$\tau$ scheduler comparison

`VERIFIED-COMPUTATIONAL / PILOT`

## Near-frontier compression signal

`NUMERICAL-EVIDENCE`

## B7 base-first activation recommendation

`EMPIRICAL-COMPUTE-POLICY`

## New rigorous lower bound

`NONE`

## $a_{\rm Leb}\ge0.8350$

`OPEN / COMPUTE-DEFERRED`

---

# 25. Assigned topic for Round 17

## AMRAL-LUC-FC-R17
### Symmetry Canonicalization and Expected-Prune-Gain Scheduler

There are two lines of work in the next round that are now very much worth doing:

### A. Formal symmetry quotient

Elevate Round 16's near-frontier paired seeds into a legitimate canonicalization:

- reflections;
- Reuleaux body rotational symmetries;
- the residual group action of the global normalization.

Goal:

$$
\boxed{
\text{proof-preserving duplicate seed elimination}.
}
$$

### B. Prune-gain scheduler

Using Round 16 pilot data, upgrade from:

$$
\text{one-step }\tau\text{ optimization}
$$

to:

$$
\boxed{
\text{predicted certificate-cost minimization}.
}
$$

If the local global long run has already started before this, Round 17 will directly ingest the real shard statistics to update the cost model.

---

# 26. Shortest handoff conclusion

Round 16 is the first time we have truly seen the shape of the complete global root.

The most important finding is not:

> there are still 10448 hard seeds.

but:

$$
\boxed{
\text{raw HARD frontier} \gg \text{true near-minimizer frontier}.
}
$$

At depth 18:

- 10448 HARD;
- only 8 centers within target + 0.002;
- 90% are estimated to require no more than 3 full refinement cycles;
- B7 still cannot close quickly on the two coarse base cells at the very tail, because of base erosion loss.

So the production strategy should be:

$$
\boxed{
\text{Base Refine}
\to
\text{Frontier Concentration}
\to
\text{B7 Lift}.
}
$$

rather than adding witness dimensions the moment HARD is seen.