# AMRAL × Lebesgue Universal Covering — Round 19
## Lazy Proof Cascade, Necessity-Marker Segmentation, and Witness Activation Frontier

**Document ID:** AMRAL-LUC-FC-R19  
**Version:** v0.1  
**Date:** 2026-09-19  
**Research status:** Round 19 / Continuation-cost calibration + base-to-lift transition  
**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  
**Prerequisite documents:** AMRAL-LUC-FC-R00 v0.2; R01–R18 v0.1  

---

# 0. Round summary and verdict

Round 18 already had a production-style checkpoint:

$$
D_3\text{ wedge}
+
\text{Tail-Window CPG}
+
\text{Rice override encoding}
$$

leaving, at depth 16:

$$
2048
$$

`PENDING` base leaves.

Round 19's two questions:

1. Is it worth continuing to compute `REP` eagerly at every hot node?
2. To what degree should base refinement actually go before it is right to trigger the $B_7$ lift?

This round obtained:

$$
\boxed{
\text{Lazy Proof Cascade}
}
$$

$$
\boxed{
\text{Necessity-Marker Theorem}
}
$$

$$
\boxed{
\text{Dual Base/Lift Frontier}
}
$$

and, for the first time, a measurement of:

$$
\boxed{
\kappa_{\rm continuation}
}
$$

together with the base-to-lift transition curve.

Main conclusions:

- under a fixed split tree, the CERT/HARD classification of `APR -> CORE -> REP` is identical to that of eager `APR+REP+CORE`;
- in the depth 16→18 A/B test, REP calls decreased by about $16.9\%$, and wall time decreased by about $5.8\%$;
- Round 18's Tail-Window continuation savings are confirmed to have paid for themselves once pushed to depth 20;
- of the 12 candidates at depth 20 with sampled-inner `center<T`, recomputing with the official exact kernel shows **0** are truly exact-sublevel;
- therefore the sampled center values from Round 16/18 can only serve as a heuristic, not as a witness-necessity proof;
- along the true $D_3$ canonical branch that actually contains the Mishra official near-minimum configuration, a center upper bracket $<0.835$ is obtained for the first time at around depth 31;
- once a cell contains a certified point whose base-hull area is $<T$, no descendant that still contains that point can ever be base-only closed;
- but the witness must not act on the entire ancestor immediately: after refining the depth-31 cell to depth 40, only 77 of the 256 pending descendants have a center outer-bracket already proved $<0.835$;
- so **necessity segmentation** should be done first, and only then should $B_7$ be lazily activated;
- the unresolved placement-volume fraction of the $B_7$ depth-12 lift shrinks as base uncertainty shrinks, gradually approaching the zero-base-uncertainty floor;
- the pilot shows that once $\tau$ enters roughly the $10^{-3}$ order of magnitude, the return from continued base refinement toward the B7 lift begins to diminish noticeably.

This round's overall verdict:

$$
\boxed{
\text{LAZY PROOF CASCADE: CLOSED}
}
$$

$$
\boxed{
\text{NECESSITY-MARKER SEMANTICS: CLOSED}
}
$$

$$
\boxed{
\text{BASE/LIFT SWITCH POLICY: PILOT-CALIBRATED}
}
$$

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

---

# 1. Lazy Proof Cascade

The base cell currently has three one-sided lower bounds:

$$
L_{\rm APR}(C),
\qquad
L_{\rm REP}(C),
\qquad
L_{\rm CORE}(C).
$$

all of which satisfy:

$$
L_i(C)
\le
\inf_{q\in C}A_{\mathcal F_3}(q).
$$

So any single one with:

$$
L_i(C)\ge T
$$

is already sufficient to close the cell.

The Round 18 emitter computed almost all three, essentially every time.

Round 19 changes this to:

$$
\boxed{
\text{APR}
\to
\text{CORE}
\to
\text{REP}
}
$$

where REP is computed only when neither of the first two has closed the cell.

---

# 2. Lazy Classification Equivalence Theorem

## Theorem 2.1

Fix a cell:

$$
C.
$$

The eager classifier accepts iff:

$$
\max
\{
L_{\rm APR},
L_{\rm REP},
L_{\rm CORE}
\}
\ge T.
$$

The lazy classifier:

1. if $L_{\rm APR}\ge T$, accept;
2. otherwise compute $L_{\rm CORE}$;
3. if $L_{\rm CORE}\ge T$, accept;
4. otherwise, only then compute $L_{\rm REP}$.

Then the Boolean `CERT/HARD` classification of Lazy and Eager is exactly the same.

### Proof

If APR or CORE is already $\ge T$, Eager must also accept.

If both APR and CORE are $<T$, Lazy will go on to compute REP, so in that case both are decided by whether REP is $\ge T$.

Q.E.D.

---

# 3. Fixed-tree A/B

Starting from Round 18's Tail checkpoint of:

$$
2048
$$

depth-16 pending roots.

To isolate the cost of proof mode, both sides were switched to the same deterministic official split, advanced to depth 18.

## Eager

$$
13206\text{ nodes},
$$

$$
4944\text{ pending},
$$

REP calls:

$$
12659,
$$

time:

$$
7.2903\text{ s}.
$$

## Lazy

Likewise:

$$
13206\text{ nodes},
$$

$$
4944\text{ pending}.
$$

REP calls:

$$
10523.
$$

time:

$$
6.8687\text{ s}.
$$

So:

$$
\boxed{
16.87\%\text{ REP calls eliminated}
}
$$

and about:

$$
\boxed{
5.78\%\text{ wall-time reduction}.
}
$$

---

# 4. Why Lazy does not simply delete REP

By this point in Round 18/19, the accepted leaves are almost all closed by CORE.

But this does not prove that:

$$
L_{\rm REP}
\le
L_{\rm CORE}
$$

holds forever.

So Round 19 does not delete REP.

It only moves it to:

$$
\boxed{
\text{be computed only after CORE fails}
}
$$

This is safe lazy evaluation, not theorem deletion.

---

# 5. Continuation cost $\kappa$

Production-like Tail-Window continuation:

## depth 16→18

input:

$$
2048
$$

pending roots.

added:

$$
12634\text{ nodes}.
$$

remaining:

$$
4599\text{ pending}.
$$

CPU time:

$$
22.7531\text{ s}.
$$

about:

$$
\boxed{
11.11\text{ ms}
}
$$

per original depth-16 pending root.

---

# 6. depth 18→20

From:

$$
4599
$$

pending roots, push two more levels.

added:

$$
26455\text{ nodes}.
$$

remaining:

$$
10114\text{ pending}.
$$

total worker CPU-s:

$$
55.3844\text{ s}.
$$

So the depth 16→20 cumulative continuation is:

$$
\boxed{
78.1375\text{ CPU-s}.
}
$$

that is:

$$
\boxed{
38.15\text{ ms / original depth-16 pending root}.
}
$$

---

# 7. Round 18 cost model now closes empirically

Round 18 Tail-Window's extra emitter cost relative to Official,

converted into a break-even cost per saved pending leaf:

$$
14.41\text{ ms}.
$$

Round 19's actual measurement, pushed to depth 20:

$$
38.15\text{ ms / original pending root}.
$$

is above the threshold by:

$$
\boxed{
2.65\times.
}
$$

So, for a workload where "typically at least four more levels still need to be pushed":

$$
\boxed{
\text{Tail-Window has already actually paid for itself}.
}
$$

---

# 8. Correction to the sampled center-sublevel reading

Of the:

$$
10114
$$

pending cells at depth 20,

using the 40-direction sampled inner center area:

$$
12
$$

have:

$$
A_{\rm sample}(q_c)<0.835.
$$

Minimum sampled inner:

$$
0.83395139.
$$

Round 19 recomputed these 12 centers using the Mishra official exact support kernel.

Result:

$$
\boxed{
0/12
}
$$

are truly exact center areas below:

$$
0.835.
$$

The minimum exact value is still about:

$$
\boxed{
0.83586503.
}
$$

---

# 9. Correction ledger

Therefore, if the earlier stage read:

> sampled inner center < target

as meaning:

> the center configuration itself is already sublevel

that reading is not valid.

Round 19 logs:

`CORRECTION-R19-001`

Correct semantics:

$$
\boxed{
A_{\rm sample}<T
\Rightarrow
\text{only LIFT-CANDIDATE / SEARCH SIGNAL}.
}
$$

Only an exact / rigorous outer upper bound:

$$
A(q)<T
$$

can serve as a necessity-proof trigger.

---

# 10. Official near-minimum branch canonicalization

The Mishra exhibited base placement, mapped by Round 17's residual $D_3$ action, onto the canonical wedge.

canonical configuration:

$$
q^\dagger
\approx
(
0.0060794482,
0.0105298389,
0.4188548915,
-0.0114949286,
-0.0199175943
).
$$

official exact-kernel area:

$$
\boxed{
A(q^\dagger)
=
0.8347809459115.
}
$$

Same as the original configuration.

---

# 11. Track the actual valley, not the center ranking

Round 19 no longer picks, at every level:

> the current lowest cell center

because a cell center is not a minimizer tracker.

Instead it switches to:

> tracking, along the production split tree, the one unique branch that actually contains $q^\dagger$.

The exact center area of this branch gradually approaches the official valley.

---

# 12. First center-sublevel trigger

On the tracked branch:

## depth 30

exact center:

$$
0.8352534195.
$$

20k support bracket outer:

$$
0.8352704487>T.
$$

## depth 31

exact center:

$$
\boxed{
0.8349033738.
}
$$

20k support bracket outer:

$$
\boxed{
0.8349184947<T.
}
$$

So, under the current pilot arithmetic,

depth 31 has already produced:

$$
\boxed{
\texttt{CENTER-SUBLEVEL-CERTIFIED-PILOT}.
}
$$

publication arithmetic is not yet complete, so this is not promoted to a final theorem certificate.

---

# 13. Necessity-Marker Theorem

## Theorem 13.1

Let base cell:

$$
C
$$

contain:

$$
q_\star.
$$

If it has been proved that:

$$
A_{\mathcal F_3}(q_\star)<T,
$$

then any descendant still containing:

$$
q_\star
$$

— call it:

$$
C'\subseteq C
$$

— cannot have a valid base-only lower certificate:

$$
L(C')\ge T.
$$

### Proof

Since:

$$
q_\star\in C',
$$

we have:

$$
\inf_{q\in C'}
A_{\mathcal F_3}(q)
\le
A_{\mathcal F_3}(q_\star)
<T.
$$

and any sound base-only lower bound must satisfy:

$$
L(C')
\le
\inf_{q\in C'}A_{\mathcal F_3}(q).
$$

Hence it is impossible that:

$$
L(C')\ge T.
$$

Q.E.D.

---

# 14. Necessity marker

Store:

$$
(q_\star,\ U(q_\star)<T)
$$

as a:

$$
\boxed{
\text{necessity marker}.
}
$$

The marker contains at least:

```text
configuration q
verified upper area
geometry-kernel hash
arithmetic-policy hash
target
```

---

# 15. Marker propagation

If a marked cell:

$$
C
$$

splits:

$$
C\to C_0\cup C_1,
$$

then:

- the child that contains $q_\star$ inherits the marker;
- the sibling that does not contain $q_\star$ is not automatically marked purely by ancestry.

So:

$$
\boxed{
\text{witness necessity can be localized along the branch}.
}
$$

---

# 16. Why the entire ancestor cannot be lifted immediately

Running B7 directly on the depth-31 necessity cell:

depth-12 lift:

$$
1051\text{ nodes},
$$

$$
227\text{ unresolved leaves}.
$$

unresolved witness placement-volume fraction:

$$
\frac{227}{4096}
\approx
5.54\%.
$$

Still not small.

---

# 17. Necessity segmentation experiment

First base-refine this same depth-31 cell to depth 40.

cost:

$$
\boxed{
561\text{ base nodes}
}
$$

and:

$$
\boxed{
1.69\text{ CPU-s}.
}
$$

producing:

$$
256
$$

pending descendants.

---

# 18. Exact-center segmentation result

For all:

$$
256
$$

descendants, compute the exact center hull.

Of these:

$$
85
$$

have exact centers:

$$
<0.835.
$$

Then, checking these 85 with the 20k support outer bracket:

$$
\boxed{
77
}
$$

outer brackets are still strictly:

$$
<0.835.
$$

So:

$$
\boxed{
\text{witness necessity shrinks from the whole ancestor to at least 77 explicit descendants}.
}
$$

The other cells cannot therefore claim they do not need a witness,

but they should not be crudely lifted all together either.

---

# 19. Dual Frontier

Round 19 splits the global pending state into two queues.

## Base Frontier

`BASE-UNRESOLVED`

does not yet have a certified sublevel point.

Main operations:

- base refine;
- APR/CORE/REP;
- symmetry prune.

## Necessity / Lift Frontier

`NECESSITY-MARKED`

already contains a certified:

$$
A_{\mathcal F_3}(q)<T.
$$

These branches must eventually either:

- add a witness;
- or split so as to localize the marker.

---

# 20. Logical activation gate vs performance activation gate

Two things must be kept separate.

## Logical necessity

Once there is a certified sublevel point:

$$
\boxed{
\text{witness is eventually necessary along that point's branch}.
}
$$

## Performance activation

does not mean:

> it is now cheapest to run the full $B_7$ tree on the ancestor immediately.

One may first continue base splitting in order to:

- shrink base erosion;
- localize witness necessity;
- reduce the number of lift cells.

---

# 21. B7 switch curve

For the same official near-minimum branch,

holding B7 lift depth fixed at 12.

base depth / $\tau$ / unresolved:

- depth 20:
  $$
  \tau\approx0.02448,\quad483;
  $$
- depth 24:
  $$
  \tau\approx0.01412,\quad330;
  $$
- depth 28:
  $$
  \tau\approx0.00964,\quad252;
  $$
- depth 31:
  $$
  \tau\approx0.00482,\quad227;
  $$
- depth 34:
  $$
  \tau\approx0.00353,\quad198;
  $$
- depth 37:
  $$
  \tau\approx0.00241,\quad183;
  $$
- depth 40:
  $$
  \tau\approx0.00153,\quad161;
  $$
- depth 46:
  $$
  \tau\approx0.000602,\quad135;
  $$
- depth 50:
  $$
  \tau\approx0.000383,\quad125.
  $$

---

# 22. Zero-base-uncertainty witness floor

Directly fixing the base placement to:

$$
q^\dagger
$$

i.e. base halfwidth:

$$
0.
$$

the B7 depth-12 probe still has:

$$
\boxed{
108
}
$$

unresolved leaves.

volume fraction:

$$
\boxed{
\frac{108}{4096}
\approx2.64\%.
}
$$

This is, for the current B7 depth-12 lower machinery, its:

$$
\boxed{
\text{witness-only floor}
}
$$

it is not the ultimate mathematical floor — only the floor of the current verifier / lift depth.

---

# 23. Diminishing base-refinement return

Comparing the depth-12 unresolved volume:

depth 31:

$$
5.54\%.
$$

depth 40:

$$
3.93\%.
$$

depth 46:

$$
3.30\%.
$$

base-zero:

$$
2.64\%.
$$

So once:

$$
\tau
$$

enters about the:

$$
10^{-3}
$$

order of magnitude,

continuing to shrink the base box brings the B7 lift tree's improvement gradually close to the witness-only limit.

Round 19 records:

$$
\tau\sim10^{-3}
$$

as:

`LIFT-EFFICIENCY-REGIME-PILOT`

not a theorem constant.

---

# 24. Empirical switch model

If the base-zero depth-12 unresolved fraction:

$$
f_0
=
108/4096
$$

is taken as the floor,

the excess in the pilot data:

$$
f(\tau)-f_0
$$

can roughly be described by:

$$
\boxed{
\sim k\tau^{0.70}
}
$$

This is only a scheduler model,

not a mathematical bound.

---

# 25. Recommended activation state machine

```text
BASE-UNRESOLVED
    |
    | certified point upper area < T
    v
NECESSITY-MARKED
    |
    | split to localize marker / reduce base uncertainty
    v
NECESSITY-SEGMENTED
    |
    | cost model says lift cheaper
    v
LIFT-ACTIVE
    |
    | complete witness tree
    v
LIFT-CERTIFIED
```

---

# 26. Lift activation policy v0.1

proof gate:

$$
\boxed{
\exists q\in C:
U_{\rm base}(q)<T.
}
$$

performance gate:

give priority to comparing:

$$
\operatorname{Cost}_{\rm base-refine}
$$

with:

$$
\operatorname{Cost}_{B_7\text{-lift}}.
$$

Current pilot recommendation:

- once necessity appears, do not lift the whole coarse ancestor;
- do segmentation first;
- when:
  $$
  \tau\lesssim10^{-3}
  $$
  or the projected further-base cost is already higher than lift,
  only then promote the branch to high-priority `LIFT-ACTIVE`.

---

# 27. B7 still not closed

Even with the base fixed exactly to:

$$
q^\dagger,
$$

the B7 shallow lift still has not fully closed at depth 18.

Therefore:

$$
\boxed{
\text{Round 19 has not proved that }B_7\text{ alone suffices to globally close }0.835.
}
$$

Still needed:

- a deeper B7 lift;
- a stronger witness lower bound;
- or witness switching.

---

# 28. Why this is still major

Round 07–08 already knew:

> base sublevel cells need to be lifted.

Round 19 is the first time to truly obtain:

$$
\boxed{
\text{when it can be proved that "base-only can never close this branch"}
}
$$

as well as:

$$
\boxed{
\text{how to avoid lifting the entire ancestor all at once}.
}
$$

This is a move from:

`lazy dimension activation`

further into:

$$
\boxed{
\text{proof-triggered dimension activation}.
}
$$

---

# 29. Production update

New global stack:

$$
\boxed{
D_3\text{ quotient}
+
\text{Tail-Window CPG}
+
\text{Lazy APR/CORE/REP}
+
\text{Necessity Markers}
+
\text{Dual Frontier}
+
\text{Selective B7 Lift}.
}
$$

---

# 30. Status ledger

## Lazy cascade

`PROVED + A/B PASS`

## Continuation $\kappa$

`MEASURED`

## Sample-inner sublevel interpretation

`CORRECTED`

## Official exact-kernel center classification

`VERIFIED-COMPUTATIONAL-PILOT`

## Necessity-Marker Theorem

`PROVED`

## Necessity segmentation

`IMPLEMENTED / PILOT PASS`

## $\tau\sim10^{-3}$ lift-efficiency regime

`NUMERICAL-EVIDENCE / SCHEDULER PARAMETER`

## B7 global closure

`COMPUTE-DEFERRED`

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

`NOT YET CERTIFIED`

---

# 31. Round 20 assigned topic

## AMRAL-LUC-FC-R20
### Necessity-Marked Lift Shards and Adaptive Witness Escalation

The next round should directly:

1. write the necessity marker into the shard/checkpoint grammar;
2. have the production emitter output, within the base tree:
   - `BASE-PENDING`
   - `NECESSITY-MARKED`
   - `LIFT-ATTACHED`
3. build B7 lift shards for the marked descendants;
4. if B7 fails within budget:
   - refine the base;
   - deepen B7;
   - or switch to a $B_9/B_{11}$/Fourier witness;
5. build a witness-escalation cost model;
6. use the official near-minimum marked branches to build the first batch of genuine mixed base/lift certificate shards.

---

# 32. Shortest handoff conclusion

Round 19's most important change is not a few fewer nodes.

Rather, it turns:

> when is a witness needed?

from a heuristic into two clearly separated layers of logic.

The first layer:

$$
\boxed{
A_{\mathcal F_3}(q)<T
\Rightarrow
\text{base-only closure impossible along descendants containing }q.
}
$$

The second layer:

$$
\boxed{
\text{necessity}
\ne
\text{immediate lift of whole ancestor}.
}
$$

First use base refinement to localize necessity,

then only raise dimension on the truly marked descendants.

So AMRAL now has a genuine:

$$
\boxed{
\text{proof-triggered dimensional activation mechanism}.
}
$$