# AMRAL × Lebesgue Universal Covering — Round 20
## Necessity-Marked Mixed Lift Shards and Adaptive Witness Escalation

**Document ID:** AMRAL-LUC-FC-R20  
**Version:** v0.1  
**Date:** 2026-09-19  
**Research status:** Round 20 / Mixed base-lift proof state / Witness escalation  
**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–R19 v0.1  

---

# 0. Round Summary

Round 19 established the Necessity Marker and the Dual Base/Lift Frontier.

Round 20 writes it properly into the shard grammar, and turns witness selection into evidence-driven escalation.

This round completes:

1. the mixed base/lift shard grammar;
2. the witness negative-counterexample theorem;
3. `UNRESOLVED != DISQUALIFIED` fail-closed semantics;
4. the two-dimensional separation of search forcing / certificate efficiency;
5. horizon-dependent witness ranking;
6. a state-dependent Pareto witness frontier;
7. 4 necessity-marked base cells × B7/B11/B19 partial lift trees;
8. independent replay;
9. the first actual witness disqualification: B23;
10. a B7 ranking reversal in deep lift.

This round's determination:

$$
\boxed{\text{MIXED SHARD GRAMMAR: CLOSED}}
$$

$$
\boxed{\text{WITNESS NEGATIVE FILTER: CLOSED}}
$$

$$
\boxed{\text{ADAPTIVE ESCALATION: PILOT-CALIBRATED}}
$$

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

---

# 1. Mixed base/lift shard

A base leaf can now carry:

```text
base_cell_id
base_path
base_box
necessity marker
lift attachments
negative witness evidence
```

A final shard can only reach `COMPLETE` once every base obligation is either `BASE-CERTIFIED` or fully covered by `LIFT-CERTIFIED`.

---

# 2. State semantics

- `BASE-CERTIFIED`: the base-only lower bound reaches the target.
- `NECESSITY-MARKED`: the cell contains a verified base configuration with area $<T$.
- `LIFT-PROBE`: a finite witness probe is in progress.
- `LIFT-ACTIVE`: a witness tree has been attached, but closure has not yet been reached.
- `LIFT-CERTIFIED`: every relevant witness placement root has reached full closure.
- `WITNESS-DISQUALIFIED`: a concrete witness placement counterexample already exists.
- `PENDING`: not yet complete.

`NECESSITY-MARKED` is not, by itself, closure.

---

# 3. Unresolved does not mean witness failure

If the $B_7$ tree still has pending nodes at depth $d$:

$$
P_d>0,
$$

the only conclusion is:

$$
\boxed{\text{CURRENT CERTIFICATE INCOMPLETE}.}
$$

It cannot be concluded that:

$$
J_{B_7}(H)<T.
$$

The cause may simply be that the box is too coarse, base erosion is too large, the common-core bound is too weak, or lift depth is insufficient.

---

# 4. Witness Counterexample Theorem

Let base cell $C$ contain configuration $q$.

If, for witness $K$, there exists a legal placement $g$ with one-sided upper bound:

$$
U(\operatorname{conv}(H_{\mathcal F_3}(q)\cup gK))<T,
$$

then:

$$
\boxed{
J_K(H_{\mathcal F_3}(q))<T.
}
$$

Therefore $K$ cannot be the sole closing witness for $C$.

This is a proof-safe negative filter.

---

# 5. Actual B23 disqualification

At reference `cell-00`, a 3-D placement search found a $B_{23}$ placement:

$$
\phi\approx0.22598188,
$$

$$
x\approx-5.68\times10^{-5},
\qquad
y\approx-3.14\times10^{-4}.
$$

exact-kernel area:

$$
0.83492540.
$$

20k support outer:

$$
\boxed{
0.83494439<0.835.
}
$$

So:

$$
\boxed{
B_{23}\text{ is pilot-disqualified as a sole closer for this marked cell}.
}
$$

This is currently still pilot floating policy, not a publication arithmetic theorem.

---

# 6. Absence of counterexample has no positive proof value

If a numerical search does not find:

$$
A<T,
$$

the only thing that can be written is:

`NO-COUNTEREXAMPLE-FOUND`.

It cannot be written as:

`VIABLE-PROVED`.

The only genuine positive proof is:

$$
\boxed{
\text{complete witness placement certificate}.
}
$$

---

# 7. Shallow certificate efficiency

8 depth-40 marked cells, lift depth 12:

$$
B_7:\ \text{mean pending}\approx160.6,
$$

$$
B_9:\ 154.0,
$$

$$
B_{11}:\ 119.1.
$$

The larger odd-gon scan over the first 4 cells:

$$
B_{13}:89,
\quad
B_{15}:68,
\quad
B_{17}:69,
\quad
B_{19}:64,
\quad
B_{21}:64,
\quad
B_{23}:64.
$$

If one looks only at shallow pending fraction, higher $n$ appears to keep getting better.

The B23 counterexample proves that this single ranking is wrong.

---

# 8. Why disk-like witnesses look easy

As $n$ increases:

$$
B_n\to D.
$$

the rotation radius approaches $1/2$, and box erosion loss becomes smaller.

Most translated placements are easily proved to lie above the target because of protrusion.

Only a small near-alignment placement tail remains.

So:

$$
\boxed{
\text{certificate tree easier}
\not\Rightarrow
\text{witness geometrically stronger}.
}
$$

---

# 9. Placement-search forcing signal

The same `cell-00` 3-D witness placement search:

$$
B_7\approx0.83707775,
$$

$$
B_9\approx0.83562957,
$$

$$
B_{11}\approx0.83617312,
$$

$$
B_{13}\approx0.83592291,
$$

$$
B_{15}\approx0.83529107,
$$

$$
B_{17}\approx0.83527109,
$$

$$
B_{19}\approx0.83540886,
$$

$$
B_{21}\approx0.83514310,
$$

$$
B_{23}\approx0.83492540.
$$

Apart from B23's negative certificate, the rest are only search evidence.

---

# 10. Witness selection is Pareto

There are at least two main axes:

## Geometric forcing

the margin between the search minimum and the target.

## Certificate efficiency

the shallow / current-depth pending fraction and node cost.

So the witness pool cannot be written as a fixed:

```text
B7 -> B11 -> B13 -> ...
```

Instead, a state-dependent Pareto frontier must be maintained.

---

# 11. Horizon-dependent ranking

The tracked necessity branch was refined down to base depth 50:

$$
\tau\approx3.83\times10^{-4}.
$$

lift depth 18:

- B7 pending fraction $\approx0.249\%$;
- B11 $\approx0.297\%$;
- B19 $\approx0.281\%$;
- B21 $\approx0.276\%$.

lift depth 20:

- B7 $\approx0.0968\%$;
- B11 $\approx0.1894\%$;
- B13 $\approx0.1868\%$;
- B19 $\approx0.1472\%$;
- B21 $\approx0.1405\%$.

So:

$$
\boxed{
B_7\text{ is costlier shallow, but its deep tail contracts faster}.
}
$$

---

# 12. B7 deep tail

For the same depth-50 base cell:

$$
u_{18}\approx2.491\times10^{-3},
$$

$$
u_{20}\approx9.680\times10^{-4},
$$

$$
u_{22}\approx2.494\times10^{-4},
$$

$$
u_{24}\approx8.327\times10^{-5},
$$

$$
u_{26}\approx3.193\times10^{-5}.
$$

Closure is still not complete, but the rate of decrease is clear.

---

# 13. Horizon-aware escalation

Witness ranking must therefore depend on the projected horizon:

- shallow probe: a certificate-efficient witness has value;
- long tail: a witness with a large forcing margin may later overtake it.

The scheduler should estimate:

$$
\rho_{K}(d_1,d_2)
=
\frac{u_{K,d_2}}{u_{K,d_1}}.
$$

It cannot look only at:

$$
u_{K,d_1}.
$$

---

# 14. Escalation State Machine

## N0 — Necessity

The base cell already has a marker.

## N1 — Negative Filter

Search for a witness placement.

If a candidate $<T$ is found, confirm it with a one-sided outer bound.

On success:

`WITNESS-DISQUALIFIED`.

## N2 — Shallow Lift Probe

Run a fixed small depth on the survivors.

Collect:

- nodes;
- pending fraction;
- time.

## N3 — Tail Estimate

Estimate the contraction rate across multiple depths.

## N4 — Base vs Lift

If base $\tau$ is still large, perform base refinement / marker segmentation first.

If $\tau$ is small and the witness tail is contracting quickly, deepen the witness.

## N5 — Switch / Batch

If the tail plateaus or is eliminated by the negative filter:

- switch witness;
- or Round 07 batch;
- and only as a last resort, joint multi-witness placement.

---

# 15. Actual mixed-shard reference

Round 20 actually establishes 4 depth-40 necessity-marked base cells.

Each cell has attached:

- a B7 depth-12 partial lift tree;
- B11 depth-12;
- B19 depth-12.

All markers were re-verified with the center outer bracket $<0.835$.

All witness trees retain pending leaves.

So:

$$
\boxed{\texttt{theorem\_ready=false}.}
$$

This is the correct fail-closed semantics.

---

# 16. Independent mixed-shard replay

The verifier replays:

1. the marker;
2. lift topology;
3. deterministic witness splits;
4. common cores;
5. the hand-written monotone-chain inner hull;
6. every CERT leaf;
7. every PENDING leaf, kept unresolved;
8. stream exhaustion;
9. the B23 negative counterexample.

Result:

$$
\boxed{
\texttt{REFERENCE-MIXED-SHARD-VERIFIED}
}
$$

But:

$$
\boxed{
\texttt{theorem\_ready=false}.
}
$$

"The format and declared evidence are correct" does not mean "the entire proof is complete."

---

# 17. Counterexample cache

Negative evidence:

$$
(q,g,U<T)
$$

should go into a content-addressed evidence cache.

As long as a future base cell still contains the same $q$, the same witness's negative filter can be reused directly.

A counterexample is also a proof asset.

---

# 18. Regular odd-gon ladder lesson

The following cannot be used as a witness strength ordering:

$$
n\uparrow
$$

Regular odd-gons simultaneously change:

- forcing geometry;
- rotational sensitivity;
- erosion loss;
- near-disk counterexample risk.

So this is a multi-objective family, not a single monotone ladder.

---

# 19. Fourier fallback

The Round 02 asymmetric dictionary is placed after:

1. the regular witness Pareto pool;
2. the negative filter / tail plateau;
3. the hard marked cells;

and is activated only then.

At that point, the target oracle should work backward from the geometry of the unresolved lift tail to infer an asymmetric witness, rather than blindly scanning the entire dictionary.

---

# 20. Production rule

Currently, the safest witness escalation is:

$$
\boxed{
\text{Necessity Marker}
\to
\text{Negative Filter}
\to
\text{Shallow Probe}
\to
\text{Tail Estimate}
\to
\text{Pareto Choice}
\to
\text{Deepen / Switch / Batch}.
}
$$

---

# 21. Status ledger

- Mixed shard grammar: `IMPLEMENTED + REPLAYED`
- Necessity markers in mixed shard: `IMPLEMENTED`
- Partial lift semantics: `IMPLEMENTED`
- Witness negative theorem: `PROVED`
- B23 negative example: `VERIFIED-COMPUTATIONAL-PILOT`
- Absence-of-counterexample -> viability: `REJECTED AS INFERENCE`
- Horizon-dependent ranking: `NUMERICAL-EVIDENCE`
- B7 full closure of marked cell: `NOT YET ACHIEVED`
- Global $0.8350$: `COMPUTE-DEFERRED`

---

# 22. Round 21

## Lift-Tail Geometry and Counterexample-Guided Witness Synthesis

The main thrust of the next round:

1. project B7's depth-22/24 unresolved witness leaves into a $(\phi,x,y)$ placement atlas;
2. analyze active-contact / near-alignment patterns;
3. analyze the shared geometry between the B23 counterexample and the B7 tail;
4. work backward from the unresolved tail to an asymmetric constant-width witness;
5. use the Round 02 Fourier dictionary to perform counterexample-guided witness synthesis;
6. stop blindly scanning odd-gon $n$.

---

# 23. Shortest Handoff

Round 20's most important conclusion:

$$
\boxed{
\text{witness selection is itself also an adaptive proof problem}.
}
$$

A single fixed witness leaderboard cannot be used.

The correct state machine is:

$$
\boxed{
\text{Necessity}
\to
\text{Negative Counterexample Filter}
\to
\text{Certificate Efficiency}
\to
\text{Tail Contraction}
\to
\text{Pareto Escalation}.
}
$$

Conditional Lift is hereby formally upgraded from "attach one guessed witness" to:

$$
\boxed{
\text{evidence-driven witness escalation}.
}
$$