# AMRAL × Lebesgue Universal Covering — Round 21
## Lift-Tail Geometry and Counterexample-Guided Witness Synthesis

**Document ID:** AMRAL-LUC-FC-R21  
**Version:** v0.1  
**Date:** 2026-09-19  
**Research status:** Round 21 / Lift-tail atlas / Adversarial witness design  
**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–R20 v0.1  

---

# 0. Round Summary

Round 20 already showed that $B_7$ is currently the most promising long-horizon regular witness, but the deep lift still leaves a very small unresolved tail.

Round 21 is the first to turn these unresolved leaves into structured adversarial geometry, and uses them, in reverse, to do witness synthesis.

This round obtains:

1. the B7 depth 22/24/26 tail atlas;
2. the two-lobe phase-lock law;
3. an area-minimizing placement ridge;
4. the Counterexample-Guided Witness Synthesis cutting-plane protocol;
5. the Finite-Tail Optimism Theorem;
6. a legal 7-fold-preserving support family;
7. synthesis tests for a smooth 7-fold wave, an interiorized B7, and nearby irregular Reuleaux7 shapes;
8. an important negative result: none of the currently tested families robustly surpasses regular B7 under a fresh placement oracle.

This round's verdict:

$$
\boxed{\text{B7 LIFT-TAIL ATLAS: CLOSED}}
$$

$$
\boxed{\text{COUNTEREXAMPLE-GUIDED SYNTHESIS LOOP: CLOSED}}
$$

$$
\boxed{\text{NEW SUPERIOR WITNESS: NOT FOUND}}
$$

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

---

# 1. Tail Scope

Fixing the depth-50 base cell of the near-minimum branch tracked in Round 19:

$$
\tau_{\rm base}
\approx3.82517\times10^{-4}.
$$

So the lift tail currently seen is now mainly witness-placement geometry, rather than coarse base uncertainty.

---

# 2. B7 Deep-Tail Volume

Lift depth 22:

$$
1046
$$

pending boxes,

$$
1046/2^{22}
\approx2.494\times10^{-4}.
$$

Lift depth 24:

$$
1397/2^{24}
\approx8.327\times10^{-5}.
$$

Lift depth 26:

$$
\boxed{
2143/2^{26}
\approx3.193\times10^{-5}.
}
$$

The pending leaf count can increase, but the placement volume keeps shrinking rapidly.

---

# 3. Phase-Lock Coordinate

Definition:

$$
\boxed{
\delta_{\rm lock}
=
\operatorname{wrap}
(
\arg t_7-7\phi_7
).
}
$$

If:

$$
\delta_{\rm lock}
\approx0,
$$

this represents the translation direction being phase-locked with the heptagonal orientation.

---

# 4. Phase-Lock Contraction

The median of $|\delta_{\rm lock}|$:

$$
0.1886
\to
0.1372
\to
0.1081
$$

for depth:

$$
22\to24\to26.
$$

95th percentile:

$$
0.4785
\to
0.3330
\to
0.2446.
$$

So:

$$
\boxed{
\arg t_7\approx7\phi_7
}
$$

becomes more pronounced as the certificate tail deepens.

---

# 5. Two-Lobe Atlas

The 2143 tail boxes at depth 26 split almost entirely into two highly concentrated clusters.

## Lobe A

About:

$$
1000
$$

boxes.

$$
7\phi\approx2.5255,
$$

$$
\arg t\approx2.5536,
$$

$$
|t|\approx0.01213.
$$

## Lobe B

About:

$$
1143
$$

boxes.

$$
7\phi\approx5.8819,
$$

$$
\arg t\approx5.8989,
$$

$$
|t|\approx0.01248.
$$

The circular concentration in both cases is close to:

$$
0.98\text{–}0.99.
$$

So the unresolved tail is not a three-dimensional scattered cloud, but two narrow placement lobes.

---

# 6. Area-Minimizing Ridge

Fixing $\phi_7$, performing full area minimization only over $(x_7,y_7)$.

The lowest primary ridge location also falls in the same two orientation channels.

So the numerical evidence supports:

$$
\boxed{
\text{deep certificate tail}
\approx
\text{thickened near-minimizing placement ridge}.
}
$$

This matters for witness synthesis:

> unresolved boxes can serve as an adversarial placement atlas.

---

# 7. The B7 Tail and the B23 Counterexample Are Not the Same Mechanism

B7 deep tail:

$$
|t_7|\sim0.012.
$$

Round 20's B23 counterexample:

$$
|t_{23}|
\approx3.19\times10^{-4}.
$$

The latter is close to directly coinciding with an existing disk.

So there are at least two failure geometries:

1. a finite-translation phase-locked channel;
2. a near-disk hiding channel.

Witness synthesis must distinguish the adversarial mechanism, and cannot look only at the pending fraction.

---

# 8. Witness Design as Max-Min

Let $\mathcal S$ be the legal witness shape family, and $\mathcal P$ the full placement domain.

Define:

$$
F(K,p)
=
A(
\operatorname{conv}
(H_{\rm base}\cup pK)
).
$$

The truly robust witness design:

$$
\boxed{
V^\star
=
\max_{K\in\mathcal S}
\min_{p\in\mathcal P}
F(K,p).
}
$$

This is a minimax between a shape-player and a placement-adversary.

---

# 9. Cutting-Plane Synthesis

Maintain a finite adversarial placement set:

$$
P_m=\{p_1,\ldots,p_m\}.
$$

Shape step:

$$
\boxed{
K_m
\in
\arg\max_K
\min_{p\in P_m}
F(K,p).
}
$$

Fresh oracle:

$$
\boxed{
p_{m+1}
\in
\arg\min_{p\in\mathcal P}
F(K_m,p).
}
$$

Then:

$$
P_{m+1}=P_m\cup\{p_{m+1}\}.
$$

Any synthesized witness must pass through the fresh placement oracle, and cannot simply report the training score on the known tail.

---

# 10. Finite-Tail Optimism Theorem

Define:

$$
v_m
=
\max_K
\min_{p\in P_m}F(K,p).
$$

Because:

$$
P_m\subseteq\mathcal P,
$$

finite-tail training is always more optimistic than the true robust problem.

And:

$$
\boxed{
v_{m+1}\le v_m.
}
$$

because the adversarial set can only grow.

This formally explains the overfitting phenomenon in Round 21.

---

# 11. Legal 7-Fold-Preserving Family

Test:

$$
\boxed{
h_{\lambda,a,\psi}
=
(1-\lambda)h_{B_7}
+
\lambda/2
+
a\cos(7\theta+\psi).
}
$$

Constant width is kept equal to 1.

Interiorization gives a curvature floor $\lambda/2$.

The $k=7$ harmonic curvature amplitude is $48|a|$.

So a sufficient legality condition:

$$
\boxed{
|a|\le\lambda/96.
}
$$

And 7-fold rotation symmetry is still preserved, with the orientation period still:

$$
2\pi/7.
$$

---

# 12. Regular B7 Baseline

On the official near-minimum base placement, a fresh placement search for regular B7 gives about:

$$
\boxed{
0.83712806.
}
$$

This is search evidence, not a lower certificate.

---

# 13. Smooth 7-Fold Family

Test:

$$
h(\theta)=1/2+a\cos7\theta.
$$

Even with:

$$
a=0.0103
$$

already very close to the convexity boundary, the fresh search minimum is about:

$$
0.8369528,
$$

still below the regular B7 baseline.

---

# 14. Tail-Targeted Overfitting: An Empirical Test

The first round optimizes the shape targeting only the known B7 tail lobes.

The training score can be pulled up to:

$$
0.8377708.
$$

But the fresh placement oracle immediately finds:

$$
\boxed{
0.8369560.
}
$$

a new low.

Continuing the cutting-plane process:

- Iter 1 oracle:
  $$
  0.8370455;
  $$
- Iter 2:
  $$
  0.8371292;
  $$
- Iter 3:
  $$
  0.8371286.
  $$

Finally returning to near the regular B7 baseline.

---

# 15. Interpretation

This is not a B7 optimality theorem.

But it indicates:

> Within the currently tested 7-fold-preserving legal deformation family, regular B7 is fairly robust against the placement adversary.

After patching the existing tail, the placement oracle will open a new hole somewhere else.

---

# 16. Nearby Irregular Reuleaux7

This round also runs an exploratory search over nearby irregular Reuleaux7 shapes:

- 30 random, projected constant-width candidates;
- the coarse top 6 are then re-searched robustly over the full orientation range.

The best robust value is about:

$$
0.83697744,
$$

which does not exceed regular B7.

So there is currently also no improvement from nearby irregular Reuleaux7 shapes.

---

# 17. Early Symmetry-Breaking Pilot

An initial $k=3$ symmetry-breaking deformation loses the 7-fold orientation quotient.

The coarse-top candidates are re-searched with the placement oracle over the full:

$$
\phi\in[0,2\pi)
$$

range; the best tested value is about:

$$
0.83628622.
$$

Still does not exceed B7.

This is not a refutation of symmetry-breaking as such, only an indication that naive single-mode breaking is insufficient.

---

# 18. Shape Gain Must Pay Its Proof Cost

A future witness objective cannot look only at:

$$
V(K)=\min_pF(K,p).
$$

It must also account for certificate cost:

$$
\boxed{
\mathcal J(K)
=
V(K)
-
\lambda_{\rm cert}
\operatorname{Cost}_{\rm lift}(K).
}
$$

Because symmetry-breaking may expand the orientation fundamental period from:

$$
2\pi/7
$$

to:

$$
2\pi.
$$

so even if the forcing becomes only slightly stronger, the formal certificate cost may increase greatly.

---

# 19. Current Synthesis Conclusion

This round did not find a superior witness.

But it rules out three directions that are easy to misjudge:

1. a smooth single $k=7$ wave;
2. a single fit targeting only the existing tail;
3. nearby irregular Reuleaux7 shapes without a fresh full placement oracle.

So:

$$
\boxed{
B_7
\text{ remains the primary deep-tail witness}.
}
$$

---

# 20. Next Search Space

The new degree of freedom truly worth opening is multi-tail symmetry-breaking odd Fourier modes:

$$
\boxed{
k=3,5,7,9,11.
}
$$

This will require:

1. Round 02 legality;
2. a full-orientation placement oracle;
3. a proof-cost penalty;
4. a fresh adversarial placement loop;
5. joint training across multiple base necessity markers.

---

# 21. Status Ledger

- B7 lift-tail atlas: `VERIFIED-COMPUTATIONAL-PILOT`
- Two-lobe phase lock: `NUMERICAL-EVIDENCE`
- Cutting-plane witness synthesis: `FORMALIZED + IMPLEMENTED`
- Finite-tail optimism theorem: `PROVED`
- 7-fold legal deformation family: `PROVED`
- superior 7-fold witness: `NOT FOUND`
- nearby irregular Reuleaux7 improvement: `NOT FOUND`
- regular B7 local minimax interpretation: `NUMERICAL-EVIDENCE`
- global $0.8350$: `COMPUTE-DEFERRED`

---

# 22. Round 22

## Symmetry-Breaking Fourier Witness Oracle and Proof-Cost-Aware Shape Search

Next round:

1. use the two-lobe atlas plus multiple necessity markers as the adversarial training set;
2. allow $k=3,5,7,9,11$;
3. enforce Round 02 curvature / constant-width legality;
4. run the fresh placement oracle over the full $[0,2\pi)$ range;
5. have the objective weigh forcing and certificate cost simultaneously;
6. any candidate that passes the search then proceeds to the Round 02 finite compiler and the Round 08 generic support-core verifier.

---

# 23. Shortest Handoff

Round 21's most important result is not a new witness,

but turning:

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

into:

$$
\boxed{\text{structured adversarial atlas}}.
$$

The B7 tail is a two-lobe, phase-locked placement ridge.

But:

$$
\boxed{
\text{tail fitting without fresh oracle = overfitting}.
}
$$

So witness synthesis formally becomes:

$$
\boxed{
\text{shape player}
\leftrightarrow
\text{placement adversary}
}
$$

a cutting-plane minimax process.