# AMRAL × Lebesgue Universal Covering — Round 17
## Residual $D_3$ Symmetry Quotient and Certified-Prune-Gain Scheduling

**Document ID:** AMRAL-LUC-FC-R17  
**Version:** v0.1  
**Date:** 2026-09-19  
**Research status:** Round 17 / Global quotient + scheduler-cost pilot  
**Research mode:** Human-Directed + Semi-Autonomous AI Mathematical Research  
**Research initiation and methodology source:** Neo.K  
**AI research collaborator and primary executor:** Aletheia / ChatGPT, GPT-5.6 Sol  
**Prior documents:** AMRAL-LUC-FC-R00 v0.2; R01–R16 v0.1

---

# 0. Summary Judgment for This Round

Round 16's global master-root pilot showed: the raw `HARD` frontier is far larger than the true near-minimizer frontier, and the one-step uncertainty that is optimal for the exact-$\tau$ split is not the same as overall certificate-cost optimality.

Round 17 completes two new layers:

$$
\boxed{\text{Residual }D_3\text{ exact symmetry quotient}}
$$

and:

$$
\boxed{\text{Certified-Prune-Gain scheduling}}
$$

Core conclusions:

- the normalized $D+B_3+B_5$ placement space still has residual $D_3$ symmetry;
- it can be exactly restricted to a $60^\circ$ $t_3$ canonical wedge;
- the `SYM` half-space prune can be added directly to the existing axis-aligned verifier;
- at depth 16, the $D_3$ quotient relative to the full-root official run: nodes decrease by about **47.32%**, HARD by about **32.73%**;
- unrestricted CPG is very good at shrinking the tree, but can be myopic;
- tail-only Windowed CPG, at depth 16, further reduces nodes by **19.10%** and HARD by **25.96%** relative to $D_3$ official, with an override density of about **10.72%**.

This round's judgment:

$$
\boxed{D_3\text{ SYMMETRY QUOTIENT: CLOSED}}
$$

$$
\boxed{\text{TAIL-WINDOW CPG: PILOT-RECOMMENDED}}
$$

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

---

# 1. Residual $120^\circ$ rotation symmetry

The current gauge fixes the disk at the origin and uses a global rotation to fix the $B_3$ orientation. Configuration:

$$
q=(x_3,y_3,\phi_5,x_5,y_5),
\qquad
\phi_5\in[0,2\pi/5).
$$

Let:

$$
r=R_{2\pi/3}.
$$

Since the regular Reuleaux triangle has three-fold rotational symmetry:

$$
R_{2\pi/3}B_3=B_3.
$$

Therefore:

$$
t_3\mapsto R_{2\pi/3}t_3,
$$

$$
t_5\mapsto R_{2\pi/3}t_5,
$$

$$
\phi_5\mapsto\phi_5+2\pi/3\pmod{2\pi/5}.
$$

The whole convex hull is only subject to a common rigid-body rotation, so:

$$
\boxed{A(q)=A(rq)}.
$$

---

# 2. Residual reflection symmetry

Let:

$$
s(x,y)=(x,-y).
$$

The current canonical $B_3$ is mirror-symmetric about the $x$-axis, so:

$$
(x_3,y_3)\mapsto(x_3,-y_3),
$$

$$
(x_5,y_5)\mapsto(x_5,-y_5),
$$

$$
\phi_5\mapsto-\phi_5\pmod{2\pi/5}.
$$

Hence:

$$
\boxed{A(q)=A(sq)}.
$$

We also have:

$$
r^3=e,\qquad s^2=e,\qquad srs=r^{-1}.
$$

so the residual group is:

$$
\boxed{G\cong D_3},
\qquad |G|=6.
$$

---

# 3. Canonical-Wedge Theorem

For $t_3\neq0$, let its polar angle be $\theta_3$. The $D_3$ action consists of $120^\circ$ rotations plus reflections, so a standard fundamental sector can be taken as:

$$
\boxed{0\le\theta_3\le\pi/3}.
$$

## Theorem 3.1

Every relevant normalized configuration orbit has at least one representative satisfying:

$$
0\le\arg t_3\le\pi/3.
$$

Therefore:

$$
\boxed{
\min_{q\in\mathcal Q_{\rm full}}A(q)
=
\min_{q\in\mathcal Q_{D_3}}A(q)
}.
$$

Reasoning: if $t_3=0$ it is already on the wedge boundary; if $t_3\neq0$, at least one of the six dihedral angular images must fall within the fundamental sector of length $\pi/3$. The same group element acts on $t_5$ only via rigid rotation/reflection, which preserves radial legality, while $\phi_5$ modulo $2\pi/5$ still returns to the normalized interval. The area is preserved by rigid-body invariance.

---

# 4. Axis-aligned implementation and `SYM`

For phase-master radius $t_3^\star$, the canonical wedge is contained in:

$$
0\le x_3\le t_3^\star,
$$

$$
0\le y_3\le\frac{\sqrt3}{2}t_3^\star.
$$

Add the canonical half-space:

$$
\boxed{y_3\le\sqrt3\,x_3}.
$$

For an axis-aligned cell, if:

$$
\boxed{y^-_3>\sqrt3\,x^+_3},
$$

then the cell does not intersect the canonical wedge, and can be marked:

`SYM`

and stopped.

`SYM` is not an area lower-bound leaf but a quotient-domain exclusion leaf. The final verifier must list the $D_3$ quotient theorem as root semantics.

---

# 5. Symmetry implementation sanity

Round 17 uses a 120-direction inner-polygon grid for the numeric cross-check, to avoid the directional bias that Round 16's $N=40$ sampling grid produced against $120^\circ$ rotation.

Samples: 20.

Maximum area discrepancy under reflection:

$$
6.661e-16.
$$

Maximum discrepancy under $120^\circ$ rotation:

$$
8.882e-16.
$$

Both are at the level of floating-point noise. This is only an implementation sanity check; the symmetry theorem itself is analytic.

---

# 6. Full-root to $D_3$ pilot

Round 16 full root + official scheduler, depth 16:

$$
23455\text{ nodes},\qquad4112\text{ HARD}.
$$

Round 17 $D_3$ root + official:

$$
\boxed{12357\text{ nodes}},
$$

$$
\boxed{2766\text{ HARD}}.
$$

So:

$$
\boxed{47.32\%\text{ fewer nodes}},
$$

$$
\boxed{32.73\%\text{ fewer HARD}}.
$$

This is enough to promote the $D_3$ quotient to the production root baseline.

---

# 7. Split-Choice Soundness

For any cell $C$, as long as the actual split children satisfy:

$$
C\subseteq C_0\cup C_1,
$$

the choice of axis only changes tree topology / cost / metadata, not proof soundness.

The scheduler is therefore a performance layer; the verifier only needs to know the actual split axis and to verify that the children cover the parent.

---

# 8. Certified-Prune-Gain

For a HARD parent $C$ and candidate axis $a$, define:

$$
P(a)
=
\mathbf1[C^-_a\text{ certified}]
+
\mathbf1[C^+_a\text{ certified}].
$$

So:

$$
P(a)\in\{0,1,2\}.
$$

Prototype primary objective:

$$
\boxed{\max_aP(a)}.
$$

On a tie, compare in order:

1. $\min(L_-,L_+)$;
2. $L_-+L_+$;
3. child Hausdorff radius;
4. deterministic axis order.

---

# 9. Unrestricted CPG is myopic

$D_3$ root, depth 14, full CPG:

$$
2975\text{ nodes},\qquad1210\text{ HARD},
$$

lower-bound evaluations:

$$
14719.
$$

It pushes node count very low, but pilot-depth HARD is actually higher than under the windowed policy.

Therefore:

$$
\boxed{
\text{maximize immediate prune}
\not\equiv
\text{minimize long-run unresolved frontier}
}.
$$

---

# 10. Windowed CPG

Round 17 sets a performance gate:

$$
\boxed{L(C)\ge T-\gamma},
$$

$$
\gamma=0.025.
$$

In this phase:

$$
L(C)\ge0.81.
$$

Alternate axes are evaluated only when the official split immediately closes 0 children and the parent has already entered the proof-near window.

---

# 11. Depth-14 full vs windowed CPG

full CPG:

$$
N=2975,\quad H=1210,\quad E=14719.
$$

windowed CPG:

$$
N=4173,\quad H=863,\quad E=8424.
$$

windowed:

- more nodes, but fewer unresolved HARD;
- lower-bound evaluations decrease by about **42.77%**;
- wall time decreases by about **42.45%**.

So unrestricted CPG is not adopted as the production default.

---

# 12. Full-window CPG at depth 16

$D_3$ official:

$$
12357\text{ nodes},\qquad2766\text{ HARD}.
$$

Starting Windowed CPG from the root:

$$
8625\text{ nodes},\qquad1862\text{ HARD}.
$$

Relative to official:

$$
\boxed{30.20\%\text{ fewer nodes}},
$$

$$
\boxed{32.68\%\text{ fewer HARD}}.
$$

But emitter lower-bound evaluations rise to about:

$$
1.82\times.
$$

---

# 13. Tail-only Windowed CPG

Round 16 already showed that the real bottleneck sits in the deeper tail, so production is better served by:

- depth $<10$: purely official;
- depth $\ge10$: CPG override is allowed only if $L(C)\ge0.81$ and the official split closes 0 children.

Depth 16 pilot:

$$
\boxed{N=9997},
$$

$$
\boxed{H=2048}.
$$

Override density:

$$
\boxed{10.72\%}.
$$

Relative to $D_3$ official:

$$
\boxed{19.10\%\text{ fewer nodes}},
$$

$$
\boxed{25.96\%\text{ fewer HARD}}.
$$

This is currently the most balanced production candidate.

---

# 14. Search cost vs replay cost

Tail-window CPG's emitter lower-bound evaluations run at about, relative to official:

$$
2.02\times.
$$

So it is not a free optimization. But the certificate will afterward be replayed repeatedly by the canonical verifier, lagged AI, and other independent verifiers.

The real optimization objective should be:

$$
\boxed{
C_{\rm total}
=
C_{\rm emit}
+
\lambda C_{\rm replay}
}.
$$

---

# 15. Sparse override semantics

Production retains `MISHRA-FIRST-ORDER` as the implicit default axis.

Only when tail-window CPG selects a non-default axis does the certificate store an extra override.

The verifier:

- does not recompute CPG;
- reads the declared override axis;
- only verifies that the child boxes cover the parent.

Since the default is known among the five axes, only four remain as override axes, so in principle 2 bits suffice to represent the axis; the node-location codec will be fixed in a later round.

---

# 16. Theorem-critical vs performance-only

## Theorem-critical

- $D_3$ group action;
- canonical wedge theorem;
- `SYM` prune;
- child coverage.

## Performance-only

- $\gamma=0.025$;
- activation depth 10;
- CPG score;
- override codec;
- scheduler version.

So future scheduler updates should not stale the symmetry/root theorem.

---

# 17. Production stack after Round 17

The best known combination at present:

$$
\boxed{
D_3\text{ wedge}
+
\text{official implicit split}
+
\text{tail-window CPG override}
+
\text{Round 16 tail-first shard priority}
}.
$$

---

# 18. COMPUTE-DEFERRED

- add `SYM` leaves to the production grammar;
- add sparse split overrides to the production grammar;
- recalibrate $\gamma$ and the activation depth on real tail shards;
- measure emitter CPU / compressed bytes / verifier replay time / multi-AI audit total cost.

At present:

$$
\gamma=0.025,
\qquad
d_{\rm activate}=10
$$

are both only pilot performance parameters.

---

# 19. Assigned topic for Round 18

## AMRAL-LUC-FC-R18
### Production Shard Emitter, Sparse Override Codec, and Replay-Cost Optimization

The next round's main focus:

1. have the formal production emitter / verifier use the $D_3$ root;
2. actually encode `SYM` leaves;
3. actually encode the sparse override stream;
4. compare raw / RLE / delta-index / compression codecs;
5. actually measure emit vs replay economics;
6. turn tail-window CPG into the production shard format.

---

# 20. Reproducibility checklist

- Residual $D_3$ group — `PROVED`
- 60° canonical wedge — `PROVED`
- `SYM` prune — `PROVED`
- Symmetry implementation sanity — `PASS`
- $D_3$ global pilot — `PASS`
- CPG — `IMPLEMENTED`
- Windowed CPG — `PILOT-PASS`
- Tail-window CPG — `PILOT-PASS / PRODUCTION CANDIDATE`
- Global $0.8350$ theorem — `STILL OPEN / COMPUTE-DEFERRED`

---

# 21. Shortest handoff conclusion

Round 17 shrinks the global search by two further layers.

The first layer is exact:

$$
\boxed{\mathcal Q_{\rm full}/D_3}
$$

only needs to search:

$$
\boxed{0\le\arg t_3\le\pi/3}.
$$

The second layer is performance:

$$
\boxed{
\text{official split}
+
\text{tail-window CPG override}
}.
$$

The global long-run stack should now be updated to:

$$
\boxed{
D_3\text{ quotient}
+
\text{base-refine-first}
+
\text{tail-window CPG}
+
\text{tail-first shard scheduling}
}.
$$