# AMRAL × Lebesgue Universal Covering — Round 24
## Portfolio Residual Core and Joint-Witness Admission

**Document ID:** AMRAL-LUC-FC-R24  
**Version:** v0.1  
**Date:** 2026-09-19  
**Research status:** Round 24 / Residual collapse / Joint-witness necessity gate  
**Research mode:** Human-Directed + Semi-Autonomous AI Mathematical Research  
**Research initiator and methodology source:** Neo.K  
**AI collaborating researcher and primary executor:** Aletheia / ChatGPT, GPT-5.6 Sol  
**Prior documents:** AMRAL-LUC-FC-R00 v0.2; R01–R23 v0.1  

---

# 0. Round Summary

Round 23, under 12 necessity-marked cells × 5 witnesses at lift depth 16, obtained:

$$
\boxed{0\text{ strict COMPLETE edges}.}
$$

Looking only at this result, it is easy to draw the premature inference:

> the single-witness portfolio is not enough; the next step should be to open a joint multi-witness tree.

Round 24 is dedicated to examining this inference.

The result is the opposite.

For the first four hardest reference cells, after continuing to deepen with the single witness $B_7$:

- cell 0:
  $$
  \boxed{\text{COMPLETE at depth }30};
  $$
- cell 1:
  $$
  \boxed{\text{COMPLETE at depth }32};
  $$
- cell 2:
  $$
  \boxed{\text{COMPLETE at depth }32};
  $$
- cell 3:
  at depth 30 still partial, but the unresolved volume has already shrunk to:
  $$
  \boxed{1.49\times10^{-7}},
  $$
  with no plateau evidence.

So Round 23's residual is mainly:

$$
\boxed{
\text{certificate depth / bound residual}
}
$$

rather than the currently-proven:

$$
\boxed{
\text{joint-witness necessity}.
}
$$

This round accomplishes:

1. the first actual strict $B_7$ complete-edge certificate;
2. residual-core collapse analysis;
3. Single-Witness Sufficiency Dominance;
4. the pointwise joint-witness necessity theorem;
5. a cell-level joint admission gate;
6. a `JOINT-REDUNDANT / DEFERRED / CANDIDATE / NECESSARY` state machine;
7. an explicit prohibition on jumping straight to a 6-D joint placement tree just because there are "many partial tails."

This round's determination:

$$
\boxed{
\text{FIRST REFERENCE STRICT LIFT EDGE: CLOSED}
}
$$

$$
\boxed{
\text{JOINT-WITNESS ADMISSION LOGIC: CLOSED}
}
$$

$$
\boxed{
\text{JOINT NECESSITY ON TESTED CELLS: NOT ESTABLISHED}
}
$$

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

---

# 1. Single-witness value

Fix the exact base hull:

$$
H(q).
$$

For witness:

$$
K,
$$

define:

$$
\boxed{
J_K(H(q))
=
\min_g
\operatorname{Area}
\operatorname{conv}
(
H(q)\cup gK
).
}
$$

If:

$$
J_K(H(q))\ge T,
$$

then the single witness $K$ already suffices to force that point to the target.

---

# 2. Joint portfolio value

For a portfolio:

$$
\mathcal B
=
\{K_1,\ldots,K_m\},
$$

define:

$$
\boxed{
J_{\mathcal B}(H(q))
=
\min_{g_1,\ldots,g_m}
\operatorname{Area}
\operatorname{conv}
(
H(q)\cup g_1K_1\cup\cdots\cup g_mK_m
).
}
$$

---

# 3. Joint Dominance Theorem

## Theorem 3.1

For any:

$$
K\in\mathcal B,
$$

we have:

$$
\boxed{
J_{\mathcal B}(H(q))
\ge
J_K(H(q)).
}
$$

Hence:

$$
\boxed{
J_{\mathcal B}(H(q))
\ge
\max_{K\in\mathcal B}J_K(H(q)).
}
$$

### Proof

For any joint placement tuple:

$$
(g_1,\ldots,g_m),
$$

the joint hull contains:

$$
\operatorname{conv}(H(q)\cup g_KK).
$$

So the joint area is at least equal to that single-witness area.

After taking the minimum over all tuples, it is still at least:

$$
\min_{g_K}
\operatorname{Area}\operatorname{conv}(H(q)\cup g_KK)
=
J_K(H(q)).
$$

This holds for every $K$.

Q.E.D.

---

# 4. Consequence: strict single edge makes joint proof redundant

If some base cell:

$$
C
$$

already has a strict edge:

$$
I_{C,K}=1,
$$

that is:

$$
J_K(H(q))\ge T
\qquad
\forall q\in C,
$$

then the joint portfolio automatically also reaches at least:

$$
T.
$$

So, for the purpose of "proving this cell," we have:

$$
\boxed{
\text{joint witness is mathematically redundant}.
}
$$

It might still yield a smaller certificate,

but it is no longer a necessity.

---

# 5. Round 23 residual was budget-relative

Round 23 at depth 16:

$$
I_{jK}^{(16)}=0
$$

for all 60 pilot edges.

This means only:

$$
\boxed{
\text{no COMPLETE edge at that budget}.
}
$$

It cannot be used to conclude:

$$
\boxed{
\text{no single witness can close}.
}
$$

Round 24 actually demonstrates that this distinction matters.

---

# 6. Cell 0 tail collapse

$B_7$'s unresolved absolute placement volume:

depth 16:

$$
9.34\times10^{-4}.
$$

depth 22:

$$
5.17\times10^{-5}.
$$

depth 24:

$$
6.94\times10^{-6}.
$$

depth 26:

$$
1.77\times10^{-6}.
$$

depth 28:

$$
9.39\times10^{-8}.
$$

depth 30:

$$
\boxed{0}.
$$

So:

$$
\boxed{
I_{0,B_7}^{(30)}=1
}
$$

under the current reference arithmetic semantics.

---

# 7. First actual complete-edge certificate

Round 24 builds a preorder topology certificate for:

$$
(\text{cell }0,B_7,d=30)
$$

Result:

$$
18555\text{ nodes},
$$

$$
9278\text{ CERT leaves},
$$

$$
0\text{ PENDING}.
$$

forest identity:

$$
18555
=
2(9278)-1.
$$

deterministic replay:

`PASS`.

This is:

$$
\boxed{
\text{REFERENCE-COMPLETE-EDGE-PILOT}
}
$$

not a publication-grade interval certificate.

---

# 8. Cells 1 and 2

cell 1:

at depth 30 only:

$$
5
$$

pending leaves remain,

unresolved volume:

$$
6.79\times10^{-10}.
$$

depth 32:

$$
\boxed{\text{COMPLETE}.}
$$

cell 2:

depth 30:

$$
152
$$

pending,

$$
2.06\times10^{-8}
$$

unresolved volume.

depth 32:

$$
\boxed{\text{COMPLETE}.}
$$

So:

$$
\boxed{
I_{1,B_7}^{(32)}
=
I_{2,B_7}^{(32)}
=
1.
}
$$

---

# 9. Cell 3

cell 3's B7 unresolved volume:

depth 16:

$$
9.70\times10^{-4}.
$$

depth 22:

$$
5.85\times10^{-5}.
$$

depth 30:

$$
\boxed{
1.49\times10^{-7}.
}
$$

So from depth 22 to 30 it shrank again by about:

$$
3.93\times10^{2}
$$

times.

Currently:

`PARTIAL`

but clearly:

`CONTRACTING`.

No tail plateau evidence.

---

# 10. Residual-core interpretation

Among the first four hardest pilot cells:

after testing deep budgets:

$$
\boxed{
3/4
}
$$

have already produced a strict $B_7$ edge.

The remaining cell 3:

$$
\boxed{
\text{single-witness contracting residual}.
}
$$

So this reference residual does not support:

> joint witnesses are already necessary.

Rather, it supports:

> finish running the strongest single-witness tail first.

---

# 11. Pointwise Joint-Necessity Theorem

## Theorem 11.1

Fix an exact base configuration:

$$
q.
$$

Fix a portfolio:

$$
\mathcal B
=
\{K_1,\ldots,K_m\}.
$$

If for every:

$$
K_i
$$

there is a verified counterexample:

$$
g_i
$$

such that:

$$
\operatorname{Area}
\operatorname{conv}
(
H(q)\cup g_iK_i
)
<T,
$$

then:

$$
\boxed{
J_{K_i}(H(q))<T
\quad
\forall i.
}
$$

So:

> any proof route relying on only a single one of the witnesses in the portfolio cannot reach $T$ at this exact $q$.

If one still wishes to prove:

$$
T,
$$

using only that portfolio, one must make use of:

- simultaneous joint-witness interaction;
- or a new theorem that is logically equivalent to joint interaction.

This is called:

$$
\boxed{
\text{JOINT-NECESSARY-RELATIVE-TO-PORTFOLIO AT }q.
}
$$

---

# 12. Important limitation

The previous theorem does not say:

$$
J_{\mathcal B}(H(q))\ge T.
$$

It only says:

> single-witness reduction fails entirely.

The joint family itself might still be:

$$
<T.
$$

So joint necessity and joint sufficiency are two different things.

---

# 13. Cell-level witness switching comes before joint

Even if a coarse cell:

$$
C
$$

has no single witness able to uniformly close the whole cell,

it may still be possible to split it:

$$
C=C_1\cup\cdots\cup C_r
$$

so that different subcells are closed by different witnesses.

This is Round 07's witness switching.

So:

$$
\boxed{
\text{cell has no strict edge}
\not\Rightarrow
\text{joint necessary}.
}
$$

One must also first allow for:

$$
\boxed{
\text{base refinement + witness switching}.
}
$$

---

# 14. Joint admission states

Round 24 formally adds:

## `JOINT-REDUNDANT`

Already has a strict single-witness edge.

## `JOINT-DEFERRED-CONTRACTING`

Currently no strict edge,

but at least one not-yet-disqualified single-witness tail is still contracting rapidly.

## `JOINT-CANDIDATE`

Single tails start to plateau,

or the main single witnesses already have negative evidence.

One can begin estimating joint cost / synergy.

## `JOINT-NECESSARY-RELATIVE-TO-PORTFOLIO`

At the same exact base point,

every single witness in the portfolio has a verified counterexample.

In this case, if one still wants to use only that portfolio,

one must make use of joint interaction.

---

# 15. Current status

## cell 0

$$
\boxed{\texttt{JOINT-REDUNDANT}}
$$

via B7 depth 30.

## cell 1

$$
\boxed{\texttt{JOINT-REDUNDANT}}
$$

via B7 depth 32.

## cell 2

$$
\boxed{\texttt{JOINT-REDUNDANT}}
$$

via B7 depth 32.

## cell 3

$$
\boxed{\texttt{JOINT-DEFERRED-CONTRACTING}}
$$

at tested depth 30.

---

# 16. Why joint tree is expensive

single-witness placement:

$$
(\phi,x,y)
$$

is:

$$
3D.
$$

two witnesses:

$$
(\phi_1,x_1,y_1,\phi_2,x_2,y_2)
$$

is:

$$
6D.
$$

If the two single-witness trees each need:

$$
L_1,L_2
$$

terminal placement boxes respectively,

naive Cartesian joint refinement may approach:

$$
\boxed{
L_1L_2
}
$$

rather than:

$$
L_1+L_2.
$$

So, before there is necessity evidence,

opening a joint tree prematurely could be an extremely costly mistake.

---

# 17. Joint certificate grammar

After genuine admission,

the joint box:

$$
C_K\times C_L.
$$

Each witness separately builds a common core:

$$
G_K(C_K),
\qquad
G_L(C_L).
$$

base core:

$$
G_{\rm base}(C).
$$

joint one-sided lower hull:

$$
\boxed{
L_{\rm joint}
=
\operatorname{Area}
\operatorname{conv}
(
G_{\rm base}
\cup
G_K
\cup
G_L
).
}
$$

If:

$$
L_{\rm joint}\ge T,
$$

then the joint box is certified.

This is the basic legitimate prune for a future joint tree.

---

# 18. Joint counterexample

In the opposite direction,

if an exact base point is found:

$$
q
$$

together with simultaneous placements:

$$
g_K,g_L
$$

such that the outer hull:

$$
U(
H(q)\cup g_KK\cup g_LL
)
<T,
$$

then the joint pair itself is also disqualified.

So a joint certificate must also maintain

- a positive complete tree;
- a negative simultaneous counterexample;

two one-sided semantics.

---

# 19. Residual Collapse Principle

Round 24 yields a scheduler principle:

> Before considering raising the dimension, first check whether the strongest surviving single witness's unresolved volume is still showing sustained contraction.

Define the performance ratio:

$$
\rho_{d,\Delta}(C,K)
=
\frac{
V_{C,K}^{(d+\Delta)}
}{
V_{C,K}^{(d)}
}.
$$

If, over consecutive budgets:

$$
\rho\ll1,
$$

then:

`JOINT-DEFERRED-CONTRACTING`.

This is not a theorem,

it is only an admission performance gate.

---

# 20. Plateau gate

Conversely,

only if:

- there have been multiple increases in depth;
- the unresolved volume is barely decreasing;
- the tail geometry is stable;
- or there are already single-witness counterexamples;

does it escalate to:

`JOINT-CANDIDATE`.

Round 24 does not fix a universal plateau constant.

It only fixes the semantics and the evidence requirement.

---

# 21. Strict edge append-only benefit

Once:

$$
I_{jK}=1,
$$

under the same theorem semantics,

that cell can be permanently removed from the joint-admission queue.

So deep single-witness compute is not wasted.

It directly shrinks the:

$$
\boxed{
\text{portfolio residual core}.
}
$$

---

# 22. Current reference residual

Round 23's first-four hardest cells:

$$
\{0,1,2,3\}.
$$

After Round 24's deep B7:

$$
\boxed{
\{0,1,2,3\}
\to
\{3\}
}
$$

at tested budgets.

This is:

$$
\boxed{
75\%
}
$$

reference residual-count reduction.

Not a full 12-cell / 77-cell result.

---

# 23. What remains untested

cells:

$$
4,\ldots,11
$$

have not yet been systematically rerun at depth 30/32.

the full:

$$
77\times5
$$

also still remains for local runtime.

So one cannot extrapolate the:

$$
3/4
$$

reference result into:

> 75% of global residual will close.

At present this is only a strategic evidence:

> deep B7 deserves priority over joint expansion.

---

# 24. Production queue update

New priority:

1. `STRICT-CLOSED`: remove from queue;
2. `SINGLE-CONTRACTING`: deep single witness;
3. `SWITCHABLE-BY-REFINE`: base split + witness switching;
4. `SINGLE-DISQUALIFIED / PLATEAU`: joint candidate;
5. `POINTWISE-ALL-DISQUALIFIED`: joint necessary relative to portfolio.

---

# 25. Round 25

## Deep Single-Witness Closure Wave and Joint-Admissible Residual Extraction

The next round should:

1. run adaptive B7 continuation on the entire 12-cell pilot;
2. use a budget that is not a fixed depth, but rather:
   - complete;
   - plateau;
   - node cap;
3. remove each cell from the residual as soon as it produces a strict edge;
4. then expand to the 77-cell atlas;
5. run, for genuinely residual cells only:
   - H19 crossover;
   - counterexample pool;
   - joint-admission test;
6. only build an actual 6-D joint pilot once the first `JOINT-CANDIDATE` appears.

---

# 26. Shortest Handoff

Round 24's biggest conclusion is:

$$
\boxed{
\text{partial portfolio}
\not\Rightarrow
\text{joint necessity}.
}
$$

Round 23 at depth-16 appeared to have no strict edge at all,

but simply running $B_7$ deeper already gave:

$$
\boxed{
3/4
}
$$

of the hardest reference cells a COMPLETE edge.

So the correct order is:

$$
\boxed{
\text{deep single}
\to
\text{refine/switch}
\to
\text{joint admission}
}
$$

rather than:

$$
\boxed{
\text{partial}
\to
\text{immediately joint}.
}
$$