# AMRAL × Lebesgue Universal Covering — Round 26
## Full Necessity-Atlas $B_7$ Wave and Production Arithmetic Contract

**Document ID:** AMRAL-LUC-FC-R26  
**Version:** v0.1  
**Date:** 2026-09-20  
**Research status:** Round 26 / Full 77-cell necessity-atlas checkpoint / Production arithmetic migration  
**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–R25 v0.1  

---

# 0. Round Summary

Round 25 found that, on the 12-cell reference subset, $12/12$ could all be closed by a single $B_7$ finite-depth closure.

Round 26 pushes the same strategy to completeness for the first time:

$$
\boxed{77}
$$

Round 19 necessity-marked base cells.

To prevent different cells from being contaminated by different maximum depths, this round's canonical checkpoint fixes a common budget:

$$
\boxed{d=36}.
$$

Result:

$$
\boxed{51/77}
$$

cells have achieved strict reference $B_7$ closure,

leaving:

$$
\boxed{26/77}
$$

residual.

coverage:

$$
\boxed{66.23\%}.
$$

More importantly:

$$
\boxed{\text{All 26 residuals are still contracting.}}
$$

From the previous common checkpoint to depth 36, the contraction ratio for all residuals is $<1$; the maximum:

$$
\boxed{\rho_{\max}\approx0.57615}.
$$

Therefore the full atlas currently still has no joint-witness necessity evidence.

At the same time, this round formally establishes a production arithmetic ABI: reference floating/exact-kernel evidence cannot be directly promoted to publication proof; all reference COMPLETE leaves are uniformly treated, before migration, as:

$$
\boxed{\texttt{ARITHMETIC-PENDING}}.
$$

This round's determination:

$$
\boxed{\text{FULL 77-CELL CHECKPOINT THROUGH }d=36:\ \text{COMPLETED}}
$$

$$
\boxed{B_7\text{ STRICT REFERENCE CLOSURE AT }d=36:\ 51/77}
$$

$$
\boxed{\text{RESIDUAL}:26/77,\ \text{ALL CONTRACTING}}
$$

$$
\boxed{\text{JOINT NECESSITY: NOT ESTABLISHED}}
$$

$$
\boxed{\text{PRODUCTION ARITHMETIC ABI: FORMALIZED}}
$$

$$
\boxed{\text{INTERVAL BACKEND / FULL REPLAY: OPEN}}
$$

$$
\boxed{a_{\mathrm{Leb}}\ge0.835:\ \text{NOT CERTIFIED}}
$$

---

# 1. Full necessity atlas

Round 19 necessity segmentation produced 77 depth-40 base cells, each of which already has a reference outer marker:

$$
U_{\rm base}(q_j)<0.835.
$$

So every cell is `NECESSITY-MARKED`; ultimately none can be closed by base-only proof.

---

# 2. Why a depth wave

If a worker pushes a single hard cell all the way to the bottom in one go, a small number of hard cells will occupy the queue, and the easy cells behind them cannot advance fairly.

Round 26 changes this to:

$$
\boxed{\text{common depth wave}}
$$

All residual cells pass through, in sequence:

$$
22\to24\to26\to28\to30\to32\to34\to36.
$$

After each wave, COMPLETE cells are permanently removed; the rest keep their pending boxes, and the next round only extends the portions not yet closed.

---

# 3. Crash/restart state

Each cell persists:

```text
cell_id
base_path
last_depth
pending_boxes
cumulative_nodes
cumulative_cert_leaves
unresolved_volume
previous_unresolved_volume
contraction_ratio
status
```

Therefore:

$$
\boxed{\text{77-cell workload is resumable}.}
$$

---

# 4. Budget monotonicity

If, for fixed cell / witness / theorem semantics, we have:

$$
I_{jK}^{(d)}=1,
$$

then a larger budget $d'>d$ will not invalidate an old complete proof:

$$
\boxed{I_{jK}^{(d)}\le I_{jK}^{(d')}}.
$$

Therefore:

$$
\Gamma_K(d)=\#\{j:I_{jK}^{(d)}=1\}
$$

is monotonically non-decreasing.

---

# 5. Full-atlas closure curve

For $B_7$:

$$
\Gamma_{B_7}(22)=0,
$$

$$
\Gamma_{B_7}(24)=0,
$$

$$
\Gamma_{B_7}(26)=0,
$$

$$
\Gamma_{B_7}(28)=0,
$$

$$
\boxed{\Gamma_{B_7}(30)=7},
$$

$$
\boxed{\Gamma_{B_7}(32)=19},
$$

$$
\boxed{\Gamma_{B_7}(34)=38},
$$

$$
\boxed{\Gamma_{B_7}(36)=51}.
$$

So the common depth-36 reference coverage:

$$
\boxed{\frac{51}{77}\approx66.23\%.}
$$

---

# 6. Canonical residual at depth 36

Under the common budget:

$$
\boxed{|\mathcal R_{36}|=26}.
$$

This round does not feed information from running only some residuals to depth 38/40 back into the depth-36 curve.

---

# 7. Residual volume distribution

Unresolved placement volume of the 26 residuals:

$$
V_{\min}\approx6.36\times10^{-12},
$$

$$
Q_{25}\approx2.31\times10^{-9},
$$

$$
Q_{50}\approx1.64\times10^{-8},
$$

$$
Q_{75}\approx3.33\times10^{-8},
$$

$$
Q_{90}\approx4.97\times10^{-8},
$$

$$
\boxed{V_{\max}\approx6.99\times10^{-8}}.
$$

---

# 8. Residual classes

The scheduler divides these into three classes:

## Terminal dust

$$
V\le10^{-9},
$$

count:

$$
\boxed{5}.
$$

## Thin

$$
10^{-9}<V\le10^{-8},
$$

count:

$$
\boxed{7}.
$$

## Broad

$$
V>10^{-8},
$$

count:

$$
\boxed{14}.
$$

---

# 9. Residual contraction

Definition:

$$
\rho=\frac{V_{\rm current}}{V_{\rm previous\ wave}}.
$$

Median of the 26 residuals:

$$
\approx0.30094.
$$

75%:

$$
\approx0.42133.
$$

90%:

$$
\approx0.51170.
$$

maximum:

$$
\boxed{0.57615}.
$$

Therefore:

$$
\boxed{\rho<1\quad\text{for all 26 residual cells}.}
$$

This is a numerical measurement, not a universal theorem.

---

# 10. Joint-witness implication

The Round 24 admission gate should only open a joint tree when single tails plateau, or single witnesses are eliminated by a pointwise counterexample.

Round 26 depth 36:

$$
\boxed{\text{Not one residual shows a volume plateau}.}
$$

So:

$$
\boxed{\text{joint-witness queue remains deferred}.}
$$

The current bottleneck remains the cost of deep single-witness certificates.

---

# 11. Pending count is not difficulty

Deep cells may have on the order of $10^4$ pending leaves, yet unresolved volume of only $10^{-8}$–$10^{-7}$.

So the production plateau / priority metric must use:

$$
V_d,\qquad
\rho_{d,\Delta},
$$

rather than the pending-count ratio.

---

# 12. Hierarchical ancestor opportunity

The 77 paths share a large number of base prefixes.

In theory, if a shallower base ancestor's $B_7$ lift can COMPLETE, Certified Ancestor Contraction could cover all descendants in one shot.

Round 26's brute-force ancestor experiment was halted because the exact-kernel compute cost was too high; there is no ancestor closure claim.

This optimization is deferred until the cache / interval kernel matures.

---

# 13. Arithmetic becomes the next theorem gate

The reference verifier currently still mixes binary64 / reference-pad semantics.

So:

$$
\boxed{\text{reference COMPLETE}\neq\text{publication COMPLETE}.}
$$

All active reference leaves are uniformly marked, before production migration:

`ARITHMETIC-PENDING`.

---

# 14. Why a larger global pad fails

Among the already-audited complete trees, the smallest observed leaf slack is only:

$$
\boxed{2.8982\times10^{-9}}.
$$

If crudely switched to a:

$$
10^{-8}
$$

global pad, multiple currently complete trees would reopen.

Therefore:

$$
\boxed{\text{global pad migration is rejected}.}
$$

---

# 15. Per-leaf arithmetic ledger

Each leaf stores:

```text
nominal_raw_lower
reference_pad
reference_slack
interval_lower
interval_slack
precision_bits
backend_hash
```

If interval replay:

$$
L_{\rm int}^{\rm lo}\ge T,
$$

promote to:

`INTERVAL-PASS`.

If:

$$
L_{\rm int}^{\rm lo}<T\le L_{\rm int}^{\rm hi},
$$

mark:

`INTERVAL-INCONCLUSIVE`.

This is not a proof failure.

---

# 16. Selective Arithmetic Revalidation Theorem

Fix a legitimate partition tree covering the root.

If some terminal leaves directly PASS under current interval arithmetic, while the remaining inconclusive leaves are legitimately subdivided only within their own subtree, then as long as these refined descendants all eventually close, overall root coverage still holds.

Other subtrees that have already interval-passed do not need to be recomputed.

Therefore:

$$
\boxed{\text{arithmetic upgrade can be local}.}
$$

---

# 17. Backend contract

The production backend requires:

$$
\boxed{\text{correctly-rounded arbitrary-precision interval / ball arithmetic}.}
$$

Precision ladder:

$$
128\to192\to256\to384\text{ bits}.
$$

If the interval straddles the decision boundary:

1. increase precision;
2. if still inconclusive, split the geometry;
3. never accept by midpoint.

---

# 18. Exact layer

The following remain exact:

- prefix paths;
- split sides;
- node counts;
- forest identities;
- dyadic child fractions;
- certificate topology;
- hashes.

These do not need to enter interval complexity.

---

# 19. Motion interval

For:

$$
\delta
=
\sqrt{h_x^2+h_y^2}
+
2R\sin(h_\phi/2),
$$

compute an outward:

$$
\delta_{\rm hi}.
$$

common-core radius:

$$
\boxed{\rho_{\rm lo}=1-\delta_{\rm hi}.}
$$

---

# 20. Preserve eroded-Reuleaux geometry

Round 26's audit tested a coarse radial/disk-like fallback.

On cell0's 9278 complete leaves, it could preserve only about a few percent of the closure capability.

So the production primary core cannot degrade into coarse disk-like samples.

It preserves the eroded-Reuleaux geometry of:

$$
\boxed{\bigcap_jD(V_j,\rho_{\rm lo})}
$$

---

# 21. Interval common-core constructor

The primary constructor uses directed intervals to complete:

1. $\rho_{\rm lo}>0$;
2. nonemptiness;
3. circle-circle intersection radicand;
4. branch separation;
5. all-disk membership.

This is more local than making the entire support-envelope crossing system interval-valued.

---

# 22. Tight-leaf polygon audit

For a certain worst-case reference leaf of cell0:

nominal exact-style lower:

$$
0.835000042174202.
$$

reference lower after a $10^{-10}$ pad:

$$
0.835000042074202.
$$

slack approximately:

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

uniform contact polygon:

4096 directions:

$$
0.83499981953
$$

did not pass.

8192 directions:

$$
0.83500000604
$$

only barely passed.

So:

$$
\boxed{\text{coarse finite polygonization is fallback only}.}
$$

---

# 23. Necessity-marker arithmetic

Reference margin of the 77 markers:

minimum:

$$
\boxed{2.885\times10^{-7}},
$$

median:

$$
\approx6.806\times10^{-5}.
$$

The upper marker usually has more margin than the thinnest lower leaf, but directed upper replay is still required.

A production marker remains active only when

$$
U_{\rm interval}^{\rm hi}<T
$$

holds.

---

# 24. Arithmetic semantic migration

Round 14's semantic revalidation now genuinely applies to proof arithmetic:

$$
\boxed{
\text{reference leaf}
\to
\text{interval replay}
\to
\begin{cases}
\text{PASS: promote in place}\\
\text{INCONCLUSIVE: local resplit}\\
\text{AUDIT FAIL: reopen dependency cone}
\end{cases}
}
$$

There is no need to recompute the whole tree just because a hash changes.

---

# 25. Theorem-ready gate

Even if geometry reference eventually reaches $77/77$, `theorem_ready=true` still cannot be declared until:

1. every active lower leaf: `INTERVAL-PASS`;
2. every active marker: directed upper PASS;
3. root / symmetry / split: current exact/interval semantics;
4. independent replay: PASS.

Currently:

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

---

# 26. Round 27

## Residual Tail Prioritization and First Directed-Interval Replay

Next round:

1. extend only the 26 depth-36 residuals;
2. priority:
   - 5 terminal dust;
   - 7 thin;
   - 14 broad;
3. resumable waves to 38/40+;
4. simultaneously implement the first interval eroded-Reuleaux constructor;
5. replay first:
   - cell0 complete-edge;
   - weakest-slack leaves;
   - weakest necessity markers;
6. measure the direct interval migration fraction versus the selective resplit fraction;
7. only restart the joint-witness queue if a residual plateau occurs.

---

# 27. Shortest Handoff

Round 26 obtains, for the first time, the common-budget state of the complete necessity atlas:

$$
\boxed{51/77\text{ B7-complete at }d=36.}
$$

remaining:

$$
26
$$

residuals,

but:

$$
\boxed{26/26\text{ still contracting}.}
$$

so there is currently no joint-witness necessity.

The proof bottleneck is shifting from geometry search toward:

$$
\boxed{\text{production arithmetic migration}.}
$$

The next step does not redo completed geometry, but instead:

$$
\boxed{
\text{interval-replay pass leaves}
+
\text{selectively resplit only thin-margin leaves}.
}
$$