> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification.

# LESR Paper 09V — Reconstructed Text
## Tail A_ref Campaign: Exact Eroded Core, Adaptive Frontier and the Finite Parent-Cell Ledger

**Version:** reconstructed-v1 · 2026-10-08  
**Nature:** A research narrative reconstructed from surviving checkpoints, the 125-file snapshot manifest, BCODR-LUC-06 and the final handoff. The title and body are this edition's editorial reconstruction, not a recovered original Paper 09V manuscript.

## Abstract

09U sealed the microscopic segment; 09V's next work domain is root 6217's tail [0.003,π/120]. The goal is cell-by-cell exclusion of configurations below a specified A_ref, not a direct global Lebesgue lower bound. The surviving records show three shifts: correcting the parent-cell ledger, compressing the hard core by deterministic look-ahead subdivision, and handling thin margins with exact eroded cores and robust cores.

## 1. Target and parameter domain

Every state is bound to

```text
root = 6217
rho-domain = [0.003, pi/120]
target = tail A_ref ≈ 0.8349865
authority = research interval / declared error contract
```

This A_ref is a local threshold. It differs from the original target A3⁺ ≈ 0.834781191, so their existing closure lists are not interchangeable.

## 2. Correcting the parent-cell ledger

Paper 08's 200-root ledger contained five earlier A_ref parent cells and was once described as 195 effective cells. Subsequent BCODR-LUC-06 work and the handoff rechecked the semantics as

```text
200 → 192 → 187 → 182
```

This sequence corrects research-stage and target classifications; it does not measure four numerical searches. Intersecting the corrected effective parent cells with the tail yields 173 active parent cells.

## 3. Look-ahead subdivision and the twenty-nine-cell hard core

The surviving first-stage checkpoint records 173 parent cells intersecting the tail: 136 already have finite A_ref closure and 37 remain open. K64/RING256 deterministic look-ahead subdivision closes eight more, leaving 29 hard-core cells.

Each parent-cell state preserves its queue, node count and minimum margin. A package that records only a summary at a given level cannot reconstruct all its leaves; this text does not rewrite summaries as per-leaf certificates.

## 4. Exact eroded-core layer

The checkpoints name the new layer `EXACT_ERODED_CORE_SUPPORT`. Its geometric intent is to use the actual support arcs of the eroded witness body rather than obtain a lower bound only from an inscribed K1024 polygon.

The historical record reports no regression against the old lower bound on 337 hard-core boxes, no violation in 640 spot checks of actual placements, and immediate closure of 24 boundary boxes. These are numbers from the original work records, not numerical experiments rerun for this reconstruction.

The publication layer still requires outward enclosures or independent backend audits for every support-function branch, parameter box and interval-arithmetic operation; the word `EXACT` does not automatically discharge these obligations.

## 5. Nine traceable snapshot closures

The manifest embedded in the 2026-09-24 snapshot lists

```text
1,13,45,47,95,143,169,178,186
```

These nine have snapshot provenance. Checkpoint milestones include 14,607 new nodes for parent 186, 97,297 nodes for parent 1, and subsequent closure of parents 13 and 169. Extremely thin margins are around 10^-10 to 10^-9, so replay must preserve precision and the original arithmetic contract.

This delivery does not replay the complete approximately 359 MB state tree. The website retains the published manifest and checkpoints; the research archive holds the complete snapshot. This reconstructed body does not invent those state trees.

## 6. The tenth parent cell in the final handoff

The final handoff and MCC-OE-27 add parent 0, making the textual ledger 10/29; the remaining nineteen are

```text
46,63,81,92,93,94,124,125,128,129,131,142,150,152,159,160,161,168,177
```

The surviving snapshot of parent 0 still has an unprocessed queue. This edition therefore labels its closure a later textual report, not a recovered final certificate.

## 7. Robust cores and backend replay

Later descriptions also use an anisotropic/robust AABB core to strengthen witness-body containment within a parent box. The minimum replay responsibility chain is

```text
parameter box
→ enclosing witness geometry
→ containment / erosion certificate
→ support bound
→ area lower bound
→ complete leaf coverage
```

If an arrow survives only as a textual description, leave that layer pending verification. If a verification script's core dependency is missing, record `DEPENDENCY_MISSING`; an unexecuted check must not be reported as PASS.

## 8. How to continue after the handoff

First validate the artifacts, then replay the nine parent cells with snapshot provenance. List the tenth textual closure separately before tackling the remaining nineteen cells. The scheduler must retain checkpoints throughout execution so a deeper DFS timeout does not leave only a description.

Keep 09A–09U's local atlas, target evolution and intermediate corrections alongside 09V. Even completion of the finite-tail campaign would leave positive rotation, all root-6217 shapes, other roots and global sufficiency to be addressed separately.

## Status

| Item | Status supported by this reconstructed edition |
|---|---|
| Research narrative and target alignment | RECONSTRUCTED |
| Nine snapshot closure IDs | Source-backed; complete trees not replayed in this round |
| Parent 0's tenth closure | TEXT_REPORTED_ONLY |
| Nineteen tail cells | OPEN |
| Root-6217 positive rotation and all shapes | OPEN |
| Global Lebesgue problem | OPEN |

## Sources

- Eight LESR_CHECKPOINT_2026-09-23/24_09V*.txt files.
- LESR_09V_Tail_Campaign_Snapshot_2026-09-24_manifest.json.
- BCODR_LUC_06_Frontier_Spine_Closure_for_Tail_Aref_v0.1.md.
- The final LESR handoff and MCC-OE-27.

===END OF RECONSTRUCTED EDITION===